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MANUAL  OF 
SEDIMENTARY  PETROGRAPHY 


The  Century  Earth   Science   Series 

KiRTLEY  F.  Mather,  Editor 


MANUAL  OF  SEDIMENTARY 
PETROGRAPHY 

I.  Sampling,   Preparation  for  Analysis,   Mechanical 
Analysis,  and  Statistical  Analysis 

BY 

W.  C.  KRUMBEIN 

II.  Shape  Analysis,  Mineralogical  Analysis,  Chemical 
Analysis,  and  Mass  Properties 

BY 


F.  J.  PETTIJOHN 


Department  of  Geology,  University  of  Chicago 


D.  APPLETON-CENTURY  COMPANY 

INCORPORATED 


NEW  YORK 


LONDON 


COPVRIGHT,    1938,    BY 

D.  APPLETON-CEXTURY  COMPANY.  IN'C. 


All  rights  reserved.  This  book,  or  parts 
thereof,  must  not  be  reproduced  in  any 
form  without  permission  of  the  publisher. 


PEIXTED    IX    THE    UNITED    STATES    OF    AMERICA 


EDITOR'S  PREFACE 

When  Sorby  and  his  contemporaries  were  laying  the  foundations  for 
modern  petrographic  research,  back  in  the  middle  of  the  nineteenth 
century,  the  importance  of  microscopic  examination  of  sedimentary 
rocks  was  stressed  on  at  least  one  occasion.  Nevertheless,  for  more  than 
half  a  century  the  scientific  study  of  that  type  of  rocks  received  rela- 
tively  slight  attention.  Many  petrographers  peered  through  their  micro- 
scopes at  thin  sections  of  igneous  rocks;  many  paleontologists  measured 
minutely  the  shape  of  fossils;  many  stratigraphers  debated  the  sys- 
tematic position  and  correlation  of  sedimentary  formations ;  but  few 
geologists  gave  any  really  serious  thought  to  the  accurate  and  detailed 
study  of  the  mineralogic  characteristics  of  sediments  and  the  rocks 
formed  therefrom. 

During  the  last  twenty  years,  however,  there  has  been  a  notable  in- 
crease in  interest  iji  this  phase  of  geology,  with  a  very  gratifying  ex- 
pansion of  knowledge  as  a  result  of  extraordinary  improvements  in  the 
techniques  of  study.  This  is  in  part  a  by-product  of  the  a]:)plication  of 
geology  in  the  petroleum  industry,  in  part  a  result  of  the  discovery  that 
sedimentary  rocks  provide  a  field  for  pure  research  unexcelled  by  any 
other  field  within  the  broad  area  of  earth  science. 

The  closely  related  sciences  of  sedimentary  petrography  and  sedi- 
mentary petrology  are  to-day  established  on  a  firm  base  of  technical 
procedure  and  deductive  theory.  They  have  become  worthy  of  a  life- 
time of  specialization  which  will  well  repay  the  devotion  of  a  consider- 
able minority  of  geologists. 

Most  of  the  data  which  such  specialists  should  use  are  widely  scat- 
tered through  a  large  number  of  memoirs  and  periodicals  which  have 
been  published  within  the  last  fifteen  years.  Many  of  these  are  to  be 
found  only  in  the  journals  devoted  to  physics,  soil  science,  statistical 
method,  and  colloidal  chemistry,  and  in  other  sources  which  like  those 
journals  are  not  ordinarily  available  in  geologic  laboratories.  No  ade- 
quate handbook  of  sedimentary  petrography  has  hitherto  been  published 
in  this  country.  Drs.  Krumbein  and  Pettijohn  have  therefore  rendered 
a  signal  service  to  the  student  and  worker  in  this  field  by  preparing 
this  very  useful  volume. 

V 


vi  EDITOR'S  PREFACE 

It  is  primarily  concerned  with  the  methods  of  petrographic  analysis 
of  the  sedimentary  rocks,  including  the  unconsolidated  sediments.  It 
covers  every  step  of  the  process,  from  the  field  sampling  to  the  final 
graphic  and  statistical  analysis,  with  due  regard  for  theory  as  well  as 
method.  It  will  serve  admirably  not  only  as  a  textbook  for  students  but 
also  as  an  indispensable  aid  for  the  professional  worker  dealing  as  in 
petroleum  geology  with  sedimentary  rocks  and  the  valuable  resources 
which  they  contain. 

KiRTLEY  F.  Mather. 


PREFACE 

I  often  say  that  if  you  measure  that  of  which  you  speak,  you  know  something 
of  your  subject;  but  if  you  cannot  measure  it,  your  knowledge  is  meager  and 
unsatisfactory. — Lord  Kelvin. 

The  recognition,  long  overdue,  of  the  value  of  laboratory  analysis  in 
the  study  of  sediments  is  beginning  to  be  apparent.  From  the  colloidal 
chemist,  the  ceramist,  the  ore-dressing  engineer,  the  pedologist.  the 
mineralogist,  the  statistician,  and  others  the  geologist  has  adapted 
methods  which  will  eventually  go  far  toward  making  the  study  of  sedi- 
ments a  more  exact  science. 

The  writers  of  this  volume  do  not  depreciate  field  study.  Such  studies 
are  a  necessary  prelude  to  the  laboratory  work  and  are  fundamental  to 
the  science  of  geology.  Realization,  however,  that  sediments,  like  all 
other  rocks,  are  a  product  of  definite  physical  and  chemical  processes 
and  are  capable  of  definite  analysis  on  the  basis  of  carefully  gathered 
quantitative  data  has  been  very  slow  indeed  among  geologists  as  a  group. 
We  believe  that  any  consideration  of  the  origin  of  a  deposit  which 
neglects  the  analysis  of  the  material  itself  is  quite  incomplete. 

It  has  been  pointed  out  by  some  that  our  ability  to  interpret  analytical 
data  lags  greatly  behind  our  ability  to  make  the  analyses  and  that  there- 
fore further  refinement  in  technique  and  greater  accuracy  in  description 
are  superfluous.  It  is  even  argued  that  laboratory  analyses  only  confirm 
what  the  field  geologist  already  knows,  and  such  work  is  therefore 
regarded  only  as  a  refinement  and  not  as  a  new  contribution.  The  au- 
thors are  not  in  sympathy  with  this  view.  It  may  be  evident,  even  to 
the  naked  eye,  that  the  sand  along  a  beach  decreases  in  size  in  the 
direction  of  transport,  but  whether  the  rate  of  decrease  in  size  is  expon- 
ential or  conforms  to  some  other  law  is  not  evident.  To  discover  some 
underlying  law  or  relationship  introduces  a  new  element  into  geological 
theory  and  opens  up  new  avenues  of  thought. 

In  addition  to  the  establishment  of  new  principles  and  the  interpreta- 
tion of  rock  origins,  laboratory  study  of  sediments  has  important  eco- 
nomic applications.  Well  known  is  the  study  of  "heavy  minerals,"  which 
has  proved  its  worth  in  the  correlation  of  sedimentary  formations.  The 
technologist  has  long  recognized  the  necessity  for  physical  analysis  of 


viii  PREFACE 

the  materials  with  which  he  is  concerned.  The  geologist  is  often  called 
upon  to  prospect  for  and  estimate  the  worth  of  pottery  clays,  brick 
earths,  fire-clay,  fullers'  earth,  molding  sand,  etc.  He  must  therefore  be 
able  to  use  such  methods  of  analysis  as  will  serve  to  indicate  the  use- 
fulness of  a  deposit  for  the  purpose  intended.  In  fact,  any  one  engaged 
in  the  study  of  the  particulate  substances,  natural  or  artificial  (cement, 
paint  pigments,  etc.),  will  find  valuable  the  methods  of  study  of 
particle  size  and  particle  shape  and  the  optical  methods  of  identification. 

In  so  far  as  the  geologist  is  involved  in  problems  of  petroleum  pro- 
duction and  reserves,  or  engaged  in  mapping  where  the  soils  are  the 
only  clue  to  the  nature  of  the  subjacent  fonnations,  or  engaged  in  pros- 
pecting for  or  estimating  the  worth  of  alluvial  deposits,  or  involved  in 
a  study  of  the  problems  of  soil  erosion  and  reservoir  silting,  he  will  find 
the  methods  described  herein  pertinent. 

The  science  of  sedimentary  petrolog}%  or  sedimentology,  has  now 
reached  a  stage  of  development  which  involves  a  large  number  of  tech- 
niques unique  to  this  science  and  distinct  from  those  employed  in  the 
study  of  the  igneous  and  metamorphic  rocks.  These  techniques  are 
described  in  a  widely  scattered  literature — in  the  literature  of  ceramics, 
pedolog}',  petroleum  technology,  hydrolog}-,  etc.  Growing  interest  in 
sedimentology,  as  evidenced  by  the  increasing  number  of  courses  de- 
voted to  the  subject,  the  establishment  of  a  journal  devoted  exclusively 
to  this  field,  and  the  increasing  use  of  its  methods  and  principles  in  the 
exploitation  of  petroleum  and  other  mineral  resources,  has,  we  believe, 
justified  the  attempt  to  bring  together,  for  the  benefit  of  the  geologist 
and  other  students  of  sedimentary  materials,  methods  of  analysis  ap- 
plicable to  these  substances. 

The  compilation  of  material  from  an  extensive  literature  has  raised 
nimierous  perplexing  problems.  Sciences  vary  widely  in  their  termi- 
nology and  in  their  use  of  mathematics.  Fundamental  principles  common 
to  one  field  are  largely  unknown  to  other  fields.  The  authors  have  ac- 
cordingly decided  to  write  this  book  primarily  from  the  point  of  view 
of  the  geologist,  with  the  hope  that  it  will  be  of  value  to  workers  in 
other  fields,  at  least  to  the  extent  of  marshaling  some  of  the  literature 
for  them.  Geologists  as  a  group  are  not  mathematically  inclined,  but 
among  geologists  are  many  who  have  a  command  of  mathematics  and 
physics.  The  problem  of  writing  a  volume  of  interest  to  both  extreme 
groups  has  been  difficult. 

No  pretense  is  made  of  making  the  volume  complete  or  exhaustive. 
It  is  inevitable  that,  in  a  work  as  broad  as  the  present  one.  the  authors 
should  give  most  space  to  those  fields  and  methods  with  which  they  are 


PREFACE  ix 

most  familiar.  Nevertheless  we  believe  the  allotment  of  space  to  the 
various  techniques  and  fields  reflects  fairly  well  present-day  interests 
and  needs.  Where  methods  are  well  established  and  generally  familiar, 
as  are  the  optical  methods,  summaries  suffice ;  where  newer  and  less 
well  known  procedures  are  involved,  more  detail  has  been  sui)]:)lied. 

It  is  perhaps  inevitable  also  that  there  should  be  some  omissions  of 
important  material,  owing  to  the  wide  literature  involved.  The  authors 
would  appreciate  advices  concerning  such  omissions.  Some  selection  has 
had  to  be  made  by  the  authors,  but  as  far  as  possible  references  are 
given  to  further  details  elsewhere.  No  apology  is  made  for  using  per- 
sonal material  for  illustrations  and  examples ;  the  greater  convenience 
of  working  with  familiar  material  is  its  own  justification. 

The  present  volume  is  largely  the  joint  efi^ort  of  the  two  authors, 
but  fortunately  their  fields  of  specialization  adapt  themselves  to  a  divi- 
sion of  the  book  into  two  parts.  This  division  is  more  apparent  than 
real.  Individually  the  authors  assume  responsibility  for  their  separate 
portions;  jointly  they  assume  responsibility  for  the  apportionment  of 
space  and  the  thread  of  continuity  which  runs  through  the  book. 

This  manual  is  written  for  a  person  of  average  training  in  the 
methods  of  science.  It  is  assumed  only  that  the  worker  has  had  an 
elementary  training  in  laboratory  technique — such  that  he  can  handle  a 
chemical  balance  intelligently — and  that  he  has  a  working  knowledge 
of  elementary  physical  and  mathematical  theory  and  some  knowledge 
of  crystallography. 

We  believe  the  book  will  be  found  suitable  as  a  textbook  for  courses 
in  sedimentary  petrography.  We  have  therefore  attempted  to  explain 
both  principles  and  objectives  of  the  various  techniques  of  analysis  and 
to  raise  in  the  student's  mind  a  critical  attitude  toward  the  purposes 
and  methods  of  sedimentary  analysis. 

The  authors  are  indebted  to  many  writers  and  workers  for  the  final 
design  and  content  of  this  book.  Gessner's  excellent  treatise,  Dcr 
Schldmmanalysc,  Johannsen's  Manual  of  Pctrographic  Methods,  I.arsen 
and  Berman's  Microscopic  Determination  of  the  Non-opaque  Minerals. 
and  Boswell's  Mineralogy  of  Sedimentary  Rocks  have  been  of  inesti- 
mable value,  and  numerous  essays  and  comprehensive  articles  have  fur- 
nished inspiration  and  information.  Credit  has  been  given  in  many  of 
these  cases. 

The  authors  are  indebted  to  numerous  individuals  for  advice  and 
criticism.  Dr.  Carl  Eckart  of  the  Department  of  Physics  of  the  Uni- 
versity of  Chicago  has  helped  in  the  mathematical  treatment  of  the 
theoretical  parts  of  mechanical  analysis;  Dr.  M.  W.  Richardson  of  the 


X  PREFACE 

Dqjartment  of  Psychology  has  critically  read  the  chapters  on  statistics ; 
and  Mr.  Paul  Reiner  has  criticized  several  portions  of  the  text.  Many 
of  the  illustrations  were  prepared  by  Messrs.  H.  HoUoway,  A.  Lundahl, 
and  W.  C.  Rasmussen,  of  the  Universit}-  of  Chicago.  Among  our  col- 
leagues in  the  Department  of  Geolog}*,  Drs.  J.  H.  Bretz,  Carey  Croneis, 
and  A.  Johannsen  have  made  valuable  suggestions  as  to  style  and  con- 
tent. Among  other  geologists  and  sedimentar}-  petrologists  who  have 
read  portions  of  the  text  are  Dr.  W.  W.  Rubey  of  the  United  States 
Geological  Survey,  J.  L.  Hough  and  Dr.  Gordon  Rittenhouse  of  the 
United  States  Soil  Conservation  Service,  and  Mr.  G.  H.  Otto  of  the 
Soil  Conservation  Laborator}-,  Pasadena,  California.  Dr.  Kirtley  Mather, 
Editor  of  the  Centurv-  Earth  Science  Series,  has  been  unfailing  in  his 
encouragement  during  the  preparation  of  the  text.  Messrs.  D.  H.  Ferrin 
and  F.  S.  Pease,  Jr.,  of  D.  Appleton-Century  Company  have  smoothed 
many  difficulties  in  the  arduous  task  of  seeing  the  book  through  the  press. 

\V.  C.  Krumbeix. 
F.   T.  Pettijohx. 
Chicago,  Illinois 


CONTENTS 


PART  I 

SAMPLING.  PREPARATION  FOR  ANALYSIS.  MECfTANI- 
CAL  ANALYSIS,  AND  STATISTICAL  ANALYSIS 

PACE 

Chapter  i.    Introduction 3 

Definitions.  Properties  of  component  ^Mviins.  At- 
tributes of  grains  in  the  aggregate.  Properties  of  the 
aggregate.  Preh'minary  field  and  laboratory  sched- 
ules. Field  observations  during  sampling. 

Chapter  2.    The  Collection  of  Sedimentary  Samples  . 

Introduction.  Purposes  of  sampling.  Outcrop 
samples,  discrete,  serial,  channel,  and  compound. 
Sub-surface  samples.  Bottom  samples.  The  problem 
of  weathering.  The  problem  of  induration.  The 
collection  of  oriented  samples.  Size  of  samples. 
Containers  for  samples.  Capacities  of  sample  con- 
tainers. Labeling  and  numbering  of  samples.  Theory 
of  sampling  sediments. 


Chapter  3.    Preparation  of  Samples  for  Analysis   . 

Introduction.  Preliminary  disaggregation.  Sample 
splitting.  Preparation  for  mechanical,  mineralogical, 
shape,  and  surface  texture  analysis.  Physical  dis- 
persion procedures.  Chemical  dispersion  procedures. 
Theory  of  coagulation.  General  critique  of  disper- 
sion. Generalized  dispersion  routine. 

Chapter  4.    The  Concept  of  a  Grade  Scale 

Introduction.  Modern  grade  scales.  Problems  of 
unequal  class  intervals.  Functions  of  grade  scales, 
descriptive   and   analytic.    Choice  of   a  grade   scale. 

Chapter  5.    Principles  of   Mechanical  Analysis 

Introduction.  Classification  of  disperse  systems. 
Concept  of  size  in  irregular  solids.  Settling  veloci- 
ties of  small  particles.  Stokes'  law  and  its  assump- 


II 


43 


76 


91 


CONTEXTS 


tions.  Other  laws  of  settling  velocities.  Theory-  of 
sedimenting  systems.  Oden's  general  theorv-.  Prin- 
ciples of  modem  methods.  Principles  of  older  meth- 
ods. Theory  of  sie%nng.  Theory-  of  microscopic 
methods  of  analysis.  Summar}-. 

Chapter  6.  Methods  of  Mechanical  Analysis  .... 
Introduction.  Sieving  methods.  Direct  measure- 
ment of  large  particles.  Decantation  methods.  Rising 
current  elutriation.  Air  elutriation.  The  sedimenta- 
tion balance.  Continuous  sedimentation  cj^linders. 
The  pipette  method.  The  hydrometer  method.  Photo- 
cell method.  ^Microscopic  methods  of  analysis.  Com- 
parisons of  methods  of  mechanical  analysis. 

Chapter  7.  Graphic  Presextatiox  of  Analytical  Dat.\  . 
Introduction.  General  principles  of  graphs.  Choice 
of  dependent  and  independent  variables.  Graphs  in- 
voh-ing  t^vo  variables.  Histograms,  cumulative 
curxes,  and  frequenc\-  cur\'es.  Graphs  with  distance 
or  time  as  independent  -v-ariable.  Scatter  diagrams. 
Graphs  invohnng  three  or  more  variables.  Isopleth 
maps  and  triangle  diagrams.  Mathematical  analysis 
of  graphic  data.  Linear,  power,  and  exponential 
functions. 

Chapter  8.  Elements  of  Statistical  An.xlysis  .... 
Introduction.  The  concept  of  a  frequenc}*  distri- 
bution. Histograms  and  ctunulative  cur\-es  as  sta- 
tistical devices.  Introduction  to  statistical  measures 
of  the  central  tendencx',  the  degree  of  scatter,  and 
degree  of  as}Tnmetr\-.  Arithmetic  and  logarithmic 
frequenc)'  distributions.  Ouartile  and  moment  meas- 
ures. The  question  of  frequeno*. 

Chapter  9.    Application  of  Statistic.\l  Measl-res  to   Sedi- 
ments       

Introduction.  Ouartile  measures.  arithmetic 
geometric,  and  logarithmic.  Moment  measures, 
arithmetic,  geometric,  and  logarithmic.  Special  sta- 
tistical measures.  Sorting  indices.  Choice  of  statisti- 
cal dexices.  Statistical  correlation.  Chi-square  test. 
Theor}-  of  control.  The  probable  error. 


135 


182 


212 


228 


CONTENTS 


Xlll 

PACE 


Chapter  io.    Orientation   Analysis   of    Sedimentary    Par- 
ticles      268 

Introduction.  Collection  of  oriented  samples. 
Laboratory  analysis  of  particle  orientation.  Presen- 
tation of  analytical  data.  Statistical  analysis. 


PART  II 

SHAPE  ANALYSIS.  MINERALOGICAL  ANALYSIS, 

CHEMICAL  ANALYSIS,  AND  MASS 

PROPERTIES 

Chapter  ii.    Shape  and  Roundness 277 

Introduction.  Review  of  (juantitatiye  methods. 
Choice  of  method.  Procedure  of  analysis.  Method 
for  large  fragments.  Wadell's  method  for  sand 
grains. 

Chapter  12,    Surface  Textures  of  Sedimentary  Fragments 

and  Particles 303 

Introduction.  Surface  textures  of  large  frag- 
ments. Surface  textures  of  small  fragments. 

Chapter  13.    Preparation  of  Sample  for  Mineral  Analysis     309 
Introduction.     Disaggregation.     Clarification     of 
grains.   Special  preparation  problems. 

Chapter  14.    Separation  ]\Iethods 319 

Preliminary  concentration  of  heavy  minerals. 
Separation  on  basis  of  specific  gravity.  Heavy 
liquids.  Standardization  of  heavy  liquids.  Separa- 
tion apparatus.  Use  of  centrifuge.  Analytical 
procedure.  Separation  on  basis  of  magnetic  per- 
meability. Separation  on  basis  of  dielectric  proper- 
ties. Separation  on  basis  of  electrical  conductivity. 
Separation  on  basis  of  visual  properties.  Separation 
on  basis  of  shape.  Separation  on  basis  of  surface 
tension.  Separation  on  basis  of  chemical  proper- 
ties. Errors  in  separation.  Systematic  schemes  of 
separation. 

Chapter  15.    Mounting  for  Microscopic   Study    ....     357 
Splitting.  Mounting.  Preparation  of  thin  sections. 
Film  method  of   study. 


XIV 

Chapter 


Chapter 


Chapter 


Chapter 


Chapter 


Chapter 


Author 
Subject 


CONTENTS 

PAGE 

i6.    Optical  Methods  of  Identification   of  Min- 
erals       366 

Introduction.  The  polarizing  microscope.  Meas- 
urements of  small  particles.  Fundamental  optical 
constants.  Observations  in  ordinary  light.  Observa- 
tions in  plane  polarized  light  (crossed  nicols). 
Observations  in  convergent  light.  Special  methods 
for  the  study  of  clays.  Preparation.  Identification. 

17.  Description     of     Minerals     of     Sedimentary 
Rocks 412 

Introduction.  Mineral  descriptions.  Determina- 
tive tables.  Miscellaneous  tables.  Record  forms. 

18.  Mineral  Frequencies  and  Computation    .      .     465 

Pebble  counts.  Thin-section  analysis.  Mineral 
frequencies.  Presentation  of  results.  Calculation  of 
mineral  frequencies  based  on  analysis  of  several 
fractions.  Statistical  methods.  Mineral  variations. 
Statistical  correlation. 

19.  Chemical  Methods  of  Study 490 

Introduction.  Quantitative  analysis.  Methods. 
Computations  based  on  quantitative  analysis.  Mi- 
crochemical  methods.  Organic  content.  Insoluble 
residues.  Staining  methods. 

20.  Mass  Properties  of   Sediments 498 

Introduction.  Color  of  sedimentary  materials. 
Specific  gravity  of  mineral  grains  and  of  sedimen- 
tary rocks.  Porosity.  Definitions.  Determination  of 
porosity.  Methods  of  porosity  measurement.  Per- 
meability. Plasticity.  Definitions.  Methods  of 
measurement.  Hygroscopicity.  Miscellaneous  mass 
properties. 

21.  The  Laboratory,   Equipment,   and   Organiza- 
tion OF  Work 522 

The  laboratory.  Apparatus.  Reference  books.  Or- 
ganization. 

Index 533 

Index 539 


PART  I 

SAMPLING,  PREPARATION  FOR  ANALYSIS, 

MECHANICAL  ANALYSIS,  AND 

STATISTICAL  ANALYSIS 


CHAPTER    1 

INTRODUCTION 

SCOPE     OF     SUBJECT 

The  study  of  sediments  is  concerned  with  Xl)  the  physical  conditions 
of  deposition  of  a  sediment,  whether  glacial,  fluvial,  marine,  etc.;   (2) 
thTTime  ojTormatjon  or  age  of  the  deposit;  and  (3)  the  provenance^' 
or^area  of  denudation  that  furnished  the  material  composing  the  sedi-"" 
ment.  All  of  the  analytical  methods  described  in  this  volume  have  as  their^ 
common  aim  the  elucidation  of  these  points. 

^^arious  names  have  been  applied  to  the  detailed  study  of  sediments, 
ranging  from  sedimentation  through  sedimentary  petrology  to  sedimen- 
tology.  The  latter  word  has  not  come  into  general  use,  despite  its  con- 
ciseness and  clear  meaning ;  it  may  be  said  that  usage  favors  the  second 
term.  Whatever  name  may  be  ultimately  chosen,  there  is  no  doubt  that 
the  subject  involves  a  complete  study  of  sediments  from  the  point  of 
view  and  with  the  methods  of  pure  science.  Here  are  included  not 
only  geological  methods  of  study,  as  typified  by  field  work,  but  also  the 
methods  of  the  chemist,  the  physicist,  and  the  statistician.  In  short, 
the  complete  study  of  sediments  must  make  use  of  any  and  all  devices 
which  lead  to  an  understanding  of  the  nature  and  origin  of  the  sedi- 
ment in  question. 

This  broad  viewpoint  means  that  the  study  of  sediments  may  be 
approached  from  various  angles.  PYom  one  angle  it  may  be  a  study  of 
the  size  attributes  of  sediments  as  physical  mixtures  of  particles ;  from 
another  it  may  be  a  study  of  mineral  suites  which  by  depositional  con- 
ditions have  been  united  into  a  single  deposit ;  or,  the  sediment  may  be 
considered  as  a  composite  of  sizes,  shapes,  and  minerals  controlled  by 
complex  environmental  conditions,  and  the  investigation  may  seek  to 
evaluate  the  conditions  of  that  environment.  All  of  these  points  of  view 
are  related,  and  in  their  ultimate  end  are  directed  to  the  elucidation  of 
geological  problems,  many  in  direct  connection  with  historical  geology. 

Whatever  the  point  of  view  applied  to  sedimentary  investigations, 
laboratory  studies  will  become  an  increasingly  important  source  of  data. 

3 


4         SEDIMENTARY  PETROGRAPHY 

Not  alone  do  laboratory  analyses  supplement  and  refine  field  observa- 
tions, but  often  they  afford  data  which  cannot  be  gleaned  by  field  meth- 
ods alone.  Criticisms  are  often  leveled  against  the  application  of  refined 
methods  of  analysis  to  geological  problems,  either  on  the  ground  that 
they  give  a  specious  air  of  preciseness  to  fundamentally  approximate 
data  or  on  the  ground  that  geolog}'  is  completely  studied  in  the  field 
and  laboraton.-  studies  should  be  left  to  chemists  and  physicists.  The 
first  criticism  has  been  more  pertinent  in  the  past  than  it  is  now,  because 
even  the  rather  poor  data  afforded  by  early  laborator}-  studies  of  sedi- 
ments have  paved  the  way  for  improvements  in  technique  and  interpre- 
tation, as  well  as  for  tests  to  determine  the  degree  of  accuracy  of  the 
data.  The  second  criticism  needs  no  answer :  tEe  world  is  the  geologist's 
domainpand  he  is  justified  in  using  whatever  techniques  he  requires  to 
solve  problems  fundamentally  geological.  True,  much  remains  to  be  im- 
proved in  the  laboratory  study  of  sediments,  but  there  can  be  no  doubt 
that  interest  is  growing  in  the  subject,  and  will  continue  at  an  accelerated 
pace  during  the  next  decade  at  least. 

Among  soil  scientists  there  is  a  fairly  standardized  routine  of  analysis, 
but  this  stage  of  development  has  by  no  means  been  reached  in  sedi- 
mentary studies.  One  finds  in  a  single  year  papers  prepared  on  methods 
of  analysis  or  presentation  of  data  as  remote  as  the  poles,  and  it  is  no 
small  problem  to  determine  the  relative  dependability  of  the  methods  or 
the  degree  to  which  the  results  are  comparable.  It  is  perhaps  too  early 
to  advocate  the  adoption  of  standardized  routines  for  sedimentary 
analysis,  inasmuch  as  it  is  not  clear  in  all  cases  whether  current  data  are 
the  most  valuable  for  the  ends  toward  which  they  are  directed.  Natural 
phenomena  are  exceedingly  complex  when  examined  in  detail,  and 
anahtical  procedures  must  be  developed  which  do  not  destroy  the  very 
data  being  sought.  It  is  at  least  fortunate  that  much  current  work  is  done 
\\-ith  modem  techniques,  based  on  sound  theory,  but  there  still  are  numer- 
ous aspects  of  the  subject  where  a  state  bordering  on  chaos  prevails. 

The  scientific  study  of  sediments  may  be  divided  into  two  broad  divi- 
sions. The  first  of  these  is  the  field  and  laboratory  investigation  of  sedi- 
ments, which  yields  data  that  lead  to  their  description  and  classification. 
The  second  part  of  the  subject  is  concerned  with  the  laws  of  sedimenta- 
tion and  the  origin  of  sedimentar}'^  deposits.  To  the  first  aspect  may  be 
applied  the  term  sedimentary  petrography  or  sedimentography.  The 
second  division  is  properly  designated  as  sedimentary  petrology  or  sedi- 
vientology. 

The  distinction  between  petrography  and  petrology  is,  according  to 


INTRODUCTION  5 

T3'rrell,^  that  petrography  is  the  study  of  rocks  as  specimens,  whereas 
petrology  is  the  science  of  rocks,  that  is.  of  the  more  or  less  detinite 
units  of  which  the  earth  is  huilt.  These  general  terms  may  apply  equally 
well  to  igneous,  sedimentary,  or  metamorphic  rocks.  Specifically,  Mil- 

ner  -  has  defined  sedimentary  petrology  as  follows : 

[Sedimentary]  petrolojry  connotes  something:  more  than  mere  description 
of  rock-types  based  on  microscopical  analysis,  and  in  its  wider  sense  embraces 
comprehensive  investijrations  of  their  nature,  origin,  mode  of  deposition, 
inherent  structures,  mineralogical  composition,  mechanical  constitution,  tex- 
tural  analysis,  various  chemical  and  physical  properties,  in  short,  all  data 
leading  to  an  understanding  of  the  natural  history  of  the  formations  under 
review. 

In  practice,  one  seldom  distinguishes  between  sedimentary  petrography 
and  sedimentary  petrology.  JMost  studies  of  sediments,  perhaps,  are  di- 
rected toward  the  petrological  aspects  of  the  problem:  the  clarification 
of  details  of  origin,  transportation,  deposition,  or  diagenesis.  Actually, 
of  course,  the  petrographic  aspects  precede  the  petrological.  because  it 
is  first  necessary  to  assemble  facts  about  the  sediments,  both  from  the 
field  and  the  laboratory. 

It  is  only  with  the  methods  of  petrographic  analysis  that  this  volume 
is  concerned.  The  purpose  of  the  book  is  to  present  theories  and  methods 
of  examining  sediments,  from  the  field  sampling  to  the  final  graphic 
and  statistical  analysis.  Petrological  aspects  are  only  touched  upon  as 
they  apply  to  certain  details  of  analytical  methods  and  to  indicate  the 
underlying  purposes  of  the  laboratory  investigations. 

The  point  of  view  which  this  book  presents  is  based  on  the  premise 
that  every  sediment  is  a  response  to  a  defiiiite^et  of  environmental  con- 
ditions. WHiatever  the  conditions  may  be,  there  are  characteristics  of 
the  sediment  which  may  be  measured  in  the  laboratory  and  which  reflect 
the  environmental  factors  that  produced  them.  A  change  in  the  environ- 
ment (pressure,  temperature,  chemical  associations)  results  in  a  corre- 
sponding adjustment  of  the  rock  to  its  new  conditions.  Owing  to 
incomplete  adjustments,  however,  the  rock  materials  may  have  some 
characteristics  inherited  from  previous  states  and  some  due  to  conditions 
existiiig  at  the  moment.  Refinements  of  technique  and  interpretation, 
however,  make  possible  the  unraveling  of  even  such  complexities.  This 
point  of  view  requires  a  careful  consideration  of  all  techniques  used,  a 
critical  evaluation  of  data,  and  a  reliance  on  the  methods  of  pure  science. 

^  G.  W.  Tyrrell,  The  Principles  of  Petrology  (London,  1926).  p.  i. 
-  H.  B.  IMihier,  Report  of  researcii  on  sedime<itary  rocks  by  British  petrologists 
for  the  year  1927:  Rely.  Com.  Sed.,  Nat.  Research  Council,  1928.  p.  0. 


6         SEDIMENTARY  PETROGRAPHY 

Mineralogical,  as  well  as  size  and  shape  studies,  throw  light  on  the 
physical  conditions  of  deposition,  but  it  is  largely  on  minerals  alone  that 
we  have  to  rely  for  the  determination  of  age  (petrographic  correlation) 
and  for  knowledge  of  the  provenance  of  the  deposit.  The  problem  of 
provenance  or  of  source  of  the  sediments  involves  a  knowledge  of  the 
composition  of  the  parent  rock  tvpes,  a  knowledge  of  mineral  stability  in 
respect  both  to  climate  and  to  mechanical  wear ;  and  an  understanding  of 
the  relations  between  mineral  frequencies  and  the  transportation  of  the 
sediment. 

Twenhof el  ^  has  defined  sedimentation  as  follows  : 

Sedimentation  includes  that  portion  of  the  metamorphic  cyclefromthe  de- 
struction  of  the  parent  rock,  no  maner  what  its  origin  or  constitution,  toOie 
consolidation  of  the  products  derivedTfrom  that  destruction  (with  any  addi- 
tions from  other  sources)  into  pother  rock. 

The  term  sedivieutatioji  thus  connotes  a  process  and  is  to  be  distin- 
guished from  the  products  of  that  process,  the  sedimentar)-  rocks.  Twen- 
hofel  -  has  also  defined  a  sediment  as 

...  a  deposit  of  solid  material  (or  material  in  transportation  which  may 
be  deposited;  made  from  any  medium  on  the  earth's  surface,  or  in  its  outer 
crust  imder  conditions  of  temperature  approximating  those  normal  to  the 
surface. 

PROPERTIES     OF     SEDIMENTS 

Every-  sedimentar)'  deposit  has  certain  fundamental  characteristics  or 
properties,  some  of  which  are  associated  with  the  individual  panicles  and 
others  \\-ith  the  aggregate  of  all  the  particles.  In  some  cases  there  is  an 
overlap,  but  the  following  classification  indicates  the  principal  character- 
istics. 

Properties  of  component  grains.  The  fundamental  properties  of  the 
component  grains  of  a  sediment  are  (i)  sizes,  (2)  shapes.  (3^  surface 
textures,  and  (4)  mineralogical  composition.  The  last  characteristic  de- 
termines such  attributes  as  densit)-.  hardness,  color,  and  the  like,  of  each 
grain.  Each  of  these  four  fundamental  properties  may  be  examined  in  the 
laborator}-. 

The  f  imdamental  properties  of  the  particles  are  important  because  they 
reflect  either  directly  or  indirectly  many  of  the  vicissitudes  through  which 
the  sediment  has  passed.  Size  is  related  to  the  medium  of  transportation 

iW.  H.  Twenhofel,  Treatise  on  Sedimentation,  2nd  ed  (Baltimore,  1932),  P- 
xxvii. 

-  hoc.  cit. 


INTRODUCTION  7 

and  its  velocity ;  shape  is  related  in  part  to  the  medium  of^jtrans]2ortation 
and  lo  the  distance  and  rigor  of  transport ;  surface  texture  may  reflect 
sul)sc(|ucnt  clKingcs  due  lo  sohilion,  or  il  nia_\-  furnisli  clues  ti>  \\\c  luclliod 
of  transijurtutiuji.  iMnally,  the  mincralogical  c<^)_mi"  '^'' '' '"  indi rales  pos- 
sihle  source  rocks,  as  well  as  any  post-depositional  chaiiL^es  tliat  may  have 
occurred. 

^Attributes  of  component  grains  in  the  aggregate.  Interest  in  the  compo- 
nent grains  of  a  sediment  often  involves  the  frequency  distributions  of 
grain  properties  in  the  aggregate ;  for  example,  size  is  expressed  in  terms 
oTTsize  frequency  distnbufion  (mechanical  analysis)  rather  than  by 
cataloging  the  individual  size  of  each  i:)article.  In  similar  manner,  shape, 
mineral  composition,  and  other  properties  may  be  considered  statistically 
as  distributions  of  grain  properties.  Each  of  these  distributions  may  then 
be  studied  in  terms  of  their  own  characteristics,  such  as  average  size, 
average  density,  average  degree  of  sorting  or  sizing,  and  the  like. 

Another  important  attribute  of  the  component  grains  in  terms  of  their 
aggregate  properties  is  the  orientation  of  the  particles  in  space  (the  "fab- 
ric" of  the  rock).  The  orientation  of  the  particles,  considered  statistically, 
may  indicate  among  other  things  whether  deposition  was  subaqueous  or 
subaerial. 

PKopcrties  of  thejwgregate.  In  addition  to  the  attributes  of  the  indi- 
vidual particles,  there  are  various  aggregate  properties  of  the  sediment 
which  are  important.  These  include  (i)  the  ccnienlation  of  the  jjarticles 
in  the  specimen,  (2)  structures,  such  as  bedding,  concretions,  and  the 
like,  and  (3)  the  color  of  the  sednriciit.  These  properties  also  furnish 
information  about  the  history  of  the  sediment.  The  color  of  the  sediment 
in  the  aggregate,  including  the  nature  of  the  cement,  may  help  determine 
conditions  of  deposition,  or  post-depositional  changes.  Some  of  these 
aggregate  properties  may  be  controlled  in  large  part  by  projK^rties  of  the 
component  grains.  The  orientation  of  the  particles  helps  determine  such 
structures  as  bedding  and  in  addition  may  be  a  factor  in  such  aggregate 
properties  as  porosity  and  permeability. 

To  a  large  extent  the  aggregate  properties,  as  they  are  defined  above, 
may  be  studied  in  the  field,  whereas  the  properties  of  the  comi)onent 
grains  and  their  distribution  in  the  sediment  may  best  be  considered  in 
the  laboratory.  A  thorough  examination  of  sediments  therefore  involves 
a  combination  of  field  and  lal.ioratory  work,  and  in  the  modern  develop- 
ment of  the  science  neither  is  comjilcte  without  the  other.  Laboratory 
methods  must  be  quantitative,  inasmuch  as  quantitative  data  are  necessary 
to  the  development  of  complete  theories  of  sediment  transportation  and 


8         SEDIMENTARY  PETROGRAPHY 

deposition.  In  the  future  development  of  the  science  there  can  be  little 
doubt  that  this  quantitative  and  theoretical  aspect  of  the  science  will  be 
increasingly  emphasized. 


PRELIMINARY  FIELD  AND  LABORATORY  SCHEDULE 

It  is  appropriate  that  a  schedule  outlining  the  examination  of  sediments 
be  given  here  as  a  preliminary  outline  by  which  the  scope  of  the  present 
book  may  be  indicated.  The  schedule  is  divided  into  two  parts :  the  first 
lists  the  features  of  the  rock  that  may  be  observed  in  the  field  by  ordinary' 
geological  methods,  and  the  second  includes  those  characteristics  which 
are  best  determined  quantitatively  in  the  laboratory.  This  volume  is 
directly  concerned  with  the  details  of  the  second  section  of  the  schedule. 

The  following  organization  of  data  on  sedimentary'  rocks  is  adapted, 
with  some  changes,  from  the  excellent  report  on  the  field  description  of 
sedimentary  rocks  by  Goldman  and  Hewett.^ 

Field  Schedule 

External  form  of  the  rock  unit 

Dimensions,  persistence,  regularitj- 

Color 

\\"et  or  dn,',  on  basis  of  accepted  color  scheme  - 

Bedding 

Sharp  or  transitional 

Plane,  midulatorj-,  or  ripple-marked 

Thickness 

Constant  or  variable 

Rhythmic  or  random 
Attitude  and  direction  of  bedding  surfaces 

Horizontal,  inclined,  or  curved 

Parallel,  intersecting,  or  tangential  to  other  beds 

Relation  of  particle  properties  to  attitude  and  direction  ^ 
Markings  of  bedding  surfaces 

Mudcracks,  rain  prints,  footprints,  etc. 
Disturbances  of  bedding 

Folding  or  crumpling 

Intraformational  conglomerates 

1  M.  I.  Goldman  and  D.  F.  Hewett,  Schedule  for  field  description  of  sedimentary 
rocks :  National  Research  Council,  Committee  on  Sedimentation,  Washington,  D.  C. 

2  M.  I.  Goldman  and  H.  E.  Merwin,  Color  chart  and  explanation  of  the  color  chart 
for  the  description  of  sedimentary  rocks,  prepared  imder  the  auspices  of  the  Divi- 
sion of  Geologv  and  Geography,  of  the  National  Research  Council,  Washington. 
D.  C,  1928. 

3  Such  properties  as  porosit>'  and  permeabilin-  may  be  determined  in  the  labora- 
tor>-  from  oriented  field  samples. 


INTRODUCTION  9 

I    Concretions 
1        Kinds,  size 

Condition  and  distribution 
.        Orientation  with  respect  to  bedding 
I        Form,  size,  composition 

Internal  structure 

Boundary  against  country  rock 
1  Sharp  or  transitional 

Relation  to  bedding 

Distribution 
I  Random  or  regular 

Organic  constituents  ^ 

Kinds,  size 
[      Condition 
•  Whole  or  broken 

Distribution 
I      Orientation  with  respect  to  bedding 

Laiioratoky  Schedule  2 

Preparation  of  sample  for  analysis 
Sample  splitting 
Disaggregation  and  dispersion 

Particle  size  analysis 

Shape  analysis 
Roundness 
I       Sphericity 

Surface  texture  analysis 

Mineralogical  analysis 
\      Separation  of  heavy  minerals 
Microscopic  examination 

Orientation  of  particles  in  sample 

Mass  properties  of  sediment 
Porosity  and  permeability 
.       Specific  gravity 

'  Chemical  analysis 

Graphic  presentation  of  data 

Statistical  analysis  of  data 

It  may  be  mentioned  that  although  the  above  schedules  present  little 
overlap,  in  actual  practice  some  of  the  quantitative  data  arc  obtained 

1  This  refers  only  to  the  megascopic  remains.  Microfossils  demand  specialized 
laboratory  techniques  not  included  in  tliis  volume. 

-  Tliis  outline  presupposes  that  samples  have  been  collected  in  the  field.  For  a 
field  outline  of  sampling  routine,  see  below;  the  subject  of  sampling  forms  Chapter 
2  of  this  volume. 


1 

V. 


lo        SEDIMENTARY  PETROGRAPHY 

directly  in  the  field  and  part  of  the  field  data  are  secured  in  the  laboratory. 
For  example,  the  study  of  pebble  orientation  may  be  conducted  at  the 
exposure,  and  if  the  rock  being  investigated  is  consolidated,  some  field 
observations,  such  as  bedding  and  other  structures,  may  be  observed  from 
the  sample.  On  the  other  hand,  if  the  material  is  incoherent,  such  features 
as  bedding,  grain  orientation,  and  the  like  are  not  preserved  during  sam- 
pling, and  steps  must  be  taken  to  complete  such  observations  in  the  field. 
It  may  be  said  as  a  general  rule  that  too  much  data  cannot  be  collected. 
This  is  true  especially  in  a  field  such  as  sedimentary  petrology,  where 
research  has  not  yet  advanced  to  the  stage  where  it  may  be  predicted 
whether  a  given  set  of  observational  data  are  pertinent  to  the  study  or  not. 
Schedule  of  field  observations  during  sampling.  In  addition  to  the 
general  observations  to  be  made  on  the  formation  as  a  whole,  as  outlined 
above,  there  are  several  specific  observations  to  be  made  in  the  field  at 
the  time  a  sample  is  collected.  These  specific  data  include : 

1.  Location  of  the  sampling  locality,  either  as  a  point  on  a  map  or  with  ref- 
erence to  some  easily  located  landmark. 

2.  Nature  of  the  sampling  point,  as  an  outcrop,  roadcut,  or  ditch. 

3.  Nature  of  the  material  sampled,  including  type  of  rock,  portion  of  bed 
sampled,  and  so  on. 

4.  Nature  of  the  sample,  as  from  a  single  point,  a  composite  sample  from 
several  parts  of  the  bed,  as  a  channel  througli  the  bed,  etc. 

5.  Relation  of  sample  to  surrounding  rock,  as,  for  example,  from  just  beneath 
a  stained  zone,  whether  cut  by  joints,  and  the  like. 

6.  Topography  of  sampling  site,  as  river  bottom,  terrace,  top  of  hill,  etc. 

7.  Depth  of  sample  beneath  immediate  surface  at  point  of  sampling. 

8.  Zone  of  weathering  from  which  sample  is  taken,  if  this  can  be  determined. 

9.  A  field  evaluation  of  the  total  condition  of  the  sample  for  the  purpose 
desired,  as  excellent,  good,  fair,  poor.  This  is  desirable  when  many  samples 
are  collected  and  the  laboratory  work  may  involve  using  scattered  samples 
to  outline  the  scope  of  the  study. 


It  cannot  be  emphasized  too  strongly  that  the  detailed  investigation  of 
sediments  should  not  be  a  hurried  process.  The  investigator  should  spend 
adequate  time  in  the  field,  examining  the  general  set-up  of  the  problem, 
locating  sampling  sites,  measuring  sections,  and  in  general  accumulating 
sufficient  data  so  that  work  need  not  be  delayed  during  the  ensuing  labora- 
tory season  owing  to  failure  to  observe  adequately  in  the  field.  As  a  gen- 
eral rule  it  is  better  to  acquire  too  many  samples  and  field  observations 
than  not  enouirh  of  either. 


CHAPTER   1 
THE  COLLECTION  OF  SEDIMENTARY  SAMPLES 

INTRODUCTION 

The  physical  impossibility  of  analyzing  an  entire  sedimentan'  formation, 
or  even  an  appreciable  part  of  one.  renders  it  necessary-  to  work  \\-ith 
samples.  A  sample  is  assumed  to  be  a  representative  part  of  the  formation 
at  the  point  of  sampling,  or  sometimes  of  the  entire_formation.  How 
nearly  it  is  representative  determines  in  large  measure  the  validity  of  the 
final  conclusions,  assuming,  of  course,  that  the  methods  of  analysis  cor- 
rectly describe  the  sample. 

Interest  in  a  sediment  may  arise  in  a  \-ariet}-  of  ways.  It  may  be  a 
question  of  the  economic  exploitation  of  a  limestone  or  a  tire-clay ;  it  may 
be  merely  a  desire  to  supplement  general  geological  field  work  with 
some  quantitative  data.  On  the  other  hand,  the  study  may  involve  consid- 
eration of  the  conditions  of  sedimentation,  the  agents  that  formed  the 
deposit,  the  possible  source  rocks,  and  the  like.  One  may  thus  argue  that 
the  process  of  sampling  may  be  either  casual  or  precise,  depending  upon 
the  ends  in  view.  This  is  an  attitude  with  which  the  authors  cannot  wholly 
concur.  It  seems  reasonable  that  if  a  sample  is  worth  collecting,  it  is 
worth  collecting  well. 

PURPOSES      OF      S  .A  M  P  I.  I  N  G 

Samples  for  display.  In  unconsolidated  material  a  display  sample  may 
consist  merely  of  a  small  vial  of  the  material — sand.  silt,  or  clay — or  it 
may  consist  of  a  selection  of  pebbles  in  a  tray.  The  collection  of  such 
samples  affords  no  particular  difficulties  unless  structures,  such  as  bed- 
ding and  grain  orientation  are  to  be  preserved. 

If  the  structure  of  unconsolidated  sediments  is  to  be  preserved,  and  if  the 
material  is  sufficiently  fine-grained  to  be  cohesive,  an  undisturbed  sample  may 
be  collected  by  a  routine  procedure.  Antevs  ^  describes  the  process  for  collect- 
ing unconsolidated  varved  clay  samples  as  follows : 

1  E.  Antevs,  Retreat  of  the  last  ice-sheet  in  eastern  Canada :  Canadian  Geol.  Sur- 
vey, Memoir  146,  p.  12,  1925. 

II 


12        SEDIMENTARY  PETROGRAPHY 

"The  samples  are  taken  in  tight  troughs  of  zinc  plate,  conveniently  195^ 
inches  long,  2  inches  wide,  and  ^  inch  high.  The  face  of  the  clay  bank  is 
carefully  smoothed  and  the  trough  is  cautiously  pressed  in,  a  knife  being  used 
to  cut  away  the  clay  just  outside  the  edges,  until  the  trough  is  entirely  filled 
with  clay.  The  troughs  are  then  cut  out  from  the  bank,  and  the  projecting 
clay  is  removed." 

If  the  rock  is  indurated,  the  display  sample  may  consist  of  a  chip  or  a 
trimmed  hand  specimen.  The  hand  specimen  should  be  about  3x4  in. 
in  size  and  from  i  to  13^  in.  thick.  The  smallest  dimension  is  usually 
chosen  at  right  angles  to  the  bedding.  The  corners  of  the  specimen  should 
be  rectangular  and  not  rounded,^  so  that  they  conform  to  the  standards 
set  for  hand  specimens  of  igneous  rocks. 

Samples  for  commercial  analysis.  Samples  of  sediments  collected  for  com- 
mercial analysis  present  a  number  of  problems  peculiar  to  the  purposes  for 
which  they  are  used.  In  general,  however,  the  methods  of  sampling  are  similar 
to  those  used  for  the  detailed  laboratory  study  of  sediments  for  scientific 
purposes. 

Commercial  analyses  may  be  made  for  such  diverse  purposes  as  the  deter- 
mination of  the  CaO  or  I\IgO  content  of  limestone ;  the  fuel  value  or  the 
determination  of  special  constituents  of  coal ;  the  value  of  gravel  for  use  as 
road  materials ;  or  the  value  of  silica  sand  for  glass-making.  Regardless  of 
the  purpose  of  the  analysis  or  the  state  of  induration  of  the  material,  the 
prime  requisite  is  that  the  sample  must  be  representative  of  the  formation.  A 
specialized  aspect  of  commercial  sampling  is  the  prospecting  of  economically 
useful  deposits.  This  topic  does  not  properly  come  within  the  scope  of  the 
volume,  and  interested  readers  are  referred  to  standard  texts  on  the  subject.^ 

Samples  for  detailed  laboratory  investigations.  A  critical  choice  of 
samples,  necessary  in  any  detailed  study  of  sediments,  should  take  into 
consideration  as  many  elements  of  the  problem  as  may  be  evaluated,  so 
that  the  final  results  are  not  weakened  by  poor  samples,  collected  without 
regard  to  the  purposes  of  the  study. 

Sediments  may  vary  in  terms  of  the  coarseness  of  their  particles,  in  the 
degree  of  sorting  or  homogeneity,  in  their  manner  of  bedding  or  arrangement 
of  particles,  in  their  degree  of  induration,  and  in  their  degree  of  alteration. 
In  any  given  formation,  one  must  also  consider  the  vertical  and  lateral  varia- 
tions in  size  of  the  formation,  the  presence  or  absence  of  bedding,  changes 
in  the  thickness  of  the  formation  or  its  individual  beds,  and  changes  in  the 
shape,  size,  and  arrangement  of  its  particles.  Further,  some  sediments  are 
exposed  to  view  in  extensive  outcrops,  and  others  are  hidden  witlnn  drilled 
wells  or  are  covered  by  bodies  of  water.  Each  of  these  cases  presents  its  own 
problems,  some  of  which  are  far  from  being  solved. 

1  A.  Johannsen,  Manual  of  Petrographic  Methods,  2nd  ed.  (New  York,  1918), 
p.  607. 

-C.  Raeburn  and  H.  B.  Milner,  Alluvial  Prospecting  (London,  1927). 


COLLECTION  OF  SAMPLES  13 

Unfortunately,  there  is  at  present  no  general  mathematical  theory  of 
sampling  sediments  which  enables  one  in  every  case  to  determine  the 
technique  of  sampling  a  priori;  the  science  of  sampling  is  still  in  the 
stage  where  "rule  of  thumb"  procedures  predominate.  These  practical 
rules  are  based  on  experience  and  thus  are  satisfactory  in  an  em[)irical 
sense;  happily,  they  are  supported  by  favorable  results,  and  to  some 
extent  they  may  be  checked  by  statistical  theory.  In  a  later  section  of  the 
chapter  some  of  the  elementary  aspects  of  sampling  theory  will  be  dis- 
cussed. 


OUTCROP     SAMPLES 

Sedimentary  formations  exposed  in  outcrops  are  the  most  convenient 
to  sample  because  the  sampling  site  may  be  examined  in  detail  and  some 
judgment  may  be  used  in  choosing  the  particular  point  of  sampling. 
Given  such  an  outcrop,  the  problem  involves  the  number  of  samples  to 
be  taken,  the  size  of  the  samples,  and  the  desirability  of  preserving 
structures  or  particle  orientations. 

Spot  samples.'^  An  isolated  sample  taken  at  a  particular  point_on_tiie 
outcrop  may  be  termed  a  spot  sample,  or  a  discrete  sample.  Such 
samples  are  collected  separately  and  kept  separately,  being  thus  distin- 
guished from  composite  samples. 

The  decision  to  collect  a  spot  sample  may  be  based  on  the  apparent 
homogeneity  of  the  deposit  as  exposed  to  view.  If  the  outcrop  represents 
a  bank  of  unbedded  sand  or  silt,  or  even  glacial  till,  with  no  changes  in 
composition  detectable  by  eye,  a  single  sample  may  be  taken  from  any 
convenient  point  along  the  outcrop.  Unless  the  object  of  the  study  is  the 
investigation  of  weathering,  the  principal  precaution  to  be  followed  is 
that  no  weathered  or  altered  phases  of  the  formation  be  included.  This 
necessitates  taking  the  sample  at  some  distance  below  the  soil  horizon, 
and  beneath  the  surface  of  the  outcrop  face. 

An  area  on  the  outcrop  face  is  first  cleaned  or  scraped,  and  a  sample 
taken  by  scooping  out  a  limited  amount  from  a  square  or  circular  zone. 
In  general  it  may  be  desirable  to  have  the  depth  of  penetration  about  as 
great  as  the  width  of  the  face  sampled,  so  that  a  roughly  cubical  or 
cylindrical  volume  is  obtained.  The  sediment  may  be  removed  with  a 
scoop,  the  ])oint  or  chisel  of  a  hammer,  or  a  small  i)ick.  A  bag  or  other 
receptacle  should  be  at  hand  so  that  none  of  the  sample  is  spilled  or  lost. 

1  The  term  grab  sample  is  often  used  for  individual  samples  collected  at  a  given 
point.  The  term  may  imply  a  degree  of  carelessness  in  the  collection  of  the  sample, 
and  the  substitute  term  spot  sample  is  used  here. 


14        SEDIMENTARY  PETROGRAPHY 

When  the  exposure  is  horizontal,  such  as  the  surface  of  a  dune  or  a 
beach,  the  sample  may  be  collected  in  various  ways.  A  simple  method  is 
to  dig  a  shallow  hole  with  vertical  walls,  and  to  take  the  sample  from 
one  of  the  walls  so  exposed.  A  more  convenient  method  is  to  have  a  short 
section  of  downspout  pipe  with  fluted  sides, ^  which  may  be  forced  down 
into  the  sand  for  a  distance  of  six  inches  or  so.  The  sand  around  the 
tube  is  then  dug  away,  and  by  inserting  the  hand  beneath  the  tube,  the 
entire  sample  may  be  preser^-ed,  even  in  dry  sand. 

A  spot  sample  is  strictly  valid  only  for  the  point  being  sampled.  A 
single  sample,  used  to  generalize  about  the  material  exposed  in  the  out- 
crop, should  be  relied  upon  only  when  the  sediment  is  quite  homogeneous, 
or  when  a  limited  part  of  the  formation  is  to  be  studied.  If  variations 
occur  vertically  or  laterally,  or  if  an  extensive  area  is  to  be  studied,  it  is 
better  to  rely  on  a  series  of  spot  samples. 

Serial  samples.  Spot  samples  which  are  part  of  a  related  set  of  samples 
may  be  called  serial  samples.  They  are  collected  in  accordance  with  some 
predetermined  plan,  involving  an  arbitrary  but  usually  equal  interval  of 
spacing.  Each  spot  sample  is  kept  in  a  separate  container  and  is  usually 
handled  as  a  imit  during  the  study. 

Serial  samples  may  be  arranged  along  a  line  of  traverse  across  a  formation, 
or  they  may  represent  a  set  of  samples  collected  at  intervals  along  a  river 
or  beach.  Likewise,  the  serial  set  may  extend  vertically  across  the  thickness 
of  a  formation.  When  the  series  is  arranged  along  a  line  in  this  manner,  the 
series  is  linear,  and  it  is  not  necessary  that  the  line  be  straight. 

In  contrast  to  linear  series  of  samples,  either  horizontal  or  vertical,  is  a 
grid  series  of  samples  collected  over  an  area,  or  over  the  face  of  a  vertical 
outcrop.  The  grid  may  represent  a  square  pattern  of  lines  superimposed  over 
the  area,  spaced  according  to  the  detail  with  which  the  work  is  to  be  carried 
on.  Samples  are  collected  at  the  points  of  intersection  of  the  crossing  lines. 
In  some  cases  section  line  roads  or  even  township  lines  may  be  used  for  grid 
patterns.  From  this  extreme  the  grid  spacing  may  range  down  to  a  foot  or 
so  for  very  detailed  studies.  As  a  first  approach  one  chooses  a  grid  interval 
having  sampling  points  spaced  equally  over  the  area,  at  such  distances  as  the 
detailed  nature  of  the  study  suggests.  As  samples  are  collected  at  the  grid 
intersections,  attention  is  paid  to  any  noticeable  variations  that  occur  from 
one  sample  to  the  next.  If  occasional  changes  are  noted,  it  may  be  well  to 
collect  an  intermediate  sample  at  half -grid  interval,  to  cover  the  transition; 
on  the  other  hand,  if  prominent  changes  occur  between  each  succeeding 
sample,  it  may  be  desirable  to  halve  the  interval  over  the  entire  area. 

The  primary  function  of  the  grid  method  of  samphng  is  to  assure  one- 
self of  random  samples,  distributed  more  or  less  evenly  over  the  area 

1  This  device  was  developed  by  G.  H.  Otto ;  its  use  is  described  in  W.  C.  Krum- 
bein,  The  probable  error  of  sampling  sediments  for  mechanical  analysis:  Am.  Jour. 
Sci.,  vol.  2~,  pp.  204-214,  1934. 


COLLECTION  OF  SAIMPLES 


15 


considered.  If  a  grid  pattern  is  not  used,  and  samples  are  taken  at  scat- 
tered intervals,  personal  bias  may  influence  one  in  the  location  of  the 
samples. 

In  most  cases  a  simple  rectangular  grid,  as  illustrated  in  Figure  i,  is 
probably  to  be  preferred,  although  other  patterns  will  suggest  themselves. 
In  the  study  of  an  alluvial  fan.  for  example,  one  may  choose  a  series 
of  concentric  arcs  of  circles  to  space  the  samples  equally  from  the  apex 
of  the  fan ;  the  cross  lines  may  be  radial  from  the  apex  or  may  cross 


0         O O— — 0 0 0 o 

o o o 0 6 6 6 


Fig.  I. — Simple  rectangular  grid 
for  sample  control.  The  distance 
between  sampling  points  may  be 
chosen  as  a  function  of  the  size  of 
the  area  and  the  number  of  samples 
to  be  collected. 


Fig.  2. — Radial  logarithmic  sam- 
pling grid,  designed  to  furnish  more 
detailed  data  near  source  of  sediment. 
Grids  of  this  type  may  apply  to 
studies  of  alluvial  fans. 


the  arcs  at  equal  intervals.  In  some  sampling  problems  it  may  be  logical 
to  space  the  grid  lines  on  logarithmic  intervals,  as  shown  in  Figure  2. 
For  example,  if  it  is  suspected  that  some  property  of  a  sediment  varies 
exponentially  from  its  source  outward,  a  closer  spacing  near  the  source 
will  afford  more  critical  data  in  the  steep  part  of  the  exponential  curve. 
Such  exponential  functions  (see  Chapter  7)  may  be  expected  in  some 
cases  of  average  size  or  thickness  of  deposits. 

The  collection  of  serial  samples,  whether  linear  or  grid,  assumes  that 
the  formation  will  outcrop  at  the  grid  points.  If  this  is  not  so,  one  must 
choose  between  taking  a  sample  from  the  nearest  outcrop  or  drilling  at 
the  exact  sampling  point. 

Some  of  the  situations  in  which  serial  samples,  either  linear  or  grid,  are 
indicated  may  be  mentioned.  If  a  beach  is  to  be  studied,  the  samples  may  be 
arranged  as  a  linear  series  near  the  strand  line.  If,  however,  only  a  limited 
stretch  of  the  beach  is  being  investigated,  a  grid  may  be  used  involving  one 
set  of  samples  along  the  strand,  another  in  the  shallow  off-shore  water,  and 
a  third  higher  up  on  the  beach.  Rivers  afford  anodier  instance.  A  linear  set 
may  be  taken  to  study  the  changes  in  the  sediment  downstream,  or  a  grid 
may  be  laid  over  a  terrace  or  river  bar  to  study  detailed  variations.  Vertical 


i6        SEDIMENTARY  PETROGRAPHY 

series  of  samples  are  indicated  where  there  are  several  formations  in  the 
study  but  each  formation  is  itself  a  more  or  less  homogeneous  unit.  In  general, 
serial  samples  are  indicated  whenever  the  variations  from  point  to  point  along 
or  over  a  deposit  are  the  subject  of  study.  Soil  surveys  have  long  used  the 
principle  of  grid  sampling,  in  which  the  data  from  numerous  closely  spaced 
samples  are  used  to  prepare  maps  of  soil  tj'pes.^ 

When  collecting  serial  samples  it  may  be  difficult  to  decide  whether 
the  interval  between  samples  should  be  relatively  long  or  short.  One 
solution  is  to  collect  the  samples  at  the  shorter  intervals,  but  in  the 
laboratory  to  analyze  only  alternate  samples.  If  these  suffice  to  bring 
out  the  variations  adequately,  the  intermediate  samples  may  be  discarded. 
Where  abrupt  changes  occur,  the  intermediate  samples  may  be  included 
to  cover  the  transitions. 

The  same  principles  of  laying  out  grids  apply  to  samples  collected  from 
bodies  of  water.  The  grid  may  be  made  to  conform  with  the  configuration 
of  a  bay,  or  it  may  be  based  simply  on  a  series  of  sailing  courses,  more  or 
less  parallel,  and  extending  at  right  angles  from  the  shore.  By  an  appropriate 
spacing  of  the  sampling  intervals  in  relation  to  the  sailing  courses,  any  type 
of  grid  may  be  followed. 

As  far  as  the  authors  are  aware,  the  first  clear  statement  of  tlie  value 
of  grid  patterns  in  the  control  of  sampling  was  made  by  Pratje  in  1932.- 
He  pointed  out  that  by  laying  a  closely  spaced  network  over  the  region 
to  be  studied  a  sufficient  number  of  related  samples  could  be  obtained 
to  evaluate  the  environment.  He  designated  liis  approach  as  the  regional- 
statistical  method  of  studying  sediments. 

Channel  samples.  A  cliannel  sample  may  be  defined  as  an  elongated 
sample  taken  from  a  relatively  narrow  zone  of  an  outcrop.  The  channels 
involve  a  continuous  strip  of  the  material  from  top  to  bottom  of  the 
channel  zone.  Cliannel  samples  are  important  whenever  the  average  char- 
acteristics of  the  formation  are  to  be  detennined.  Consequently  such 
samples  are  widely  used  in  commercial  sampling,  as  for  road  gravel, 
fire-clays,  molding  sand,  and  the  like.  For  scientific  studies  channel 
samples  are  to  be  avoided  if  they  extend  through  zones  of  weathering 
or  alteration  or  in  any  other  manner  introduce  complexities  into  the 
sample  collected.  There  are  cases,  however,  in  which  channel  samples 

1  Samples  for  soil  surveys  are  usually  collected  from  the  center  point  of  the  rec- 
tangular grid  pattern,  instead  of  from  the  points  of  intersection.  Either  approach  is 
equally  logical. 

-  O.  Pratje,  Die  marinen  Sedimente  als  Abildung  ihrer  Umwelt  und  ihre  Aus- 
wertung  durch  regional-statistische  Methode :  fortschritte  der  Geol.  u.  Palaon.,  voL 
II,  pp.  220-245,  1932. 


COLLECTION  OK  SAMPLES  17 

are  indicated  for  detailed  work  on  sediments,  and  these  will  be  con- 
sidered below. 

The  channel  sampling  method  may  be  illustrated  by  a  common  pro- 
cedure used  in  obtaining  samples  of  road  gravels.  The  more  or  less 
vertical  face  of  the  gravel  pit  is  cleared  of  surface  material  for  a  width 
of  about  a  foot  or  eighteen  inches,  extending  from  top  to  bottom  of  the 
exposure.  The  fallen  material  is  cleared  from  the  base  of  the  wall,  and  a 
shallow  indentation  is  dug  at  the  base  of  the  cleared  strip,  for  the  in- 
sertion of  the  edge  of  a  tarpaulin  beneath  the  channel  zone.  After  these 
preliminaries,  a  small  sugar  scoop  is  used  to  scrape  the  material  from  the 
channel,  allowing  it  to  fall  on  the  tarpaulin  below.  The  depth  of  the 
channel  is  made  equal  to  the  diameter  of  the  largest  pebble  in  the  sam- 
pling zone.  This  insures  a  representative  sample,  on  the  whole. 

After  the  material  has  been  removed  from  the  face  of  the  outcrop, 
there  will  be  a  channel  extending  from  top  to  bottom,  about  a  foot  wide 
and  two  or  three  inches  deep.  The  material  from  this  strip  lies  on  the 
tarpaulin  below.  The  tarpaulin  with  its  pile  of  gravel  is  pulled  away 
from  the  face,  and  the  gravel  is  thoroughly  mixed  with  a  shovel,  and 
spread  into  a  roughly  conical  heap.  This  heap  may  then  be  divided  into 
four  quarters  with  the  shovel,  and  two  alternate  quarters  discarded,  if 
the  size  of  the  sample  is  too  large. 

Channel  samples  for  general  testing  purposes  are  usually  collected 
normal  to  the  bedding  of  the  deposit.  The  purpose  is  to  obtain  a  wide 
range  of  the  material  to  be  tested,  so  that  both  its  average  composition 
and  its  extremes  of  size  may  be  known.  For  the  detailed  study  of  sedi- 
ments, however,  the  inclusion  of  separate  beds  in  a  single  sample  may  be 
a  distinct  disadvantage.  Channel  samples  tend  to  mask  details,  because 
they  furnish  no  data  on  the  range  of  sizes  in  individual  beds ;  they  j-ield 
composite  data  made  up  of  several  sets  of  individual  data,  and  furnish 
no  information  whatever  on  the  degree  of  sorting,  or  mineralogical  com- 
position of  individual  beds.  Thus  where  the  sediment  is  composed  of 
numerous  thin  members  having  a  wide  range  of  characteristics  (a  glacial 
outwash  deposit  is  an  example),  it  is  doubtful  whether  a  channel  sample 
taken  through  the  deposit  will  yield  any  detailed  knowledge  about  the 
conditions  of  deposition. 

On  the  other  hand,  where  the  material  is  unbedded  and  apparently 
homogeneous,  as  in  loess,  it  may  be  advantageous  to  choose  channel 
samples  instead  of  spot  samples.  In  the  absence  of  any  evidence  of 
heterogeneity,  it  may  be  argued  that  a  channel  sample  is  more  repre- 
sentative than  a  sample  from  a  point  in  the  outcrop.  Samples  inter- 
mediate between  a  spot  sample  and  a  channel  sample  may  also  be  con- 
sidered. That  is,  instead  of  taking  a  single  channel  from  top  to  bottom 
of  the  deposit,  two  or  three  separate  shorter  channels  may  be  used,  each 


i8        SEDIMENTARY  PETROGRAPHY 

collected  separately.  In  this  manner  hidden  variations  are  disclosed  in 
the  analyses,  which  would  remain  masked  in  a  single  channel. 

The  decision  between  spot  samples,  short  channels,  and  long  chan- 
nels must  depend  partly  on  the  judgment  of  the  collector.  It  is  not  pos- 
sible to  make  rigid  rules,  inasmuch  as  the  purposes  of  the  study  influence 
the  decision.  The  important  point,  perhaps,  is  that  the  collector  should 
be  aware  of  the  choice  at  his  disposal  in  a  given  case,  so  that  his  judg- 
ment may  be  sound. 

Compound  samples.'^  A  compound  sample  is  a  mixture  of  a  number  of 
spot  samples  combined  to  form  an  aggregate  single  sanipH^  For  ex- 
ample, a  number  of  small  pieces  of  limestone  are^^TTected  at  various 
points  within  a  quarry  and  combined  into  a  single  composite.  This 
sample  is  analyzed  for,  say,  its  AlgO  content,  and  the  value  obtained  is 
taken  as  an  average  value  for  the  quarry.  Compound  samples  find  wide 
application  in  commercial  sampling,  because,  like  the  channel  sample, 
they  afford  average  values. 

An  advantage  of  compound  samples  over  channel  samples  for  furnish- 
ing average  data  arises  in  some  cases  from  the  fact  that  the  sampling 
localities  may  be  spread  widely  over  the  formation  studied,  instead  of 
being  confined  to  a  single  vertical  cut.  The  same  effect  may  be  obtained 
by  taking  a  number  of  channel  samples  and  mixing  them  all  into  one 
composite.  Again  the  nature  of  the  material  and  the  purposes  of  the 
study  will  control  the  choice  of  method  to  be  followed. 

If  the  detailed  variation  of  sediment  characteristics  is  being  studied, 
the  same  disadvantages  which  apply  to  channel  samples  may  be  extended 
to  compound  samples,  which  afford  average  values  and  merge  varia- 
tions into  a  single  value.  There  is  at  least  one  important  type  of  com- 
pound sample,  however,  which  is  of  importance  in  detailed  studies. 
It  was  mentioned  earlier  that  any  single  spot  sample  is  rigorously  valid 
only  for  the  exact  point  of  sampling.  There  is  a  possibility  that  this 
single  sample  may  actually  represent  a  deviation  or  departure  from  the 
composition  round  about  rather  than  represent  the  general  character- 
istics of  the  sediment  in  the  sampling  vicinity.  Such  possible  deviation  of 
a  sample  from  the  average  is  referred  to  as  the  probable  error  -  of  the 
sample. 

1  Both  compound  samples  and  channel  samples  belong  to  the  general  group  of 
composite  samples,  in  which  more  than  a  single  set  of  characteristics  may  be  com- 
bined. 

-  The  probable  error  is  a  statistical  measure  of  the  chance  deviation  of  a  given 
sample  from  the  average  value  of  the  material  being  sampled.  It  is  that  error  which 
will  not  be  exceeded  by  half  the  samples  collected,  and  hence  it  serves  as  a  measure 
of  the  reliability  of  the  sample. 


COLLECTION  OF  SAMPLES  19 

The  subject  of  probable  errors  in  sampling  is  discussed  later  (page  41), 
but  an  introduction  to  the  topic  is  necessary  here  to  clear  the  discussion.  The 
general  concept  of  the  probable  error  may  be  illustrated  with  a  specific 
example.  It  is  desired  to  sample  an  exposure  of  unbedded  sand,  about  10  ft. 
thick  and  30  ft.  long.  A  single  sample  is  to  be  collected  from  the  bank, 
to  represent  that  locality  in  a  set  of  serial  samples.  One  decides  to  take  the 
sample  from  the  center  point  of  the  exposure.  How  would  his  results  have 
differed  if  he  had  taken  the  sample  from  one  of  tlie  sides,  at  tlie  top,  or  at 
the  bottom  of  the  cut  ?  For  simplicity  it  will  be  assumed  that  the  sand  appears 
to  be  homogeneous  to  the  eye  and  that  it  is  equally  convenient  to  take  the 
sample  anywhere  in  the  exposure. 

The  only  answer  that  can  be  given  to  the  question  at  this  point  is  a  general 
one :  it  depends  on  the  actual  variation  of  the  sand  from  point  to  point, 
regardless  of  how  the  eye  may  appraise  it.  Studies  have  been  made  of  the 
probable  error  in  a  few  cases,  and  it  is  found  that  even  the  most  apparently 
homogeneous  sands  actually  do  vary  slightly  from  point  to  point.  In  beach 
sands,  for  example,  the  variation  in  the  average  grain  size  ranges  from 
0.8  to  4.1  per  cent  over  distances  of  a  few  hundred  feet.^  An  important  point 
that  emerges  from  a  study  of  the  probable  error  is  that,  regardless  of  its 
magnitude,  the  error  may  be  reduced  to  any  desired  value  by  mixing  a  num- 
ber of  discrete  samples  into  a  single  compound  sample.  Actually  a  mixture 
of  four  discrete  spot  samiples  into  one  compound  sample  will  reduce  the  error 
by  about  half.  Ten  samples,  on  the  other  hand,  will  reduce  it  to  about  0.3  of 
itself.  Thus  the  maximum  reduction  for  practical  purposes  is  obtained  by 
mixing  four  discrete  samples  into  one  composite.  In  the  example  given  above, 
then,  any  concern  about  the  efifect  of  the  precise  sampling  point  may  be  halved 
by  taking  four  samples  scattered  over  the  exposure  and  mixing  tlaem  into  a 
single  compound  sample. 

From  the  point  of  view  of  the  probable  error  of  sampling,  compound 
samples  may  deserve  consideration  in  any  studies  where  apparently 
homogeneous  sediments  are  involved.  Serial  samples  along  a  beach,  for 
example,  may  be  composed  of  a  series  of  compound  samples,  each  made 
by  combining  four  spot  samples  collected  within  a  short  distance  of  each 
other  at  each  of  the  main  sampling  points.  In  this  manner  the  accidental 
sampling  of  an  unsuspected  deviation  in  the  sand  may  to  a  large  extent 
be  avoided.  The  precaution  to  be  followed  in  this  type  of  sampling  is  to 
make  certain  that  the  distance  between  the  individual  samples  which 
make  up  each  composite  is  small  compared  to  the  distance  between  the 
main  serial  points.  The  need  for  this  precaution  is  that  each  composite 
of  four  samples  must  represent  essentially  a  point  on  the  scale  at  which 
the  successive  serial  samples  are  spaced,  so  that  complexities  are  not 
introduced  into  the  study  by  averaging  actual  sediment  variations  rather 
than  random  deviations  within  small  areas. 

1  W.  C.  Krumbein,  loc.  cit.,  1934. 


20        SEDIMENTARY  PETROGRAPHY 

Single  vs.  composite  sauiplcs.  From  the  preceding  sections  it  may  be 
noted  that  there  are  no  fixed  rules  regarding  the  relative  merits  of  the 
several  types  of  samples.  In  every  case  qualifications  must  be  included, 
and  these  qualifications  are  partly  a  reflection  of  the  lack  of  quantitative 
data  on  the  problem.  There  is  a  pressing  need  for  more  research  on  the 
problems  of  sampling,  so  that  quantitative  data  may  be  available  for  the 
development  of  a  general  theory  of  sampling  sediments.  One  function 
of  composite  sample's  (citluT  compound  or  channel  samples)  is  to  reduce 
the"nc)n-systcniatic  Aariaiions  that  may  be^presenTat  anY^iven  sampling 
point,  so  that  when  a  series  of  samples  is  considered  as  a  whole,  the 
errOT  due  to  sampling  may  be  kept  smaller  than  the  actual  range  of  "sedi- 
ment variation  from  point  to  point  along  the  series.  Another  function"  of 
composite  samples  is  to  reduce  the  extremes  of  sediment  variation  at 
any~given  locality  by  combining  the  properties  of  all  the  maTeriai  at  that 
point.  The  functions  of  discrete  or  spot  samples  include  the  preservation 
of  individual  differences  among  adjacent  samples.  Chance  may  in  some 
cases  introduce  an  appreciable  error  in  these  samples,  but  at  least  the 
individuality  is  preserved  and  important  data  are  not  obscured  by  a  gen- 
eral average  composition.  It  is  obvious  that  if  sediment  variations  are  to 
be  studied  within  a  single  outcrop,  discrete  samples  will  bring  the  vari- 
ations to  light,  whereas  composite  samples  will  tend  to  suppress  them. 
The  decision  to  use  compound  samples  as  against  discrete  samples  thus 
depends  in  part  on  the  scale  of  the  investigation. 

An  attempt  to  establish  sound  principles  for  sampling  sediments  was 
made  very  recently  by  Otto,^  who  classified  sampling  techniques  into 
four  groups  in  terms  of  the  purposes  to  be  fulfilled.  These  groups  are 
samples  for  engineering  uses,  for  descriptive  purposes,  for  environmental 
studies,  and  for  correlation  studies.  His  discussion  of  environmental 
sampling  is  of  especial  interest.  For  this  purpose  he  developed  the  con- 
cept of  a  sedimentation  unit,  defined  as  that  thickness  of  sediment,  at  a 
given  sampling  point,  which  was  deposited  under  essentially  constant 
physical  conditions.  Chance  deviations  about  average  values  may  be 
present,  but  these  deviations  should  themselves  form  a  unimodal  distri- 
bution. Otto's  classification  and  analysis  offer  a  basis  for  a  generalized 
theory  of  sediment  sampling,  which  may  be  applied  in  the  field  to  a 
variety  of  problems. 

1  G.  H.  Otto,  The  sedimentation  unit  and  its  use  in  field  sampling :  Jour.  Geology, 
vol.  41,  pp.  569-582,  1938.  Through  Mr.  Otto's  kindness  the  authors  were  privileged 
to  read  the  manuscript. 


COLLECTION  OF  SAMPLES  21 

The  Problem  of  Weathering 

L'p  to  the  present  the  assumption  has  been  tacitly  made  that  only 
unweathered  sediments  were  involved  in  the  sampling  process.  In  gen- 
eral, sediments  collected  for  studies  of  the  conditions  of  deposition  or 
the  nature  of  source  rocks  should  be  unweathered.  For  other  studies, 
involving  the  alteration  history  of  the  sediments,  it  may  be  necessary  to 
collect  samples  from  the  w^eathered  zones.  Sample  collectors  should  be 
able  to  recognize  weathered  sediments  in  the  field  and  should  understand 
some  of  the  changes  which  are  introduced  into  the  sediment  by  w^eather- 
ing  changes.  Leighton  and  MacClintock/  and  more  recently  Grim,  Bray, 
and  Leighton,-  have  shown  that  weathering  involves  the  development 
of  four  horizons,  each  of  which  is  characterized  by  certain  features.  The 
studies  were  conducted  on  glacial  till  and  loess,  but  the  principles  found 
are  of  general  application. 

The  fir£t_change  that  occurs  is  oxidation,  which  affects  mainly  tlie  iron- 
bearing  mineraIs.Tlie^iierFesuItTs~a~cTTange  in  color  inclining  toward  brown. 
Following  tlie  oxidation  comes  a  leaching  of  the  more  soluble  minerals,  such 
as  the  carbonates,  notably  calciteTTRe  Ihird  stage  is  the^  decomposition_of 
the  silicates,  during  which  feldspars  and  similar  minerals  are  decompose'3. 
Finally,  near  the  surface,  the  soil  zone  proper  is  evolved,  with  only  the  more 
resistant  minerals  remaining,  notably  quartz.  The  chemical  changes  are 
accompanied  by  changes  in  the  size  distribution  and  other  physical  attributes 
of  the  sediment.  Thus  a  calcareous  sediment,  which  includes  primary  grains 
of  calcite,  has  a  different  size  distribution  after  it  has  been  subjected  to 
leaching.  Similarly  the  breakdown  of  the  feldspars  into  clays  and  colloids 
involves  significant  changes  in  the  physical  properties.  The  drainage  con- 
ditions at  the  site  of  weathering  also  influence  the  process  of  decomposition, 
so  that  dift'erent  end-products  result  from  well  and  poorly  drained  situations. 

The  authors  of  the  present  volume  have  found  that  in  some  sediments, 
such  as  sand,  loess,  and  glacial  till,  there  are  no  significant  differences  in 
the  general  properties  of  unweathered  samples  and  oxidized  samples  as 
far  as  routine  size  analysis  is  concerned.  Usually  changes  become  notice- 
able in  the  leached  zone  and  are  generally  striking  in  the  silicate-decom- 
position zone.  Quartz  sand,  with  relatively  few  and  resistant  hea\y 
minerals,  appears  to  remain  essentially  unatTected  by  weathering  condi- 
tions. For  most  present-day  studies  it  seems  possible  to  use  material  from 
the  oxidized  zone  if  unweathered  samples  are  not  available.  This  does 
not  imply  that  oxidation  is  negligible,  but  rather  that  present-day  ex- 

1  M.  M.  Leighton  and  P.  MacClintock,  Weathered  zones  of  the  drift  sheets  of 
Illinois :  J  our.  Geology,  vol.  38,  pp.  28-53,  1930- 

2  R.  E.  Grim,  R.  H.  Bray  and  M.  M.  Leighton,  Weathering  of  loess  in  Illinois : 
Geol.  Sac.  America,  Proceedings,  1936,  p.  76. 


22        SEDIMENTARY  PETROGRAPHY 

perimental  errors  appear  to  be  at  least  of  the  order  of  magnitude  of  the 
oxidation  changes. 

When  studies  are  undertaken  specifically  on  the  effects  of  weathering, 
the  sampling  procedure  should  involve  first  the  identification  of  the  sev- 
eral weathering  zones,  if  they  are  fully  developed,  and  a  collection  of 
samples  from  each  zone  as  well  as  from  the  transitions  from  one  zone 
to  the  next.  The  authors  have  relied  on  vertical  series  of  spot  samples 
rather  closely  spaced,  rather  than  a  channel  sample  through  each  zone. 
The  use  of  discrete  samples  here  offers  an  opportunity  for  a  more  de- 
tailed picture  of  the  changes  from  point  to  point  in  the  weathering 
profile. 

The  Problem  of  Induration 

Just  as  weathering  introduces  complexities  into  the  sedimentary  pic- 
ture, induration  complicates  the  study  by  virtue  of  the  physical  and 
chemical  changes  involved.  If  the  induration  is  due  simply  to  the  cemen- 
tation of  non-calcareous  grains  by  calcite,  no  special  problems  are  intro- 
duced, because  presumably  the  original  material  may  be  recovered  by 
leaching  in  acid.  Where  secondary  material  has  been  introduced  in- 
timately into  the  rock,  or  where  changes  have  taken  place  in  the  origi- 
nal material  itself,  the  problem  of  original  constitution  may  be  quite 
complex.  For  the  most  part,  however,  the  problem  belongs  to  the 
laboratory,  where  the  successful  breaking-down  of  the  rock  may  be  a 
difficult  problem. 

The  sampling  process  in  the  case  of  indurated  rock  differs  from  that 
with  unconsolidated  rocks  mainly  in  the  greater  difficulty  of  obtaining 
representative  samples.  Channel  samples,  for  instance,  may  have  to  be 
literally  chiseled  from  the  outcrop.  Grout  ^  discussed  the  sampling  of 
igneous  rocks  for  chemical  analysis,  and  the  principles  he  developed 
appear  to  apply  also  to  indurated  sediments. 

The  Collection  of  Oriented  Samples 

For  certain  types  of  investigations  it  is  important  that  the  exact 
orientation  of  rock  specimens  or  sedimentary  particles  be  known.  If  the 
rock  is  a  sandstone  or  other  consolidated  sediment,  the  dip  and  strike 
may  be  painted  on  the  rock  face  with  a  quick-drying  enamel  before  the 
specimen  is  broken  off.  With  these  lines  of  reference  it  is  possible  to 
prepare  sections  at  any  given  orientation  for  studies  of  the  rock  fabric. 

1  F.  Grout,  Rock  sampling  for  chemical  analysis:  Atn.  Jour.  Sci.,  vol.  24,  pp.  394- 
404,  1932. 


COLLECTION  OF  SAMPLES  23 

If  the  sediment  is  loose  sand,  it  may  be  impregnated  with  paraffin  in  some 
instances,  or  with  a  dikite  bakeHte  varnish. 

Among  coarse  sediments  Hke  gravel,  the  individual  pebbles  may  be 
large  enough  to  have  their  orientation  marked  directly  on  them.  Wadell  ^ 
developed  a  technique  in  which  horizontal  and  vertical  lines  were  drawn 
on  the  pebbles  with  red  and  black  enamel  respectively,  so  that  in  the 
laboratory  the  exact  orientation  of  each  pebble  in  space  may  be  repro- 
duced. The  detailed  procedure  used  by  Wadell  is  described  in  Chapter  10 
of  this  volume. 


S  U  B  -  S  t?  R  I'  A  C  E     S  A  M  P  L  E  S 

The  sampling  procctlurcs  outlined  in  the  preceding  sections  assumed 
complete  exposure  of  the  deposits  or,  at  worst,  sediments  concealed  be- 
neath a  thin  veneer  of  surface  materials,  so  that  shallow  pits  expose 
the  unaltered  sediment.  Beyond  a  depth  of  several  feet,  reliance  must 
be  placed  on  mechanical  devices  for  obtaining  the  samples.  More- 
over, different  methods  apply  with  depth,  because  of  the  mechanical 
difficulty  of  penetrating  very  far  beneath  the  surface  with  hand-operated 
tools. 

Hand  auger  sa})iplcs.  Hand  augers  may  be  used  for  obtaining  samples 
down  to  a  depth  which  seldom  exceeds  20  or  30  ft. ;  for  most  practical 
purposes  the  labor  involved  excludes  this  device  for  depths  much  greater 
than  10  ft. 

Fland  augers  were  developed  in  connection  with  soil  sampling,  where 
the  sampling  depth  is  often  limited  to  about  30  in.  For  such  shallow 
depths  augers  are  excellent,  and  many  types  have  been  devised  for  gen- 
eral and  special  purposes.  A  simple  hand  auger  of  general  utility  for 
sedimentary  purposes  -  may  be  constructed  from  an  ordinary  steel  bit 
about  2  in.  in  diameter,  of  the  type  used  for  drilling  w^ood.  The  screw 
at  the  end  of  the  bit  and  the  small  flanges  on  the  first  whorl  are  filed  off. 
The  bit  is  welded  to  a  hollow  steel  tube  ^  in.  in  diameter  and  about  3  ft. 
long.  Additional  lengths  of  tubing  with  threaded  connections  are  also 
prepared,  so  that  the  assembled  auger  is  about  12  or  14  ft.  long.  The 
handle  is  made  from  an  i8-in.  length  of  the  same  tubing,  with  a  threaded 
connection  at  the  middle,  so  that  it  acts  as  a  crosspiece. 

^  H.  \\'adell.  Volume,  shape,  and  shape  position  of  rock  fragments  in  openwork 
gravel :  Ccoijrafiska  Ainialcr,  1936,  pp.  74-92. 

-  This  device  was  developed  by  the  Illinois  State  Geological  Survey.  It  has  been 
extensively  used  for  a  variety  of  purposes,  such  as  test  holes,  collecting  samples,  and 

the  like. 


24        SEDIMENTARY  PETROGRAPHY 

In  using  the  auger,  the  bit  is  turned  down  to  about  its  own  depth  in 
the  soil  and  withdrawn.  The  coiled  sample  is  unwound  and  laid  on  a 
square  of  canvas  provided  for  the  purpose.  The  process  is  repeated,  in 
each  case  turning  the  bit  only  so  far  that  the  labor  of  withdrawing  it  is 
not  excessive.  The  coiled  samples  of  sediment  are  placed  end  to  end  on 
the  canvas,  in  the  order  of  their  removal. 

Certain  precautions  should  be  followed  in  using  any  auger,  to  avoid 
too  great  a  contamination  of  the  sample.  The  surface  material  about  the 
drill  hole  should  be  scraped  away  so  that  loose  fragments  do  not  fall  into 
the  hole.  Also,  in  dropping  the  auger  into  the  hole,  some  material  is  in- 
variably scraped  from  the  walls.  This  is  tamped  down,  and  when  the  next 
sample  is  withdrawn,  it  will  be  found  at  the  top  of  the  bit.  By  examining 
the  unwound  coil  it  is  usually  possible  to  determine  how  much  should  be 
discarded. 

Augers  of  this  general  type  are  most  effective  with  silt  or  clay ;  also 
such  deposits  as  loess,  lake  clay,  and  glacial  till  may  be  quite  conveniently 
sampled.  Sand  generally  does  not  form  a  coil  around 
the  bit,  but  fine  sand,  if  moist,  can  often  be  sampled. 
Water  may  be  poured  into  the  hole  to  facilitate  the 
sampling.  Very  wet  material  like  quicksand,  on  the 
other  hand,  merely  flows  from  the  bit. 

Sampling  devices  which  permit  the  collection  of  un- 
disturbed samples  at  comparatively  shallow  depths  have 
also  been  described.  A  recent  device,  suitable  for  soil 
samples,  was  developed  by  Heyward.^ 

Mitscherlich  -  described  a  special  type  of  boring  tube 

for  small  samples  at  shallow  depths.  The  instrument  is 

a  hollow  tube  with  a  slot  along  one  side  (Figure  3),  so 

that  in  cross  section  it  is  like  the  letter  C.  The  diameter 

of  the  tube  is  about  an  inch.  The  lower  end  is  pointed, 

and  at  the  upper  end  is  a  crossbar  for  a  handle.  The 

Fig.    3. —        tube  is  forced  into  the  ground,  twisted  through  a  single 

small    sam-        turn,  and  withdrawn.  The  sample  so  obtained  is  a  small 

pies  of  soil  or        core  of  the  sediment. 

Mitscherlich  Other  auger-like  devices,  suitable  for  shallow  depths, 

are  post-hole  diggers  and  golf -hole  drills.  The  former  is 

a  common  farm  implement  consisting  of  two  handles  with  semicylindrical 

blades.  The  instrument  is  twisted  into  the  soil  and  a  cylindrical  sample 

withdrawn.  It  may  be  used  for  depths  up  to  about  5  ft.  The  golf -hole 

1  F.  Heyward,  Soil  sampling  tubes  for  shallow  depths:  Soil  Science,  vol.  41,  pp. 
357-360,  1936. 

-  E.  A.  Mitscherlich,  Bodcnkunde  fiir  Land-  und  Forstwirte  (Berlin,  1905),  pp. 
314-315. 


COLLECTION  OF  SAMPLES  25 

drill  is  a  hollow  steel  cup  about  3  iu.  iu  ilianictc-r,  with  a  plunger  within 
it  for  expelling  the  plug  obtained  when  the  drill  is  forced  into  the 
ground. 

Augers  suitable  for  loose  sand  have  been  deseribetl  by  \'eateh.^  Such 
devices  consist  essentially  of  an  auger  or  cutting  tool  surrounded  by  a 
metal  cylinder  which  retains  the  loose  material  picked  up  by  the  bit. 

Drive-pipe  samples.  A  device  commonly  usetl  t\)r  sampling  cla\  s  and 
other  fme-grained  sediments  is  an  ortlinary  iron  pipe  about  an  inch  or 
two  in  diameter,  which  is  driven  vertically  into  the  tleposit.  A  heavy 
metal  collar  shouKl  be  fastened  to  the  upper  end  of  such  pipes  to  pre- 
vent spreailing,  and  also  to  facilitate  subsecpient  renu)val. 

For  s!iaIK)w  depths,  up  to  about  5  ft.,  the  pipe  may  be  set  upright  in  a 
small  pit  and  driven  down  with  a  sletlge  hammer.  A  wooden  box  is  jiro- 
vided  for  standing  upon  until  the  pipe  is  down  far  enough  to  be  struck 
from  ground  level.  After  the  jiipe  has  been  driven  in,  it  is  raised  by  a  jack 
or  with  block  and  tackle,  and  the  core  removed. 

When  dei^ths  up  to  15  or  20  ft.  are  involved,  a  tripod  and  heavy  weight 
may  be  used  for  driving  the  pipe.  The  tripod,  also  made  of  jiipc.  is  set 
uj)  above  the  sampling  site,  and  the  pipe  itself  is  set  into  a  shallow  pit  in 
an  uiM-igbt  position.  An  iron  weight  of  about  50  lbs.  is  used  as  the  driver. 
The  weight  is  a  solid  cylinder  with  a  ring  on  one  end  and  a  thin  metal 
rod  about  4  ft.  long  on  the  other.  A  rope  is  tied  to  the  ring  and  run  over 
a  pulley  in  the  tripod.  The  thin  rod  is  inserted  into  the  drive  pipe  and  the 
weight  is  lifted  about  3  ft.  above  the  pipe.  The  weight  is  released,  and 
the  rod  guides  it  onto  the  drive-jiipe.  By  repeating  the  blows,  the  jiipe  is 
driven  into  the  ground.  To  remove  the  pipe,  a  block  and  tackle  is  used. 

The  core  in  the  pipe  is  removed  with  a  pressure  screw,  which  is 
mounted  on  a  rigid  frame.  The  screw  is  turned  into  the  pipe,  forcing  the 
core  out  at  the  other  end. 

Drive-pipes  may  also  consist  of  an  outer  driving  pipe  with  an  inner 
collar,  in  which  two  halves  of  a  tin  cylinder,  split  lengthwise,  are  inserted. 
After  withdrawing  the  drive-pipe,  the  inner  tube  is  removed  and  the 
sample  readily  obtained. 

A  modification  of  the  drive-pipe  is  the  so-called  sampling  rod.  de- 
scribed by  Simpson.-  It  consists  of  a  pipe  about  2  in.  in  diameter  and 
7  ft.  long,  with  a  narrow  vertical  slit  extending  from  the  bottom  nearly 
to  the  top  of  the  i^ipe.  The  slit  allows  the  pipe  to  yield  slightly  as  it  is 
forced  down  and  enables  it  to  lu)ld  the  sample  by  tightening  when  it  is 
withdrawn. 

^  A.  C.  Voatch,  (icology  and  uiKlori;rouinl  water  resources  of  northern  Louisiana 
and  soutliern  Arkansas:  ('/.  .V.  (,'col.  Siinry,  I'rof.  Puller  ./o.  pp.  03  IT.,  u)o(). 

-  D.  Simpson.  Sand  sampling  in  eyanidc  works:  Trans.  Inst.  Miii.  (uui  Met.,  vol. 
16,  pp.  30-41,  1 906- 1 907. 


26        SEDIMENTARY  PETROGRAPHY 

Drilled  Well  Samples 

Samples  from  drilled  wells  differ  considerably  among  themselves  de- 
pending upon  the  method  of  drilling  employed.  Three  methods  are  used, 
and  because  of  the  wide  variation  in  the  quality  of  samples  obtained  the 
discussion  will  be  based  on  the  method  employed. 

Diamond  core  drilling.  This  method  affords  the  best  type  of  samples, 
because  a  solid  core  of  the  material  is  preserved  during  drilling.  The 
drilling  is  accomplished  by  means  of  a  bit  made  of  a  hollow  steel  cyl- 
inder, along  the  lower  edge  of  which  are  set  black  diamonds  to  act  as  the 
cutting  edge.  The  bit  is  attached  to  a  core  shell  and  core  barrel,  the  latter 
in  turn  being  connected  to  a  series  of  hollow  steel  rods  which  extend 
to  the  surface.  The  rods  and  tools  are  rotated,  and  a  stream  of  clear 
water  is  run  down  through  the  rods  and  core  barrel.  The  water  serves 
to  keep  the  cutting  edge  cool,  and  as  it  flows  upward  between  the  rods 
and  the  wall  of  the  bore  hole,  it  carries  away  the  rock  cuttings. 

As  the  bit  penetrates  into  the  rock,  a  core  is  cut  out,  which  gradually  fills 
the  bit  and  the  core  barrel.  When  the  core  barrel  is  filled,  the  entire  mechanism 
is  pulled  to  the  surface  and  the  core  removed.  In  this  manner  a  continuous 
record  is  had  of  the  rocks  penetrated  during  the  drilling.  The  cores  may  be 
taken  to  the  laboratory  for  a  complete  examination,  or  portions  of  each  type 
of  rock  may  be  removed  as  samples.  In  general,  the  core  is  excellently  pre- 
served, although  occasionally  soft  materials  may  be  washed  away  by  the 
circulating  water. 

Percussion  drilling.  In  this  method  of  drilling  a  series  of  rock  cuttings 
are  obtained  which  in  general  are  satisfactory  for  laboratory  study.  Cer- 
tain characteristics  of  the  particles,  such  as  size  and  shape,  may  suffer, 
but  mineral  content  and  micro  fossils  may  be  examined. 

For  drilling,  a  steel  cutting  tool  is  attached  to  a  string  of  heavy  steel 
cylinders  which  are  suspended  from  the  drilling  rig  with  a  rope  or  cable. 
The  string  of  tools  is  alternately  lifted  and  dropped,  and  the  repeated  blows 
of  the  bit  serve  to  cut  into  the  rock.  The  product  is  a  granular  or  powdery 
material,  which  is  kept  wet  either  by  seepage  into  the  hole  or  by  pouring  water 
into  it.  At  intervals,  the  string  of  tools  is  withdrawn  and  a  bailing  tube  is 
lowered  to  withdraw  the  cuttings.  The  bailer  is  a  hollow  cylinder  with  a 
valve  at  the  bottom.  The  contents  of  the  bailer  are  dumped  at  the  side  of 
the  rig,  and  the  rock  debris  which  constitutes  the  sample  is  left  behind  as  the 
water  runs  off. 

There  is  considerable  danger  of  contamination  of  the  sample  from 
percussion  drilling,  due  to  material  caving  from  the  hole,  or  by  the  mix- 
ture of  two  formations  if  the  bailer  is  not  used  often  enough.  With 


COLLECTIOX  OF  SAMPLES  2- 

reasonable  care,  however,  fair  samples  may  be  obtained.  A  sample  taken 
from  the  bailer  contents  is  roughly  equi\-alent  to  a  channel  sample  taken 
through  tlie  depth  penetrated  between  two  successive  bailings.  It  thus 
represents  an  average  sample  from  a  given  thickness.  There  is  also  some 
danger  due  to  selective  losses,  inasmuch  as  the  finer  material  is  swept 
away  with  the  w-ashings  from  the  bailer. 

Rotary  driU'mg.  From  the  point  of  \-iew  of  sampling,  rotary  drilling 
furnishes  the  poorest  kind  of  well  samples. 

In  this  method  of  drilling,  a  bit.  s;.2.ped  like  a  nslitail.  is  rotated  at  the 
end  of  a  stem  of  hollow  rods.  A  stream  of  mud-laden  water  is  circulated 
downward  through  the  hollow  stem,  bodi  to  lubricate  the  cutting  tool  and  to 
cool  it.  As  die  mud  returns  to  the  surface,  it  carries  widi  it  cuttings  of  die 
rock  penetrated.  It  is  from  the  mixture  of  mud  and  cuttings  that  the  sample 
is  taken. 

The  continued  use  of  tlie  same  driUing  mud  results  in  considerable 
contantination  of  samples  in  rotarj-  driUing.  Various  methods  have  been 
de\-ised  for  obtaining  satisfactory-  samples.  The  mud  may  be  passed  over 
screens  to  collect  the  cuttings,  or  fliunes  may  be  fined  with  wiers  to 
allow  the  cuttings  to  settle.  A  detailed  discussion  is  given  bv  Whiteside.^ 


BOTTOM      SAMPLES 

The  collection  of  sedimentary  samples  from  die  bottom  of  bodies  of 
standing  water  is  a  somewhat  specialized  procedure  which  requires  ap- 
paratus of  one  sort  or  another,  depending  upon  several  factors,  such  as 
the  depth  of  water,  the  nature  of  the  bottom,  whether  an  undisturbed 
sample  is  required,  and  whether  the  sample  should  be  large  or  small. 
Numerous  devices  have  been  developed,-  but  many  of  them  represent 
slight  modifications  of  a  few  fundamental  t\-pes.  Tliese  tA-pes  include  ap- 
paratus which  is  dragged  along  the  bottom,  tubes  wliich  are  driven 
vertically  into  the  bottom  deposits,  or  mechanical  dcA-ices  which  snap 
a  sample  of  the  sediment  between  spring-operated  jaws.  Of  these  t\-pes, 
the  tube  samplers  are  perhaps  the  most  extensively  used. 

Bottom  dredges  and  drag  buckets.  Dredges  and  drag  buckets  are  of  several 
types.  Among  die  earlier  forms,  of  which  modified  versions  are  still  in  use. 
are  dredges  of  the  Challenger  type.'  The  Challenger  dredge  consists  of  two 

1 R.  M.  Whiteside,  Geolc^c  interpretations  from  rotary  well  cuttings :  Bull. 
A.  A.  P.  G.,  vol.  16.  1932,  pp.  653-674. 

-  F.  M.  Soule.  Oceanographic  instruments  and  methods :  Nat.  Research  Council, 
Bull.  85.  pp.  411-454,  19J2. 

3  J.  Murray.  Report  on  the  Scientific  Results  of  the  Voyage  of  H.M.S.  Challenger 
(London,  18S5),  vol.  i,  pp.  73  fiF. 


28        SEDIMENTARY  PETROGRAPHY 

parts,  the  iron  framework  which  skims  the  surface  of  the  deposit,  and  a 
bag  or  sack  which  collects  and  retains  the  skimmings.  Another  t>-pe  of 
sampler  which  is  pulled  along  the  bottom  is  the  Gilson  sampler.^  developed  in 
1906.  It  consists  of  a  hemispherical  bowl  attached  in  the  center  to  an  iron 
rod.  The  Mann  sampler,  according  to  Trask.-  consists  of  a  cylindrical  iron 
tube  about  4  in.  in  diameter  and  6  in.  long.  It  is  closed  at  one  end,  and  at 
the  open  end  is  attached  to  a  sounding  line.  The  sampler  is  dragged  along  the 
bottom  until  filled. 

A  sampling  device  similar  to  the  bucket  type  is  the  cup  lead  described  in 
the  Challenger  report.^  This  consists  of  a  hollow  cone,  fitted  with  a  sliding 
lid,  and  fastened  to  a  weighted  spike.  The  lid  prevents  loss  of  the  sample  in 
the  cone. 

A  bottom  sampler  devi.sed  by  Lugn  *  for  collecting  sediments  from 
the  Mississippi  River  belongs  in  this  classification.  The  device  consists 
of  two  weights,  rigidly  attached  to  a  central  stem,  and  a  loose-fitting  cup 
which  rests  around  a  shoulder  on  tlie  lower  weight.  The  dimensions  are 
such  that,  when  the  instrument  hes  on  its  side,  the  cup  inclines  easily 
without  falling  from  the  shoulder,  and  it  shps  back  to  vertical  when  the 
instrument  is  pulled  up  to  the  surface.  In  use,  the  instrument  is  dragged 
along  the  bottom ;  when  it  is  hauled  up,  the  cup  fits  tightly  enough  about 
the  lower  collar  to  prevent  losses. 

Bottom  sampling  tubes.  These  bottom  sampling  devices  consist  es- 
sentially of  a  tube  of  varying  length,  with  weights  attached.  As  it  strikes 
the  bottom,  the  tube  settles  into  the  deposit  and  fills  with  a  core  of  the 
sediment.  !Many  variations  of  this  instrument  have  been  used,  and  it  is 
perhaps  the  most  widely  used  of  bottom  sampling  devices. 

Among  the  earliest  of  such  instruments  was  the  Baillie  Rod.^  It  was  made 
of  an  iron  pipe  about  23/3  in.  in  diameter,  beveled  at  the  bottom  and  fitted 
with  a  butterfly  valve.  A  more  effective  device,  fitted  with  a  comb-valve  to 
prevent  the  loss  of  sediment,  was  the  Buchanan  Combined  Water  Bottle  and 
Sampling  Tube.^ 

Among  modern  apparatus  the  Ekman  Sampler  '  and  its  modifications 
are  the  most  important.  Trask  *  used  an  instrument  essentially  like  the 

1  Stina  Gripenberg,  A  study  of  the  sediments  of  the  North  Baltic  and  adjoining 
seas:  Fcmiia,  vol.  60,  no.  3,  p.  11,  1934. 

2  P.  D.  Trask,  Origin  and  Environment  of  Source  Sediments  of  Petroleum  (Hous- 
ton, Texas,  1932),  p.  14. 

3  J.  Murray,  op.  cit.,  1885,  p.  69. 

*A.  L.  Lugn,  Sedimentation  in  the  Mississippi  River:  Augustana  Library  Pub- 
lications, no.  II,  1927. 

5  J.  Murray,  op.  cit.,  1885,  pp.  59  ff. 

^  Ibid.,  pp.  117  ff. 

"  V.  \V.  Ekman,  An  apparatus  for  the  collection  of  bottom  samples :  Publications 
de  Circonstances,  no.  27,  Copenhagen,  1905.  (Reference  from  Gripenberg,  loc.  cit., 

1934- ) 

8  P.  D.  Trask,  op.  cit.,  1932,  pp.  11-13. 


COLLECTION  OF  SAMPLES 


29 


original,  and  its  description  will  serve  to  indicate  the  general  pattern. 
The  apparatus  (Figure  4)  consists  of  a  galvanized  iron  pipe,  3  ft.  long 
and  iJ/2  in.  in  diameter.  The  lower  end  of  the  pipe  is  open, 
and  attached  to  the  top  is  a  vertical  check  valve  with  a  per- 
forated reducer,  which  in  turn  is  attached  to  a  stem  with 
lead  weights.  The  stem  is  fastened  to  a  sounding  line,  and 
the  apparatus  is  allowed  to  fall  to  the  bottom,  where  the 
tube  is  driven  into  the  sediment  a  distance  dependent  upon 
the  softness  of  the  deposit.  After  hauling  the  sampler  to 
the  surface,  the  collected  sediment  is  driven  from  the  pipe 
into  a  container.  If  it  is  desired,  cardboard  cylinders  may 
be  placed  inside  the  tube,  to  act  as  receptacles  for  the  scdi 
mcnt. 

The  Ekman  type  sampler  is  very  effective  in  water  to 
any  depth,  especially  with  silt  or  clay.  Cores  up  to  120  cm. 
in  length  may  be  obtained.  It  is  unsuccessful  with  coarser 
sediments,  however,  because  of  the  absence  of  a  valve  at 
the  base  of  the  tube. 

The  most  recent  modification  of  the  Ekman  type  of  sam- 
pler is  the  Piggot  sampler,^  which  introduces  a  new  prin- 
ciple into  the  design  of  such  instruments.  This  device  was 
designed  to  apply  an  impulse  to  the  sampling  tube  when  it 
strikes  the  bottom,  so  that  the  tube  will  be  driven  farther 
into  the  mud  than  is  the  case  with  gravity-settling  alone. 
Essentially  the  sampler  consists  of  a  sampling  tube  which 
is  attached  to  a  heavy  mass  acting  as  a  gun.  Within  the 
gun  is  a  charge  of  powder  and  a  cap,  set  off  by  the  impact 
of  the  device  on  the  bottom.  The  tube  is  driven  into  the 
mud  by  the  force  of  the  exi)losion  and  is  prevented  from 
escape  by  an  auxiliary  cable,  which  permits  the  two  parts 
to  be  withdrawn  to  the  surface. 

In  detail  the  assembled  instrument  is  about  15  ft.  long 
and  weighs  about  400  lbs.  Adcf|uate  hoisting  equipment  is 
accordingly  necessary  for  its  use.  However,  the  instrument 
has  been  successful  in  depths  greater  than  15,000  ft.,  and 
it  is  capable  of  taking  cores  10  ft.  long.  The  core  is  col- 
lected in  a  brass  tube  within  the  outer  casing,  to  facilitate 
removal  and  storage  of  the  sample.  A  schematic  diagram  of  the  sampling 


F  I  G.  4  — 
Mmlificd  Ek- 
m  a  n  bottom 
sampler.  The 
valve  at  B 
prevents  loss 
of  the  core 
in  tube  C 
during 
withdrawal 
of  the  instru- 
ment. A  is  a 
heavy  mass 
to  assure 
p  c  n  e  tration 
of  the  tube 
into  the 
sediment. 
(A  d  a  p  t  e  d 
from  Trask, 
1930.) 


^  C.   S.  PipfRot,   Apparatus  to  secure  cores  from  the  ocean  bottom :    Ccol.   Soc. 
America,  Bulletin,  vol.  47,  pp.  675-684,  10.36. 


30 


SEDIMENTARY  PETROGRAPHY 


. 


I 


tube  is  shown  in  Figure  5 ;  a  more  detailed  drawing  may  be  found  in  the 
original  paper. 

Hydraulic  coring  tubes.  \"amey  and  Redwine  ^  have  re- 
cently developed  a  hydraulic  coring  instrument  which  suc- 
cessfully applies  the  principle  that  the  differential  pressure 
due  to  high  water  pressure  outside  the  instrument,  and 
low  air  pressure  inside,  may  be  used  to  drive  a  coring  tube 
'^^  into  the  sea  bottom.  The  apparatus  consists  of  a  coring 

n  tube  passing  through  and  attached  to  a  piston  sliding  in  a 

^  °  cylinder.  The  piston  is  supported  near  the  upper  end  of  the 

cylinder  by  trigger  arms,  which  release  it  when  the  bottom 
is  struck  and  permit  the  piston  to  move  downward  under 
water  pressure,  driving  the  core  barrel  with  it.  The  ap- 
paratus was  used  in  water  from  50  to  300  ft.  deep,  with 
penetration  var\'ing  from  3  to  7  ft. 

Another  tj'pe  of  tube  sampler,  which  relies  on  mechanical 
force  to  obtain  a  longer  sample,  is  the  Knudsen  sampler,  de- 
scribed by  Trask.-  It  consists  of  a  tube  fastened  to  a  drum, 
around  which  latter  are  a  number  of  turns  of  sounding  line. 
A  catch  is  released  on  impact,  and  as  the  line  is  pulled  upward 
the  drum  rotates  and  operates  a  pump  which  draws  the  water 
from  the  collecting  tube.  Pressure  differences  cause  the  tube 
to  settle  farther  into  the  mud.  Trask  reports  difficulty  in  the 
use  of  the  instrument. 

Clam-shell  snappers  and  other  closing  types  of  sampler. 
Among  the  more  recent  sampling  devices  which  are  espe- 
cially suitable  for  medium-grained  sediments  are  the  clam- 
shell snappers  and  related  devices.^  These  instruments  con- 
sist of  two  jaws  which  are  held  open  by  a  trigger,  as  shown 
in  Figure  6.  When  the  device  strikes  the  bottom  a  spring 
is  released  which  snaps  the  jaws  shut.  Snapping  devices 
may  have  several  jaws  which  open  in  orange-f)eel  fashion. 

Larger  devices  of  the  closing  type,  which  are  operated 
by  cables  which  draw  the  jaws  together,  instead  of  with 
springs,  are  represented  by  the  Peterson  Dredge.*  This 
consists  of  two  jaws  hinged  at  their  intersection  and  closed 
by  means  of  a  chain  which  pulls  them  together. 

1  F.  M.  Varney  and  L.  E.  Redwine,  A  hydraulic  coring  instrument  for  submarine 
geologic  investigations:  Rep.  Com.  Sed.,  Nat.  Research  Council,  1937,  pp.  107-113. 

-  P.  D.  Trask.  op.  cit.,  1932,  p.  if. 

3  F.  M.  Soule,  loc.  cit.,  1932. 

*0.  Pratje.  Die  Sedimente  des  Sudaltlantischen  Ozeans:  Wiss.  Ergeb.  d.  Dcutsch 
Atlantischcn  Exped.  auf  d.  Meteor,  vol.  3,  part  2,  p.  12,  1935. 


F  I  G.  ;.— 
Piggot  bot- 
tom sampler. 
The  gxon  A 
drives  the 
core  bit  C 
into  the  sedi- 
ment on  im- 
pact. B  is  a 
"water  exit 
port"  to  facil- 
itate p  e  n  e- 
tration  and  to 
prevent  loss 
of  the  sam- 
p  1  e.  The 
"stirrup"  D 
permits  r  e- 
covery  of  the 
core  bit. 


COLLECTION  OF  SAMPLES 


31 


Miscellaneous  devices.  A  simple  instrument  which  may  be  used  to 
obtain  information  about  bottom  deposits  is  a  small  sounding  lead  with 
an  indentation  in  the  bottom,  into  which  wax  is  placed.  When  the  lead 
strikes  the  bottom,  some  of  the  sediment  clings  to  the  wax  and  furnishes 
information  about  the  nature  of  the  bottom. 

Another  sampling  device  used  to  determine  the  nature  of  bottom  de- 
posits is  the  sampling  *'spud."  ^  This  consists  of  a  long  rod, 
along  which  grooves  have  been  machined  to  form  a  series 
of  cups  with  lips  directed  upward.  The  cups  are  spaced  a 
little  more  than  an  inch  apart  along  the  length  of  the  rod. 
The  rod  is  forced  down  into  the  deposit,  and  as  it  is  with- 
drawn each  cup  catches  a  small  amount  of  sediment  from 
the  depth  to  which  it  penetrated.  A  vertical  section  is  thus 
disclosed.  The  spud  is  operated  by  hand  and  is  used  in  rela- 
tively shallow  water. 


SIZE     OF     COLLECTED     SAMPLES 

The  size  of  sample  to  be  collected  in  any  given  case  de- 
pends mainly  upon  two  considerations :  the  coarseness  of 
the  sediment  and  the  uses  to  which  the  sample  is  to  be  put. 
What  is  desired  is  the  smallest  sample  that  will  adequately 
represent  the. material.  As  a  general  rule,  field  samples  of 
medium-  and  fine-grained  sediments  are  much  larger  than      q  \  ^  m-shdl 
the  amount  required  for  a  single  laboratory  determination,      snapper, 
but  as  the  material  becomes  coarser,  the  field  sample  tends 
to  be  about  suf^cient  for  a  single  detailed  analysis,  owing  to  the  labor 
of  transporting  very  large  samples.  From  the  point  of  view  of  sample 
size,  it  is  immaterial  whether  the  sample  is  discrete  or  composite;  the 
essential  point  is  that  enough  material  be  present  to  give  adequate  repre- 
sentation to  the  largest  sizes  present. 

Wentworth  -  has  investigated  the  relation  of  coarseness  to  sample  size, 
and  his  practical  rule  is  that  the  samples  should  be 


lar^e  enough  to  include  several  fragrnents  which  fall  in  the  largest  grade 
present  in  the  deposit.  Several  fragments  may  be  interpreted  as  a  number 
sufficiently  large  so  that  the  probability  of  a  serious  accidental  deviation  from 

^  H.  M.  Kakin,  Silting  of  reservoirs:  U .  S.  Dcf't.  Agric,  Tech.  Bull.  524,  p.  27, 
1936. 

-  C.  K.  Wentworth,  Methods  of  mechanical  analysis  of  sediments :  Univ.  lozcu 
Studies  in  Nat.  Hist.,  vol.  11,  no.  11,  IQ26. 


32 


SEDIMENTARY  PETROGRAPHY 


the  normal  number  of  such   fragments  in  a   sample  collected  by  a  reliable 
random  method  is  small. 

At  the  other  extreme,  according  to  Wentworth,  it  is  hardly  advisable  to 
collect  less  than  about  125  g.  of  any  sediment  regardless  of  its  fineness. 
Wentworth  summarized  his  findings  in  a  table,  a  modified  form  of 
which  is  given  in  Table  i.  The  sediment  size  is  expressed  in  terms  of  the 
coarsest  material  present,  which  is  indicated  in  the  first  column.  The 

Table  i 
Practical  Sample  Weights 


Diameter  of 

Coarsest  Size 

(nun.) 

Suggested  Weight 
of  Sample 

Approximate 
Volume  of  Sample 

Cobbles 

128-64  mm 

Pebbles 

64-4  mm 

Granules 

4-2  mm 

Sand 
2-Mg  mm       

32  kg. 

16  to  2  kg. 

I  kg. 

500  to  125  gm. 
125  gm. 

125  gm. 

I  "liter 
500  cc. 

Silt 

250  cc. 

Clay 
Under  %56  mm 

250  cc. 

second  column  shows  the  range  of  suggested  sample  weights  for  the 
given  sizes,  and  the  third  column  indicates  the  approximate  volume  of 
material  which  coincides  with  the  given  weight. 

The  table  affords  a  basis  for  judging  sample  size  in  terms  of  the 
coarseness  of  material.  In  addition  it  is  necessary  to  consider  the  types 
of  analyses  to  be  performed  on  the  sample.  Wentworth's  values  are  for 
mechanical  analysis,  but  the  general  values  hold  for  most  purposes.  With 
coarse  sediments,  from  sand  upward,  the  same  sample  may  be  used  for 
size,  shape,  and  mineral  analyses,  but  among  the  silts  and  clays  some 
methods  of  size  analysis  (the  pipette  method,  Chapter  6)  prevent  the 
re-use  of  the  material ;  however,  as  only  20  or  30  g.  are  used  for  this 
purpose,  the  suggested  sample  size  is  adequate  to  cover  a  number  of  anal- 
yses. Chemical  analysis  likewise  destroys  the  sample.  If  storage  or  dis- 
play material  is  to  be  made  of  part  of  the  sample,  that  must  be  allowed 


COLLECTION  OF  SAMPLES  33 

for  except  in  cases  where  the  material  analyzed  itself  becomes  the  display 
or  storage  material. 

An  interesting  approach  to  the  problem  of  sample  size  was  made  by 
Knight  ^  in  connection  with  ceramic  materials.  The  principle  introduced 
involves  sample  sizes  proportional  to  the  square  of  the  diameters  of  the 
particles,  starting  with  a  half -gram  sample  of  200-mesh  material. 

CONTAINERS      FOR      SAMPLES 

Sample  containers  may  be  used  for  transportation,  storage,  or  display. 
In  some  cases  the  same  container  serves  more  than  a  single  purpose,  but 
for  convenience  the  treatment  is  based  on  function.  Indurated  specimens 
may  be  wrapped  in  newspaper  for  shipment,  and  displayed  in  trays  or 
boxes.  The  greater  part  of  the  discussion,  accordingly,  will  be  devoted 
to  unconsolidated  material. 

Containers  for  collecting  and  shipping.  Current  usage  varies  consider- 
ably in  the  choice  of  containers  for  field  samples.  The  commonest  con- 
tainer is  undoubtedly  a  bag.  W'entwonh  -  recommends  cloth  bags  for 
most  general  use  with  dry  sediments.  The  authors  experimented  exten- 
sively with  brown  kraft  paper  bags,  and  found  them  suited  to  more  pur- 
poses than  has  generally  been  believed.  Even  wet  sand  may  be  trans- 
ported in  them  if  certain  precautions  of  packing  the  bags  are  followed. 
The  most  commonly  used  containers  for  wet  or  damp  sediments  have 
been  the  familiar  glass  Mason  jars.  Ice-cream  cartons  of  the  cylindrical 
type  appear  to  be  increasingly  used  for  the  same  purpose;  their  light 
weight  when  empty  and  the  absence  of  a  breakage  risk  suit  them  ad- 
mirably for  the  purpose. 

Storage  containers.  It  is  common  practice  to  retain  parts  of  each  field 
sample  for  later  reference.  The  most  convenient  storage  containers  are 
the  cylindrical  cartons  mentioned  earlier,  because  of  their  compact  size 
and  the  convenience  with  which  they  may  be  stacked  one  above  the  other. 
The  next  most  convenient  storage  container  is  a  brown  kraft  paper  bag. 
These  also  are  compact  and  may  be  stacked.  These  two  t\-pes  of  con- 
tainers apply  to  medium-  and  fine-grained  sediments.  If  gravels  are  to  be 
stored,  small,  square  corrugated  paper  cartons  may  be  used,  or  in  ex- 
ceptional cases,  cement  bags. 

If  the  storage  samples  are  to  be  used  extensively,  as  in  assigning  them 
to  classes  for  analysis,  the  cylindrical  cartons  are  best.  Paper  bags  do  not 

^  F.  P.  Knight,  Jr.,  The  importance  of  accurate  sampling  in  the  production  and 
use  of  ceramic  materials:  Jour.  Am.  Ccram.  Soc,  vol.  15,  pp.  444-451,  1932. 
-C.  K.  Wentworth,  loc.  cit.,  1926. 


34        SEDIMENTARY  PETROGRAPHY 

stand  much  handling,  but  have  proved  their  worth   for  dead  storage. 

When  the  storage  of  samples  involves  the  preservation  of  their  orig- 
inal moisture,  Mason  jars  are  perhaps  the  most  useful  containers. 

Display  containers.  Display  containers  may  be  of  several  types,  de- 
pending partly  on  the  size  of  material  involved.  Ordinary  cardboard 
trays,  2x3x5^  in.  or  3x4x1  in.,  are  convenient  for  small  displays  of 
pebbles.  Other  sizes  of  trays  are  available,  some  fitted  with  glass  covers 
to  exclude  dust.  Finer  sediments  may  be  displayed  very  conveniently  in 
small  round  glass  vials  of  about  2-oz.  (liquid)  capacity  and  measuring 
about  an  inch  in  diameter  and  2^  in.  in  height.  These  may  be  obtained 
either  with  corks  or  with  screw  tops.  Another  convenient  display  jar  is 
the  inverted  type  of  bottle,  round-topped,  and  fitted  with  a  cork  at  the 
base.  These  are  available  in  a  variety  of  sizes. 

Capacities  of  Sample  Containers 

Table  2  lists  the  more  common  types  of  sample  containers  and  indicates 
their  capacities  and  dimensions.  The  following  discussion  supplements 
the  data  in  the  table. 

Clotli  bags.  Cloth  bags  are  available  in  many  varieties.  The  mouths  may 
have  tying  strings  attached,  or  drawstrings.  The  weave  or  weight  of  the  bag 
may  be  chosen  in  terms  of  the  fineness  of  the  sediment ;  a  general  rule  is  that 
the  cloth  mesh  should  be  finer  than  the  smallest  particles.  Sized  cloth  is  sat- 
isfactory for  dry  sediments,  but  wet  samples  may  soften  the  sizing. 

Paper  bags.  Where  cost  is  a  factor,  and  where  most  of  the  samples  are 
medium  to  fine-grained,  paper  bags  have  a  wide  applicability.  The  cost  is 
practically  negligible,  and  sizes  i  to  3  may  be  had  for  15  to  20c  per  hundred. 
The  authors  have  used  paper  bags  with  many  kinds  of  sediments  in  the  past 
several  years,  and  the  loss  due  to  breakage  or  leakage  has  been  negligible. 
In  one  instance  wet  sands  were  carried  400  mi.  by  automobile  with  no  losses. 
Certain  precautions,  however,  must  be  followed  in  using  paper  bags  for 
transportation.  The  bag  is  filled  about  half  full  of  sediment,  and  is  tamped 
by  jostling  the  bag  on  a  plane  surface.  The  bag  is  then  closed  by  folding  the 
upper  part  into  a  series  of  underfolds  until  the  package  forms  a  rectangular 
solid.  The  bags  need  not  be  tied  with  string.  After  the  bags  have  been  folded, 
they  are  placed  upside  down  in  a  corrugated  paper  box,  and  packed  tightly 
together.  The  reason  for  packing  them  upside  down  is  that  dampness  may 
loosen  the  bottom  seams,  and  if  the  bags  are  later  removed  while  still  damp, 
no  loss  is  occasioned  by  the  bottoms  dropping  out.  If  the  bags  remain  packed 
until  they  are  dry,  the  seams  reglue  themselves.  Added  strength  is  gained  if 
the  bags  are  used  double. 

For  fine  sediments  like  silt,  loess,  or  clay,  especially  when  the  sediment  is 
dr>-,  the  #1  bags  are  suitable,  and  they  may  be  packed  in  larger  kraft  bags. 
For  example,  the  #1  bags  fit  very  snugly  crosswise  in  #8  bags,  in  rows  of 


COLLECTION  OF  SA^IPLES  35 


Table  2 
Capacities  of  Sample  Containers 

cloth  bags 

Dimensions  Capacity 

i^in.)  (g.  of  sand) 

i2xiS    10  kg. 

9x14    5    " 

/x   9    2    " 

5x   8   I    •• 

4x   6   500  gm. 

3  X   4H    -00     •• 

2x   4   100     " 

BROWN    KRAFT   PAPER    BAGS 

N'umbcrartd 
Capacity  Dimcnsiotus    {in.) 

Ub.)  Length  ll'idth  Breadth 

i^* 13^4  7H  4li 

10* 13  6%  414 

8* i2y2  6  4 

6  iiH  6  3J^ 

5  II  554  3M 

4   9H  5  334 

3** 8H  4^  3 

2** 8  4H  2H 

I  ** T'A  S'A  2 

*  Suitable  for  packing  smaller  bags. 

**  Suitable  for  sand  and  fine  sediments. 

GLASS    JARS    AND    VIALS    (^CYLINDRICAL) 

Di))iensions  Capacity 

(in.)  J'olione  {g.  of  sand) 

7y2X4'A   I  qu^'ift  1.400 

5^x3^    I  pint  700 

4/i^3y2    y2  pint  400 

CYLINDRICAL   ICE-CREAM    CARTONS 

Dimensions  Capacity 

(in.)  J'olitme  {g.  of  sand) 

7  X  3^    I   quart  1.600 

4x3^^    I  pint  800 

2^x3/^    y2  pint  400 


36        SEDIMENTARY  PETROGRAPHY 

three.  Thus  about  nine  of  the  smaller  bags  may  be  assembled  in  larger  bags,  a 
decided  convenience  when  large  numbers  of  samples  are  collected.  Paper 
bags  are  not  suitable  for  gravel,  because  the  pebbles  wear  through.  In  general, 
sizes  I,  2,  and  3  are  used  for  individual  samples,  and  sizes  8,  10,  and  12  for 
repackaging. 

Glass  jars.  Mason  jars  have  found  wide  use  for  collecting  wet 
samples,  where  there  may  be  as  much  water  as  sediment.  They  are  eminently 
satisfactory  for  this  purpose,  but  there  are  inconveniences  attached  to  them. 
The  risk  of  breakage  is  present,  but  may  be  largely  eliminated  by  carrying 
them  always  in  their  cartons.  Present  tendencies  seem  to  be  to  replace 
these  jars  with  cylindrical  cartons,  which  are  less  expensive  and  come  in 
the  same  sizes. 

Cylindrical  ivaterproof  cartons.  Damp  or  wet  samples  are  most  conveniently 
carried  in  these  cartons,  which  are  supplied  with  friction  tops  to  prevent 
leakage.^  ^Modern  cartons  of  this  type  are  free  from  wax,  so  that  none  is 
rubbed  off  by  the  sediment.  The  largest  size  is  suitable  for  fine  gravel,  and  for 
smaller  particles  the  cartons  range  down  in  size  to  a  half  pint.  For  greatest 
all-around  adaptability  in  sampling,  it  is  difficult  to  choose  between  cartons 
and  cloth  bags;  the  bags  have  the  advantage  of  occupying  less  room  when 
empty,  but  the  cartons  have  the  advantage  that  they  pack  more  readily 
when  filled.  The  cartons  may  be  used  for  either  wet  or  dry  sediments,  but 
cloth  bags  have  limitations  when  used  for  wet  material. 

Paper  envelopes.  Another  type  of  container,  not  mentioned  in  the  table,  is 
the  small  paper  envelope,  fitted  with  an  aluminum  strip  across  the  top,  so 
that  the  edge  may  be  rolled  over  and  the  envelope  sealed  securely  against 
losses.  These  envelopes  are  usually  too  small  for  field  samples,  but  they  are 
convenient  for  storing  small  samples  of  laboratory  materials.  For  example, 
the  sieve  separates  obtained  during  mechanical  analysis  may  be  stored  in  this 
manner,  and  during  heavy  mineral  work,  the  light  and  heavy  separates  may 
be  kept  in  them. 


LABELING     AND     NUMBERING     OF      SAMPLES 

Every  sample  should  be  numbered  or  labeled  at  the  time  of  sampling. 
A  convenient  plan,  suggested  by  Wentworth,-  and  used  by  the  authors 
as  a  standard  procedure,  is  to  number  all  samples  serially  during  a  given 
sampling  expedition,  regardless  of  their  nature  or  locality.  The  serial 
number  alone  is  marked  on  the  sample,  and  at  the  same  time  a  notation 
is  made  in  the  field  book,  giving  all  necessary  data.  Suggested  field 
observations  to  be  made  at  the  time  of  sampling  are  given  elsewhere 
(Chapter  i).  It  is  also  convenient  to  indicate  the  location  and  number 
of  each  sample  on  a  map.  In  this  manner  samples,  notebook,  and  map 
are  all  coordinated. 

1  A  satisfactory  brand  of  carton  is  the  "Titelok,"  manufactured  by   Sutherland 
Paper  Co.,  Kalamazoo,  Michit^an. 
-  C.  K.  Wentworth,  loc.  cit.,  1926. 


COLLECTION  OF  SAMPLES  37 

If  it  is  necessary  to  keep  two  or  more  projects  separate  during  sampling,  an 
appropriate  number  o£  capital  letters  may  be  used  to  designate  tlie  several 
projects,  and  tlie  samples  under  each  project  may  be  numbered  serially, 
as  Ai,  A2,  etc.  In  some  instances,  also,  it  may  be  desirable  to  num- 
ber the  samples  in  accordance  with  a  predetermined  grid  pattern,  regardless 
of  the  order  in  which  the  samples  are  collected.  This  may  be  accomplished  by 
labeling  one  set  of  coordinates  with  capital  letters  and  the  other  with  num- 
bers. In  this  manner  any  sample,  as  Ci  or  D3.  may  at  once  be  lo- 
cated. The  authors'  experience  suggests,  however,  that  the  grid  keying  may 
just  as  conveniently  be  accomplished  in  the  laboratory  from  serially  numbered 
samples. 

Individual  practices  vary,  but  in  the  labeling  of  specimens  the  authors 
have  found  that  large  numbers,  legibly  written  directly  on  the  bag  or 
container  with  an  indelible  pencil,  are  adequate.  It  is  often  advisable 
to  write  the  number  in  more  than  one  place  on  the  container  to  avoid 
erasure  by  friction.  Likewise,  large  figures  do  not  blur  into  illegibility  as 
smaller  ones  often  do,  due  to  damp  samples.  If  the  rocks  are  consoli- 
dated, the  number  may  be  wTitten  with  indelible  pencil  on  a  small  square 
of  adhesive  tape,  which  is  fastened  to  the  specimen.  The  same  number 
may  also  be  written  on  the  outside  of  the  wrapping,  to  facilitate  sorting 
in  the  laboratory. 

For  storage  and  other  laboratory  purposes  it  is  advisable  to  have  some 
system  of  distinguishing  among  samples  from  different  projects.  Went- 
worth  ^  suggested  a  series  of  ciphers  and  key  digits  which  could  be 
written  in  front  of  the  field  serial  number,  to  distinguish  the  projects. 
The  authors  follow  a  plan  of  entering  the  samples  in  an  accession  cata- 
logue, approximately  following  the  suggestion  of  Johannsen  -  in  connec- 
tion with  igneous  rocks.  In  the  accession  catalogue  a  new  serial  niunber 
is  given  to  each  sample.  The  record  itself  is  an  ordinary  daybook,  ruled 
into  the  following  columns : 

Accession  I  Field      I  Type  of     I ,.  I  ^  I  t-»  .^    1  la 

T-T  K'  L-  LocATiox    Collector    Date     Remarks 

Number      [  dumber  |  bEDiMEXT  |  |  |  | 

The  accession  numbers  are  serial,  regardless  of  the  project.  The  original 
field  number  is  placed  in  the  second  cokmm.  and  a  short  descriptive  term, 
as  "dune  sand,"  "glacial  till."  or  the  like  is  placed  in  the  third  cokimn. 
The  location  may  be  indicated  in  full  detail  or  roughly  by  county  or 
locality.  The  collector's  initials  are  usually  sufficient,  as  is  the  year  of 

1  C.  K.  Wentworth,  he.  cit.,  1026. 

-A.  Johannsen.  }faiiua[  of  Pctrographic  Methods,  2nd  ed.  (Xew  York.  IQ18), 
pp.  609  ff. 


38        SEDIMENTARY  PETROGRAPHY 

collection.  Under  "Remarks"  a  number  of  items  may  be  included.  A 
given  project  may  be  bracketed  with  a  reference  to  the  field  notes ;  if 
some  of  the  data  from  the  samples  are  published,  that  ma\-  be  indicated 
with  a  reference. 

The  accession  catalogue  is  used  for  storage  purposes.  For  reference, 
a  card  catalogue  may  be  made,  cross-referring  types  of  sediments  with 
localities,  and  the  like.  The  following  examples  show  a  sediment-type 
card  and  a  locality  card  : 


BEACH  SAND 


14     North  Chicago,  111. 
21     Benton  Harbor,  Mich. 
42-57     Waverly  Beach,  Ind. 


WISCONSIN 
Door  Count\' 


131     River  sand 
153     Beach  gravel 
160-177     Beach  gravel 


More  elaborate  numbering  systems  and  cross-reference  schemes  may 
be  devised,  based  on  a  complete  classification  of  sedimentary  materials. 
Milner  ^  follows  a  classification  of  letters  and  numbers,  but  the  unsatis- 
factory' state  of  present  sedimentary  classifications  suggests  that  a  simple 
descriptive  scheme  be  followed.  As  genetic  classification  develops,  cards 
may  be  prepared  with  references  to  the  sample  numbers  that  belong  in 
each  classification. 


THEORY     OF      SAMPLING      SEDIMENT  S 

Statisticians  have  paid  considerable  attention  to  the  theory  of  sampling, 
but  a  direct  application  of  the  principles  to  sedimentary  petrology  is  not 
apparent  without  careful  consideration.  Problems  of  sampling  have  re- 
ceived very  little  formal  treatment  from  a  strictly  sedimentary  point  of 
view.  This  may  be  attributed  to  the  difficult  nature  of  the  problem  and 
to  the   fact  that  virtually  no  sedimentary  petrologists  are   trained   in 

1  H.  B.  Milner,  Sedimentary  Petrography,  2nd  ed.  (London,  1929),  pp.  266  ff. 


COLLECTION  OF  SAMPLES  39 

mathematical  statistics.  The  present  section  will  not  attempt  to  establish 
a  general  theory  of  sampling  sediments  but  instead  will  discuss  the  nature 
of  the  problem  the  types  of  work  that  have  already  been  done. 

In  the  discussion  of  compound  samples  (page  18)  it  was  mentioned 
that  practically  every  sampling  procedure  was  subject  to  some  error. 
This  may  be  illustrated  by  supposing  a  population  or  universe  of  10,000 
spherical  pebbles,  from  which  samples  of  100  pebbles  each  are  to  be 
drawn.  The  pebbles  range  in  size  continuously  from  ri  to  r2,  where  rz 
may  be  taken  as  about  twice  ri.  The  population  has  an  average  radius  rav, 
which  is  not  known.  Suppose  five  samples  of  100  pebbles  are  drawn 
from  the  population  at  random,  and  the  average  radius  of  each  sample 
determined  by  measurement.  By  the  law  of  error,  the  chances  are  that 
no  two  of  these  averages  would  be  identical.  Similarly,  it  is  likely  that 
not  one  of  the  individual  averages  would  be  identical  with  ray.  However, 
if  additional  samples  are  withdrawn  and  measured,  the  average  values 
of  the  samples  would  tend  to  distribute  themselves  as  a  symmetrical  bell- 
shaped  distribution  about  the  value  of  r^y.  Furthermore,  the  peak  of 
the  distribution  would,  for  all  practical  purposes,  coincide  with  the  value 
of  rav. 

This  general  principle  suggests  a  simple  definition  of  a  satisfactory 
sample  from  any  given  sedimentary  deposit :  a  random  sample  may  be 
defined  as_one  in  which  the  characteristics  of  the  sample  show  no  sys- 
tematic variations  from  the  characteristics  ofjhe^deposr^at  the  sampling 
locality.^  This  aft'ords  a  basis  for  collecting  representative  samples  in  the 
field.  In  order  that  deviations  of  the  sample  be  random,  the  material 
sampled  must  be  homogeneous,  and  this  suggests  that  individual  beds  or 
strata  be  used  as  fundamental  units  in  sampling,  inasmuch  as  in  a  major- 
ity of  cases  a  given  bed  will  have  approximately  the  same  characteristics 
throughout  its  thickness  at  the  sampling  site.- 

On  this  basis,  samples  for  detailed  investigations  should  not  transect 
more  than  a  single  bed.  Here,  however,  the  question  of  defining  a  single 
bed  arises,  and  superimposed  on  that  is  a  question  of  practicability  in 
terms  of  the  labor  and  inconvenience  involved.  By  starting  out  from  this 
fundamental  concept,  however,  the  individual  worker  may  form  his  own 
judgment  as  to  the  sampling  unit  he  should  use.  The  principle  of  beds 
or  strata  as  units  in  sampling  has  been  implied  in  the  body  of  the  chapter, 

1  This  discussion  illustrates  the  parallelism  of  the  authors'  and  Otto's  independent 
approaches  to  the  problem  of  sampling.  By  introducing  the  term  sedimentation  unit 
(page  20),  and  by  classifying  sampling  techniques  more  fully,  Otto  developed  the 
present  notions  more  rigorously. 

-  Graded  bedding  is  a  marked  exception,  but  the  simpler  case  is  used  here  as  a 
first  approach. 


40        SEDIMENTARY  PETROGRAPHY 

in  connection  %\'ith  the  choice  of  sample  in  a  given  case.  It  applies  most 
strongly  in  studies  directed  toward  an  elucidation  of  the  detailed  history 
of  a  sediment,  rather  than  to  studies  involving  only  average  characteristics. 

An  example  may  be  given  here  of  the  point  of  view  involved.  Suppose 
an  outwash  terrace  composed  of  numerous  beds  of  varying  thicknesses 
and  degrees  of  coarseness.  How  shall  the  terrace  be  sampled  to  determine 
its  characteristics  in  detail  ?  One  answer  is  to  treat  each  bed  as  a  homo- 
geneous population  and  to  sample  it  individually.  Each  sample  could 
thereupon  be  analyzed  individually,  and  from  the  series  of  average  values 
obtained  a  frequency  distribution  of  average  sizes  could  be  constructed 
which  would  reflect  the  general  characteristics  of  the  terrace.  Not  only 
average  conditions,  but  the  spread  of  the  individual  beds  would  be 
brought  out  by  such  a  study.  The  same  reasoning  applies  not  only  to 
size  characteristics,  but  to  mineral  content,  shape  attributes,  and  the  like. 

There  is  a  further  question  involved,  however,  even  in  the  case  of  a 
sample  from  a  given  stratum :  how  may  one  know  whether  his  sample 
of  that  bed  is  actually  a  random  sample?  There  are  several  tests  by 
which  this  may  be  learned,  and  all  of  them  rest  on  the  principle  that  a 
random  sample  will,  on  the  average,  show  no  significant  systematic  de- 
viations from  the  true  characteristics  of  the  bed.  These  tests  include  the 
probable  error  method,  the  chi-square  test,  and  the  theory  of  a  state  of 
control,  all  of  which  have  been  applied  to  sedimentary-  problems  and  are 
mentioned  in  Chapter  9  of  this  volume.  Each  of  the  methods  involves 
technical  statistical  operations,  and  hence  they  are  appropriately  deferred 
to  the  chapter  on  statistics.  The  probable  error  method  has  been  applied 
specifically  to  sampling  problems  by  Krumbein,^  the  chi-square  test  was 
used  by  Eisenhart  ^  in  a  discussion  of  geological  correlation,  and  the 
theor}'  of  control  was  used  by  Otto  ^  in  connection  with  test  samples 
split  from  a  field  sample. 

The  sampling  error  is  a  funaion  of  the  homogeneity  of  the  sediment, 
of  the  precise  locality  in  which  the  sample  is  collected,  and  of  the  manner 
in  which  it  is  collected.  Regardless  of  the  magnitude  ot  the  error,  tiow- 
ever,  it  was  pointed  out  earlier  that  the  sampling  error  could  be  reduced 
to  any  value  desired  by  securing  compound  samples.  This  arises  from  the 
nature  of  the  error  function  and  may  be  illustrated  as  follows :  suppose 
a  series  of  numbers,  r^,  r,,  . .  .  r^,  and  their  average  value  r^v  If  any  two 

1  \V.  C.  Krumbdn,  loc.  cit.,  1934. 

-  C.  Eisenhart,  .\  test  for  the  significance  of  lithological  variations :  Jour.  Sed. 
Petrology,  vol.  5,  pp.  I37-I45,  1935- 

3  G.  H.  Otto,  The  use  of  statistical  methods  in  effecting  improvements  on  a  Jones 
sample  splitter:  Jour.  Scd.  Petrology,  vol.  7,  pp.  101-133,  1937. 


COLLECTION  OF  SAMPLES 


41 


VALUES    OF 


of  the  numbers  are  chosen  at  random,  they  will  generally  differ  from 
Tav  by  some  fixed  amount,  which  may  be  referred  to  as  their  error.  How- 
ever, if  the  pair  of  numbers  are  themselves  averaged,  the  value  of  their 
average  will  never  be  farther  from 
Tav  than  the  extreme  of  the  paired 
values. 

It  has  been  found  ^  that  the  error 
of  the  mean  of  a  set  of  observations 
varies  inversely  as  the  square  root  of 
the  number  of  observations.  E^i- 
pressed  mathematicallv.  this  is  Em  = 
£/>/»,  where  £,„  is  the  error  of  the 
mean,  E  is  the  error  of  a  single  ob- 
servation, and  ;;  is  the  number  of 
observations  made.  This  equation  may  bo  expressed  as  a  ratio, 
En,/E=  i/\/n.  In  this  latter  case  it  is  possible  to  study,  by  means  of  a 
simple  graph,  the  behavior  of  the  function  as  n  increases.  By  choosing 
values  of  ;/  from  i  to  10,  the  corresponding  values  of  E^/E  are  found, 
as  shown  in  Table  3.  Figure  7  is  a  graph  of  the  function,  which  demon- 
strates that  as  ;;  increases  the  error  decreases  rapidly  at  first  and  then 
more  slowly.  The  point  where  the  curve  begins  to  flatten  out  is  at  about 
n  =  4,  where  E,„/E  =  o.5.  As  the  curve  is  followed  out  to  h^io, 
Em/E^  0.316.  so  that  the  rate  of  change  has  decreased  appreciably. 


^-i* 


Fig.    7. — Graph    of    the    function 
E,„/E=  i/V».  Data  from  Table  3. 


Table  3 


Values  of  PEn/E  and  n  from  ?£„/£=  i/Vh 


n 

PE„yE 

I    

2    

0.707 

3    

c    

0.447 
0.40S 
0.378 
0.354 

0.^16 

6    

8    

0    

10    

H.  L.  Rietz,  Handbook  of  ^[atllcmatical  Statislics  (Boston.  19^4),  p.  77. 


42        SEDIMENTARY  PETROGRAPHY 

The  greater  relative  reduction  in  the  error  when  four  observations  are 
combined,  together  with  the  added  effort  required  to  collect  a  large 
number  of  samples,  suggests  as  a  first  approximation  that  in  sampling 
sediments  where  the  sampling  error  may  be  a  factor,  four  discrete  sam- 
ples be  combined  into  a  single  composite.  It  was  for  this  reason  that 
composites  of  four  were  suggested  in  the  discussion  of  compound  samples. 

It  is  possible  to  apply  the  error  equation  to  the  problem  of  reducing  the 
sampling  error  to  any  given  value.  Suppose  the  error  of  the  individual  sample 
is  4.5  per  cent,  as  has  been  found  in  certain  beach  sands.  It  is  desired  to  reduce 
this  error  to  0.5  per  cent.  How  many  samples  must  be  combined  into  a  com- 
posite? The  solution  is  found  by  first  determining  the  decimal  value  of  the 
ratio  0.5:4.5.  This  is  o.iii.  Hence  E^/E  =  cm  =  1/ V  h.  Solving  this  for 
Vn  yields   V w  =  i/o.iii  =  9.0,  and  hence  n  =  8i  samples. 

The  present  discussion  of  sampling  theory  as  applied  to  sediments  is 
far  from  exhaustive,  and  there  are  numerous  problems  which  have  not 
even  been  touched  upon.  For  example,  an  important  problem  connected 
with  the  areal  study  of  sediments  involves  the  determination  of  the  change 
in  sediment  characteristics  as  the  formation  is  followed  away  from  its 
source.  Among  the  questions  which  arise  is  whether  the  variations  ob- 
served are  due  to  an  actual  change  in  sedimentary  characteristics  or 
whether  they  are  due  to  other  causes,  such  as  sampling  errors  and  the 
like.  Problems  of  this  sort  have  not  been  investigated  in  detail  from  a 
sedimentary  point  of  view,  but  analogous  problems  have  been  studied 
in  agricultural  science.  In  this  instance  the  problem  was  whether  observed 
variations  in  crops  were  due  to  actual  variations  among  the  species 
studied  or  whether  they  were  due  to  variations  in  soil  fertility  over  the 
experimental  plot.  The  nature  of  the  problem  and  methods  of  attacking 
it  statistically  are  given  by  Fisher.^  The  method  applied  by  Fisher  is 
called  analysis  of  variance,  and  it  seems  likely  that  applications  of  the 
technique  to  sedimentary  problems  will  yield  significant  resuUs. 

1  R.  A.  Fisher,  The  Design  of  Experiments  (Edinburgh,  1935),  Chap.  4. 


CHAPTER  3 

PREPARATION  OF  SAMPLES  FOR  ANALYSIS 

INTRODUCTION 

The  preparation  of  sedimentary  materials  for  study  is  a  usual  prelimi- 
nary to  their  detailed  analysis.  Field  samples  commonly  are  larger  than 
the  laboratory  test  sample,  and  in  addition  the  sediment  may  be  in  a  state 
of  aggregation  unsuitable  for  direct  analysis.  Several  steps  may  be  in- 
volved in  the  preparation  of  the  samples,  and  the  nature  of  the  treatment 
depends  upon  the  sediment  and  on  the  study  to  be  made. 

Although  there  is  no  universal  method  of  treatment  for  all  possible 
types  of  sedimentary  analysis,  there  are  several  steps  in  the  process  which 
have  more  or  less  in  common.  The  first  treatment  is  a  preliminary  dis- 
aggregation of  the  field  sample  into  smaller  aggregates  suitable  for  split- 
ting off  test  samples.  The  test  samples  are  then  further  disaggregated  or 
dispersed  to  a  state  suitable  for  the  type  of  analysis  to  be  performed. 
In  mechanical  analysis  the  sample  may  be  treated  with  mild  chemical 
agents  which  effect  a  separation  of  the  aggregates  into  individual  particles 
and  supply  each  particle  with  electrical  charges  which  prevent  reaggre- 
gation  during  analysis.  In  mineralogical  analysis,  on  the  other  hand, 
strong  treatment  with  acids  or  bases  may  be  required  to  clean  the  surfaces 
of  the  grains  and  to  prepare  them  for  microscopic  study. 

Care  should  be  exercised  during  splitting  to  obtain  a  representative 
part  of  the  field  sample.  It  is  not  sufficient  to  pour  a  quantity  of  the  field 
sample  from  the  container,  especially  with  medium-  or  coarse-grained 
sediments,  because  properties  such  as  size,  shape,  density,  magnetic  prop- 
erties, coefficient  of  friction,  and  elasticity  may  cause  a  selective  error.^ 
Such  errors  may  have  a  serious  elTect  on  the  final  results,  especially  if 
very  small  samples  are  to  be  split  off,  as  in  heavy  mineral  studies.  With 
very  fine-grained  material  no  serious  errors  may  be  involved  if  the 
material  is  thoroughly  mixed  and  portions  extracted  with  a  spatula,  as 
is  often  done  for  chemical  analysis,  but  even  here  formal  methods  of 
splitting  are  to  be  preferred. 

1  G.  H.  Otto,  Comparative  tests  of  several  methods  of  sampling  heavy  mineral 
concentrates:  Jour.  Sed.  Petrology,  vol.  3.  pp.  30-39,  1933. 

4.3 


44        SEDIMENTARY  PETROGRAPHY 

If  the  sediment  is  composed  of  loose  grains  or  particles,  the  splitting 
process  may  be  undertaken  without  preliminaries.  However,  if  the  mate- 
rial occurs  as  aggregates,  a  preliminary  disaggregation  should  be  per- 
formed, to  avoid  having  too  large  lumps  of  material  in  the  test  sample. 

SAMPLE     SPLITTING 

To  avoid  selective  errors,  the  field  sample  should  be  separated  into 
individual  grains  or  small  aggregates  before  splitting  off  a  test  sample, 
n  the  sediment  is  a  loosely  cemented  sandstone,  the  preliminary  treatment 
may  consist  of  gently  crushing  the  rock  with  a  rubber  pestle  or  a  wooden 
rolling  pin.  Clays  and  silts  often  harden  during  drying,  and  the  lumps 
may  be  broken  in  a  similar  manner,  so  that  no  aggregates  larger  than  a 
pea  are  present,  li  the  rock  is  partially  indurated,  as  a  shale,  some  crush- 
ing device  may  be  used  to  obtain  pea-sized  fragments  if  it  is  possible  to 
do  so  without  destro}ing  individual  particles.  Gravels  which  are  cemented 
may  sometimes  be  separated  into  pebbles  by  acid  leaching,  but  before 
any  chemical  methods  are  used,  precautions  should  be  taken  to  see  that 
no  material  which  may  be  needed  for  the  analysis  is  lost.  The  purpose 
of  preliminary  disaggregation  is  merely  to  obtain  the  material  in  such 
form  that  it  may  be  quartered  into  smaller  samples. 

Quartering  by  hand.  The  simplest  method  of  splitting  samples  is  to 
pour  the  field  sample  into  a  conical  pile  on  a  large  sheet  of  smooth- 
surfaced  paper  and,  with  a  spatula  or  other  device  (the  hands  may  be 
used  with  coarse  material),  to  separate  the  heap  into  four  quarters  by 
cutting  it  pie  fashion  along  two  normal  diameters.  Alternate  quarters  are 
retained,  and  the  others  are  laid  aside.  H  the  remaining  quarters  are  still 
too  large,  they  may  be  recombined  into  a  smaller  pile  and  the  process 
repeated. 

An  adaptation  of  hand  quartering  was  made  by  Pettijohn,^  who  used 
four  rectangular  sheets  of  paper  and  overlapped  them  to  form  a  square 
composed  of  one-fourth  of  each  sheet.  The  sample  was  poured  on  the 
center  of  the  square,  spread  into  a  circular  heap,  and  the  papers  pulled 
apart.  Opposite  quarters  were  recombined  and  the  process  repeated  until 
a  sufficiently  small  split  was  obtained. 

Knife-edge  splitters.  Several  years  ago  Krumbein  -  experimented  with  a 
common  splitting  device  which  consisted  of  a  conical  hopper,  the  lower  open- 
ing of  which  was  superimposed  over  two  crossed  knife  edges,  which  separated 

1  F.  J.  Pettijohn,  Petrography  of  the  beach  sands  of  southern  Lake  Michigan: 
Jour.  Gcologx,  vol.  39,  pp.  432-455,  1931. 

2  G.  H.  Otto,  loc.  cit.,  1933- 


PREPARATION  OF  SAMPLES 


45 


the  stream  of  grains  into  four  divisions.  Otto,^  in  comparing  several  splitting 
methods,  improved  on  the  original  device  but  found  that  for  certain  types  of 
samples  the  deviation  was  larger  than  in  other  splitting  devices. 

Jones  sample  splitter.  One  of  the  most  widely  used  devices  for  splitting 
samples  is  the  Jones  sample  splitter,  which  consists  of  a  series  of  inclined 
chutes  leading  alternately  to  two  pans  placed  on  opposite  sides  of  the 
apparatus.  The  sample  is  poured  into  a  hopper,  using  a  rectangular  pan, 
the  width  of  which  is  equal  to  the  width  of  the  set  of  chutes.  The  sample 
is  split  to  the  desired  size  by  resplitting  the  right- 
and  left-hand  halves  alternately. 

The  commercial  model  of  the  Jones  splitter  (Fig- 
ure 8)  has  splitting  compartments  about  i  cm.  in 
width,  so  that  the  use  of  the  device  is  limited  to 
particles  smaller  than  about  a  centimeter  in  diam- 
eter. Wentworth  -  studied  the  error  involved  in  the 
use  of  the  Jones  splitter,  using  sandy  gravel  having 
a  range  of  sizes  from  about  8  mm.  to  Y^  mm.  diam- 
eter. His  results  showed  that  the  error  was  larger 
with  the  larger  sizes,  but  the  relation  between  par- 
ticle size  and  magnitude  of  the  error  was  not  con- 
stant. More  recently  Otto  ^  made  similar  tests,  in- 
volving a  method  of  statistical  control.  The  method 
involves  a  study  of  the  performance  characteristics 
of  the  instrument  in  terms  of  the  deviations  from  expected  theoretical 
results.  The  study  indicated  among  other  things  that  the  personal  ele- 
ment affected  results,  in  the  manner  in  which  the  material  was  poured 
into  the  hopper.  Otto  thereupon  designed  a  modified  Jones-type  splitter 
and  applied  the  control  method  to  it.  The  new  device  showed  satisfactory 
agreements  between  expected  and  attained  results  and  was  not  influenced 
by  the  personal  element. 

Otto's  modified  Jones-type  splitter  dififers  from  commercial  models  in 
several  respects.  The  hopper  is  so  designed  that  the  sample  can  only  be 
poured  in  a  standard  manner ;  the  pans  have  lugs  on  them,  which  elimi- 
nates the  personal  element  both  in  pouring  and  receiving.  The  receiving 
pans  also  have  dust  covers  to  prevent  losses  during  splitting.  Complete 
working  drawings  for  the  instrument  are  given  in  Otto's  paper. 

1  G.  H.  Otto,  loc.  cit.,  1933. 

-  C.  K.  Wentworth,  The  accuracy  of  mechanical  analysis:  .Un.  Jour.  Sci.,  vol.  13, 
pp.  399-408,  1927. 

3  G.  H.  Otto,  The  use  of  statistical  methods  in  effecting  improvements  on  a  Jones 
sample  splitter:  Jour.  Scd.  Petrology,  vol.  7,  pp.  101-133,  1937. 


Fig.  8. — Jones  Sam- 
ple Splitter.  (Cour- 
tesy W.  S.  Tyler 
Company,  Cleveland, 
O.) 


46        SEDIMENTARY  PETROGRAPHY 

At  an  earlier  date  Otto  also  designed  a  miniature  form  of  the  Jones 
splitter,  called  a  "Microsplit."  ^  This  device  is  described  more  fully  in 
Chapter  15. 

Rotary  type  sample  splitters.  Wentworth,  Wilgus  and  Koch  -  developed 
a  rotar)-  tj^pe  of  sample  spUtter  which  was  made  in  two  sizes,  one  for 
large  samples  and  the  other  for  small  samples.  The  device  consists  of  a 
set  of  cylindrical  tubes  arranged  around  the  peripher}'  of  a  rotating  table. 
The  sample  is  fed  into  a  hopper  which  delivers  it  through  a  funnel  to  a 
position  above  the  rotating  set  of  cylinders.  As  the  table  rotates,  the 
grains  are  distributed  among  the  tubes.  The  speed  of  rotation  may  be 
varied  and  the  tubes  made  to  pass  as  often  as  desired  beneath  the  funnel. 
The  splits  could  be  used  individually  as  sixteenths  of  the  original  amount, 
or  opposite  tubes  could  be  combined  in  various  ways.  A  series  of  tests 
were  conducted  on  the  relative  accuracy  of  the  rotarv-  splitter  and  a  Jones 
splitter ;  the  rotarj^  splitter  showed  a  marked  superiority. 

Oscillatory  sample  splitters.  Several  years  ago  J.  E.  Appel  developed 
an  oscillatory  t>-pe  sample  splitter  at  the  University  of  Chicago.  The 
device  consists  of  a  vertical  brass  funnel  suspended  on  an  axis  and 
rocked  over  a  knife  edge  through  a  small  arc.  The  device  is  more  fully 
described  in  Chapter  14. 

Significance  of  statistical  tests  on  sample  splitters.  In  all  the  investigations 
made  of  the  relative  merits  of  sample  splitters,  the  data  were  based  on  the 
grades  recovered,  rather  than  on  the  effect  of  the  splitter  on  the  statistical  para- 
meters of  the  frequencv'  curve.  In  other  connections  it  has  been  noted  that 
comparative  analyses  of  sediments  may  show  fluctuations  in  the  amounts  of 
material  collected  by  the  several  sieves,  and  yet  the  statistical  values  of  the 
sediment  as  a  whole  will  be  only  slightly  affected.  It  would  be  instructive  to 
have  a  study  made,  eitlier  in  terms  of  probable  error  or  in  connection  with  the 
theor>'  of  control,  on  the  size  distribution  as  a  whole  with  various  splitting 
devices.  The  only  data  on  this  question  known  to  the  authors  are  contained  in 
an  incidental  study  conducted  by  Krumbein  ^  in  connection  with  field  sampling 
errors.  The  probable  error  of  splitting  and  sieving  combined  was  found  to 
range  from  0.75  to  142  per  cent,  as  computed  with  respect  to  the  median  grain 
size  of  beach  sands.  The  sample  splitter  was  a  commercial  Jones  splitter,  and 
the  magnitude  of  the  total  laboratorj'  error  suggests  that  for  most  analyses 
the  error  is  perhaps  not  unduly  large. 

1  G.  H.  Otto,  loc.  cit.,  1933. 

2C.  K.  Wentworth,  \N'.  L.  Wilgus  and  H.  L.  Koch.  A  rotary  type  of  sample 
splitter :  Jour.  Sed.  Petrology,  vol.  4,  pp.  127-138,  1934. 

3W.  C.  Krumbein,  The  probable  error  of  sampling  sediments  for  mechanical 
analysis:  Am.  Jour.  Sci,  vol.  27,  pp.  204-214,  1934. 


PREPARATION  OF  SAMPLES  47 

DISAGGREGATION     OF     TEST     SAMPLES 

Further  disaggregation,  after  splitting  test  samples  from  the  original 
material,  must  be  undertaken  with  an  understanding  of  the  efifect  of 
each  process  on  the  characteristics  of  the  sediment.  Of  primary  impor- 
tance is  the  principle  that  no  method  should  be  used  which  alters  or 
destroys  any  of  the  data  to  be  obtained  in  the  subsequent  analysis,  with- 
out an  evaluation  of  the  error  introduced.  Unconsolidated  sediments  may 
receive  so  little  pretreatment  that  their  properties  are  not  appreciably 
affected.  As  the  degree  of  cementation  or  induration  increases,  however, 
methods  of  disaggregation  become  progressively  more  "violent"  and 
there  is  greater  likelihood  that  some  of  the  characteristics  of  the  sediment 
may  be  modified.  When  it  appears  that  disaggregation  can  only  be  had 
at  the  expense  of  greatly  altered  properties,  it  may  be  better  to  rely  on 
thin-section  methods  of  analysis. 

It  may  be  pertinent  here  to  contrast  the  kinds  of  treatment  which  may 
be  resorted  to  in  the  several  types  of  analysis : 

Preparation  for  mechanical  analysis.  In  disaggregating  sediments  for  me- 
chanical analysis  the  precautions  to  be  followed  are  that  the  grains  should  not 
be  broken  and  that  none  of  the  primary  constituents  should  be  removed  by  the 
disaggregation  process.  Breakage  of  grains  results  in  an  error  in  the  average 
size  and  in  other  statistical  constants,  because  the  larger  grains  are  reduced 
in  number  and  the  smaller  ones  increased.  The  removal  of  primary  material 
will  yield  results  which  are  biased  in  the  degree  to  which  the  removed  material 
is  an  integral  part  of  any  given  grade  size. 

The  treatment  accorded  to  samples  for  mechanical  analysis  depends  also 
on  the  coarseness  of  the  sediment.  Coarser  particles  are  usually  sieved,  whereas 
fine-grained  sediments  are  usually  separated  in  terms  of  their  settling  velocities 
in  water.  For  sieving  it  is  only  necessary  to  obtain  a  state  of  disaggregation 
such  that  each  particle  is  individual ;  for  finer  sediments  precautions  must  also 
be  taken  to  see  that  the  individual  grains  do  not  reaggregate  during  analysis. 
This  second  requisite  of  the  fine-grained  sediments  automatically  rules  out 
certain  chemical  procedures,  which  may  coagulate  the  particles  during  the 
analysis. 

Preparation  for  shape  and  surface  texture  analysis.  Owing  to  limitations 
in  the  practical  application  of  techniques  of  shape  and  surface  texture  anal- 
ysis, most  work  of  this  kind  is  limited  to  coarse-  and  medium-grained  sedi- 
ments. The  principal  precautions  that  must  be  followed  relate  to  the  breakage 
of  the  grains,  which  strongly  affects  their  roundness  although  it  may  not 
seriously  change  their  degree  of  sphericity.  In  like  manner,  methods  of  dis- 
aggregation which  alter  surface  textures,  either  by  solution  or  abrasion,  must 
be  avoided  in  surface  texture  analysis.  Tests  to  be  applied  in  such  cases  may 
involve  the  examination  of  particular  grains  before  and  after  disaggregation, 
to  determine  whether  changes  have  occurred. 


48        SEDIMENTARY  PETROGRAPHY 

Preparation  for  viincralogical  analysis.  For  certain  kinds  of  mineralogical 
studies  it  is  not  necessary  to  prevent  breakage  of  grains,  and  individual  sur- 
face textures  usually  need  not  be  preserved.  Attention  is  generally  focused  on 
the  heavy  minerals,  and  hence  a  wider  choice  of  disaggregation  procedures  is 
available  for  mineralogical  studies.  When  it  is  necessary  to  count  mineral 
frequencies  for  statistical  comparisons,  however,  the  factor  of  grain  breakage 
must  be  considered.  Inasmuch  as  mineralogical  analysis  is  a  complete  subject 
within  itself,  further  details  of  the  disaggregation  processes  are  given  in 
Chapter  13. 


COARSE-     AND     MEDIUM -GRAIN  ED     SEDIMENTS 

Unconsolidated  gravel  and  sand  present  no  disaggregation  problems. 
With  such  materials  one  may  proceed  to  the  analysis  as  soon  as  a  test 
sample  has  been  split  from  the  field  sample.  If  on  the  other  hand  the 
sediment  is  consolidated,  several  devices  are  available  for  disaggregation. 
Several  fragments  should  be  examined  under  a  binocular  microscope 
before  treating  the  rock,  so  that  the  proper  choice  of  method  may  be 
made. 

Removal  of  cement.  Among  the  more  common  cementing  materials  in 
sedimentary  rocks  are  calcite,  iron  oxide,  quartz  (silica  in  general)  and 
organic  materials  like  bitumen.  The  simplest  of  these  to  eliminate  is  the 
calcite  cement,  which  may  be  removed  by  gently  heating  the  rock  frag- 
ments in  dilute  hydrochloric  acid.  Before  resorting  to  acid  treatment, 
however,  it  is  well  to  note  whether  any  primary  calcite  fragments  are 
present  in  the  sediment.  In  gravel  there  may  be  limestone  pebbles,  and 
some  sands  contain  calcite  and  other  carbonates  as  an  integral  part  of 
the  size  frequency  distribution.  If  acid  leaching  is  resorted  to  indis- 
criminately, the  mechanical  analysis  will  be  inaccurate  to  the  extent  that 
primary  carbonate  particles  are  present.  In  the  absence  of  any  alternative, 
the  rock  may  be  leached  in  acid  and  some  method  used  to  correct  for 
lost  material.  The  data  for  the  corrections  may  be  obtained  microscopi- 
cally in  many  cases. 

Iron  oxide  cement  may  be  removed  with  stannous  chloride.  A  solution 
of  the  salt  is  added  to  dilute  hydrochloric  acid,  and  the  rock  fragments 
are  heated  in  the  solution.  Tester  used  15-18  per  cent  HCl  with  about 
10  per  cent  of  stannous  chloride.^  Because  of  the  acidic  nature  of  this 
solvent,  the  same  precautions  should  be  taken  about  primary  carbonate 
grains  as  in  the  case  of  acid  treatment  alone. 

Silica  cements  are  in  general  the  most  difficult  to  remove.  If  the  cement 

1  A.  C.  Tester,  The  Dakota  stage  of  the  type  locality.  Appendix  A,  Laboratory 
Methods:  Iowa  Geol.  Survey,  vol.  35,  p.  305,  1931. 


PREPARATION  OF  SAMPLES  49 

is  quartz  and  shows  secondary  enlargement  of  the  primary  quartz  grains 
in  the  sediment,  disaggregation  is  practically  hopeless.  Thin-section 
methods  of  mechanical  analysis  (Chapter  6)  may  be  used  in  such  cases. 
When  the  cement  is  opal  or  amorphous  silica,  the  use  of  concentrated 
alkalies  is  sometimes  sufficient  to  remove  the  material.^  Such  alkalies  also 
affect  some  of  the  mineral  grains,  however,  and  should  be  used  with 
recognition  of  this  fact. 

Occasionally  pyrite  occurs  as  a  cementing  agent.  By  boiling  the  speci- 
men in  dilute  nitric  acid  the  pyrite  may  commonly  be  dissolved.- 

Organic  cements,  such  as  bitumen,  are  most  effectively  treated  by 
using  such  solvents  as  ether,  acetone,  benzol,  or  gasoline.^  Needless  to 
say,  the  specimen  should  not  be  boiled  in  such  solvents  unless  provisions 
are  taken  to  avoid  fires  or  explosions.  Reflux  condensers  attached  to 
Pyrex  flasks  are  usually  suf^cient. 

For  the  removal  of  colloidal  binding  materials  in  soils,  such  as  organic 
matter,  iron  oxide,  or  colloidal  silica,  Truog  and  others  ■*  recently  de- 
veloped a  procedure  involving  the  use  of  oxalic  acid  and  sodium  sulphide. 
The  nascent  hydrogen  sulphide  liberated  in  the  soil  suspension  dissolved 
the  cementing  materials  and  effected  a  completed  dispersion  of  the  soil. 

Disruption  of  rock  specimens.  In  some  cases  the  cementing  material 
may  defy  all  attempts  to  remove  it,  or  the  rock  may  be  highly  indurated. 
W'hen  dealing  with  conglomerates,  individual  pebbles  may  be  chiseled 
from  the  matrix,  but  for  mechanical  analyses  care  must  be  exercised  to 
obtain  the  complete  size  range  of  the  pebbles,  as  well  as  a  representative 
number  of  the  several  sizes.  If  the  rock  material  is  not  coarse-grained 
enough  for  individual  treatment  of  the  particles,  various  disrupting  de- 
vices may  be  tried. 

The  rock  may  be  heated  to  redness  and  plunged  into  water, ■'^  whereupon 
some  of  the  grains  may  be  loosened.  This  method  is  extremely  effective 
in  altering  minerals,  however,  and  should  not  be  resorted  to  as  a  general 
rule  for  mechanical  analysis.  Grain  breakage  is  also  a  common  accom- 
paniment of  such  violent  treatment.  A  less  destructive  method  of  treat- 
ment involves  the  use  of  saturated  solutions  of  various  chemicals,  which 

1  G.  L.  Taylor  and  N.  C.  Georgcscn,  Disaggregation  of  clastic  rocks  by  use  of  a 
pressure  chamber :  Jour.  Scd.  Pctrolotjy,  vol.  3,  pp.  40-43.  1933- 

2  E.  S.  Dana.  Tcvtbook  of  Mineralogy,  3rd  rev.  ed.   (New  York,  1922),  p.  377. 

3  E.  AI.  Spickcr,  Bituminous  sandstone  near  Vernal,  Utah:  U.  S.  Geol.  Survey 
Bull.  Sssc,  pp.  77-98,  1930. 

*  E.  Truog.  J.  R.  Taylor,  R.  W.  Pearson,  M.  E.  Weeks  and  R.  W.  Simonson, 
Procedure  for  special  type  of  mechanical  and  mineralogical  soil  analysis :  Proc.  Soil 
Sci.  Soc.  America,  vol.  i,  pp.  101-112,  1936. 

5  This  procedure  was  used  as  early  as  1863  by  R.  Ulbricht :  Ein  Bcitrag  zur 
Methode  der  Bodcn-analyse,  Landivirts.  Vcrsuchs-Stat.,  vol.  5,  pp.  200-209,  1863. 


50 


SEDIMENTARY  PETROGRAPHY 


are  allowed  to  permeate  the  rock  by  prolonged  soaking.  The  fragments 
are  then  removed  and  allowed  to  dry,  whereupon  the  force  of  crystalliza- 
tion will  sometimes  disrupt  the  rock.  Sodium  sulphate  ^  and  sodium  hypo- 
sulphite (hypo)-  have  been  used  for  this  purpose.  A  somewhat  analogous 
method  is  to  saturate  the  rock  with  sodium  carbonate  solution  and  then 
plunge  the  rock  in  acid.^  The  pressure  of  the  escaping  COo  aids  in  freeing 
the  grains  from  each  other. 

Neumaier  "*  used  ammonium  nitrate  in  an  ingenious  manner  to  effect  a 
disaggregation  of  sedimentary  grains.  The  salt  is  soluble  to  the  extent 

of  177  g.  per  liter  at  20°  C,  and 
1,011  g.  per  liter  at  100°  C.  The 
sediment  was  accordingly  placed  in 
a  saturated  solution  of  the  salt  at 
110°  for  five  minutes,  whereupon 
the  solution  was  rapidly  cooled  to 
room  temperature.  The  resulting 
crystallization  of  the  excess  ammo- 
nium nitrate  forced  apart  the  ag- 
gregates of  the  sediment.  The 
process  was  repeated,  and  finally 
the  salt  was  removed  by  washing 
with  distilled  water. 

Pressure-chamber  disaggrega- 
tion. Taylor  and  Georgesen  ^  de- 
veloped a  pressure  chamber  which 
proved  very  effective  in  disaggre- 
gating indurated  rocks.  The  cham- 
ber consists  of  a  12-in.  length  of 
lo-in.  steel  casing,  with  a  plate  of 
half-inch  steel  welded  to  one  end  to  form  the  base.  A  flange  of  the  same 
material  was  welded  to  the  open  end,  to  afford  a  means  of  fastening  a 
cover  of  half -inch  steel  to  it.  A  stopcock  and  pressure  gauge  were  added 
to  the  cover  (Figure  9).  Within  the  chamber  was  a  wooden  rack  on 
which  several  beakers  could  be  set.  The  specimen  to  be  disaggregated  is 
placed  in  a  beaker  and  covered  with  an  appropriate  solution.  Additional 
1  M.  Morris,  Unsoundness  of  certain  types  of  rocks:  loiva  Acad.  Sci.  Proc,  vol. 
38,  pp.  175-181,  1931. 

-  I.   Tolmachofif,   Crystallization  of  certain  salts  used  for  the  disintegration  of 
shales :  Science,  vol.  76,  pp.  147-148,  1932. 

3  A.  Mann,  Proc.   U.  S.  Nat.  Mus.,  vol.  60,  pp.   1-8,   1932. 

4  F.  Neumaier,  t)ber  Vorbehandlungsverfahren  der  Sedimente  zur  Sclammanalyse : 
Zeutr.  f.  Min.,  Abt.  A,  pp.  78-95,  1935- 

5  G.  L.  Taylor  and  N.  C.  Georgesen,  loc.  cit.,  1933. 


Fig.  9. — ^Taylor  and  Georgesen  pres- 
sure disaggregator. 


PREPARATION  OF  SAMPLES  51 

solution  is  poured  into  the  bottom  of  the  chamber,  the  hd  is  fastened  on, 
and  the  vessel  heated  with  blowtorches  until  a  desired  pressure  is  regis- 
tered on  the  gauge.  The  maximum  pressure  used  was  350  lb.  to  the 
square  inch.  The  chamber  proved  successful  with  conglomerates,  grits, 
sandstones,  siltstones,  and  shales,  cemented  with  calcium  carbonate,  iron 
oxide,  silica,  or  combinations  of  the  three.  In  most  cases  the  specimens 
were  either  completely  disaggregated  or  so  weakened  that  they  crumbled 
under  a  rubber  mallet. 

FiXE-GRAIXED    SeDIMEXTS 

Considerable  attention  has  been  devoted  to  the  question  of  disaggre- 
gating and  dispersing  fine-grained  sediments  for  mechanical  analysis.  Be- 
cause of  the  difficulty  of  determining  in  all  cases  the  effect  of  various 
agents  on  the  extremely  small  particles,  it  has  been  considered  safest  to 
avoid  the  more  rigorous  methods  used  with  coarse  sediments  and  in 
general  to  avoid  the  use  of  harsh  chemicals.  In  addition  to  the  actual 
chemical  changes  which  may  accompany  drastic  treatment,  there  is  the 
factor  that  the  clay  minerals  may  be  so  thoroughly  coagulated  that  they 
cannot  be  dispersed  without  considerable  effort.  Among  fine-grained 
sediments  the  processes  of  disaggregation  and  dispersion^  are  usually 
carried  out  simultaneously,  either  by  the  use  of  physical  methods  alone 
or,  more  commonly,  by  a  combination  of  physical  and  chemical  methods. 

Mud  and  silt  require  very  little  treatment  and  usually  offer  no  difficul- 
ties. Partially  or  completely  indurated  rocks,  however,  or  sediments  with 
abundant  soluble  salts,  may  yield  to  no  methods.  Experimentation  with 
small  samples  is  often  necessary  before  suitable  techniques  are  found,  and 
various  tests  are  available  to  determine  whether  dispersion  is  complete 
or  not. 

It  is  of  fundamental  importance,  theoretically,  that  the  dispersive  treat- 
ment should  be  vigorous  enough  to  separate  the  aggregates  into  individual 
particles,  but  should  not  break  the  crystal  fragments.  The  development 
of  any  universal  technique  which  lies  between  these  limits  may  be  im- 
possible in  practice,  but  one  may  approach  it  to  varying  degrees  in  given 
cases. 

1  Disaggregation,  as  the  term  is  used  here,  refers  to  the  breaking-down  of  aggre- 
gates into  smaller  clusters  or  into  individual  grains.  Dispersion  refers  to  the  process 
of  actually  separating  and  dispersing  the  particles  throughout  some  fluid  medium,  so 
that  each  grain  acts  as  an  individual  when  settling. 


52        SEDIMENTARY  PETROGRAPHY 

PHYSICAL     DISPERSION     PROCEDURES     FOR 
MECHANICAL    ANALYSIS 

Prolonged  soaking  in  zi'ater.  The  sample  is  crushed  into  small  lumps 
and  allowed  to  soak  either  in  water  or  in  dilute  solutions  of  electrolytes.^ 
The  period  of  soaking  allows  each  particle  to  be  surrounded  by  a  film 
of  water  or  fills  the  pores  of  the  rock  and  so  loosens  the  grains  and  aids 
in  their  dispersion.  The  method  is  particularly  suitable  for  partially  in- 
durated sediments.  Rubey,-  using  it  in  his  study  of  Cretaceous  shales, 
reviewed  earlier  work  and  emphasized  that  the  ease  of  disintegration 
varies  with  the  moisture  content  of  the  sample.  Rubey  soaked  his  samples 
in  dilute  ammonia  for  eight  weeks,  but  he  pointed  out  that  prolonged 
soaking  may  dissolve  fine  particles  and  hydrate  minerals.  In  general,  the 
period  of  soaking  depends  on  the  degree  of  consolidation  of  the  sediment. 
Dragan^  recommended  a  24-hr.  period  of  soaking  in  distilled  water  as 
advantageous  to  the  dispersion  of  soils. 

Rubbing  or  trituration  in  water.  The  sample  is  made  into  a  paste  with 
water  and  rubbed  with  the  finger  or  a  stiff  brush  or  triturated  with  a 
rubber  pestle.  Water  is  added  from  time  to  time,  and  the  dispersed 
material  poured  into  a  beaker,  until  all  the  aggregates  are  destroyed. 
Whittles*  emphasized  the  necessity  of  wetting  the  samples  gradually  so 
that  the  water  penetrates  them  throughout.  The  general  method  has  found 
considerable  favor  among  analysts. 

Shaking  in  water.  Shaking  the  sediment  in  water  is  a  widely  used  dis- 
persion procedure.  Reciprocating,  end-over-end,  and  rotary  shakers  are 
commonly  used,  Joseph  and  Snow  ^  considered  reciprocating  shakers 
preferable.  Periods  of  shaking  varying  from  i  to  24  hr.  have  been  rec- 
ommended. Richter  ®  observed  some  breaking  of  grains  during  shaking, 
and  Nolte  ^  compared  the  size-reduction  effects  of  shaking  and  boiling 

1  While  the  use  of  electrolytes  is  a  chemical  procedure,  it  is  mentioned  here  as  a 
variation  which  often  accompanies  this  and  following  physical  procedures.  In  the 
discussion,  however,  the  effects  of  the  electrolyte  are  not  considered,  inasmuch  as 
that  subject  will  be  treated  under  a  separate  head. 

-  W.  W.  Rubey,  Lithologic  studies  of  fine-grained  Upper  Cretaceous  sedimentary 
rocks  of  the  Black  Hills  region :  U.  S.  Gcol.  Siirz'ey,  Prof.  Paper  165A,  pp.  1-54, 
1930. 

3  I.  C.  Dragan,  Die  Vorbehandlung  der  Bodenproben  zur  mechanischen  Analyse : 
Landziirts.  Jahrb.,  vol.  74,  pp.  27-46,  1931. 

*  C.  L.  Whittles,  Methods  for  the  disintegration  of  soil  aggregates  and  the  prepa- 
ration of  soil  suspensions :  Jour.  Agric.  Sci,  vol.  14,  pp.  346-369,  1924. 

5  A.  F.  Joseph  and  O.  W.  Snow,  The  dispersion  and  mechanical  analysis  of  heavy 
alkaline  soils:  Jour.  Agric.  Sci.,  vol.  19,  pp.  106-120,  1929. 

s  G.  Richter,  Die  Ausfiihrung  mechanischer  und  physikalischer  Bodenanalysen : 
Int.  Mitt,  fiir  Bodcnktmdc,  vol.  6,  pp.  193-208;  318-346,  1916. 

^  O.  Nolte,  Der  Einfluss  des  Kochens  und  des  Schuttelns  auf  seine  Mineralteil- 
chen :  Landivirts.  Versuchs-Stat.,  vol.  93,  pp.  247-258,  1919. 


PREPARATION  OF  SAMPLES 


53 


on  carefully  separated  grades.  He  found  less  reduction  from  shaking 
than  from  boiling,  although  he  warned  against  too  long  a  period  of 
shaking.  Olmstead,  Alexander,  and  Middleton  ^  considered  the  breaking 
of  particles  to  be  negligible  even  after  i6  hr.  of  shaking.  Hissink  ^ 
contended  that  rubbing  with  a  brush  had  as  great  a  grinding  effect  as 
shaking. 


There  are  many  types  of  shaking-  machine  on  the  market,  and  others  may  be 
constructed  at  little  cost.  A  simple  end-over-end  sliaker,  described  by  Puri  and 
Keen  (see  below),  consists  of  a  wheel 
to  which  two  bottles  are  fastened.  The 
wheel  is  rotated  by  a  motor.  A  recip- 
rocating shaker  of  the  type  used  by 
the  United  States  Bureau  of  Soils  ^  is 
illustrated  in  Figure  lo.  It  consists  of 
a  box  divided  into  compartments  for 
nursing  bottles,  which  lie  lengthwise 
in  the  device.  A  motor  drives  a  gear 
system  which  imparts  a  to-and-fro 
motion  to  the  box.  This  type  of  shaker 
has  been  used  with  considerable  suc- 
cess at  the  laboratories  of  the  Univer- 
sity  of    Chicago.    Rotary   shakers   are 

available  at  various  laboratory  supply  houses;  they  consist  of  a  metal  plate 
with  clamps  to  hold  flasks ;  during  the  shaking  the  flasks  are  swung  through 
an  elliptical  motion  similar  to  hand  shaking. 


Fig.  10. — Reciprocating  shaker, 
adapted  from  Briggs,  Martin,  and 
Pearce,  1904. 


A  detailed  study  of  shaking  was  made  by  Puri  and  Keen  ^  in  1925. 
They  shook  soil  samples  in  water  for  varying  lengths  of  time  and  meas- 
ured the  degree  of  dispersion  by  the  percentage  of  fine  material  set  free. 
This  percentage,  called  the  "dispersion  factor,"  increased  rapidly  at  first 
and  then  more  slowly,  as  shown  in  Figure  ii,  adapted  from  the  original 
paper.  The  curve  was  found  to  agree  with  the  empirical  equation 
d  =  a-\-  klog  t,  where  d  is  the  dispersion  factor,  a  and  k  are  constants,  and 
t  is  the  time.  The  test  periods  extended  over  intervals  as  great  as  100  hr., 
but  in  no  case  did  the  dispersion  factor  reach  an  upper  limit.  Disj^ersion 
was  thus  shown  to  be  a  continuous  function  of  time,  but  a  24-hr.  period 


iL.  B.  Olmstead,  L.  T.  Alexander  and  H.  E.  Middleton,  A  pipette  method  of 
mechanical  analysis  of  soils  based  on  improved  dispersion  procedure:  U.  S.  Dcpt. 
Agric.  Tech.  Bull.  170,  1930. 

2D.  J.  Hissink,  Die  Mcthode  der  mechanischen  Bodenanalyse :  Int.  Mitt,  fi'tr 
Bodcitkuudc,vo\.  Ii,  pp.  i-ii,  1921. 

3  L.  J.  Briggs,  F.  D.  Martin  and  J.  R.  Pearce,  The  centrifugal  method  of  me- 
chanical soil  analysis :  U.  S.  Dcpt.  Agric,  Bur.  of  Soils,  Bull.  24,  1904- 

4  A.  N.  Puri  and  B.  A.  Keen,  The  dispersion  of  soil  in  water  under  various  con- 
ditions:  Jour.  Agric.  Sci.,  vol.  15,  pp.  147-161,  1925. 


54 


SEDniENTARY  PETROGRAPHY 


10 

§^ 

„a^^ 

i 

/ 

% 

6 

0 

■            !            ' 

of  shaking  was  found  sufficient  to  carry  the  degree  of  dispersion  over  the 
steep  part  of  the  curve.  The  original  moisture  content  of  the  sample  and 
the  concentration  of  the  suspension  were  found  materially  to  affect  dis- 
persion. 

Experiments  by  Krumbein  ^  showed  that  the  effectiveness  of  shaking 
also  depends  on  the  presence  of  coarse  material  in  the  sediment.  A  lake  clay, 

having  no  particles  larger  than  0.03 
mm.  in  diameter,  contained  undis- 
persed  clay  pellets  after  6  hr.  of  shak- 
ing, whereas  a  glacial  till  with  consid- 
erable sand  was  fully  disaggregated 
within  an  hour.  Davis  ^  used  rubber 
balls  to  hasten  the  dispersion  of  fine 
material,  and  glass  beads  have  also 
been  used.  The  effect  is  similar  to  the 
sand  in  till,  reducing  the  time  of 
shaking  to  a  fraction  of  its  previous 
length.  Comparative  analyses  of  two 
samples  of  the  lake  clay,  one  of  which  was  shaken  for  12  hr.  without 
glass  beads,  and  the  other  i  hr.  with  them,  showed  no  differences  in  the 
size  distribution  beyond  the  limits  of  experimental  error,  so  that  the 
grinding  eft'ect  of  the  glass  beads  appears  to  be  neghgible. 

Stirring  in  water.  In  1927  Bouyoucos^  used  an  electric  drink  mixer 
for  dispersing  soils.  Due  to  the  high  speed  of  the  stirrer,  wire  baffles 
were  placed  in  the  cup  to  prevent  circular  motion  of  the  suspension. 
Bouyoucos  also^  compared  stirring  with  shaking  and  found  that  10  min. 
of  the  former  were  more  effective  than  16  hr.  of  the  latter.  In  sandy 
soils  he  noticed  some  apparent  breaking  of  the  sand  grains  from  more 
prolonged  stirring.  The  stirrer  is  one  of  the  most  effective  physical  dis- 
persion devices. 

Vibratian  in  zivter.  Among  the  newer  dispersion  procedures  is  the  use 
of  rapid  vibrations  to  achieve  dispersion.  In  1924  \\'hittles  ^  developed  a 
mechanism  in  which  a  rapidly  vibrating  hammer  struck  the  celluloid 


Fig.  II. — Graph  of  dispersion  fac- 
tor as  a  function  of  time.  After 
Puri  and  Keen,   1925. 


1  W.  C.  Krumbein,  The  dispersion  of  fine-grained  sediments  for  mechanical  analy- 
sis:  Jour.  Sed.  Petrology,  vol.  3,  pp.  121-135,  1933. 

-R.  O.  E.  Davis,  Colloidal  determination  in  mechanical  analysis:  Joj<r.  Am.  Soc. 
Agron.,  vol.  17,  pp.  275-279,  1935. 

3  G.  J.  Bouyoucos.  The  hydrometer  as  a  new  and  rapid  method  for  determining 
the  colloidal  content  of  soils:  Soil  Science,  vol.  23,  pp.  319-331,  1927. 

4  G.  J.  Bouyoucos,  Studies  on  the  dispersion  procedure  used  in  the  hydrometer 
method  for  making  mechanical  analysis  of  soils :  Soil  Science,  vol.  33,  pp.  21-26, 
1932. 

s  C.  L.  Whittles,  loc.  cif.,  1924. 


PREPARATION  OF  SAMPLES  55 

bottom  of  a  glass  cylinder  at  controlled  frequencies  and  amplitudes.  A 
fair  degree  of  dispersion  was  obtained  by  trituration  with  a  rubber  pestle, 
followed  by  vibration  for  one  hour  at  10.000  vibrations  per  minute. 

In  1 93 1  Olmstead  ^  used  supersonic  waves  for  dispersion.  The  vibra- 
tions are  produced  by  a  piezoelectric  quartz  crystal  immersed  in  an  oil 
bath  and  energized  by  a  vacuum  tube  oscillator.  The  energy  is  transmitted 
through  the  oil  into  a  flask  containing  the  soil  suspension.  The  sample  is 
vibrated  for  several  2-min.  periods,  followed  by  decantation  of  the  dis- 
persed material.  Olmstead  compared  his  method  with  rubbing  and  found 
a  close  agreement,  but  the  supersonic  method  was  much  more  rapid. 

Ignition.  Heating  the  samples  in  the  dry  state  has  been  used  by  some  work- 
ers. Nolte  -  mentioned  the  far-reaching  physical  and  chemical  effects  of  such 
treatment,  and  Richter  ^  found  that  among  several  procedures,  ignition  gave 
the  least  satisfactory  results  due  to  the  destruction  of  colloids  and  the  possible 
fusion  of  grains. 

Boiling  in  wafer.  The  boiling  of  samples  in  water  or  in  dilute  electro- 
lytes is  a  procedure  about  which  there  has  been  considerable  controversy. 
In  19 19  Nolte  *  furnished  a  summary  of  the  situation.  He  concluded  that 
boiling  simultaneously  reduced  the  size  of  the  larger  particles  and  coag- 
ulated the  smaller  ones.  In  the  same  year  Oden  ^  observed  that  boiling 
destroyed  aggregates  larger  than  10  microns,  while  particles  smaller  than 
I  micron  coagulated  to  aggregates  between  i  and  2  microns  in  diameter. 
In  1927  von  Hahn  ^  referred  to  boiling  as  a  "barbaric  practice"  because 
of  its  physical  and  chemical  eft'ects  on  the  suspension. 

The  most  important  contribution  on  the  subject  was  made  by  W'iegner  " 
in  1927.  He  compared  the  effects  of  boiling  on  samples  in  which  the 
water-soluble  salts  were  either  washed  out  or  left  in.  In  the  washed 
samples  dispersion  was  increased  by  boiling,  whereas  the  salts  in  the  un- 
washed soils  caused  coagulation.  He  thus  showed  that  the  same  treatment 
may  either  prevent  or  aid  dispersion,  depending  on  the  presence  or 
absence  of  appreciable  amounts  of  foreign  electrolytes. 

Periods  of  boiling  have  varied  from  about  10  min.  to  more  than  40  hr. 

1  L.  B.  Olmstead,  Dispersion  of  soils  by  a  supersonic  method:  Jour.  Agric.  Re- 
search, vol.  42,  pp.  841-852,  1931. 
-  O.  Nolte,  loc.  cit..  1919. 

3  G.  Richter,  loc.  cit.,  1916. 

4  O.  Nolte,  loc.  cit.,  1919. 

5  S.  Oden,  t'ber  die  Vorbehandlung  der  Bodenproben  zur  mechanischen  Analyse : 
Bull.  Geol.  Inst.  Utsahi,  vol.  16,  pp.  125-134.  1919. 

^  F.-V.  von  Hahn,  Dispcrsoidanalysc  (Leipzig  u.  Dresden,  1927). 

"  G.  Wiegner,  Method  of  preparation  of  soil  suspension  and  degree  of  dispersion 
as  measured  by  the  Wiegner-Gessner  apparatus :  Soil  Science,  vol.  23,  pp.  377-390, 
1927.  (Translated  by  R.  M.  Barnette.) 


56        SEDIMENTARY  PETROGRAPHY 

It  is  doubtful  whether  boihng  for  24  hr.  or  more  aids  dispersion,  even  if 
no  foreign  electrolytes  are  present.  Wiegner  considered  an  hour  to  be 
sufficient.  Krumbein  ^  followed  the  procedure  of  heating  the  suspensions 
to  the  boiling  point  but  not  allowing  them  to  boil.  In  this  manner  the 
agitation  due  to  heating  is  able  to  perform  its  function  of  dispersion 
without  the  disadvantages  that  may  follow  a  more  prolonged  application 
of  heat. 

A  variation  of  the  usual  procedure  of  boiling  the  samples  was  used  in 
1933  by  Postel,-  who  used  steam  agitation  to  effort  dispersion.  The  clay 
was  placed  in  a  flask,  and  a  copper  pipe  was  inserted  through  a  stopper. 
The  pipe  was  connected  with  a  boiler  which  supplied  steam  at  30  lbs. 
pressure.  The  steam  condensed  in  the  flask,  but  in  the  10  min.  during 
which  it  flowed,  the  clay  was  sufficiently  dispersed  to  permit  elutriation. 

Removal  of  icater-solnble  salts.  The  washing-out  of  foreign  electro- 
lytes, as  a  preliminary  treatment  of  samples,  has  received  increasing  at- 
tention during  the  last  decade.  Although  the  procedure  is  essentially 
physical  in  nature,  detailed  discussion  is  deferred  to  a  later  section  of 
this  chapter. 

CHEMICAL      DISPERSION      PROCEDURES      EOR 
MECHANICAL     ANALYSIS 

Leaching  in  acids  or  alkalies.  Leaching  in  dilute  acids  as  a  preliminary 
treatment  has  long  been  followed  by  soil  scientists.  From  the  viewpoint 
of  sedimentary  petrolog)',  however,  it  cannot  be  emphasized  too  strongly 
that  all  primary  carbonate  particles  in  a  sediment  should  be  retained 
during  mechanical  analysis,  because  they  are  an  integral  part  of  the  size 
frequency  distribution.  When  only  secondary  carbonates  are  present 
they  may  be  removed,  but  when  both  primary  grains  and  secondary 
cement  are  present,  the  one  cannot  be  removed  without  destroying  the 
other,  and  the  decision  to  remove  all  or  none  must  depend  upon  the 
problem  at  hand. 

The  digestion  of  samples  in  strong  alkalies,  such  as  sodium  hydroxide, 
has  also  been  practised.  Such  treatment  is  in  the  same  categor}-  as  acid 
treatment  from  the  point  of  view  of  sedimentary  petrolog}'. 

Peptization  zvith  very  dilute  electrolytes.  The  use  of  small  amounts 
of  peptizing  electroh-tes,   such  as  ammonium  hydroxide,   sodium  car- 

1  W.  C.  Krumbein,  loc.  cit.,  1933. 

2  A.  W.  Postel,  The  preparation  of  clay  samples  for  elutriation  by  steam  agita- 
tion: Jour.  Sed.  Petrology,  vol.  3,  pp.  1 19-120,  1933. 


PREPARATION  OF  SAMPLES  57 

bonate,  or  sodium  oxalate,  is  very  widespread,  and  few  techniques  do 
not  include  one  or  another  of  them.  The  purpose  of  adding  these  elec- 
trolytes is  to  disperse  the  sediment  into  individual  ^^articles  and  to  pre- 
vent the  particles  from  coagulating  during  the  subsequent  analysis.  The 
subject  is  of  such  importance  in  the  conduct  of  mechanical  analysis  that 
the  following  theoretical  considerations  are  included  here  as  an  intro- 
duction to  the  process  of  coagulation.^ 

Stable  and  Unstable  Suspensions 

When  a  single  particle  settles  in  water,  it  does  so  at  a  rate  which  depends 
in  part  on  its  size  and  shape,  and  on  the  nature  of  the  fluid  (Chapter  5). 
When  a  system  of  particles  settles,  they  may  descend  as  individuals  essentially 
uninfluenced  by  their  neighbors,  or  they  may  coagulate  and  settle  as  aggre- 
gates. It  is  obvious  that  the  results  of  a  given  analysis  are  sound  only  when 
the  former  condition  holds,  since  mechanical  analysis  is  an  attempt  to  deter- 
mine the  frequency  distribution  of  the  primary  particles  in  the  sediment.  It  is 
of  paramount  importance,  therefore,  that  the  factors  affecting  coagulation 
be  known,  to  guard  against  introducing  errors  of  considerable  magnitude  into 
the  results. 

Colloidal  suspensions  differ  from  true  solutions  in  that  the  latter  are  per- 
manently stable,  whereas  the  former  are  not  necessarily  so.  It  is  generally 
accepted  that  in  a  stable  colloidal  suspension  each  particle  has  an  electric 
charge  arranged  about  it  in  a  double  layer.  The  charge  may  be  positive  or 
negative,  depending  in  part  on  the  nature  of  the  colloid.  The  nature  of  the 
double  layer  may  be  visualized  as  follows :  In  a  solid  particle  the  atoms  are 
held  in  a  crystalline  structure,  and  within  the  interior  of  the  particle  each 
atom  is  balanced,  in  terms  of  its  valence,  with  corresponding  atoms  of  other 
elements.  The  atoms  at  the  boundary  of  the  particle,  however,  are  only  par- 
tially satisfied  in  terms  of  their  valence,  and  they  are  therefore  capable  of 
attracting  a  swarm  of  ions  f  rom  tlie  surrounding  fluid.  Of  this  swarm  of  ions, 
either  the  positive  or  negative  ions  (depending  upon  the  nature  of  the  particles 
and  of  the  ions  in  solution)  arrange  themselves  alongside  the  solid  particle 
and  constitute  the  inner  layer,  which  gives  the  particle  its  charge.  Meanwhile, 
the  oppositely  charged  ions  in  solution  swarm  about  the  inner  layer.  Thus  is 
built  up  a  double  layer  of  ions.  ]\Iore  technical  details  of  tlie  double  layer  may 
be  found  in  standard  reference  books  on  colloids. - 

Under  the  influence  of  Brownian  movement  the  charged  colloidal  particles 
are  brought  into  the  vicinity  of  others,  but  as  long  as  the  charges  are  above  a 
critical  potential  (Figure  12)  the  particles  repulse  each  other  and  adherence 
is  prevented.  If  the  charges  are  below  the  critical  potential,  or  zero,  the  par- 
ticles may  adliere  when  tliey  collide,  with  the  result  that  aggregates  are 
formed.  These  aggregates  begin  to  settle  and  eventually  tlie  entire  dispersed 
phase  may  settle  out  of  suspension  as  a  flocculent  precipitate.  The  rate  of 

1  The  terms  coagulation  and  flocailation  appear  to  be  used  synon\-mously  by  many 
authors.  The  former  has  been  chosen  here  for  the  sake  of  consistency. 

2  See  for  example  H.  R.  Kruyt,  Colloids,  translated  by  H.  S.  Van  Klooster,  2nd 
ed.  (New  York,  1930),  pp.  no  ff. 


CONCENTRATION    Of   ELECTROl 


58  SEDIIMENTARY  PETROGRAPHY 

coagulation  may  be  either  slow  or  rapid,  depending  upon  whether  or  not  any 
charges  are  present  on  the  particles. 

The  magnitude  of  the  charge  varies  appreciably  with  slight  changes  in  the 
electrolyte  content  of  the  suspension,  and  the  effect  of  a  given  electrolyte  ap- 
pears to  vary  with  the  nature  of  the  colloid.  In  clays,  for  example,  the  charges 
may  be  reduced  by  adding  calcium  chloride.  Such  electrolytes  are  called  coag- 
ulants. Other  electrolytes,  as  sodium  carbonate,  increase  the  charges  on  the 
clay  particles  and  are  called  peptizers.  Beyond  certain  limits  of  concentration 
the  peptizing  electrol>tes  also  cause  coagulation,  so  that  part  of  the  distinction 

between  the  two  types  may  be  due  to  the 
relative  concentrations  necessary  to  pro- 
duce coagulation. 

Two  types  of  coagulation  are  recognized, 
perikinctic  and  orthokinctic.  Perikinetic 
coagulation  occurs  in  systems  where  essen- 
tially no  sedimentation  is  taking  place  and 
where  the  probability"  of  collision  is  equally 
likely  in  any  direction,  due  only  to  the 
chaotic  Brownian  movement.  Orthokinetic 
Fig.  12.— Diagram  of  boundary  coagulation  occurs  in  sedimenting  svstems 
charge   as  a  function  of   electro         j  j         .Qbabilitv  of  contact  is  greater 

lyte   content    in   colloidal    suspen-      .  ,.'.',.,, 

sions,  after  Kru>t.  When  the  m  some  directions  than  m  others,  due  to 
curve  is  above  the  critical  poten-  the  downward  motion  of  the  settling  par- 
tial, the  suspension  is  stable.  tides. 

Perikinctic  coagulation.  Inasmuch  as 
perikinetic  coagulation  involves  no  sedimentation  of  the  individual  particles, 
it  is  not  as  important  from  the  point  of  view  of  mechanical  analysis  as  ortho- 
kinetic  coagulation.  However,  it  is  not  out  of  place  to  consider  the  subject 
briefly.  Von  Smoluchowski  ^  was  the  first  to  develop  a  mathematical  theory  of 
coagulation.  He  considered  the  factors  that  must  be  present  before  coagula- 
tion takes  place.  It  is  clear  that  particles  can  only  adhere  if  they  collide,  and 
hence  the  probability  of  collision  is  of  primary  importance.  Once  they  have 
collided  it  is  necessary  to  consider  the  conditions  under  which  they  adhere, 
so  that  the  probability  of  adherence  is  the  second  important  factor.  The  prob- 
ability of  collision  is  controlled  by  the  Brownian  movement,  and  the  probabil- 
ity of  adherence  is  controlled  by  the  electric  charges  on  the  particles.  Von 
Smoluchowski  first  considered  the  progress  of  coagulation  in  a  monodisperse 
system  where  no  electrical  charges  were  present  on  the  particles,  so  that 
every  collision  resulted  in  adherence.  Here  the  probability  of  adherence  is  i. 
In  setting  up  his  theory,  he  considered  that  each  particle  of  radius  r  has 
about  it  a  sphere  of  attraction  of  radius  R,  such  that  any  other  particle  whose 
center  enters  this  sphere  of  attraction  is  united  to  the  first.  Now  the  proba- 
bility Tt'  that  another  particle  will  move  into  the  sphere  of  attraction  of  a  given 
particle,  the  latter  being  considered  motionless,  is  tc;  =  4TDR,  where  D  is  the 
displacement  due  to  Brownian  movement.  From  this  starting  point,  von 
Smoluchowski  developed  an  equation  showing  the  number  of  primary  particles 

1  M.  von  Smoluchowski,  Versuch  einer  mathematischen  Theorie  der  Koagula- 
tionskinetik  kolloider  Losungen :  Zeits.  Pliys.  Cliem.,  vol.  92,  pp.  129-168,  1916-1918. 


PREPARATION  OF  SAMPLES 


59 


t 

«<0.50 


remaining  after  a  given  time  t.  Similarly,  he  considered  the  formation  of 
aggregates  having  two,  tliree,  and  more  primary  particles  and  the  change 
in  their  number  with  time.  Figure  13,  taken  from  his  paper,  shows  the  vari- 
ation in  the  number  of  all  particles,  (2h),  of  primary  particles  (Hi),  of 
dyads  (wa).  and  o*  triads  ("3).  The  ratio  h/hq  is  plotted  as  ordinate,  and  t/T 
as  abcissa.  The  symbol  n^,  represents  the  original  number  of  particles  present, 
and  T  is  a  measure  of  the  rate  of  coagulation.  It  is  clear  that  n  and  h^  start  at 
the  point  n/)iQ  =  i  and  decrease  continually.  The  number  of  dyads,  however, 
is  zero  at  the  start,  but  it  rapidly  rises  to  a  maximum  and  then  decreases  as  the 
number  of  triads  becomes  prominent.  Similarly,  the  more  complex  aggregates 
all  show  a  maximum  at  points  succes- 
sively farther  to  the  right  along  the 
X-axis.  This  simple  case  of  von  Smolu- 
chowski's  theory  was  experimentally 
verified  by  several  workers. 

Von  Smoluchowski  next  considered 
the  case  in  which  the  charges  on  the 
particles  were  below  the  critical  poten- 
tial but  not  equal  to  zero.  In  this  case 
every  collision  does  not  result  in  ad- 
herence, so  that  the  rate  of  coagulation 
is  slower  than  in  the  first  instance.  The 
net  effect  of  the  slower  rate  of  coag- 
ulation on  the  mathematical  theory 
was  the  insertion  of  the  probabilit\-  ^, 
less  than  i,  for  the  certainty  which 
distinguished  the  rapid  coagulation. 

In  1926  iMiiller^  developed  a  theory 
for   the    rapid   perikinetic   coagulation 

of  bidisperse  systems  in  which  two  sizes  of  particles  were  present.  Miiller's 
theory  was  experimentally  verified  by  Wiegner  and  Tuorila.-  The  conclusions 
drawn  were  that  the  coagulation  of  bidisperse  systems  is  more  rapid  than  that 
of  monodisperse  systems,  and  that  when  the  number  of  one  size  of  particles 
is  great  compared  to  the  number  of  the  other,  the  rate  of  coagulation  ap- 
proaches that  of  a  monodisperse  system  composed  of  the  dominant-sized 
particle. 

Orthokinetic  coagulation.  In  mechanical  analysis  larger  particles  are  con- 
tinually overtaking  smaller  ones  by  virtue  of  their  greater  settling  velocities. 
Hence  the  probability  of  collision  is  large,  and  it  may  be  expected  that  coag- 
ulation would  proceed  at  a  rapid  pace.  The  theory  underlying  this  type  of 
coagulation  was  developed  by  Tuorila.^  Wiegner  *  published  an  excellent  sum- 
mary of  the  work  done  in  this  field,  including  also  the  earlier  work  of  Von 


1^ 

\  \ew 

V 

V 

'~^~--_ 

^ 

fe 

---_i__ 

"  12  3  4 

Fig.  13. — Progress  of  coagulation  in 
a  monodisperse  system,  after  von 
Smoluchowski,  191 6. 


1  H.  Miiller,  Die  Theorie  der  Koagulation  polydispersen  Systeme :  Kolloid  Zcits., 
vol.  38,  pp.  1-2,  1926. 

2  G.  Wiegner  and  P.  Tuorila,  Ueber  die  rasche  Koagulation  polydisperser  Sys- 
teme :  Kolloid  Zcits.,  vol.  38,  pp.  3-22,  1926. 

3  P.  Tuorila,  Ueber  orthokinetische  und  perikinetische  Koagulation :  KoHoidchcm. 
Beihcfte,  vol.  24,  pp.  1-122,  1927. 

^G.  Wiegner,  Ueber  Koagulationen :  Kolloid  Zcits.,  vol.  58,   pp.  157-168,   1932. 


6o        SEDIMEXTARY  PETROGRAPHY 

Smoluchowski  and  Miiller.  The  present  discussion,  including  parts  of  the 
preceding  material,  is  based  on  these  papers. 

In  developing  his  theory-,  Tuorila  considered  suspensions  in  which  there 
\vere  less  than  lo^  particles  per  c.c,  so  that  perikinetic  coagulation  did  not 
take  place  during  the  time  of  observation.  He  set  up  the  assumptions  that  large 
particles  attract  smaller  ones  in  an  attraction  volume  having  a  cross-sectional 
area  of  w  (A-  —  R-),  where  A=  (R  +  r),  the  sum  of  the  radii  of  large 
and  small  particles.  In  an  element  of  time,  the  large  particles  settle  a  distance 
L,  so  that  the  volume  occupied  by  the  attraction  zone  is  ir  (A-  —  R-)L.  This 
volume  was  designated  by  Tuorila  as  the  Hautraumz-ohimen  of  the  particles, 
and  was  represented  by  b.  Since  each  large  particle  has  this  attraction  volume, 
X  particles  would  have  "Sb  =  ir( A-  —  R-)LX  =  B  for  their  total  attraction 
volume.  Into  this  formula  Tuorila  inserted  Stokes's  Law  (Chapter  5) 
of  the  radius,  and  obtained  as  his  final  result  the  expression 
B  =  KMr  (2+S  — 2S  — S3) 

where  S  =  r/R,  K  is  a 
constant,  and  M  is  the  mass  of  large  particles  per  c.c.  It  is  clear  from 
this  expression  that  the  total  attraction  volume,  B.  is  proportional  to  the 
weight  of  large  particles  in  the  suspension,  and  dependent  on  the  function 
r(2+S  —  2S-  —  S').  Since  S  =  r/R,  B  is  dependent  on  a  function  of  the 
ratio  of  the  radii  present.  When  r  —  o,  B  =  o,  and  the  system  is  monodis- 
perse.  Likewise  when  r/R=i,  B=o.  The  value  of  B  as  a  whole  may  thus 
var\'  bet\veen  r/R  —  o,  and  r/R  =  i.  By  plotting  the  function  (2  +  S— 2S- 
— S^)  against  r/R,  Tuorila  foimd  the  values  were  nearly  constant  between 
values  of  r/R  from  0.0  to  0.4. 

Tuorila  reasoned  that  of  «  small  particles  per  c.c.  of  suspension,  the  number 
«B  would  be  in  the  attraction  volume  B,  and  hence  swept  along.  Thus,  the 
decrease  in  small  particles,  dti,  per  unit  of  time  dt  would  be  d)i/dt  =  — 71B, 
where  the  minus  sign  indicates  a  decreasing  function.  This  differential  equa- 
tion yields  the  negative  exponential. 

where  n  is  the  number  of  small  particles  in  suspension  at  any  instant,  and  no 
is  the  original  number.  This  function  requires  that  B  be  a  constant;  Tuorila's 
analysis  of  the  function  (2  +  S  —  2S-  —  S^)  indicates  that  this  condition  is 
satisfied  over  the  range  indicated,  i.e.,  r/R < 0.4.  As  sedimentation  proceeds, 
the  larger  particles  settle  to  successively  deeper  zones  in  the  suspension,  so 
that  their  effects  on  the  small  particles  in  a  given  zone  are  limited  to  the 
time  required  for  the  large  particles  to  settle  through  the  zone. 

Tuorila  confirmed  his  theories  in  a  series  of  experiments,  and  much  of  the 
material  that  came  to  light  is  pertinent  to  the  subject  of  mechanical  analysis. 
The  experiments  showed  that  polydisperse  systems  in  which  the  charges  on 
the  particles  were  above  a  given  critical  potential  showed  no  coagulation 
effects,  even  in  concentrations  as  high  as  150  g.  of  solid  per  liter.  When  the 
charge  was  below  the  critical  potential,  orthokinetic  coagulation  took  place, 
during  which  the  larger  particles  swept  the  smaller  ones  along  with  them. 
The  eft'ects  of  orthokinetic  coagulation  were  also  found  to  increase  rapidly 
with  an  increase  in  the  concentration  of  the  suspension. 

In  quartz  suspensions,  particles  larger  than  20  microns  in  radius  (0.04  mm. 


PREPARATION  OF  SAMPLES 


6i 


Fig.    14. — Progress   of  coagulation 
as  a  function  of  time,  after  Tuorila. 


diameter)  did  not  enter  into  the  coagulation.  Thus  an  upper  limit  of  coag- 
ulation was  estahlished,  which  appears  to  vary  with  the  nature  of  the  material 
being  studied,  inasmuch  as  in  a  clay  the  upper  limit  was  higher.  Quartz  par- 
ticles between  10  and  20  microns  acted  only  slightly  on  the  smaller  particles; 
between  6  and  10  microns  the  effect  was  nmch  stronger,  and  it  reached  a 
maximum  at  5-6  microns.  Particles  under  4  microns  in  radius  were  completely 
swept  out  of  suspension  during  the  process. 

Tuorila  pointed  out  that  in  many  instances  coagulation  is  very  slight  at  the 
beginning  and  hence  hardly  noticeable.  As  time  goes  on  the  effect  increases 
rapidly,  and  toward  the  end  of  the  process  it  slows  down  again.  Thus  an 
S-shaped  curve  results  ;. Figure  14,  taken 
from  Tuorila's  paper,  illustrates  the 
case.  This  effect  may  also  occur  in 
monodisperse  systems,  where  perikinetic 
coagulation  produces  large  aggregates 
which  begin  to  settle  and  thus  exert  an 
orthokinetic  effect  on  the  remaining 
smaller  particles.  This  orthokinetic  ef- 
fect naturally  increases  as  the  number 
of  larger  aggregates  increases,  until 
most  of  the  particles  are  coagulated, 
when  the  process  slows  down  again. 

Summary  of  coagulation.  The  work 
of  Tuorila  and  others  has  an  important 
bearing  on  mechanical  analysis.  Inasmuch  as  sediments  are  polydisperse  sys- 
tems, the  phenomena  of  orthokinetic  coagulation  may  manifest  themselves. 
In  this  type  of  coagulation  the  larger  particles  drag  along  the  smaller  ones 
and  thus  hasten  coagulation  effects.  Likewise,  orthokinetic  coagulation  pro- 
ceeds at  an  accelerated  pace  in  concentrated  suspensions.  This  suggests  that 
dilute  suspensions  may  be  preferable  to  concentrated  suspensions  for  analysis. 

General  colloidal  theory  also  indicates  that  coagulation  does  not  take  place 
if  the  particles  are  charged  above  a  critical  potential,  and  this  state  appears 
to  be  associated  cither  with  certain  peptizing  electrolytes  or  with  suspensions 
entirely  free  from  electrolytes.  As  a  result,  much  thought  has  been  devoted 
to  the  elimination  of  all  electrolytes  from  the  suspension  or  to  the  discovery 
of  a  peptizing  electrolyte  that  may  be  applied  to  the  widest  possible  range 
of  sedimentary  types. 

A  number  of  tests  have  been  devised  for  determining  whether  coagulation 
has  occurred  in  a  suspension.  These  tests  are  discussed  in  detail  in  the  gen- 
eralized dispersion  routine  given  at  the  end  of  this  chapter. 

Peptization  procedures.  Among  chemical  agents  which  have  long  been 
used  for  dispersing  soils  and  sediments  are  ammonium  hydroxide  and 
sodium  carbonate.  The  former  was  adoj^ted  by  Briggs,  Martin,  and 
Pearce^  for  the  United  States  Bureau  of  Soils  in  1904,  and  sodium  car- 
bonate appears  to  have  been  introduced  by  Beam-  in  191 1.  These  pep- 

1  L.  J.  Briggs,  F.  O.  Martin  and  J.  R.  Pearce,  loc.  cit.,  1904. 

-  W.  Beam,  The  mechanical  analysis  of  arid  soils :  Abst.  in  Exp.  Sta.  Record, 
vol.  25,  p.  513,  191 1. 


62 


SEDIMENTARY  PETROGRAPHY 


tizers  found  wide  favor  among  analysts  until  comparatively  recent 
years,  and  they  are  still  used  at  present,  despite  the  competition  offered 
by  numerous  other  agents. 

The  use  of  ammonium  hydroxide  for  dispersion  was  systematized  by 
Oden  ^  in  1919,  when  he  developed  his  "normal  method,"  which  involves 
rubbing  the  soil  or  sediment  with  a  stiff  brush,  adding  ammonium  hy- 
droxide to  a  concentration  of  X/ioo,  and  shaking  the  suspension  for 
24  hr. 

Oden's  "normal  method"'  was  used  by  numerous  workers.  Correns 
and  Schott.-  in  1932,  found  it  preferable  to  other  methods  for  general 

work.  For  recent  marine  sediments, 
however,  they  recommended  dialysis 
(see  page  66). 

A  detailed  study  of  the  effect  of 
peptizers  on  soil  suspensions  was 
made  by  Puri  and  Keen  ^  in  1925. 
This  study  marked  a  new  epoch  in  the 
investigation  of  peptizers  by  compar- 
ing the  eft'ects  of  a  number  of  electro- 
lytes on  dispersion,  and  it  paved  the 
way  for  further  detailed  studies 
which  are  being  continued  to-day. 

Puri  and  Keen  studied  the  effects 
of  several  electrolytes  on  soil  sus- 
pensions which  had  previously  been 
washed  free  of  soluble  salts.  \^arying 
amounts  of  the  electrolyte  were  added 
and  the  degree  of  dispersion  meas- 
ured and  plotted.  It  was  found  that  the  sodium  carbonate  curve  displayed 
a  prominent  plateau,  which  indicated  that  quite  a  range  of  concentration 
caused  approximately  the  same  degree  of  dispersion.  The  plateau-effect 
is  a  distinguishing  feature  of  good  peptizers  because  it  allows  some 
flexibility  in  the  concentration  that  may  be  used.  Other  electrolytes,  as 
KCl,  had  a  much  smaller  dispersive  effect  and  displayed  a  sharp  peak 
at  the  optimum  concentration.  The  effects  of  several  electroh-tes  are 
shown  in  Figure  15,  adapted  from  Puri  and  Keen.  Puri  and  Keen  con- 
cluded that  the  effects  of  electrolytes  on  suspensions  are  not  abrupt  but 


CC     NORMAL     SOLUTION     IN    SOO    CC 
OF  SUSPeNSION 

Fig.  15. — The  dispersing  effect  of 
several  electrohtes,  after  Puri  and 
Keen.  1925.  Relatively  small  concen- 
trations of  these  electrolj-tes  cause 
coagulation. 


1  S.  Oden,  loc.  cit.,  1919. 

-  C.  W.  Correns  and  W.  Schott,  Vergleichende  Untersuchungen  iiber  Schlamm- 
und  Aufbereitungsverfahren  von  Tonen:  KoUoid  Zeits.,  vol.  61,  pp.  68-80,  1932. 
3  A.  N,  Puri  and  B.  A.  Keen,  loc.  cit.,  1925. 


PREPARATION  OF  SAMPLES  63 

cause  a  continuous  change  in  the  degree  of  dispersion  as  the  electrolyte 
concentration  is  varied.  They  related  their  results  to  the  phenomenon  of 
base  exchange,  but  no  detailed  explanation  was  attempted  at  the  time. 

Winters  and  Harland  ^  also  studied  the  effects  of  sodium  carbonate 
on  dispersion.  Their  results  agreed  with  Puri  and  Keen  and  showed  that 
the  dispersion  effects  vary  somewhat  with  the  soil  horizon. 

Olmstead,  Alexander,  and  Middleton  -  compared  several  pcjDtizers 
and  decided  that  sodium  oxalate  was  the  most  satisfactory.  They  pointed 
out  that  sodium  and  ammonium  hydroxides  yield  good  results  when  the 
calcium  and  magnesium  carbonates  have  been  removed  by  acid  treat- 
ment and  thorough  washing.  I  f  calcium  carbonate  is  present,  sodium  car- 
bonate is  better  than  either  hydroxide  because  the  carbonate  decreases 
the  solubility  of  the  calcium  carbonate,  while  the  hydroxides  produce 
coagulating  calcium  ions.  Sodium  oxalate,  they  found,  was  even  better 
than  sodium  carbonate,  because  the  calcium  ions  are  completely  removed 
by  the  oxalate.  A  comparison  of  the  four  peptizers  on  four  soils  showed 
that  the  oxalate  had  the  greatest  dispersive  effect  in  every  case. 

Loebe  and  Kohler  ^  also  studied  the  dispersive  effects  of  sodium 
oxalate  and  found  it  best  suited  for  general  work. 

Krumbein  *  performed  a  series  of  experiments  with  sodium  oxalate 
to  determine  whether  it  had  a  plateau-effect  like  sodium  carbonate. 
Water  suspensions  containing  2.5  per  cent  of  an  unconsolidated  cal- 
careous Pleistocene  lake  clay  were  prepared  by  brush  rubbing,  and  vary- 
ing amounts  of  N/5  sodium  oxalate  or  N/5  sodium  carbonate  were 
added.  The  percentage  of  material  under  i  micron  in  the  suspension  was 
then  measured  by  pipetting.  Figure  16  shows  the  resulting  curves.  It  is 
clear  that  sodium  oxalate  has  both- a  greater  dispersive  effect  and  a  wider 
range  of  safety.  In  both  cases  a  concentration  of  about  N/ioo  is  op- 
timum, as  deduced  from  these  curves. 

Ungcrer  °  conducted  a  detailed  investigation  of  several  methods  of 
preparing  soils  for  mechanical  analysis.  Tests  were  made  with  lithium 
chloride  and  lithium  carbonate,  by  comparing  the  amounts  of  material 
smaller  than  2  microns  in  the  suspensions.  Lithium  chloride  was  found 

1  E.  Winters,  Jr.,  and  M.  B.  Harland,  Preparation  of  soil  samples  for  pipette 
analysis:  Jour.  Am.  Soc.  Agrun.,  vol.  22,  pp.  771-780,  1930. 

2  L.  B.  Olmstead,  L.  T.  Alexander,  and  H.  E.  Middleton,  lac.  cif.,  1930. 

3  R.  Loebe  and  R.  Kohler,  Beitrage  zur  Praxis  der  Schlammanalyse :  Mitt.  a.  d. 
Lab.  Prcuss.  Gcol.  Landcsanst.,  vol.  11,  Berlin,  1932. 

*  W.  C.  Krumbein,  lac.  cit.,  1933. 

^  E.  lingerer,  K()rngr(')sscnl)estiinmungen  nacli  dem  Dckantier-  und  Pipcttvcrfah- 
ren  unter  dcm  Einfluss  verschicdcner  Vorbebandlungsmctboden :  Zcits.  j.  Pflanccucr- 
ndhruni}.,  Diiiig,  u.  Bodcnk.,  vol.  26A,  pp.  330-336,  1932. 


64 


SEDIMENTARY  PETROGRAPHY 


0:30 


^St__A 

K 

V    \ 

If 

N 

\ 

\ 

v. 

1 

\ 

1^" — -^ 

N>-^_. 

5  10 

cc  n/5  solution  per  I 


15  20 

cc      SUSPENSION 


Fig.  1 6, 
(A)   and  sodium  carbonate   (B)   on  the 
dispersion  of  a  lake  clay. 


well  suited  to  soils,  whereas  lithium  carbonate  was  not  recommended. 
Vinther  and  Lasson  ^  studied  the  effects  of  several  electrolytes  on  the 
dispersion  of  kaolin,  including  sodium  carbonate,  ammonium  hydroxide, 
lithium  carbonate,  calcium  citrate,  potassium  silicowolframate,  and 
sodium  pyrophosphate.  The  sodium  pyrophosphate  in  0.002  m.  concen- 
tration was  found  most  suitable. 
Lithium  carbonate  was  next  best 
after  the  sodium  pyrophosphate.  In 
all  cases  a  17-hr.  period  of  shaking 
was  used  to  effect  dispersion. 

Among  recent  detailed  studies  of 
dispersion,  the  work  of  Foldvari  ^ 
is  of  considerable  importance.  He 
compared  the  effects  of  ammonium 
hydroxide,  sodium  oxalate,  and  so- 
dium metasilicate  on  a  wide  variety' 
Effects  of  sodium  oxalate  of  soils  and  sediments.  The  choice 
of  these  three  dispersing  agents  re- 
spectively was  determined  by  Fold- 
vari's  classification  of  peptizers  into  three  broad  groups:  (i)  those  which 
supplied  (OH) —  ions  to  the  suspension;  (2)  those  which  depended  for 
their  effects  in  part  at  least  upon  the  removal  of  coagulating  ions  (nota- 
bly Ca+  +  )  from  the  suspensions;  and  (3)  those  which  supplied  the 
particles  with  a  "protective  cover"  and  thus  prevented  the  coagulating 
ions  from  reaching  the  particles.  As  a  result  of  his  comparative  analysis, 
Foldvari  concluded  that  0.005  ^  sodium  oxalate  gave  the  best  results 
in  most  routine  cases  but,  for  sediments  with  a  high  content  of  gypsum 
or  calcite,  sodium  metasilicate  afforded  the  best  dispersion.  The  concen- 
tration of  the  latter  dispersing  agent  was  i  cc.  of  waterglass  (36°-38° 
Be)  per  liter  of  suspension. 

The  influence  of  base  exchange  phenomena  on  dispersion  was  inves- 
tigated by  Thomas.^  He  found  that  soils  in  which  the  exchangealile  bases 
were  replaced  by  sodium  were  most  readily  dispersed  by  sodium  car- 
bonate, while  a  magnesium  soil  flocculated  immediately  on  the  addition 
of  sodium  carbonate,  due  probably  to  the  release  of  magnesium  ions. 
Because  of  the  difficulty  of  preparing  sodium  soils  for  routine  purposes, 

1  E.  H.  Vinther  and  M.  L.  Lasson,  Uber  Korngrossemmessungen  von  Kaolin-  und 
Tonarten :  Ber.  Dcittsch.  Kcram.  Gcs.,  vol.  14,  pp.  259-279,  1933. 

2  A.  Foldvari,  Uber  die  Wirkung  einiger  Tonstabilisitoren :  Kolloid-Beihejte,  vol. 
44,  pp.  125-170,  1936. 

3  M.  D.  Thomas,  Replaceable  bases  and  the  dispersion  of  soil  in  mechanical  analy- 
sis :  Soil  Science,  vol.  25,  pp.  419-427,  1928. 


PREPARATION  OF  SAMPLES  65 

Thomas  recommended  freeing  the  sample  of  exchangeable  bases  by  acid 
treatment,  followed  by  dispersion  with  sodium  carbonate. 

Soil  scientists  have  devoted  considerable  attention  to  the  subject  of 
base  exchange  during  the  last  decade,  and  numerous  dispersion  pro- 
cedures make  use  of  chemical  agents  which  effect  an  elimination  of  coag- 
ulating ions  by  removing  them  by  base  exchange  reactions. 

Robinson  ^  investigated  the  subject  in  a  comprehensive  treatise  in 
1933.  He  compared  several  methods  of  dispersion  which  involved  base 
exchange  phenomena,  notably  the  International-A  method,  which  relied 
on  the  use  of  HQ  to  remove  the  exchangeable  bases  and  subsequent 
dispersion  with  ammonium  hydroxide ;  the  Sudan  method,-  which  in- 
volves the  direct  use  of  0.05  per  cent  sodium  carbonate  to  effect  dis- 
persion ;  and  the  Puri  method,^  which  consists  essentially  of  removing 
the  exchangeable  bases  with  sodium  chloride  to  yield  sodium  clay. 
Sodium  hydroxide  is  added  if  necessary,  to  obtain  an  alkaline  suspension. 
Robinson  modified  the  International-A  method  by  treating  the  soil  with 
4  c.c.  of  X  XaOH  per  10  g.  of  soil  after  acid  treatment,  instead  of  with 
ammonium  hydroxide.  The  modified  technique  was  called  the  Inter- 
national-soda method,  and  Robinson's  work  showed  it  to  yield  the  best 
results  for  most  types  of  soil.  Sodium  oxalate,  on  the  other  hand,  while 
satisfactory  in  many  cases,  eft'ected  only  incomplete  disj^ersion  with 
lateritic  and  ferruginous  soils. 

A  problem  of  considerable  importance  in  sedimentary  petrology,  and 
one  which  is  not  involved  in  most  cases  of  soil  analysis,  is  the  content 
of  primary  carbonate  particles  in  the  size  distribution  of  sediments. 
Sedimentary  analysis  is  usually  performed  on  unaltered  or  unweathered 
samples,  in  which  the  carbonate  particles  may  represent  an  appreciable 
part  of  the  size  distribution.  Acid  treatment  in  such  cases  will  seriously 
aft'ect  the  analytical  results  by  removing  acid-soluble  material  and  dis- 
torting the  resulting  statistical  data.  Alethods  involving  base  exchange, 
and  relying  on  the  formation  of  hydrogen  clay,  thus  appear  to  be  un- 
suitable to  sediments  in  general,  although  they  are  applicable  to  non- 
calcareous  sediments. 

A  more  recent  study  by  Puri  *  involved  a  modification  of  his  original 

1  G.  \V.  Robinson,  The  dispersion  of  soils  in  mechanical  anah^sis:  Imp.  Bur.  Soil 
Sci.,  Tcchn.  Comvi.  26,  1933- 

2  A.  F.  Joseph  and  F.  J.  Martin,  The  determination  of  clay  in  heavy  soils:  Jour. 
Agric.  Sci.,  vol.  11,  pp.  293  ff.,  1921.  A.  F.  Joseph  and  O.  W.  Snow,  loc.  cit.,  1929. 

3  A.  N.  Puri,  A  new  method  of  dispersing  soils  for  mechanical  analysis :  India 
Dcpt.  Aaric.  Mc^u,  Chew.  Scries,  vol.  10,  pp.  209-220,  1929. 

*  A.  N.  Puri.  The  ammonium  carbonate  method  of  dispersing  soils  for  mechanical 
analysis:  Soil  Science,  vol.  39,  pp.  263-270,  1935. 


66        SEDIMENTARY  PETROGRAPHY 

dispersion  technique  by  the  use  of  ammonium  carbonate  to  effect  base 
exchange.  In  his  newer  technique,  the  exchangeable  bases  are  replaced 
with  ammonia  by  boiling  the  soil  with  N  ammonium  carbonate  solution. 
Boiling  is  continued  until  the  volume  of  the  solution  is  reduced  by  half, 
whereupon  4  to  8  c.c.  of  X  sodium  hydroxide  or  X  lithium  hydroxide 
is  added  per  10  g.  of  soil,  to  effect  dispersion.  Puri  favored  the  lithium 
hydroxide  in  preference  to  sodium  hydroxide  because  of  its  greater  dis- 
persive effect, 

Puri's  new  method  is  somewhat  more  drastic  than  others  in  terms  of 
the  boiling  involved,  and  for  sediments  it  may  be  slightly  modified  as 
will  be  discussed  later  (page  75).  The  great  advantage  of  Puri's  ap- 
proach is  that  it  eliminates  the  need  for  acid  treatment  and  makes  avail- 
able for  sediments  a  dispersion  technique  which  applies  the  advantages 
of  base  exchange  phenomena. 

Removal  of  water-soluble  salts.  Most  sediments  and  soils  con- 
tain water-soluble  salts  in  varying  amounts,  and  it  is  to  be  expected 
that  the  foreign  electrolytes  thus  introduced  into  the  suspension 
may  have  a  marked  effect  on  dispersion.  Cations  that  appear  to  be 
commonly  present  in  sediments  are  varying  amounts  of  Ca++,  Na+, 
Fe+++,  and  Mg"^"^.  Common  anions  are  S04=,  COz=,  and  Cl~.  Gyp- 
sum is  an  important  salt  in  non-calcareous  sediments,  whereas  cal- 
cium carbonate  or  bicarbonate  is  more  common  in  the  calcareous 
types. 

Perhaps  the  most  important  study  of  the  eff'ects  of  foreign  electro- 
lytes was  made  by  Wiegner  ^  in  1927.  Wiegner  pointed  out  that  if  very 
small  amounts  of  soluble  salts  are  present,  the  charges  on  the  particles 
are  above  the  critical  potential,  whereas  if  appreciable  amounts  are  pres- 
ent, the  charges  are  below  the  critical  potential,  and  dispersion  may  be 
seriously  hindered.  Dispersion  procedures,  such  as  shaking  or  boiling, 
increase  the  agitation  of  the  particles  and  thus  increase  the  number  of 
collisions  among  them.  If  the  charges  are  above  the  critical  potential 
this  added  movement  increases  dispersion,  whereas  if  they  are  below  the 
critical  potential  the  added  collisions  increase  the  rate  of  coagulation  and 
thus  slow  down  or  prevent  dispersion. 

Wiegner  compared  the  effects  on  soils  of  (i)  shaking  for  6  hr., 
(2)  rubbing  for  an  hour  with  a  brush,  and  (3)  boiling  for  an  hour  with 
a  reflux  condenser.  The  analyses  were  conducted  in  a  Wiegner  tube 
(Chapter  6),  in  which  the  same  sample  could  be  used  after  various 
treatments,  because  none  of  it  is  removed  during  the  analysis.  It  was 

1  G.  Wiegner,  loc.  cit.,  1927. 


PREPARATION  OF  SAMPLES 


6? 


found  that  boiling  was  more  etTective  than  rubbing  or  shaking  on  washed 
soils,  while  shaking  was  the  most  effective  on  unwashed  soils.  The  effect 
of  N/io  ammonia  was  tested  on  washed  and  unwashed  soils,  and  it  was 
found  that  the  washing-out  of  the  foreign  electrolytes  was  more  effective 
than  the  use  of  the  peptizer  on  unwashed  soils. 

Numerous  other  workers  have  discussed  the  removal  of  water-soluble 
salts  either  as  a  standard  procedure  in  the  routine  analysis  of  soils  or 
sediments  or  in  connection  with  samples  which  do  not  respond  to  direct 
dispersion  with  peptizers.  Among  writers  who  have  included  the  pro- 
cedure in  their  methods  are  Olmstead,  Alexander,  and  Middleton ;  ^ 
Correns  and  Schott,-  who  recom- 
mended it  for  recent  marine  sedi- 
ments ;  Gessner,^  Gallay,*  Robinson,^ 
and  others. 

Several  methods  are  available  for 
washing  the  sediments,  and  one  of  the 
simplest  is  by  means  of  Pasteur- 
Chamberland  filters,  (See  Figure 
17.)  Olmstead,  Alexander,  and  Mid- 
dleton's  technique  in  this  connection 
is  effective : 


Fig.  17. — Apparatus  for  removing 
water-soluble  salts  by  suction  tiltra- 
tion. 


The  lower  12  cm.  of  the  filter  is 
sawed  off,  and  fitted  with  a  removable 
stopper.  The  suspension  of  sediment 

is  placed  in  a  beaker,  and  the  filter,  attached  to  a  suction  pump,  is  im- 
mersed within  it.  The  suction  is  continued  until  as  much  as  possible  of 
the  liquid  is  removed.  The  liquid  within  the  filter  is  then  removed  by 
extracting  the  stopper,  and  the  fiher  core  is  filled  ^^^th  distilled  water. 
The  stopper  is  re-inserted  and  back  pressure  is  applied  by  means  of  a 
rubber  bulb,  to  remove  the  material  adhering  to  the  outside  of  the  filter. 
Additional  distilled  water  is  added  to  the  beaker,  and  the  process  of 
washing  is  repeated.  Usually  six  washings  are  sufficient  to  remove  the 
soluble  salts. 

Robinson  used  a  Buchner  funnel  fitted  with  p-cm.  diameter  hardened 
filter  paper  (Whatman  50).  The  paper  was  fixed  to  the  funnel  with 
cellulose  cement,  after  etching  the  funnel  with  hydrofluoric  acid  to  in- 
sure adhesion.  The  funnels  were  fitted  to  filter  flasks  arranged  in  a  bat- 
tery of  four  attached  to  a  single  pump.  Three  washings  with  20-30  c.c, 
of  water  each  were  generally  vised  on  soils. 


1  L.  B.  Olmstead,  L.  T.  Alexander  and  H.  E.  Middleton,  loc.  cit., 

2  C.  W.  Correns  and  W.  Schott.  loc.  cit..  1932. 

3  H.  Gessner,  Die  Schliimmanalysc  (Leipzig,  1931^ •  PP-  164  ff. 
*  R.  Gallay.  KoUoid-Bcihejte,  vol.  21,  pp.  431  ff.,  1925. 

5  G.  W.  Robinson,  loc.  cit.,  1933. 


1930. 


68        SEDIMENTARY  PETROGRAPHY 

The  washing-out  of  foreign  electrolytes  is  at  best  a  tedious  process, 
and  some  workers  have  taken  a  pragmatic  view  of  the  problem.  That  is, 
for  average  sediments  without  a  large  content  of  water-soluble  salts, 
dispersion  is  effected  by  the  use  of  a  peptizer  such  as  sodium  carbonate 
or  oxalate,  or  lithium  chloride  or  hydroxide.  Only  in  those  cases  where 
coagulation  occurs  despite  this  treatment  is  washing  resorted  to. 


GENERAL     CRITIQUE     OF     DISPERSION 

The  extensive  literature  on  dispersion  demonstrates  that  the  problem 
is  no  simple  one.  Among  the  variables  that  enter  the  situation  with 
respect  to  the  dispersive  effect  of  a  given  electrolyte  arFits  cblTiposttion, 
its  concentration,  associated  base  exchange  phenomena,  and  the  presence 
of  foreign  electrolytes.  Moreover,  the  general  problem  is  further  com- 
plicated by  the  fact  that  all  sizes  of  particles  are  not  equally  sensitive  to 
dispersion  or  coagulation.  Tuorila  showed  that  in  quartz  suspensions  the 
effects  of  coagulation  begin  to  manifest  themselves  at  diameters  of  about 
0.04  mm.,  and  become  very  pronounced  between  5  and  10  microns,  while 
particles  under  4  microns  are  completely  removed  from  suspension.  Thus 
sediments  made  up  predominantly  of  particles  in  the  most  sensitive  range 
may  be  exj^ected  to  be  strongly  affected  by  slight  changes  in  the  dis- 
persion technique. 

The  interplay  of  numerous  variables,  some  of  which  are  independent 
of  the  others,  strongly  suggests  that  there  can  be  no  single  dispersion 
technique  for  all  tyi>es  of  materials.  This  statement  has  been  repeated 
by  numerous  writers,  and  current  researches  appear  increasingly  to 
verify  it.  Soil  scientists  have  made  considerable  progress  toward  stand- 
ardized routines  which  apply  to  a  wide  range  of  soils,  but  among  sedi- 
ments there  are  problems  of  degrees  of  alteration  and  of  induration 
which  greatly  complicate  the  problem. 

Dispersion  has  been  found  to  be  a  continuous  process,  rapid  at  first 
and  slower  later  on.  Puri  and  Keen  showed  this  in  their  experiments  on 
shaking,  and  Olmstead  noted  the  same  effect  in  his  work  on  vibration. 
If  this  is  universally  true,  it  appears  that  complete  dis^^ersion  can  never 
be  effected,  or  that  there  is  a  continuous  increase  of  fine  material  due 
to  the  disruption  or  attrition  of  individual  grains.  Clark  ^  raised  the 
question  whether  there  exists  in  soils  any  unique  size  frequency  dis- 

1  C.  L.  Clark,  The  dispersion  of  soil-forming  aggregates :  Soil  Science,  vol.  35,  pp. 
291-294,  1933- 


PREPARATION  OF  SAMPLES  69 

tribution  or  whether  the  distribution  is  not  a  function  of  the  dispersion 
process.  The  difficuhy  of  setthng  the  question  Hes  in  the  fact,  as  Clark 
pointed  out,  that  the  dispersion  of  aggregates  cannot  readily  be  dis- 
tinguished from  the  disruption  of  crystal  fragments  by  their  end- 
products.  This  appears  to  be  particularly  true  of  the  finer  particles  in 
the  sediments. 

Among  fine-grained  sediments  in  which  few  authigenic  changes  have 
taken  place  there  should  theoretically  be  little  difficulty  in  effecting  dis- 
persion into  the  individual  particles,  but  even  here  there  is  an  increase 
of  fine  material  with  an  increase  of  vigor  or  time  in  the  dispersion  pro- 
cess. If  the  sediment  is  indurated  or  altered  by  weathering,  the  original 
size  distribution  may  have  suffered  considerable  change  due  to  secondary 
growth,  dehydration,  the  introduction  of  secondary  minerals,  or  the 
leaching-out  of  certain  constituents.  The  reconstruction  of  the  original 
distribution  may  accordingly  be  nearly  impossible.  Clark's  point  thus 
applies  in  part  to  sediments  as  well,  and  the  problem  raised  is  not  one 
that  can  be  readily  solved. 

Soil  scientists,  working  in  cooperation  on  several  dispersion  pro- 
cedures, discovered  that  results  obtained  in  different  laboratories  on  the 
same  soils  were  not  consistent.  As  a  result,  considerable  effort  has  been 
expended  to  develop  standard  methods  wliich  are  free  of  subjective 
errors,  and  which  may  yield  comparable  results.  In  a  recent  paper 
Xovak  ^  compared  several  dispersion  procedures  and  commented  on  the 
lack  of  uniformity  among  various  laboratories.  Sedimentary  petrologists 
have  not  yet  united  in  an  attempt  to  adjust  such  difficulties  in  connection 
with  sediments.  Unquestionably  the  complexity  of  the  general  problem 
and  the  unknown  influence  of  some  of  the  variables  account  in  part  for 
the  difficulties  encountered. 

The  relative  merits  of  chemical  and  physical  methods  of  dispersing 
sediments  have  recently  been  investigated  by  Neumaier,-  who  reached 
the  conclusion  that  chemical  methods  should  not  be  used,  but  that  re- 
liance should  be  placed  entirely  upon  physical  means,  Neumaier,  how- 
ever, included  mainly  earlier  papers  in  his  critique. 

The  relative  merit  of  analyzing  samples  in  their  natural  moist  con- 
dition and  analyzing  air-dried  samples  has  also  been  the  subject  of  con- 
troversy. Some  writers  maintain  that  dried  sediments  undergo  changes 
which  seriously  aft'ect  the  subsequent  analysis  of  the  sample.  Correns  and 

^  W.  Novak,  Vorbehandlung  der  Bodenproben  zur  mechanischen  Bodenanalyse : 
Proc.  2i\d  Int.  Congr.  Soil  Sci.,  vol.  i,  pp.  14-39,  1932. 
-  F.  Neumaier,  loc.  cit.,  1935- 


70        SEDIMENTARY  PETROGRAPHY 

Schott^  investigated  the  problem  in  1933  and  reached  the  conclusion 
that  the  samples  should  not  be  dried.  Foldvari-  contended,  however, 
that  in  the  case  of  ancient  sediments  the  vicissitudes  through  which  the 
material  has  passed  render  relatively  meaningless  the  accidental  mois- 
ture state  in  which  the  sample  may  have  been  found  at  the  time  of 
sampling.  Among  other  writers  who  have  expressed  themselves  on  the 
question  of  damp  vs.  dry  samples  are  von  Sigmond,^  Richter,*  Hissink,^ 
and  Xeumaier,® 

GENERALIZED     DISPERSION      ROUTINE 

The  entire  subject  of  dispersion  revolves  about  a  point  earlier  men- 
tioned :  the  aggregates  must  be  destroyed  without  affecting  the  sizes 
of  the  individual  particles.  There  is  a  further  condition_iin2lied  in  this 
process:  the  dispersed  particles  should  not  form  aggregates  again  dur- 
ing the  course  of  the  analysis.  To  satisfy  these  conditions,  shearing 
stresses  and  abrasive"action  on  the  individual  particles  should  be  kept 
at  a  minimum,  while  disaggregation  and  dispersion  should  be  at  a 
maximum. 

Gessner '  devoted  considerable  space  to  the  dispersion  of  samples 
and  presented  a  general  routine  which  involved  several  procedures,  each 
followed  by  tests  for  dispersion  and  coagulation,  so  that  the  treatment 
given  depends  on  the  difficulty  of  dispersing  the  material.  In  his  routine 
the  sample  is  shaken  and  then  tested  for  coagulation.  The  test  shows 
either  complete  dispersion,  incomplete  dispersion  without  coagulation, 
or  coagulation.  The  first  case  is  analyzed,  the  second  is  boiled,  the  third 
is  washed.  The  boiled  sample  is  tested  and  either  analyzed  or  washed. 
The  washed  samples  are  either  analyzed  or  rewashed. 

As  a  result  of  a  series  of  comparative  tests  made  on  certain  of  the 
dispersion  procedures  received  "earlier,  Krumbein  *  developed  a  routine 
for  dispersing  fine-grained  sediments  in  which  the  procedures  become 
successively  more  vigorous,  so  that  only  the  more  resistant  sediments 
receive  the  most  vigorous  treatment.  Figure  1 8  shows  the  routine  graph- 

1  C.  W.  Correns  and  W.  Schott,  Uber  den  Einfluss  des  Trockens  auf  die  Kom- 
grossenverteilungen  von  Tonen :  Kolloid  Zeits.,  vol.  65,  pp.  196-203,  1933. 

2  A.  Foldvari,  loc.  cit.,  1936. 

3  A.  A.  J.  von  'Sigmond,  Bericht  iiber  den  Int.  Kom.,  u.  s.  w:  Int.  Mitt.  f.  Bo- 
denk.,  vol.  4,  pp.  25-27,  1914. 

*  G.  Richter,  loc.  cit.,  1916. 
5  D.  J.  Hissink,  loc.  cit.,  1921. 
8  F.  Neumaier,  loc.  cit.,  193.5. 
''  H.  Gessner,  op.  cit.,  p.  167,  1931. 
8W.  C.  Krumbein,  loc.  cit.,  1933. 


PREPARATION  OF  SAMPLES  71 

ically.  The  foreign  electrolytes  are  washed  out  only  when  the  need  is 
indicated.  In  every  case  the  sample  is  soaked  in  dilute  peptizer  for  a 
preliminary  period.  This  is  followed  by  one  of  two  sequences,  depending 
on  the  amount  of  material  above  0.06  mm.  diameter  in  the  sediment.  I'>y 
grinding  a  fragment  between  the  teeth  or  rubbing  between  finger  and 


PRELIMINARY 
SOAKING 

1 

BRUSH     OR 
PESTLE 

;     \ 

SHAKE     OR 
STIR 

WASH  OUT 
ELECTROLYTES 

HEAT   TO 
BOILING 

1 

MECHANICAL 
ANALYSIS 

Fig.  18. — Flow-sheet  of  generalized  dispersion  procedure. 


thumb,  the  amount  of  sand  present  may  be  estimated.  No  fixed  pro- 
portion is  involved,  but  the  authors  have  found  that  in  most  cases,  if 
only  a  trace  of  coarse  material  is  present,  brushing  or  pestling  is  more 
effective  than  shaking;  while,  if  there  is  an  appreciable  amount  of  sand, 
shaking  or  stirring  is  preferable.  This  is  particularly  true  of  uncon- 
solidated sediments,  but  in  partially  indurated  cases  it  is  desirable  to 
brush  the  sample  before  shaking  it.  The  dashed  line  in  the  figure  in- 


y2  SEDIMENTARY  PETROGRAPHY 

dicates  this  possibilin-.  The  several  arrows  illustrate  the  paths  that  may- 
be followed.  An  attempt  was  made  to  allow  some  flexibility  and  yet  to 
pass  from  gentle  to  more  vigorous  steps  as  the  need  was  felt. 

-\mong  the  many  diipersing  agents  that  have  been  used  for  soils  and 
sediments,  sodiimi  oxalate  appears  to  be  most  favorable.  In  comparative 
tests  it  usually  ranks  among  the  best  for  routine  purposes,  and  it  has 
been  adopted  by  the  authors  as  the  standard  agent.  The  following  routine 
accordingly  describes  dispersion  in  terms  of  sodium  oxalate,  but  the 
reader  may  substitute  other  dispersing  agents  if  he  w^ishes.  Among  the 
older  dispersers  sodium  carbonate  (in  concentrations  from  X/25  to 
X/ioo)  is  ver)-  effective  in  some  cases.  The  writers  have  found  it  pref- 
erable to  use  sodium  carbonate  for  weathered  phases  of  glacial  till. 
Among  the  newer  dispersing  agents  lithium  hydroxide  has  proved  suc- 
cessful in  many  cases,  and  may  be  used  in  concentrations  of  about  X/50 
or  X/ioo.  In  connection  with  dispersion  procedures  involving  base 
exchange,  the  authors  favor  sodium  hydroxide,  but  lithium  hydroxide 
has  also  found  considerable  favor  among  some  workers. 

The  following  description  of  the  several  procedures  is  arranged  in 
the  order  of  the  two  main  sequences,  but  any  steps  common  to  both  are 
described  only  once. 

The  air-dried  ^  sediment  is  crushed  with  a  rubber-tipped  pestle,  a  roll- 
ing pin,  or  a  wooden  mallet  until  the  fragments  are  reasonably  small.  A 
test  sample  is  then  quartered  and  weighed.  The  weight  of  the  sample 
depends  on  the  range  of  sizes  present.  It  is  desirable  to  have  a  suspension 
of  2  to  3  per  cent  concentration  for  analysis  by  modern  methods,  so  that 
about  25  g.  are  optimum  for  a  Hter  of  suspension.  If  the  sample  contains 
say  25  per  cent  of  sand,  the  test  sample  should  weigh  about  30  g.  \\'hen 
the  sand  is  later  sieved  oflF,  the  remaining  fine  material  will  yield  a  sus- 
pension of  about  2.3  per  cent  concentration. 

The  quartered  sample  is  placed  in  a  250-c.c.  Erlenmeyer  flask  with 
100  c.c.  of  X/ioo  sodium  oxalate  solution-  and  allowed  to  soak  for  a 
period  depending  on  the  rate  at  which  the  lumps  disaggregate.  A  mini- 
mum of  24  hr.  and  a  maximum  of  eight  weeks  are  possible  limits  to  the 
time.  Some  Pennsh-\-anian  shales  required  ten  days  of  soaking  before 
they  yielded  to  the  brush,  while  others  needed  only  24  hr.  By  occasional 
shaking  the  process  of  disintegration  may  be  observed,  and  if  the  sample 
is  taken  out  too  early  it  may  be  returned  to  the  flask  after  brushing  or 
pestling  is  found  ineffective  on  the  lumps. 

1  If  it  is  preferred  to  work  with  sediments  in  their  natural  moist  condition,  the 
moisture  content  may  be  determined  from  a  separate  sample. 

-  It  is  convenient  to  have  an  X/5  solution  of  sodium  oxalate  on  hand,  prepared 
by  dissolving  13.4  g.  of  the  salt  in  a  liter  of  water.  The  X/ioo  solution  is  made  as 
needed  by  adding  5  c.c.  of  this  solution  to  95  cc  of  water. 


PREPARATION  OF  SAMPI.KS  73 

After  the  preliminary  soaking  the  sample  is  jXHired  into  an  cvaix)rating 
dish  and  ruhbcd  with  a  stiff  brush  or  nihher  pestle.  The  writers  favor  a 
brush  for  the  purix)se.  As  the  lumps  disa,y;grcgate,  water  is  added  and 
the  dispersed  material  poured  into  a  beaker.  The  water  used  during 
brushing  should  have  a  concentration  of  N/ioo  sodium  oxalate.  The 
brushing  process  often  requires  the  better  part  of  an  hour,  but  the  final 
results  justify  the  use  of  ample  time.  Whittles^  prepared  a  rubber  jiestle 
by  fding  down  a  stopix'r  and  attaching  it  to  a  glass  rod.  Such  pestles 
are  advantageous  as  an  aid  to  brushing,  because  the  more  resistant  lumps 
may  be  gently  crushed  before  brushing. 

The  volume  of  sus]XMision  after  brushing  may  be  about  400  c.c.  It  is 
tested  for  complete  dispersion  by  placing  a  drop  on  a  slide  under  a  cover- 
glass,  allowing  it  to  rest  for  a  few  minutes,  and  examining  it  under  the 
microscope.  If  each  grain  stands  out  as  an  individual,  and  the  smaller 
ones  display  Brownian  movement,  the  dispersion  is  proi)ably  complete. 
If  bead-like  strings  and  clusters  of  individual  particles  extend  through 
the  field,  the  susi)ension  is  coagulated.  If  the  field  shows  a  mixture  of 
individual  grains  and  aggregates,  not  clustered  together,  coagulation  is 
probably  al)sent,  but  dispersion  is  incomplete.  In  ])ractice  there  is  con- 
siderable gradation  among  these  three  situations,  and  it  is  often  difficult 
to  distinguish  among  them.  I'^niher,  if  slow  coagulation  is  present,  the 
susiKMision  may  remain  ai)parently  (lis])ersed  for  several  hours.  As  a 
final  test  the  writers  allow  the  apparently  fully  dispersed  suspension  to 
stand  overnight ;  if  visible  coagulation  has  not  set  in  within  that  time, 
it  will  not  occur  during  the  analysis.  In  a  very  few  cases  suspensions 
have  remained  apparently  dispersed  for  two  or  three  days  and  finally 
coagulated. 

If  coagulation  is  rapid,  a  flocculent  precipitate  settles  out,  leaving 
essentially  clear  li(|uid  behind.  This  i)recipitate  differs  from  the  sediment 
normally  accumulating  in  that  it  behaves  in  a  quasi-li(|uid  fashion  and 
"flows"  as  tlie  container  is  tilted,  whereas  the  normal  sediment  adheres 
rigidly  to  the  b(jttom  of  the  vessel.  In  cases  of  slow  coagulation  this 
effect  may  require  several  hours  to  manifest  itself.  In  extremely  slow 
coagulation  the  effect  may  be  delayed  for  several  days.  Tuorila-  has 
suggested  a  critical  test  for  slow  coagulation,  which  follows  as  a  corollary 
from  his  researches.  Inasmuch  as  coagulation  increases  rajiidly  with  con- 
centration, the  sediment  may  be  analyzed  twice,  in  dilute  and  concen- 
trated suspensions.  If  the  results  check  within  reasonable  limits,  no 
coagulation  is  present. 

If  the  test  shows  complete  dispersion,  the  suspension  should  be  diluted 
to  a  liter  with  N/ioo  sodium  oxalate  and  analyzed.  If  dispersion  is  in- 
complete, the  suspension  is  diluted  to  about  800  c.c.  with  N/ioo  sodium 
oxalate  and  heated  to  the  boiling  point.  As  st)on  as  the  li(iuid  boils  it  is 
withdrawn  from  the  flame.  After  cooling,  the  suspension  is  tested,  and 
in  every  case  studied  by  the  writers  the  samples  were  either  fully  dis- 
persed or  coagulation  had  set  in. 

1  C.  L.  Whittles,  Iflc.  cit.,  1924. 
-  P.  Tuorila,  loc.  cit.,  1927. 


74        SEDIMENTARY  PETROGRAPHY 

If  coagulation  is  present  at  any  stage  of  the  routine,  the  suspension, 
diluted  to  about  a  liter,  is  poured  into  a  tall  beaker  and  the  liquid  filtered 
through  a  Pasteur-Chamberland  filter  with  suction.  The  set-up  involves 
the  filter  connected  to  a  filter  bottle,  with  the  outlet  of  the  bottle  con- 
nected to  the  filter  pump.  The  suspension  passes  into  the  filter  (  which  is 
a  cylindrical  tube)  and  collects  in  the  bottle.  \\'hen  the  residue  in  the 
beaker  is  like  paste,  the  filter  is  cleaned  and  the  sediment  again  made  up 
to  a  liter  with  X/ioo  sodium  oxalate,  since  most  of  the  original  peptizer 
has  been  removed  by  the  filtration.  The  suspension  is  again  heated  to 
boiling,  and  usually  it  will  be  dispersed.  In  some  cases  more  than  one 
washing  is  necessar\'.  It  is  conceivable  that  some  samples  may  be  quite 
obstinate  in  their  resistance  to  dispersion. 

When  the  sediment  has  an  appreciable  amount  of  sand  in  it,  the  pre- 
liminar}'  soaking  is  followed  by  shaking  or  stirring,  unless  the  sediment 
is  indurated,  in  which  case  brushing  should  precede  the  shaking  or  stir- 
ring. The  sample  is  shaken  for  an  hour,  preferably  in  a  reciprocating 
shaker,  or  it  is  stirred  in  an  electric  drink  mixer  for  5  or  lo  min.  It  is 
important  to  use  wire  bafl^es  in  the  cup.  After  the  shaking  or  stirring, 
the  suspension  is  tested  for  dispersion  as  described  above. 

If  dispersion  is  incomplete,  the  suspension  is  diluted  to  about  800  c.c. 
and  heated  to  boiling,  as  described  above.  If  coagulation  is  present,  the 
foreign  electrolytes  are  washed  out  as  described  above. 

The  material  above  1/16  mm.  diameter  is  sieved  from  the  sample 
before  analysis.  The  suspension  is  poured  through  the  sieve  and  the 
liquid  collected  in  a  beaker.  The  residue  is  washed  with  a  gentle  stream 
of  water.  The  total  volume  of  suspension  should  be  less  than  a  liter, 
including  the  wash  water.  The  suspension  may  be  sieved  after  brushing 
or  shaking,  but  any  convenient  point  in  the  routine  may  be  used.  The 
material  above  1/16  mm.  is  sieved  drj-  into  grades  if  it  is  appreciable 
in  amount.  It  should  also  be  examined  to  determine  whether  it  consists 
of  individual  grains  or  undisintegrated  aggregates. 

In  many  instances  small  undisintegrated  fragments  are  found  in  the 
sieve,  and  in  some  sediments  without  coarse  grains  there  are  numerous 
small  ironstone  pellets.  It  is  not  easy  in  every  case  to  decide  what  disposition 
should  be  made  of  such  materials.  Undisaggregated  lumps  may  be 
treated  further,  unless  they  are  so  firmly  cemented  that  the  fragments 
tend  to  break  rather  than  disintegrate.  Ironstone  pellets  commonh-  are 
secondan,'  materials,  and  a  correction  may  be  applied  to  the  sample 
weight  to  allow  for  them.  At  best  a  pragmatic  attitude  suggests  that 
small  amounts  of  such  materials  be  discounted  by  correcting  for  them ; 
if  appreciable  parts  of  the  sediment  remain  as  undisaggregated  lumps, 
the  particular  dispersion  routine  was  not  successful. 

The  experience  of  the  authors  and  students  in  the  laboratories  of  the 
Universit}-  of  Chicago  has  shown  that  the  generalized  dispersion  routine 
described  above  is  successful  in  a  large  majority  of  cases.  Obstinate 
samples  are  encountered,  however,  in  which  induration  prevents  even 
an  approximate  disintegration  or  dispersion ;   in   such   instances  thin- 


PREPARATION  OF  SAMPLES  75 

section  mechanical  analysis  may  be  resorted  to  (Chapter  6).  If  the 
difficulty  arises  from  a  high  content  of  soluble  salts,  especially  calcium 
ions,  washing  usually  proves  adequate.  In  this  connection,  however,  it 
seems  desirable  that  the  base  exchange  method  of  Puri^  should  be 
investigated  for  its  general  applicability  to  difficult  sediments.  Puri's 
method  involves  boiling  the  sample  with  ammonium  carbonate  for  pro- 
tracted periods  of  time,  but  this  may  be  modified  to  a  period  of  soaking 
in  the  electrolyte  or  to  merely  heating  the  suspension  to  the  boiling  point. 
The  addition  of  sodium  hydroxide  to  a  concentration  of  0.004  N  to 
replace  the  ammonia  results  in  a  sodium  clay  with  its  high  stability. 

The  authors  have  adopted  the  practice  of  experimenting  on  small 
samples  in  "test-tube"  dispersion  to  determine  the  relative  effects  of 
various  procedures  and  various  dispersing  agents.  Likewise,  qualitative 
tests  on  the  nature  of  the  foreign  electrolytes  are  of  value  in  deter- 
mining the  advisability  of  departing  from  the  generalized  routine.  For 
"test-tube"  dispersion,  various  amounts  of  the  several  peptizers  are 
added  to  small  volumes  of  suspension  and  the  effects  noted  qualitatively 
by  allowing  the  tubes  to  stand  for  several  hours  or  over  night. 

1  A.  N.  Puri,  loc.  cit.,  1935. 


CHAPTER   4 
THE  CONXEPT  OF  A  GRADE  SCALE 

INTRODUCTION 

In  most  types  of  sedimentary  analysis  the  data  are  arranged  on  some 
kind  of  size  scale  (which  may  be  diameter,  area,  or  volume)  for  con- 
venience both  in  conducting  the  analysis  and  in  tabulating  the  analytical 
data.  This  is  especially  true  of  mechanical  analysis,  but  the  general  topic 
of  grade  scales  is  in  itself  so  important  that  a  separate  chapter  is  de- 
voted to  it  here. 

A  grade  scale  may  be  defined  as  an  arbitrary-  division  of  a  continuous 
scale  of  sizes,  such  that  each  scale  unit  or  grade  may  serve  as  a  con- 
venient class  interval  for  conducting  the  analysis  or  for  expressing  the 
results  of  an  analysis.  Against  this  grade  scale  may  be  plotted  the 
amount  of  material  in  each  grade  (a  size  frequency  diagram),  or  the 
amount  of  some  particular  mineral  in  the  sediment  (a  mineral  fre- 
quency diagram),  or  the  average  sphericity  or  roundness  of  material 
in  each  grade  (a  shape  frequency  diagram),  and  so  on.  In  these  cases 
size  is  usually  chosen  as  the  independent  variable,  and  the  grade  scale 
is  therefore  arranged  along  the  horizontal  axis  of  the  diagram,  whereas 
the  frequency  is  plotted  along  the  vertical  axis. 

One  of  the  commonest  types  of  frequency  diagram  is  the  histogram, 
which  is  drawn  verv*  simply  by  setting  a  vertical  block  above  each  grade, 
proportional  in  height  to  the  value  of  the  other  variable  (amount  of 
material,  average  sphericity,  etc.)  in  each  grade.  Convention,  in  Amer- 
ica at  least,  has  been  to  draw  each  grade  equal  in  width,  whether  it  is  so 
in  fact  or  not.  The  subject  of  histograms  receives  more  detailed  con- 
sideration in  Chapter  7:  for  the  present  only  their  relation  to  grade 
scales  need  be  discussed. 

Unfortunately,  the  shape  of  a  particular  histogram  will  vary  according 
to  the  grade  scale  used  in  the  analysis.  That  is,  the  same  identical  sedi- 
ment, if  analyzed  on  the  basis  of  two  different  grade  scales,  will  yield 
figures  which  may  be  quite  unlike  each  other.  Recognition  of  this  fact 
has  led  to  a  considerable  discussion  of  grade  scales  and  methods  of 

76 


GRADE  SCALES  yy 

presenting  data,  in  which  one  or  another  scale  was  proposed  for  all 
analyses  to  avoid  the  unfortunate  variation  of  the  histograms.  An  even 
greater  volume  of  literature  has  discussed  the  relative  merits  of  the 
several  grade  scales,  and  in  some  of  these  papers  it  was  shown  that  one 
or  another  grade  scale  was  more  logical,  more  convenient,  or  more 
"natural"  for  sedimentary  purposes.^ 

It  seems  that  even  to-day  it  is  not  universally  recognized  that  the 
choice  of  grade  scale  is  perfectly  arbitrary.  Except  for  minor  differences 
in  the  statistical  values  obtained,  the  unique  frequency  curve  is  inde- 
pendent of  any  particular  grade  scale,  whether  equal  or  unequal  in  class 
interval. 


MODERN     GRADE     SCALES 

Uddcn  grade  scale.  The  first  true  geometric  scale  for  soils  or  sedi- 
ments, as  far  as  the  authors  are  aware,  was  introduced  by  Udden  -  in 
1898.  In  choosing  his  grade  limits,  Udden  mentioned  his  indebtedness 
to  soil  scientists  but  departed  from  their  choice  of  grade  scale  because 
of  the  absence  of  a  fixed  geometric  interval.  In  order  to  achieve  a  fixed 
ratio,  he  changed  from  the  values  i,  ^,  54>  i/io  mm.  to  the  values  i, 
/^,  /4.  H-  By  applying  the  same  ratio  of  ^  (or  2,  depending  upon  the 
sense  of  direction)  Udden  developed  his  original  grade  scale  of  twelve 
grades,  extending  from  16  mm.  diameter  to  1/256  mm.,  with  the  fol- 
lowing limiting  diameters:  16,  8,  4,  2,  i,  3/2,  %,  ^i,  1/16,  1/32,  1/64, 
1/128,  1/256  mm.  Later,  in  1914,  Udden  extended  his  scale  in  both 
directions,  to  include  coarser  and  finer  materials.^  The  introduction  of 
the  Udden  grade  scale  marked  the  beginning  of  the  modern  period  of 
grade  scale  development,  although  even  to-day  there  are  grade  scales  in 
wide  use  which  do  not  follow  the  principle  of  strict  geometric  intervals. 

The  concept  of  a  geometrical  grade  scale  should  be  made  explicit  for  the 
non-mathematical  reader.  A  geometric  series  is  defined  as  a  progression  of 
numbers  such  that  there  is  a  fixed  ratio  between  successive  elements  in  the 

1  Among  papers  on  the  subject  may  be  listed  the  following:  A.  Atterberg,  Die 
mechanische  Bodenanalyse  und  die  Klassifikation  der  Mincralboden  Schwedens :  Int. 
Mitt,  fiir  Bodcnkiindc,  vol.  2,  pp.  312-342,  1912.  C.  W.  Correns,  Grundsatzliches  zur 
Darstellung  der  Korngrossenverteilung :  Ccntralbl.  f.  Min.,  GcoL,  u.  Pallion.,  Abt. 
A.,  pp.  321-331,  1934.  G.  Fischer,  Gedanken  zur  Gestcinssystematik :  Jahrb.  d. 
Preuss.  Gcol.  Landcsanst.,  vol.  54,  pp.  553-584,  1933.  C.  K.  Wentworth,  Fundamental 
Umits  to  the  sizes  of  clastic  grains :  .Science,  vol.  77,  pp.  633-634,  1933. 

-J.  A.  Udden,  Mechanical  composition  of  wind  deposits:  Augiistana  Library  Pub- 
lications, no.  I,  1898. 

3  J.  A.  Udden,  Mechanical  composition  of  clastic  sediments :  Geol.  Soc.  America, 
Bulletin,  vol.  25,  pp.  655-744,  1914. 


78        SEDIMENTARY  PETROGRAPHY 

series.  Thus  the  series  i,  j^,  ^,  ^  is  such  a  series  because  each  number  is 
one-half  as  large  as  the  preceding  one,  so  that  any  successive  number  may 
be  found  by  multiplying  its  predecessor  by  Yz.  Any  series  of  numbers  may 
readily  be  tested  for  a  fixed  ratio  by  dividing  any  term  by  its  successor.  If  the 
quotient  is  the  same  for  all  pairs,  the  series  is  geometric.  In  the  series  i,  Y^, 
y^,  i/io,  the  ratio  i :  J4  =  2,  but  the  ratio  J4  •  i/io  =  2.5,  so  the  ratio  is  not 
fixed.  In  some  cases  the  geometric  nature  of  the  series  is  not  immediately 
apparent  by  inspection,  as  in  the  series  i.ooo,  0.707,  0.500,  0.354,  0.250.  A  test 
indicates  that  the  ratio  is  fixed,  however :  i :  0.707  =  0.707  :  0.500  =  0.500 :  0.354 
=  0.354:0.250=  1.414.  The  number  1.414  is  the  square  root  of  2,  and  these 
numbers  are  on  the  grade  scale  based  on  V2.  Another  simple  test  for  a 
geometric  series  is  that  the  logarithms  of  the  numbers  to  any  base  form  an 
arithmetic  series,  i.e.,  a  series  of  numbers  differing  by  a  fixed  amount. 

Hopkins  grade  scale.  In  1899,  the  year  following  Udden's  work, 
Hopkins  ^  pleaded  for  a  scientific  basis  for  the  division  of  particles  into 
grades,  suggesting  a  true  geometric  grade  scale  based  on  \/iO-  Hop- 
kins's grade  scale  was  not  adopted  by  the  United  States  Bureau  of 
Soils ;  -  nevertheless,  the  scale  forms  an  excellent  basis  for  mechanical 
analysis. 

Bureau  of  Soils  grade  scale.  The  grade  scale  used  by  the  United  States 
Bureau  of  Soils  is  shown  in  Table  4. 

Table  4 
Size  Classification  of  United  States  Bureau  of  Soils 


Grade  Limits 
{Diameters) 

Name 

2-1  mm 

Gravel 
Coarse  sand 

Medium  sand 

%-i/io  mm 

Fine  sand 

1/10-1/20  mm 

Verv  fine  sand 

1/20-1/200  mm 

Silt 

Below  i/'oo  mm 

Clav 

Atterherg  grade  scale.  In  1905  Atterberg  -  advanced  the  subject  by 
seeking  for  fundamental  physical  properties  as  a  basis  for  erecting  a 
grade  scale.  His  class  intervals  were  based  on  the  unit  value  2  mm.  and 

1  C.  G.  Hopkins,  A  plea  for  a  scientific  basis  for  the  division  of  soil  particles  in 
mechanical  analysis :  U.  S.  Dept.  Agric,  Dept.  Chetti.,  Bull.  j6,  pp.  64-66,  1899. 

-  Briggs,  Martin,  and  Pearce,  loc.  cit. 

3  A.  Atterberg,  Die  rationelle  Klassifikation  der  Sande  und  Kiese :  Chem.  Zcitung, 
vol.  29,  pp.  195-198,  1905. 


GRADE  SCALES  79 

involved  a  fixed  ratio  of  10  for  each  successive  grade,  yielding 
the  limiting  diameter  200.  20,  2.0,  0.2,  etc.  Each  of  these  major  grades 
was  divided  into  two  subgrades,  chosen  at  the  geometric  mean  of 
the  grade  limits.  Thus,  the  division  between  the  20  and  2  mm.  limits 
was  found  by  taking  the  square  root  of  the  product  of  the  grade 
limits:  20  X  2  =  40;  and  V  4°  =  6.32.  The  value  6.32  was  rounded 
otT  to  6.00  for  convenience.  This  rounding-off  process  destroyed  the 
geometric  simplicity  of  the  subgrades  but  did  not  affect  the  funda- 
mental geometric  nature  of  the  main  classes.  Correns/  in  discussing 
Atterberg's  scale,  pointed  out  that  the  subgrades,  as  well  as  the  main 
classes,  should  be  kept  as  geometric  intervals.  Atterberg's  scale,  with 
the  descriptive  names  applying  to  each  principal  grade,  is  shown  in 
Table  5. 

Table  5 
Atterberg's  Size  Classification 


Grade  Limits 
(Diameters) 

Name 

Blocks 

Cobbles 

Pebbles 

Silt 

Clay 

In  choosing  his  grade  limits,  Atterberg  observed  that  sand  coarser  than 
2  mm.  diameter  does  not  hold  water,  whereas  sand  with  smaller  grains  does 
to  some  extent,  depending  upon  capillarity.  The  next  significant  boundary 
was  found  to  be  0.2  mm.,  where  a  distinction  was  drawn  between  truly  "wet 
sand"  and  relatively  dry  sand.  Another  change  was  noted  at  0.02  mm.,  below 
which  the  individual  grains  could  not  be  seen  with  the  unaided  eye,  and  in 
material  finer  than  which  root-hairs  were  not  able  to  penetrate  the  pores.  At 
the  next  grade  limit,  0.002  mm.,  Atterberg  pointed  out  that  Brownian  move- 
ment began.  This  correlation  of  physical  properties  with  critical  grain 
diameters  is  the  outstanding  characteristic  of  Atterberg's  work,  and  his  grade 
scale  has  been  widely  adopted  by  European  workers  with  both  soils  and 
sediments.  In  1927  the  grade  scale  was  adopted  by  the  International  Com- 
mission on  Soil  Science  as  the  standard  for  all  soil  analyses.  The  United  States 
Bureau  of  Soils,  however,  did  not  adopt  the  scale. 

iC.  W.  Correns,  GrundsatzHches  zur  Darstellunc:  der  Korngossenverteilung- : 
Cciitr.  f.  Min.,  Abt.  A,  pp.  321-331,  1934. 


8o 


SEDIMENTARY  PETROGRAPHY 


Wentzi'orth  grade  scale.  In  America,  sedimentary  petrologists  favor 
the  Udden  grade  scale,  as  it  was  modified  in  1922  by  Wentworth.^ 
WentAvorth  compared  the  usage  of  such  terms  as  cobble,  coarse  sattd, 
and  the  hke,  and  on  the  basis  of  the  usages  he  modified  and  extended 
Udden's  scale,  retaining,  however,  the  geometric  interval  introduced  in 
1898.  This  grade  scale,  justly  called  the  \\'entworth  scale,  has  been 
adopted  by  practically  all  American  workers.  The  full  grade  scale  is 

Table  6 
Wextworth's  Size  Classification 


Grade  Limits 
(Diameters) 


Name 


Above  256  mm.  . 

256-64  mm 

64-4  mm 

4-2  mm 

2-1  mm 

1-J/2  mm 

^2-%  mm 

%-}i  mm 

^-1/16  mm.    .  . 
1/16-1/256  mm. 
Below  1/256  mm. 


Boulder 

Cobble 

Pebble 

Granule 

Very  coarse  sand 

Coarse  sand 

Medium  sand 

Fine  sand 

Verv   fine   sand 

Silt' 

Clav 


given  in  Table  6.  In  1933  Wentvvorth  -  examined  the  limits  of  his  grades 
in  terms  of  the  physical  properties  involved  in  grain  transportation.  He 
showed  that  given  class  Hmits  in  the  grade  scale,  far  from  being  ar- 
bitrar}-,  agreed  well  with  certain  distinctions  between  suspension  and 
traction  loads. 

Emphasis  has  been  placed  here  on  the  Atterberg  and  Wentworth 
scales,  largely  because  there  is  an  increasing  tendency  for  workers  in 
sediments  to  use  one  or  the  other  of  them.  It  will  be  shown  later  that 
the  rivalr}-  between  the  grade  scales  is  more  apparent  than  real  and  that 
from  the  point  of  view  of  statistical  analysis  either  scale  is  equally 
convenient.  It  is  not  to  be  assumed,  however,  that  these  two  scales  are 
the  only  ones  that  have  been  entertained  by  analysts.  On  the  contrary, 
a  great  many  grade  scales  have  been  proposed  and  used  for  soils  and 


1  C.  K.  Wentworth,  A  scale  of  grade  and  class  terms  for  clastic  sediments :  Jour. 
Geology,  vol.  30,  pp.  377-392,  1922. 
-  C.  K.  Kentworth,  loc.  cit.,  1933. 


GRADE  SCALES 


8i 


sediments,  and  some  of  these  are  discussed  and  compared  by  Fischer  ^ 
and  Zingg.-  To  a  large  extent  these  grade  scales  are  similar  to  that  of 
the  Bureau  of  Soils  or  to  the  Atterberg  scale,  \\-ith  more  grades  or  with 
slightly  different  limits  and  class  names.  The  Wentworth  scale,  also,  has 
been  modified,  largely  from  the  point  of  view  of  decreasing  the  class 
interval  by  using  the  ratio  \/Tor  ^JXinstead  of  2. 

Table  7 
A.ST.M.  Sieve  Scale 


Mesh 


Opening 
(;/im!) 


6 

8 
10 
12 

14 
16 
18 
20 

25 
30 
Z-:> 
40 

45 

;o 

60 

70 

80 

100 

120 

140 

170 

200 

230 

270 


4.00 
3-36 
2.83 
2.38 
2.00 
1.68 
1.41 
1.19 
i.oo 
0.84 
0.71 

0.59 
0.50 
0.42 

0.35 

0.297 

0.250 

0.210 

0.177 

0.149 

0.125 
0.105 
0.088 
0.074 

0.062 
0.053 

0.044 


Engineering  grade  scales.  In  addition  to  tlie  types  of  grade  scales  used  by 
sedimentary  petrologists  and  soil  scientists,  there  is  a  wide  variety  of  grade 


1  G.  Fischer,  loc.  cit.,  1933. 

-  Th.  Zingg,  Beitrag  zur  Schotteranalyse :  Sclizi 
pp.  39-140,  1935- 


ic.  Miti.  u.  Pet.  Mitt.,  vol.  15, 


82 


SEDIMENTARY  PETROGRAPHY 


scales  based  on  the  mesh  system,  which  are  extensively  used  in  engineering 
and  commercial  testing.  Among  the  best  known  of  these  are  the  scale  adopted 
by  the  American  Society  for  Testing  Materials.^  The  scale  was  based  on  the 

fixed  ratio  v  2.  Table  7  shows  the  relation  bet>veen  the  sieve  openings  in 
millimeters  and  the  corresponding  mesh  number.  It  may  be  noticed  that  every 
fourth  value  in  this  table  agrees  with  the  Wentvvorth  class  limits,  starting 
with  4.00  mm. 

Another  well-known  system  based  on  mesh  was  that  adopted  in  1907  by  the 
Institute  of  Mining  and  Metallurgy  of  England.^  There  is  no  fixed  ratio  be- 
tween the  successive  sieve  openings  in  the  I.M.M.  series,  and  hence  it  is  not  a 
true  geometric  scale.  Table  8  lists  the  mesh  numbers  and  corresponding  open- 
ings in  millimeters  for  this  series.  The  I.M.M.  series  has  been  used  widely  in 
England  for  mechanical  analysis  of  sediments. 

Table  8 

I.M.M.  Sieve  Scale 


Mesh 


Opening 
(mm.) 


3 
8 
10 
12 
16 
20 
25 
30 
35 
40 
50 
60 
70 
80 
90 
100 
120 
150 
200 


2.540 

1-574 
1.270 
1.056 
0.792 

0.635 
0.508 
0.421 
0.416 

0.317 
0.254 
0.21 1 
0.180 

0.157 
0.139 
0.127 
0.107 
0.084 
0.063 


A  criticism  which  may  in  general  be  directed  against  the  mesh  system  of 
nomenclature  is  that,  imless  the  openings  in  millimeters  or  some  other  vmit 
are  also  given,  it  is  not  possible  to  convert  the  values  to  their  metrical  equiv- 
alents. A  comparison  of  the  openings  corresponding  to  the  various  meshes  in 

1  American  Societ>-  for  Testing  materials,  A.S.T.M.  Standards  (1930),  part  2, 
p.  1 120. 

-  Original  reference  not  available.  Data  from  W.  S.  Tyler  Company,  Catalog  53, 
p.  14. 


GRADE  SCALES  83 

Tables  7  and  8  will  indicate  that  they  differ  widely  enouj^h  to  be  sif^nificant. 
Furthermore,  when  sieves  are  purchased  merely  in  terms  of  so  many  meshes 
to  the  inch,  without  specify  ins?  a  particular  standard  set,  there  may  be  no 
definite  relation  between  openinj^  and  mesh,  inasmuch  as  the  numl)er  of  meshes 
to  the  inch  may  be  fixed,  but  the  openings  will  vary  according  to  the  diameter 
of  the  wire  or  cloth  used  in  weaving  the  sieve. 

Robinson  grade  scale.  In  the  preceding  discussion,  grade  scales  have 
been  based  on  the  diameters  of  the  particles  being  classified.  There  are 
other  types  of  grade  scale,  however.  Robinson,^  for  example,  considers 
expressions  of  size  of  irregular  particles  to  be  unsatisfactory  in  reporting 
mechanical  analyses,  and  he  recommends  the  direct  use  of  settling  vel- 
ocities or  their  logarithms.  By  thus  expressing  size  in  terms  of  settling 
velocities,  and  the  latter  in  terms  of  their  logarithms,  Robinson  intro- 
duced the  first  logarithmic  transformation  scale.  The  great  advantage  of 
such  scales  is  that  they  convert  unequal  geometrical  intervals  into  equal 
arithmetical  intervals  and,  with  a  suitable  choice  of  logarithms,  introduce 
integers  instead  of  fractions  as  the  grade  limits.  Such  transformations 
are,  of  course,  more  appropriate  for  true  geometric  scales  than  for  ir- 
regular unequal-interval  scales,  because  the  latter  will  not  yield  an  arith- 
metic series  of  integers. 

Table  9 

Rubey's  Size  Classification 
Based  on  Settling  Velocities 


Grade 

Settling  Velocity 
{in   microns/sec.) 

Very  fine  sand   

>  3,840 
960-3,840 
240-960 
60-240 
15-60 
3-75-15 
0-9375-3-75 
<  0.9375 

Coarse  silt   

Medium  silt  

Fine  silt 

Very  fine  silt 

Coarse  clay   

Medium  clay   . 

Fine  clay ... 

Ruhey  grade  scale.  Rubcy  ^  followed  Robinson  in  the  use  of  settling 
velocities  instead   of   diameters  directly,   but  carried   the   work   to  the 

1  G.  W.  Robinson,  The  forms  of  mechanical  composition  curves  of  soils,  clays, 
and  other  granular  substances:  Jour.  Aqric.  .Sci.,  vol.  14,  pp.  626-633,  1924. 

-  W.  W.  Rubcy,  Lithologic  studies  of  fine-grained  Upper  Cretaceous  sedimentary 
rocks  of  the  Black  Hills  region:  U.  S.  Gcol.  Siinrv,  Prof.  Paper  165A,  pp.  1-54, 
1930. 


84 


SEDIMENTARY  PETROGRAPHY 


development  of  an  actual  grade  scale  based  on  velocities.  Rubey  plotted 
the  settling  velocities  and  diameters  of  particles  on  double  log  paper  and 
then  drew  in  the  size  limits  according  to  Atterberg.  Wentworth,  and 
Udden,  on  the  same  scale.  A  straight  line,  based  on  average  settling 
velocity  and  size  limit,  was  drawn  through  the  graph,  yielding  Rubey 's 
grade  scale,  which  is  shown  in  Table  9. 

It  will  be  noticed  that  the  Hmiting  velocities  between  the  successive 
size  fractions  in  Rubey 's  scale  decrease  by  the  constant  ratio  i  to  4.  In 
terms  of  diameters  this  means  that  the  ratios  are  i  to  2,  because  by 
Stokes's  law  (Chapter  5)  it  may  be  shown  that  the  settling  velocity  varies 
as  the  square  of  the  diameter.  Thus  Rubey's  grade  scale  conforms  to 
the  principles  of  fixed  geometric  intervals. 

Table  id 
Phi  a.vd  Zeta  Grade  Scales 


U'cnin'orth 
Grades 

* 

Atterberg 
Grades 

f 

~5 
-4 
-3 
—2 

—  I 
0 
+  1 
+2 
+3 
+4 
+  5 
+6 
+  7 
+8 
+9 
+  10 

-3 

—2 

4  mm 

—  I 

0 

0  2  mm 

Yi  mm 

+  1 

0  02  mm           ... 

1/32  mm 

+2 

1/64  mm               

0  002  mm             .  . 

1/128  mm           

+3 

1/256  mm     

0  0002  mm     

1/5 1 2  mm     

+4 

1/1024  mm     

Phi  and  zcta  scales.  In  1934  Krumbein  ^  applied  a  logarithmic  trans- 
formation equation  to  the  \\'entworth  grade  scale  and  obtained  a  "phi 
scale"  which  had  integers  for  the  class  limits  and  increased  with  de- 
creasing grain  size.  This  grade  scale  was  developed  specifically  as  a 
statistical  device  to  permit  the  direct  application  of  conventional  statistical 

1  W.  C.  Krumbein,  Size  frequency  distributions  of  sediments :  Jour.  Sed.  Petrol- 
ogy, vol.  4,  pp.  65-77,  1934-  See  also  The  application  of  logarithmic  moments  to  size 
frequency  distributions  of  sediments :  Jour.  Scd.  Petrology,  vol.  6,  pp.  35-47,  1936. 


GRADE  SCALES  85 

practices  to  sedimentary  data.  More  recently,  Krumbein  ^  also  applied 
negative  logarithms  to  the  Atterberg  scale  and  obtained  a  "zeta  scale" 
with  properties  similar  to  the  phi  scale.  The  theory  on  which  these  trans- 
formed scales  is  based  is  that  any  true  geometric  scale  may  be  con- 
verted to  an  equivalent  scale  with  arithmetic  intervals  if  logarithms  of 
the  scale  limits  are  substituted  for  the  diameter  values.  Krumbein  chose 
the  transformation  equation  </>=  —  log2  i,  (where  |  is  the  diameter 
in  millimeters)  for  the  Wentworth  scale,  and  C  =  0.301'—  logio  ^  for  the 
Atterberg  scale.  The  resulting  phi  and  zeta  scales  are  shown  with  their 
equivalent  Wentworth  and  Atterberg  scales  in  Table  10. 

PROBLEMS     OF     UNEQUAL     CLASS     INTERVALS 

It  may  be  noted  that  without  exception  the  grade  scales  proposed  for 
soils  and  sediments  have  been  based  on  unequal  class  intervals.  To  some 
extent  this  is  due  to  necessity,  inasmuch  as  the  range  of  sizes  in  sedi- 
mentary particles,  even  within  the  same  sediment  in  some  cases,  is  so 
great  that  equal  intervals  are  a  practical  impossibility.  Thus,  sandy  shales 
may  range  in  particle  size  from  i  mm.  to  less  than  0.00 1  mm.  in  di- 
ameter. If  an  interval  such  as  o.i  mm.  were  used  for  the  classes,  the 
result  would  be,  perhaps,  that  more  than  half  the  distribution  would  be 
in  the  smallest  grade.  To  give  full  significance  to  the  smaller  sizes,  it 
would  be  necessary  to  use  a  class  interval  of  G.ooi  mm.  Two  practical 
difficulties  arise ;  one  is  that  some  thousand  classes  would  be  necessary, 
and  the  other  is  that  it  is  virtually  impossible  to  distinguish  between 
grains  of  0.999  ♦i"^  i.ooo  mm.  diameter,  esj:>ecially  when  the  particles 
are  not  regular  geometrical  solids.  Finally,  ditTerences  of  o.ooi  mm. 
in  the  diameters  of  large  particles  are  negligible,  whereas  the  difference 
between  particles  of  o.ooi  and  0.002  mm.  diameter  may  be  significant. 

The  obvious  conclusion  to  draw  from  these  observations  seems  to  be 
that  a  grade  scale  should  be  devised  such  that  each  grade  bears  a  fixed 
size  ratio  to  preceding  and  succeeding  grades.  This  is  the  principle  in- 
troduced by  Udden  and  exemplified  by  the  Atterberg  and  Wentworth 
scales.  This  is  not  to  imply  that  unequal  interval  scales  are  not  satis- 
factory unless  they  have  a  fixed  ratio ;  on  the  contrary,  any  grade  scale 
is  satisfactory  for  descriptive  pur^xjses  if  it  is  mutually  agreed  upon  by 
a  sufficient  number  of  workers.  From  an  analytical  point  of  view,  on  the 
other  hand,  an  irregular  interval  in  the  grade  scale  may  interfere  with 

^  W.  C.  Krumbein.  Korngrosseneinteilungen  und  statistische  Analyse :  Ncucs 
Jahrb.  f.  Mm.,  etc.,  Beil.-Bd.  73,  Abt.  A,  pp.  137-150,  1937. 


86        SEDIMENTARY  PETROGRAPHY 

the  convenient  application  of  statistical  analysis  to  the  data.  The  recog- 
nition that  two  separate  and  distinct  functions  are  associated  with  any 
grade  scale  ^  has  not  been  sufficiently  emphasized  by  soil  scientists  and 
sedimentary  petrologists,  and  the  topic  deserves  detailed  discussion. 


FUNCTIONS     OF     GRADE      SCALES 

Descriptive  function.  The  first  and  perhaps  the  most  important  func- 
tion of  a  grade  scale  is  a  descriptive  function,  which  serves  to  place 
nomenclature  and  terminology  on  a  uniform  basis.  If  one  reads  the 
term  coarse  sand  in  a  rejxDrt,  he  would  prefer  to  understand  by  the 
term  exactly  what  the  writer  intended  to  convey.  As  long  as  there  is  no 
uniform  terminology,  each  writer  coins  his  own  meanings,  which  may 
or  may  not  be  precisely  defined.  If,  however,  the  reader  knows  that  the 
writer  is  using  the  Atterberg  classification,  he  may  understand  that 
material  having  a  range  of  sizes  from  2.0  mm.  to  0.6  mm.  diameter  is 
meant.  Likewise,  if  the  Wentworth  scale  is  being  used,  the  term  refers 
to  material  from  i  mm.  to  ^  mm.  in  diameter. 

Obviously,  no  "justification"  whatever  is  required  for  the  descriptive 
function  of  a  grade  scale.  The  particular  choice  of  such  terms  as  coarse 
sand,  fine  sand,  clay,  and  the  like  need  be  based  on  no  other  criterion 
than  mutual  agreement.  If  the  limits  chosen  for  each  grade  are  also 
related  to  the  physical  properties  of  sediments,  that  fact  may  be  taken 
as  an  added  advantage. 

Analytic  function.  In  addition  to  the  use  of  grade  scales  to  establish 
uniformity  of  terminology,  the  classes  or  grades  are  used  as  units  in 
performing  various  kinds  of  analyses  on  the  sediment.  The  classes  are 
used,  for  example,  in  determining  the  mechanical  composition  of  the 
sediment,  and  in  addition  they  may  be  used  as  units  during  statistical 
analysis.  It  is  in  connection  with  these  analytic  functions  of  grade  scales 
that  most  of  the  confusion  arose  regarding  the  merits  of  one  or  another 
of  the  proposed  scales.  The  recognition  of  the  fact  that  histograms  vary 
in  form  according  to  the  grade  scale  used  has  led  various  writers  -  to 
the  conclusion  that  some  single  scale  should  be  used  for  all  analyses,  so 
that  the  unfortunate  variation  of  the  histogram  could  be  avoided. 

Unfortunately,  there  can  be  no  single  "correct"  grade  scale  for  all 
mechanical  analyses,  because  the  concept  of  discrete  grades  is  absent 

iW.  C.  Krumbein,  loc.  cit.,  1934- 

2  See,  for  example,  L.  Dryden,  Cumulative  curves  and  histograms :  Am.  Jour.  Sci., 
vol.  27,  pp.  146-147,  1934- 


GRADE  SCALES  87 

from  any  continuous  size  range.  That  is,  where  the  sizes  change  by 
infinitesimals  along  the  entire  range  of  diameters,  one  has  a  continuous 
function,  in  which  any  class  interval  whatever  is  purely  artificial.  From 
the  nature  of  the  particles  comprising  sediments  it  is  clear  that,  with  few 
exceptions,  the  diameters  vary  by  infinitesimals  along  the  entire  range 
of  sizes  present,  rather  than  by  abrupt  steps  from  one  size  to  the  next. 
In  studying  these  distributions,  however,  convenient  units  are  desirable ; 
but  the  units  themselves  are  quite  arbitrary,  and  for  practical  purix)ses 
they  need  not  be  related  to  the  descriptive  aspects  of  the  grade  scale. 
If  the  grade  scale  is  flexible  enough  to  permit  its  use  both  for  descriptive 
and  for  analytical  purposes,  a  strong  advantage  of  convenience  is 
gained.  Both  the  Atterberg  and  the  Wentworth  grade  scales  have  such 
flexibility. 

The  recognition  that  a  sediment  is  really  a  continuous  size  frequency 
distribution  of  particles,  without  any  implication  of  a  "natural"  group- 
ing of  the  material  into  classes,  frees  mechanical  analysis  from  the  con- 
fines of  any  single  grade  scale.  From  this  point  of  view,  the  class  in- 
tervals used  in  the  actual  analysis  may  be  so  chosen  that  they  bring  out 
most  clearly  the  characteristics  of  the  distribution  itself.  This  continuous 
size  frequency  distribution  may  then  be  described  in  conventional  sta- 
tistical terms,  which  themselves  may  be  related  to  any  of  a  number  of 
descriptive  grade  scales.  Two  common  statistical  methods  of  analysis 
are  available,  based  either  on  the  moments  of  the  distribution  or  on  the 
quartilcs  and  median  (see  Chapters  8  and  9).  In  the  moment  method 
the  class  intervals  are  preferably  chosen  on  a  fixed  geometric  ratio,  to 
facilitate  computation. When  quartile  measures  are  used,  however,  it  is 
immaterial  what  class  intervals  are  used,  or  even  whether  or  not  they 
are  based  on  a  fixed  geometric  ratio.  This  high  degree  of  independence 
with  quartile  measures  depends  on  the  fact  that  purely  graphic  methods 
are  used,  so  that  the  choice  of  class  interval  may  be  determined  by  the 
preciseness  with  which  the  analyst  wishes  to  construct  his  curves. 

Although  there  is  a  growing  recognition  of  the  continuous  nature  of 
most  sedimentary  data,  many  current  analyses  are  still  conducted  in 
terms  of  the  descriptive  units  of  which  the  sediment  is  comix)sed.  That 
is,  instead  of  considering  the  sediment  as  a  whole,  one  may  be  interested 
in  the  percentages  of  specific  grades  present.  Soil  scientists,  for  example, 
are  more  interested  in  the  amounts  of  sand  or  colloid  present  in  a  soil 
than  in  the  nature  of  the  distribution  as  a  whole.  Likewise  in  commercial 
testing,  specified  amounts  of  sand,  silt,  or  clay  are  desired,  and  the 
analyses  are  directed  toward  testing  the  material  with  this  in  view.  In 


88        SEDIMENTARY  PETROGRAPHY 

such  cases  the  obvious  technique  is  to  analyze  the  material  with  the 
class  units  of  the  descriptive  grade  scale,  so  that  results  are  obtained 
directly  in  terms  of  the  grade  scale  being  used. 

It  is  to  be  explicitly  pointed  out,  however,  that  when  the  mechanical 
analysis  data  are  secured  in  terms  of  the  continuous  size  frequency  dis- 
tribution, it  is  always  possible  to  express  the  analysis  in  terms  of  specific 
grades  on  any  grade  scale,  whereas  when  a  fixed  grade  scale  is  used 
indiscriminately  on  all  analyses,  it  may  not  be  convenient  to  determine 
the  nature  of  the  continuous  distribution,  especially  if  the  fixed  scale  has 
too  few  points  along  it,  or  if  the  intervals  are  not  based  on  a  fixed  ratio. 


CHOICE     OF     A     GRADE     SCALE 

The  wide  choice  of  viewpoint  possible  in  the  analysis  of  sediments 
and  other  particulate  substances  suggests  that  for  general  purposes  it 
may  be  preferable  to  choose  some  analytical  class  interval  which  would 
furnish  data  adequate  for  all  the  purposes  outlined.  The  authors  believe 
that  a  grade  scale  based  on  a  fixed  geometric  interval  and  flexible  enough 
to  afford  a  number  of  relatively  small  subgrades  is  to  be  preferred. 
Either  the  Atterberg  or  the  Wentworth  scale  is  suitable  as  a  base,  and 
it  is  immaterial  which  is  used,  because  the  results  obtained  from  the  one 
scale  may  readily  be  expressed  in  terms  of  the  other,  if  desired.  In  this 
connection,  Hopkins's  grade  scale,  mentioned  earlier,  deserves  consid- 
eration because  it  also  aflfords  a  convenient  basis  for  analysis.  The 
possibility  of  converting  values  from  one  descriptive  scale  to  another 
depends  simply  on  the  fact  that  any  continuous  distribution  is  inde- 
pendent of  the  class  intervals  used  in  analyzing  it,  and,  within  the  rela- 
tively small  range  of  experimental  error  due  to  particular  grade  limits, 
the  characteristics  of  the  sample  are  constant. 

It  is  in  connection  with  the  statistical  manipulation  of  sedimentary  data  and 
with  the  conversion  of  the  data  from  one  descriptive  scale  to  another  that  the 
logaritlmiic  type  of  grade  scale  is  most  useful.  In  the  Wentworth  grade  scale, 
the  use  of  a  logarithmic  notation,  such  as  the  phi  scale  (Table  lo),  yields 
integers  which  mark  the  limits  of  each  grade.  If  an  analysis  based  on  the 
Wentworth  limits  direcUy  does  not  yield  adequate  data  for  the  complete  study 
of  the  sediment,  the  scale  limits  may  be  changed  to  half-phi  units,  which 
yields  double  the  number  of  experimentally  determined  points.  The  analytical 
scale,  however,  still  remains  arithmetic,  except  that  the  phi  intervals  change 
from  I,  2,  3,  ...to  i,  1.5,  2.0,  2.5,  3.0,  ...In  short,  the  substitution  of  the  V2 
grade  scale  does  not  affect  the  convenience  of  the  phi  notation.  Likewise,  even 
the  use  of  the  *V2  scale  merely  results  in  the  units  along  the  phi  scale  be- 


GRADE  SCALES  89 

coming  0/4,  so  that  the  series,  still  arithmetic,  is  i.o,  1.25,  1.50,  1.75,  2.0, ...  In 
a  similar  manner  the  limits  of  the  Atterherg  scale  may  be  too  large  for  con- 
venient analysis,  and  by  choosing  additional  points  at  half-  or  quarter-zeta 
values  (Table  11)  any  detail  whatever  may  be  brought  out,  limited  only  by 
imperfections  of  technique. 


Table  ii 

Logarithmic  Grade  Scales  Based  on 
f/4  AND  </>/4  Class  Intervals 


atterberg  scale 

WENTWORTH   SCALE 

Grades 
(mm.) 

f 

Grades 
(vim.) 

V  2-Scale 
(null.) 

0 

—  1. 00 

-0.75 

—0.50 

-0.25 

0.00 

+0.25 
+0.50 
+0.75 
+  1.00 
+  1.25 
+  1.50 

+  1.75 
+2.00 

2 

I 

'A 

M 

H 

Vm 

2.00 
1.69 
I.4I 

i.ig 
1. 00 
0.84 
0.71 

0.59 
0.50 
0.420 

0.351 
0.297 
0.250 
0.210 
0.177 
0.149 
0.125 
0.105 
0.088 
0.074 
0.06.? 

II. 2t;    

-0.75 
—0.50 
-0.25 

6.^2    

-tee    

+  0.25 
+  0.50 

+  0.75 
+  1  00 

2.000    

1.12c;    

+1.25 

+  1.50 

+  1-75 
+2.00 

+2.25 
+2.50 

+2.75 
+  3.00 

+3-25 
+350 
+3.75 
+4.00 

0.6'?2      

o.ic:6    

0.200    

0.112    

0.06^    

co^q    

0.020    

The  conversion  of  statistical  values  from  one  descriptive  scale  to  another 
is  a  relatively  simple  matter  as  long  as  the  several  scales  are  based  on  geo- 
metric intervals  and  expres.sed  as  logarithms.  The  general  approach  may  be 
illustrated  in  terms  of  the  Wentwortii  and  Atterberg  scales,  expressed  in  the 
phi  and  zeta  notations  respectively.  Suppose  a  statistical  measure  of  average 
size  is  computed  for  a  sediment  and  expressed  in  phi  terms.  It  is  desired  to 


90 


SEDIMENTARY  PETROGRAPHY 


convert  this  measure  to  its  equivalent  value  on  the  Atterberg  scale,  as  ex- 
pressed in  zeta  terms.  The  phi  scale  is  based  on  the  fact  that  any  diameter 
value  may  be  expressed  in  terms  of  the  Wentworth  scale  as  €  =  2~<P,  where 
^  is  the  diameter  in  millimeters  and  </>  is  a  value  along  the  phi  scale.  By  taking 
logs  of  this  expression,  one  obtains  0  =  —  logo  I.  In  like  manner,  any  value  on 
the  Atterberg  scale  may  be  expressed  as  I  =  2  X  10— i',  where  I  is  the  diameter 
in  millimeters,  as  before,  and  <:  is  a  value  on  the  zeta  scale.  When  logs  are 
taken  of  this  last  expression,  there  results  f  =  0.301 — logiol.  To  convert 
values  from  one  scale  to  the  other,  use  is  made  of  the  general  logarithmic 
equation  for  change  of  base :  log^o  ^  —  logio  2  logo  I.  For  the  factor  logo  I  is 
substituted  the  expression  — 0,  and  for  the  term  log^o  I  is  substituted  the 
expression  0.301  —  '(.  These  substitutions  yield  the  final  transformation  equa- 
tion, f  =  0.301  (0  +  i).  By  means  of  this  last  equation  any  scale  value  in  the 
zeta  notation  may  be  converted  to  its  equivalent  in  the  phi  notation,  and  vice 
versa.  The  net  effect  is  that  analyses  may  be  conducted  on  any  convenient  scale 
with  true  geometric  intervals,  and,  by  the  choice  of  appropriate  transformation 
equations,  the  statistical  values  may  be  expressed  in  terms  of  any  other  geo- 
metric scale.  The  relation  between  diameter,  (I)  and  tlie  <t>  and  f  scales  is 
shown  graphically  in  Figure  19. 

ZETA     SCALE — ► 

•' 0  1 


+-r 


10  9  8  7    6     5      4        3  2  1.0  9  8  7  -6    -5    -4        3 

II  I  II    II 

• DIAMETER       IN     MM 

J 1 \ I   [    I    I 


Fig.  19. — Relations  between  logarithmic  grade  scales  and  diameters  in  millimeters. 
The  "zeta  scale"  is  adapted  to  Atterberg  grades,  and  the  "phi  scale"  to  Wentworth 
grades. 

A  full  discussion  of  statistical  methods  available  for  sedimentary  data 
is  given  in  Chapters  8  and  9.  These  methods  are  illustrated  by  examples 
of  mechanical  analyses,  and  statistical  computations  based  on  the  use 
of  conventional  and  logarithmic  grade  scales  are  described.  The  principal 
purpose  of  the  present  chapter  has  been  to  lay  the  foundation  for 
mechanical  analysis  by  indicating  that  some  kind  of  grade  scale  is  im- 
portant in  the  analysis  of  sediments,  but  that  the  particular  choice  may 
depend  upon  the  convenience  with  which  it  may  be  used. 


CHAPTER   5 

PRINCIPLES  OF  .AIECHANICAL  ANALYSIS 

INTRODUCTION 

Mechanical  analysis  is  the  quantitative  expression  of  the  size  fre- 
quency distribution  of  particles  in  granular,  fragmental,  or  powdered 
material.  It  does  not  necessarily  involve  the  actual  separation  of  the 
substance  into  grade  sizes,  nor  does  it  require  unconsolidated  material. 

[Methods  of  mechanical  analysis  may  be  divided  into  two  broad  groups, 
the  modern  precision  methods  and  the  older  routines.  The  fundamental 
ditiferences  between  these  groups  are  first,  that  the  newer  methods  are 
underlain  by  a  single  mathematical  theory  of  sedimenting  systems  which 
unifies  the  field  and  coordinates  the  methods ;  second,  the  older  methods 
seek  to  separate  the  material  into  grade  sizes,  whereas  the  new  techniques 
do  not. 

All  methods  of  analysis  are  supported  by  several  underlying  princi- 
ples, such  as  the  settling  velocities  of  particles,  the  dispersion  and  coagu- 
lation of  suspensions,  and  theories  of  their  operation.  The  factors  in- 
volved in  dispersion  and  coagulation  were  discussed  in  Chapter  3  :  the 
present  chapter  will  concern  itself  with  the  remaining  principles  of 
mechanical  analysis. 

The  historical  development  of  methods  of  mechanical  analysis  is  a 
topic  which  deser\'es  consideration  by  workers  in  the  field,  but  limitations 
of  space  prevent  its  treatment  here.  The  interested  reader  is  referred  to 
Krumbein's  paper  ^  for  a  short  history  of  the  subject. 

CLASSIFICATION       OF      DISPERSE      SYSTEMS 

Suspensions  of  solids  in  liquids  are  called  disperse  systenis  when  the 
solid  is  so  thoroughly  distributed  that  the  individual  particles  may  no 
longer  be  of  primary  im^XDrtance  and  interest  is  focused  on  the  totality 
of  the  particles  as  a  system.  The  solid  constitutes  the  dispersed  pJiase, 

^  W.  C.  Krumbein,  A  history  of  the  principles  and  methods  of  mcclianical  analy- 
sis :  Jour.  Scd.  Petrology,  vol.  2,  pp.  89-124,  1932.  An  error  in  the  original  paper  is 
corrected  in  Jour.  Sed.  Petrology,  vol.  3,  p.  95,  1933. 

91 


92        SEDIMENTARY  PETROGRAPHY 

and  the  liquid  is  the  dispersion  medium.  Such  systems  are  divided  into 
monodispcrsc  and  polydispersc  systems,  dej)ending  upon  whether  the 
particles  are  all  of  the  same  size  or  of  various  sizes.  Both  tyi>es  of 
systems  may  be  classified  according  to  the  size  of  particles  present ; 
polydisperse  systems  may  of  course  belong  to  more  than  a  single  size 
classification. 

Coarse  disperse  systems.  Particles  larger  than  o.i  micron  ro.oooi  mm.)  in 
diameter  constitute  coarse  disperse  systems.  This  lower  limit  is  generally 
accepted  as  the  upper  limit  of  the  colloidal  state,  but  it  is  recognized  that  there 
is  a  transition  zone  in  which  particles  may  have  some  of  the  attributes  both 
of  coarse  systems  and  of  colloids.  This  transition  zone  may  extend  above  i 
micron. 

Coarse  disperse  systems  may  be  divided  into  three  groups:  fa)  macroscopic 
systems,  in  which  the  individual  grains  may  be  resolved  by  the  unaided  eye; 
this  group  includes  particles  larger  than  about  lo  microns  (o.oi  mm.)  diam- 
eter; (b)  microscopic  systems,  in  which  the  individual  grains  may  be  resolved 
with  a  compound  microscope ;  this  group  extends  down  to  about  0.2  micron 
(0.0002  mm.)  diameter;  and  (c)  the  nltramicroscopic  system,  in  which  par- 
ticles are  no  longer  seen  as  individuals  under  the  microscope. 

Colloidal  disperse  systems.  Colloidal  particles  range  in  size  from  about  o.l 
micron  to  i  mumu  (0.000,001  or  IQ— ^  mm.)  in  diameter.  Only  the  larger 
sizes  within  this  category  belong  even  to  the  ultramicroscopic  group.  The  col- 
loidal state  may  be  defined  as  that  state  in  which  the  dispersed  phase  is  so 
finely  divided  that  properties  depending  upon  the  surface  area  control  its 
behavior.  Such  phenomena  as  dispersion  and  coagulation  are  outstanding 
phenomena  among  colloids. 

Molecular  disperse  systems.  Disperse  systems  containing  only  particles 
smaller  than  about  i  mumu  are  true  solutions. 

It  should  be  understood  that  this  classification  of  disperse  systems  is 
arbitrary,  because  the  attributes  of  particles  show  complete  transitions 
from  one  state  to  another.  Especially  is  this  true  of  polydisperse  sys- 
tems, in  which  it  is  not  uncommon  to  find  a  range  of  sizes  from  very 
coarse  particles  through  colloids  and  into  soluble  material.  Mechanical 
analysis  is  concerned  mainly  with  the  first  two  types  of  disperse  systems, 
the  coarse  systems  and  colloids.  Methods  of  analysis  depend  very  largely 
upon  the  predominant  sizes,  or  the  range  of  sizes  in  the  material.  Present 
methods  of  analysis,  in  fact,  are  most  efifective  for  diameters  larger  than 
I  micron  (o.ooi  mm.),  so  that  essentially  it  is  in  connection  with  coarse 
disperse  systems  only  that  most  mechanical  analyses  are  conducted. 
Material  smaller  than  i  micron  is  frequently  grouped  into  a  single  size 
class. 

In  order  to  illustrate  the  influence  of  size  on  the  methods  of  analysis 


PRINCIPLES  OF  SIZE  ANALYSIS  93 

commonly  used,  the  following  outline  indicates  the  subdivisions  of  dis- 
I>erse  systems  and  the  analytical  methods  commonly  used  for  each : 

Coarse  disperse  systems.  Particles  larger  than  10  mm.:  direct  measurement 
by  macroscopic  methods,  sieving  methods. 

Particles  between  10  and  0.05  mm.  diameter:  sieving  methods,  direct  meas- 
urement by  microscopic  methods  in  part. 

Particles  between  0.05  and  o.ooi  mm.  diameter:  indirect  sedimentation 
methods  (pipette,  Oden  balance,  Wiegner  tube,  etc.),  but  in  some  cases  this 
group  is  subdivided  as  follows : 

0.05  to  o.oi  mm.  diameter:  rising  current  elutriation,  docantation  methods 
o.oi  to  o.ooi  mm.  diameter:  indirect  sedimentation  mediods 

Particles  between  o.ooi  and  o.oooi  mm.  diameter:  centrifugal  methotls. 
Colloidal  Disperse  systems.  Centrifuge,  ultramicroscope,  turbidity,  various 
optical  methods. 


CONCEPT     OF     SIZE     IN     IRREGULAR     SOLIDS 

If  all  s(m1s  or  sediments  were  composed  of  perfect  spheres,  a  dolinition 
of  size  would  be  simple.  The  fact  that  natural  materials  are  seldom 
regular  in  shape,  combined  with  the  fact  that  the  particles  composing  a 
given  mi.xture  may  range  widely  in  their  shapes,  gives  rise  to  a  problem 
wdiich  has  engaged  the  attention  of  numerous  workers.  In  some  cases 
the  definition  of  size  has  depended  upon  the  magnitude  of  the  particle: 
large  particles  that  could  be  conveniently  handled  were  defined  in  terms 
of  one  set  of  criteria,  and  smaller  particles  were  defined  on  entirely  other 
bases.  To  a  large  extent  definitions  of  size  have  been  based  on  the  most 
convenient  and  immediately  applicable  maimer  of  obtaining  a  number 
which  could  be  used  for  the  purpose  at  hand. 

One  of  the  most  thorough  investigations  of  the  concept  of  size  of 
irregular  particles  has  been  made  by  Wadell,^  and  the  following  dis- 
cussion is  largely  based  on  his  work.  Wadell's  thesis  is  that  "size"  of  a 
particle  is  best  expressed  by  its  simple  volume  value,  because  the  volume 
is  independent  of  its  shape.  The  use  of  long,  intermediate,  and  short 
diameters,  or  of  the  arithmetic  or  geometric  mean  of  these,  is,  according 
to  Wadell,  relatively  meaningless  as  a  definition  of  size.-  It  is  quite 
possible,  for  example,  that  the  mean  of  three  diameters  of  an  irregular 
solid  may  be  numerically  equal  to  the  diameter  of  a  given  sphere,  and 
yet  the  volumes  will  be  entirely  diiYerent.  The  term  diameter  has  a 

1  H.  Wadell,  Volume,  shape,  and  roundness  of  rock  particles :  Jour.  Ccohniy,  vol. 
40,  pp.  443-451,  1932. 

-See,  however,  the  discussion  of  definitions  on  page  127  of  this  chapter,  under 
microscopic  methods  of  analysis. 


94        SEDIMENTARY  PETROGRAPHY 

definite  significance  only  in  connection  with  a  sphere ;  in  that  case  diam- 
eter and  size  are  synonymous,  and  calculations  of  surface  area  or  volume 
may  readily  be  made  from  the  value  of  the  diameter.  For  any  other 
£hai:>e,  however,  the  term  diameter  will  not  serve  these  purposes  and 
hence  cannot  be  used  for  any  fundamental  investigations  of  physical 
properties. 

This  line  of  reasoning  impelled  ^^■adcll  to  define  the  size  of  irregular 
solids  in  terms  of  a  true  nominal  diameter,  which  is  equal  to  the  diameter 
of  a  sphere  of  the  same  volume  as  the  particle.  The  true  nominal  diam- 
eter has  become  a  concept  of  great  significance  in  sedimentary  work, 
not  only  because  of  its  adaptability  to  theoretical  investigations,  but 
also  because  of  its  immediate  use  in  the  laboratory  study  of  sediments. 
Further  details  of  its  application  to  shape  studies  of  particles  will  be 
given  in  Chapter  1 1 . 

In  mechanical  analysis  various  terms  have  been  developed  to  express 
the  size  of  irregular  particles  in  terms  of  their  settling  velocities. 
Schone  ^  in  1868  introduced  the  term  hydraidischer  Werth  (hydraulic 
value)  to  define  the  diameter  of  a  quartz  sphere  having  the  same  settling 
velocity  as  a  given  particle  in  water.  The  hydraulic  value  has  no  bearing 
on  the  actual  size  of  the  particle  in  terms  of  its  volume,  but  it  was  used 
to  express  "size"  in  numerical  terms.  Oden  -  in  191 5  improved  the  con- 
cept by  introducing  the  term  equivalent  radius  as  the  radius  of  a  sphere 
of  the  same  material  as  the  particle  and  having  the  same  settling  velocity. 
More  recently  Wadell  ^  sharpened  the  definition  by  introducing  the  term 
sedimentation  radius  as  "the  radius  of  a  sphere  of  the  same  specific 
gravity  and  of  the  same  terminal  uniform  settling  velocity  as  a  given 
particle  in  the  same  sedimentation  fluid."  Robinson,*  previous  to  Wadell's 
work,  recognized  the  apparently  meaningless  use  of  the  term  radius  in 
connection  with  irregular  particles  studied  by  sedimentation  and  chose 
to  ignore  any  expression  of  size  in  his  mechanical  analyses.  Instead,  he 
expressed  his  values  directly  as  the  logarithms  of  setding  velocity. 

Wadell's  sedimentation  radius  will  be  accepted  as  standard  in  this 
text,  and  in  any  context  referring  to  the  size  of  irregular  particles  as 
determined  by  sedimentation  methods,  the  sedimentation  radius  will  be 
either  explicit  or  implied.  It  will  be  developed  shortly  that  two  laws 

1  E.  Schone,  Neber  einen  neuen  Apparat  fur  die  Schlammanalyse :  Zcits.  f.  anal. 
Chemie,  vol.  7,  pp.  29-47,  1868. 

2  S.  Oden,  Eine  neue  methode  zur  mechanischen  Bodcnanalyse :  Int.  Mitt.  f. 
Bodenkitnde,  vol.  5,  pp.  257-311,  1915. 

3  H.  Wadell,  Some  new  sedimentation  formulas:  Physics,  vol.  5,  pp.  281-291,  1934. 
*  G.  W.  Robinson,  The  form  of  mechanical  composition  curves  of  soils,  clays,  and 

other  granular  substances :  Jour.  Agric.  Sci.,  vol.  14,  pp.  626-633,  1924- 


PRINCIPLES  OF  SIZE  ANALYSIS  95 

of  settling  velocities  are  generally  applicable  to  sedimentary  studies, 
either  Stokes'  law  directly  or  a  modification  introduced  by  Wadell,  de- 
signed to  correct  for  the  non-spherical  particles  in  sediments.  When 
Stokes'  law  is  used  directly,  the  computed  size  values  may  be  called 
Stokes'  sedimentation  radius  (or  diameter),  and  when  Wadell's  prac- 
tical formula  is  used,  they  may  be  called  the  practical  sedimentation 
radius  (or  diameter). 


THE    SETTLING    VELOCITIES    OF    SMALL     PARTICLES 

One  of  the  fundamental  principles  on  which  mechanical  analysis  is 
based  is  that  small  particles  will  settle  with  a  constant  velocity  in  water 
or  other  fluids.  It  is  universally  true  that  small  particles  reach  this  con- 
stant velocity  in  a  fluid  medium  as  soon  as  the  resistance  of  the  fluid 
exactly  equals  the  downward  constant  force  (gravity)  which  acts  on 
the  panicle.  In  general  the  settling  velocity  of  the  particle  depends  on 
its  radius,  its  shape,  its  density,  its  surface  texture,  and  the  density  and 
viscosity  of  the  fluid.  A  number  of  mathematical  expressions  have  been 
developed  to  show  the  relations  among  these  factors,  some  based  on 
empirical  grounds  and  others  on  theoretical  grounds.  Several  of  these 
laws  will  be  discussed  in  varying  detail,  depending  upon  their  applica- 
bility to  mechanical  analysis. 

Stores'  Law  of  Settling  Velocities 

The  classic  formula  for  settling  velocities,  and  the  best  known,  is  that 
of  Stokes,^  which  confines  itself  to  spheres.  Since  this  equation  is  of 
such  fundamental  importance  and  of  such  widespread  application,  it 
will  be  considered  in  detail. 

Theory  of  Stokes'  /ate.  Stokes  first  considered  the  resistance  which  a 
fluid  ofifers  to  the  movement  of  a  sphere  suspended  in  it,  and  arrived  at 
the  equation 

R  =  6  .^r^;y      .      .      .      .      (i) 

where  R  =  resistance  in  g.  cm./sec- 

r=  radius  of  the  sphere  in  cm. 

V  —  viscosity  of  the  fluid 

Z'  =  velocity  of  the  sphere  in  cm.  sec.~^ 

When  a  small  sphere  settles  in  a  fluid,  it  is  acted  on  by  the  force  of 

gravity.  —  ^if'-^dig,  acting  downward ;  and  by  the  buoyant  force  of  the 

1  G.  G.  Stokes,  On  the  effect  of  the  internal  friction  of  fluids  on  the  motion  of 
pendulums:  Trans.  Cambridge  Philos.  Soc,  vol.  9,  part  2,  p.  8-106,  1851. 


96        SEDIMENTARY  PETROGRAPHY 

liquid,  — nr '(/o(7,   given   by  Archimedes'   principle,   and   acting   upward, 

which  results  in  a  net  force  — nr'fJi  —  d..)g  acting  downward.  At  the 

instant  when  the  resistance  R  exactly  equals  this  net  force,  the  velocity 
becomes  constant  and  remains  so.  When  this  uniform  state  is  reached 
there  results : 

6nn]v=- —  nr"(di  —  d.)g  ....      (2) 

3 
where  the  additional  symbols  are  d^  =  density  of  the  sphere 

rf,  —  density  of  the  fluid 
g   =  acceleration  due  to  gravit>' 
C980  cm.  sec.— 2) 

Bv  solving  equation  (2)  for  v,  one  obtains 

'^=f^r^'  •  ■  ■  •  (3) 

the  equation  of  Stokes'  law. 

In  general,  if  standard  conditions  are  assumed,  that  is,  a  constant 
temperature,  a  given  fluid,  and  a  known  specific  gravity  of  the  sphere, 
equation  (3)  may  be  expressed  as 

v=Cr-     ....      (4) 

where  C  is  a  constant,  and  equal  to ^ ^-^.  Tables  for  the  value 

^  9  V 

of  this  constant  under  various  conditions  have  been  computed  and  will 

be  referred  to  later.  It  may  be  mentioned,  however,  that  for  water  at 

20°  C,  and  particles  with  a  specific  gravity  of  quartz,  2.65,  the  value 

of  the  constant  is  C  =  3.57X  10*. 

The  assumptions  of  Stokes'  la-u.'.  Several  assumptions  underlie  Stokes'  law, 
and  it  is  important  to  consider  them  in  the  light  of  mechanical  analysis.  The 
following  assumptions  are  generally  recognized : 

(i)  The  particle  must  be  spherical,  it  must  be  smooth  and  rigid,  and  there 

should  be  no  slipping  between  it  and  the  medium. 
(2)  The  medium  may  be  considered  homogeneous  in  comparison  to  the  size 

of  the  particle. 
("3)   The  particle  should  fall  as  it  would  in  a  medium  of  unlim.ited  extent. 

(4)  A  constant  settling  velocity  must  have  been  reached. 

(5)  The  settling  velocity  should  not  be  too  great. 

Each  of  these  assumptions  deser\-es  some  consideration  from  the  point 
of  view  of  mechanical  analysis  to  determine  whether  it  is  in  fact  satisfied 
in  practice.  Assumption  i  is  satisfied  to  the  extent  that  there  is  no  slip  between 
the  particles  and  the  fluid,  inasmuch  as  they  are  wetted  by  the  liquids  com- 


PRINCIPLES  OF  SIZE  ANALYSIS 


97 


monly  used  in  mechanical  analysis.  Similarly,  since  the  particles  are  solid, 
the  assumption  of  rigidity  is  satisfied,  but  it  is  seldom  true  that  tlie  grains 
are  perfectly  smooth.  Arnold  ^  has  shown  that  pitted  surfaces  do  not  appre- 
ciably affect  the  settling  velocities  of  small  spheres,  and  consequently  this 
factor  may  not  be  of  paramount  importance.  The  condition  that  the  particle 
be  a  sphere  is  perhaps  least  satisfied,  and  this  introduces  several  difficulties. 
The  same  sediment  may  have  grains  varying  in  shape  from  almost  true 
spheres  to  irregular  grains,  plates,  and  laths.   Experiments  have  been  per- 


a 
^ 


<n4 


—  stokes'  Law,  Calculated 

4 

A  Schbne        ^ 

0  Hilgard 

+  Owens          I   Observed 

X  Atterberg    /      Values 

D  Boswell       \ 

/         [ 

•  Richards 

J 

/ 

^   • 

-  iif* 

• 

_^^^:^ 

^^ 

0.02 


0.04  0.06 

DIAMETER  IN    MM. 


0.08 


0.1 


Fig.  20. — Comparison  of  some  observed  settling  velocities  with  Stokes'  law,  in  the 
size  range  o  to  o.i  mm.  diameter. 


formed,  however,  to  determine  the  agreement  between  the  settling  velocities 
of  powders,  soils,  and  sediments  in  terms  of  observation  on  the  one  hand  and 
the  expected  theoretical  values  of  Stokes'  law  on  the  other.  The  degree 
of  agreement  is  quite  remarkable  within  the  range  significant  in  most  methods 
of  mechanical  analysis.  Andreason  and  Lundberg,-  for  example,  found  a  very 
satisfactory  agreement  between  grades  separated  by  an  elutriator  (Schone's) 

1  H.  D.  Arnold,  Limitations  imposed  by  slip  and  inertia  terms  upon  Stokes'  law 
for  the  motion  of  spheres  through  liquids:  Phil.  Mag.,  vol.  22,  pp.  755-/75.  IQH- 

2  A.  H.  M.  Andreason  and  J.  J.  V.  Lundberg,  Ueber  Schlammgeschwindigkeit  und 
Korngrosse :  Kolloid  Zeits.,  vol.  49,  pp.  48-51,  1929. 


98        SEDIMENTARY  PETROGRAPHY 

and  Stokes'  law  for  grades  ranging  upward  to  0.088  mm.  diameter,  although 
they  mentioned  tliat  the  larger  grades  showed  some  deviation.  The  general 
extent  of  the  agreement  may  be  shown  by  a  comparison  of  the  observed 
values  of  several  experimenters  and  the  theoretical  values  as  given  by  Stokes' 
law.  Figure  20  is  such  a  curve  based  on  values  from  Schone,^  Hilgard,- 
Owens,^  Atterberg,*  Boswell,^  and  Richards.^  Stokes'  law  is  shown  as  a 
solid  line  computed  for  quartz  settling  in  water  at  a  temperature  of  15°  C. 
These  conditions  are  chosen  as  an  average  because  of  the  lack  of  definite 
data  regarding  the  exact  conditions  under  which  the  experiments  were  carried 
on. 

It  will  be  noted  that  there  is  a  fairly  close  agreement  between  observed  and 
computed  velocities  until  a  diameter  of  about  0.05  mm.  is  reached.  Richard's 
values  begin  to  depart  at  about  0.04  mm.,  Schone's  at  about  0.06  mm.,  and 
Hilgard's  at  about  0.07  mm.  These  deviations  may  of  course  depend  upon 
varying  experimental  conditions  as  well  as  upon  shape  differences.  It  seems 
fairly  safe,  however,  to  consider  the  agreement  satisfactory  up  to  a  diameter 
of  at  least  0.05  mm.,  but  probably  not  beyond  0.07  mm.  It  will  be  seen  later 
that  these  upper  limits  agree  quite  well  with  the  theoretical  upper  limits  of 
Stokes'  law  for  true  spheres. 

Because  of  the  difficulty  of  defining  the  size  of  the  irregular  particles  com- 
monly found  in  soils  or  sediments,  the  sizes  are  usually  defined  in  terms  of 
their  settling  velocities  according  to  Stokes'  law,  by  such  terms  as  hydraulic 
radius,  eqim'alcnt  radius,  or  sedimentation  radius,  as  described  on  page  94. 

Assumption  2  merely  states  that  the  distances  between  the  molecules  of  the 
fluid  must  be  small  compared  with  the  sizes  of  the  particles.  This  condition 
is  fully  satisfied  down  to  the  borders  at  least  of  the  colloidal  state,  and  perhaps 
well  within  it.  For  most  practical  purposes  it  may  be  ignored. 

Assumption  3  is  involved  to  a  considerable  extent  in  mechanical  analysis, 
inasmuch  as  all  methods  of  analysis  involve  vessels  of  finite  size.  Since  the 
assumption  states  that  the  medium  should  be  of  unlimited  extent,  it  is  neces- 
sary to  consider  the  error  introduced  by  sedimentation  cylinders  and  tubes  of 
particular  diameters.  Several  equations  have  been  developed  to  express  the 
influence  of  wall  nearness  on  settling  velocities.  Lorentz  ^  set  up  an  equation 
for  the  resistance  met  by  a  particle  settling  parallel  to  a  plane  wall,  and  by 
using  his  terms  instead  of  Stokes'  R  one  finds  that  the  ratio  of  the  "true" 
velocity  to  Stokes'  velocity  is  as  follows : 

v,/v,=^i—9r/i6L (5) 

1  E.  Schone,  loc.  cit.,  1868. 

2  E.  W.  Hilgard,  On  the  silt  analysis  of  soils  and  clays :  Am.  Jour.  Set.,  vol.  6, 
pp.  288-296,  333-339,  1873- 

3  J.  S.  Owens,  Experiments  on  the  settlement  of  solids  in  water:  Gcog.  Journal. 
vol.  27,  PP-  59-79,  191 1- 

4  A.  Atterberg,  Die  mechanische  Bodenanalyse  und  die  Klassifikation  der  Mi- 
neralbodcn  Schwedens:  Int.  Mitt.  f.  Bodcnknndc,  vol.  2,  pp.  312-342,  1912. 

5  P.  G.  H.  Boswell,  A  Memoir  on  British  Kcsources  of  Refractory  Sands  for  Fur- 
nace and  Foundry  Purposes,  Part  i   (London,  1918). 

c  R.  H.  Richards  and  C.  E.  Locke,  A  Textbook  of  Ore  Dressing,  2nd  ed.  (New 
York,  1925). 
7  H.  A.  Lorentz,  Abhandliingen  iiber  theoret.  Physik  (Leipzig,  1911),  vol.  i,  p.  40. 


PRINCIPLES  OF  SIZE  ANALYSIS 


99 


where  L  is  tlie  distance  between  the  sphere  and  the  wall.  Figure  21  shows 
tlie  value  of  this  ratio,  expressed  as  a  percentage,  for  spheres  of  radius 
o.ooi  cm.  and  0.0025  cm.,  at  various  distances  from  the  wall.  The  values  of 
the  ratio  are  all  smaller  tlian  i,  indicating  that  the  effect  of  the  factor  is  to 
reduce  the  velocity.  The  curves  are  hyperbolic,  and  the  effects  of  wall 
nearness  decrease  very  rapidly.  The  two  curves  also  show  that  the  effects 
vary  with  the  size  of  the  particles,  being  larger  for  larger  particles.  The 
suggestion  ottered  by  these  data  is  that  vessels  of  some  appreciable  radius 
(a  minimum  of  about  2  cm.  radius)  should  be  used  in  mechanical  analysis, 
to  render  wall  nearness  eft'ects  essentially  negligible. 

Ladenburg  ^  attacked  the  problem  from  the  point  of  view  of  a  sphere  of 

radius  r  settling  in  a  cylinder  of  length  

L  and  radius  R.  Experiments  by  Ar- 
nold- in  this  case  showed  that  the 
velocity  according  to  Stokes'  law  is  not 
appreciably  aft'ected  until  the  radius  of 
the  particle  equals  i/io  the  radius  of 
the  cylinder.  It  would  seem  from  this 
that  tubes  of  very  small  radius  may  be 
used  in  mechanical  analysis,  but  another 
factor  may  enter  when  a  system  of 
particles  is  present.  In  such  cases  each 
particle  is  influenced  by  its  neighbors, 
so  that  an  extremely  complicated  situa- 
tion develops,  which  has  not  been  fully 
elucidated  mathematically.  In  dilute  sus- 
pensions these  eft'ects  are  apparently  not 
serious.  The  authors  know  of  no  quanti- 
tative data  on  the  subject,  but  it  is  per- 
haps best  to  use  suspensions  containing 
not  more  than  about  25  g.  of  solid  to  the 
liter.  To  be  conservative,  also,  vessels  of  reasonably  large  diameter  should  be 
used,  say  5  cm.  or  larger. 

Assumption  4  states  that  the  constant  velocity  of  fall  must  have  been 
reached.  It  is  clear  that  at  time  t  =  0,  the  velocity  is  zero,  so  that  the  particle 
increases  its  velocity  until  the  resistance  of  the  fluid  exactly  counterbalances 
the  downward  force  on  the  particle.  There  is  thus  an  interval  of  time  before 
the  constant  velocity  is  reached,  and  it  is  necessary  to  consider  the  order  of 
magnitude  of  this  interval.  Weyssenhoff  ^  has  developed  an  equation  which 
permits  a  computation  of  this  interval.  The  equation  is  rather  complex  and 
need  not  be  considered  in  detail ;  computations  for  a  sphere  of  diameter  0.05 
mm.  (which  is  near  the  upper  limit  of  applicability  of  the  law)  indicate 
tliat  it  requires  about  0.003  sec.  to  achieve  constant  velocity.  Hence  assump- 
tion 4  need  not  concern  practical  mechanical  analysis. 


A 
3 


0  1  0  2  0  3  0  4         0.5 

DISTANCE    FROM    WALL    IN  CU 

Fig.  21. — Effect  of  wall-nearness 
on  settling  velocities  of  spheres. 
Curve  A,  spheres  of  radius 
0.0025  mm. :  curve  B,  spheres  of 
radius  o.ooi  mm. 


1  R.  Ladenburg,  Tber  den  Einfluss  von  Wanden  auf  die  Bewegung  einer  Kugel 
in  einer  reibenden  Flussigkeit :  Ann.  dcr  Physik,  vol.  2Z,  pp.  447-458,  1907. 

-  H.  D.  Arnold,  loc.  cit..  191 1. 

3  J.  Weyssenhoff.  Betrachtungen  iiber  den  Giiltigkeitsbereich  der  Stokesschen 
und  der  Stokes-Cunninghamschen  Formel:  Ann.  der  Physik.  vol.  62,  pp.  1-45,  1920. 


lOO        SEDIMENTARY  PETROGRAPHY 

Assumption  5  provides  that  the  motion  should  be  slow.  This  condition  im- 
poses certain  limits  on  the  range  of  sizes  that  may  be  studied  by  Stokes'  law, 
and  it  is  important  to  consider  it  in  some  detail.  The  assumption  is  based 
on  the  fact  that  the  viscosity  of  the  medium  should  furnish  all  of  the  re- 
sistance which  the  sphere  meets  in  its  descent.  When  the  sphere  is  so  large 
that  this  no  longer  holds,  the  particle  drags  some  of  the  liquid  with  it,  and 
the  radius  no  longer  holds  the  same  simple  relation  to  the  velocity  as  before. 
The  limiting  size  in  any  given  case  will  depend  on  a  number  of  factors.  If 
the  liquid  is  particularly  viscous,  the  particle  may  be  larger  than  in  less  viscous 
liquids.  Similarly,  when  the  difference  between  the  density  of  the  sphere 
and  that  of  the  liquid  is  slight,  the  particle  may  be  larger  than  in  the  reverse 
case.  Allen, 1  discussing  the  upper  limit  of  Stokes'  law,  pointed  out  that  the 
law  was  valid  as  long  as  the  inertia  terms  are  neglected  in  comparison  with 
viscosity.  This  requires  that  the  velocity  times  the  radius  times  the  density 
of  the  fluid  must  be  small  compared  to  the  viscosity: 

vd.^-<r) (6) 

In  seeking  an  upper  limit,  Allen  defined  as  the  critical  radius  that  value  of 
r  which  establislied  equality  of  the  two  sides  of  the  expression.  By  setting 
this  equation  up  in  the  form  z'  =  v/dnV,  and  substituting  v  =  Cr-  from 
Stokes'  law,  one  obtains 

d.,r 


from  which  it  is  clear  that 


-I 


ir "' 


Arnold  2  subsequently  showed  tliat  the  inertia  terms  begin  to  manifest  them- 
selves when  a  radius  of  0.6  the  value  of  the  critical  radius  is  reached. 

By  considering  a  sphere  of  quartz  (specific  gravity  =  2.65)  settling  in 
water  at  a  temperature  of  20°  C.  (the  value  of  C  in  Stokes'  law  in  this  case 
is  3.57  X  lO"*)*  't  may  be  shown  that  the  uncorrected  critical  radius  has  a 
value  of  about  0.006  cm.,  which  is  a  diameter  of  0.12  mm.  Six-tenths  of  this 
value  is  about  0.08  mm.  diameter,  which  sets  an  upper  limit  to  the  application 
of  Stokes'  law  in  ordinary  mechanical  analysis.  This  corresponds  to  a  grain 
slightly  larger  than  Vm  mm.,  and  involves  a  settling  velocity  of  about  5  mm. 
per  second.  It  will  be  noted  that  this  value  is  of  the  same  order  of  magnitude 
as  the  experimental  data  shown  in  Figure  20. 

The  problem  of  the  lower  limit  of  Stokes'  law  has  also  received  the 
attention  of  several  workers.  Perrin's  work  ^  in  this  connection  is  particularly 
noteworthy.  He  prepared  essentially  monodisperse  systems  of  very  small  par- 
ticles by  centrifuging  gamboge  suspensions.  The  radii  of  the  particles  were 
determined  by  three  methods,  one  of  which  was  tlieir  settling  velocities.  The 
values  found  by  Stokes'  law  agreed  strikingly  with  the  values  obtained  by 

1  H.  S.  Allen,  The  motion  of  a  sphere  in  a  viscous  fluid :  Phil.  Mag.,  vol.  50,  pp. 
323-338,  519-534,  1900. 

2  H.  D.  Arnold,  loc.  cit.,  1911. 

3  J.  Perrin,  Atoms,  translated  by  D.  L.  Hammick   (London,   1920). 


PRINXIPLES  OF  SIZE  ANALYSIS  loi 

the  other  two  methods.  Inasmuch  as  the  particles  ranged  in  radius  from  about 
0.15  micron  to  0.5  micron.  Perrin  concluded  that  Stokes'  law  held  despite 
Brownian  movement  and  that  it  was  valid  within  the  borders  of  the  colloidal 
state.  Van  Halin.^  after  considering  the  results  of  several  observers,  con- 
curred with  Perrin  in  this  conclusion.  That  there  must  be  a  lower  limit  seems 
obvious,  or  colloidal  suspensions  would  not  be  essentially  permanent,  as 
they  are. 

For  the  purposes  of  mechanical  analysis  we  may  consider  it  sufficient  that 
the  law  holds  to  a  diameter  of  o.i  micron,  which  is  at  the  lower  limit  of 
coarse  disperse  systems  as  earlier  defined.  Thus  both  the  upper  and  lower 
limits  of  Stokes'  law  fall  at  convenient  points,  as  far  as  mechanical  analysis 
is  concerned,  because  tlie  upper  limit  of  tlie  law  occurs  just  about  at  the 
lower  limit  of  sieving,  and  the  lower  limit  occurs  within  the  colloidal  state, 
at  sizes  smaller  than  ordinary  methods  of  analysis  can  separate.  This  permits 
a  composite  analysis  to  be  made  by  sieving  down  to  about  0.06  mm.  diameter, 
and  below  that  by  settling  velocities  computed  from  Stokes'  law. 

Summary  of  Stokes'  laze.  In  summary  of  the  discussion  of  Stokes' 
law  it  may  be  said  that  the  highly  practical  results  that  follow  from  its 
application  to  mechanical  analysis  warrant  its  use  as  a  fundamental 
equation  in  the  development  of  any  method  of  analysis.  In  applying  the 
law  there  are  certain  precautions  to  be  obserxed  as  to  the  sizes  of  par- 
ticles to  be  separated  and  the  variables  in  the  equation  that  may  aftect 
the  results. 

The  limits  of  the  sizes  which  may  be  studied  by  Stokes'  law  have 
already  been  discussed ;  among  the  other  variables  are  the  density  of 
the  particles  and  the  viscosity  of  the  liquid.  In  a  pigment,  where  all  the 
particles  have  the  same  specific  gravity,  the  first  problem  does  not  enter, 
but  among  sediments  and  soils  there  is  a  mixture  of  particles  ranging 
rather  widely  in  specific  gravity.  The  results  of  heavy  mineral  separa- 
tions show,  however,  that  in  the  average  case  more  than  95  per  cent  of 
the  material  is  quartz  or  feldspar.  The  specific  gravity  of  quartz  is  2.65, 
and  that  of  feldspar  about  2.6.  Hence  by  far  the  greatest  amount  of 
material  is  under  2."/,  so  that  the  two  most  common  values  to  be  adopted 
are  either  2.65  or  2.7,  depending  upon  the  percentage  of  hea\y  minerals 
in  the  sediment.  In  special  cases,  as  where  magnetite  concentrates  occur, 
the  value  assumed  will  have  to  be  adjusted  to  the  special  merits  of  the 
case.  The  effect  of  specific  gravity  on  the  settling  velocity  may  be  seen 
from  a  simple  example.  For  a  sphere  of  diameter  0.05  mm.  in  water  at 
20°  C.  an  increase  in  specific  gravity  from  2.6  to  2.j  causes  an  increase 
of  nearly  6  per  cent  in  the  rate  of  settling. 

The  x-iscosity  of  water  varies  rather  considerably  in  small  ranges  of 

1  F.-V.  von  Hahn,  Dispcrsoidanalyse   (Leipzig,  1928),  pp.  270  ff. 


102 


SEDIMEXTARY  PETROGRAPHY 


temperature.  Figure  22  shows  the  curve  of  viscosity  of  water  from  0° 
to  30°  C.  The  value  drops  about  50  per  cent  in  this  range.  As  in  the 
case  of  varying  specific  gravities,  changes  in  the  viscosity  of  the  water 
cause  considerable  changes  in  the  settling  velocity  of  the  particle.  A 
quartz  sphere  of  diameter  0.05  mm.  has  a  settling  velocity  of  0.196 
cm./sec.,  in  water  at  15°,  and  0.223  cm. /sec.  at  20°  C.  This  is  an  increase 
of  1 1.4  per  cent  for  five  degrees,  or  an  average  of  2.3  per  cent  per 
degree.  From  this  it  is  clear  that  some  temperature  control  should  be 

exercised,  so  that  no  considerable 
fluctuations  occur  during  an  analysis. 
In  addition  to  the  changes  in 
viscosity  which  accompany  changes 
of  temperature,  convection  currents 
may  be  set  up,  which  materially  af- 
fect the  normal  settling  of  the  par- 
ticles. 

Reference  was  made  earlier  to  a 
form  of  Stokes'  law  (equation  4), 
in  which  all  variables  except  the 
velocity  and  the  radius  were  com- 
bined into  a  single  constant.  By 
choosing  standard  conditions  of  temperature  and  average  specific  gravity 
of  the  material  being  analyzed,  the  velocities  for  spheres  of  any  given 
radii  may  easily  be  computed.  The  most  common  temperatures  at  which 
analyses  are  made  are  i^""  and  20°  C,  and  the  average  specific  gravities 
usually  chosen  for  sediments  are  2.65  and  2.70.  Table  12  (page  no) 
shows  the  values  of  the  constant  in  Stokes'  law  under  these  conditions, 
and  the  velocities  of  particles  ranging  in  diameter  from  0.06  mm.  to  0.5 
micron.  This  table  furnishes  basic  information  used  in  several  methods 
of  analysis  and  will  be  referred  to  later.  It  should  also  be  mentioned  that 
the  velocities  in  cm./sec.  of  any  intermediate-sized  particles  may  be 
fovmd  by  multiph-ing  the  value  of  the  constant  by  the  square  of  the 
radius  in  centimeters. 


10         IS         20 
TEMPERATURE,    'C. 


Fig.  22. — Viscosity  of  water  as  a 
function  of  temperature. 


Rubey's  Formula 
In  1933.  Rubey^  developed  a  general  formula  for  settling  velocities 
which  agrees  with  observed  values  over  a  wider  range  than  Stokes'  law. 
In  extending  the  law  of  settling  velocities  beyond  the  critical  value  of 

1  W.  W.  Rubey,  Settling  velocities  of  gravel,  sand,  and  silt  particles:  Am.  Jour. 
Sci,  vol.  25,  pp.  325-338,  1933- 


PRINCIPLES  OF  SIZE  ANALYSIS 


103 


Stokes'  law,  Rubey  conceived  that  the  total  force  acting  on  a  large  par- 
ticle was  the  sum  of  the  forces  due  to  viscous  resistance  and  the  impact 
of  the  fluid.  By  equating  these  forces  to  the  eiitective  weight  of  the 
particle,  Rubey  obtained  the  expression 


—  7ir^ (di  —  d.2,)(j  =  6jir)]v  -\-  nr-v'-d^  . 
from  which  he  obtained  his  formula  by  solving  for  v\ 


-gd.,(d,-d,)r'-^(j,j--\-^)] 


]"A- 


(8) 


(9) 


Figure  23,  adapted  from  Rubey's  paper,  shows  the  gradual  transition 
between  the  ranges  of  viscous  resistance  and  fluid  impact.  The  heavy 


I02 


LU 
00    10 


10- 


10 


1        ' 

'^^ 

/ 

^ 

^ 

" 

sh 

^^ 

^1 

,^ 

^ 

ss*^ 

/^ 

^y 

^^ 

f  / 

r 

^ 

^S^ 

^^ 

'     // 

/ 

•^-^ 

General  Fo 

rmula 

\^^ 

'f 

f 

V 

1 

" 

/ 

/ 
/ 

f 

/ 

/ 

/ 

/ 

^ 

/ 

10" 


10' 


10' 


10 


10* 


DIAMETERS    IN    MM. 


Fig.  23. — Rubey's  general  formula  for  settling  velocities.  The  heavy  line  agrees 
well  with  observed  data. 


line  agrees  well  with  observed  settling  velocities  of  quartz  or  galena, 
as  Rubey's  original  figure  indicates. 

Analysis  of  Rubey's  formula,  as  expressed  in  eqtiation    (8).  shows 
that,  when  the  velocity  of  the  particle  is  so  small  that  the  inertia  terms 


104       SEDIMENTARY  PETROGRAPHY 

(nr^v-dz)  may  be  neglected,  the  expression  simplifies  to  Stokes'  law. 
It  is  thus  a  generalization  of  the  latter. 

Wadell's  Sedimentation  Formula 

The  most  recent  work  on  settling  velocities  has  been  done  by  Wadell,^ 
who  opened  a  new  approach  to  the  problem  by  examining  the  functional 
relationship  between  the  coefficient  of  resistance,  Cr,  and  Reynolds  num- 
ber. Re. 

The  coefficient  of  resistance  is  defined  by  equating  tlie  force  producing 
motion  of  a  sphere  to  the  force  resisting  motion,  expressed  as  the  coefficient 
of  resistance  times  the  dynamic  pressure  acting  on  the  cross-sectional  area  of 
the   sphere : - 

^^r-{d^  —  d^)g  =  C, j-^ (10) 

from  which  Cr  =  } ^^-  The  symbols  have  the  same  significance  as 

3      d^v- 

in  the  case  of  Stokes'  law,  equation  3. 

Reynolds  number  is  defined  in  terms  of  the  radius  of  the  sphere,  its  velocity, 
and  the  density  and  viscosity  of  the  fluid,  as  follows :  ^ 

■^-^  <■■> 

Reynolds  number  is  dimensionless,  i.e.,  it  is  a  pure  number. 

Wadell  chose  as  his  starting  point  an  equation  relating  the  coefficient 
of  resistance  to  Reynolds  number  in  the  following  manner : 

C,Z!!^_/(R.)f^ (,2) 

from  which  Cr  =  /(Re).  By  plotting  a  number  of  observed  settling 
velocities  and  radii  in  terms  of  Re  and  Cr,  with  Re  as  abscissa  and  Cr 
as  ordinate,  on  double  log  paper,  Wadell  developed  an  empirical  formula 
for  settling  velocities  which  not  only  extends  the  range  of  practical  set- 
tling velocities  to  much  larger  diameters  than  those  afforded  by  Stokes' 
law,  but  in  addition  enabled  him  to  elucidate  the  influence  of  shapes  of 

1  H.  Wadell,  The  coefficient  of  resistance  as  a  function  of  Reynolds  number  for 
solids  of  various  shapes:  Jour.  Franklin  Inst.,  vol.  217,  pp.  459-490,  1934. 
H.  Wadell,  Some  new  sedimentation  formulas:  Physics,  vol.  5,  pp.  281-291,  1934. 
H.  Wadell,  Some  practical  sedimentation  formulas :  Geol.  Forcn.  Forhdndl.,  vol. 
58,  pp.  397-408,  1936. 

2  J.  E.  Christiansen,  Distribution  of  silt  in  open  channels :  Trans.  Am.  Geophysical 
Union,  part  II,  pp.  478-485,  1935. 

3  H.  Wadell,  loc.  cit.,  1934. 


PRINCIPLES  OF  SIZE  ANALYSIS  105 

particles  on  their  settling  velocities.  jMoreover,  the  opening  of  this  new 
approach  to  the  problem  of  settling  velocities  affords  important  means 
of  studying  actual  sedimentary  problems  in  terms,  for  example,  of 
deposition  in  water  as  against  air.  The  theoretical  aspects  of  these 
appHcations  do  not  belong  in  the  present  volume,  which  considers 
only  petrographic  applications.  In  the  latter  connection,  Wadell's 
formula  extends  the  use  of  pure  sedimentation  methods  to  particles 
which  at  present  are  generally  sieved,  and  it  may  be  expected  that  such 
sedimentation  methods  of  analysis  may  be  developed  for  practical  work. 
Likewise  shape  factors  may  be  included,  an  important  item  when  it  is 
recalled  that  truly  spherical  particles  are  practically  non-existent  among 
sediments. 

Wadell's  formula,  expressed  in  terms  of  a  correction  to  be  applied  to  Stokes' 
radius,  is 

r,  =  rSi  +  o.oS(2r,vJ,/v)'''''>'j (13) 

where  r^  is  the  actual  radius,  r^  is  the  radius  according  to  Stokes'  law,  and 
c'a  is  the  actual  settling  velocity.  The  exponent  0.69897  =  log^oS  was  deter- 
mined by  statistical  methods  from  the  observational  data. 

Figure  24,  adapted  from  Wadell,  shows  a  portion  of  the  range  covered  by 
the  original  graph.  The  heavy  line  is  Wadell's  curve,  which  was  shown  to 
agree  quite  closely  with  observed  settling  velocities  of  splieres  up  to  values 
of  Reynolds  number  of  about  3.000.  In  the  same  figure  are  shown  Stokes', 
Oseen's,  and  Goldstein's  laws  (see  below),  to  indicate  their  departure  from 
Wadell's  curve  in  the  higher  ranges.  Stokes'  law  is  valid  to  values  of  about 
Rg  =  0.2,  Oseen's  and  Goldstein's  laws  apply  to  about  Rg  =  0.5.  Rubey's  for- 
mula, according  to  Christiansen  ^  agrees  with  observed  values  on  quartz  and 
galena  to  values  of  Re  =  1000. 

Rouse  -  recently  extended  the  curve  developed  by  Wadell  to  values  of  Rg 
up  to  one  million.  Beyond  10^  the  curve  for  spheres  shows  an  abrupt  change 
in  slope,  due  to  the  onset  of  turbulence  in  the  boundary  layer  at  the  front  of 
the  spliere. 

Wadell  ^  also  derived  a  modified  form  of  Stokes'  law  which  has  im- 
mediate application  to  mechanical  analysis.  Arguing  on  the  basis  that 
sedimentary  particles  are  not  true  spheres,  but  a  mixture  of  particles 
of  varying  shape,  he  developed  a  resistance  formula  for  a  hypothetical 
particle  intermediate  in  shape  between  a  disc  and  a  sphere.  By  applying 
his  reasoning  to  Stokes'  resistance  equation,  R  ==  6w)]V,  Wadell  obtained 

1  J.  E.  Christiansen,  loc.  cit.,  1935. 

-  H.  Rouse,  Nomogram  for  the  settling  velocity  of  spheres :  Report  of  the  Com- 
mittee on  Sedimentation  1936-37,  pp.  57-64,  Nat.  Research  Council,  1937. 
3  H.  Wadell,  loc.  cit.,  1936. 


io6  SEDIMENTARY  PETROGRAPHY 

the  value  Rw  =  g.44^n]v.  By  equating  this  value  to  the  effective  weight 
of  the  particle,  -  Tir^C^i  —  f/o)^,  Wadell  obtained 
I  (d,  —  d.)gr^- 


(14) 


7  V 

where  v^  is  the  "practical  settling  velocity,"  and  r^  is  the  "practical  sedi- 
mentation radius."  It  may  be  noted  from  a  comparison  of  this  expression 


103 


10= 


CO 
CO 


10 


LiJ 

o 


1 

— I— 

-^ 

\ 

\ 

s 

s. 

\ 

s 

- 

\ 

\ 

s 

s 

\ 

^ 

N 

\^ 

s> 

r  ^ 

Oseen  ^ 

*i     I 

-- 

^^ 

S[^v 

•-- 

— L 

k\ 

"v. 

Goldstein 

^     i     1 

\ 

fS 

iiwadell 

\      '■ 

"X 

1 

^  1 

^ 

ii 

^- 

• . 

stokes^ 

i\ 

_ 

|°-;k^ 


10 


I0-*  10"'  I  10 

REYNOLDS   NUMBER 


10 


103 


Fig.  24. — Comparison  of  several  laws  of  settling  velocities  of  spheres,  expressed 
in  terms  of  Reynolds  number  and  the  co-efficient  of  resistance.  The  heavy  line 
(Wadell's  formula)  agrees  closely  with  observed  data.  (Adapted  from  Wadell, 
1 934-) 

with  Stokes'  law,  (equation  3),  that  the  only  difference  betiveen  the 
two  equations  is  the  numerical  constants.  Stokes'  law  has  the  fraction 
%,  whereas  Wadell's  value  is  34- 

This  simple  relation  offers  an  immediate  method  of  correcting  Stokes' 
law  to  allow  for  shape  variations  in  the  sediment,  namely,  by  finding 
the  value  of  the  ratio  v^/v^,  where  v^  refers  to  Stokes'  velocity.  This 
ratio  may  be  found  by  dividing  equation  (14)  by  equation  (3)  : 


PRINCIPLES  OF  SIZE  ANALYSTS  107 

Thus  the  practical  settling  velocity  of  a  given  sedimentary  particle  is  64 
per  cent  of  the  theoretical  settling  velocity  of  the  corresponding  sphere. 
Likewise  the  ratio  r^/r^  may  be  computed  from  the  same  equations  and 
is  found  to  be  rJrs  =  \/y^\/q/2  =  \/i^/c)=  1.2^,  and  hence  for  a 
given  settling  velocity  the  practical  radius  is  1.25  as  large  as  the  radius 
of  the  corresponding  sphere.  These  results  are  in  strict  accord  with 
theory,  because  an  irregular  particle  of  the  same  volume  as  a  sphere 
will  have  a  greater  surface  area  and  hence  a  smaller  settling  velocity; 
likewise  for  a  given  settling  velocity  an  irregular  particle  will  be  larger 
than  the  corresponding  sphere. 

As  in  the  case  of   Stokes'  law,  the  practical  sedimentation   formula 

may  be  expressed  as  v^  =  Krp-,  where  K  =    — ^ =-^^.  For  quartz 

particles  (sp.g.  ^2.65)  at  20°  C,  K  has  the  value  2.28  X  lo*.  From  the 
relation  Vp/zf^  =  Kr-/Cr-,  it  follows  that  K/C  should  also  equal  0.64, 
so  that  it  becomes  a  simple  matter  to  compute  K  from  published  tables 
for  the  value  of  C. 

Oseen's  Law  of  Settling  Velocities 

Oseen  ^  developed  a  resistance  formula  which  differs  from  Stokes' 
resistance  (equation  i)  in  that  the  latter  is  the  first  term  of  Oseen's 
series 

R  =  67rn;z.(i+|^|z.|) (15) 

where  the  symbols  have  the  same  meaning  as  in  Stokes'  law,  and  \v\  is 
the  absolute  value  of  the  velocity.  By  equating  this  expression  to  the 

force  acting  downward  on  the  particle,— 7ir^(c?i  —  d2)g,  and  solving 
for  V,  Oseen  obtained  his  settling  velocity  equation 


^+V^P+3^/.(^.-4).^ 


(16) 


^d. 


4 

From  the  nature  of  equation   (16),  it  follows  that  Oseen's  equation 
becomes  identical  with  Stokes'  law  when  all  but  the  first  term  of  equation 

1  C.  W.  Oseen,  Ueber  den  Gultigkeitsbereich  der  Stokes'schen  Widerstandformel : 
Ark.  Mat.,  Astron.  Fys.,  vol.  6,  1910;  vol.  7,  191 1;  vol.  9,  1913. 


io8 


SEDIMENTARY  PETROGRAPHY 


(15)  is  neglected.  In  this  case  v  becomes  so  much  smaller  than  unity 
that  V-  may  be  neglected  in  comparison  with  v.  This  relation  may  readily 
be  seen  by  solving  equation  (16)  for  r  and  dropping  powers  of  v  higher 
than  i.^ 

Oseen's  law  is  directly  applicable  to  mechanical  analysis  by  sedimen- 
tation for  diameters  above  the  upper  limit  of  Stokes'  law.  Figure  25 

shows  the  curves  of  Stokes'  and 
Oseen's  laws,  and  indicates  how  the 
latter  departs  from  Stokes'  curve  be- 
yond diameters  of  about  0.0 1  mm.  For 
the  smaller  sizes  of  particle,  in  the  silt 
and  clay  ranges,  the  greater  ease  of 
computing  diameters  according  to 
Stokes'  law  or  Wadell's  practical  for- 
mula renders  these  laws  more  con- 
venient for  general  use. 

Goldstein's  Law  of  Settling 
Velocities 


i 

.*^ 

1 

1 

A 

^ 

/ 

/ 

J 

7 

Goldstein  -   began  his   consideration 


DIAMETtRS  IN  MM. 

Fig.    25. — Departure    of    Stokes' 
law   from   Oseen's   law.    Below   di- 
ameters of  about  0.05  mm.,  Stokes'  ..u       ^       1      •    i.     r 
law    is    a    special    case   of    Oseen's     of  the  problem  from  the  standpomt  of 
more  general  equation.                        Oseen's   resistance    formula    (equation 

15),  in  which  terms  depending  upon 
the  square  of  the  velocity  were  neglected,  and  solved  the  equation 
for  the  complete  series  introduced  by  Oseen,  restricting  himself,  how- 
ever, to  small  values  of  Reynolds  number.  Goldstein's  solution  was  ex- 
pressed entirely  in  terms  of  Reynolds  number  and  a  "drag  coefficient" 

kn.  which  he  defined  as  k^=--^^  ,  where  D  is  the  "drag"  and  the 
other  symbols  have  their  previous  meaning.  In  his  final  solution  Gold- 
stein obtained  the  expression 


^-f(^  + 


3  r__^r:+_7i_r; 


1,280 


20,480 


(17) 


for  the  complete  law.   In  this   series   Stokes'  law  is   /eD==i2/Re  and 
Oseen's  law  is  i^D=  (12/Re)  +  2.25.  Goldstein  pointed  out  that   for 


1  H.    Gessner,    Die    Schldmmanalysc    (Leipzig,    1931),    P-    20,    shows   the    steps 

2  S.  Goldstein,  The  steady  flow  of  viscous  fluid  past  a  fixed  spherical  obstacle  at 
small  Reynolds  numbers:  Proc.  Roy.  Soc.  London,  pp.  225-235,   1929. 


PRINCIPLES  OF  SIZE  ANALYSIS  109 

values  of  Re  less  than  1.6  it  was  unnecessary  to  consider  the  influence  of 
corrections  beyond  Oseen's  expression. 

It  is  interesting  in  connection  with  the  abbreviated  forms  of  Stokes'  and 
Oseen's  laws  just  given,  that  Stokes'  law  and  Rubey's  equation  may  also  be 
expressed  very  briefly  in  terms  of  the  coefficient  of  resistance,  Cp,  and 
Reynolds  number  as  Cf  =  24/Re  and  Cf  =  (24/Re)  +2,  respectively.^ 

To  a  large  extent  mechanical  analysis  by  sedimentation  methods  is 
performed  on  silt  and  clay,  but  as  methods  are  developed  for  sedimen- 
tation studies  of  sand  and  gravel,  Oseen's  law  and  perhaps  Goldstein's 
further  modifications  may  be  more  extensively  used.  These  same  con- 
siderations affect  the  more  extensive  use  of  Wadell's  theoretical  law, 
which  agrees  with  observed  settling  velocities  up  to  values  of  Reynolds 
numbers  of  about  3,000.  Moreover,  the  allowance  for  shape  factors 
which  may  be  included  in  Wadell's  formulas  becomes  increasingly  im- 
portant as  large  particles  are  studied  by  their  settling  velocities. 

Summary  of  Laws  of  Settling  Velocities 

The  preceding  discussion  of  settling  velocities  indicates  that  for  prac- 
tical purposes  Stokes'  law  (equation  3)  and  Wadell's  practical  sedi- 
mentation formula  (equation  14)  are  most  applicable  to  mechanical 
analysis.  Stokes'  law  and  Wadell's  formula  both  extend  to  values  of 
Reynolds  number  of  at  least  0.2  (diameter  about  0.06  mm.  for  quartz 
spheres  at  20°  C). 

Stokes'  law  is  widely  used  in  mechanical  analysis  and  affords  good 
values  even  for  sedimentary  particles,  although  it  is  generally  recognized 
that  perfect  agreement  is  not  theoretically  possible  because  of  the  non- 
spherical  shapes  involved.  Wadell's  practical  formula  is  a  modification 
of  Stokes'  law,  designed  to  adjust  this  limitation,  and  it  undoubtedly  is 
the  most  accurate  expression  for  general  use.  It  may  be  mentioned  that 
the  consideration  of  the  assumptions  of  Stokes'  law  is  pertinent  also 
to  the  practical  sedimentation  formula,  with  the  exception  of  the  as- 
sumption of  sphericity. 

For  practical  purix)ses,  the  most  convenient  manner  of  using  Stokes' 
equation  or  Wadell's  equation  is  from  tabulated  values  of  the  settling 
velocities.  Table  12  furnishes  these  values  for  Stokes'  law,  and  Table 
13  makes  available  the  corresponding  values  from  Wadell's  formula 
for  the  same  conditions  of  sedimentation.  It  may  be  noted  that  any  two 
corresponding  values  in  Tables  12  and  13  satisfy  the  ratio  K/C  =  o.64. 

^  J.  E.  Christiansen,  loc.  cit.,  1935. 


no 


SEDIMENTARY  PETROGRAPHY 


Table  12 


Settling  Velocities  of  Spheres  of  Specific  Gravity  2.65  and  2.70 
AT  Temperatures  of  15°  and  20°  C,  Computed  from  Stokes'  Law  * 


DIAMETERS  IN 
MILLIMETERS 

SETTLING  VELOCITIES  IN  CM./SEC. 

Specific   Gravity   of 
Particles  =  2.65 

Specific   Gravity   of 
Particles  =  2.J0 

15°  c. 
C=3.i4Xio* 

20°  C. 
C=3-57Xio* 

15°  c. 

C-3^24Xio* 

20^  C. 
€=3.67X10* 

1/16  0.0625 

.0442 
T^/l^ 0312 

.0221 
1/64 0156 

.0110 
I/128 0078 

•0055 
1/256 0039 

.00276 
I/512 00195 

.00138 
1/1024 00098 

.00069 
1/2048 00049 

0-305 
•153 
.0764 
.0382 
.0191 
.0096 
.0048 
■00239 
.00120 
.00060 
.00030 
.000148 
.000075 
.000038 
.000019 

0.374 
•174 
.0869 

•0435 
.0217 
.0109 

•00543 

.00272 

.00136 

.00068 

•00034 

.000168 

.000085 

.000043 

.000021 

0-315 
•158 
•079 
•0394 
.0197 
.0099 

.00493 

.00247 

.00123 

.00062 

.00031 

.000153 

.000077 

.000039 

.000019 

0-357 
.179 
.0893 

.0447 

.0223 

.0112 

.00558 

.00280 

.00140 

.00070 

-00035 

.000173 

.000087 

.000044 

.000022 

*    Computations  by  slide  rule. 


THEORY     OF     SEDI  MEN  TING     SYSTEMS 

Laws  of  settling  velocities  confine  themselves  to  the  settling  rates  of 
individual  particles  falling  through  a  fluid  of  infinite  extent.  In  mechan- 
ical analysis,  however,  one  deals  with  a  system  of  particles,  and  it  is 
important  to  consider  how  the  system-  behaves  as  a  whole  during  sedi- 
mentation. Such  systems  of  particles  may  be  all  of  one  size  (mono- 
disperse)  or  of  various  sizes  (poly disperse). 

In  the  following  discussion  it  will  be  assumed  that  the  concentration 
of  the  system  is  so  dilute  that  the  particles  do  not  interfere  with  one 
another  during  descent,  that  the  particles  are  small  spheres  (to  permit 
the  direct  application  of  Stokes'  law),  and  that  no  coagulation  phe- 
nomena are  present.  In  actual  mechanical  analysis,  of  course,  these 
simplifications  may  not  hold  rigorously,  but  it  will  be  shown  that  for  all 
practical  purposes  the  theory  affords  a  sound  basis  for  a  number  of  new 
techniques  of  mechanical  analysis. 


PRINCIPLES  OF  SIZE  ANALYSIS 
Table  13 


III 


Settling  Velocities  of  Spheres  of  Specific  Gravity  2.65  and  2.70  at 

Temperatures  of  15°  axd  20°  C,  Computed  from  Wadell's  Practical 

Sedimentation  Formula* 


DIAMETERS  IN 

SETTLING  VELOCITIES  IN  CM./SEC. 

Specific   Gravity   of 
Particles  =  2.65 

Specific   Gravity   of 
Particles  =  2.yo 

15°  c. 

K  =  2.0lXlO* 

20°  C. 
K  =  2.28XlO'' 

15°  c. 

K=2.o7Xio^ 

20°  C. 
K=2.35Xio* 

I/16  0.0625 

.0442 
1/32  0312 

.0221 
1/64  0156 

.0110 
1/128 0078 

.0055 
1/256 0039 

.00276 
I/512 00195 

.00138 
I/1024 00098 

.00069 
1/2048 00049 

0.195 
.098 
.0487 
.0244 

.0122 

.0061 

.00306 

■00153 

.00077 

.000384 

.000192 

.000096 

.000048 

.000024 

.000012 

0.221 
.III 
•0554 
.0278 
.0139 
.0070 
.00348 
.00174 
.00087 
.000435 
.000217 
.000108 
.000054 
.000027 
.000013 

0.200 
.101 
.050 
.0251 
.0126 
.0063 
.00316 
.00158 
.00079 
•000395 
.000197 
.000098 
.000049 
.000024 
.000012 

0.228 
.114 
•057 
.0286 

•0143 

.0072 

.00358 

.00179 

.00089 

.000448 

.000224 

.0001X2 
.000056 
.000028 
.000014 

*  Computations  by  slide  rule. 

Sedimentation  of  monodisperse  systems.  The  laws  g-overning  the  sedimen- 
tation of  monodisperse  systems  are  quite  simple,  inasmuch  as  all  the  particles 
are  of  one  size.  Assuming  a  dilute  suspension  of  spheres  settling  in  a  fluid 
at  constant  temperature,  it  follows  that  each  particle  will  settle  with  the  same 
velocity  v,  and  in  time  t  will  have  reached  a  point  /;  units  lower  in  the  column 
of  fluid,  on  the  basis  of  the  relation  v  =  h/t.  If  one  considers  a  cylinder  of 
height  /;,  which  has  the  particles  uniformly  distributed  through  the  liquid  at 
time  ^0'  it  is  clear  that  the  entire  suspension  will  settle  with  a  uniform  speed 
and  collect  on  the  bottom  of  the  vessel.  Now  the  amount  of  material  p  which 
has  settled  out  at  any  given  time  t  is  dependent  on  the  total  amount  of  dis- 
persed material  P,  on  the  time,  on  the  velocity,  and  inversely  on  the  depth  h. 
This  permits  the  setting-up  of  the  relation 


kVvt 
h 


in  which  k  is  the  constant  of  proportionality. 


(18) 


112        SEDLMEXTARY  PETROGRAPHY 

At  the  time  when  all  the  material  has  settled  to  the  bottom,  />  =  P,  and 

kit 

but  /i  =  zt  from  the  relation  v^h/t,  so  that  k=i,  and  the  equation   (i8) 
may  be  written  simply  as 

^  =  -v ^^9) 

Sedimentation  of  polydispcrse  systems.  In  polydisperse  systems  the 
problem  may  be  considered  from  the  point  of  view  of  a  series  of  mono- 
disperse  systems,  with  the  radii  of  the  successive  groups  differing  by 
infinitesimals  from  each  other.  This  was  the  approach  made  by 
Oden  ^  in  1915,  and  it  forms  the  first  clear  expression  of  a  mathematical 
theory  of  sedimenting  systems.  Oden  later  generalized  his  theory  in 
collaboration  with  Fisher,-  and  this  latter  form  of  the  theor}-  furnishes 
the  foundation  of  all  the  modern  precision  methods  of  mechanical  anal- 
ysis. Oden's  original  theory  is  an  excellent  example  of  the  application  of 
mathematical  analysis  to  the  solution  of  a  complicated  problem,  and  an 
abbreviated  form  of  the  theor\'  is  presented  here. 

Oden's  original  theory.  Consider  a  polydisperse  suspension  with  its  particles 
uniformly  distributed  through  the  liquid.  As  sedimentation  proceeds,  each  frac- 
tion having  a  given  radius  settles  as  a  unit,  and  at  any  given  time  the  amount  of 
material  wliich  settles  to  the  bottom  consists  of  fractions  which  have  com- 
pletely settled  from  the  suspension,  plus  some  part  of  the  fractions  which 
have  not  completely  settled  out,  because  their  velocities  are  not  great  enough 
to  carry  them  from  the  top  to  the  bottom  of  the  cylinder  in  the  time  involved. 
The  total  amount  settled  on  the  bottom  may  be  indicated  by  P(0.  ^nd  this 
is  to  be  divided  into  the  two  parts  mentioned.  The  fraction  that  has  com- 
pletely settled  from  the  suspension  has  a  velocity  greater  than  h/t,  and  the 
portion  of  the  partially  sedimented  fractions  that  have  settled  out  has  a 
velocity  less  than  /)//.  The  value  of  this  portion  is  given  by  equation  (19). 

The  letter  P  represents  a  mixture  of  particles  in  which  the  successive  radii 
differ  from  each  other  by  infinitesimals.  It  may  therefore  be  written  as 
P  =  F(r)£fr.  In  order  to  express  />  as  a  function  of  the  radii,  we  may  sub- 
stitute this  value  for  P  in  equation  (-19)  : 

_  F(r)dr  •  it 

^ h 

and  since  by  equation  (4),  v  =  Cr-, 

_  F(r)r-dr  •  Q 


(20) 


where  C  is  the  constant  of  Stokes'  law. 

1  S.  Oden,  he.  cit.,  191 5. 

2  R.  A.  Fisher  and  S.  Oden,  The  theory  of  the  mechanical  analysis  of  sediments 
by  means  of  the  automatic  balance:  Proc.  Roy.  Soc.  Edinburgh,  vol.  44,  pp.  98-115, 
1923-24- 


PRINCIPLES  OF  SIZE  ANALYSIS 


113 


When  a  fraction  is  completely  sedimented,  p  =  F,  so  that  i 


r-Ct 


;  and 


-V? 


(21) 


The  value  for  r  in  equation  (21)  is  the  critical  radius  for  any  time  t,  which 
determines  tlie  fractions  that  completely  settle  to  the  bottom  and  those  which 
only  settle  in  part.  We  may  now  use  tliese  values  in  setting  up  an  expression 
for  P(0: 

\'li  Ct  00 

P(^)^  rF(;-);--W;--a    ^     f     p^^.y^         (22) 

0  vhTcT 

In  this  equation  the  first  integral  is  the  sum  of  the  portions  of  the  partially 
sedimented  fractions,  whose  radii  are  smaller  than  V/j/C/;  and  the  second 
integral  is  tlie  sum  of  the  completely  sedimented  fractions,  whose  radii  are 
greater  than  this  critical  radius. 

By  ditTerentiating  equation    (22)   there  results: 

Vh/Ct 


dF(t) 
dt 


/¥{r)r-dr 
h 


(23) 


It  will  be  noticed  that  equation  (23)  is  exactly  like  the  first  term  on  the  right- 
hand  side  of  equation    (22),  except  that  t  is  absent.   If   equation    (23)    is 
multiplied  through  by  t,  we  obtain: 
Vh/Ct 


dF{t) 
dt 


C¥{r)r-dr 


C^ 


(24) 


Thus  the  important  point  has  been  established  that  the  sum  of  the  partially 
sedimented  fractions  is  equal  to  tlie  first  derivative  of  P(0.  multiplied  by  /. 


A_^ 

""TTTT 

^rr-^^^^^ 

C 

/ 

( 

t 

Fig.   26. — Principles   of  graphic  analysis  of  Oden  curves.  See  text  for  details. 

This  result  may  now  be  used  in  interpreting  the  experimental  curve  and  ob- 
taining the  frequency  distribution  of  the  analyzed  material.  Figure  26  shows 


114 


SEDIMENTARY  PETROGRAPHY 


the  P(f)  cune  as  a  function  of  the  time,  obtained  by  weighing  the  amount 

of  material  that  has  accumulated  at  the  bottom  of  a  cylinder  of  suspension.  At 

the  point  /  an  ordinate  has  been  erected,  and  at  A,  where  this  intersects  the 

P(t)  curve,  a  tangent  has  been  drawn,  intersecting  the  Y-axis  at  B.  Two 

horizontal  lines.  AD  and  BC,  are  also  drawn  as  indicated.  We  know  that  the 

distance  OD  is  the  total  weight  of  sediment  at  time  t,  and  it  is  required  to 

prove  that  tlie  distance  OB  is  the  amount  of  material  having  radii  greater 

than  that  radius  which  is  obtained  by  substituting  the  particular  value  of  t 

in  equation  (21). 

The  derivative  of  the  P(0  cune  at  anv  point  is  its  slop>e  at  that  point.  This 

.^'^^         ,         dP(J)        AC 
is  represented  by  tan  a.  But  tan  a 


BC- 


AC/BC,  so  that 
dP{t) 


dt 


BC 


dt 


AC 


Xow  the  distance  BC  represents  the  time  of  sedimentation,  and  hence  by 
substituting  /  for  BC  we  obtain 

dPif)        _      ^^ 

(25) 


dt 


AC  =BD 


We  have  already  seen  from  equation  (24)  that  t 


dP{t) 
dt 


represents  the 


partially  sedimented  fractions,  and  by  equation  (25)  it  was  shown  that  this 
equals  the  line  segment  BD.  This  means  graphically  that  of  the  total  amount 
OD  sedimented  in  a  particular  time  /,  the  distance  OB  represents  the  portion 


^^ 

1 

_^^ — " 

^.r-"""^^^''^ 

J^--^""^ 

.^-"""^^Z^^y^ 

PYRAMID          ! 

// 

// 

1 

TIME.   RADIUS,     OR     VELOCITY 

Fig.  2-j. — Relation  between  Oden's  P(/)  curv'c,  the  cumulative  curve,  and  the  fre- 
quency pyramid  or  histogram  of  a  sediment. 


which  has  completely  settled  out  of  suspension  and  the  distance  BD  represents 
the  portion  only  partially  sedimented. 

If  we  now  choose  a  series  of  time  intervals  corresponding  to  the  settling 
velocities  or  radii  of  particular  grade  sizes,  it  is  necessary  only  to  draw 
tangents  to  the  P(0  curve  at  those  points  and  to  read  off  the  intercepts  (the 


PRINCIPLES  OF  SIZE  ANALYSIS  115 

values  of  OB)  on  the  Y-axis.  The  data  so  obtained  are  cumulative,  and  by 
subtracting  one  value  from  the  next  the  amount  of  material  in  any  grade  is 
directly  obtained.  Figure  27  illustrates  an  original  V{t)  curve,  the  cumulative 
curve  obtained  from  it,  and  the  histogram  derived  from  the  cunuilative  curve. 

The  modified  Odcn  theory.  It  was  pointed  out  that  the  theory  devel- 
oped by  Odcn  in  191 5  was  generalized  in  1923-1924  by  Fisher  and 
Oden  to  include  all  possible  methods  of  determining  the  frequency  dis- 
tributions of  soils  and  sediments  by  indirect  methods.  In  1925  Oden  ^ 
summarized  the  general  theory,  and  his  last  pai>er  is  followed  in  the 
accompanying  discussion. 

It  is  assumed  that  a  suspension  of  particles  which  has  complete  dis- 
persion and  a  uniform  distribution  of  the  particles  at  time  f  ==  o  is  at  a 
constant  temperature  and  has  a  concentration  so  dilute  that  the  jxirticles 
do  not  interfere  with  each  other  during  their  descent.  If  G  is  the  total 
weight  of  the  particles  suspended  in  V  c.c.  of  water  and  .s-  is  the  six^cific 
gravity  of  the  particles,  then  at  time  to  every  cubic  centimeter  of  the 
suspension  contains  G/V  g.,  or  G/V.y  c.c,  of  solid  particles,  and  there- 
fore (1  —  G/V,y)  c.c.  of  water.  Hence  the  uniform  specific  gravity  of 
the  suspension  at  the  start  is 

</>o=i  +  -^(-^-i) (26) 

Let  z  represent  the  fraction  or  percentage  by  weight  of  particles  hav- 
ing a  velocity  less  than  z'  =  x/t.  Then  at  time  t  and  depth  .r  there  will 
remain  ^G  g.  of  particles  per  V  c.c,  because  at  that  time  all  jxirticles 
with  a  velocity  greater  than  x/t  will  have  settled  below  this  depth,  and 
those  particles  whose  velocity  is  less  than  x/t  will  continue  in  the  same 
concentration  as  at  the  start.  Hence  at  time  t  and  depth  x  the  specilic 

gravity  of  the  suspension  will  be  </>=  i  +-^77  (^' —  0'  ov 

</>=i  +  /^'" (27) 

where  k  = — -^ ,  a  constant  under  the  given  exi^erimental  condi- 
tions. Equation  (27)  is  fundamental  to  the  derivation  of  the  several 
methods  which  may  be  used  to  determine  the  frequency  curve  of  the 
sediment.  By  definition  .c  is  a  function  of  v,  and  it  is  an  ordinate  of  the 
cumulative  curve  of  the  sediment.  If  we  let  Y  =  dc/dv,  or  dc=Ydv, 
it  is  clear  that  Y  must  be  an  ordinate  of  the  frequency  ctn"ve,  since  da 
is  the  proportion  of  particles  between  z'  and  ■:'-{■  dx<  and  is  equal  to  Y 

^  S.  Odcn,  The  size  distribution  of  particles  in  soils  and  the  experimental  methods 
of  obtaining  them:  Soil  Science,  vol.  19,  pp.  1-35,  1925. 


Ii6        SEDLMENTARY  PETROGRAPHY 

times  dv.  Here  Y  is  the  frequency  function  /(f)  of  the  sediment. 
It  is  usually  more  convenient  to  represent  settling  velocities  as  log 
V  or  radii  as  log  r  for  purposes  of  graphing  them.  In  the  former 
case  we  may  let  dz^Ydv/v=Y •d{\ogev).  Since  by  Stokes'  law 
v==Cr^,  and  hence  dv  =  2Crdr,  we  may  substitute  these  values  for  v 
and  dv  in  the  preceding  equation  and  obtain  ds  =  2Y dr/r  =  2Y-d  (loger) , 
where  Y  is  now  the  frequency  function  /(loger).  Either  of  these  values 
of  Y  may  be  used  in  the  following  treatment,  depending  on  the  method 
of  presentation  desired. 

For  the  sake  of  completeness  a  summary  of  the  important  equations  and 
their  significance  is  included  here,  but  the  interested  reader  is  referred  to  the 
writers  mentioned  for  a  full  discussion  of  the  mathematical  details.  In  de- 
veloping the  equations  which  follow,  equation  (27)  is  fundamental,  and  in 
addition  the  two  partial  derivatives  of  Z'  —  x/t,  Sz-/5x=i/t  and  Sv/5t  = 
—  x/t-,  are  of  importance. 

There  are  four  general  methods  by  which  the  frequency  function  /(loge^') 
or  /(logg/')  may  be  obtained  from  equation  (zj).  Changes  in  the  density  of 
the  suspension  may  be  considered  as  a  function  of  the  time  at  a  constant  depth,  or 
as  a  function  of  the  depth  at  a  constant  time.  Similarly,  changes  in  the 
hydrostatic  pressure  may  be  measured  as  a  function  of  the  time  or  of  the 
depth.  In  addition  to  these  four  general  methods,  Oden  discussed  two  others. 
One  concerns  the  change  in  weight  of  an  immersed  body,  and  the  other 
considers  the  weight  of  sediment  accumulating  at  the  base  of  a  suspension. 

If  we  consider  first  changes  in  the  density,  equation  (27)  may  be  differ- 
entiated with  respect  to  x.  ^Multiplication  by  5iy5i'  yields  an  expression  in 
which  several  substitutions  may  be  made  to  obtain 

^=f^     (^«) 

By  diiTerentiating  equation  (27)  witli  respect  to  t,  and  multiplying  by  ^v/ov, 
an  expression  is  obtained  in  which  substitution  yields 

^=-x^     (^') 

In  order  to  consider  changes  in  the  hydrostatic  pressure  of  the  suspension, 
it  should  be  recalled  that  if  p  denotes  the  hydrostatic  pressure  at  depth  x  in 
the  suspension,  then  at  depth  x  +dx,  the  increment  to  p,  ^p,  is  equal  to  ^^x, 
because  it  is  the  density  times  the  added  depth  which  equals  the  added  weight, 
and  hence  measures  the  added  pressure.  Consequently,  4>  =  ^p/^x,  and 
S(p/5x  =5-p/5x-.  We  may  substitute  the  value  of  Y  from  equation  (28)  for 
S(p/dx  in  the  last  equation,  and  obtain 

^-X^         (30) 

Equation  (30)  measures  the  hydrostatic  pressure  as  a  function  of  the  depth 
at  a  constant  time.  In  order  to  consider  the  decrease  as  a  function  of  the  time 


PRINCIPLES  OF  SIZE  ANALYSIS  117 

at  a  constant  depth,  a  more  lengthy  mathematical  process  is  necessary,  which 
yields  as  its  final  result  the  equation 

Y  =  ^ii^ (3.) 

\Ve  may  next  consider  the  change  in  weight  of  an  immersed  body.  A  cyl- 
inder of  weight  \V  in  air  and  cross-sectional  area  /  is  counterpoised  in  the 
suspension  by  a  weight  A,  which  just  balances  when  the  upper  end  of  the 
cylinder  is  at  the  surface  of  the  liquid.  As  the  density  of  the  suspension  de- 
creases, A  will  increase.  The  relation  that  holds  for  any  time  then  is 
A  =  \\'  —  fp,  or  5A/5/  =  ~-fdp/5t.  We  may  take  the  second  derivative  of  the 
same  expression,  and  by  substitution  from  equation  (31)  gain  the  result 

--i,^     <-) 

Finally,  the  rate  of  accumulation  of  sediment  on  a  plate  of  area  a  suspended 
at  a  depth  x  below  the  surface  of  the  suspension  may  be  considered.  To  obtain 
the  value  of  Y  in  this  case  it  is  necessary  to  compute  the  pressure  exerted 
on  the  pan  by  the  column  of  water  of  height  x,  and  the  downward  force 
exerted  by  the  particles  remaining  in  suspension.  The  relation  obtained  is 
differentiated  with  respect  to  t,  and  substitution  in  the  resulting  derivative 
yields,  after  some  simplification,  the  equation 

^^--X;    W (33) 

where  Ai  represents  the  total  weight  of  all  the  particles  in  the  suspension. 


MODERN     METHODS    OF     MECHANICAL    ANALYSIS 

Any  of  the  four  general  equations,  (28),  (29),  (30),  or  (31),  may 
be  used  to  determine  the  ordinate  of  the  frequency  curve.  The  possible 
methods  fall  into  two  groups.  Either  the  density  or  the  hydrostatic  pres- 
sure is  measured  as  a  function  of  the  time  at  a  constant  depth,  or  they 
are  measured  as  a  function  of  the  depth  at  a  constant  time.  It  is  clear 
that  the  apparatus  needed  for  the  latter  group  would  be  more  unwieldy 
than  for  the  former,  because  it  would  be  necessary  to  take  measurements 
at  a  number  of  points  in  the  suspension  simultaneously.  Specifically, 
equation  (28)  requires  the  simultaneous  observation  of  the  density  at 
various  depths  in  the  suspension  at  a  given  time  t.  From  these  data 
S<f>/Sx  for  the  different  values  of  x  may  be  computed,  and  their  sub- 
stitution in  the  equation  furnishes  a  series  of  values  for  Y.  Each  value 
of  Y  corresponds  to  a  certain  value  of  v  or  of  r,  and  the  frequency  curve 
may  be  plotted  with  the  Y's  as  ordinates  and  loge^'  or  loger  as  abscissae 
In  the  case  of  equation  (30)  it  is  necessary  to  measure  the  hydrostatic 
pressure  simultaneously  at  various  depths.  From  the  data  obtained,  the 


Ii8       SEDIMENTARY  PETROGRAPHY 

second  derivative  of  the  pressure  with  respect  to  the  depth  may  be 
computed,  and  by  substitution  the  corresponding  vakies  of  Y  may  be 
found. 

Oden^  developed  an  apparatus  for  the  appHcation  of  equation  (30). 
He  used  a  long  sedimentation  tube,  to  which  ten  capillary  tubes  were 
attached.  The  capillary  tubes  were  filled  with  pentane,  and  at  any  instant 
the  values  measured  by  them  presented  a  curve  across  the  tubes  in  the 
rack.  The  method  was  discarded  by  Oden  because  of  a  number  of  sources 
of  error  which  could  not  be  corrected  by  any  simple  device. 

By  far  the  greatest  number  of  methods  of  analysis  utilize  equations 
(29)  and  (31).  Equation  (29)  is  the  foundation  of  the  pipette  method,- 
but  in  practice  the  equation  itself  is  seldom  used,  because  sufficient  data 
for  cumulative  curves  or  histograms  can  be  obtained  directly  from  the 
successive  weights  of  the  pipette  residues.  The  hydrometer  method  is 
also  based  on  this  equation.  Equation  (31)  forms  the  basis  of  such 
methods  as  Wiegner's  tube.  Equation  (32)  affords  a  method  of  deter- 
mining the  frequency  curve  from  the  apparent  change  in  weight  of  a 
plummet  counter jx)ised  in  the  suspension.  Equation  (33)  was  developed 
especially  for  the  Oden  sedimentation  balance.  For  the  application  of 
centrifugal  force  to  modern  methods,  see  page  123. 

Siinmwry  of  modern  methods:  the  inherent  error.  The  discussion  of 
methods  of  analysis  based  on  Oden's  theory  would  not  be  complete 
without  a  discussion  of  the  sources  of  error  involved  in  their  application. 
The  theory  itself  raised  mechanical  analysis  to  new  high  levels  of  de- 
velopment, but  in  the  practical  application  of  the  theory  various  com- 
plicating factors  arise.  Several  writers  ^  have  discussed  the  errors  in- 
volved. Coutts  and  Crowther  showed  that  an  "inherent  error"  is  in- 
volved in  Oden's  method  due  to  currents  set  up  in  the  suspension,  owing 
to  differences  below  and  at  the  edges  of  the  suspended  balance  pan.  These 
currents  interfere  with  normal  settling  of  the  particles.  Shaw  and 
Winterer  pointed  out  that  another  source  of  error  is  due  to  electrical 
charges  on  the  wall  of  the  vessel  which  tend  to  draw  the  smaller  particles 
to  the  outer  portions  of  the  suspension.  Gessner  pointed  out  that  in 
Wiegner's  tube  clear  water  enters  the  suspension  from  the  manometer 
and  interferes  with  sedimentation.  The  pipette  method  is  also  subject 

1  S.  Oden,  loc.  cit.,  1925. 

2  Details  of  this  and  other  methods  of  analysis  are  given  in  Chapter  6. 

3  J.  R.  H.  Coutts  and  M.  Crowtlier,  A  source  of  error  in  the  mechanical  analysis 
of  sediments  by  continuous  weighing:  Trans.  Faraday  Soc,  vol.  21,  pp.  374-380, 
1925-1926.  C.  F.  Shaw  and  E.  V.  Winterer,  A  fundamental  error  in  mechanical 
analysis  of  soils  by  the  sedimentation  method :  Proc.  ist  Int.  Congr.  Soil  Sci.,  vol. 
I,  pp.  385-391,  1928.  H.  Gessner,  op.  cit.,  1931,  p.  96. 


PRINCIPLES  OF  SIZE  ANALYSIS  119 

to  error,  inasmuch  as  the  withdrawal  of  the  sample  afifects  a  spherical 
rather  than  a  thin  horizontal  zone. 

Despite  these  practical  defects,  modern  methods  may  still  be  referred 
to  as  precision  methods  in  the  sense  that  they  are  based  on  sound  prin- 
ciples, and  the  errors  may  in  many  cases  be  evaluated  sufficiently  closely 
so  that  their  limitations  may  be  known.  Kohn,^  for  example,  showed 
that  the  error  in  the  pij^ette  method  is  only  a  fraction  of  i  per  cent  in 
a  lO-c.c.  sample  taken  from  a  depth  of  10  cm.  Likewise,  various  other 
workers  -  have  argued  that  the  "inherent  errors"  may  in  many  cases  be 
small  enough  to  neglect.  Correns  and  Schott  questioned  their  importance, 
and  Vendl  and  Szadeczky-Kardoss  found  that  errors  of  the  type  men- 
tioned by  Coutts  and  Crowther  are  generally  quite  small. 


OLDER     METHODS     OF      ANALYSIS 

Sedimentation  and  Elutriation  Methods 

The  preceding  section  discussed  the  theories  underlying  modern  meth- 
ods of  analysis,  but  the  use  of  older  routine  techniques  is  still  current, 
and  it  may  be  well  briefly  to  review  the  principles  on  which  they  are 
based. 

Theory  of  decantation  methods.  Decantation  methods  of  mechanical 
analysis  are  among  the  oldest  techniques.  With  them  a  separation  of  the 
several  grades  is  effected  by  allowing  the  suspension  to  stand  until 
particles  larger  than  a  given  radius  have  settled  to  the  bottom  of  the 
vessel.  At  that  instant  the  supernatant  liquid  is  decanted  or  siphoned 
off,  and  clear  water  poured  in.  The  sediment  is  resuspended.  and  an 
equal  interval  of  time  is  allowed  to  elapse,  so  that  all  the  larger  particles 
may  completely  settle  again,  and  the  supernatant  liquid  is  drawn  off  as 
before.  By  a  repetition  of  this  process  a  practically  complete  separation 
of  the  material  into  grades  may  be  made.  The  decanted  liquid  may  itself 
be  put  through  the  same  process  to  separate  still  smaller  grades. 

No  complete  matliematical  theory  has  been  developed  for  decantation 
methods,  but  it  is  possible  to  set  up  a  relation  which  will  indicate  the  course 
of  the  process.  We  may  consider  a  bidisperse  system,  such  that  the  settling 

1  M.  Kohn,  Beitrage  zur  Theorie  und  Praxis  der  mcchanischen  Bodenanalyse : 
Landwirts.  Jahrb.,  vol.  67,  pp.  485-546,  1928. 

-  C.  W.  Correns  and  W.  Scott,  Vergleichende  Untersuchungen  iiber  Schlamm- 
und  Aufbereitungsverfahren  von  Tonen :  Kolloid  Zcits.,  vol.  61,  pp.  6S-80,  1032. 
M.  Vendl  and  E.  V.  Szadeczky-Kardoss,  Uber  den  sogennanten  grundsatzlichen 
Fehler  der  mechanischen  Analyse  nach  dem  Oden'schen  Prinzip :  Kclloid  Zcits.,  vol. 
67,  pp.  229-233,  1934. 


SEDIMENTARY  PETROGRAPHY 


velocit>-  of  the  larger  particles  is  twice  that  of  the  smaller.  In  successive 
decantations  the  amount  of  the  smaller  grade  remaining  will  obviously  be  J/2, 
14,  yi This  series  may  be  written  as  J/^,  J^z',  J^^,  - .  -  >4°,  where  the  ex- 
ponent indicates  the  number  of  decantations  involved.  In  the  general  case, 
if  the  larger  particles  have  a  velocit>-  p  times  as  great  as  the  smaller,  the 
series  is  i/p,  i/p^,  i/p^,...  i//>°.  This  relationship  enables  us  to  e.xpress  the 
amount  of  fine  material  remaining  after  any  number  of  decantations  as  a 
proportion  of  the  original  amount  of  fine  material : 

v:/w^  =  p-^ C34) 

where  w  is  the  amount  of  fine  material  left  after  «  decantations.  and  ti'o  is 
the  original  amount  of  fine  material.  Since  by  Stokes'  law  the  velocity  varies 
as  the  square  of  the  radius,  equation  (34)  may  be  expressed  in  terms  of 
radii  as 

w/7Vq  =  p-'-'- (35) 

where  p  now  represents  the  relation  beuveen  the  radii,  instead  of  between 
the  velocities.  Figure  28  illustrates  tn'O  curves  of  the  type  developed  by  equa- 
tion (35).  The  amount  of  fine  material 
remaining  is  plotted  as  ordinate  and  the 
number  of  decantations  as  abcissa.  The 
upper  curve  represents  the  case  in 
which  the  radius  of  the  larger  particles 
is  10/9  that  of  the  smaller  (/>=  10/9), 
and  in  the  lower  curve  twice  that  of  the 
smaller  (p  =  2).  The  steepness  of  the 
cur\-e  is  thus  determined  by  the  value 
of  p. 

By  means  of  equation    (35)   it  is 
possible  to  compute  the  number  of 
decantations    necessar\-    to    effect    a 
separation  of  the  two  grades  in  a  bidi- 
sperse  system  to  any  desired  degree 
of  accu^ac)^  From  the  nature  of  the 
equation  it  is  clear  that  a  complete 
separation  can  never  be  made,  but  we 
may  consider  the  case  in  which  not 
more    than     i     per    cent    of    finer 
material  remains.  If  p  =  2,  and  zf/u'o  =  o.oi,  these  values  may  be  sub- 
log  o.oi 
2  log-  2 


■^M 

s. 

^ 

i    \^ 

0 

\      \^                                                         ' 

z 

\          ^v 

z 

75 

1             ^\ 

■< 

\                \ 

s 

\                       X 

sc 

\           Tv 

_, 

\           1  \ 

< 

\                 X 

CE 

iO 

\ 

•£ 

2 

\                     ^\^ 

z 

2S 

0 

\ 

3 

C 

^^^ 

Fig.  28. — Progress  of  decantation 
methods  in  separation  of  bidisperse 
systems.  Ciir\-e  A  represents  the  sys- 
tem R/r  =  10/9 :  curve  B  represents 
the  system  R/r  ^  2,  where  R  and  r 
are  the  large  and  small  radii,  respec- 
tively. 


stituted  in  the  equation.  After  taking  logs,  we  obtain  n  = 

from  which  the  value  of  n  is  found  to  lie  between  3  and  4.  In  actual 
practice  the  suspensions  are  polydisperse,  which  introduces  complexities 
because  there  is  a  continuous  decrease  in  size  of  radius  from  one  grade 
to  the  next  and  the  nature  of  the  frequency  distribution  determines  to 


PRINCIPLES  OF  SIZE  ANALYSTS  121 

some  extent  the  successive  weights  of  material  left.  To  determine  the 
exact  error  involved  in  any  given  number  of  decantations  it  v^ould  be 
necessary  to  know  the  frequency  distribution  of  the  sediment.  The  ai>- 
proximate  number  of  decantations  necessary  to  effect  a  practically  com- 
plete separation  of  the  fine  material  may  be  determined,  however.  If  it 
is  desired  to  have  not  more  than  i  per  cent  of  material  finer  than  9/10 
of  the  critical  radius  at  which  the  separation  is  to  be  made,  the  value  of 
n  from  equation  (35)  is  found  to  be  22. 

It  would  seem  from  these  ix:)ints  that  decantation  methods  are  subject 
to  very  definite  errors  in  practice,  and  in  addition  they  are  affected  by 
errors  of  the  same  nature  as  those  discussed  under  modern  methods. 
That  is,  when  the  liquid  is  decanted  or  siphoned  off,  a  noticeable  amount 
of  the  sedimented  material  is  often  carried  over.  This  of  course  acts  as 
a  comiDensating  error,  but  it  renders  more  difficult  the  exact  evaluation 
of  the  total  error. 

Rising  current  clutriation.  Rising  current  elutriation  was  more  ex- 
tensively used  in  the  past  than  it  is  at  present,  although  the  elutriator 
devised  by  Schone  ^  is  still  used  to  a  considerable  extent.  In  the  general 
method  a  current  of  water  is  sent  up  through  a  vertical  tube,  and  all  the 
particles  whose  settling  velocities  in  quiet  water  are  less  than  the  ui> 
ward  velocity  of  the  water  are  carried  away.  By  varying  the  strength  of 
the  current,  or  by  introducing  several  tubes  of  varying  dimensions,  a 
separation  into  several  grades  may  be  effected. 

The  theory  underlying  elutriation  by  rising  currents  involves  first  the  rela- 
tion between  the  settling  velocity  of  tlie  particle  and  the  upward  velocity  of 
the  water.  It  is  clear  tliat  the  downward  velocity  7'  of  the  particle  at  any 
instant  is  the  settling  velocity  Z'q  of  the  same  particle  in  quiet  water,  decreased 
by  the  upward  velocity  z',„  of  the  current : 

^'  =  ^'o-^'. (36) 

If  the  upward  velocity  of  tlie  water  is  greater  than  the  settling  velocity  of 
the  particle,  the  latter  is  carried  upward  with  a  velocity  equal  to  that  of  the 
current,  diminished  by  the  settling  velocity  of  the  particle.  In  this  case  the 
V  of  equation  (36)  has  a  negative  value. 

The  velocity  of  the  particles  carried  upward  by  the  current  depends  on  the 
difference  between  7',,,  and  v^.  In  separating  the  smaller  particles  of  a 
bidisperse  system,  the  velocity  of  the  water  is  made  equal  to  the  settling 
velocity  of  the  larger  particles,  so  that  they  remain  suspended,  while  the 
smaller  ones  are  borne  away.  In  polydisperse  systems  the  separation  between 
two  grades  is  made  in  an  analogous  manner.  The  current  is  so  adjusted  that 
it  is  die  same  as  the  settling  velocity  of  the  particles  having  the  critical 
radius,  and  all  the  smaller  material  is  carried  off  by  the  rising  current. 

1  E.  Schone,  he.  cit..  1868. 


122       SEDIMENTARY  PETROGRAPHY 

The  time  required  to  separate  the  smaller  panicles  from  a  sediment 
depends  on  the  length  of  the  vessel  and  on  the  velocity  with  which  they 
rise  through  the  tube.  Gessner,^  computing  the  time  necessar)-  to  effect 
complete  separations  between  grades  of  various  sizes,  found  that  it 
increased  verj-  rapidly  for  the  finer  sizes.  The  time  element,  therefore, 
is  one  of  the  fartors  which  limits  the  usefulness  of  rising  current  elu- 
triators. 

Another  of  the  difticulties  met  in  rising  current  elutriators  is  the  lack 
of  uniformity  of  the  current  throughout  the  cross  section  of  the  tube. 
This  is  due  to  wall  friction,  and  in  many  instances  particles  are  seen  to 
be  carried  upward  in  the  central  part  of  the  tube,  only  to  settle  down 
again  along  the  sides.  It  is  necessary,  therefore,  to  assume  an  average 
velocity  of  the  current  and  to  base  separations  on  that  average  value. 

Rising  current  elutriation  has  been  applied  to  a  wide  range  of  sizes, 
but  the  large  volumes  of  water  required  and  the  time  necessary  to  sepa- 
rate the  smaller  grades  impose  practical  limits.  Another  factor  which 
should  be  kept  in  mind  is  tliat  the  dispersing  electroljle  becomes  greatly 
diluted  as  fresh  water  enters  the  tube;  when  the  concentration  of  the 
dispersing  agent  approaches  zero,  a  coagulation  of  the  smaller  particles 
may  follow.  Thus  the  method  may  be  strictly  applicable  only  to  sizes 
above  the  limit  of  coagulation;  in  general  it  is  best  suited  to  particles 
above  o.oi  mm,  in  diameter.  When  used  within  the  range  of  sizes  to 
which  it  is  suited,  rising  current  elutriation  appears  to  afford  a  con- 
venient and  practical  method  of  sorting  sediments  into  grades. 

Air-current  elutriation.  The  theor}'  of  rising  current  elutriation  dis- 
cussed in  the  preceding  section  applies  equally  well  to  fluids  other  than 
water.  Rising  currents  of  air  have  been  used  extensively  for  the  sepa- 
ration of  fine  powders,  but  the  method  has  not  been  used  by  sedimentary 
petrologists.  In  general,  the  same  theoretical  considerations  apply  in  the 
separation  of  the  grades  by  air,  except  that  the  much  lower  density  of 
air  permits  a  more  thorough  separation  of  ver)'  fine  grades,  lessens  the 
effects  due  to  wall  friction,  and  renders  less  important  variations  in 
temperature.  Instead  of  coagulation  phenomena,  which  commonly  ac- 
company water  elutriation,  air  elutriation  often  develops  electrical 
charges  on  the  particles,  w^hich  must  be  eliminated.  Roller  -  has  recently 
shown  that  air  elutriators  may  be  designed  to  eliminate  most  of  the 
difficulties  commonly  encountered.  Details  of  his  apparatus  and  tech- 
nique are  given  in  Chapter  6. 

1  H.  Gessner,  op.  cit.,  1931,  p.  118. 

-  P.  S.  Roller,  Separation  and  size  distribution  of  microscopic  particles :  U.  S. 
Depf.  Commerce,  Bur.  of  Mines,  Tech.  Paper  490,  1931. 


PRINCIPLES  OF  SIZE  ANALYSIS  123 

Application  of  centrifugal  force  to  mccluuiical  analysis.  A  number  of 
workers  have  applied  centrifugal  force  to  sedimenting  systems  in  order 
to  hasten  the  settling  of  small  particles.  Among  the  earliest  users  of 
centrifugal  force  for  this  purpose  was  the  United  States  Bureau  of 
Soils.i 

The  general  theory  of  centrifugal  force  as  applied  to  the  sedimentation 
of  small  particles  was  developed  by  Svcdbcrg  and  Nichols,-  who  used 
the  method  to  determine  the  size  of  colloidal  particles.  Other  workers 
applied  the  theory  to  modern  methods  of  analysis ;  among  these  are 
Trask,^  who  developed  a  centrifugal  modification  of  Oden's  method, 
and  Steele  and  Bradfield,"*  who  developed  a  centrifugal  modification  of 
the  pipette  method. 

Full  details  of  tlie  theory  of  centrifu,c:ins'  may  be  had  from  Svcdbcrg-  and 
Nicliols's  paper;  the  treatment  here  will  briefly  indicate  the  mctliod  of 
approach.  Under  centrifugal  acceleration  the  force  applied  to  cause  movement 
of  a  particle  in  a  lluid  is  i-iTrr^{d^' — d.,)i^'-(.v  +  a).  The  symbols  here  have  tbc 
same  meaning  as  the  riglit-hand  term  of  equation  (2),  except  that  for  g  has 
been  substituted  the  value  w-(.r  +  a),  where  w  is  the  angular  velocity,  a  tbe 
distance  of  the  particle  from  the  axis  of  rotation  before  fall,  and  x  the  dis- 
tance of  fall.  By  setting  tbc  above  equation  equal  to  Stokes'  resistance,  one 
obtains  O-m-T)':'  =  %irr\dj^  —  d.,) 01- (x  +  o ) ,  and  solving  tbe  equation  for  z', 
there  results 

2(d,—d.,)co-^(x  +  a)r- 
V        —  ;„  {3/ ) 

9  V 

Inasmuch  as  tbe  velocity  of  fall  through  tbe  distance  x  may  be  expressed 
as  dx/dt,  equation  (37)  can  be  written  as  a  differential  equation  in  which 
dx/dt  is  substituted  for  tbe  v  on  tlie  left.  By  rearranging  tbe  terms  for  in- 
tegration one  obtains 

X 


J  "'        2.-^{d,~d,)J  x  +  a 


This  expression  yields,  upon  integration, 

r't  = — y T-r-  '"§^e (38) 

Equation  (38)  may  be  solved  for  r  or  for  /.  Trask  applied  the  theory 
to  Oden's  method,  using  the  equation  to  solve  for  r;  Steele  and  Brad- 

1  L.  J.  Brings,  F.  O.  Martin  and  J.  R.  Pcarce,  loc.  cit.,  1904. 

2T.  Svcdbcrg  and  J.  B.  Nichols,  Dotcrmiiiation  of  size  and  distribution  of  size 
of  particle  by  centrifugal  niotliods :  Jour.  Am.  Clicm.  Soc,  vol.  45,  pp.  2910- 2917, 
1923. 

=' P.  D.  Trask,  Mechanical  analysis  of  sediments  by  centrifuge:  Ecou.  Ccoloiiy, 
vol.  25,  pp.  581-599.  1930. 

*  J.  G.  Steele  and  R.  Bradfield,  The  significance  of  size  distribution  in  the  clay 
fraction:  Rep.  Am.  Soil  Survey  Assn.,  Bull.  15,  pp.  88-93,  1934. 


124        SEDIMENTARY  PETROGRAPHY 

field  solved  for  t,  to  determine  the  time  of  centrifuging  for  the  pipette 
method.^ 

Theory  of  Sieving 

The  use  of  sieves  to  separate  the  coarser  portions  of  soils  or  sediments 
dates  back  to  the  early  da3S  of  mechanical  analysis,  and  the  simplicity 
and  convenience  of  the  method  have  been  the  greatest  factors  in  its 
continued  use.  In  practice,  sieving  is  exceedingly  simple.  The  material 
to  be  sieved  is  placed  in  a  sieve  and  shaken  until  the  particles  smaller 
than  the  mesh  openings  fall  through.  By  repeating  the  process  with 
successively  smaller  meshes,  the  material  may  be  separated  into  any 
given  number  of  grades. 

The  theory  of  sie\-ing  is  not  so  simple  as  the  practice,  and  if  due  con- 
sideration is  given  to  all  the  factors  involved,  a  number  of  complexities 
are  found  to  enter  which  limit  the  accurac}'  of  the  usual  operations  of 
sieving.  A  number  of  opponents  to  the  use  of  sieves  as  instruments  of 
mechanical  analysis  have  written  critiques  of  the  method;  one  of  the 
most  antagonistic  was  Mitscherlich,^  who  pointed  out  a  number  of  years 
ago  that  sieves  sort  grains  not  only  according  to  size,  but  also  according 
to  shape.  This  may  be  illustrated  by  considering  spherical  and  lath- 
shaped  grains.  The  largest  sphere  that  can  pass  through  a  given  sieve  has 
a  diameter  equal  to  the  mesh,  whereas  a  lath  of  any  length,  theoretically, 
can  pass  through  the  sieve,  providing  only  that  its  two  smaller  dimen- 
sions are  less  than  the  miximum  dimensions  of  the  mesh,  including  its 
diagonals.  A  long  lath  may  have  a  much  larger  volume  than  a  sphere  of 
the  same  cross  section,  and  hence  if  size  is  defined  in  terms  of  the 
nominal  diameter  (i.e.,  based  on  volume j,  the  sieving  process  does  not 
sort  according  to  size. 

If  size  is  defined  in  terms  of  some  average  diameter,  sieves  again  fail 
to  make  a  sharp  distinction,  because  for  non-spherical  shapes  the  max- 
imum length  has  no  direct  bearing  on  passage  through  the  sieve.  Instead, 
the  intermediate  and  shortest  diameters  are  the  deciding  factors,  and 
hence  it  may  be  seen  that  sieves  sort  grains  on  the  basis  of  the  least 
cross-sectional  area,  which  may  or  may  not  have  any  fixed  relation  to 
the  volume  of  the  particle. 

Despite  the  general  validit}-  of  these  criticisms,  sieving  is  a  well  es- 
tablished procedure  in  mechanical  analysis,  and  it  is  possible  to  use 
sieve  data  for  a  number  of  purposes  in  sedimentary  studies.  The  fact 

1  Steele  and  Bradneld  used  a  different  notation,  but  the  results  are  the  same. 

-  E.  A.  Mitscherlich,  Bodenkunde  fur  Land-  und  Forstwirte  (Berlin,  1905),  PP-  37  ff- 


PRINCIPLES  OF  SIZE  ANALYSIS  125 

that  many  sands,  separated  into  size  frequeney  distributions  l)y 
sieves,  plot  as  straight  hncs  on  logarithmic  ])rol)al)ility  i)aper  (page 
189),  indicates  that  in  general  all  the  significant  data  arc  not  ob- 
scured by  sieving.  Of  course,  for  precise  shape  studies  and  their 
influence  on  the  properties  of  sediments  sieving  may  be  merely 
a  preliminary  procedure  of  separating  the  sample  into  convenient  size 
units. 

In  developing  a  theory  of  sieving,  the  simplest  case  may  be  con- 
sidered: that  of  a  mixture  of  two'  sizes  of  spheres,  one  of  which  is 
slightly  larger  and  the  other  slightly  smaller  than  the  sieve  openings. 
If  a  mixture  of  such  spheres  be  placed  in  a  sieve  to  an  appreciable 
depth,  so  that  a  number  of  layers  of  spheres  are  involved,  it  may  be 
seen  that  as  the  sieve  is  shaken,  the  separation  of  the  smaller  spheres 
depends  upon  the  number  of  such  spheres  which  come  into  contact  with 
the  screen  at  a  given  moment.  During  early  stages  of  the  sieving  the 
number  of  smaller  spheres  falling  through  is  relatively  large  because  of 
the  high  projxjrtion  of  small  spheres  in  the  mixture,  but  as  the  remaining 
small  spheres  decrease  in  proportion  in  the  mixture,  the  number  falling 
through  in  any  given  instant  is  reduced. 

As  a  first  approximation  one  may  assume  that  the  numl)er  of  small  spheres, 
dy,  falling:  tin-ough  tlie  sieve  in  any  small  interval  of  time,  dt,  is  proportional 
to  the  number  of  small  spheres  present  at  that  moment.  This  assumption  leads 
to  the  differential  equation 

dy/dt  =  —  ay 

where  y  is  the  number  of  small  spheres  in  the  mixture  at  time  t  and  a  is  a 
constant  oi  i)roportionality.  The  negative  sign  indicates  that  the  function 
decreases  with  time. 

Integration  of  the  differential  equation  yields  log  y  =  — at  +  log  C,  and  by 
evaluating  log  C  at  time  ^  =  o,  it  is  found  that  log  C  =  log  y,  so  that  C=  y,„ 
the  original  number  of  small  spheres  in  the  mixture.  Substituting  y^  for  C 
and  rearranging  yields  the  equation 

y/yo  —  c-''' (39) 

For  spheres  this  function  would  assume  various  values  of  the  constant  a; 
in  general,  the  finer  the  sieve,  the  smaller  the  value. 

In  practice  it  may  be  doubtful  whether  so  simple  a  relation  holds,  be- 
cause of  the  complexity  introduced  by  the  shape  factor.  Not  only  nnist 
the  grains  come  into  contact  with  the  screen,  but  in  order  to  fall  through 
they  must  assume  the  proper  position  so  that  the  short  and  intermediate 
diameters  are  approximately  normal  to  the  plane  of  the  screen.  In  prac- 
tice it  is  also  common  to  have  a  finite  difference  between  successive  sieve 


126 


SEDIMENTARY  PETROGRAPHY 


meshes,  so  that  on  any  sieve  there  is  a  relatively  vi^ide  range  of  particle 
sizes.  The  smallest  particles  vv^ill  readily  fall  through  the  meshes,  hut  the 
larger  ones,  closer  to  the  critical  radius,  w^ill  suffer  delays  due  to  the 
need  for  proper  orientation.  Thus  it  is  likely  that  the  function  may 
show  a  more  marked  decrease  at  first  than  the  exponential  function,  and 
a  slower  decrease  later. 

Wentworth^  investigated  the  question  of  sieving  quantitatively  and 
found  that  an  empirical  equation  of  the  tyi>e  3;  =  at-""  -\-  b  fitted  the  data 
fairly  well.  Figure  29,  adapted  from  Wentworth,  indicates  the  progress 
of  sieving  on  a  ^-mm.  sieve.  Time  is  shown  along  the  .r-axis,  and  the 


id  55  72 


ZO 
um70 

Li      69 

68 


20          30  40  50 

TIME    IN    MINUTES 


Fig.  29. — Progress  of  sieving  on  sieve  with  meshes   of   ^   mm.    (Adapted  from 
Wentworth,  1927.) 


percentages  remaining  above  the  sieve  are  plotted  along  the  y-axis. 
Wentworth's  study  also  showed  that  the  separation  of  the  grains  is  prob- 
ably never  quite  complete,  especially  among  the  smaller  sizes.  In  general, 
however,  he  concluded  that  a  five-  or  ten-minute  period  of  shaking  in 
an  automatic  shaker  is  usually  sufficient. 

Theory  of  Microscopic  Methods  of  Analysis 

The  microscope  has  been  used  by  numerous  workers  for  determining 
particle  size  and  particle-size  distribution.  A  number  of  techniques  have 
been  developed,  ranging  from  direct  measurement  of  diameters  by  means 
of  micrometer  oculars  to  microprojections  of  the  particles  onto  screens 
or  photographic  plates,  followed  by  measurement  of  the  enlarged  images. 
The  wide  variety  of  techniques  is  in  part  a  response  to  the  large  number 
of  materials  which  have  been  studied  microscopically.  Pigments,  dust, 
ceramic  products,  sensitivity  of  photographic  emulsions,  pulverized  coal, 

1  C.  K.  Wentworth,  The  accuracy  of  mechanical  analysis :  Am.  Jour.  Sci.,  vol.  13, 
pp.  399-408,  1927. 


PRINCIPLES  OF  SIZE  ANALYSIS  127 

and  many  other  materials  ^  have  been  investigated.  Sedimentary  petrolo- 
gists  have  not  utilized  microscopic  methods  as  extensively  as  sieving  and 
sedimentation  methods,  but  there  appears  to  be  an  increasing  tendency 
to  adapt  the  microscope  to  measurements  of  size  attributes. 

The  microscope  is  a  convenient  instrument  for  measuring  grains  from 
diameters  of  about  0.5  mm.  down  to  the  limit  of  resolving  power  of  the 
microscope.  The  lower  limit  varies  with  different  instruments  and  with 
the  wave  length  of  the  light  being  used.  By  a  combination  of  oil  immer- 
sion and  blue  light,  particles  as  small  as  0.0002  mm.  diameter  (0.2 
micron)  have  been  resolved.-  For  most  general  purposes,  however,  micro- 
scopic methods  may  be  used  down  to  diameters  of  about  0.00 1  mm. 
(i  micron). 

Definition  of  "size"  of  microscopic  particles.  Inasmuch  as  most  nat- 
ural particles  are  irregular  in  shape,  the  influence  of  shape  factors  on 
detinitions  of  size  should  be  considered.  In  addition,  it  is  necessary'  to 
consider  the  orientation  of  the  particles  on  the  microscopic  sHde.  If  the 
grains  are  sprinkled  on  the  slide  in  a  dry  state,  the  tendency  will  be 
for  them  to  assume  positions  of  rest  such  that  the  shortest  diameter  will 
be  approximately  vertical.^  The  section  exposed  to  view  will  accordingly 
represent  approximately  the  long  and  intermediate  dimensions  of  the 
grain.  If  the  grains  are  mounted  in  balsam  or  another  medium,  the 
orientation  of  the  three  axes  may  be  random,  with  the  result  that  it  is 
difficult  to  determine  whether  the  grain  is  viewed  along  an  edge  or  broad- 
side. If  the  grains  are  nearly  equidimensional  the  influence  of  the  ran- 
dom orientation  is  usually  negligible,  but  with  flat  or  lath-shaped  grains 
it  may  influence  the  values  obtained. 

Regardless  of  the  manner  of  mounting  the  grains,  several  definitions 
of  size  are  possible.  One  may  define  size  as  the  arithmetic  mean  of  the 
diameters  exposed  to  view,  or  one  may  add  in  the  estimated  thickness  of 
the  grain  as  a  third  diameter.  For  rapid  work  one  may  express  size  in 
terms  of  the  intermediate  diameter  only,  or  if  the  grains  are  oriented 
at  random,  the  maximum  horizontal  intercept  through  the  grain  may  be 
used.'*  For  more  detailed  work  Wadell  ^  has  recommended  that  the  area 
of  the  grain  image  (in  a  camera  lucida  drawing,  a  photograph,  or  a 

1  A  complete  bibliography  of  non-sedimentary  studies  of  particle  size  is  given  by 
E.  M.  Chamot  and  C.  W.  Mason,  Handbook  of  Chemical  Microscopy  (New  York, 
1931).  vol.  I,  Chap.  12. 

-'  F.-V.  von  Hahn,  op.  cit.  (1928),  p.  38. 

3  H.  Wadell.  Volume,  shape,  and  roundness  of  quartz  particles:  Jour.  Geology. 
vol.  43,  pp.  250-279,  1935. 

*  W.  C.  Krumbein,  Thin  section  mechanical  analysis  of  indurated  sediments :  Jour. 
Geologx,  vol.  43,  pp.  482-496,  1935- 

5  H.  Wadell,  loc.  cit.,  1935. 


128        SEDIMENTARY  PETROGRAPHY 

screen  projection  of  the  images)  be  measured  with  a  planimeter,  and  ex- 
pressed as  the  diameter  of  a  circle  having  the  same  area.  This  diameter 
is  called  by  Wadell  the  "nominal  sectional  diameter."  For  grains  of 
marked  elongation  or  flattening,  the  harmonic  mean  of  the  diameters 
may  be  used.^  If  the  three  diameters  of  the  grain  are  a,  b,  and  c,  the 
harmonic  mean  is  defined  as  dh=  (T,abc)/(ab  -{-  be  -{-  ac).  Perrot  and 
Kinney  -  considered  this  diameter  the  most  logical  to  be  used  for  pig- 
ments and  ceramics  because  of  its  relation  to  the  specific  surface  of  the 
materials.^  Roller  ^  pointed  out  that,  in  addition,  diameters  defined  in 
this  manner  are  closely  related  to  diameters  calculated  from  Stokes' 
law.  For  practical  purposes,  however.  Roller  also  showed  that  in  the 
average  case  the  harmonic  mean  lies  within  about  6  per  cent  of  the  value 
of  the  arithmetic  mean,  and  that  the  arithmetic  mean  may  therefore  be 
used  directly  for  computing  mean  surface  diameters.  The  arithmetic 
mean  of  the  three  diameters  is  simply  da.=  (a-{-b-\-  c;/3. 

In  addition  to  the  arithmetic  and  harmonic  mean  diameters,  the  geo- 
metric mean  diameter  of  the  particle  may  be  used.  This  is  com- 
puted by  finding  the  cube  root  of  the  product  of  the  three  diameters, 
c/g  =  ^  abc. 

Determination  of  size  distribution  from  microscopic  counts.  What- 
ever definition  of  particle  size  is  chosen,  the  succeeding  step  is  to  meas- 
ure a  number  of  grains  and  arrange  them  into  grades  or  classes  to 
determine  the  distribution  of  sizes  within  the  sample.  Practice  varies 
considerably ;  the  grades  may  be  chosen  as  equal  arithmetic  intervals  or 
on  a  geometric  basis,  and  the  number  of  grains  counted  ranges  from  a 
few  hundred  to  more  than  a  thousand.  For  most  routine  analyses,  in- 
volving grains  more  or  less  equidimensional  and  having  a  restricted 
range  of  sizes  (quartz  sand,  for  example),  a  few  hundred  grains  prob- 
ably suffice. 

It  should  be  recognized  that  the  frequency  data  obtained  by  micro- 
scopic measurement  are  expressed  in  terms  of  numbers  of  grains  rather 
than  by  weights,  as  in  sieving  and  sedimentation.  If  the  material  is 
homogeneous,  the  weight  frequency  may  be  computed  from  the  number 
frequency,   but  as   a  general   rule  microscopic   size  determinations  by 


1  H.  Green,  A  photomicrographic  method  for  the  determination  of  particle  si 
of  paint  and  rubber   pigments :   Juitr.  1-ranklin  Inst.,  vol.    192,   pp.   637-606,    19: 

2  G.  St.  J.  Perrot  and  S.  P.  Kinney,  The  meaning  and  microscopic  measurer 
of  average  particle  size :  Jour.  Am.  Ceram.  Soc,  vol.  6,  pp.  417-439,  1923. 

3  For  a  sphere  the  specific  surface  S  is  defined  as  follows  in  terms  of  the  diam 
d  and  the  density  p  :  i/d  =  (1/6)  pS. 

•*  P.  S.   Roller,   Separation  and  size  distribution  of  microscopic  particles :   V 
Dept.  Commerce,  Bur.  of  Mines,  Tech.  Paper  490,  1931. 


ze 
1921. 


PRINCIPLES  OF  SIZE  ANALYSIS  129 

number  should  not  be  directly  compared  with  sieve  determinations  by 
weight.^ 

The  usual  statistical  methods  may  be  applied  to  microscopic  data  for 
expressing  the  average  size  of  sediments,  as  well  as  other  statistical 
values.  These  methods  are  discussed  in  Chapters  8  and  9;  practical 
examples  of  microscopic  size  analyses  are  given  in  Chapter  6. 

Mechanical  analysis  of  thin  sections  of  indurated  sediments.  For  the 
study  of  grain  size  distributions  of  indurated  sediments,  in  which  the 
particles  are  too  firmly  cemented  to  be  disaggregated,  microscopic 
methods  afford  practically  the  only  method  of  attack.  There  are  several 
precautions  which  must  be  taken,  however,  when  size  is  estimated  from 
thin  sections,  because  in  detail  the  problem  is  quite  complex. 

Tlae  problem  of  thin-section  mechanical  analysis  has  been  attacked  from 
several  angles  by  sedimentary  petrologists,  but  the  mathematical  theory  of 
random  sections  through  groups  of  spheres  has  also  been  treated  by  astron- 
omers and  biologists.  Krumbein,-  working  independently,  approached  the 
problem  from  tlie  moments  of  the  grain  distribution,  but  after  publication 
learned  througli  correspondence  tliat  Hagerman  ^  had  previously  attacked  the 
problem  in  a  similar  manner,  but  from  a  different  mathematical  approach. 
The  essential  features  of  Krumbein's  mathematical  analysis  had,  however, 
been  used  by  astronomers  in  the  study  of  globular  star  clusters.*  To  increase 
the  complexity,  it  was  further  learned  that  similar  but  more  rigorous 
mathematical  treatment  had  been  applied  by  Wicksell  to  the  study  of  spherical 
corpuscles  embedded  in  tissues.^  To  cap  the  situation,  Fisher^  had  also 
approached  the  proljlem  of  indurated  sediments,  but  from  a  point  of  view 
different  from  that  of  Hagerman  or  Krumbein.  As  a  result  there  are  several 
methods  of  approach  to  the  study  of  thin-section  mechanical  analysis,  and 
all  are  complex  in  terms  of  ordinary  methods  (sieving  or  sedimentation) 
because  of  the  restrictions  placed  upon  analysis  by  the  sectioning  of  the 
grains.  The  discussion  here  will  follow  essentially  the  method  used  by  Krum- 

1  The  relation  between  number  frequency  curves  and  weight  frequency  curves 
has  been  studied  by  Hatch  for  distributions  which  are  symmetrical  on  a  logarithmic 
size  scale.  See  T.  Hatch,  Determination  of  "average  particle  size"  from  the  screen 
analysis  of  non-uniform  particulate  substances:  Jour,  franklin  Inst.,  vol.  215,  pp. 
27-38,  1933.  The  question  of  number  vs.  weight  frequencies  is  discussed  further  in 
Chapter   8. 

2  W.  C.  Krumbein,  loc.  cit.,  1935. 

3  T.  H.  Hagerman,  Ein  metod  for  bedomning  av  kornstorleken  och  sorterings- 
graden  inom  finkorniga  mekanist  sedimentara  bergarter :  Gcol.  Forcning,  Fork.,  vol. 
46,  pp.  325-353,  1924- 

•*  S.  D.  Wicksell,  A  study  of  the  properties  of  globular  distributions :  Arkn>  f. 
Matcniatik.  Astron.,  och  Fysik,  vol.  18,  1924. 

5  S.  D.  Wicksell,  The  corpuscle  problem :  Biomctrika,  vol.  17,  pp.  84-99,  1925. 
S.  D.  Wicksell,  The  corpuscle  problem  (ellipsoidal  case)  :  ibid.,  vol.  18,  pp.  152-172, 
1926. 

«  G.  Fisher,  Die  Petrographie  der  Grauwacken:  Jahrb.  d.  prcnss.  gcol.  Landcs- 
aust,  vol.  54,  pp.  320-343,  1933- 


I30 


SEDIMENTARY  PETROGRAPHY 


bein,  inasmuch  as  it  is  most  familiar  to  the  authors,  and  because  it  is  based 
on  the  mathematical  foundation  developed  by  Wicksell. 


It  is  common  knowledge  that  if  random  grain  diameters  are  measured 
from  thin  sections,  the  distribution  of  observed  diameters  will  not  be 
an  accurate  indication  of  the  grain  diameters  themselves.  This  is  because 
in  only  a  very  small  number  of  cases  will  the  random  sections  be  exactly 
through  the  center  of  the  grain.  Generally  the  average  size  of  the  grain 
sections  will  be  less  than  the  average  size  of  the  grains.  However,  with 
spherical  grains  there  is  a  definite  relation  between  the  random  section- 
ing and  the  true  size  distribution,  so 
that  mathematical  analysis  may  de- 
termine what  corrections  must  be 
applied  to  the  random  sections. 

A  complete  solution  of  the  prob- 
lem is  not  simple,  but  fortunately 
the  most  important  statistical  values 
of  the  sediment,  including  the  aver- 
age grain  size  and  the  standard  de- 
viation, may  be  determined  satisfac- 
torily even  though  the  grains  are  not 
true  spheres. 

If  a  number  of  lead  shot,  all  of 
the  same  radius,  are  embedded  at 
random  in  sealing  wax  and  ground 
down  to  a  polished  section,  the  sec- 
tion wmU 'disclose  a  number  of  lead 
circles  of  varying  radius.  The  radii 
of  the  observed  circular  sections  may  be  measured,  and  the  data  arranged 
in  classes  equal  to  tenths  of  the  true  radius.  The  observed  radii  range 
from  zero  to  unity,  and  when  the  observations  are  plotted,  as  in  Figure 
30,  the  result  is  striking.  A  smoothed  curve  passed  through  the  histogram 
yields  a  frequency  curve  in  which  the  line  rises  to  the  right  but  does  not 
descend  again.  The  average  radius,  computed  from  the  observed  data,  is 
0.763  of  the  actual  radius  of  the  shot. 

If  one  confined  himself  to  the  observed  data  he  would  assume  that  he 
had  spheres  of  various  sizes,  and  he  would  find  that  the  obser\'ed  average 
size  is  some  24  per  cent  smaller  than  the  actual.  This  serves  to  illustrate 
the  general  effect  of  sectioning  through  a  rock,  and  emphasizes  that 
one  cannot  argue  from  observation  alone  that  he  has  either  the  true 
value  of  the  average  size,  or  even  an  approach  to  the  true  frequency 


J    .4    .5    .6    .7 

OBSERVED  RADII 


Fig.  30. — Frequency  distribution  of 
observed  sectional  radii  of  uniform 
lead  shot. 


PRINCIPLES  OF  SIZE  ANALYSIS 


131 


curve.  In  practice  the  picture  is  more  complex  than  in  the  example  given. 
The  grains  of  sediments  are  not  all  of  one  size,  nor  are  they  true  spheres. 


Krumbein  considered  the  mathematical  relations  between  the  moments  ^  of 
the  grain  distribution  and  the  moments  of  the  observed  sectional  distribution. 
For  spheres  of  one  size,  as  in  the  example  given, 
the  treatment  is  fairly  simple,  and  the  same  ap- 
proach was  made  by  both  Hagerman  and  Krum- 
bein: 

Consider  a  random  section  through  a  sphere  of 
radius  r;  the  problem  is  to  determine  the  fre- 
quency function  of  observed  radii  which  occurs 
when  a  number  of  spheres  of  this  same  radius  are 
sectioned  at  random.  Fortunately  a  two-dimen- 
sional analysis  suffices.  In  Figure  31  is  a  sphere  to 
be  cut  by  a  random  section  anywhere  along  the 
y-axis  from  —  r  to  -|-  r,  with  an  equal  likelihood 
for  any  point  of  intersection.  One  may,  in  fact, 
restrict  himself  to  the  range  from  o  to  r,  since 
the  circle  is  symmetrical.  The  equal  likelihood  of 

cutting  the  section  between  y  and  y  -f-  dy  may  then  be  stated  as  P^  =  i/r. 
Call  the  observed  radius  of  the  random  section  x,  a  variable  measured  along 
the  .r-axis.  To  find  P(-r),  the  distribution  of  observed  radii,  let  y  =  H(.i-)  and 
dy  =  'ii' {x)dx.  From  the  circle,  .r- +  y- =  r-,  so  that 


Fig.  31. — Part  of  a 
sphere  to  be  cut  by  a  ran- 
dom section  along  the  3; 
axis  from  — r  to  -\-r. 


y  =  H(.r) 
dy  =  YL'{x)dx  = 


Vr- 


-  X-  . 
xdx 


Vr-- 


(40) 
(41) 


For  transforming  the  probabilities  the  usual 
relation  P(.r)(/.r  =  Pi[H(.r)]H'(.t-)rf.v  may  be 
used;  substitution  yields 


P(.r)J.i 


I  . 

r  V7^ 


dx 


(42) 


It  will  be  noted  that  the  sign  of  dx  has  been 
changed,  to  keep  the  probabilities  positive. 

Fig.  32  shows  the  form  of  the  P(.r)  curve. 
It  is  the  same  as  the  curve  obtained  experi- 
mentally in  Figure  30.  The  curve  is  asymptotic 
to  the  line  .r  =  r,  but  the  area  is  finite. 

In  the  mathematical  treatment  of  die  general 
problem  Hagerman  and  Krumbein  followed 
different  lines  of  reasoning.  The  latter  consid- 
ered a  frequency  distribution  of  radii  F(r), 
where  ¥{r)dr  is  the  probability  that  r  lies  between  r  and  r -\- dr.  From  the 

1  The  moments  of  a  frequency  distribution  are  parameters  whicli  describe  the 
properties  of  the  distribution.  The  theory  of  moments  is  discussed  in  Chapter  8. 


Fig.  32. — Form  of  the  P(.v) 
curve. 


132       SEDIMENTARY  PETROGRAPHY 

random  sections  is  obser\ed  a  distribution  of  jr's  or  apparent  radii,  which  may 
be  called  Q(x).  Here  Q(x)dx  represents  the  probability  that  x  lies  between 
X  and  x-{-dx.  Now  the  probability-  that  r  is  bet^veen  r  and  r -i;- dr,  and  the 
probability  that  x  is  betiveen  x  and  x-\-dx,  is  P(x')F (r)drdx.  Since  r  may 
have  any  value  greater  than  x,  Q(jr)  extends  over  all  the  possible  values  of 
r,  and  hence  is  the  integral  of  the  expression  from  x  to  infinity : 


J  Jr.r—x- 


Q(x)=   j   P{x)F(r)dr  =  x    I  "^JT^IT^r^ ^''  •      '      •      •      "^43; 

X  z 

where  Jx  has  been  dropped  from  both  sides  of  the  equation. 

Equation  (43;  is  an  integral  equation,  and  in  practice  Q(x)  will  be  a 
set  of  empirical  data,  F(r;  will  be  entirely  unknown,  and  V(x)  will  apply 
strictly  only  when  perfect  spheres  are  involved.  For  these  reasons  the  solution 
is  restricted  to  a  consideration  of  the  moments  of  the  distributions,  because 
from  the  obsened  moments  of  Q(x)  may  be  computed  and  converted  into 
the  corresponding  moments  of  F(rJ. 

To  solve  equation  (43)  several  steps  are  necessarj-.^  The  solution  is  ob- 
tained in  such  manner  that  the  integrals  are  in  the  form  of  the  «th  moments 
of  the  distributions  Q  and  F  about  the  origins  of  x  and  r  respectively : 

\yi-^U)dx^C\r-Y(r)dr (44; 

o  o 

where  C  is  a  constant  which  depends  upon  n.  By  letting  n  =  i,  2,  3 ,  the 

moments  of  F(r)  may  be  obtained  in  terms  of  Q(-r).  The  final  solutions  for 
the  first  four  moments  are: 

«ii  =  -— Mri Us) 

4 

nx2  =  -T-"r2 (46; 


«I3=   T^«r3 (47) 


(48) 


In  practice  the  analysis  is  performed  by  determining  the  moments  of  the 
observed  distribution  Q(t)  and  correcting  them  by  means  of  the  preceding 
equations  for  at  least  the  first  two  moments,  which  afford  the  average  size 
and  "degree  of  sorting"  of  the  sediment.  Examples  of  sediments  studied  from 
this  approach  are  given  in  Chapter  6. 

The  above  mathematical  approach  is  the  same  as  that  made  by  Wick- 
sell  in  the  study  of  globular  star  clusters.  In  his  later  paper  on  corpuscles 

1  The  steps  are  given  in  Knunbein's  paper,  \oc.  ctt.,  1935. 


PRINCIPLES  OF  SIZE  ANALYSIS  133 

Wicksell  attacked  the  more  general  case  of  sectioning  spheres  and  in- 
cluded the  probability  that  a  sphere  of  radius  r  to  r-\-  dr  would  be  cut 
by  the  sectioning  plane.  Thus  Krumbein's  solution  is  a  special  case  of 
the  general  problem,  based  on  the  assumption  that  the  thin  section  is  a 
typical  sample  of  the  grain  distribution.  This  assumption  has  been 
accepted  in  most  thin-section  work  in  geolog}'.  but  it  may  require  closer 
scrutiny.  If  the  probability  of  sectioning  is  included  in  the  problem,  it 
is  found  that  the  probability  of  a  sphere  being  sliced  is  the  ratio  of  its 
radius  to  the  mean  radius  of  the  spheres,  r/r^. 

Thus  in  order  to  generalize  Krumbein's  solution  it  is  necessary-  to  introduce 
this  probability  of  slicing,  Pq  =  r/r^  into  equation  (42),  with  the  result  that 
the  r  in  the  denominator  is  canceled  and  replaced  by  r^.  With  this  change, 
the  mathematical  operations  proceed  essentially  as  before,  resulting  finally  in 
the  expression 


X  00 

j  .r"Q(.r)J.r=-^  J  r"  +  1  F{r)dr 


(49) 


for  equation  (44).  This  general  solution  differs  from  the  particular  case  in 
that  the  «th  moment  of  the  .r-distribution  is  equated  to  the  (h  +  i)xh  moment 
of  the  r's.  Hence  tlie  relations  between  the  moments  differ  in  the  final  results. 
The  general  equations  corresponding  to  (45),  (46),  (47),  and  (48)  become, 
in  the  general  case: 

•j-i-hm  =  «ri (50) 

"xl  =  ---— - (51) 

4       «rl 

«x.=-f^ (5-^) 

3      «rl 

where  x^^  of  equation  (50)  is  the  harmonic  mean  of  the  .r-distribution. 

The  equations  of  the  general  solution  were  applied  by  Krumbein  to  the 
data  used  for  his  particular  solution,  and  it  was  found  that  the  agreement 
between  observed  and  e.xpected  values  was  better  by  a  considerable  per- 
centage with  the  original  solution  than  with  the  more  general  theor>-.  The 
general  theory  implies  that  the  spheres  are  suspended  in  a  medium,  and  thus 
presumably  are  not  in  contact.  In  sediments,  on  the  other  hand,  the  grains  are 
in  actual  contact,  and  it  may  be  that  when  the  spheres  are  packed  closely 
together  the  chances  of  slicing  are  nearly  equal  for  all  sizes.  From  the 
evidence  thus  far  in  hand  it  appears  that  the  original  solution  (equations  45 
to  48)  has  a  more  direct  application. 


134        SEDIMENTARY  PETROGRAPHY 

An  example  of  an  analysis  by  thin  section  is  given  in  Chapter  6,  using  the 
original  simplified  theory  for  the  calculations. 


SUMMARY     OF     PRINCIPLES     OF     MECHANICAL 
ANALYSIS 

The  increasing  emphasis  on  the  quantitative  aspects  of  sedimentar>^ 
studies  requires  that  workers  in  the  field  be  informed  regarding  the  fun- 
damental principles  upon  which  so  much  of  their  technique  depends. 
The  reaching  of  sound  conclusions  about  anah-tical  data  and  the  de- 
velopment of  theories  to  account  for  sedimentary  phenomena  require 
that  the  influence  of  technique  on  the  resultant  data  be  known.  It  is  for 
these  reasons  that  the  principles  tmderhing  mechanical  analysis  have 
received  detailed  discussion  in  the  present  chapter.  In  practice  many  of 
the  finer  points  are  ignored  b)*  common  consent,  but  when  a  particularly 
precise  and  exhaustive  study  is  to  be  made,  a  knowledge  of  the  under- 
l}-ing  principles  may  be  of  considerable  aid. 

The  complete  shift  of  emphasis  in  anahlical  techniques  which  was 
introduced  by  Oden's  theorj'  of  sedimenting  systems  illustrates  the  ad- 
\-antages  that  accrue  from  an  investigation  of  underlying  principles,  as 
well  as  the  greater  precision  possible  because  methods  can  be  developed 
which  are  in  accord  with  the  demands  of  the  theory.  The  result  of 
Oden's  w^ork  has  been  that  older  routine  methods  of  analysis,  such  as 
decantation  and  elutriation,  are  gradually  being  displaced  by  more  mod- 
em precision  methods.  One  cannot,  how^ever,  ignore  the  older  methods, 
which  are  capable  of  yielding  good  results  in  the  range  of  sizes  directly 
suited  to  them,  and  it  seems  likely  that  by  a  more  careful  consideration 
of  their  underlying  principles  significant  improvements  may  be  made  in 
their  usefulness. 


CIIArrKK   6 

METHODS  OI''  MI<X:ilANR"AL  ANALYSIS 

I  N  T  R  ()  I)  U  C:  T  I  ()  N 

The  number  of  methods  at  prescnf  avriilrihlc  for  nieclianical  analysis  is 
so  great  that  an  entire  volume  cduld  he  devolcd  to  their  enumeration 
and  description.'  Many  of  the  metliods  represent  minor  variations  of 
fundamental  techniques,  some  de[)end  upon  slight  changes  in  established 
apparatus,  and  some  vi^ere  developed  for  the  analysis  of  special  materials. 
It  is  virtually  impossible  for  any  single  worker  t()  have  i>ersonal  ex- 
perience with  every  device  known,  and  it  l)ecomes  necessary  to  choose 
from  among  the  wide  variety  a  few  methods  which  may  be  adapted  to 
sediments  primarily  on  the  basis  of  their  soundness  and  secondarily  on 
the  basis  of  convenience  or  cost. 

In  the  present  chapter  both  old  and  new  methods  will  be  descriljed, 
but  special  emphasis  will  be  given  to  mclhods  based  on  Oden's  theory 
of  sedimenting  systems.  Among  these  latter,  the  pipette  method  will  be 
stressed  as  one  of  practically  universal  application  to  fine  sediments. 
This  emphasis  on  the  pipette  method  is  a  natural  consequence  of  the 
authors'  greater  familiarity  with  the  method,  which  they  have  adoi>ted 
for  all  laboratory  work  at  the  University  of  Chicago. 

In  contrast  to  the  wide  variety  of  methods  available  for  the  finer 
sediments,  the  process  of  sieving  has  remained  by  far  the  most  i)opular 
method  for  material  in  the  sand  ranges  and  above.  To  some  extent 
microscopic  methods  may  take  the  place  of  sieving  in  the  future,  but 
certainly  at  present  sieve  techniques  are  j)raclieally  universal  in  America 
at  least.  Mention  was  made  in  Chapter  5  of  a  few  of  the  theoretical 
objections  to  sieving.  Some  workers  prefer  to  use  elutriation  methods 
exclusively  for  analysis,  to  avoid  composite  data  based  partly  on  sieving 
and  partly  on  sedimentation.   With   this  end  in  view,   elutriators  and 

1  Perhaps  the  best  known  and  most  complete  volumes  on  the  subject  are  F.  V. 
von  Hahn,  Dispcrsoidanalysc  (Leipzig,  1928),  and  H.  Gessncr,  Die  Schliimmanalysc 
(Leipzig,  1931).  The  former  is  wriUcn  primarily  from  the  point  of  view  of  the 
colloid  chemist;  the  latter  treats  the  subject  from  the  soil  scientist's  point  of  view. 

135 


136       SEDIMEXTARY  PETROGRAPHY 

settling  tubes  for  sand  have  been  developed  as  substitutes  for  sieving 
(seepage  157). 

Xumerous  sediments  have  a  range  of  sizes  from  coarse  to  fine.  Among 
these  are  sandy  silt,  loess,  glacial  till,  sandy  shale,  and  the  like.  For  such 
sediments  a  composite  method  of  analysis  is  necessary,  involving,  usu- 
ally, a  splitting  of  the  sample  at  some  convenient  size,  so  that  the  coarser 
material  may  be  sieved  and  the  finer  material  analyzed  by  a  sedimenta- 
tion method.  Purely  for  convenience,  the  line  between  coarse  and  fine 
sediments  will  be  chosen  at  %6  nun.  This  is  the  lower  Umit  of  the 
sand  size  in  \\'enrworth's  classification,  and  sieves  may  readily  be  ob- 
tained with  meshes  fine  enough  to  separate  material  at  about  this  dimen- 
sion. Further,  14 6  "OTI-  is  near  the  upper  limit  of  applicability  of  Stokes' 
law  or  Wadell's  practical  sedimentation  formula  and  so  furnishes  a 
convenient  line  of  demarcation.^ 

With  composite  types  of  sediments  the  coarse  and  fine  portions  are 
analyzed  separately,  and  the  anahtical  data  combined  into  a  single  size 
frequency  distribution.  Such  composite  analyses  often  show  an  abrupt 
"break"  in  the  distribution  in  the  vicinity  of  Vie  nim.,  because  the 
principle  on  which  sieves  separate  the  particles  is  not  the  same  as  the 
principles  operative  in  sedimentation  analysis.  In  many  cases  this  hiatus 
between  the  two  methods  is  not  serious,  but  in  general  any  unusual 
features  of  the  data  in  the  vicinity  of  %6  mm.  should  be  examined 
from  the  point  of  view  of  possible  experimental  errors. 


COARSE     SEDIMENTS 

In  Chapter  5  sedimentary  particles  were  classified  in  terms  of  the 
disperse  systems  to  which  they  belong.  The  outline  on  page  92  includes 
all  particles  larger  than  o.i  micron  diameter  as  coarse  disperse  systems. 
For  mechanical  analysis  the  limit  ^4 6  mrn-  (0.0625  mm.)  has  been 
chosen  as  a  convenient  point  to  distinguish  between  techniques  for  coarse 
and  fine  sediments.  Coarse  particles  may  be  further  subdivided ;  the 
smaller  group  includes  sand  and  pebbles  conveniently  analyzed  by  siev- 
ing, and  the  larger  group  comprises  pebbles  and  cobbles  large  enough 
to  be  handled  as  individual  particles. 

The  fine  between  the  two  groups  of  coarse  particles  may  be  chosen, 
merely  for  convenience  of  discussion,  at  16  mm.  diameter.  Pebbles  of 

1  See  C.  K.  Wentworth,  Methods  of  mechanical  analysis  of  sediments :  Univ. 
lou-a  Studies  in  \at.  History,  vol.  11,  no.  11,  1926,  for  a  more  detailed  discussion 
of  the  advantages  of  the  lower  limit  of  sand  as  a  limit  for  sieving.  With  other 
grade  scales,  such  as  the  Atterberg  scale,  the  limiting  value  will  not  be  precisely 
1/16  mm.,  but  it  will  be  of  the  same  order  of  magnitude. 


METHODS  OF  SIZE  ANALYSIS 


137 


this  diameter  have  a  sufficiently  large  volume  to  render  that  measure 
convenient,  and  they  are  large  enough  to  be  handled  individually. 

Analysis  by  Sieving 

Sieve  analysis  is  rather  well  standardized,  and  in  the  following  dis- 
cussion the  instructions  will  involve  sieves  arranged  on  the  Wentworth 
scale  or  related  scales,  such  as  \/2  or  ^^2  scales.  The  reader  may 
understand  from  the  discussion  in  Chapter  4  on  grade  scales,  however, 
that  any  convenient  set  of  sieves  may  equally  well  be  used. 

TAliLK    14 

Wentworth  Grade  Scale,  V"2  Scale,  V2  Scale,  and  Corresponding 
Tyler  Sieve  Openings 


Wcnhvoyih   Grade  Scale 

VT  Scale 

^3  Scale 

Tyler  Screens 

(tinii.) 

{mill.) 

{iiiiii.) 

(mm.) 

4  

4.00 

4.00 

3-96 

3.36 

3-33 

2.83 

2.83 

2.79 

2.38 

2.36 

2  • 

2.00 

2.00 

i.g8 

1.68 

1.65 

1.41 

1.41 

1.40 

1. 19 

1. 17 

I   

1. 00 

1. 00 
0.840 

0.991 

•833 

0.707 

.707 

.701 

•595 

•589 

IX    

0.500 

.500 

•495 

/2      

.420 

.417 

•354 

•354 

•351 

.297 

•295 

M  

.250 

.250 

.246 

.210 

.208 

.177 

.177 

•  175 

.149 

.147 

Vo  

.125 

.125 

.124 

/o    

.105 

.104 

.088 

.088 

.088 

.074 

.074 

1/10    

.062 

.062 

.062 

There  arc  many  kinds  of  sieves  on  the  market,  including  bolting  cloth 
sieves,  j^lates  with  round  holes  punched  through  them,  and  woven  wire 


138       SEDIMENTARY  PETROGRAPHY 

sieves  with  square  meshes.  The  preferred  type  for  general  analyses  are 
those  with  woven  vs-ire  meshes,  double  crimped  to  prevent  distortion. 
Sieves  are  available  in  brass  rings  of  several  diameters,  and  either  6-  or 
8-in.  diameters  are  commonly  used. 

Unfortunately  most  commercial  types  of  sieves  have  been  developed 
in  connection  with  engineering  uses,  so  that  they  may  not  agree  pre- 
cisely ^\-ith  the  Wentworth  grade  scale,  which  is  most  commonly  used 
by  sedimentar}-  petrologists  in  America.  Among  the  better  known  prod- 
ucts are  Tyler  Standard  Screen  Scale  Sieves,^  which  are  based  on  a 
200-mesh  sieve  ha\-ing  openings  of  0.0029  in.,  and  increasing  uniformly 
of  the  ^J/'2  scale.  The  millimeter  equivalents  of  these  sieves  do  not  agree 
fH-ecisely  with  the  W'entworth  grade  limits,  but  they  lie  so  close  that 
the  difterence  is  well  \\'ithin  the  United  States  Bureau  of  Standards 
limit  of  tolerance  for  sieves. 

Table  14  lists  the  grade  limits  of  the  Wentworth,  x/T,  and  Nj/Fgrade 
scales  and  indicates  the  corresponding  sizes  of  the  Tyler  screen  meshes. 
In  the  first  column  the  W'entworth  scale  is  shown  alone:  the  second 
column  lists  the  \/~2  scale,  and  the  third  colurrm  has  the  complete  -^fz 
scale  from  the  range  of  4  mm.  to  %6  m™- 

In  actual  practice  it  is  largely  immaterial  what  particular  meshes  of 
sieves  are  used  as  long  as  a  sufficiently  small  internal  is  involved  between 
sieves  to  bring  out  the  continuous  nature  of  the  frequency  distribution. 
The  discussion  of  grade  scales  in  Chapter  4  covers  this  topic.  For  most 
routine  analyses,  however,  it  is  customary-  to  use  either  the  \\'ent«orth 
internals  directly,  or  V^  inter\'als.  When  the  data  are  to  be  plotted  as 
cumulative  curves,  it  is  usually  convenient  to  sieve  the  material  first 
with  \/2  sieves  and,  by  inspection  of  the  sieve  residues,  to  resieve  the 
heaviest  loaded  sieves  through  the  intermediate  ■\f2  sieves.  An  example 
will  indicate  the  procedure :  A  beach  sand  is  sieved  with  the  \/2  scale, 
yielding  the  following  weights  of  material  on  each  sieve 


0.701  mm- 

o.oi  g. 

0495 

0.35 

0.351 

2.90 

0.246 

14-75 

0.175 

5-75 

0.1 24 

0.45 

O.OS3 

0.06 

24-36  g. 

1  Manufactured  bj-  the  W.  S.  Tyler  Qmipan}-,  Qe^•eland,  Ohio. 


METHODS  OF  STZIC  ANALYSTS  I3() 

In  comparison  with  the  other  sieve  resichies,  the  (|uantity  ()n  the 
0.246-inni.  sieve  is  so  large  that  it  would  he  convenient  to  divide  the 
amount  into  two  i^rades,  inasmuch  as  it  represents  more  than  60  per 
cent  of  the  sample.  \\y  usin^  the  o.jS(j  sieve,  on  the  %y^  scale,  this 
grade  was  separated  into  the  sul)-grades  0.351-0.289  =  4.91  g.,  and 
0.289-0.246  =  9.84  g.  Thus  an  additional  point  on  the  cumulative  curve 
results  in  a  more  accurate  smoothing  ol'  the  data.  If  the  data  :\rv  to  he 


Fic.  33. — Kd-Tap  Automatic  Sliakiiig  Machine.    (Courtesy  of  W.  S.  Tyler  Co., 
Cleveland,  O.) 


used  in  computing  the  moments  of  the  distrihution  (Chapter  9),  cither 
the  Wentworth  or  the  \/'2  grades  may  be  used  directly,  without  the 
necessity  of  resieving. 

Most  commonly  sieves  are  furnished  with  flanges  so  that  one  sieve 
may  be  fitted  above  another.  In  this  manner  an  entire  column  of  sieves 
may  be  used  simultaneously.  In  hand  sieving  the  colmnn  of  sieves  may 
be  set  up  in  decreasing  mesh  downward  from  the  to]),  and  with  a  \x\n 
at  the  base  of  the  column.  The  material  is  i>oured  into  the  top  sieve, 
and  the  entire  column  rocked  and  tapped  with  the  Hat  of  the  hand  until 
sieving  is  completed  in  the  top,  coarsest  sieve.  This  may  then  be  re- 
moved from  the  column,  and  the  process  re^xiatcd  until  sieving  is  com- 


I40 


SEDIMEXTARY  PETROGRAPHY 


plete.  In  this  manner  the  finer  sieves  are  worked  for  a  longer  period 
than  the  coarser,  and  the  critical  sieve  is  always  open  to  view. 


A  simple  test  for  the  completeness  of  sieving  is  to  shake  the  sieve 
over  a  large  sheet  of  glazed  paper.  As  long  as  any  appreciable  number 
of  grains  pass  through,  the  sieving  should  be  continued. 

^^'hen  many  sieve  analyses  are  to  be  made,  it  is  convenient  to  liave 
an  automatic  shaking  machine.  Such  devices  are 
marketed  and  take  an  entire  set  of  sieves.  They 
rLEB    >  \'«i\  are  operated  by  electricity,  and  may  combine  a 

rotary  motion  with  a  tapping  effect.  The  Tyler 
automatic  "Ro-Tap"  shaker  is  such  a  machine, 
[lllll  2Lnd  it  may  be  equipped  with  an  automatic  clock 

;;^:MH|||^y^  which  times  the  sieving  interval.  The  shaker  and 
""^HiP        clock  are  shown  in  Figures  33  and  34. 

Other  shaking  devices  are  on  the  market,  and 
p.-  _    j^utomatic     simple  but  efficient  hand-operated  machines  can 

timer,  f  Courtesy  of  W.  S.     be  constructed  at  low  cost.  Andreason  ^  described 
Tyler  Compan>.)  ^  shaking  machine  constructed  by  placing  a  nest 

of  sieves  in  a  support  which  itself  was  mounted  to  a  baseboard  by  means 
of  flexible  bands,  so  that  during  operation  the  nest  of  sieves  is  agitated 
to  and  fro  by  an  eccentric  shaft.  Figure  35  is  a  diagram  of  this  ap- 
paratus, from  Andreason's  paper. 
WentAvorth's  study  of  sieving,  de- 
scribed in  Chapter  5,  indicated  that 
an  interval  of  about  10  min.  in  an 
automatic  shaker  is  usually  sufficient  lt=I  o 
for  approximately  complete  separa- 
tions, and  that  inter\-al  has  been 
fairly  widely  adopted  by  sedimentary 
petrologists  in  America.  The  weight  , — c 
of  sample  to  be  sieved  depends  upon 
the  sizes  of  material  present,  but  in  J^^;  35.-^dreason's  shaking  ma- 
general  a  sample  of  25  g.-  is  suffi- 
cient for  material  between  J/^  and  Yxc,  mm.  diameter.  The  test  sample  is 
split  from  the  field  sample  by  one  of  the  methods  described  in  Chapter  3, 
weighed,  and  placed  in  the  top  of  a  column  of  sieves.  After  shaking, 
each  sieve  may  be  emptied  in  turn  onto  a  large  sheet  of  glossy  paper 

1  A.  H.  M.  Andreason,  Zur  Kenntnis  des  Mahlgutes :  Kolloidchem.  Beihefte,  vol. 
27,  pp.  349-458,  1928. 

2  C.  K.  Wentworth,  loc.  cit.,  1926. 


METHODS  OF  SIZE  ANALYSTS  141 

(12  X  12  or  16  X  16  in.)  and  the  separate  transferred  to  the  balance  pan 
for  weighing.  When  a  number  of  sami)les  are  to  be  worked,  time  is 
saved  by  having  as  many  sheets  as  there  are  sieves,  each  labeled  with  a 
corresponding  sieve  opening.  The  material  from  each  sieve  is  removed  to 
its  corresponding  sheet,  and  in  this  manner  the  sieves  are  freed  for  the 
next  sample.  The  weighing  of  one  set  of  separates  may  then  be  jjer- 
formed  while  the  next  sample  is  being  sieved. 

In  many  cases,  when  the  grains  are  angular,  it  may  be  found  that  a 
number  of  grains  remain  lodged  in  the  sieve,  and  cannot  be  removed 
readily.  The  use  of  a  fairly  stifif  brush,  rubbed  over  the  bottom  of  the 
sieve,  is  often  useful  in  loosening  the  grains,  and  a  similar  effect  can  l)e 
had  by  tapping  the  rim  of  the  sieve  with  a  wooden  mallet,  taking  care 
to  strike  the  rim  along  the  general  diagonals  of  the  wire  mesh,  to  prevent 
distorting  the  sieve. 

The  scale  used  in  weighing  sieve  separates  need  not  be  an  expensive 
analytical  balance.  Any  good  beam  scale,  with  sliding  weights,  and 
sensitive  to  o.oi  g.,  may  be  used.  It  is  seldom  necessary  to  weigh  the 
sieve  separates  to  more  than  two  decimal  places.  Each  sample  should 
be  recorded  on  a  separate  sheet  as  the  weighing  progresses.  The  accom- 
panying report  of  an  analysis  indicates  a  convenient  method  of  setting 
up  the  form. 


Report  of  Sievic  Analysis 
Samfylc  Number     33  Analyacd  by     WCK  Date     1/5/38 

Description  of  Sample  Beach   sand 

10  min.    shaking 
Weight  of  Test  Sample    28.54   g.     Method  of  Analysis  in  Ro-Tap 

Screen  Opening              Grade  Siae          Weight             Jl'eiglit  Cuuinlati'.'e 

Retained          Per  Cent  Per  Cent 

0.701 1-0.707              0.03  g.              O.I  O.I 

0.495 O.707-.5OO             0.06                          0.2  0.3 

•351 O.5OO-.354              1.02                           3.6  3.9 

•246 O.354-.25O          16.37                        57.5  61.4 

.175 O.250-.I77           10.22                        35.8  97.2 

•  124 0.177-.125         0.74                 2.6  99.8 

.088 0.125-.088         0.03                  O.I  99.9 

28.47  g-  99-9 

Sieve  Loss   0.07  o.  i 

28.54  S-  loo.o 


142        SEDIMENTARY  PETROGRAPHY 

When  it  is  desired  to  sieve  coarse  pebbles  or  cobbles  of  a  size  larger 
than  is  commonly  handled  with  sieves,  several  devices  are  available.  Metal 
squares  or  rings  may  be  used,  made  of  heavy  wire,  to  extend  the  sieve 
sizes  as  far  as  necessary.  Squares  are  probably  preferable  inasmuch  as 
common  sieves  have  square  meshes,  and  the  same  shaped  rings  insure 
uniformity  of  the  basis  on  which  separations  are  effected.  A  simple  and 
convenient  device  may  be  made  from  a  sheet  of  zinc  measuring  about 
12  X  12  in^  Square  holes  may  be  cut  into  it,  ranging  from  8  to  64  mm. 
on  the  ^2  scale.  The  sieving  is  accomplished  by  dropping  the  pebbles 
through  corresjxDnding  openings  one  by  one,  and  either  placing  them  in 
separate  piles  or  tallying  them.  The  method  is  slow,  but  is  effective  when 
sieving  is  to  be  done  directly  in  the  field.  The  pebbles  in  each  group 
may  be  weighed  or  counted,  depending  upon  the  manner  in  which  fre- 
quency is  to  be  expressed. 

Wet  sieving  of  coarse  particles.  Wet  sieving  is  sometimes  resorted  to 
for  sieving  coarse  material,  but  for  general  purposes  it  is  not  as  satis- 
factory as  dry  sieving,  because  the  separation  is  not  complete.  It  appears 
that  the  film  of  water  in  the  wet  sieve  prevents  some  small  particles 
from  passing  through.  However,  when  the  sediment  is  dirty  or  partially 
aggregated,  wet  sieving  often  aids  in  separating  the  aggregates  and 
obtaining  a  cleaner  product.  In  all  cases,  however,  the  wet  sieve  sepa- 
rates should  be  resieved  through  the  same  sieves  when  dry,  to  remove 
the  finer  particles.^ 

For  wet  sieving  two  general  procedures  may  be  used.  The  loaded 
sieve  may  be  agitated  in  a  pan  of  water,  taking  care  that  none  of  the 
material  is  washed  over  the  edge  of  the  sieve,  or  a  spray  of  water  may 
be  directed  into  the  sieve  or  a  nest  of  sieves. 

Preliminary  splitting  of  composite  sediments.  Thus  far  the  assump- 
tion has  been  tliat  the  grains  of  the  sediment  were  all  within  the  sieving 
range,  namely  larger  than  %6  mm.  diameter.  If  there  is  any  material 
smaller  than  Yiq  mm.,  it  collects  in  the  pan  and  may  be  grouped  into  a 
single  class,  smaller  than  Yiq  mm.,  or  it  may  be  analyzed  further  by  a 
sedimentation  method.  Many  sediments  have  particles  both  coarser  and 
finer  than  %6  nim.,  so  that  composite  analyses  are  necessary.  The  usual 
procedure  is  to  split  the  sample  at  the  YiQ-mm.  point.  The  following 
routine  is  suggested,  to  prevent  aggregates  of  fine  material  from  re- 
maining on  the  sieves : 

A  weighed  test  sample  of  the  sediment  is  disaggregated  by  means  of 
the  routine  suggested  in  Chapter  3,  either  by  shaking  in  a  dilute  solution 
of  a  peptizer  or  by  rubbing  with  a  brush.  Disaggregation  should  be  con- 
tinued until  all  the  grains  are  clean  and  no  more  aggregates  can  be  de- 

1  H.  Gessner,  op.  cit.  (1931),  p.  145. 


METHODS  OF  SIZE  ANALYSIS  143 

tected.  The  entire  suspension  is  then  poured  through  a  sieve  with 
1 1,3 -mm.  mesh,  and  the  fine  material  caught  in  a  beaker.  A  convenient 
device  is  to  have  a  tin  funnel  large  enough  to  accommodate  the  sieve, 
so  that  the  wash  water  is  not  lost.  A  fine  stream  of  water  from  a  rubber 
tube  is  directed  over  the  sieve  residue  until  it  remains  clear.  The  sieve 
material  is  then  washed  into  filter-paper  and  dried.  The  residue  is  sieved. 
care  being  taken  to  use  a  pan  below  the  smallest  sieve  to  catch  any  fines 
that  were  not  separated  by  the  washing.  These  fines  are  added  to  the 
coarsest  grade  separated  by  sedimentation.  Methods  for  the  analysis  of 
the  fine  material  are  given  in  later  portions  of  this  chapter. 

Direct  Measurement  of  Large  Particles 
Sedimentan,-  particles  larger  than  about  16  mm.  diameter  may  be 
measured  indix-idually,  and  from  a  compilation  of  the  size  data  the 
frequency  may  be  determined.  The  measurements  to  be  performed  on 
the  particles  depend  upon  the  definition  of  size  adopted.  The  concept  of 
size  of  irregular  particles  was  discussed  in  Chapter  5  (pages  93.  127)  ; 
in  the  present  discussion  three  common  measurements  of  size  will  be 
included  :  the  nominal  diameter,  the  mean  diameter,  and  the  intermediate 
diameter. 

Measurement  of  the  nominal  diameter.  The  nominal  diameter  of  a 
particle  is  found  by  determining  the  diameter  of  a  sphere  having  the 
same  volume  as  the  particle.  The  equipment  required  consists  of  a  cyl- 
inder graduated  with  metric  divisions.  The  diameter  of  the  cylinder 
should  be  large  enough  to  accommodate  the  pebbles,  and  the  scale  divi- 
sions should  be  sufficiently  small  so  that  the  volume  of  the  pebble  may 
be  read  with  a  reasonable  degree  of  accuracy.  The  graduate  is  partially 
filled  with  water,  and  a  rubber  stopper  is  dropped  in  to  prevent  breakage 
as  the  pebbles  are  introduced.  The  initial  volume  of  water  is  recorded,  a 
pebble  is  dropped  in.  and  the  new  reading  made.  The  diflterence  is  the 
volume  of  the  pebble  in  cubic  centimeters.  The  process  is  repeated  with 
successive  pebbles  until  the  graduate  is  filled.  Care  should  be  exercised 
to  avoid  having  air  bubbles  on  the  pebbles,  especially  the  smaller  ones. 
By  wetting  the  pebbles  in  a  beaker  of  water  before  introducing  them 
into  the  graduate  the  air  bubbles  may  be  eliminated. 

For  pebbles  appreciably  smaller  than  16  mm.  diameter  an  ordinary- 
burette  may  be  used,  with  scale  divisions  of  o.i  c.c.  In  such  a  tube 
pebbles  as  small  as  about  4  mm.  diameter  may  be  measured, 

A  number  of  special  devices  have  been  used  for  measuring  volumes. 
Fancher,  Lewis,  and  Barnes  ^  describe  several  of  them. 

After  the  volumes  of  the  pebbles  have  been  found,  the  corresponding 

1  G.  H.  Fancher,  J.  A.  Lewis,  and  K.  B.  Barnes,  Some  physical  characteristics 
of  oil  sands :  Pcnn.  State  College,  Bull.  12,  p.  72,  1933. 


144 


SEDIMENTARY  PETROGRAPHY 
I  2  3 


3        4      5     6    7  8910 
DIAMETER,  CM. 

500  cc 


METHODS  OF  SIZK  ANALYSTS  145 

diameter  of  a  sphere  of  equal  vcjlunie  may  Ije  calculated  Ijy  the  C(iuatioii 

(/_^^  W-^=  >y  1.9-V,  where  V  is  the  volume  in  cubic  centimeters  and 
dn  is  the  nominal  diameter  in  centimeters.  It  may  he  converted  to  milli- 
meters by  multiplying  rf„  by  lO.  The  computations  may  he  ix;r formed  on 
a  slide  rule,  but  a  graph  showing  volume  along  one  axis  and  diameter 
along  the  other  is  convenient.  The  graph  is  constructed  by  plotting 
several  corresponding  values  for  d  and  V  on  double  logarithmic  pai>er 
and  drawing  a  straight  line  through  the  ^xDints.  Figure  36  is  such  a  grai>h 
for  the  range  i.o  to  lo.o  cm.  diameter. 

A  convenient  meth(jd  (;f  recording  the  results  of  nominal  diameter 
measurements  is  to  write  the  oljserved  volume  in  the  first  ccjlumn  and 
the  calculated  nominal  diameter  in  the  seccjnd  column.  If  the  number 
of  pebbles  to  be  measured  is  not  ioa  large,  it  is  often  desirable  to 
number  each  pcl)ble  with  a  ^Mjucil,  for  later  reference  and  comparison, 
inasmuch  as  pebbles  of  apparently  different  "sizes"  to  the  eye  may  have 
very  similar  nominal  diameters,  depending  upon  their  shape.  The  re- 
corded size  data  may  be  arranged  into  grades  if  frequency  data  are 
desired.  In  some  studies  involving  shapes  (Chapter  1 1)  the  volumes  may 
be  used  directly  for  the  assembling  of  frequency  classes.  In  such  cases 
the  nominal  diameter  need  not  be  calculated,  the  volumes  being  arranged 
into  grade  sizes  at  once. 

Measurement  of  long,  intenncdiatc,  and  short  diameters.  For  some 
purix)ses  it  may  be  desirable  to  know  the  longest,  intermediate,  or  short- 
est diameters  of  particles,  or  their  arithmetic  mean,  the  mean  diameter. 
These  lengths  may  be  measured  with 
a  caliper,  but  a  convenient  device  is 
available  for  rapid  work.  This  device, 
developed  by  Api)el  in  the  laboratories 
of  the  University  of  Chicago,  consists 
of   a   board   aliout    12    in.    long,    3   in.        Y\g.  37.— Device  for  measuring  di- 
broad,  and  I  in.  deep,  on  which  a  cen-    amcters  of  pebbles, 
timeter  scale  is  eml)edded  by  counter- 
sinking. One  end  of  the  base  is  fastened  to  a  block  of  wood,  flush  with 
the  end  of  the  centimeter  rule.  Another  block,  mounted  on  side  runners, 
slips  over  the  base  as  shown  in  Figure  37.  The  pebble  is  laid  against 
the  end  block,  in  proper  orientation,   and  the   sliding  block  is  moved 
against  it.  The  scale  reading  yields  the  appropriate  diameter  directly. 
Three  measurements  are  made  on  each  pebble,  the  longest,  intermediate, 
and  shortest  diameters,  each  of  which  can  be  found  by  simple  insi)ection 
or  trial.  The  arithmetic  mean  of  the  three  values  is  the  mean  diameter. 
It  is  computed  according  to  the  equation 


146 


SEDIMENTARY  PETROGRAPHY 


where  a,  h,  and  c  are  the  three  measured  diameters.  If  the  intermediate 
diameter  itself  is  to  be  used  as  a  measure  of  size,  the  other  diameters 
need  not  be  measured. 

Theoretically,  much  criticism  may  be  directed  against  the  mean  and 
intermediate  diameters  of  irregular  particles,  but  in  actual  practice  it 
is  often  found  that  these  values  closely  approach  the  value  of  the  nomi- 
nal diameter.  They  may,  accordingly,  be  used  as  approximations  of  the 
latter  if  the  departure  of  the  particle  from  true  sphericity  is  not  too 
great.^  As  long  as  the  particles  are  nearly  spherical,  the  mean  diameter 
will  be  very  nearly  equal  to  the  nominal  diameter,  but  with  increasing 
departure  from  sphericity  the  mean  diameter  becomes  larger  than  the 
nominal  diameter.  For  very  flat  disc-like  pebbles  there  may  be  no  direct 
relationship  between  the  two  values,  but  within  limits  the  agreement 
may  be  close  enough  for  practical  purposes.  To  test  this,  a  random 
sample  of  100  pebbles  was  taken  from  a  beach  and  all  three-types  of 
diameter  measured.  The  pebbles  were  approximately  disc-shaped,  had  an 
average  sphericity  of  0.705,  and  an  average  roundness  of  0.678.  The 
ratio  of  the  shortest  to  the  longest  diameter  was  1/2.1,  and  the  ratio 
of  the  shortest  to  the  intermediate  diameter  was  1/1.5. 


Table  15 

Comparison  of  Average  Mean  and  Ixtermediate  Diameters  with 

Average  Nominal  Diameter  of  100  Beach  Pebbles  from  Little  Sister 

Bay,  Wisconsin 


Xominal 

Mean 

Intermediate 

Data  Compared 

Diameter,   d^ 

Diameter,  d^ 

Diameter,  rf. 

(mm.) 

(mm.) 

(»jm.) 

Largest  pebble 

Smallest  pebble 

Arithmetic  mean  size 

46 
23 

45 
23 

49 
22 

of  100  pebbles  .... 
Ratio 

Geometric   mean   size 

30.0 

dn/dn=  1. 00 

30.9 

30.6 
djd^  =  1.02 

of  100  pebbles  

Ratio 

22.4 

d„/da=i.oo 

21.2 

d^/d,,  =  0.95 

21.6 
d,/d,  =  o.97 

Table  15  shows  that  the  range  of  sizes  is  roughly  the  same  in  all  three 
cases,  and  that  the  departure  of  the  arithmetic  average  of  the  mean  and 

1  It  was  pointed  out  in  Chapter  5  that  Roller  demonstrated  the  usefulness  of  the 
mean  diameter  as  an  approximation  of  the  harmonic  mean  diameter.  Thus  the  mean 
diameter  may  have  practical,  if  not  rigorous  theoretical,  significance. 


METHODS  OF  SIZE  ANALYSIS  147 

intermediate  diameters  of  the  pebbles  from  the  nominal  diameter  is 
within  3  per  cent.  With  the  geometric  means  the  agreement  is  somewhat 
less  satisfactory,  although  here  the  values  agree  within  5  per  cent. 
Roller  ^  considered  an  agreement  of  6  per  cent  between  the  harmonic 
and  arithmetic  means  to  be  satisfactory  for  most  purposes. 

Although  the  agreement  between  average  values  of  the  pebbles  ap[)ears 
to  be  satisfactory,  comparisons  of  the  frequency  distributions  as  histo- 
grams may  yield  figures  which  ditifer  widely  in  appearance. 


FINE     SEDIMENTS 

Direct  Separation  into  CIrades 

Sediments  with  particles  smaller  than  Y^  mm.  may  be  separated 
directly  into  grades  by  either  of  two  general  methods,  dccaiitatioii,  or 
rising  current  clutriation:  in  the  former  the  centrifuge  may  be  used  to 
hasten  sedimentation,  giving  rise  to  one  tyi^e  of  centrifugal  separation. 
Among  the  disadvantages  of  the  direct  methods  are  that  complete  se^xi- 
ration  is  practically  never  accomplished  and  that,  except  for  centrifugal 
methods,  the  techniques  are  not  practical  for  diameters  smaller  than 
about  o.oi  mm.  Indirect  methods  are  generally  more  precise  and  apply 
without  difficulty  to  particles  at  least  as  small  as  0.001  mm.  diameter. 

Dccantation  methods.  In  Chapter  5  it  was  ix)inted  out  that  decantation 
methods  include  all  methods  of  mechanical  analysis  in  which  the  grades 
are  seixirated  by  starting  with  a  thoroughly  mixed  suspension,  allowing 
sufficient  time  to  elapse  for  particles  above  a  given  diameter  to  settle 
to  the  bottom  and  at  that  moment  drawing  oflf  the  sui>ernatant  liquid, 
including  the  smaller  jxirticles  still  in  suspension. 

The  theory  of  decantation  methods  indicates  that  complete  seixiration 
of  the  grades  is  seldom  effected  and  that  for  precise  work  the  number 
of  decantations  is  high.  Decantation  methods  are  among  the  simplest 
techniques,  however,  in  terms  of  apparatus  and  ease  of  o^^eration.  The 
apparatus  required  may  be  merely  a  set  of  beakers,  a  liter  graduate, 
or  a  si>ecially  constructed  settling  tube  with  a  side  outlet.  The  following 
simple  procedure  requires  only  a  liter  graduate  and  a  large  beaker ;  it  is 
based  essentially  on  the  technique  recommended  by  Wentworth  in  1926.- 

A  weighed  quantity  of  the  sediment,  disaggregated  and  dispersed  in 
accordance  with  the  techniciues  given  in  Chapter  3,  and  with  all  particles 

1  P.  S.  Roller,  Separation  and  size  distribution  of  microscopic  particles :    U.  S. 
Dcpt.  Commerce,  Bur.  of  Mines,  Tech.  Paper  490,  1931. 
-  C.  K.  Wentworth,  loc.  cit.,   1926. 


148 


SEDIMENTARY  PETROGRAPHY 


above  Yiq  mm.  sieved  off,  is  diluted  to  a  sufficient  volume  so  that  it 
fills  the  liter  graduate  to  a  depth  of  30  cm.  To  separate  the  %6-%2  mm. 
grade,  sufficient  time  is  allowed  after  thorough  shaking  for 
particles  larger  than  ^732  mm.  to  settle  to  the  bottom. 
From  Stokes'  law  ^  it  is  found  that  particles  with  a  diam- 
eter of  y^o  mm.  settle  0.0869  cm. /sec,  or  require  11.5 
sec.  to  settle  i  cm.  This  is  equivalent  to  345  sec,  or  5^ 
min.,  for  30  cm.  At  the  end  of  that  time  the  supernatant 
suspension  is  withdrawn  with  a  rubber  tube  as  a  siphon, 
taking  care  to  avoid  drawing  up  any  of  the  bottom  sedi- 
ment. The  siphoned  liquid  is  drawn  into  a  beaker.  The 
original  graduate  is  filled  with  clear  water,  shaken  thor- 
oughly, and  the  settling  process  is  repeated  for  the  same 
length  of  time,  after  which  the  drawn-off  liquid  is  com- 
bined with  the  previously  decanted  suspension.  The  process 
is  repeated  three  or  four  times,  until  the  supernatant  liquid 
is  clear  after  the  settling  period.  The  sediment  in  the 
graduate    is    then    collected    in    filter-paper. 


F  I  G.  38- 
Kiihn's  set 
tling  tube. 


and    weighed 


tlie    yi6-V32    nmi. 


n 


dried, 

grade. 

For  the  separation  of  the  ^2"%4  mm. 
grade,  the  decanted  water  is  combined  and  poured  into 
liter  beakers  to  a  depth  of  10  cm.  The  settling  period  for 
particles  %4  mm.  in  diameter  for  this  depth  of  suspension 
is  46  sec.  per  centimeter  or  7  min.  40  sec.  for  10  cm.  The 
separation  below  ^04  mm.  is  seldom  carried  on,  so  that  the 
siphoned  suspension  may  be  run  directly  into  the  drain 
and  material  below  %4  mm.  computed  by  difiference.  The 
decantations  are  continued  until  the  water  is  clear,  the 
residue  is  filtered  as  before,  and  weighed.  The  result  of  the 
■ys2  mm. ;  1/32-% 4  mm. ;  and 
YQ^  nun.  my  difference.  If  the  %4-^/l2s 
mm.  grade  is  to  be  recovered,  the  time  required  to  settle  10 
cm.  is  31  min.  The  time  required  thus  increases  by  a  factor 
of  four  for  each  smaller  grade,  and  the  volume  of  water  to 
be  handled  also  increases  by  a  factor  of  three  or  four, 
depending  upon  the  number  of  decantations  for  each  grade. 

Numerous  workers  have  introduced  special  apparatus 
for  decantation.  The  tube  and  siphon  were  originally  used 
by  Wagner  in  1891 ;  ^  at  a  still  earlier  date  Kiihn  ^  used  a 
cylinder  with  a  side  opening  to  facilitate  withdrawing  the 

1  Wentworth  used  a  series  of  empirical  settling  velocities  based  on  experiment.  It 
is  preferable  to  use  Stokes'  law  or  Wadell's  practical  formula  to  compute  settling 
velocities.  See  Table  16  (page  166)  for  numerical  data. 

-E.  Wolff,  Die  Bodenuntersuchung :  Landwirts.  Vcrsuchs-Stat.,  vol.  38,  pp.  290- 
292,  1891. 

3  H.  W.  Wiley,  Principles  and  Practice  of  Agriculfnral  Analysis,  2nd  ed.  (Easton, 
Pa.,  1906),  vol.  I,  p.  203. 


METHODS  OF  SIZE  ANALYSIS 


149 


supernatant  liquid.  Appiani  ^  introduced  a  siphon  with  stopcock  near  the 
base  of  the  tube  to  facihtate  withdrawal  of  the  liquid.  Atterberg  -  in 
1914  raised  the  method  to  the  peak  of  its  develop- 
ment by  introducing  an  efficient  side  outlet.  Trask  ^  ,'-'"^^~'^> 
subsequently  introduced  a  removable  bottom  cup  to 
facilitate  removal  of  the  collected  sediment.  In  all  of 
these  de\'ices  the  underlying  method  of  operation  is 
similar,  although  various  authors  use  different  set- 
tling  times,    depending   upon   the   hydraulic   \^ues 
which  seemed  most  logical  to  them.  The  t}-pes  of 
apparatus   mentioned   above   are   shown  in   Figures 
38-41. 

Atterberg's  apparatus  has  been  ex- 
tensively used  by  European  workers, 
and  several  writers  *  have  discussed  it 
at  some  length.  Kohn  ^  studied  the 
streamhning  and  turbulent  effects  in 
the  withdrawal  of  the  suspension  from 
the  outlet.  Figure  42.  adapted  from  his 
photograph,  shows  the  eddy  set  up  dur- 
ing the  flow. 

Centrifugal  decaritation  methods.  Owing  to  the  time  re- 
quired for  small  particles  to  settle,  especially  in  decanta- 
tion  methods  where  a  nimiber  of  settling  periods  must  be 
allowed,  \-arious  workers  have  hastened  the  process  by  the 
use  of  centrifugal  force.  Among  the  earliest  de\'ices  for 
F  r  G.  41. —    this  purpose  was  a  centrifugal  cream  separator  of  the  Bab- 
Trask's    de-    ^q^]^  ^^^  ^^^  ^y  Whitney  to  hasten  sedimentation."^  Per- 
cantation  -^  -^  -^ 

tube.  haps  the  most  extensive  use  ot  the  centntuges  was  bv 


r 


Fig.     40. — Atter- 
berg's sedimentation 

cylinder. 


1 G.  Appiani,  Ueber  einen  Schlammapparat  fiir  die  Analyse  der  Boden  und 
Thonarten:  Forsch.  Geb.  Agrik.  Physik,  vol.  I",  pp.  291-.207,  1S94. 

-  A.  Atterberg,  Die  mechanische  Bodenanalyse  und  die  Klassitikation  der  Mine- 
ralboden  Schwedens  :  Int.  Mitt,  fiir  Bod<rnkundt\  vol.  2,  pp.  312-342,  1912. 

3  P.  D.  Trask,  Sedimentation  tube  for  mechanical  analysis:  Science,  vol.  71,  pp. 
441-442,  1930. 

*  G.  Richter,  Die  Ausfiihrtrng  mechanischer  und  physikalischer  Bodenanalysen : 
Int.  Mitt,  fiir  Bodenkimdc,  vol.  6,  pp.  198-208,  31S-346,  IQ16.  J.  P.  Van  Zyl,  Der 
.A.tterbergsche  Schlammzylinder :  Int  Mitt,  fiir  Bcdcnkundc,  vol.  S,  pp.  1-32,  lOiS. 
A.  A.  J.  von  'Sigmond,  Ueber  die  Methoden  der  mechanischen  und  physikalischen 
Bodenanalyse:  Publik.  der  k.  ungar.  Geol.  Reichscinstalt.  Budapest,  1016. 

5  M.  Kohn,  Beitrage  zur  Theorie  und  Praxis  der  mechanischen  Bodenanalyse: 
Landwirts.  Jahrb..  vol.  67,  pp.  485-346,  192S. 

6  L.  J.  Briess.  F.  O.  Martin  and  T.  R.  Pearce.  The  centrifugal  method  of  mechani- 
cal soil  analysis :  U.  S.  Dept.  Agric,  Bur.  of  Soils,  Bull.  34,  1904. 


SEDIMEXTARY  PETROGRAPHY 


Fig.  42. — Streamlines 
i  n  Atterber^'s  tube. 
Sketched  from  a  photo- 
graph by  Kohn,  1928. 


Briggs,  Martin,  and  Pearce  in  the  United  States  Bureau  of  Soils.^ 
Their  method  involved  the  removal  of  the  coarsest  particles  with  a 
sieve,  after  which  the  finer  material  was  dis- 
persed in  a  sterilizer  bottle.  Sedimentation  was 
effected  in  the  same  bottle  by  shaking  it  and  al- 
lowing the  suspension  to  rest  imtil  all  the  sand 
settled  out,  as  determined  by  microscopic  exami- 
nation. The  silt  and  clay  were  decanted  off  into  a 
centrifuge  tube,  and  the  material  was  centrifuged 
until  all  the  silt  settled  out,  lea\-ing  the  clays  and 
colloids  still  in  suspension.  In  each  case  the  de- 
cantations  were  repeated  until  satis  factor)'  sepa- 
rations were  made,  using  the 
microscope  to  check  results.  In  this 
manner  three  grades  were  ob- 
tained: the  sand  remained  in  the 
sterilizer  bottles,  the  silt  in  the 
centrifuge,  and  the  clays  and  col- 
loid in  the  decantations  from  the 
latter. 
More  recently  Truog  and  others  -  made  a  detailed  study 
of  centrifugal  decantation  methods,  developing  procedures 
for  the  more  complete  separation  of  the  finer  clay  frac- 
tions. 

Rising  current  elutriation.  The  separation  of  particles 
into  grades  by  rising  currents  may  be  accomplished  either 
with  water  or  with  air.  and  either  in  a  single  vessel  or  in  a 
series  of  vessels  of  different  sizes.  The  simplest  t>-pe  of 
apparatus  is  the  single-tube  water  elutriator,  the  best 
known  of  which  is  that  of  Schone.^  introduced  in  1867. 
Schone's  elutriator  is  a  conical  vessel  about  20  in.  tall, 
with  an  inlet  tube  at  the  base  and  an  opening  at  the  top 
for  the  passage  of  the  water.  As  indicated  in  Figure  43- 
a  glass  tube  is  inserted  at  the  top.  which  acts  simultane-  p  i  g.  43  — 
ously  as  an  outlet  and  a  piezometer.  The  sediment,  properly  j^""^^'^/^; 
dispersed,  is  placed  in  the  vessel,  and  a  current  of  water  is    elutriator. 

iz^T^^^f^R.  Tavlor,  R.  W.  Pearson,  M.  E  \yeeks  and  R.  W.  Simonson 
Procedure  for  special  t>-pe  of  mechanical  and  mineralogical  soil  analysis:  Froc.  5 01/ 
Sci.  Soc.  America,  vol.   i,  pp.  101-112,   1936  ^^ 

3  E.  Schone,  Ueber  einen  neuen  Apparat  fur  die  Schlammanalyse :  Zeits.  /.  anal. 
Chemie,  vol.  7,  PP-  29-47,  1867. 


METHODS  OF  SIZE  ANALYSIS 


151 


[:dO 


Hilgard's  clu- 


sent  through  with  a  velocity  just  greater  than 

the  setthng  velocity  of  the  smallest  grade  to  be 

removed.  The  flow  is  continued  until  the  water 

flows  out  clear.  The  finest  grade  is  then  sepa- 
rated from  the  collected  water  by 
filtration,  dried,  and  weighed. 
The  next  grade  is  obtained  by  in- 
creasing the  velocity  to  a  value 
just  above  the  settling  velocity  of 
the  next  larger  grade  size,  and 
the  process  repeated  until  the  de- 
sired number  of  grades  is  re- 
moved. Various  modifications 
have  been  made  of  Schone's  ap- 
paratus, principally  in  connection 
with  the  piezometer  and  the  tube 
connections,  but  in  principle  the 
apparatus  still  follows  Schone's 
original. 

There  is  a  strong  tendency  for 
aggregates  of  particles  to  form  in 

rising  current  elutriators,  and  these  interfere  with  the 
separation  of  the  smaller  particles.  To  overcome  this  diffi- 
culty, Hilgard  ^  devised  his  churn  elutriator,  illustrated  in 
iMgure  44.  In  principle  this  operates  like  the  Schone  elu- 
triator, except  that  near  the  base  of  the  tube  is  an  auto- 
matic stirring  device  which  aids  in  breaking  up  the  floc- 
cules  and  preventing  streaming  of  the  fluid. 

One  of  the  most  recent  elutriators,  designed  to  over- 
come the  disadvantages  of  the  older  type,  is  Andrews's 
kinetic  elutriator,-  introduced  in  1927.  This  elutriator  con- 
sists of  several  upright  tubes  in  a  vertical  column,  as 
shown  in  Figure  45.  The  upper  tube  feeds  the  undispersed 
sediment  to  a  restricted  zone  in  the  main  vessel,  where  an 
upward  current  of  water  is  directed  against  a  stationary 
cone.  The  impact  of  the  particles  against  the  cone  disag- 

1  E.  W.  Hilgard,  On  the  silt  analysis  of  soils  and  clays:  Am. 
Jour.  Sci.,  vol.  6,  pp.  288-206,  333-339.   1873. 

-L.  Andrews,  Elutriation  as  an  aid  to  enfrinccring  inspection: 
/;(/.  Engineering  Inspection,  Separate.  1927.  Andrews's  elutriator 
is  obtainable  from  Internal  Combustion,  Ltd.,  Aldwvch,  London 
W.  C.  2,  Rnsland. 


F  I  .;.  45.- 
A  n  d  r  c  w  s' 
kinetic  elutri- 
ator. 


152 


SEDIMENTARY  PETROGRAPHY 


grcgates  them,  and  the  fine  material  is  carried  off  through  a  spout.  The 
coarse  particles  settle  to  the  bottom  of  the  vessel  where  they  are  again 
picked  up  by  the  current  and  redirected  against  the  cone.  This  bombard- 
ment is  continued  until  the  aggregates  are  destroyed,  after  which  the  re- 
maining coarse  particles  are  separated  into  grades  in  the  lower  vessels  of 
the  apparatus.  Andrews's  elutriator  thus  combines  the  efficiency  of  a 
single-tube  elutriator  with  the  separation  effects  of  multiple-tubed  devices 
and  simultaneously  eliminates  the  tendency  toward  flocculation  which  is 
so  common  to  most  devices. 

Another  device  designed  to  overcome  the  disadvantages  of  early-type 
elutriators  was  introduced  in  1929  by 
Gross,  Zimmerley,  and  Probert.^  A  rotary 
rubber  impeller  provided  a  mobile  bed  for 
the  material  to  be  separated.  Gum  arable 
was  used  to  aid  dispersion. 

Among  multiple-tubed  elutriators  the 
Kopecky  -  apparatus  is  also  well  known. 
This  apparatus  consists  of  three  tubes 
arranged  side  by  side,  as  shown  in  Figure 
46.  The  sediment  is  placed  in  the  smallest 
tube,  and  the  water  enters  from  the  bot- 
tom of  this  vessel.  The  narrow  diameter 
of  the  first  tube  results  in  a  fairly  high 
velocity  of  the  water  within  it,  so  that  all 
particles  except  the  coarsest  are  removed.  In  the  second  vessel,  with  its 
larger  diameter,  an  intermediate  grade  is  removed,  because  the  same 
volume  of  water  passing  through  it  results  in  a  lower  net  velocity. 
Finally,  in  the  third  and  largest  vessel,  all  but  the  finest  particles 
remain.  The  result  of  the  analysis  is  to  obtain  four  grades,  the  limits 
of  which  depend  upon  the  original  volume  of  water  per  second  and  on 
the  respective  diameters  of  the  tubes.  As  in  the  case  of  the  single- 
vessel  elutriators,  the  water  is  allowed  to  flow  until  it  becomes  clear. 
Studies  of  Kopecky's  apparatus  ^  showed  that  the  largest  vessel  is 
rather  poorly  designed  for  the  continuous  flow  of  water  at  a  uniform 
velocity.  Tests  with  colored  fluids  showed  distinct  "streaming"  at  the 
center.  Andreason  ^  attacked  the  problem  of  multiple-tubed  elutriators 


Fig.  46. — Kopecky's   elutriator. 


1  J.  Gross,  S.  R.  Zimmerley  and  A.  Probert,  A  method  for  the  sizing  of  ore  by 
elutriation:  U.  S.  Bur.  Mines.  Reps,  of  Invcstujalions,  Serial  2931,  1929. 

-J.  Kopecky,  Die  Bodenuntersuchung  ziim  Zwccke  dcr  Drainage-arbeiien  (Prag, 
1901). 

3  H.  Gessner,  op.  cit.   (1931),  P-  203. 

*A.  H.  M.  Andreason,  he.  cit.,  1928. 


METHODS  OF  SIZE  ANALYSIS 


153 


by  using  three  vessels  shaped  similarly  to  Schone's,  as  shown  in  Figure 
47.  x\ndreason  devised  his  three  vessels  so  that  the  successive  ratios  of 
the  velocities  would  be  i.oo,  2.08,  and  5.40.  With  a  water  flow  of  y^ 
liter  per  minute,  the  velocities  in  the  three  vessels  were  0.27,  0.13,  and 
0.05  cm. /sec.  Thus  Andreason  separated  particles  at  the  size  limits  0.06, 
0.04,  and  0.02  mm.  diameter.  Among  multiple-tubed  elu- 
triators  of  the  Kopecky  type,  Andreason's  apparatus  ap- 
pears to  furnish  the  best  general  results  because  of  the 
more  efficient  shape  of  the  vessels. 

A  multiple-tube  rising  current  elutriator  which  achieved 
considerable  popularity  in  England,  was  developed  by 
Crook.^  It  consists  of  a  cylindrical  tube  surmounted  by  a 
larger  vessel.  In  operation,  the  water  velocity  is  so  adjusted 
that  the  sand  remains  in  the  lower  tube,  the  silt  remains  in 
the  upper  vessel,  and  the  clay  is  carried  through  an  outlet 
tube  into  a  beaker.  A  constant  head  apparatus  for  water 
flow  was  made  by  using  a  funnel  as  an  overflow  in  a  large 
bottle.  Full  instructions  for  the  operation  of  his  elutriator 
are  given  by  Crook  in  the  reference  cited. 

Gollan  -  developed  two  models  of  rising  current  elutria- 
tors.  The  first  consisted  of  a  single  conico-cylindrical  vessel 
with  auxiliary  tubes  which  could  be  inserted  within  the 
main  tube  to  eflfect  the  separation  of  additional  grades. 
The  second  model  consisted  of  three  vertical  tubes  and 
is  essentially  a  modification  of  Kopecky's  apparatus. 

A  slightly  modified  Kopecky  apparatus  was  used  by 
Rhoades  ^  for  the  aggregate  analysis  of  soils.*  Other  types  of  elutriators 
are  discussed  by  Gessner ;  ^  more  recent  devices  include  Rauterberg's  ^ 
single-vessel  elutriator,  constructed  very  simply  from  a  cylindrical  sepa- 
ratory  funnel. 

1  T.  Crook,  The  systematic  examination  of  loose  detrital  sediments.  Appendix 
to  Hatch  and  Rastall's  Pctrolo(iy  of  Scd'uncniary  Rocks  (Lundun,  1913),  pp.  348  ff. 

~  J.  Gollan,  Nouvel  Appareil  de  levigation  pour  I'analyse  mecaniqut  des  sols : 
Analcs  dc  la  Sci.  Agron.,  pp.  145  ff.,  1930.  J.  Gollan,  L.  Hervot  and  V.  Nicollier, 
Analisis  mecanico  de  Suelos :  Rev.  Fac.  Qiiiin.  Ind.  Agr.,  vol.  2,  19.52. 

3  H.  F.  Rhoades,  Aggregate  analysis  as  an  aid  in  soil  structure  studies :  Rep.  Am. 
Soil  Survey  Assn.,  Bull.  13,  pp.  165-174,  1932. 

4  Aggregate  analysis  is  a  term  applied  to  a  form  of  mechanical  analysis  in  which 
an  attempt  is  made  to  preserve  the  soil  structure  during  analysis.  A  comparison 
of  such  an  analysis  with  a  mechanical  analysis  based  on  complete  dispersion  affords 
a  means  of  determining  the  nature  of  soil  aggregates.  Aggregate  analysis  has  not 
been  used  widely  in  connection  with  sediments,  but  it  appears  to  afford  a  means  of 
studying  coagulation  effects  during  and  after  deposition. 

^  H.  Gessner,  op.  cit.  (1031).  pp.  115  ff. 

*  E.  Rauterberg,  Ein  einfacher  Schlammapparat :  Zcits.  J.  Pflancenenidli,  Di'ing., 
u.  Bodenkunde,  vol.  15  A,  pp.  263-269,  1930. 


F  I  G.  47.— 
Andrea- 
son's  elutri- 
ator. 


154 


SEDIMENTARY  PETROGRAPHY 


Stokes'  law  may  be  used  directly  in  connection  with  rising  current 
elutriators  if  the  apparatus  is  calibrated  for  the  range  of  velocities  de- 
sired. The  volume  of  the  tube,  assumed  to  be  cylindrical,  is  \'  =  qh, 
where  q  is  the  cross  section  and  h  is  the  height.  The  velocity  of  a  column 
of  water  may  be  expressed  as  v==h/t,  where  h  is  the  height.  From  these 
two  relations  one  may  derive  the  following  expression  for  the  velocity 
of  the  water  when  a  given  volume  flows  through  the  tube:  v=\'/qt. 
To  calibrate  the  tube,  the  piezometer  is  set  in  place  and  a  liter  beaker 
is  set  below  the  outlet.  The  water  is  turned  on  to  a  given  extent,  and 
the  time  is  measured  until  the  beaker  is  filled.  Knowing  Y,  q,  and  t,  the 
velocity  v  can  be  computed.  By  trial  and  error  the  volume  per  second 
is  controlled  until  the  desired  value  of  v  is  obtained.  When  this  is 
accomplished,  the  level  of  water  in  the  piezometer  is  marked  on  the  tube. 
After  the  several  velocities  have  been  established  (assuming  a  single- 
vessel  elutriator),  the  water  flow  may  thereafter  be  adjusted  until  the 
piezometer  stands  at  the  required  height. 


Air  chitriation.  The  use  of  air  currents  to  separate  fine  particles  has 
received  considerable  attention,  especially  in  its  application  to  the  analysis 
of  pigments,  cement,  and  ceramic  materials.  Air  analyzers  or  elutriators 
have  not  been  applied  extensively  for  the 
.      '      ,       . ,  study  of  sediments,  however.  Most  gener- 

ally the  apparatus  used  consists  of  an  up- 
right cylindrical  vessel  through  which  an 
upward  current  of  air  passes,  carrj-ing 
with  it  the  finer  particles. 

The  use  of  air  elutriation  dates  back  to 
the  early  years  of  the  present  centur)'.  In 
1906  Gary  ^  described  an  air  separator 
which  consisted  essentially  of  a  conical 
container  for  the  powder  charge  sur- 
mounted by  a  tall  cylinder.  The  air  blast 
impinged  vertically  on  the  powder  and  blew  out  the  fine  particles.  An- 
other early  elutriator  was  introduced  by  Cushman  and  Hubbard  in 
1907.^  Five  percolating  jars  were  arranged  in  series,  as  shown  in  Figure 
48,  the  first  of  three-gallon,  the  second  of  t^vo-gallon,  and  the  last  three 
of  one-gallon  capacity.  The  powder  was  placed  in  the  largest  jar,  at  the 
bottom  of  which  an  air  blast  entered  the  system.  A  series  of  tubes  con- 
nected each  jar  with  the  next,  and  a  suction  device  at  the  end  of  the 


J  L 


Fig.  48. — Cushman  and  Hub- 
bard's air  elutriator. 


1  M.  Gar}-,  Determination  of  a  uniform  method  for  the  separation  of  the  finest 
particles  in  Portland  cement  by  liquid  and  air  processes:  Int.  Assn.  Testing  Ma- 
terials, Brussels  Congr.,  1906. 

-  A.  S.  Cushman  and  P.  Hubbard,  Air  elutriation  of  fine  powders :  Jour.  Am. 
Chem.  Soc,  vol.  29,  pp.  589-596,  1907. 


METHODS  OF  SIZE  ANALYSIS 


4 


Fig.  4p.— 
G  o  n  e  1  i  '  s 
air  elutriator. 


system  was  so  adjusted  that  none  of  the  particles  was  carried  beyond 
the  last  jar.  Pearson  and  Sligh  ^  developed  an  analyzer  in  1915  similar 
in  principle  to  Gary's  but  with  an  automatic  device  for 
tapping  the  cylinder  during  separation.  Among  the  more       ^^-^-^ 
recent   devices   is   that   of   Gonell,-   introduced   in    1929. 
Gonell's    apparatus    consists    of    three    main    parts,    as 
sketched  in  Figure  49.  The  bottom  vessel  contains  an  air 
blast  inlet  which  terminates  just  above  the  bottom,  where 
the  powder  charge  is  placed.  Above  this  is  a  conical  vessel 
which  supports  a  c\-lindrical  tube.  A  glass  plate  near  the 
top  of  the  apparatus  serves  to  support  a  bell  jar  N\-ith  a 
cone    inverted    over    the    cylinder.    As    the 
analysis  proceeds,  the  fine  material  is  carried 
into  the  bell  jar,  some  settling  in  the  cone 
and  part  falling  on  the  glass  plate.  Through 
an  opening  in  the  top  of  the  bell  jar,  part  of 
the  fine  material  is  withdra\\Ti  to  a  collecting 
vessel.  Gonell  used  Stokes'  law  in  determin- 
ing the   velocities   required   for  separation. 
Perhaps  the  most  intensive  study  of  air 
elutriators  was  made  by  Roller  in  1931.^  He 
reviewed  earlier  work  and  developed  his  own  apparatus, 
which  consists  of  a  cylindrical  separator  with  a  conical  top 
and  bottom.  The  sample  is  placed  in  a  U-tube  at  the  bot- 
tom of  the  main  chamber,   and  as  the  air  blast  passes 
through    the    tube,    an    automatic    hammer    agitates    the 
U-tube  to  expose  fresh  charges  of  powder  to  the  current. 
Figure  50  shows  the  general  set-up.  without  details  of  the 
hammer.  The  particles  carried  upward  by  the  air  current 
are  collected  as  fractions  in  a  paper  thimble  at  the  top 
of  the  apparatus,  and  by  proper  control  of  the  current 
Roller  was  able  to  eflfect  separations  down  to  about  3 
microns.  Roller  used  Stokes'  law  for  computing  velocities 
and  checked  his  results  with  microscopic  measurements.  Full  details  for 
the  operation  of  the  apparatus,  sample  analyses,  and  an  excellent  dis- 
cussion of  methods  of  graphic  presentation  and  statistical  analysis  of 
the  data  are  given  in  Roller's  report. 

1  J.  C.  Pearson  and  W.  H.  Sligh,  An  air  analyzer  for  determining  the  fineness 
of  cement:  U.  S.  Dcpt.  Comni..  Bur.  Standards  Tech.  Paper  4S,  I9I5- 

-  H.  \V.  Gonell,  Determination  of  size  distribution  of  powder,  especially  cement : 
Zement.  vol.  17,  pp.  1786  ff..  1929. 

3  P.  S.  Roller,  Separation  and  size  distribution  of  microscopic  particles :  U.  S. 
Dept.  Comm.,  Bur.  of  Mines,  Tech.  Paper  490.  1931. 


Fig.  50. — 
Roller's 
a  i  r  elutria- 
t  o  r.  The 
hammer  i  s 
shown  at  A- 


156        SEDIMENTARY  PETROGRAPHY 

Other  direct  separation  methods.  In  the  usual  decantation  methods  the 
sediment  is  uniformly  distributed  through  the  suspension  at  the  start 
of  the  separation,  but  in  contrast  to  this  is  a  technique  in  which  the 
sediment  is  introduced  into  the  settling-  tube  as  a  unit.  The  principle  here 
is  that  if  a  mixture  of  various-sized  particles  is  introduced  at  the  top 
of  a  column  of  water  at  the  start  of  the  separation,  the  differential  set- 
tling velocities  of  the  particles  will  result  in  the  segregation  of  the 
several  grades  during  their  fall  through  the  tube. 

Numerous  devices  have  been  developed  to  operate  on 
this  principle,  which  was  first  described  by  Rham  ^  in 
1840.  Perhaps  the  best  known  of  the  devices  is  Bennig- 
sen's  silt  flask,-  introduced  circa  i860  and  shown  in  Fig- 
ure 51,  The  sample  is  introduced  into  the  flask  and 
shaken.  A  cork  is  inserted  in  the  neck,  and  the  flask  in- 
verted. The  material  settles  out  according  to  its  size,  and 
the  respective  amounts  present  are  read  in  cubic  centi- 
meters from  graduations  on  the  neck  of  the  flask. 

Clausen  ^  modified  Bennigsen's  flask  by  separating  the 
bulb   of   the   flask   from  the  volumetric   settling  tube.   A 
rubber  tube  connects  the  two  parts  of  the  apparatus.  Tube 
F  I  G.  51.—     ^"<J  flask  are  filled  with  water,  and  the  sediment  is  placed 
B  e  11  n  i  g-     in  the  flask.  The  flask  itself  is  agitated  at  the  side  and 
flasl-  then  superimposed  over  the  tube.  This  permits  the  mate- 

rial to  enter  the  tube  at  a  given  instant  and  results  in  a 
more  effective  separation  of  sizes.  A  more  recent  modification  of 
Clausen's  tube  was  introduced  by  Lober  ^  in  1932.  Instead  of  relying  on 
a  volumetric  reading  of  the  grades,  Leber's  tube  was  closed  with  the 
finger  at  the  bottom.  As  the  successive  grades  settled  (based  on  com- 
puted settling  velocities),  the  tube  was  dipped  into  a  dish  of  water,  the 
finger  removed,  and  the  grade  collected.  A  separate  dish  was  used  for 
each  grade,  and  by  collecting  the  separates  the  weight  composition  of 
the  material  was  readily  obtained. 

Other  types  of  apparatus,  utilizing  the  principle  of  the  silt  flasks,  are 
described  by  Gessner.^  Most  recent  of  the  separatory  techniques  in  this 

1  W.  L.  Rham,  An  essay  on  the  simplest  and  easiest  mode  of  analyzing  soils : 
Jour.  Roy.  Agric.  Soc.  England,  vol.  i,  pp.  46-59.  i840- 

-  F.  Wahnschaffe  and  F.  Schucht,  Anleitimg  sur  zcisscnschaftlichcn  Bodcmintcr- 
suchnnq.  4th  ed.    (Berlin,   1924),  p.  24. 

3F.  Wahnschaffe  and  F.   Schucht,  op.  cif.   (1924),  P-  25. 

4  H.  Lober,  Ein  besonders  einfaches  Verfahren  der  Schliimmanalyse :  CcntralbJatt 
filr  Mincralogie,  Abt.  B,  pp.  364-368,  1932. 

^^H.  Gessner,  op.  cit.   (193O,  PP-  73-7^- 


METHODS  OF  SIZE  ANALYSIS 


157 


category  is  that  of  Emery,^  who  used  a  glass  tube  5  ft.  long,  with  one 
end  tapered  and  connected  with  a  stopcock.  The  tube  is  filled  with  water 
and  the  sand  introduced  at  the  top.  Emery  proposed  his  tube  as  an 
alternate  method  to  the  sieving  of  sands. 


Indirect  Determination  of  Sizes 

Oden's  scdinwntation  balance.  In  191 5  the  principles  underlying  the 
mechanical  analysis  of  fine-grained  sediments  were  abruptly  placed  on 
an  entirely  new  foundation  by  the  publication  of  Oden's  theory  of  sedi- 
menting  systems.^  As  a  practical  ap- 
plication of  his  theory,  Oden  devel- 
oped his  continuous  sedimentation 
balance,  the  essential  principles  of 
which  are  shown  in  Figure  $2.  The 
apparatus  consists  of  a  balance  pan, 
suspended  near  the  bottom  of  a 
cylinder  of  soil  suspension,  upon 
which  the  falling  particles  accumu- 
late. The  pan  is  counterpoised  with 
another  in  such  a  manner  that  when 
the  sediment  lowers  the  pan  below 
a  critical  level,  the  scale  beam  oper- 
ates an  electrical  contact  which  re- 
leases shot  into  the  counterpoise.  By  recording  the  number  of  shot  and 
the  times  of  adjustment,  a  curve  is  constructed  showing  the  weight  of 
accumulated  sediment  as  a  function  of  the  time. 

]\Iore  than  any  other  device,  perhaps,  Oden's  balance  has  been  studied 
and  modified  by  numerous  workers.  Oden  himself  developed  elaborate 
controls  for  his  method,  and  in  1924  Coutts,  Crowther,  Keen,  and  Oden  ^ 
developed  an  automatic  recording  balance,  which  represents  the  ultimate 
yet  developed  in  such  apparatus. 

Figure  53  is  a  diagram  of  the  automatic  balance.  The  activating 
mechanism  is  the  rod  I\l  in  the  solenoid  S,  which  controls  the  recording 
pen  on  the  drum  H.  The  weights  which  drop  into  the  pan  R  serve  to 


Fig.  52. — Diagram  of  Oden's  con- 
tinuous sedimentation  balance. 


1  K.  O.  Emery,  Rapid  method  of  sand  analysis:  Gcol.  Soc.  America,  50th  Ann. 
Meeting,  Abstracts,  p.  15,  1937- 

-  S.  Oden,  Eine  neue  Methode  zur  mechanischen  Bodenanalyse :  Inf.  Mitt,  fiir 
Bodcnknnde,  vol.  6,  pp.  257-311,  1915.  Oden's  original  theory  is  given  in  Chapter  5, 
which  also  describes  the  graphic  method  devised  by  Oden  for  evaluating  his  ana- 
Ij'tical  data. 

■''  J.  R.  n.  Coutts,  E.  M.  Crowther,  B.  A.  Keen  and  S.  Oden,  An  automatic  and 
continuous  recording  balance:  Proc.  Roy.  Soc,  vol.  106A,  pp.  33-51,  1924. 


158 


SEDIMENTARY  PETROGRAPHY 


lower  M  into  the  solenoid  at  intervals  during  the  procedure,  and  result 
in  a  record  as  shown  on  the  drum  in  the  diagram.  Full  details  and  addi- 
tional data  on  the  theory-  of  the  apparatus  are  given  by  Keen.^ 

Other  modifications  of  Oden's  balance  were  made  by  various  workers. 
Johnson-  substituted  an  ingenious  recording  device  for  Oden's  weight- 
dropping  attachment.  The  recording  device  punctures  holes  in  a  record  on  a 
revolving  drum  by  sending  electric  sparks  through  at  timed  intervals.  After 


^ 


Fig.  53. — Diagram  of  Oden-Keen  self-recording  apparatus.   (After  Keen.) 


the  analysis,  the  paper  record  shows  a  series  of  holes  which  can  be  connected 
with  a  continuous  line  to  furnish  an  Oden  sedimentation  curve.  Werner  ^ 
developed  a  simplified  apparatus  in  which  a  volumetric  method  replaces  the 
continuous  weighing  technique  of  Oden.  The  sedimented  material  is  collected 
in  a  graduated  tube,  and  a  record  kept  of  the  volume  of  material  deposited  as 
a  function  of  time.  Werner's  device  has  the  advantage  that  it  is  easily  con- 
structed and  requires  no  expensive  equipment.  The  essential  features  are 
sketched  in  Figure  54.  Vendl  and  Szadeczky-Kardoss  ^  described  another 
modification  of  Oden's  method  in  which  a  delicate  spring  balance  is  used  for 
determining  the  weight  of  sedimented  material. 

1  B.  A.  Keen,  The  Physical  Properties  of  the  Soil  (London,  1931),  PP-  82  ff- 

2  W.  H.  Johnson,  A  new  apparatus  for  mechanical  analysis  of  soils :  Soil  Science, 
vol.  16,  pp.  363-366,  1923. 

3  D.  Werner,  A  simple  method  of  obtaining  the  size  distribution  of  particles  in 
soils  and  precipitates:  Trans.  Faraday  Soc,  vol.  21,  pp.  38i-394,  1925-26. 

4  M.  Vendl  and  E.  v.  Szadeczky-Kardoss,  Uber  den  sogennanten  grundsatzlichen 
Fehler  der  mechanischen  Analyse  nach  dem  Oden'schen  Prinzip:  Kolloid  Zcits., 
vol.  67,  pp.  229-233,  1934- 


METHODS  OF  SIZE  ANALYSIS 


59 


Schramm  and  Scripture  ^  used  a  series  of  test-tubes  instead  of  a  Ijalance 
to  obtain  the  Oden  sedimentation  curve.  Given  volumes  of  suspension  are 
poured  into  a  series  of  tubes  and  the  liquid  above  a  mark  on  the  side  is 
drained  from  successive  tubes  at  stated  intervals.  The  sedimented  material 
in  each  tube  is  dried  and  weij^hed,  to  determine  how  much  had  accumulated 
during  the  intervals.  The  data  so  obtained  furnish  points  along  the  sedi- 
mentation curve.  Tickell  ^  describes  the  Schramm  and  Scrip- 
ture method  in  full  detail. 

In  1930  Trask^  applied  centrifugal  force  to  aliquot  portions 
of  the  sample  to  hasten  sedimentation  of  the  smaller  particles. 
The  complete  method  includes  decanting  the  sands,  separating 
the  suspension  of  fine  materials  into  a  number  of  aliquots, 
and  determining  the  weight  of  material  that  separates  from 
each  aliquot  after  centrifuging  for  definite  times  at  specified 
speeds.  The  sedimentation  curve  is  constructed  from  the  data 
obtained  from  the  aliquots.  The  method  is  rapid  and  requires 
less  expensive  apparatus  than  Oden's  original  method. 


In  the  discussion  of  Oden's  theory  of  sedimcnting  sys- 
tems (Chapter  5)  it  was  pointed  out  that  equation  (33) 
(page  117)  was  developed  for  the  Oden  balance.  In  prac- 
tice the  equation  is  seldom  used,  inasmuch  as  simple 
graphic  methods  (page  114)  are  available  for  analyzing 
the  Oden  curve.  All  of  the  techniques  described  above 
yield  Oden  curves,  which  can  be  treated  in  the  standard 
graphic  manner.  The  principal  difiference  among  the 
methods  is,  perhaps,  the  precisencss  with  which  the  curve 
is  determined. 

Continuous  sedimentation  cylinders.  In  1918  Wieg- 
ner  *  introduced  a  manometric  sedimentation  cylinder 
which  rested  upon  the  principle  that  two  columns  of  liqtior  of  difTer- 
ent  specific  gravities  will  rise  to  levels  inversely  proportional  to  their 
densities,  when  confined  in  separate  tubes  which  are  joined  at  some 
point. 

Wiegner's  apparatus  consists  of  a  long  glass  cylinder  to  which  is 
attached  a  parallel  manometric  tube  of  about  the  same  length  but  of  a 
smaller  diameter,  as  shown  in  iMgure  55.  A  sto^Kock  controls  the  point  of 


F  I  G.  S4.— 
W  c  r  n  e  r's 
m  0  (1  i  fi  c  a- 
t  i  o  11  of 
Oden's 
method. 


1  E.  Schramm  and  E.  W.  Scripture,  Jr.,  The  particle  analysis  of  clays  by  sedi- 
mentation :  Jottr.  Am.  Ccram.  Soc,  vol.  8,  pp.  243-258,  1925. 

-  F.  G.  Tickell,  The  Examination  of  Fragmcntal  Rocks  (Standard  University 
Press,  1931),  pp.  10-16. 

3  P.  D.  Trask,  Mechanical  analysis  of  sediments  by  centrifuge :  Econ.  Gcoloqy, 
pp.  581-599,  1930. 

4  G.  Wiegner,  Ueber  cine  neuc  Methode  dcr  Schlammanalyse :  Landzvirts.  Vcr- 
suchs-Stat.,  vol.  91,  pp.  41-79,  1918. 


i6o 


SEDIMENTARY  PETROGRAPHY 


juncture.  With  this  closed,  water  is  poured  into  the  manometer,  and  the 
soil  suspension  into  the  cylinder.  When  the  stopcock  is  opened,  the 
manometer  registers  the  hydrostatic  pressure  at  the  point  of  juncture, 
and  as  particles  in  the  suspension  settle  below  this  level, 
the  pressure  decreases.  By  observing  the  decrease  in  the 
height  of  water  in  the  manometer  tube  as  a  function  of 
the  time,  a  continuous  curve  is  obtained,  from  which  the 
size  frequency  distribution  may  be  determined  graphically. 
In  terms  of  Oden's  theory,  Wiegner's  apparatus  is  based 
on  equation  (31)  of  Chapter  5  (page  117).  That  is,  the 
manometer  measures  the  hydrostatic  pressure  at  a  fixed 
depth  as  a  function  of  the  time.  The  hydrostatic  pressure 
diminishes  with  time  as  the  particles  settle  below  the  junc- 
ture of  the  manometer,  and  hence  the  sedimentation  curve 
obtained  is  concave  upward,  instead  of  convex  as  in  the 
case  of  an  Oden  curve.  However,  the  Wiegner  curve  is  a 
reflection  of  an  Oden  curve  with  respect  to  the  .sr-axis,^ 
as  Figure  56  shows,  and  hence  the  frequency  distribution 
may  be  determined  in  the  same  graphic  manner  that  is  used 
with  an  Oden  curve. 

Gessner  -  added  a  major  improvement  to  Wiegner's  tube 
in  1922,  when  he  added  an  automatic  photographic  record- 
ing device.  A  sheet  of  pho- 
tographic paper  is  placed  on 
a  revolving  drum  in  a  light- 
tight  box.  A  beam  of  light 
is  directed  to  a  mirror  be- 
hind the  manometer  tube  and  reflected 
back  to  a  lens  leading  to  the  sensitive 
paper  on  the  drum.  As  the  column  of 
water  falls,  a  continuous  photographic 
record  is  obtained.  Figure  57  is  a  ver- 
tical diagram  of  the  essential  features 
of  the  apparatus. =*  With  Gessner's  ap- 
paratus it  is  not  difficult  to  analyze  the 
range  of  sizes  from  o.i  to  .CX)2  mm. 
diameter. 


Fig  55. 
— W  i  e  g- 
n  e  r  '  s  con- 
t  i  n  u  o  u  s 
s  e  d  i  in  e  n- 
tation   tube. 


Wiegner  Cu 


Fig.  56.  —  Relation  between 
Oden  and  Wiegner  curves. 
(After  Oden,  1925.) 


1  S.  Oden,  The  size  distribution  of  particles  in  soils  and  the  experimental  methods 
of  obtaining  them :  Soil  Science,  vol.  19,  pp.  1-35,  1925- 

2  H.  Gessner,  op.  cit.,  p.  98. 

3  Gessner  has  a  detailed  discussion  of  the  apparatus  in  his  book    (op.  cit.,  pp. 
191  fT.),  including  full  directions  for  its  operation. 


METHODS  OF  SIZE  ANALYSIS  i6i 

Wiegner's  tube,  like  Oden's  balance,  has  been  the  subject  of  considerable 
modification  by  other  workers.  Zunker  ^  added  an  auxiliary  manometric  tube 
near  the  top  of  the  main  cylinder  so  that 

the    difference    in    heights    could    more  "-q  "  wiegner  tube       

accurately  be  measured  by  eye   from  a        """^^--^^^  \^ 

parallel   scale   of   the   two  manometers.  ~~'<' 

One    of    the    difficulties    of    Wiegner's 


original  apparatus  was  that  small  differ-        p^^    :^7.— Vertical  view  of  Gess- 
ences  of  height  in  the  two  tubes  could     ner's    continuous    recording    device 
not  be  accurately  read  by  eye.  Zunker's     for  Wiegner's  tube, 
tube  was  designed  to  overcome  this  dif- 
ficulty. Another  modification,  also  directed  toward  this  end,  was  introduced 
by  Kelley,-  who  bent  the  manometer  tube  through  an  angle.  Kelley's  modifica- 
tion   is    shown    in    Figure    58.    By    bending   tlie    tube 
through  an  angle  0  from  the  vertical,  a  given  vertical 
difference  in  height,  Ah,  becomes  Ah /cos  6  along  the 
inclined   tube,   so   that   the   reading   may   be   made   to 
i/cos<9  of  the  original  vertical  scale,  thus  permitting 
a  much  closer  reading.  Other  modifications  of  \\'ieg- 
ner's  tube  were  made  by  Oden,^  who  introduced  an  in- 
ternal manometric  tube  having  a  liquid  with  a  lighter 
specific  gravity  than  water,  and  Von  Hahn,'*  who  de- 
veloped two  modifications,  one  of  which   involves  an 
inverted  U-tube,  one  arm  of  which  extends  downward 
into  a  vessel  of  the  suspension,  and  the  other  into  a 
vessel  of  clear  water  as  shown  in  Figure  59.  A  stop- 
cock at  the  juncture  of  the  tubes  leads  to  a   suction 
pump,  and  by  opening  the  stopcock  columns  of  liquid 
are  drawn  into  the  two  tubes  to  heights  depending  upon 
their  specific  gravities.  The  stopcock  is  closed  during 
an  analysis,  and  the  uniform  gas  pressure  within  the 
Fig.     58.— Kelley  s   ^  |^      results  in  varving  relations  between  the  heights 
modilication  of  Wieg-       .    ,  ,  -     ,?     .  ,    ^,  .     o-     .  •    .1 

ner's  tube.  of  the  two  columns  of  liquid.  The  net  effect  is  the  same 

as  in  Wiegner's  tube. 

A  more  recent  modification  of  the  Wiegner  apparatus  was  made  by  Barnes.^ 
A  needle  was  geared  to  a  dial  on  the  manometer  to  obtain  precise  readings  of 
the  water  level.  The  instrument  detected  changes  of  level  of  the  order  of 
magnitude  of  10-*  cm.  A  galvanometer  was  used  to  detect  the  point  of  contact, 
and  the  corresponding  value  was  read  from  the  dial. 

In  1934  Knapp^  described  a  patented  automatic  sedimentation  unit,  called 

1  F.  Zunker,  Die  Bestimmung  der  specifischen  Oberflache  des  Bodens :  Laudzcirts. 
Jahrb.,  vol.  58,  pp.  159-203,  1922. 

-  W.  J.  Kelley,  Determination  of  distribution  of  particle  size :  Jour.  hid.  Eng. 
Chcm.,  vol.  16,  pp.  928-930,   1924. 

3  S.  Oden,  loc.  cit.,  1925. 

*  F.-V.  von  Hahn,  0/'.  cit.,  p.  310. 

5  A  method  for  the  determination  of  size  distributions  in  soils :  Rep.  Am.  Soil 
Survey  Assn.,  Bull,  ir,  pp.  169-173,  1930. 

6  R.  T.  Knapp,  New  apparatus  for  determination  of  size  distribution  of  particles  in 
fine  powders:  Ind.  and  Eng.  Chem.  (Analytical  edition),  vol.  6,  pp.  66-71,  1934. 


1 62 


SEDIMENTARY  PETROGRAPHY 


A 


the  "Microneter,"  which  utilizes  the  principle  of  W'iegner's  tube.  The  sedi- 
mentation cell  is  mounted  in  a  brass  casting  which  contains  a  small  pressure- 
measuring  orifice,  connected  to  a  pressure  cell.  The  pressure  cell  is  a  metal 
bellows  so  arranged  that  a  change  of  pressure  moves  a  small  mirror,  which 
reflects  a  beam  of  light  to  a  photographic  plate.  The  cur\-e  obtained  is  ana- 
lyzed graphically  in  the  manner  of  Oden  or  Wiegner  curves. 

In  1927  Crowther  ^  developed  a  continuous  sedimentation  tube  with  a 
manometer  which  measured  the  hydrostatic  pressure  at  two  points  in  the 
suspension.  The  essential  details  are  shown  in  Figure  60,  which  illus- 
trates the  manometric  attachment.  Aniline  (specific  gravity  =  1.02)  was 
used  for  the  manometer  liquid.  The  three-way 
stopcocks  are  added  for  convenience  in  washing 
the  side  tubes  without  losing  the  aniline. 

Although  the  differential  manometer  of  Crow- 
ther's  tube  actually  measures  the  excess  of  the 
hydrostatic  pressure  between  the  two  points  in 
the  suspension  over  that  of  an  equal  column  of 
water,  this  pressure  difference  is  equal  to  the 
density  of  the  suspension  half  way  between  the 
entry  tubes.  Keen-  thus  points  out  that  Crow- 
ther's  tube  may  be  used  in  connection  w4th 
Oden's  theory-  of  sedimenting  systems,  speci- 
fically equation  (29)  of  Chapter  5  (page  116). 
It  will  be  noted  that  although  both  Crowther's 
tube  and  Wiegner's  tube  are  continuous  sedimen- 
tation de\'ices,  they  do  not  operate  on  identical 
principles,  inasmuch  as  \\'iegner's  apparatus  is 
related  to  equation  31  of  Chapter  5. 

The  method  of  Crowther  is  unique  among 
sedimentation  tube  methods  because  of  the  fact  that  it  furnishes  a 
cumulative  curv-e  instead  of  an  Oden-type  curve.  When  the  manome- 
ter readings  are  plotted  directly  against  diameters  of  particles  (as 
determined  from  settling  velocities),  an  S-shaped  cur\-e  is  obtained, 
which  can  be  interpreted  as  the  cumulative  curve,  because  the  manome- 
ter readings  are  directly  proportional  to  the  ordinates  of  the  ciunulative 


Fig.  59. — Xon  Hahn's 
double  sedimentation 
tube. 


The  pipette  method.  Among  all  the  methods  of  mechanical  analysis 
related  to  Oden's  theory,  the  pipette  method  has  received  by  far  the 

1  E.  M.  Crowther,  The  direct  determination  of  distribution  curves  of  particle  size 
in  suspension:  Jour.  Soc.  Chem.  Ind.,  vol.  46,  pp.  105T-107T,  1927. 

2  B.  A.  Keen,  op.  cit.  (1931),  p.  60. 


METHODS  OF  SIZE  ANALYSTS 


163 


widest  official  recognition,  both  because  of  its  convenience  and  because 
of  the  sinipHcity  of  the  required  apparatus.  Actually  no  more  equipment 
is  required  than  a  10-  or  20-c..c.  pipette,  a  liter  graduate,  several  50-c.c. 
beakers,  a  hot-plate,  and  an  analytical  balance — equipment  to  be  found 
in  almost  any  laboratory. 

The  pipette  method  was  developed  independently  in  1922-1923  by 
Robinson  ^  in  England,  Jennings,  Thomas,  and 
Gardner  -  in  the  United  States,  and  Krauss  ^  in 
German}-.  In  theory  the  pipette  method  is  based 
on  equation  (29)  of  Chapter  5  (page  116),  inas- 
much as  the  method  actually  determines  the 
density  of  the  suspension  at  a  fixed  depth  as  a 
function  of  the  time.  For  practical  purposes  of 
analysis,  however,  the  principles  on  which  the 
pipette  method  is  based  may  be  considered  as  fol- 
lows. If  a  suspension  is  thoroughly  shaken  so  that 
the  particles  are  uniformly  distributed  and  is  then 
set  at  rest,  all  particles  having  a  settling  velocity 
greater  than  h/t  will  have  settled  below  a  plane 
of  depth  h  below  the  surface,  at  the  end  of  an  in- 
terval of  time  t.  All  particles  having  a  velocity 
less  than  h/t,  however,  will  remain  in  their  original  concentration  at 
depth  h,  because  they  will  have  settled  only  a  fraction  of  this  distance  in 
time  t.  A  small  sample  is  taken  from  depth  h  at  time  t  and  evaporated 
to  dryness.  The  weight  of  the  residue,  multiplied  by  a  proiX)rtionality 
factor  based  on  the  ratio  of  the  pipette  volume  to  the  total  suspension 
volume,  will  represent  the  total  amount  of  material  having  settling 
velocities  less  than  h/t. 

After  the  first  pij^ette  sample  has  been  withdrawn,  the  susjjcnsion  is 
again  shaken  and  a  greater  period  of  time  is  allowed  to  elapse,  so  that 
particles  of  a  next  smaller  size  may  settle  below  depth  h.  The  second 
pipette  sample  will  then  contain  a  residue  smaller  than  that  of  the  first 
sample  by  an  amount  equal  to  the  weight  of  material  lying  between  the 
two  chosen  sizes  or  settling  velocities.  The  process  may  obviously  be 
repeated,  and  by  simply  subtracting  the  weights  of  successive  residues 


Fic;.  60. — Detail  of 
Crowthcr's  continuous 
sedimentation  tube. 


1  G.  W.  Robinson,  A  new  method  for  the  mechanical  analysis  of  soils  ano  other 
dispersions:  Jour.  A(/ric.  Science,  vol.  12,  pp.  306-321,  1922. 

2  D.  S.  Jennings,  M.  D.  Thomas  and  W.  Gardner,  A  new  method  of  mechanical 
analysis  of  soils  :  Soil  Science,  vol.  14,  pp.  485-499,  1922. 

3  G.  Krauss,  Ueber  eine  .  . .  neue  Methode  der  mechanischen  Bodenanalysc :  Inf. 
Mitt,  jiir  Bodcnkunde,  vol.  13,  pp.  147-160.  1923. 


164 


SEDIMENTARY  PETROGRAPHY 


(each  multiplied  by  the  proix)rtionality  factor)  the  amount  of  material 

in  any  grade  may  be  determined  directly. 

Theoretically  the  pipette  sample 
should  represent  a  horizontal  stra- 
tum of  depth  h  and  essentially  in- 
finitesimal thickness.  Practically,  the 
pipette  taps  a  spherical  zone,  and 
Kohn  ^  studied  the  influence  of  this 
fact  on  the  accuracy  of  the  analysis. 


Fig.  61. — Streamlines  of  flow  in  pi- 
pette method.  (After  Kohn,  1928.) 


He  photographed  the  streamlines  of 
liquid  entering  the  pipette  and  con- 
cluded that  inasmuch  as  part  of  the 
sphere  is  above  the  theoretical  stra- 
tum and  part  below,  the  error  is 
essentially  compensa- 
tory. Figure  61  is 
dra\yn  from  Kohn's 
photograph.  Keen  ^  and  Gessner  ''  concur  with  Kohn  in  his 
conclusion  that  the  net  error  is  practically  negligible. 

Many  devices  have  been  developed  for  pipetting  the  sus- 
pension. Robinson  used  an  ordinary  pipette,  Jennings, 
Thomas,  and  Gardner  used  a  multiple-intake  pipette  fixed 
at  a  constant  depth,  as  shown  in  Figure  62,  and  Krauss 
used  a  series  of  three  pipettes  with  side  openings,  which 
were  introduced  to  the  desired  depth  by  a  rack  and  pinion. 
A  diagram  of  this  device  is  shown  in  Figure  63.  Other 
special  pipettes,  including  devices  for  raising  and  lowering 
them,  as  well  as  constant-temperature  jackets  for  the  cyhn- 
der  of  suspension,  were  developed  by  various  workers.^ 
Steele  and  Bradfield  ^  applied  centrifugal  force  to  the 
pipette  method  to  obtain  detailed  analyses  of  material 
smaller   than    5    microns    in   diameter.    Ordinary    gravity 


1  M.  Kohn,  loc.  cit.,  1928. 

2  B.  A.  Keen,  op.  cit.  (1931),  P-  73- 

3  H.  Gessner,  op.  cit.  (1931).  P-  79  ff- 

4  M.  Kohn,  loc.  cit.,  1928.  A.  H.  M.  Andreason,  loc.  cit..  1928.  L.  B.  Olmstead, 
L.  T.  Alexander  and  H.  E.  Middleton,  A  pipette  method  of  mechanical  analysis  of 
soils  based  on  improved  dispersion  procedure:  U  S.  Dcpt.  Auric.  Tech  Bull,  ijo, 
1930.  T.  M.  Shaw,  New  aliquot  and  filter  devices  for  analytical  laboratories:  Ind. 
and  Eng.  Client.,  vol.  4,  pp.  409-413,  1932- 

5  J.  G.  Steele  and  R.  Bradfield.  The  siprnificance  of  size  distribution  in  the  clay 
fraction:  Rep.  Am.  Soil  Survc\  Assn.,  Bull.  15,  pp.  88-93,  1934- 


METHODS  OF  SIZE  ANALYSIS 


165 


settling  was  used  to  diameters  of  0.5  micron,  after  which  25-c.c.  portions 
of  the  suspension  were  placed  in  tuhes  and  centrifuged  at  2,200  r.p.m. 
to  hasten  sedimentation  of  the  finest  particles.  Analyses  down  to 
0.0000625  mm.  diameter  were  successfully 
carried  on. 

The  authors  have  found  that  the  use  of  an 
ordinary  pipette,  with  a  rubber  tube  for  suc- 
tion,^ and  supported  by  hand,  yields  amply 
satisfactory  results  after  a  preliminary 
period  of  practice.  A  20-c.c.  pipette  is  used, 
on  the  stem  of  which  have  been  engraved 
marks  at  5,  10,  and  20  cm.  from  the  tip. 
The  pipette  is  held  by  both  hands,  one  rest- 
ing on  the  edge  of  the  cylinder  of  suspen- 
sion. The  pipette  is  lowered  to  the  proper  ^ 
mark,  and  an  even  suction  is  applied  with  the 
mouth.  When  the  pipette  is  filled,  the  end  of 
the  rubber  tube  is  clamped  with  the  teeth,  and  the  pipette  transferred 
above  a  50-c.c.  beaker.  By  releasing  the  tube  the  contents  are  trans- 
ferred without  loss.  A  single  rinse  of  the  pipette  with  clear  water  suffices. 
If  one  prefers,  a  simple  suction  device  suggested  by  Whittles  -  may  be 


Fig.  63. — Krauss's  pipette. 


used   (Figure  64). 


Fig. 


The  aspirator  bottle  is  attached  to  a  suction  pump, 
and  a  rubber  tube  on  the  left  leads  to 
the  pipette.  A  short  glass  tube  (A  in 
Figure  64)  extends  through  the  stop- 
per. In  taking  a  sample,  the  rubber 
tube  from  the  pipette  is  pinched,  the 
pipette  inserted  in  the  suspension,  a 
finger  is  placed  on  A,  and  the  tube  is 
released  until  the  pipette  is  filled.  At 
that  instant  the  tube  is  again  pinched 
and  the  finger  removed  from  A.  The 
suction  pump  is  operated  at  a  slow 
uniform  rate  during  the  entire  process. 


To  Pump 


withdrawing  pipette  sample. 


Considerable  experimentation  has  been  performed  on  the  pij^ette 
method  in  the  laboratories  of  the  University  of  Chicago,  and  because 
of  the  authors'  wide  experience  with  the  method,  the  following  detailed 
I)rocedures  are  given  here.  The  general  remarks  about  the  comi)utation 
of  settling  velocities,  the  preliminary  preparation  of  the  samples,  and 
similar  items  apply  to  any  method  of  analysis,  but  all  are  included  here 
for  the  sake  of  completeness. 

1  W.  C.  Krumbein,  The  mechanical  analysis  of  fine-grained  sediments :  Jour.  Sed. 

Petrol fltiv.\-o\.  2,  pp.  140-140,  IQ32. 

2  C.  h.  Whittles,  Methods  for  the  disaggregation  of  soil  aggregates  and  the 
prc])arriti()n  of  soil  suspensions:  Jour.  Agric.  Sci.,  vol.  14,  pp.  346,  369,  1924. 


[66        SEDIMENTARY  PETROGRAPHY 


PROCEDURE      FOR     PIPETTE     ANALYSIS 

Choice  of  grade  sices.  For  most  routine  analyses  workers  in  America 
probably  use  Wentworth  grades  directly.  If  the  pipette  method  is  re- 
stricted to  material  finer  than  %6  mm.  in  size,  Stokes'  law  or  Wadell's 
practical  sedimentation  formula  may  be  used  for  computing  the  settling 
velocities  of  the  limiting  sizes.  Tables  12  and  13  of  Chapter  5  include 
the  settling  velocities  of  particles  on  the  V^  grade  scale.  From  these 
tables  one  may  prepare  the  time  schedule  for  analysis.  For  example,  on 
the  basis  of  Stokes'  law,  a  quartz  particle  %2  "ini-  in  diameter  has 
a  settling  velocity  of  0.0869  cm./sec.  at  20°  C.  For  sampling  depths 
of  10  cm.,  the  time  of  settling  may  readily  be  found.  From  the  relation 
v^h/t,  where  v  =  0.0869  and  h  =  10,  one  obtains  t  =  h/-j=  10/0.0869 
=  58  sec.  In  a  similar  manner  the  time  for  any  other  limiting  diameters 
may  be_computed.  Table  16  shows  the  depths  and  times  for  grades  on 
the  V  2  scale  from  ^/^o  mm.  to  /4o4s  mm.,  based  on  Stokes'  law.  In 


Table  16 
Times  of  Settling  Computed  According  to  Stokes'  Law* 


Diameters  in  Velocity  h 

Millimeters  {cm./sec.)         (cm.) 


Hr.        Min.  Sec. 


1/16 0.0625            0.347  20  o              o              58 

.0442              .174  20  o              I              56 

1/32 0312              .0869  10  o              I              56 

.0221              .0435  10  o              3              52 

1/64 0156              .0217  10  o              7             44 

.0110              .0109  10  o            15 

1/128 0078              -00543  10  o            31 

.0055              .00272  10  I               I 

1/256 0039              .00136  10  2              3 

.00276            .00068  10  4              5 

1/512 00195            .00034  10  8             10 

.00138            .000168  10  16            21 

1/1024 00098            .000085            5  16  21 

.00069            .000043            5  32  42 

1/2048 00049            .000021            5  65  25 

*  The  values  in  this  table  are  based  on  temperature  of  20°   C.  and  an  average 

specific  gravity  of  the  sediment  equal  to  2.65.  Seconds  are  neglected  in  lower  part 
of  table. 

the  table  the  values  assume  a  temperature  of   20°    C.   and  a  specific 

gravity  of  the  sediment  equal  to  2.65.  A  similar  table  of  time  values 

based  on  Wadell's  formula  may  be  prepared  from  the  data  in  Table  13 
of  Chapter  5. 


METHODS  OF  SIZE  ANALYSIS 


167 


Experience  has  shown  that  a  standard  depth  of  10  cm.  for  sampUng  is 
inconvenient  for  some  sizes  of  material.  During  the  first  few  moments 
after  shaking  a  suspension  one  may  obsers-e  irregular  currents  in  the 
cylinder;  it  seems  desirable  to  allow  a  sufficient  length  of  time  for  these 
to  become  quiet,  and  hence  the  first  few  values  of  the  time  schedule 
have  been  computed  for  sampling  depths  of  20  cm.  Likewise  for  the 
finest  sizes  the  time  required  for  settling  10  cm.  is  quite  long :  to  elim- 
inate the  time  factor,  the  last  several  values  have  been  computed  for 
depths  of  5  cm. 

Preparation  of  samples  and  technique  of  analysis.  The  sediment  is 
dispersed  in  accordance  with  the  techniques  of  Chapter  3  (^page  yz), 
and  if  there  is  any  material  coarser  than  Y^q  mm.  present, 
it  is  removed  by  wet  sieving  as  described  on  page  142. 
The  sieve  residue  is  dried,  weighed,  and  sieved  into  grades. 
The  suspension  passing  the  sieve  is  poured  into  a  liter 
graduate  and  water  added  to  bring  the  vokmie  to  exactly 
1,000  c.c.  The  suspension  is  well  shaken  by  holding  the 
palm  of  one  hand  over  the  mouth  of  the  graduate  and 
inverting  the  vessel,  or  a  simple  stirring  device  may  be 
made.^  This  device,  illustrated  in  Figure  65,  consists  of  a 
narrow  brass  rod  about  16  in.  long,  at  the  base  of  which 
a  perforated  disc  is  fastened.  The  device  is  inserted  into 
the  graduate  and  moved  rapidly  up  and  down.  Agitation  is 
continued  until  the  material  collected  at  the  bottom  of  the 
vessel  has  been  distributed  through  the  suspension.  As 
soon  as  the  agitation  has  been  completed,  the  time  is  noted, 
or  a  stopwatch  is  started.  Exactly  i  min.  56  sec.  later  the 
pipette  is  inserted  to  a  depth  of  20  cm.,  and  a  20-c.c.  sam- 
ple withdrawn  with  a  uniform  suction.-  The  sample  is 
transferred  to  a  50-c.c.  beaker  and  set  on  a  hot-plate  to 
evaporate.  The  hot-plate  should  have  a  temperature  of 
about  100°   C,  to  prevent  boiling  or  spattering. 

After  the  first  pipette  sample  has  been  w^ithdrawn.  the 
suspension  is  again  agitated,  and  at  the  expiration  of  the 
next  time  interval  another  pipette  sample  is  withdrawn. 
Each  pipette  sample  is  taken  with  respect  to  the  new  level  of  the  sus- 
pension— no  water  should  be  added  to  the  suspension  during  the 
analysis. 

Computation  of  results.  After  the  several  pipette  samples  have  been 
taken,  and  the  beakers  evaporated  to  dryness,  the  weight  of  residue  in 
each  beaker  is  determined  with  an  anahtical  balance  to  3  or  4  decimal 
places.  For  each  beaker  the  weighing  notation  may  be  as  follows : 


1  This  device  was  called  to  the  authors'  attention  by  G.  Rittenhouse  of  the  United 
States  Soil  Conservation  Service,  Washington,  D.  C. 

-  If  there  is  no  material  coarser  than  1/16  mm.  in  the  suspension,  it  is  not  neces- 
sary to  take  a  sample  for  material  coarser  than  1/32  mm.  Some  analysts  prefer, 
however,  to  withdraw  a  sample  immediately  after  the  first  sliaking,  as  a  check  on 
the  total  amount  of  material  in  the  suspension. 


i68       SEDIMENTARY  PETROGRAPHY 

Weight  of  beaker  and  residue   17-938  g. 

Less  weight  of  beaker 17.406 

Weight  of  residue 0-53-2  g- 

The  following  example  of  the  first  several  separations  will  indicate 
the  computational  routine :  i  liter  of  suspension  contains  27.44  g.  of 
sediment  finer  than  Yiq  mm.  and  w^s  dispersed  with  X/ioo  sodium 
oxalate.  X/ioo  sodiimi  oxalate  is  equivalent  to  0.67  g.  sodium  oxalate 
per  liter  of  suspension,  or  0.013  g.  per  20-  c.c.  of  suspension.  This 
value  must  be  subtracted  from  the  weight  of  residue  in  each  beaker  to 
correct  for  the  dispersing  agent. 

The  weight  of  the  residue  in  beaker  #1,  representing  material  finer 
than  1^2  mm.,  is  0.532  g.,  and  that  in  beaker  =r2,  representing  material 
finer  thian  Yiq  mm.,  is  0.446  g.  Subtracting  0.013  g.  from  each  of 
these  yields  0.519  g.  and  0.433  S-  The  volume  of  the  pipette,  20  c.c,  is 
%o  the  volume  of  the  suspension,  so  that  each  of  the  weights  found 
are  to  be  multiplied  by  50,  to  convert  the  results  into  terms  of  the 
original  volume.  After  this  multiplication,  a  table  is  set  up  as  follows, 
showing  the  amount  of  material  in  the  successive  grades : 

Weight  of  material  finer  than  1/16  mm.  27.44  S- 

Weight  of  material  finer  than  1/32  mm 25.95 

Difi'erence  :  amount  in  1/16-1 732  mm.  grade 1.49  g. 

Weight  of  material  finer  than  1/32  mm 25.95  S- 

Weight  of  material  finer  than  1/64  mm 21.65 

4-30  g. 
Etc. 

These  weights  may  be  converted  into  percentages  of  the  total  sample 
weight  for  histograms  or  cumulative  cur\es.  If  material  coarser  than 
YiQ  mm.  was  present,  that  material  is  sieved  into  grades  and  the  com- 
bined results  expressed  as  the  size  distribution  of  the  sample. 

\'arious  time-saving  procedures  have  been  developed  for  the  pipette 
routine.  Rittenhouse^  found  by  experiment  that  for  the  finer  sizes  no 
serious  error  is  introduced  if  the  successive  pipette  samples  are  with- 
drawn without  shaking  the  suspension  between  pipettings.  The  time 
saved  by  this  procedure  is  considerable  for  such  small  sizes  as  1/512  and 
1/1024  mm.  Rittenhouse  has  also  found  that  a  battery  of  thirty-  or  fort\' 
analyses  may  be  conducted  simultaneously  by  setting  up  a  time  schedule 
which  allows  inter\-als  of  about  one  minute  between  the  sampling  times 
of  successive  suspensions.  In  this  manner  he  has  run  as  many  as  100 
analyses  in  two  or  three  days.  Rittenhouse  has  also  developed  a  short 
method  for  computing  the  percentages  in  each  grade.- 

1  G.  Rittenhouse,  A  suggested  modification  of  the  pipette  method :  Jour.  Sed. 
Petrology,  vol.  3,  pp.  44-45.  1933-  r  ^     ,         r 

-  G.  Rittenhouse,  a  laboratory  study  of  an  unusual  series  of  varved  clays  from 
northern  Ontario:  Am.  Jour.  Set.,  voL  28,  pp.  1 10-120,  1934. 


METHODS  OF  SIZE  ANALYSIS 


169 


o  o  o  o  o  000 
o  o  0000  00 
o       o     o    o  o  o  o  • 


o    o 
o   o 


DIAMETERS    IN    MM. 

Fig.  ()6. — Time  chart  for  pipette  method. 


I70 


SEDIMENTARY  PETROGRAPHY 


When  cumulative  curves  are  to  be  constructed  from  the  analytical 
data,  instead  of  histograms,  it  is  not  necessary  to  sample  the  susi>ension 
for  precise  grade  limits.  Instead,  the  number  of  determinations  to  be 
made  may  depend  ujxjn  the  detail  with  which  the  curve  is  to  be  drawn. 


9 


--  15 


--  25 


30 


20 


-ir   50 


•3        -- 


-.r   4-5 


■ir   4-0 


-h  35 


-1-3         -^   SO 
4- 
5 


10         -1 


--  20     :: 


-ir  30 

40 
50 


^t.  100       -■- 


-.r   25 


20 


-r    15 


■^v 


300  -r 
200 -i- 


fOO 


50 -ir 
>^   20  4- 


10  Tr 


3  ■# 
Z 

60 
40 

30+- 
ZO-.'r 

5-.: 

3-ir- 
2-11- 


Cma 


Q  Piam. 

Fig.  67. — Crowther's  nomogram  for  time  of  settling,  computed  for  particles  with  a 
specific  gravity  of  2.70.  Example :  Compute  the  time  for  a  particle  of  diameter  2 
microns  to  settle  10  cm.  at  a  temperature  of  20°  C.  A  straight  edge  is  laid  between 
points  20  and  2  of  scales  II  and  III,  and  the  line  extended  to  scale  IV,  which  it 
intersects  at  3.45.  This  last  point  is  connected  with  10  on  scale  I  and  the  line  ex- 
tended to  scale  V,  which  it  intersects  at  8  hours. 

A  time  chart  showing  diameters  against  settling  tiine  ^  may  be  prepared, 
as  shown  in  Figure  66.  By  means  of  this  chart  any  convenient  values 
may  be  used,  especially  along  the  steeper  parts  of  the  cumulative  curve, 
to  bring  the  slope  of  the  curve  out  in  greater  detail. 

1  W.  C.  Krumbein,  A  time  chart  for  mechanical  analyses  by  the  pipette  method : 
Joiir.  Sed.  Petrology,  vol.  5.  PP-  93-95,  1935- 


METHODS  OF  SIZE  ANALYSIS 


171 


Several  writers  ^  have  prepared  nomo<^rams  for  computing  velocities, 
radii,  or  times  of  settling  for  various-sized  i)articles,  according  to  Stokes' 
law.  Figure  67  is  a  reproduction  of  Crowther's  chart. 

One  of  the  variables  which  may  affect  the  accuracy  of  pii)ette  analyses, 
but  which  applies  equally  well  to  any  method,  is  the  temperature  at 


DISAGGREGATION 


DISPERSION 


PIPETTE 
ANALYSIS 


Fig.  68.— Flow-sheet  for  mechanical   analysis.   The  process   indicated  below  "dis- 
aggregation" refers  to  wet  sieving  through  0.061  mm.  sieve. 

which  the  analysis  is  conducted.  The  settling  velocities  given  above  as- 
sume a  temjDerature  of  20°  C,  and  the  discussion  in  Giaptcr  3  showed 
that  the  viscosity  of  water  varies  with  the  tem]XM-aturc.  Inasmuch  as  the 
settling  velocity  depends  in  i:)art  upon  the  viscosity,  it  is  clear  that  tem- 
l^erature  fluctuations  should  be  avoided  during  analysis.  Various  meth- 
ods are  available   for  maintaining  uniform  temi>eratures.   Andreason^ 

1  E.  M.  Crowther,  Nomographs  for  use  in  mechanical  analysis  calculations :  Proc. 
isf  Inf.  Cougr.  Soil  Sri..  Part  II  (1927),  pp.  300-404,  1028.  H.  Rouse.  Nomogram 
for  the  settling  velocity  of  spheres :  7?r/».  Com.  Scd.,  1936-37,  Nat.  Research  Coun- 
cil, 1937,  pp.  57-64. 

2  A.  H.  M.  Andreason,  Joe.  cit.,  1928. 


172 


SEDIMENTARY  PETROGRAPHY 


used  an  insulated  cell  for  his  analyses,  but  a  simple  and  effective  device 
is  to  have  a  water-tight  box  or  compartment  deep  enough  to  submerge 
liter  graduates  nearly  to  the  top.  The  entire  suspension  is  thus  sur- 
rounded by  water,  and  even  though  the  room  temperature  may  fluctuate 

several    degrees,    the    effect    on    the    water-bath    will    be 

negligible. 

The  steps  involved  in  routine  mechanical  analyses  by 
the  pipette  method  are  shown  in  the  accompanying  flow- 
sheet (Figure  68).  The  sheet  shows  all  the  procedures 
from  the  splitting  operation  to  the  final  analysis.  Processes 
are  shown  in  rectangles  in  the  flow-sheet,  and  materials 
are  shown  in  circles. 

The  hydrometer  method.  The  hydrometer  method  of 
mechanical  analysis  was  introduced  by  Buoyoucos  ^  in 
1927.  A  hydrometer,  calibrated  to  read  grams  of  soils  per 
liter,  is  introduced  into  the  suspension  at  intervals,  and 
readings  are  taken.  From  the  data  obtained  a  cumulative 
curve  may  be  drawn  directly.  Theoretically  the  hydrometer 
measures  the  density  of  the  suspension  at  a  given  depth 
as  a  function  of  time,  and  consequently  is  based  on  equa- 
tion (29)  (Chapter  5)  of  Oden's  theory.  As  in  most  other 
devices,  however,  the  equation  itself  is  not  used  in  practice. 

Bouyoucos's  hydrometer  is  shown  in  Figure  69.  It  con- 
sists of  a  cylindro-conical  base,  weighted  with  lead,  and 
_  a  narrow  stem  with  a  scale  calibrated  directly  in  grams. - 
""  The  rapidity  of  the  method,  compared  with  most  other 
^'  techniques,  has  led  to  an  extensive  study  of  it  in  terms 
of  its  accuracy  and  theoretical  soundness,  as  well  as  in 
the  most  effective  shape  of  the  hydrometer  bulb. 


AK 


0 


F  I  G.  6q 
B  o  u  y  o 
cos's  h 
drometer. 

terms  of 


Numerous  comments  have  been  made  for  and  against  the  hydrometer  as 
an  accurate  device.  Keen,^  Joseph,*  Gessner,"  and  Olmstead,  Alexander,  and 
Lakin®  have  criticized  the  method  from  the  point  of  view  of  accuracy  and 

1  G.  J.  Bouyoucos,  The  hydrometer  as  a  new  method  for  the  mechanical  analysis 
of  soils :  Soil  Science,  vol.  23,  pp.  343-353,  1927- 

2  The  instrument  and  glass  cylinder  are  obtainable  from  the  Taylor  Instrument 
Company,  Rochester,  N.  Y. 

3  B.  A.  Keen,  Some  comments  on  the  hydrometer  method  for  studying  soils:  Soil 
Science,  vol.  26,  pp.  261-263,  1928. 

4  A.  F.  Joseph,  The  determination  of  soil  colloids:  Soil  Science,  vol.  24,  pp. 
271-274,  1927. 

5  H.  Gessner,  op.  cit.  (1931),  p.  II4- 

«  L.  B.  Olmstead,  L.  T.  Alexander  and  H.  W.  Lakin,  The  determination  of  clay 
and  colloid  in  soils  by  means  of  a  specific  gravity  balance:  Rep.  Am.  Soil  Survey 
Assn.,  Bull.  12,  pp.  161-166,  1931. 


METHODS  OF  SIZE  ANALYSIS  173 

theoretical  soundness.  Bouyoucos  replied  to  his  critics, ^  undertook  a  series 
of  experiments  -  to  demonstrate  that  tlie  hydrometer  method  agrees  well  with 
other  standard  methods  of  analysis,  and  argued  that  the  metliod  conformed 
to  the  principles  of  Stokes'  law.^ 

More  recently  Bouyoucos  developed  a  more  sensitive  hydrometer  for 
soils.*  The  instrument  has  a  range  from  o-io  g.  per  liter.  It  has  a  large 
stream-lined  bulb  and  a  short  stem.  Numerous  other  workers  have  de- 
veloped special  types  of  hydrometers  in  the  decade  since  1927.  Puri.^ 
for  example,  introduced  a  hydrometer  with  a  short  bulb  and  a  long  stem. 
To  increase  the  accuracy  of  the  readings,  a  pin  was  mounted  on  the  top 
of  the  stem,  and  its  level  read  with  reference  to  a  burette  scale  mounted 
above  the  cylinder. 

One  of  the  most  tliorough  studies  of  the  hydrometer  method  was  made  by 
Casagrande,^  who  recognized  that  a  method  affording  the  basic  simplicity  and 
convenience  of  hydrometer  readings  as  compared  to  other  methods  justified 
an  attempt  to  place  it  upon  a  firm  foundation.  Casagrande  developed  the 
theory  of  the  hydrometer  method  in  considerable  detail,  including  the  influence 
of  hydrometer  shape  on  the  results.  The  several  sources  of  error  of  the  tech- 
nique were  evaluated,  including  such  items  as  temperature  corrections,  effects 
of  concentration  of  the  suspension,  accuracy  of  the  hydrometer  readings,  and 
the  like.  As  a  result  of  his  investigations  Casagrande  developed  a  hydrometer 
having  the  form  shown  in  Figure  70.  For  routine  purposes  a  glass  instru- 
ment was  used ;  for  precise  studies  he  used  a  hydrometer  having  a  rust-proof 
metal  stem  fitted  at  the  top  with  a  thin  horizontal  disk.  The  level  of  the 
hydrometer  was  read  by  referring  the  edge  of  the  disk  to  a  scale  mounted 
on  an  adjoining  stand.  Casagrande's  work  showed  that  the  hydrometer 
method,  when  used  with  proper  precautions,  yields  results  comparable  to 
those  obtained  with  other  precision  methods.  For  such  exact  work  the 
time  element  is,  however,  of  the  same  order  of  magnitude  as  with  other 
techniques. 

Another  detailed  study,  largely  in  terms  of  Bouyoucos's  hydrometer,  was 

1  G.  J.  Bouyoucos,  The  hydrometer  method  for  studying  soils  :  Soil  Science,  vol. 
25,  pp.  3^5-3^9,  1928. 

-  G.  J.  Bouyoucus,  The  hydrometer  method  for  making  a  very  detailed  mechanical 
analysis  of  soils  :  Soil  Scioice,  vol.  26,  pp.  233,  238,  1928.  G.  J.  Bouyoucos,  A  com- 
parison between  pipette  and  hydrometer  methods  for  making  mechanical  analyses  of 
soil :  Soil  Science,  vol.  38,  pp.  335  ff.,  1934.  G.  J.  Bouyoucos,  Further  studies  on 
the  hydrometer  method  for  making  mechanical  analyses  of  soils  and  its  present 
status:  Rep.  Am.  Soil  Survey  Assn.,  Bull.  13,  pp.  126-131,  1932. 

3  G.  J.  Bouyoucos,  Making  mechanical  analyses  of  soils  in  fifteen  minutes :  Soil 
Science,  vol.  25,  pp.  473-480,  1928. 

*  G.  J.  Bouyoucos,  A  sensitive  hydrometer  for  determining  small  amounts  of  clay 
or  colloids  in  soils :  Soil  Science,  vol.  44,  pp.  245-246,  1937. 

5  A.  N.  Puri,  A  new  type  of  hydrometer  for  the  mechanical  analysis  of  soils : 
Soil  Science,  vol.  33,  pp.  241-248,  1932. 

^  A.  Casagrande,  Die  Ariiomcter-Methode  zur  Bestiminung  der  Kornverteilung 
von  Boden  (Berlin:  Verlag  von  Julius  Springer,  1934). 


174 


SEDIMENTARY  PETROGRAPHY 


made  by  Wintermyer,  Willis,  and  Thoreen  ^  of  tlie  United  States  Bureau  of 
Public  Roads.  A  tecbnique  was  developed  in  which  readings  were  made  with 
Bouyoucos's  hydrometer  at  intervals  up  to  1,440  min.  (24  hr.).  Settling 
velocities  were  computed  according  to  Stokes'  law,  and  the  percentage  of  soil 
remaining  in  suspension,  P,  was  determined  by  the  equation  P  =  100  (R/W), 
where  R  is  the  hydrometer  reading  and  W  is  the  weight  of 
material  originally  dispersed  per  liter  of  suspension.  Various 
correction  coefficients  were  involved  in  the  final  evaluation  of 
the  results.  Subsequently  Thoreen  ^  showed  that  an  ordinary 
specific  gravity  hydrometer  could  be  used  in  place  of  Bou- 
youcos's special  instrument.  Graphic  methods  of  evaluating  the 
correction  coefficients  were  developed  by  Willis,  Robeson,  and 
Johnston,^  also  of  the  United  States  Bureau  of  Public 
Roads. 

The  widespread  interest  in  the  hydrometer  method 
reflects  the  need  for  a  simple,  rapid,  and  yet  accurate 
method  of  mechanical  analysis.  The  authors  have  not  had 
enough  experience  with  the  hydrometer  method  to  evaluate 
it  in  terms  of  general  sedimentary  analysis,  but  it  appears 
to  be  useful  for  the  preliminary  study  of  large  groups  of 
samples.  Although  minor  irregularities  of  the  cumulative 
curve  may  not  be  brought  out,  significant  differences  be- 
tween samples  may  be  evaluated ;  if  more  complete 
analyses  are  desired,  the  pipette  method  may  be  used. 

Bouyoucos  furnishes  a  simplified  instruction  sheet  with 
his  instruments.*  Essentially  the  technique  involves  pre- 
liminary dispersion,  followed  by  hydrometer  readings  at  40 
sec,  I  hr.,  and  2  hr.  These  three  readings  furnish  the  data 
for  the  following  four  classes  of  material :  sand  (coarser  than  0.05  mm. 
diameter),  silt  (0.05  to  0.005  nim.),  clay  (0.005-0.002  mm.),  and  "fine 
clay"  (finer  than  0.002  mm.).  Although  these  limits  do  not  agree  with 
the  Wentworth  grade  limits,  they  nevertheless  furnish  several  points 
along  an  approximate  cumulative  curve. 

The  plummet  method.  A  method  closely  related  to  the  hydrometer 
method  involves  the  use  of  a  plummet  suspended  within  the  suspension, 
but  near  the  surface.  The  change  in  its  apparent  weight  is  observed  as  a 


1  A.  M.  Wintermyer,  E.  A.  Willis  and  R.  C.  Thoreen,  Procedures  for  testing 
soils  for  the  determination  of  the  subgrade  soil  constants :  Public  Roads,  vol.  12, 
no.  8,  1931. 

2  R.  C.  Thoreen,  Comments  on  the  hydrometer  method  of  mechanical  analysis : 
Mimeographed  report,  U.  S.  Bureau  Public  Roads,  1932. 

3  E.  A.  Willis,  F.  A.  Robeson  and  C.  M.  Johnston,  Graphical  solution  of  the  data 
furnished  by  the  hydrometer  method  of  analysis:  Public  Roads,  vol.  12,  no.  8,  1931. 

■^  Supplied  by  the  Taylor  Instrument  Company,  Rochester,  N.  Y. 


METHODS  OF  SIZE  ANALYSIS 


175 


function  of  the  time.  The  data  so  obtained  yield  the  change  of  density 
at  a  constant  depth  as  a  function  of  the  time,  and  hence  the  method  may 
be  related  to  equation  (29)  of  Chapter  5.  Oden,  however,  developed  a 
special  equation  for  the  plummet  method  (equation  (32)  of  Chapter  5). 
Schurecht  ^  used  the  method  in  1921  ;  he  suspended  a 
small  glass  tube,  partially  filled  with  mercury,  in  the  sus-  "bll'm* 

pension.  The  plummet  is  attached  to  an  analytical  balance, 
and  weighings  are  made  at  intervals  ranging  from  a  few 
minutes  to  a  number  of  days.  The  general  principle  of  the 
apparatus  is  shown  in  Figure  71.  Ries  -  describes  the 
method  in  some  detail  and  furnishes  an  outline  of  the  com- 
putations to  be  made. 

Other  workers  who  used  or  investigated  the  plummet 
method  include  Van  Niewenberg  and  Schoutens,^  von 
Hahn,*  and  Olmstead,  Alexander,  and  Lakin.^  Von  Hahn 
used  a  ]\Iohr  specific  gravity  balance  (see  Figure  150, 
Chapter  14.  for  illustration)  and  concluded  that  the 
method  could  not  be  recommended  for  general  application. 
Olmstead  and  his  associates  used  a  small  pear-shaped 
plummet  suspended  from  a  chainomatic  balance.  Their 
study  was  made  primarily  to  find  a  rapid  method,  having  a 
convenience  equal  to  the  hydrometer,  but  with  the  accuracy 
of  the  pipette  method.  The  results  of  the  study  showed 
sources  of  error  which  prevented  the  method's  recom- 
mendation. The  paper  contains  an  excellent  discussion  of 
the  problem  involved. 

Photocell  iiicfliod.  In  1934  Richardson  ^  applied  a  photoelectric  cell  to 
the  problem  of  determining  the  size  distribution  of  soils  and  clays.  The 
method  consists  essentially  of  directing  a  beam  of  light  through  a  sedi- 
menting  system  and  against  a  photocell.  The  photocell  is  connected  to  a 
galvanometer,  which  indicates  the  intensity  of  the  beam  in  terms  of 
current.  To  obtain  a  continuous  record  of  the  change  in  light  intensity, 
Richardson  used  a  string  galvanometer,  the  motion  of  the  string  being 


Fig.  71  — 
Principle  o  f 
the  plummet 
method, 
after  Schu- 
recht. 


1  H.  C.  Schurecht,  Sedimentation  as  a  means  of  classifying  the  extremely  fine 
clay  particles:  Jour.  Am.  Ccnim.  Soc,  vol.  4,  pp.  812-821.  1921. 

-  H.  Ries,  Clays,  Their  Occurrence  Properties,  and  Uses  (New  York,  1927), 
pp.  204  flF. 

3  C.  J.  Van  Niewenberg  and  Wa.  Schoutens,  A  new  apparatus  for  a  rapid  sedi- 
mentation analysis:  Jour.  Ant.  Cerant.  Soc.,  vol.  11,  pp.  696-705,  1928. 

■*  H.-V.  von  Hahn.  o/>.  cit.,  pp.  296  fF. 

■'  L.  B.  Olmstead,  L.  T.  Alexander  and  H.  W.  Lakin,  he.  cit..  igp. 

•">  E.  G.  Richardson,  An  optical  method  for  mechanical  analysis  of  soils,  etc., 
Jour.  Agric.  Sci.,  vol.  24,  pp.  457-46S,  1934. 


1-6        SEDIMENTARY  PETROGRAPHY 

recorded  on  a  strip  of  bromide  paper  fastened  to  a  revoh-ing  drum.  The 
curve  obtained  decreased  rapidly  at  first  and  more  slowly  later  on. 

Richardson  showed  that  the  light  extinction  as  measured  by  the  cell  is 
proportional  to  ShJ-,  where  n  is  the  number  of  particles  of  diameter  d  and 
2  is  a  summation  sign.  Experiments  showed  that  this  relation  holds  down  to 
diameters  of  12  microns  (0.012  mm.)  at  least  In  theor}-,  the  settling  velocity 
of  a  particle  is,  by  Stokes'  law,  V  =  Cr-.  The  proportionality  found  above 
in  terms  of  light  intensity  is  I  =  k^nd-.  For  a  given  depth  Vi,  the  light  cut 
oflF  at  time  t^  will  be  due  to  all  particles  having  velocities  less  than  y\/t^,  or 
d-  less  than  yi/C/i.  That  is, 

y-/Ct 
I  =  k  Y»<i-. 
d-  =  o 
By  constructing  a  cur^e  of  I  against  t,  the  slope  at  any  point  /  will  be  pro- 
portional to  the  number  n  of  particles  for  which  d-  =  y\/Ct^.  Thus  plotting 
the  slope  against  y  (or  i//)  yields  the  corresponding  frequency  curve  as  n 
against  d-.  In  similar  manner  Richardson  showed  that  a  second  approach  was 
possible,  involving  the  simultaneous  measurement  of  light  extinction  over  the 
entire  cell  at  a  fixed  instant  Of  the  two  methods,  the  simpler  involves  meas- 
urements at  a  given  depth  as  a  fimction  of  the  time. 

Richardson  subsequently  -  improved  the  apparatus  used  and  developed 
an  ingenious  spiral  drum  which  records  the  log  of  time,  so  that  the  record 
)-ields  the  light  intensit}'  as  a  function  of  log  t  directly. 

Other  indirect  methods.  In  addition  to  the  techniques  described  in  the 
foregoing  section,  many  other  methods  of  analysis  have  Ijeen  described 
in  the  hterature.  Among  these  methods  are  several  based  on  the  absorp- 
tion of  X-rays,  the  use  of  ultracentriftiges,  the  turbidit}-  of  suspensions 
(T}-ndall  ettect),  and  the  like.  \'on  Hahn^  discusses  these  and  other 
techniques  in  some  detail. 

Mechanical  Analysis  under  the  Microscope 

Xiunerous  techniques  have  been  used  for  the  microscopic  measurement 
of  particles.  Perhaps  the  most  extensively  used  method  with  loose  grains 
has  been  direct  measurement  with  a  micrometer  eyepiece.  In  their 
simplest  form  such  eyepieces  are  merely  a  scale  engraved  on  glass  at 
the  focal  plane  of  the  eyepiece,  so  that  object  and  scale  are  simultaneously 
visible.  Figure  y2  illustrates  the  scale;  other  tj-pes  of  micrometer  eje- 

1  E.  G.  Richardson,  A  photo-electric  apparatus  for  delineating  the  size  frequency 
curve  of  clays  or  dusts:  Jour.  Scientific  Instruments,  vol.  13,  pp.  229-233.  1936. 
The  instrument  in  its  improved  form  is  offered  for  sale  by  A.  Gallencamp  and 
Company,  Finsburj'  Square,  London  E.  C.  2. 

2  F.-V.  von  Hahn,  op.  cit.,  1927. 


METHODS  OF  SIZE  ANALYSIS 


177 


pieces  are  described  by  Johannsen/  In  order  to  use  a  micrometer  eye- 
piece it  is  necessary  to  calibrate  it  for  the  microscopic  combination  being 
used.  This  is  accompHshed  by  placing  an  accurately  ruled  scale  on  the 
microscope  stage  and  focusing  the 
micrometer  eyepiece  on  it  so  that  a 
line  of  the  eyepiece  scale  coincides 
with  a  line  on  the  stage  micrometer. 
The  number  of  divisions  of  the  eye- 
piece scale  which  correspond  to  a 
given  number  of  divisions  of  the 
other  furnishes  the  data  for  calibra- 
tion. If,  for  example,  fifty  divisions 
of  the  eyepiece  scale  correspond  to 
eleven  divisions  of  the  stage  mi- 
crometer, the  relation  is  50X  =  i.i 
mm.,  or  X  =  0.022  mm.  Thus,  one 
eyepiece  division  equals  22  microns. 
To  measure  individual  grains 
with  the  micrometer  eyepiece,  the  grain  is  l)rought  to  proper  orientation 
along  the  scale  (a  mechanical  stage  is  highly  desirable  for  this),  and  the 
chosen  lengths  are  measured.  If  areas  are  to  be  measured,  a  grid  microm- 
eter may  be  used,  which  has  squares  ruled  on  it.  The  area  of  the  grain 
may  be  estimated  by  referring  it  to  the  size  of  the  smallest  enclosing 
set  of  scale  lines.  For  the  measurement  of  a  number  of  particles  a  tally 
sheet  is  convenient.  The  range  of  sizes  present  (as  represented,  say, 
by  long  or  intermediate  diameters)  is  roughly  determined,  and  a  series 
of  size  classes  is  set  up  covering  this  range.  The  number  of  grains  in 
each  class  interval  is  then  indicated  by  tally  marks. 


Fig. 


-Alicromoter  eyepiece. 


Measurement  of  loose  grains.  In  conducting  mechanical  analyses  by 
means  of  a  microscope,  it  is  important  that  the  sample  used  be  repre- 
sentative of  the  material  being  studied.  This  is  true  in  all  methods  of 
analysis,  of  course,  but  inasmuch  as  relatively  small  samples  are  usually 
mounted  on  the  slide  for  microscopic  measurements,  particular  care 
should  be  used  in  splitting  the  sample.  The  field  sample  may  be  split  to 
a  smaller  sample  of  20  or  25  g.  by  means  of  a  Jones  sample  splitter 
(page  45),  and  this  smaller  sample  may  be  quartered  down  by  hand  or 
preferably  with  a  microsplit  (page  358),  to  obtain  a  sample  small 
enough  to  mount  on  the  slide.  No  fixed  rules  can  be  given  for  the  size  of 
the  final  sample,  inasmuch  as  individual  practice  varies  from  counting 


1  A.   Tnhannscn,  Maintal  of  Pctro(iraphic  Methods,  2nd  ed.    (New  York,   1918), 
pp.  287  ff. 


178       SEDIMENTARY  PETROGRAPHY 

a  few  hundred  grains  to  counting  several  thousand.  One  "rule  of  thumb" 
that  may  be  used  is  to  count  about  300  grains  and  convert  the  numbers 
to  percentages  by  classes;  this  is  follovi'ed  by  a  count  of  an  additional 
hundred  or  so  grains,  and  percentages  are  recalculated  on  the  entire  num- 
ber of  grains  counted.  If  the  percentages  remain  fairly  fixed,  the  sample 
is  probably  adequate;  if  large  differences  occur,  an  additional  number  is 
counted  until  only  minor  fluctuations  remain. 

If  the  grains  are  to  be  mounted  in  the  dry  state,  the  final  sample  may 
be  transferred  to  a  clean  microscope  slide,  and  the  edges  of  the  slide 
tapped  with  a  pencil,  so  that  the  grains  assume  their  most  stable  position 
of  rest.^  This  is  especially  important  for  measurements  of  the  nominal 
sectional  diameter  (page  296),  which  are  defined  in  terms  of  the  area 
of  the  grain  section  in  the  plane  of  the  long  and  intermediate  diameters. 

The  actual  operation  of  measuring  and  tallying  the  grains  may  be 
based  on  a  count  of  all  the  grains  in  the  sample ;  or  of  all  the  grains  in 
several  random  fields  over  the  sample ;  or  along  certain  horizontal  or 
vertical  lines  through  the  sample  as  laid  off  by  traverses  with  the 
mechanical  stage. 

The  reader  is  referred  to  Table  33,  and  Figure  143  of  Chapter  11,  for 
an  example  of  a  microscopic  size  and  shape  analysis. 

The  measurement  of  numerous  grains  under  the  microscope  is  usually 
a  tedious  process,  and  various  other  techniques  have  been  developed.  A 
field  of  grains,  with  eyepiece  micrometer  in  place,  may  be  photographed, 
and  the  individual  grains  measured  from  the  print  or  an  enlargement. 
Direct  drawing  of  the  field  may  be  made  with  a  camera  lucida,  or,  much 
more  conveniently,  with  a  microscopic  projection  device,^  which  projects 
an  enlarged  image  of  the  field.  Figure  'j'^  illustrates  such  an  instrument. 
By  focusing  the  image  on  drawing  pa^^er  at  some  convenient  magnifi- 
cation, and  drawing  outlines  of  the  grains,  an  entire  field  may  be  cov- 
ered in  a  short  time.  The  resulting  images  may  be  measured  directly 
with  a  centigrade  scale  and  translated  into  correct  dimensions  in  terms 
of  the  magnification  used.  For  measurements  of  the  nominal  sectional 
diameter  a  planimetet:  may  be  used  to  determine  the  area  of  each  grain. 

Thin-section  analysis.  The  methods  of  measurement  described  here 
apply  equally  well  to  loose  grains  or  thin  sections.  In  the  former  case 
the  measurements  are  used  directly  in  tabulating  the  frequency  distribu- 
tion; if  thin  sections  are  used,  the  observed  radii  must  be  corrected  in 
accordance  with  the  theory  of  thin-section  analysis  described  in  Chapter 
5.  The  procedure  to  be  followed  involves  setting  up  the  frequency  table 

1  H.  Wadell,  Volume,  shape,  and  roundness  of  quartz  particles :  Jour.  Geology, 
vol.  43,  pp.  250-280,  1935. 

-A  very  satisfactory  device  for  this  purpose  is  the  "Promar  Microscopic  drawing 
and  projection  apparatus"  offered  by  the  Clay-Adams  Co.,  New  York,  N.  Y.  A  less 
expensive  device,  called  the  "Microprojector,"  is  offered  by  Bausch  and  Lomb, 
Rochester,  N.  Y. 


METHODS  OF  SIZE  ANALYSIS 


179 


of  observed  radii  (or  diameters)  and  computing  the  moments  ^  of  the 
obser\-ed  distribution.  The  observed  moments  are  then  corrected,  which 
yields  the  characteristics  of  the  grain  distribution  but  does  not  yield  the 
entire  distribution. 


a 


>=%. 


Fig.  y^i- — Bausch  and  Lomb's  "Microprojector." 

Table  17  shows  the  data  of  a  thin-section  mechanical  analysis  of  Palms 
^line  quartzite,  including  the  computation  of  the  first  two  moments. 


Table  17 

Distribution  of  Intercept  Diameters  in  Thin  Section  of 
Quartzite  from  Palms  Mine,   Bessemer,   Michigan  * 


Classes 
{nun.) 

Xitmber 

Frequency 

(/) 

m 

/;;; 

m- 

fm- 

0  08—0  16 

16 
87 
155 
150 
65 
3^ 
8 

4 

I 

0.12 
.20 
.28 
.36 

•44 
■52 
.60 
.68 
0.76 

1.92 

17.40 

4340 

;4.oo 

28.60 

16.64 

4.80 

2.72 

0.76 

0.014 
.040 
.078 
.130 
.192 
.271 
.360 
.463 

0.579 

0.224 

3.480 

12.100 

19.500 

12.500 

8.670 

2.880 

1.850 

0.579 

.16-  .24    

•24-  .32    

•32-  .40    

•40-  .48    

.48-  .56    

.5^  .64    

.64-  .72 

0.72-0.80    

Total 

518 

170.24 

61  rS^^ 

*  Computations  by  slide  rule. 

1  The  moments  of  the  frequency  distribution  are  described  in  Chapter  9.  In  the 
present  tjpe  of  analysis,  arithmetic  rather  than  logarithmic  moments  should  be 
used,  inasmuch  as  the  theory  of  thin-section  analysis  is  based  on  the  arithmetic  mo- 
ments directly. 


i8o       SEDIMENTARY  PETROGRAPHY 

In  determining  the  moments  of  tlie  distribution,  the  mid-point  of  each  class 
is  entered  in  the  column  headed  iii.  The  grain  frequency  /  in  each  class  is 
multiplied  by  m,  and  the  products  entered  in  the  fiu  column.  This  column  is 
totaled  and  divided  by  the  total  frequency,  here  518.  The  resulting  quotient 
is  the  observed  first  moment.  11^^  =  0.329.  For  the  second  moment  the  mid- 
point of  each  class  is  squared  and  entered  in  the  m-  column.  The  grain 
frequencies  are  now  multiplied  by  these  squared  values,  as  shown  in  the  fm- 
column.  The  total  is  divided  by  518  to  yield  the  observed  second  moment, 
"x2  ~  0.1 19.  Higher  moments  may  be  computed  by  following  a  similar  process 
with  successively  higher  powers  of  m. 

Details  of  computation  are  summarized  below  the  table ;  a  fuller  dis- 
cussion of  arithmetic  moments  is  given  in  Chapter  9.  The  diameters  used 
in  the  study  were  defined  as  the  maximum  horizontal  intercepts  through 
the  variously  oriented  grains.^ 

The  characteristics  of  the  grain  distribution  of  the  quartzite  may  be 
found  by  means  of  equations  (45J  and  (46)  of  Chapter  5  (page  132). 
The  observ^ed  first  moment  of  the  grain  sections,  Jin,  is  0.329  mm.  From 
equation  (45),  «ri  =  i-27"n  =0.418  mm.,  the  arithmetic  mean  size 
of  the  quartzite  grain  distribution.  The  observed  second  moment,  «i2, 
is  0.119.  From  equation  (46),  «r2=i-50  nx2  =  o.i79.  If  the  standard 
deviation  (page  219)  of  the  grain  distribution  is  to  be  used  as  a  measure 
of  spread,  it  may  be  obtained  from  the  relation  0-  =  «2  —  ("i)'-  In  the 
present  case  this-  is  a  =  \/o.4i8>—  (0.179)- =  0.071  "i"^- 


COMPARISONS     OF      METHODS     OF      MECHANICAL 
ANALYSIS 

The  wide  choice  of  methods  available  for  mechanical  analysis,  espe- 
cially for  fine-grained  sediments,  has  resulted  in  numerous  workers' 
comparing  the  relative  accuracy  and  convenience  of  two  or  more  tech- 
niques.   Earlier   papers  -   compared   various   decantation   methods   with 

1  W.  C.  Knimbein,  Thin-section  mechanical  analysis  of  indurated  sediments :  Jour. 
Geology,  vol.  43,  pp.  482-496,  1935. 

2  Among  the  large  number  of  such  papers  may  be  mentioned  the  following : 
G.  M.  Darby,  Determination  of  grit  in  clays :  Chcm.  and  Met.  Engineermg,  vol.  32, 
pp.  688-690,  1925.  A.  F.  Joseph  and  F.  J.  Martin,  The  determination  of  clay  in 
heavy  soil:  Jour.  Agric.  Sci.,  vol.  II,  pp.  293-303,  1921.  W.  Novak,  Zur  Methodik 
des  mechanischen  Bodenanalyse :  Int.  Slitt.  fiir  Bodenktinde,  vol.  6.  pp.  110-141, 
1916.  C.  W.  Parmelee  and  H.  W.  Moore,  Some  notes  on  the  mechanical  analysis  of 
clays:  Trans.  Am.  Ccram.  Soc,  vol.  II,  pp.  467-493,  1909.  X.  Pellegrini,  Ueber  die 
physikalisch-chemische  Bodenanalyse :  Landwirts.  Versuchs-Stat.,  vol.  25,  pp.  48-52. 
1880.  H.  Puchner,  Fin  \'ersuch  zum  Vergleich  der  Resultate  verschiedener  me- 
chanischer  Bodenanalyse:  Landwirts.  Versuchs-Stat.,  voL  56,  pp.  141-148,  1902. 


METHODS  OF  SIZE  ANALYSIS  i8i 

rising  current  elutriation.  More  recent  studies  ^  compared  the  newer 
methods  based  on  Oden's  theory  among  themselves  or  compared  them 
with  the"  older  routine  methods.  Many  of  the  studies  compared  individual 
grade  sizes,  others  compared  the  cumulative  curves  obtained  by  the 
several  methods,  and  some  included  the  effects  of  dispersing  agents  on 
the  results.  It  is  difficult  in  all  cases  to  decide  upon  the  most  favorable 
method,  owing  in  part  to  conflicting  results.  In  addition  to  differences 
in  methods  of  dispersion,  the  personal  element  enters  the  study  to  some 
extent,  inasmuch  as  familiarity  with  a  method  often  results  in  a  degree 
of  success  not  obtained  with  limited  experience.  The  most  complete  com- 
parisons were  made  by  committees  in  connection  with  soil  analysis,-  who 
decided  on  the  pipette  method  as  the  most  suitable  for  general  work. 

1  Among  such  papers  may  be  mentioned  the  following :  A.  H.  M.  Andreason,  loc. 
cit.,  1928.  G.  J.  Bouyoucos,  A  comparison  of  the  hydrometer  method  and  the  pipette 
method  for  making  mechanical  analysis  of  soil,  with  new  directions:  Jour.  Am.  Soc. 
Agron.,  vol.  2^,  pp.  747-/51.  1930.  A.  Cannes  and  H.  D.  Sexton,  A  comparison  of 
methods  of  mechanical  analysis  of  soils:  Agric.  Engineering,  vol.  13,  pp.  15  ff.,  1933. 
C.  W.  Correns  and  W.  Schott,  Vergleichende  Untersuchungen  iiber  Schlamm- 
und  Aufbereitungsverfahren  von  Tonen :  Kolloid  Zeits.,  vol.  61,  pp.  68-80,  1932.  M. 
Kohn,  loc.  cit.,  1928.  A.  Kuhn,  Die  Methoden  zur  Bestimmung  der  Teilchengrosse  : 
Kolloid  Zeits.,  vol.  37,  pp.  365-377,  1925.  L.  B.  Olmstead,  L.  T.  Alexander  and 
H.  W.  Lakin,  loc.  cit.,  1931.  O.  Pratje,  Die  Sedimente  des  Siidatlantischen  Ozeans: 
IViss.  Ergcb.  der  Deutsch.  Atlantischcn  Expedition  auf  dem  .  . .  "Meteor"  vol.  3, 
part  2,  Lief,  i,  1935. 

-  See  for  example,  Subcommittee  of  A.  E.  A.,  The  mechanical  analysis  of  soils  ; 
a  report  on  the  present  position,  and  recommendations  for  a  new  official  method: 
Jottr.  Agric.  Sci.,  vol.  16,  pp.  123-144,  1926. 


CHAPTER   7 
GRAPHIC  PRESENTATION  OF  ANALYTICAL  DATA 

INTRODUCTION 

Graphic  presentation  is  one  of  the  first  steps  in  an  analysis  of  the 
results  of  any  sedimentary  study.  Not  only  does  a  graph  present  the 
results  visually,  but  it  serves  an  additional  purpose  in  suggesting  new 
lines  of  attack. 

There  are  certain  principles  of  graphic  presentation,  which  dej^end 
upon  conventions  of  analytic  geometry,  and  which  should  be  followed 
in  order  to  introduce  uniformity  into  methods  of  presentation.  At  the 
risk  of  discussing  familiar  material,  some  of  these  elementary  principles 
will  be  presented  here. 

GENERAL     PRINCIPLES     OF     GRAPHS 

Most  graphs  involve  the  plotting  of  one  set  of  data  against  another, 
usually  by  drawing  points  on  a  plane  for  each  pair  of  observations.  It 
is  conventional  to  choose  two  coordinate  axes  at  right  angles  to  each 
other,  which  are  used  as  axes  of  reference.  The  vertical  axis  is  called 
the  y-axis,  and  the  horizontal  axis  is  called  the  .y-axis. 

Choice  of  dependent  and  independent  variables.  It  is  a  well  established 
mathematical  convention  thatJheJl'^i'/^'^g^^^  variable  shall  be  plotted 
a]ong  th^.y-axis,  and  the  dependent  variable  along  the  .-yraxis.  The  in- 
dependent variable  increases  or  decreases  by  arbitrarily  chosen  amounts, 
and  the  dependent  variable  is  measured  at  each  of  these  given  values. 
For  example,  if  the  change  in  heavy  mineral  content  is  studied  in  a 
linear  series  of  samples,  the  distance  between  the  samples  is  arbitrarily 
chosen,  and  the  mineral  content  of  these  samples  is  then  investigated. 
This  procedure  defines  distance  as  the  independent  variable  and  per- 
centage of  heavies  as  the  dependent  variable.  Similarly,  when  frequency 
is  plotted  against  diameter  of  grain,  diameters  are  arbitrarily  fixed  (by 
a  choice  of  sieve  meshes,  for  example),  and  the  frequency  of  grain  on 
each  sieve  is  then  determined.  Here  diameter  is  the  independent  variable, 
and  frequency  the  dependent  variable. 

182 


GRAPHIC  PRESENTATION  183 

In  some  instances  it  is  not  obvious  which  is  the  independent  variable, 
and  in  such  circumstances  an  arbitrary  choice  is  made,  depending  upon 
the  emphasis  which  is  to  be  given  to  the  results.  One  may  compare  the 
organic  content  of  a  series  of  samples  with  their  skewness.  These  two 
apparently  unrelated  sets  of  data  may  justifiably  be  examined  either  in 
terms  of  how  the  organic  content  varies  as  the  skewness  changes,  or 
how  the  skewness  varies  as  the  organic  content  changes.  In  the  first  case 
skewness  is  the  independent  variable,  and  in  the  second  organic  content 
is  the  independent  variable. 

Choice  of  scale  units.  ]\Iost  graphs  are  drawn  on  ordinary  arithmetic 
graph  paper,  which  is  divided  into  squares,  with  a  given  number  of 
rulings  per  inch  or  per  centimeter.  In  such  paper  the  successive  rulings 
are  equally  spaced ;  that  is,  the  actual  measured  distance  between  the 
values  I  and  2  is  the  same  as  the  distance  between  2  and  3,  and  so  on. 

A  second  type  of  ruling,  which  is  extensively  used  with  sedimentary 
data,  is  the  logarithmic  scale.  Here  the  measured  distances  between  suc- 
cessive units  are  not  equal,  but  decrease  in  geometric  intervals  through 

0123456  789  10 

I  2  3         '       4  5  6         7       8      9    10 

Fig.  74. — Relation  between  arithmetic  scale   (top)   and  logarithmic  scale   (bottom). 

cycles  of  10.  Each  cycle  is  of  equal  length,  so  that  the  measured  distance 
from  I  to  10  is  the  same  as  that  from  10  to  100,  and  so  on.  The_use_of 
logarithmic  ^cales_achieves  the  jame_  result  as  plotting  the  logarithms 
of _the j)xiginal  data_on_qrdinar}'  arjthmeti^^rapli^aper.  The  logarithmic 
ruling  is  used  when  rates  of  change  are  to  be  compared,  or  when  wide 
fluctuations  in  the  values  of  experimental  data  are  to  be  smoothed. 
Graph  paper  may  be  obtained  either  with  logarithmic  scales  along  both 
axes,  or  with  a  logarithmic  scale  along  the  y-axis  and  arithmetic  ruling 
along  the  .r-axis.  The  former  type  of  paper  is  called  double  log  paper, 
and  the  latter  is  semi-log  paper. 

The  relations  between  arithmetic  and  logarithmic  scales  are  shown  in 
Figure  74.  In  using  arithmetic  scales  the  zero  point  is  indicated  on  the 
scale,  whereas  with  a  logarithmic  scale  there  is  no  zero  point ;  instead, 
the  cycles  extend  simply  from  the  highest  to  the  lowest  values.  When  a 
range  of  values  greater  than  10  is  involved  in  logarithmic  plotting, 
multiple  cycle  paper  is  used,  in  which  more  than  a  single  cycle  is  in- 
cluded. 

Even  though  original  data  are  plotted  directly  on  a  logarithmic  scale, 


i84       SEDIMENTARY  PETROGRAPHY 

it  is  not  correct  to  consider  the  result  as  a  graph  of  the  original  data. 
The  use  of  a  logarithmic  scale  changes  the  variable  to  its  logarithm.  If 
one  plots  mineral  frequencies  on  a  logarithmic  scale  against  distance  on 
an  arithmetic  scale,  the  graph  shows  the  relation  of  log  mineral  fre- 
quency against  distance,  and  not  mineral  frequency  directly.  The  subject 
of  changing  the  variables  in  connection  with  frequency  curves  is  dis- 
cussed more  fully  in  Chapter  9. 


GRAPHS  INVOLVING  TWO  VARIABLES 

It  is  convenient  to  distinguish  between  graphs  based  on  two  variables 
and  those  based  on  three  or  more  variables  because  of  the  increasing 
complexity  of  higher-dimensional  figures.  Two-dimensional  graphs  in- 
clude a  wide  variety  of  forms — the  frequency  of  grain  diameters,  changes 
of  average  size  with  distance,  comparisons  of  size  and  degree  of  sorting, 
comparisons  of  shapes  of  grains,  mineral  compositions,  and  so  on. 

Histograms 

The  simplest  manner  of  depicting  the  results  of  mechanical  analyses 
is  to  prepare  a  histogram  ^  of  the  data.  For  this  purpose  the  results  of 
the  analysis  are  compiled  into  a  frequency  table,  which  shows  the  class 
intervals  in  millimeters  or  any  other  convenient  units,  and  the  frequencies 
of  each  class  or  grade,  usually  as  a  percentage  of  the  total  weight. 
Diameters  in  millimeters,  their  logs,  or  whatever  size-equivalent  is  used 
is  chosen  as  the  independent  variable,  and  frequency  is  the  dependent 
variable.  In  general,  the  class  intervals  are  laid  off  along  the  horizontal 
.f-axis,  and  above  each  of  the  classes  a  vertical  rectangle  is  drawn,  with  a 
width  equal  to  the  class  interval  and  a  height  proportionate  to  the 
frequency  in  the  class. 

Conventions  among  sedimentary  petrologists  have  varied  widely  in 
plotting  histograms.  Some  writers  plot  diameters  on  the  vertical  axis 
and  frequency  on  the  horizontal  axis,  but  as  long  as  the  size  is  more 
conveniently  chosen  as  the  independent  variable,  it  is  preferable  to 
standardize  the  procedure  by  plotting  size  always  on  the  horizontal  axis. 
Another  common  convention  for  sediments  is  to  plot  the  size  scale  such 
that  values  of  x  (the  diameter)  increase  to  the  left.  This  results  in  a 
reversed  scale  of  values  in  terms  of  conventional  mathematical  practice, 

1  Some  writers  in  sedimentary  petrology  have  used  the  term  frequency  pyramid 
instead  of  the  word  histogram.  The  latter  word  is,  however,  a  statistical  term  of 
common  usage,  accepted  universally  by  statisticians,  and  its  use  will  be  retained  here. 


GRAPHIC  PRESENTATION 


i8; 


but  despite  that  there  are  numerous  arguments  in  favor  of  such  a  re- 
versed scale  for  sedimentary^  data.^  Merely  for  convenience  of  compari- 
son, it  is  perhaps  desirable  that  the  reversed-scale  convention  be 
adopted  as  a  standard  practice. 

Two  general  types  of  histograms  are  used.  In  one,  the  diameters  are 
laid  off  directly  on  an  arithmetic  scale,  with  the  result  that  the  successive 
vertical  rectangles  decrease  in  width  as  the  class  intervals  become  smaller. 
In  the  other  case,  the  class  limits  are  drawn  on  a  logarithmic  scale,  either 
directly  or  by  implication,  so  that  each  vertical  bar  is  equal  in  width, 
regardless  of  the  original  difference  in  absolute  class  interval.-  In  pre- 
paring the  first  type  of  histogram  certain  precautions  must  be  followed, 
because  it  is  necessary  to  preserve  the  area  under  the  curve  as  a  constant, 
equal  to  lOO  per  cent  of  the  frequency. 


In  order  to  preserve  the  area  under  the  curve  when  a  histogram  is  drawn 
on  tlie  basis  of  class  intervals  in  millimeters,  each  histogram  block  must  be 


so 

1 

y*o 

o 

1 

£ 

uj3C 

s 

£- 

o 

^10 

-L      \ 

DIAMETER   IN  MM. 

Fig.  7S. — Incorrect  method  of 
drawing  histogram  with  diameter 
in  mm.  as  independent  variable. 


^ 

E  1  Per  cent                                                   j 

i 

f-n 

1 

t- 

1 

i 
* 

0.4  0.3  0  2  01 

DIAMETER    IN   UU. 

Fig.  76. — Histogram  of  same  data 
as  in  Figure  75,  showing  represen- 
tation by  areas. 


drawn  in  terms  of  area  rather  than  height.  Areas  involve  both  width  and 
height,  and  hence  for  a  given  frequency  percentage  the  height  of  the  block 
will  depend  on  the  class  interval.  Figures  75  and  76  illustrate  the  wrong  and 
right  way  to  draw  a  histogram  on  this  basis.  It  will  be  noted  that  the  use 

1  The  data  of  most  mechanical  analyses  are  obtained  in  order  from  coarsest  to 
finest,  and  certain  special  sedimentation  curves,  as  Oden  curves  (page  113),  are 
obtained  automatically  in  the  reversed  sense.  Choice  of  direction  of  a  scale  is  per- 
fectly arbitrary ;  in  fact,  in  astronomy  stellar  magnitudes  are  expressed  on  such 
a  reversed  scale,  as  is  hydrogen-ion  concentration  in  chemistry.  The  practice  of 
reversing  scales  for  convenience  is  therefore  no  radical  departure  from  accepted  prac- 
tices. 

-  In  the  latter  case  it  is  assumed  that  the  Wentworth  or  Atterberg  scale,  or  some 
other  true  geometric  grade  scale,  is  used  in  preparing  the  logarithmic  graph. 


1 86 


SEDIMENTARY  PETROGRAPHY 


of  a  single  scale  along  the  vertical  axis  results  in  a  histogram  with  appar- 
ently much  of  the  material  in  the  larger  classes.  The  correct  diagram  on  the 
right  is  drawn  on  an  areal  basis,  as  all  histograms  should  be,  with  the  small 
square  representing  i  per  cent.  This  histogram  is  quite  noticeably  different 
from  its  neighbor. 

It  was  noted  years  ago  that  if  each  grade  is  indicated  as  of  equal  width 
on  the  horizontal  axis,  the  histogram  becomes  much  more  symmetrical. 

Because  of  this,  presumably,  it  has 
become  customary  for  sedimentary 
petrologists  to  draw  their  histograms 
with  each  rectangle  equal  in  width, 
so  that  the  sediment  shown  in  Figure 
76  appears  like  that  in  Figure  jy. 
Actually  this  procedure  does  not  show 
diameters  directly,  because  the  scale 
has  been  transformed  to  a  logarithmic 
scale,  whether  that  fact  is  so  indicated 
on  the  figure  or  not.  It  is  important 
that  the  practical  worker  be  aware  of 
the  differences  involved  in  plotting 
data  on  arithmetic  and  logarithmic 
scales,  because  any  statistical  devices 
that  may  be  used  to  describe  the  sedi- 
ment are  strikingly  affected  by  the 
change  of  independent  variable  from  diameter  to  log  diameter. 


so 

i*° 

^30 

z 

1 

0  50    0.3S      0.2  s 

DIAMETER  IN  MM 

Fig.  'j'J. — Conventional  histogram 
of  same  data  as  in  Figure  76,  plotted 
on  an  implied  logarithmic  scale,  with 
class  intervals  shown  as  equal.  (Data 
from  Petti  John  and  Ridge,  1932.) 


That  the  intervals  become  equal  when  logs  are  used  is  easily  shown.  Thus, 
logi„2  =  0.301 ;  logioi  =  0.000;  logio(/^)= —0.301.  Here  each  class  is  equal 
in  width  (interval  =  0.301),  with  the  origin  at  i  mm.  The  logs  of  numbers 
smaller  than  i  are  negative,  and,  if  the  base  10  is  used,  the  class  limits  are 
not  marked  by  integers.  This  suggests  that  logs  be  taken  to  such  a  base  that 
the  class  limits  become  integers  and,  for  the  sake  of  convention,  so  that 
negative  numbers  may  extend  to  the  left  instead  of  to  the  right.  These  ends 
are  accomplished  by  taking  negative  logs  to  the  base  2  of  the  diameter  values, 
whereupon  the  above  class  limits  become  —  logo2  =  — i;  — log2i=o; 
—  logoCH)  "^^  +  i>  aiid  the  scale  is  transformed  to  an  arithmetic  scale  with 
equal  units.^ 

Frequency  polygons.  In  addition  to  histograms  as  frequency  diagrams, 
a  common  statistical  device  is  to  indicate  variations  in   frequency  by 

1  This  type  of  transformation  substitutes  a  new  variable  for  diameters  in  milli- 
meters. This  concept  is  discussed  in  Chapter  4,  as  the  0  scale,  and  is  treated  more 
fully  in  Chapter  9.  In  an  analogous  manner,  a  <;  scale  may  be  derived  for  the  Atter- 
berg  grades. 


GRAPHIC  PRESENTATION 


187 


050       03S 


0125    o.oaa  0  061 


Fig.  78. — Frequency  polygon  of  same 
data  as  in  Fig.  77. 


means  of  a  line  diagram  instead  of  with  rectangular  blocks.  Such  fre- 
quency diagrams  are  called  frequency  polygons,  and  they  are  prepared 
by  plotting  the  frequency  corre- 
sponding to  a  given  grade  size  on  a 
line  midway  between  the  grade 
limits.  The  resulting  points  are  then 
connected  with  a  continuous  line, 
made  of  straight  line  segments,  as 
shown  in  Figure  yS.  The  continuous 
line  is  brought  down  to  the  zero 
point  at  the  centers  of  the  grades 
just  larger  and  smaller  than  the 
limiting  grades  in  the  analysis.  Fre- 
quency polygons  are  recommended 
by  some  statisticians  ^  as  a  device  to 
be  used  when  the  data  vary  continu- 
ously, to  avoid  the  implication  that 
each  grade  is  an  individual  entity.  Frequency  polygons  have  not  been 
used  widely  by  sedimentary  petrologists,-  but  they  may  occasionally  ex- 
press the  frequency  more  clearly  than 
histograms,  and  hence  may  be  con- 
sidered for  use.  Either  an  ordinary 
arithmetic  scale  or  a  logarithmic  scale 
may  be  used,  as  in  the  case  of  histo- 
grams. 

The  use  of  histograms  for  attrib- 
utes other  than  size.  Although  histo- 
grams have  been  discussed  from  the 
point  of  view  of  size  characteristics, 
it  is  possible  to  plot  many  other  at- 
tributes of  sediments  in  that  manner. 
A  histogram  is  essentially  a  statistical 
device  used  to  represent  frequency. 
Hence  any  frequency  attribute  may 
l)e  so  expressed.  Mineral  frequencies, 
shape  frequencies,  surface  texture  frequencies,  and  others  are  included 
here.  In  general,  the  same  principles  of  construction  apply,  and  care 


O  I         .2        .3       .4        .5         6         7        .8        a     10 


ROUNDNESS 

Fig.  79. — Histogram  of  roundness 
variation  of  quartz  particles.  (After 
Wadell,  1935.) 


1  F.  C.  Mills,  Statistical  Methods  (New  York,  1924),  pp.  79-81. 

2  Miss  Gripenberg  has  used  a  combination  histogram  and  frequency  polygon  in 
expressing  the  composition  of  sediments.  See  Stina  Gripenberg,  A  study  of  the  sedi- 
ments of  the  North  Baltic  and  adjoining  seas :  Fcnnia,  vol.  60,  no.  3,  pp.  191  flF. 


i88 


SEDIMENTARY  PETROGRAPHY 


must  be  exercised  to  represent  areas  correctly,  as  previously  pointed  out. 
Figure  79  is  a  histogram  of  the  distribution  of  roundness  of  quartz 
grains  in  the  O.ooi-O.oi  cu.  mm.  volume  class,  after  \\'adell,^  to  illustrate 
the  tj'pe  of  diagram  involved. 

Cumulative  Cur\t:s 

The  cumulative  frequency  cur\'e  is  a  cur\-e  based  on  the  original  his- 
togram data,  and  is  made  by  plotting  ordinates  which  represent  the  total 

amount  of  material  larger  or  smaller 
than  a  given  diameter.  Two  types  of 
ciunulative  curves  are  possible,  the 
"more  than"  curve  and  the  "less  than" 
curv-e.  It  is  immaterial  which  is  used,  in- 
asmuch as  either  furnishes  the  same  t)-pe 
of  information.  The  commoner  type  in 
sedimentary  data  is  perhaps  the  "more 
than"  type.  It  is  made  by  choosing  a  size 
scale  along  the  horizontal  axis,  and  a 
frequency  scale  from  o  to  100  per  cent 
along  the  vertical  axis.  The  horizontal 
scale  may  be  either  arithmetic  or  loga- 
rithmic, as  in  the  case  of  the  histograms. 
In  either  case  the  procedure  is  the  same, 
and  it  is  not  necessar}-  to  consider  the 
areas  in  drawing  the  curve. 

At  the  upper  limit  of  the  first  class  in- 
terval  an   ordinate  is   erected   equal   in 
height   to   the   frequency  in   that   class. 
'''  At  the  end  of  the  next  class  another  or- 

dinate is  drawn,  equal  in  height  to  the  sum  of  the  frequencies  in  the 
first  two  classes,  and  so  on.  In  short,  the  cumulative  curve  is  equivalent 
to  setting  one  histogram  block  above  and  to  the  right  of  its  predecessor, 
so  that  the  base  of  each  block  is  the  total  height  of  all  preceding  blocks. 
Strictly  speaking,  this  would  yield  a  step  diagram,  as  shown  in  Figure 
80.  However,  it  is  common  practice  to  draw  only  points  at  the  upper 
limit  of  each  ordinate,  and  to  connect  the  points  with  short  line  seg- 
ments. It  is  also  customar\'  to  draw  a  smoothed  curve  through  the  points, 
to  obtain  a  continuous  curv'e  representing  the  continuous  distribution  of 
sizes.  Cumulative  curves  have  come  into  wide  use  in  sedimentar}-  work 

1  H.  Wadell,  Volume,  shape,  and  roundness  of  quartz  particles :  Jour.  Geology, 
vol.  43,  pp.  250-279,  1935. 


100 

»- 

( 

^80 

S 

1 

S60 

z 

/ 

-40 

/ 

5  20 

i 

5 

/ 

/ 

OIAUETER     IN    UU. 


Fig.  80. — Diagram  showing  re- 
lation bet^veen  histogram  and 
cumulative  curve.  Data  as  in  Fig. 


GRAPHIC  PRESENTATION 


189 


because  of  die  convenience  with  which  statistical  values  are  drawn  from 
them  (see  Chapter  9). 

A  type  of  graph  paper  of  considerable  value  in  analyzing  cumulative  curves 
is  logarithmic  probability  paper,^  which  has  a  logarithmic  scale  along  one 
axis  and  a  probability  scale  along  the  other.  The  probability  scale  is  so  de- 
signed that  a  symmetrical  cumulative  curve  will  plot  as  a  straight  line  on  the 
graph.  Many  sands  show  straight  lines  on  this  paper,  and  it  affords  an  ex- 
cellent method  of  comparing  sedimentary  data.  A  further  use  of  the  paper  is 
to  study  the  eft'ect  of  using  combined  sie-<-''"g  a"d  sedimentation  methods  on 


I    .8    .6       .4  .2  0.1       .06     .04  .02         .01 

DIAMETER  IN  MM. 

Fig.  8i. — Cumulative  curves  of  beach  sand  (steep  curve)  and  glacial  till  (gentle 
curve)  drawn  on  logarithmic  probability  paper.  The  till  shows  a  "break"  at  l/i6 
mm.,  due  to  change  from  sieving  to  sedimentation  method  of  analysis. 

the  same  sample.  It  was  mentioned  on  page  136  that  there  often  is  a  hiatus 
between  the  portions  sieved  and  sedimented,  and  such  an  hiatus  will  appear 
on  the  probability  paper  as  a  change  in  the  slope  of  the  line.  The  paper  is 
also  of  much  use  in  the  statistical  study  of  samples,  inasmuch  as  from  it 
the  average  size  and  degree  of  spread  may  be  directly  read  for  sediments 
which  plot  as  straight  lines.  This  latter  point  is  further  discussed  in  Chap- 
ter 9.  Figure  81  shows  two  samples  plotted  on  probability  paper,  one  of 
which  is  a  straight  line  and  the  otlier  shows  an  abrupt  change  at  ^g  mm.  When 
the  line  is  curved,  the  sediment  does  not  have  a  symmetrical  cumulative 
curve. 


9919 

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^  Probability  paper,  designed  by  Hazen,  Whipple,  and  Fuller,  may  be  obtained 
from  the  Codex  Book  Co.,  New  York,  N.  Y. 


1 90 


SEDIMENTARY  PETROGRAPHY 


Frequency  Curves 

Frequency  curves  are  smooth  curves  which  show  the  variation  of  the 
dependent  variable  as  a  continuous  function  of  the  independent  vari- 
able. Histograms,  cumulative  curves,  and  frequency  curves  are  related 
mathematically,  and  any  one  may  be  obtained  from  any  other.  The 
relation  between  histograms  and  frequency  curves  is  more  direct  than 
between  these  two  and  cumulative  curves,  but  in  the  construction  of 
frequency  curves  from  histogram  data  the  cumulative  curve  may  be 
most  conveniently  used.  The  relation  between  the  histogram  and  the 


/ 

-f- 
/ 

f 

N 

N 

/fir 

4 

\ 

J- 

\ 

J- 

\ 

-.—  DIAMETERS     IN     MILLIMETERS 
Fig.  82. — Diagram  showing  transition  from  histogram  to  frequency  curve. 

frequency  curve  is  that  the  latter  represents  the  limit  of  a  histogram 
as  the  class  intervals  become  smaller  and  smaller  and  finally  reach  zero, 
while  the  frequency  increases  without  bound.  Figure  82  illustrates  the 
transition  from  one  to  the  other,  and  in  fact  a  common  manner  of 
drawing  frequency  curves  is  simply  to  superimpose  a  smoothed  curve 
over  the  histogram  bars.  This  procedure  is  not  always  accurate,  how- 
ever, owing  to  the  relatively  large  classes  used  in  sedimentary  work, 
and  because  of  an  unfortunate  variation  in  the  histograms  of  the  same 
sediment  when  different  grade  sizes  are  used  in  the  analysis. 

There  is  a  unique  frequency  curve  which  may  be  obtained  fairly 
satisfactorily  from  the  cumulative  curve  by  a  graphic  method.  The 
smoothed  cumulative  curve  is  a  continuous  curve,  and  from  it  by 
graphic  differentiation  may  be  obtained  the  smooth  frequency  curve, 
independent  of  the  particular  grade  sizes  used  in  the  analysis. 

Graphic  Differentiation  of  Cumulative  Curves 

There  are  several  methods  by  which  an  approximation  to  the  unique 
frequency  curve  may  be  obtained  from  sieve  data.  One  such  is  obtained 


GRAPHIC  PRESENTATION  191 

by  drawing  a  smooth  curve  over  the  histogram  itself,  if  care  be  exer- 
cised to  preserve  areas  in  the  smoothing  process.  Methods  for  the 
numerical  or  semi-graphic  computation  of  frequency  curve  ordinates 
have  been  described  by  several  writers.^  However,  since  the  frequency 
curve  is  the  derivative  of  the  cumulative  curve  (see  page  215),  the  usual 
method  of  graphic  differentiation  may  be  used,-  but  cognizance  should 
be  taken  of  the  fact  that  the  ;r-axis  is  logarithmic  when  the  cumulative 
curve  is  drawn  on  the  basis  of  equal  intervals  for  each  grade. 

As  an  illustration,  data  based  on  Wentworth  class  intervals  will  be 
used,  so  that  the  final  frequency  curve  will  be  obtained  in  terms  of  that 
descriptive  grade  scale.  For  this  purpose  it  is  convenient  to  use  3-cycle 
semi-logarithmic  paper  (Eugene  Dietzgen  #340-L  310),  in  which  the 
length  of  a  cycle  is  about  8.1  cm.,  so  that  when  the  Wentworth  scale 
is  laid  off  along  the  jr-axis,  the  actual  distance  between  successive  points 
is  about  2.4  cm.  Also  by  convention,  10  per  cent  on  the  vertical  scale  is 
chosen  so  that  it  is  about  half  the  length  of  the  horizontal  scale  unit, 
or  about  1.2  cm.  This  assures  a  cumulative  curve  which  is  usually  not 
too  steep  for  convenient  handling.  By  adopting  these  conventions  the 
final  curves  are  directly  comparable,  because  they  are  all  obtained  in 
reference  to  fixed  scale  relations. 

When  the  variables  are  plotted  on  arithmetic  scales  a  pole  p  is  usually 
drawn  to  the  left  of  the  vertical  axis  at  a  distance  equal  to  a  unit  along 
the  .r-axis.  By  this  means  the  ordinates  of  both  the  cumulative  curve 
and  frequency  curve  may  be  read  from  the  same  numerical  units  of  the 
3/-axis.  In  the  present  case  a  logarithmic  ^-scale  is  used,  but  fortunately 
the  same  relations  hold  as  in  the  arithmetic  case,  and  the  pole  p  is  drawn 
to  the  left  of  the  vertical  axis  a  distance  equal  to  the  arithmetic  value 
(actual  length)  of  the  Wentworth  units  along  the  Ji'-axis,  here  2.4  cm. 
Since  in  such  logarithmic  scales  the  geometric  ratio  between  successive 
points  yields  equal  arithmetic  intervals,  the  length  of  the  interval  de- 
termines the  pole  distance. 

The  cumulative  curve  is  divided  into  any  convenient  number  of  units, 
as  shown  in  part  in  Figure  83.  At  each  of  the  points  an  ordinate  is 
erected  to  the  curve  as  at  Ji'iA,  ,roB.  These  ordinates  need  not  necessarily 
coincide  with  the  experimentally  determined  points  of  the  cumulative 

1  S.  Oden,  On  clays  as  disperse  systems :  Trans.  Faraday  Soc,  vol.  17,  pp.  327- 
348,  1921-22.  D.  S.  Jennings,  M.  U.  Thomas  and  W.  Gardner.  A  new  method  of 
mechanical  analysis  of  soils :  Soil  Science,  vol.  14,  pp.  485-499,  1922.  C.  E.  Van 
Orstrand,  Note  on  the  representation  of  the  distribution  of  grains  in  sands :  Re- 
searches in  Sedimentation  in  1924,  Nat.  Research  Council,  pp.  63-67,  1925. 

2  H.  Von  Sanden,  Practical  Matlieiiiatical  Aiialxsis,  translated  by  H.  Levy  (New 
York,  1924),  Chap.  VII. 


192 


SEDIMENTARY  PETROGRAPHY 


cun-e.  A  tangent  is  now  drawn  to  the  cumulative  curve  at  A.  The  angle 
a,  made  by  this  tangent  with  the  horizontal,  is  then  laid  off  at  P,  and  the 
line  PO  is  drawn.  From  O,  where  the  ray  of  the  angle  intersects  the 
3'-axis,  a  horizontal  line  is  drawn  to  the  ordinate  XiA  or  its  projection. 
The  intersection  of  the  horizontal  line  from  Q  with  this  ordinate  locates 
a  point  on  the  desired  frequency  curve.  The  process  is  repeated  at  x.B, 
where  the  tangent  at  B  makes  an  angle  /3  with  the  horizontal.  A\'hen  this 
angle  is  laid  off  at  P.  it  yields  the  point  R  on  the  y-axis,  and  finally 


POLE      DISTANCE 


1/4 
LOG  DIAMETER 

Fig.  83. — Graphic  differentiation  of  cumulative  curve.  See  text  for  explanation. 

the  second  point  along  the  frequency  curve.  Further  repetition  yields 
as  many  points  on  the  frequency  curve  as  the  number  of  originally 
chosen  points  along  the  cumulative  curv'e.  In  general,  these  arbitrarily 
chosen  points  of  tangency  may  not  include  the  inflection  point  ^  of  the 
cumulative  curve.  Hence  a  separate  determination  should  be  made  at 
that  point  because  the  mode  of  the  frequency  curve  lies  on  the  ordinate 
of  the  inflection  point. 

Proof  that  the  method  of  graphic  dift'erentiation  applies  in  the  case 
of  a  logarithmic  ^-scale  is  not  given  in  the  references  cited,  but  it  may 


1  The  inflection  point  of  a  curve  is  the  point  of  maximum  slope  of  the  tangent. 
It  is  the  point  where  the  tangent  to  the  curve  changes  its  direction  of  rotation.  It 
may  accordingly  be  located  by  moving  a  ruler  along  the  curve  so  that  it  is  always 
tangent  to  it,  until  the  point  is  reached  where  the  direction  of  rotation  of  the  ruler 
reverses  itself. 


GRAPHIC  PRESENTATION  193 

be  shown  that  the  method  satisfies  the  requirements  of  mathematical 
rigor :  ^ 

In  the  conventions  adopted  in  the  illustration,  it  is  possible  to  plot  directly 
the  values  of  the  experimentally  determined  points  of  the  cumulative  curve 
in  terms  of  their  logs  to  the  base  10.  At  the  same  time,  the  Wentworth  units 
determine  the  areal  relations  of  the  resulting  frequency  curve  by  fixing  the 
position  of  the  pole  P.  For  plotting  purposes  we  may  lay  ofif  the  logs  of  the 
Wentworth  scale  to  the  base  10,  in  which  case  the  distance  between  successive 
points  will  be  k  log^o  2,  where  k  is  the  length  of  a  logarithmic  cycle  on  the 
graph  paper.  If  the  convention  of  decreasing  the  values  of  the  diameters  to 
tne  right  is  followed,  the  ch.oice  for  plotting  \s  x  =  —  k  logir,  i,  where  x  is  the 
actual  distance  along  the  ^--axis  from  the  arbitrary'  origin  logj^  1  =  0,^,  is  the 
length  of  a  cycle,  and  I  is  the  numerical  value  of  the  diameter. 

The  transformation  logio  I  =  log^^  2  logo  I  may  be  used  to  locate  the  pole  P 
in  terms  of  the  \\'ent\vorth  scale.  Substitution  in  the  equation  for  x  yields 
x  =  —  k  logio  2  logo  ^.  Call  the  unit  along  the  jr-axis  X^,  and  set  k  log^o  2  =  X^. 
Now  let  0  =  —  logo  i,  and  by  substitution  we  have 

x^\'> (I) 

One  may  next  choose  y  and  y',  the  actual  heights  of  the  ordinates  of  the 
cumulative  and  frequency  curves  respectively,  such  that 

3'  =  M (2) 

y  =  v (3) 

where  1?  is  a  function  of  <t>,  and  Xy  is  the  unit  along  the  y-axis,  chosen  equal 
in  both  cases  so  that  the  same  numerical  unit  applies  to  the  ordinates  of  each 
curve. 


P  X 

Fig.  84. — Proof  of  graphic  differentiation.  See  text. 


From  these  parametric  equations,  allow  v  —  f{1>)  to  be  the  cumulative 
curve,  so  that  r  =  dv/d4>  will  be  the  corresponding  frequency  curve.  In  Fig- 
ure 84  a  tangent  is  drawn  to  the  cumulative  curve  at  y,  making  an  angle  o^ 

1 W.  C.  Krumbein,  Size  frequency  distributions  of  sediments :  Jour.  Sed.  Pe- 
trology, vol.  4,  pp.  65-77,  1934. 


194  SEDIMENTARY  PETROGR.\PHY 

with  the  horizontal.  By  the  calculus,  tan  a^  —  dy/dx,  and  from  equation  (2), 

tan  a,  =  dy/dx  =  K  dv/dx  =^  \  ^  ^  ....      (4; 
■  d'P  dx 

But  dv/d<P=r,  and  from  equation   (i),  dx/d4>  =  \^  so  that  d<f>/dx=  i/\. 
Substituting  in  equation  (4), 

tana,  =  _ll- (5) 

Also  from  Figure  84,  tan  a.^  —y'/P,  where  p  is  the  pole  distance.  From 
equation  (3),  y'=  Vr,  and  by  substitution 

X..-J 

tan  do  = (6) 

P 
But  by  construction  in  the  graphic  method,  a-j^  =  a,.  Hence   (5)   and   (6) 
may  be  equated: 

X.-:  \c 

from  which  there  results 

p-\ 
and  the  method  applies. 

In  choosing  inter\-als  along  the  cumulative  cur\'e,  it  is  well  to  include 
at  least  ten.  When  the  form  of  the  frequency  cur\e  develops,  additional 
points  may  be  chosen  to  bring  out  needed  details!  Both  cur\es  represent 
continuous  functions,  and  consequently  one  is  not  limited  to  specific 
points  along  either  curve.  The  possible  errors  introduced  by  the  smooth- 
ing process  are  discussed  belo%v. 

Figure  85  is  inserted  to  illustrate  several  types  of  frequency  cur\'es 
obtained  by  the  method  described,  and  it  shows  as  well  the  correspond- 
ing cumulative  cur\-es.  Certain  points  may  be  noted.  In  curve  A,  which 
represents  a  beach  sand,  the  highest  ordinate  of  the  frequency  curve 
extends  above  the  100  per  cent  line  of  the  cumulative  cur\'e.  This  means 
that  the  steepness  of  the  cumulative  curve  at  that  point  is  so  great  that 
the  rate  of  change  of  percentage  per  unit  of  the  grade  scale  here  is  105 
per  cent.  In  other  words,  the  grains  are  very^  higlily  concentrated  about 
this  modal  value. 

Cur\'e  B  of  Figure  85  represents  a  silt  (loess)  in  which  the  cumulative 
cur\e  is  less  steeply  inclined,  so  that  the  mode  of  the  frequency  curve 
is  much  less  accentuated  than  in  the  sand. 

These  examples  focus  interest  on  another  point,  which  is  the  accuracy 
of  the  graphic  method.  It  may  confidently  be  stated  that  if  the  curvature 
of  the  aimulative  cur\^e  is  known  at  ever)^  point,  and  if  the  tangents 
are  correctly  drawn,  the  method  yields  rigorous  results.  In  experimental 
work,  however,  interpolation  of  the  cumulative  cune  is  necessary,  and 


GRAPHIC  PRESENTATION 


195 


to  the  extent  that  the  smoothing  introduces  errors,  the  final  results  are 
inaccurate.  Also  the  tangents  to  the  curv-e  cannot  be  drawn  correctly  by 
inspection,  and  another  error  is  introduced  by  this  fact.  There  are  de- 
vices for  reading  the  tangents  to  curves/  and  hence  this  error  may  be 
made  practically  negligible.  The  correction  of  the  error  involved  in 
smoothing  the  cumulative  curve  requires  some  discussion. 

As  stated  earlier,  cumulative  curves 
are  independent  of  the  grade  scale 
used  in  the  analysis,  but  in  smoothing 
such  curves  it  is  obvious  that  the  pre- 
cise locations  of  the  known  points  will 
influence  the  smoothing  process,  so  that 
the  cumulative  curves  obtained  from 
different  grade  scales  may  not  be  com- 
pletely identical,  especially  if  too  few 
points  are  known.  Since  both  cumula- 
tive curves  and  frequency  curves  of 
sediments  are  continuous  functions  of 
the  diameters,  the  most  logical  way  of 
avoiding  these  errors  of  smoothing  is 
to  determine  by  experiment  as  many 
points  as  possible  along  the  cumulative 
curve.  It  should  be  borne  in  mind  that 
uneven  class  intervals  do  not  distort 
the  cumulative  curve,  and  consequently 
as  many  points  may  be  experimentally 
determined  as  one  wishes,  regardless 
of  the  particular  intervals  between  successive  points.  It  is  not  convenient, 
however,  to  construct  a  histogram  from  such  unequall}-  spaced  points, 
unless  the  points  have  a  fixed  ratio  to  each  other.  Thus  in  using  the 
analytic  data  to  construct  a  cumulative  curve,  one  is  independent  of  the 
need  of  a  fixed  scale  for  analvsis. 


A 

i 

' '  i 
1 

! 

i 

'Jf 

V 

z 

> 

^ 

i 

B 

/ 

J 

\ 

v_ 

I    CUY 


Fig.  85. — Examples  of  graphic 
differentiation  of  beach  sand  (A) 
and  loess  ( B ) . 


Sedimextatiox  Curves 

A  type  of  graphic  presentation  commonly  used  in  connection  with 
such  methods  of  analysis  as  the  Oden  balance  (Chapter  6)  is  included 
here  to  complete  the  classification  of  curves  representing  the  size  attri- 
butes of  sediments,  although  the   details  of  the  construction  of   such 


iThe  Richards-Roope  tangent  meter,  offered  by  Bausch  and  Lomb,  is   such  a 
device. 


196 


SEDIMENTARY  PETROGRAPHY 


curves  lias  been  given  in  Chapter  5.  The  Oden  curve  is  related  mathe- 
matically to  the  cumulative  curve,  and  in  turn  to  the  frequency  curve, 
despite  the  fact  that  time  rather  than  size  is  the  original  independent 
variable  in  the  Oden  curves.  Other  sedimentation  curves  include  the 
Wiegner  curve,  also  described  elsewhere  (Chapter  6). 

Graphs  with  Distance  or  Time  as  Independent  Variable 

Distance.  Among  graphic  devices  much  used  in  the  study  of  sediments 
are  graphs  of  the  linear  variation  in  the  average  size  of  sediments  along 
a  line  of  samples,  variations  in  thickness  with  distance,  and  in  short, 
the  linear  variation  of  any  measurable  attribute  as  a  function  of  distance 
along  a  formation,  stream,  beach,  or  the  like.  In  all  these  cases  distance 
is  used  as  the  independent  variable,  and  is  plotted  along  the  horizontal 

axis.  With  very  few  exceptions,  the  dis- 
tance scale  is  chosen  as  arithmetic,  and 
if  a  logarithmic  function  is  sought,  the 
logarithmic  scale  is  generally  used  along 
the  y-axis.  An  exception  to  this  generali- 
zation is  Baker's  use  of  a  logarithmic 
scale  for  distance  in  the  study  of  sands 
in  the  London  Basin.^  If  a  general  rule 
may  be  adduced  regarding  the  choice  of 
scale  for  the  independent  variable,  it  is 
perhaps  that  of  convenience.  With  re- 
gard to  mathematical  convenience,  it 
may  be  mentioned  that  when  parabolic 
or  hyperbolic  functions  are  suspected, 
double  log  paper  may  be  used  to  test  the  relation,  as  explained  in  a  later 
portion  of  this  chapter. 

As  illustrations  of  graphs  using  distance  as  the  independent  variable, 
Figures  86  and  87,  adapted  from  originals  -  in  the  literature,  indicate 
types  of  data  commonly  presented. 

There  is  no  general  term  applied  to  graphs  of  the  nature  just  dis- 
cussed, but  various  writers  have  coined  terms  to  describe  them.  Petti  John 
and  Ridge  ^  used  the  term  "Size  \'ariation  Series"  to  describe  the  rela- 


20 

5 

5 

z  1.0 

^    .6 

Ul 

2    .4 
|3 

i^ 

a 

bJ 

5     J 

! 

! 

A             r- 

,   i   i   1   [ 

— 1 

/\             1      1  i  1  i  1 

\             !    1 

1 

1  1   (   i-l 

l/\  .      1   ! 

1 

' 

y  Vi-AL 

1 

1 

J! 

1  !  i  i 

Yi\ 

J 

^^ 

's, 

DISTANCE 

Fig.  86. — Size  variation  dia- 
gram. The  samples  are  spaced  at 
half-mile  intervals.  (Data  from 
Petti  John  and  Ridge,  1932.) 


1  H.  A.  Baker,  On  the  investigation  of  loose  arenaceous  sediments  by  the  method 
of  elutriation,  etc.:  Geo!.  Mag.,  vol.  57,  pp.  321-332,  363-370,  411-420,  463-470,  1920. 

•  F.  J.  Pettijohn  and  J.  D.  Ridge,  A  mineral  variation  series  of  beach  sands  from 
Cedar  Point,  Ohio:  Jour.  Scd.  Petrology,  vol.  3,  pp.  92-94,  1933.  C.  Burri,  Sedi- 
mentpetrographische  Untersuchungen  an  Alpinen  Flussanden :  Schweis.  Min.  u.  Pet. 
Mitt.,  vol.  9,  205-240,  1929. 

3F.  J.  PeUijohn  and  J.  D.  Ridge,  loc.  cit.,  1933. 


GRAPHIC  PRESENTATION 


197 


tions  of  size  to  distance,  and  Krumbein^  used  the  term  "Median  Pro- 
cession Curve"  to  describe  the  variation  of  average  grain  size  (the 
median)   with  distance. 

Time.  The  direct  use  of  time  as  an  independent  variable  in  sedi- 
mentary studies  has  not  found  wide  appHcation,  perhaps  because  few 
studies  have  been  made  of  the  same  phenomenon  over  any  appreciable 
interval.  Related  studies  by  engineers,  however,  such  as  measurements 
of  the  flow  of  streams  as  a  function  of  time,  or  measurements  of  the 
silt  content  of  streams  as  a  function  of  time,  are  not  uncommon.  Like- 
wise, in  the  studies  of  cyclical  sedimentary  phenomena,  such  as  varved 
clays,  where  the  length  of  each  cycle  is  uniform    (a  year,  sayj,  time 


DISTANCE  ^— 

Fig.  87. — Mineral  variation  diagram.    (Data  from  Burri,  1929.) 

may  be  used  instead  of  distance  along  the  horizontal  axis,  because  each 
unit  along  the  x-axis  is  an  equal  unit  of  time.  Likewise  it  is  sometimes 
convenient  to  replace  thickness  by  time  if  the  rate  of  deposition  of  a 
sediment  can  be  shown  to  have  been  constant. 

For  ordinary  graphical  purposes  time  is  usually  plotted  on  an  arith- 
metic scale  rather  than  on  a  logarithmic  scale,  with  the  exception  perhaps 
of  studies  involving  possible  parabolic  or  hyperbolic  functions,  in  which 
mathematical  convenience  is  involved.  The  dependent  variable  may  be 
plotted  either  on  arithmetic  or  logarithmic  intervals. 

The  investigation  of  time  series  and  trends  is  itself  a  major  field  in 
the  study  of  economic  data,  and  most  textbooks  of  statistics  devote 
considerable  space  to  the  analysis  of  graphs  with  time  as  independent 
variable.  Inasmuch  as  the  methods  used  in  conventional  statistics  are 
identical  with  those  used  in  the  analysis  of  such  data  in  sedimentary 
studies,  the  reader  is  referred  to  standard  texts-  for  details  beyond  the 
relativelv  few  included  here. 


1 W.    C.    Krumbein,   Textural   and   lithological   variations   in  glacial   till :   Jour. 
Gcologv,  vol.  41,  pp.  382-408,  1933. 
2  F.  C.  Mills,  op.  cit.  (1924),  Chap.  7. 


198 


SEDIMENTARY  PETROGRAPHY 


The  analysis  of  time  series  usually  involves  one  or  more  of  four 
elements:  the  long-time  trend  of  the  variate,  such  as  the  increase  or 
decrease  in  average  size  over  a  long  period  of  time ;  seasonal  variations, 
such  as  the  increase  or  decrease  of  sediment  carried  by  a  stream  during 
times  of  high  and  low  water;  cyclical  movements,  such  as  recurrent 
increases  or  decreases  of  some  variable  (thickness,  size)  during  regular 
time  intervals ;  and  finally  accidental  variations  introduced  by  any  of  a 
number  of  random  causes. 


Table  18 

Computation  of  Three-year  Moving  Average  of  Thickness  of 
Varved  Slate  * 


Vai'c'c  Thickness 

Arbitrary  Year 

(mm.) 

2,-ycar  Totals 

2,-ycar  Averages 

I 

28 

2 

4 

37 

12.3 

3 

5 

26 

8.7 

4 

17 

50 

16.7 

S 

28 

59 

19.6 

6 

14 

48 

16.0 

7 

6 

44 

14.6 

8 

24 

81 

27.0 

9 

51 

98 

32.6 

10 

23 

102 

340 

II 

28 

77 

25.6 

12 

26 

78 

26.0 

13 

24 

88 

29-3 

14 

38 

69 

23.0 

IS 

7 

93 

31.0 

16 

48 

*  Data  from  F.  J.  Pettijohn.  The  varves  are  from  Lake  Mimiitaki,  Ontario.  See 
his  paper,  Early  Pre-Cambrian  varved  slate  in  northwestern  Ontario:  Gcol.  Soc. 
Am.,  Bull,  vol.  47,  pp.  621-628,  1936. 

It  is  to  be  expected  that  variations  will  occur  in  the  measured  values 
through  successive  intervals  of  time,  and  the  essential  attack  on  time 
series  problems  involves  the  smoothing  of  these  irregularities  to  disclose 
the  underlying  trend.  A  simple  method  commonly  used  for  studying 
time  trends  is  the  "moving  average"  method,  which  involves  simply  the 
taking  of  averages  over  the  data  arranged  chronologically,  in  such  a 
manner  that  each  average  involves  three,  five,  or  more  successive  values, 
and  the  succeeding  averages  are  computed  by  dropping  the  first  item 


GRAPHIC  PRESENTATION 


199 


of  each  group,  and  adding  the  next  item  from  the  table.  In  this  manner 
a  series  of  average  values  is  obtained,  each  average  representing  a  group 
of  observations,  and  thus  disclosing  the  underlying  trend. 


The  method  of  computing  moving  averages  is  illustrated  in  Table  18.  The 
data  represent  the  thickness  of  a  series  of  Pre-Cambrian  varved  slates,  re- 
ferred chronologically  to  an  arbitrary   succession  of  yearly   intervals.   The 
second  column  shows  the  observed  thicknesses 
of  the  varves.  and  the  third  column   indicates 
the   3-yr.   totals.   The   first   figure   is   found  by 
adding   the   thicknesses   of   years    i,   2,   3;    the 
second    figure    is   the    sum   of    thicknesses    for 
years  2,  3,  4,  and  so  on.  The  averages  for  the 
3-yr.    periods    are    given    in    the    last    column. 
Figure  88  includes  the  individual  varves  and  the 
moving  average,  to  indicate  how  individual  ir- 
regularities are  smoothed  by  the  average. 


Scatter  Diagrams 


ARBITRARY   YEARS 


Fig.  88. — Three-year  mov- 
ing average  of  varve  thick- 
ness. The  average  is  indi- 
cated by  the  heavy  line. 


A  scatter  diagram  is  any  graph  in  v^hich 
the  values  of  one  variable  are  plotted  against 
another.  In  the  broadest  sense,  therefore, 
scatter  diagrams  include  all  graphs  involving 
two  variables.  As  the  term  is  used  here,  scatter  diagrams  include  those 
graphs  in  which  any  two  independent  sets  of  data  taken  from  the  sedi- 
mentary study  are  plotted  against  each  other  to  learn  whether  there  may 
be  any  relation  between  them.  For  example,  a  study  of  sediments 
may  include  an  investigation  of  the  average  "degree  of  sorting"  of 
the  sediment,  and  an  analysis  of  the  carbon  content.  Each  of  these 
sets  of  data  is  obtained  from  an  independent  laboratory  investigation : 
the  problem  is  whether  there  is  any  relation  between  the  two  charac- 
teristics. 

The  choice  of  independent  variable  often  presents  a  problem  in  the 
prei^aration  of  scatter  diagrams.  This  is  not  true  when  size,  distance,  or 
time  are  used,  which  are  usually  chosen  as  independent  variables  by  con- 
vention. In  the  example  given,  however,  the  choice  of  independent  vari- 
able may  be  entirely  arbitrary  (page  183). 

In  scatter  diagrams  there  is  no  fixed  rule  regarding  the  choice  of 
scales  for  the  two  axes.  Either  or  both  may  be  arithmetic  or  logarithmic  ; 
the  essential  problem  is  to  find  the  simplest  relation  between  the  vari- 
ables, if  any  relation  exists.  Scatter  diagrams  are  usually  made  as  a 
preliminary  to  a  statistical  study  of  the  correlation  between  the  variables. 


200 


SEDIMENTARY  PETROGRAPHY 


Methods  of  computing  the  correlation  coefficient  are  given  in  Chapter  9. 

It  is  not  necessary  in  every  case  to  compute  the  correlation  coefficient, 

because  inspection  of  the  graph  often  indicates  whether  some  relation 

exists,  although  it  may  not  indicate  the  exact  nature  of  the  relation. 

It  is  usually  safe  to  conclude  that  there  is  no  fixed  relation  between 

the  variables  if  the  points  scatter  widely  over  the  field. 

Figure  89  is  included  here  as  an  example  of  a  scatter  diagram.  It 

shows  the  relation  between  the  geometric  mean  size  and  the  average 

degree  of  roundness  of  samples  of 
beach  pebbles.  The  data  in  this 
scatter  diagram  are  used  in  Chap- 
ter 9  as  an  example  for  computing 
the  correlation  coefficient. 


GRAPHS  I  N  V  O  I.  \'  I  N  G 

THREE   OR   MORE 

VARIABLES 


0 

on 

0 

0 
0 

0 

0 

s? 

0 

i 

0 

0 

t 

0 

° 

<uo 

GEOMETRIC   MEAN  SIZE  IN  MM. 


Fig.  89. — Scatter  diagram  of  average 
roundness  and  geometric  mean  size  of 
beach  pebbles. 


Strictly  speaking,  each  variable 
involved  in  a  graph  represents  a 
dimension,  and  the  problem  in- 
volved in  plotting  three  or  more 
variables  becomes  one  of  indicating  more  than  two  dimensions  on  a  sheet 
of  paper.  There  are  numerous  devices  for  accomplishing  this.  The  most 
familiar  example,  perhaps,  is  a  contour  map,  in  which  three  variables 
are  involved,  two  of  them  (length  and  breadth)  being  considered  as 
independent  variables,  and  the  third  (elevation)  being  the  dependent 
variable.  The  contour  map  is  itself  an  illustration  of  a  broader  type  of 
diagram,  the  three-variable  surface,  and  the  discussion  may  appropri- 
ately begin  wtih  it. 

Three-variable  surfaces.  Any  three  variables  may  be  plotted  against 
each  other  along  three  axes,  one  of  which  is  vertical,  and  the  other  two 
horizontal  but  at  right  angles  to  each  other.  One  of  the  varialjles  may 
be  independent  and  the  other  two  dependent,  or  two  may  be  independent 
and  the  third  dependent,  depending  upon  the  particular  situation.  In 
sedimentary  studies  perhaps  the  most  common  application  of  surfaces 
is  to  areal  studies,  in  which  length  and  breadth  are  the  independent 
variables,  and  the  other  variate  (size,  mineral  content,  organic  content, 
or  any  other  measurable  attribute)  is  the  dependent  variable.  The  usual 
procedure  in  such  cases  is  to  plot  distances  along  the  x-  and  y-axes. 


GRAPHIC  PRESENTATION 


201 


^-^ 


and  the  dependent  variable  along  the  vertical  r-axis.  This  most  general 
manner  of  plotting  the  points  must  be  modified  for  use  on  graph  paper, 
which  has  no  vertical  dimension.  Isometric  projection  paper,  which  has 
one  vertical  axis  and  two  diagonal  axes  at  120°  to  each  other,  serves 
the  purpose.  Such  figures,  illustrated  in  Figure  90,  are  not  convenient 
for  immediate  visualization,  especially  if  the  points  are  to  be  connected 
by  a  series  of  lines  of  equal  value.  For  the  latter  purpose  the  plane  of 
the  paper  is  taken  as  the  x-y  plane,  and  the  rr-axis  is  implied  by  contour 
lines  which  show  the  configuration  of  the  surface  at  ever\'  point  as  a 
projection.  The  general  term  applied 
to  such  contoured  surfaces  is  iso- 
plcth  maps,  which  may  include  any 
variable  as  the  surface  form. 

Isoplcth  maps.  An  isopleth  may 
be  defined  as  a  line  of  equal  abun- 
dance or  magnitude.  An  isopleth 
map,  therefore,  shows  the  areal  dis- 
tribution of  a  variable  in  terms  of 
lines  of  equal  magnitude.  A  common 
example  is  an  isopleth  map  of  aver- 
age size  of  sediment.^  Figure  91  is 
such  an  illustration.  The  sampling  grain  size  of  beach  sand.  The  vertical 
,  ,         ,  ...  ,     lines   represent   the   median  diameters, 

pomts  along  a  beach  are  mdicated    jhe  grid  spaces  are  10  ft.  on  edge, 
by  dots,  and  the  average  size  of  each 

sample  is  indicated  at  the  appropriate  sampling  site.  Isopleths  are  drawn 
through  points  of  equal  magnitude,  or  are  interpolated  between  the  sam- 
pling points.  The  result  is  a  surface  which  expresses  simply  and  clearly 
the  areal  variation  of  average  size. 

Isopleth  maps  may  be  prepared  with  any  variable  which  shows  a  con- 
tinuous gradation  of  value.  Practically  all  sedimentary  data,  as  far  as 
present  researches  extend,  are  continuous,  and  the  areal  variation  of 
such  items  as  average  size,  degree  of  sorting,  shape  of  particle,  hea\y 
mineral  content,  organic  content,  porosity,  and  the  like  may  be  con- 
veniently represented  in  this  manner.  The  increasing  use  of  areal  sets 
of  samples  suggests  that  sedimentary  petrologists  may  make  more  ex- 
tensive use  of  such  maps  in  the  future.  Not  only  are  isopleth  maps 
useful  for  depicting  the  areal  variation  of  sedimentary-  characteristics, 

1  Trask  {Econ.  Geology,  vol.  25.  pp.  581-599,  1930)  suggested  the  term  median 
viap  for  areal  representations  of  the  median  grain  size.  Shepard  and  Cohee  (Geol. 
Sac.  Am.  Bull.,  vol.  47,  pp.  441-458,  1936)  substituted  the  term  iso-tnegathy  map 
for  such  areal  representations  of  the  median. 


Fig.  90. — Isometric  surface  of  median 


202  SEDIMENTARY  PETROGR.\PHY 

but  they  may  be  used  for  comparative  puqioses.  By  drawing  the  maps  on 
translucent  paper,  one  map  may  be  superimposed  on  the  other  and  areal 
relations  sought.  Likewise  it  is  possible  to  prepare  ratio  isopleth  maps, 
in  which  the  ratio  of  size  to  sorting,  say,  for  each  sample  is  computed 
and  the  results  are  plotted  as  a  map.  In  such  cases  irregularities  in  the 
surface  may  furnish  clues  to  changed  conditions,  or  to  sampling  errors, 
and  thus  suggest  areas  for  more  detailed  study. 

In  the  construction  of  isopleth  maps  it  is  conventional  to  use  arith- 
metic scales  throughout.  This  is  not  necessar}%  of  course,  when  isometric 
paper  is  used  to  plot  a  perspective  view  of  the  data,  but  the  use  of  such 

LAtTE    MICHIOAN 


Fig.  91. — Portion  of  a  median  map  of  beach  sand.  Data  are  the  same  as  in  Figure  90. 
The  contour  intenal  is  o.oi  mm. 

involved  graphs  may  likely  find  its  widest  use  in  future  studies  of  the 
functional  relations  among  the  variables.  It  is  possible  to  use  a  logarith- 
mic scale  for  the  dependent  variable  by  means  of  an  indirect  device. 
That  is,  the  logarithms  of  the  variable  may  be  plotted  at  the  sampling 
points,  and  lines  of  equal  logarithmic  magnitude  drawn  through  them. 
This  device  was  used  in  a  study  of  beach  pebbles  by  Krumbein  and 
Griffith,^  who  plotted  the  logarithms  of  average  size  on  an  isopleth  map. 
Triutigle  diagrams.  An  effective  device  for  comparing  three  variables 
is  the  use  of  triangle  graph  paper.  An  illustration  may  serve  as  the 
simplest  manner  of  indicating  the  use  of  such  paper  for  mechanical 
analysis  data.  The  analjtical  data  are  arranged  into  three  groups  for 
each  sample,  percentages  respectively  of  sand,  silt,  and  clay.  The  result 
is  three  numbers  for  each  sample,  and  these  numbers  are  used  for  plot- 
ting. The  three  vertices  of  the  triangle  are  labeled,  one  for  each  of  the 
three  variables.  Assume  that  a  sample  has  the  following  composition  • 
26  per  cent  sand,  43  per  cent  silt,  and  31  per  cent  clay,  a  total  of  100 
per  cent.  To  plot  this,  one  locates  the  point  which  lies  26  units  upward 

1  W.  C.  Krumbein  and  J.  S.  Griffith,  Beach  environment  at  Little  Sister  Bay, 
Wisconsin:  Geol.  Soc.  Am.,  Bull.,  vol.  49,  pp.  629-652,  1938. 


GRAPHIC  PRESENTATION 


203 


Z 


7^ 


/\ 


/  /' 

^0/    \/\ 

/ 

^(/7\\ 

/ 

?;'°/ 

A/  /A 

Fig.  92. — Triangle  diagram,  show- 
ing method  of  locating  point. 


along  a  vertical  axis,  43  units  along  the  axis  joining  the  silt  vertex  and 
the  opposite  side  of  the  triangle,  and  31  units  along  the  third  axis.  The 
result  is  a  single  point  as  shown  in  Figure  92. 

Triangle  diagrams  have  found  wide 
use  in  sedimentary  studies,  not  only 
for  plotting  size  attributes,  but  also  for 
mineral  attributes.  The  heavy  minerals 
in  sediments  may  be  classified  accord- 
ing to  their  ultimate  origin  from 
igneous,  sedimentary,  and  metamorphic 
rocks.  A  triangle  plot  of  the  result  will 
indicate  the  relative  contributions  of 
each.  In  using  triangle  paper,  the  three 
values  must  be  expressed  as  percent- 
ages totaling  100  per  cent,  because  sin- 
gle points  will  only  result  when  parts 
per  hundred  are  plotted. 

Another  common  use  of  triangle  diagrams  is  to  subdivide  the  field 
into  classes  for  descriptive  purposes.  Gessner  ^  used  such  graphs  to 
classify  soils  into  groups,  such  as  sand,  silt,  sandy  clay,  and  the  like. 

Figure  93  illustrates  the  method 
CLAY  of  subdivision  used.  Each  vertex 

is  chosen  in  terms  of  a  primary 
constituent,  and  the  field  sub- 
divided into  groups.  It  may  be 
seen  that  such  devices  offer  log- 
ical methods  for  describing  sedi- 
ments in  terms  of  fixed  percent- 
ages of  material,  and  in  a  manner 
such  that  the  relation  of  any 
group  to  the  others  is  immedi- 
ately apparent. 

It  is  also  possible  to  use  tri- 
angle graphs  to  plot  four  vari- 
ables. For  that  purpose  the  four 
variables  are  recalculated  to  100 
per  cent,  and  three  of  them  are  plotted  as  before.  The  result  will  be  not 
a  point  in  the  field,  but  an  area  enclosed  by  three  lines,  each  perpendicu- 
lar to  the  respective  axes.  These  three  lines  form  a  small  triangle,  the 
size  of  which  indicates  the  amount  of  the  fourth  variable.  Figure  94 
1  H.  Gessner,  Die  Schldmwmialyse  (Leipzig,  1931),  p.  217. 


SAND 


SILT 


Fig.  93. — Triangular  field  showing  clas- 
sification of  soils.  After  Lakin  and  Shaw, 
1936. 


204 


SEDIMENTARY  PETROGR.\PHY 


SILT 


CLAY 


Fig.  94. — Method  of  plotting  four 
variables  on  triangle  diagram.  The 
size  of  the  small  triangle  indicates  the 
amount  of  gravel  in  the  sediment. 


illustrates  the  plotting  of  a  sediment  composed  of  16  per  cent  sand,  43 
per  cent  silt,  30  per  cent  clay,  and  1 1  per  cent  gravel.  The  first  three  are 
plotted  on  the  three  axes ;  the  fourth  variable  (gravel)  is  indicated  by  the 

small  triangle. 

Ratio  charts.  Various  types  of 
charts  and  graphs  have  been  devel- 
oped to  indicate  the  ratios  between 
variables.  One  may  wish  to  compare 
the  relative  abundance  of  garnet  and 
hornblende  in  a  series  of  samples  col- 
lected along  a  traverse.  A  simple  de- 
vice for  presenting  these  data  is  to 
choose  a  horizontal  distance  scale  and 
at  a  given  distance  above  this  line  to 
draw  a  parallel  line  which  represents 
the  amount  of  garnet  present,  called 
unity  for  convenience.  The  ratio  of 
hornblende  to  garnet  is  computed  for 
each  sample;  if  the  garnet  has  the  frequency  value  24  (percentage,  num- 
ber of  grains,  or  the  like)  and  hornblende  has  the  frequency  value  10, 
the  ratio  10/24  =  0.42  furnishes  a  point  to  be  drawn  at  the  scale  value 
0.24,  using  the  garnet  line  as  tmity.  Values  of  the  ratio  larger  than  i 

are    plotted    above    the    unit}'    line.  

The  result  is  a  curve  which  varies 
above  or  below  the  garnet  line  and 
indicates  the  variation  of  hornblende 
to  garnet,  on  the  assumption  that  the 
garnet  frequency  is  fixed.  Figure  95 
illustrates  such  a  chart,  as  used  by 
Pettijohn^  in  comparing  the  ratio 
of  hornblende  to  garnet.  Here  the 
garnet  line  was  chosen  as  100.  and 
a  logarithmic  scale  was  used  on 
the  vertical  axis.  The  logarithmic 
axis  serves  to  decrease  the  abso- 
lute   range    of    the    values    and    is 


Fig.  95. — Graph  of  hornblende-gar- 
net ratio  (garnet  =  100).  After  Pet- 
ti John,  1 93 1. 

suitable    when    wide    fluctuations 


1  F.  J.  Petti  John,  Petrography  of  the  beach  sands  of  souther 
Jour.  Geology,  vol.  39,  pp.  432-455.  I93i- 


Lake  Michigan: 


GRAPHIC  PRESENTATION 


205 


Miscellaneous  Graphic  Devices 

In  many  instances  it  is  desirable  to  show  a  number  of  related  phe- 
nomena on  a  single  chart,  without  regard  to  the  arrangement  of  the 
data  according  to  variables  or  axes.  Among 
the  wide  variety  of  such  charts  which  are 
available,  three  will  be  mentioned. 

Bar  charts.  These  charts  are  simply  con- 
structed by  choosing  a  vertical  scale  of  fre- 
quency (percentage,  number,  amount)  and 
representing  the  several  variables  to  be  com- 
pared as  vertical  bars,  all  of  the  same  width 
(Figure  96).  Bar  charts  are  not  suitable  for 
detailed  analysis,  but  they  serve  for  rapid 
comparisons. 

Pie  diagrams.  Pie  diagrams  are  used  for 
the  same  purposes  as  bar  charts,  except  that 
segments  of  a  circle  indicate  the  relative 
magnitudes.  Figure  97  shows  the  same  data 
as  the  bar  chart,  arranged  in  a  pie  diagram. 
Such  diagrams  are  useful  for  rough  presentation  and  for  popular  illus- 
trations. 


J  llnnT 


u  10 


5      3      5      i      f      S 
§     t:     d     !     5     5 

Fig.  96. — Bar  chart  of  av- 
erage composition  of  shale. 
(Data  from  Clarke,  U.  S. 
G.  S.  Bulletin  770.) 


Sailboat  aud  star  diagrams.  Sailboat  and  star  diagrams  have  been  used  to 
illustrate  the  cliemical  composition  of  igneous  rocks, ^  and  they  are  applicable 
to  a  number  of  situations  in  sedimentary 
data.  In  their  construction  a  central  point  is 
chosen,  from  wliich  radiate  as  many  lines  as 
tliere  are  variables  to  be  compared.  Along 
each  of  these  a  length  is  laid  off  proportion- 
ate to  the  magnitude,  and  the  termini  of  the 
lines  are  joined,  yielding  a  figure  similar  to 
a  sailboat. 


M  A  T  H  E  M  A  T  I  C  A  L 
OF      GRAPHIC 


A  N  A  L  Y  S  I  S 
DATA 


Fig.  97. — Pie  diagram  of  same 
data  as  in  Figure  96. 


Graphs  and  charts  are  seldom  an  end  in 
themselves  but  are  used  to  draw  conclu- 
sions from  data,  to  investigate  the  relations  or  lack  of  relations  among 
variables,  or  merely  to   simplify  text  descriptions.    Conclusions   to   be 


J.  F.  Kemp,  A  Haudbook  of  Rocks  (New  York,  191 1). 


206 


SEDIMENTARY  PETROGRAPHY 


drawn  from  data  are  often  clarified,  however,  if  a  definite  mathematical 
relation  can  be  found  between  the  variables.  When  such  a  relation  is 
found,  it  is  possible  to  investigate  the  geological  implications  more  pre- 
cisely, because  behind  each  mathematical  function  is  a  set  of  conditions 
which  must  be  true  if  the  relation  holds. 

The  detailed  examination  of  curves  based  on  experimentally  deter- 
mined points  is  beyond  the  scope  of  this  book,  and  there  are  available 
a  number  of  excellent  treatises  on  the  subject.^  However,  there  are 
several  fairly  simple  relations  which  may  be  encountered  in  sedimentary 
studies. 

Special  attention  will  be  given  to  three  common  types  of  functions  of 

two  variables,  applicable  to  scatter 
diagrams,  or  graphs  with  time  or  dis- 
tance as  independent  variables.  These 
three  functional  relationships  are  (a) 
linear  functions,  (b)  power  func- 
tions, and  (c)  exponential  functions ; 
they  are  easily  recognized  by  the  fact 
that  they  plot  as  straight  lines  respec- 
tively on  ordinary  graph  paper,  dou- 
ble logarithmic  graph  paper,  and 
semi-logarithmic  graph  paper. 

Linear  functions.  A  linear  function 
of  two  variables  may  be  defined  as 
such  a  relation  that  a  change  in  one 
variable   induces   an   exactly   propor- 
FiG.  98.— Flow  of  water  through    tional   change   in  the   other  variable. 
SrrnrMS!' .IS  """"""■""■   This  consta„.  proportionality  results 

in  a  straight  line  graph  when  one 
variable  is  plotted  against  the  other  on  ordinary  graph  paper.  An  exam- 
ple of  such  a  function  is  given  by  Plummer,  Harris,  and  Pedigo,^  in 
connection  with  the  flow  of  water  and  other  fluids  through  sandstone. 
By  plotting  the  volume  of  water  that  flowed  through  a  rock  core  per 
unit  of  time  under  given  pressure  differences,  a  straight  line  was  ob- 
tained, as  shown  in  Figure  98. 

The  fundamental  characteristic  of  a  linear  function  is  that  the  rate 
of  change  of  the  dependent  variable  with  respect  to  the  independent 
variable  is  constant.  In  the  calculus  this  is  expressed  In-  the  relation 

1  See  for  example  C.  O.  Mackey,  Graphical  Solutions  (New  York,  1936)  ;  also 
T.  R.  Running,  Empirical  Formulas  (New  York,   1917). 

-  F.  B.  Plummer,  S.  Harris  and  J.  Pedigo,  A  new  multiple  permeability  appara- 
tus :  Am.  Inst.  Min.  and  Met.  Eng.  Tech.  Pub.  578,  1934. 


PRESSURE    DIFFERENCE  IN  ATMOSPHERES 


GRAPHIC  PRESENTATION  207 

dy/dx^  const.  If.  in  the  example  cited,  volume  in  cubic  centimeters 
per  second  is  chosen  as  the  dependent  variable,  it  must  suffer  a  constant 
rate  of  change  with  respect  to  the  pressure  difference :  at  twice  a  given 
pressure  difference  the  quantity  must  be  doubled,  etc.,  for  the  range  in 
which  the  linear  function  holds.  The  fact  that  the  line  ascends  to  the 
right  indicates  that  the  constant  in  the  differential  equation  d\/dx^ 
const,  is  positive. 

Linear  functions  are  the  simplest  mathematical  relations  which  exist 
between  variables,  and  it  may  be  anticipated  that  among  sedimentary 
data  in  general  they  will  not  be  as  common  as  other  functions,  because 
of  the  large  number  of  factors  which  are  present  in  most  sedimentary 
situations.  In  the  present  case  the  occurrence  of  a  linear  function  is  due 
to  the  laminar  motion  of  the  fluid  through  the  sandstone,  in  accordance 
with  Darcy's  Law. 

If  one  is  interested  in  determining-  the  actual  analytical  expression  involved 
in  the  linear  function,  he  may  read  it  directly  from  the  graph.  The  equation 
of  any  straight  line  may  be  written  as  _v  =  ax  +  h,  where  a  is  the  slope  of 
the  line  and  b  is  the  point  where  the  line  crosses  the  v-axis.  Both  of  these 
values  are  constants.  To  determine  a,  especially  when  the  scale  units  are  not 
equal  in  length,  it  is  perhaps  best  to  choose  two  convenient  points  on  the 
line  rather  far  apart,  and  divide  the  change  in  y  by  the  increase  in  .r.  This 
quotient,  expressed  as  a  decimal,  will  at  once  furnish  the  value  of  a.  Likewise, 
h  is  determined  simply  by  reading  tlie  value  of  the  point  where  the  line 
crosses  the  v-axis.  In  the  example  given,  a  change  of  10  units  in  volume 
occurs  in  an  increase  of  0.6  units  of  pressure  difference.  Thus  the  value  of 
a  is  10/0.6  =  16.7.  The  value  is  positive  because  the  curve  rises  to  the  right. 
The  value  of  h  is  zero  because  the  curve  passes  through  the  origin,  and  hence 
the  equation  of  the  line  is  y  =  iG.jx.^ 

Pozver  functions.  A  power  function  of  two  variables  may  be  defined 
as  such  a  relation  that  if  the  independent  variable  is  changed  by  a  fixed 
multiple,  the  dependent  variable  will  also  change  by  a  fixed  multiple. 
The  general  equation  for  such  a  function  is  y  =  a.r°  where  a  is  a  con- 
stant and  n  is  either  positive  or  negative,  a  whole  number  or  a  fraction.- 
Thus  both  parabolas  and  hyperbolas  are  included,  the  latter  when  n 
is  negative.  From  the  nature  of  the  function  it  may  be  seen  that  if 
logs  are  taken  of  both  sides  of  the  equation,  the  expression  log  y^ 

1  The  actual  case  has  been  simplified  in  this  example.  The  y-intercept  has  a  small 
value  because  some  pressure  difference  is  required  in  practice  before  liquid  flows 
through  the  core.  Likewise  the  value  a  =  16.7  cannot  be  used  directly  for  the  co- 
efficient of  permeability.  The  expression  given  is  merely  an  empirical  statement  of 
the  relation  between  y  and  x. 

2  \Mien  H=  I,  the  function  is  linear,  which  is  thus  a  special  case  of  the  power 
function. 


208 


SEDIMENTARY  PETROGRAPHY 


n  log  X -\-  log  a  results.  This  means  that  if  the  original  data  are  plotted 
on  double  logarithmic  paper  the  graph  will  be  a  straight  line. 

The  presence  of  a  power  function  requires  that  the  rate  of  change 
of  the  dependent  variable  must  itself  dei^end  on  the  value  of  the  in- 
dependent variable,  or  on  some  ix)wer  of  it.  Thus  in  a  simple  parabola, 
the  rate  of  change  of  y  will  be  proportional  to  x,  dy/dx  =  mx,  where  m 
is  a  constant. 

An  example  of  a  power  function,  perhaps  not  strictly  sedimentary  in 
nature,  occurs  in  connection  with  the  probable  error  of  collecting  sam- 
ples. This  probable  error,  discussed  earlier,  was  found  to  decrease  as 
the  number  of  samples  in  a  composite  was  increased.  By  plotting  the 

values  of  Table  3  (page  41)  on 
double  logarithmic  paper,  the 
straight  line  shown  in  Figure  99  re- 
sults, demonstrating  that  the  rela- 
tion is  a  power  function.  Moreover, 
I  '[""l^jsl      the  fact  that  the  line  descends  to  the 

right  indicates  that  the  exponent  n 
is  negative,  and  hence  the  relation  is 
a  hyperbola  rather  than  a  parabola. 

The  determination  of  the  analytical 
I  2         3      4     6    6  7  8  910      expression  for  the  curve  is  more  com- 

"""  plicated  with  power  functions  in  gen- 

FiG.  99.— Double  logarithmic  graph  ^^al  than  with  linear  functions.  How- 
of  E„/E  =  1/  V  n.  (Data  from  Table  g^.^^,  in  the  present  case  the  relation  is 
3,  page  41.;  fairly  simple.    Inasmuch   as   the  loga- 

rithmic units  are  of  the  same  length  on  both  axes,  the  value  of  n  may  be  deter- 
mined very  simply  by  finding  the  tangent  of  the  angle  of  slope  of  the  straight 
line.  This  angle  is  26.5°  measured  in  a  clockwise  direction.  The  tangent  of 
26.5°  is  0.50,  and  since  the  line  descends  to  the  right,  n=  —  0.5  =  —  ^. 
Also,  the  line  intersects  the  axes  at  the  point  (i,  i),  and  hence  the  value  of  a 
is  unity.  Thus  the  equation  is  y  =  jir~^,  or  y  =  i/\/  x. 

The  implications  of  a  hyperbolic  function  are  that  the  value  of  the  de- 
pendent variable  must  decrease  in  fixed  ratio  as  the  independent  variable 
increases  in  a  fixed  ratio.  In  the  present  case  it  may  be  seen  that,  as  a  result 
of  this  property,  the  value  of  the  probable  error  decreases  rapidly  at  first  and 
then  more  slowly,  as  the  number  of  samples  in  a  composite  increases. 

In  physics  parabolic  and  hyperbolic  functions  are  numerous ;  as  an 
illustration  it  may  be  mentioned  that  Stokes'  law,  v  =  Cr-,  is  a  parabolic 
function. 

Exponential  functions.  An  exponential  function  may  be  defined  as  a 
relation  in  which  the  dependent  variable  increases  or  decreases  geometri- 


1.0 

^ 

^\ 

0.7 

ae 

05 

OA 

03 
02 

01 

\ 

\ 

K^^ 

^ 

S 

^ 

s 

s 

GRAPHIC  PRESENTATION 


209 


cally  as  the  independent  variable  increases  arithmetically.  All  exponen- 
tial functions  may  be  written  as  v  ^  mb",  where  m,  b,  and  a  are  con- 
stants. In  the  nature  of  the  case,  m  is  always  the  value  of  y  at  the  origin, 
and  b  is  usually  chosen  as  a  certain  constant  ^  =  2.7182...,  the  base 
of  natural  logarithms.  Hence  it  is  common  convention  to  express  ex- 
ponential functions  as  y  =  yoc",  where  a  may  be  either  positive  or  nega- 
tive. If  logs  are  taken  of  both  sides  of  the  expression,  there  results 
log  y  =  log  yo  -f-  ax  log  e.  By  taking  the  logs  to  the  base  e,  the  expres- 
sion simplifies  to  loge  y  =  ax-{-  log^  yo-  If  V  and  Vo  are  combined  into  a 
single  term,  it  may  be  seen  that  a  log  appears  on  the  left,  but  none  on  the 
right:  \og{y/yo)  =ax.  Thus  if  an  ex- 
ponential function  is  plotted  on  semi- 
logarithmic  paper,  the  points  will  lie  on 
a  straight  line.  Hence  it  is  only  neces- 
sary to  plot  the  observed  values  on  such 
paper  to  determine  whether  an  exponen- 
tial function  is  involved. 

Exponential  functions  may  be  found 
to  occur  rather  commonly  in  sedimen- 
tary situations.  Krumbein  ^  discussed 
several  negative  exponential  functions, 
and  the  following  example  is  repeated 
here.  The  average  size  of  beach  pebbles 
was  determined  at  several  points  along 
a  beach,  and  the  data  were  plotted  on  semi-logarithmic  paper.  A  straight 
line,  descending  to  the  right,  was  found  as  shown  in  Figure  100.  This 
demonstrates  that  the  equation  is  of  the  type  y  =  y^e—",  and  several  in- 
teresting properties  follow  from  the  nature  of  the  function.  In  any  expo- 
nential function  of  this  type,  it  is  necessary  that  the  rate  of  change  of 
the  dependent  variable  be  proportional  to  the  value  of  the  dependent 
variable  at  any  given  point,  dy/dx  =  — ay,  where  a  is  constant  and  y  is 
the  dependent  variable.  This  means,  in  a  negative  exponential,  that  if 
the  pebbles  are  being  worn  down  as  they  move  along  the  beach,  the 
rate  of  wear  is  proportional  to  the  average  size  of  the  pebble.  Com- 
plexities which  may  enter  into  the  interpretation  of  exponential  func- 
tions are  discussed  in  the  original  paper. 


1        1        1 

80 

!                      ' 

60 

'           1 

1 

t 

40 

Sx 

X 

i           1 

20 

in 

^ 

N.i       i 

200        300        400         SCO     SOO 

DISTANCE      IN  FEET 


Fig.  100.  —  Semi-logarithmic 
graph  of  geometric  mean  size  of 
pebbles  as  a  function  of  distance 
along  the  beach. 


In  considering  exponential  functions  anahtically,  the  problem  reduces  to 
the  determination  of  the  constant  a.  In  the  example  cited,  the  function  is  a 


1  W.  C.  Krumbein,  Sediments  and  exponential  curves :  Jour.  Geology,  vol.  45,  pp. 
577-601,  1937. 


2IO 


SEDIMENTARY  PETROGRAPHY 


negative  exponential,  and  the  following  treatment  indicates  the  steps  in- 
volved. 

There  are  several  methods  for  determining  a;  one  of  the  most  convenient 
is  a  simple  analytical  method  in  which  it  is  only  necessary  to  find  the  value 
of  X  at  the  point  where  y  is  reduced  to  half  its  original  value.  In  that  case 
y/\\  =  Yz,  and  the  original  equation  becomes  ^  =  c  — ".  By  taking  logs  of 
tliis  expression  to  the  base  e  and  changing  the  sign,  there  results  —  loge(H) 
=  ax.  But  —  loge(>^)  =  loge2  =  0.693,  ^^d  hence  a  =  o.693/.r,  where  the 
value  of  X  is  called  the  '"half  distance"  and  corresponds  to  the  point  where 
_V  is  half  its  original  value.  In  the  present  case  3V,  is  52  mm.,  and  the  .r-value 
at  the  point  where  y  =  26  mm.  (half  its  original  value)  is  260  ft.,  as  deter- 
mined from  Figure  100.  The  unit  of  distance  was  chosen  as  100  ft.  for  con- 
venience, so  that  the  half-distance  value  of  x  is  2.6.  Placing  this  value  in  the 
equation  just  given,  we  obtain  a  =^  0.693/2.6  =  0.26.  Hence  the  required 
equation  for  pebble  size  is  y  —  52  £'~°-*^^. 

It  is  customary  to  refer  to  a  as  the  coefficient  of  the  physical  attribute 
being  considered ;  in  the  present  case  the  value  a  =  0.26  may  be  called  the 
coefiicient  of  pebble  size. 

Table  19 
Properties  of  Linear,  Power,  and  Exponential  Functions 


Function 

Rate  of  Change  of  Dependent 
Variable 

Nature  of  Curve 

Linear 

dy/dx  =  const.,  and  hence  slope 
of  curve  is  constant 

Straight  line,  with  fixed  slope. 
Plots  as  straight  line  on  or- 
dinary arithmetic  graph  pa- 
per 

Power 

dy/dx  =  »!.rP,  and  hence  slope 
of  curve  is  dependent  on  x 

Parabolas  and  hyperbolas.  Plot 
as  straight  lines  on  double 
log  paper.  Parabolas  rise  to 
right,  hyperbolas  descend  to 
right 

Exponential 

dy/dx  =  ay,   and   hence    slope 
of  curve  is  proportional  to 
the  value  of  y  at  any  given 
point 

Exponential  curves.  Plot  as 
straight  lines  on  semi-log  pa- 
per. Positive  exponentials 
rise  to  right;  negative  expo- 
nentials descend  to  right 

Comparison  of  linear,  pozcer,  and  exponential  functions.  The  three 
functions  considered  here  do  not  by  any  means  exhaust  the  possible 
mathematical  relations  which  may  be  found  in  sedimentary  situations. 
For  example,  periodic  functions,  such  as  rhythmic  variations  in  grain 


GRAPHIC  PRESENTATION  211 

size  or  the  like,  may  be  expected  to  occur,  but  tlicir  analysis  is  usually 
complex. 

Perhaps  the  most  significant  comparison  that  may  l)e  made  of  the 
three  functions  is  in  terms  of  their  rates  of  change.  This  topic  was 
mentioned  in  connection  with  each  function,  but  for  comparison  they 
are  shown  in  Table  IQ,  which  summarizes  the  mathematical  proi>erties 
of  the  functions.^ 

In  any  given  case,  the  physical  significance  of  the  constants  in  the 
equations  depends  partly  on  the  geological  set-up  of  the  data.  The 
mathematical  implications  are  fixed,  but  the  interpretation  placed  on 
the  data  depends  on  the  individual  case.  In  any  event,  the  physical  in- 
terpretations must  not  violate  any  of  the  mathematical  principles  in- 
volved. 

[Many  workers  in  sedimentary  petrology  question  the  value  of  mathe- 
matical analysis,  owing  to  the  large  number  of  variables  and  errors 
which  are  involved  in  the  simplest  situation.  In  general,  this  may  be 
granted  in  tlie  present  stage  of  development  of  the  science,  but  it  is 
equally  true  that  an  approximate  determination  of  the  nature  of  the 
functional  relationship  may  point  the  way  for  more  rigorous  studies  of 
problems  and  eventually  establish  underlying  principles  of  universal 
application.  Work  in  sediments  has  already  reached  a  fair  state  of  rig- 
orous analysis  in  some  connections,  and  it  seems  that  elementary  tyi:>es 
of  mathematical  analysis  at  least  may  be  applied  in  cases  where  graphic 
presentation  suggests  simple  functional  relationships. 

1  An  excellent  table  which  includes  numerous  empirical  functions  and  the  con- 
ditions under  which  they  plot  as  straight  lines  is  given  by  C.  O.  Mackey,  op.  cit. 
(1936),  p.  117. 


CHAPTER   8 
ELEMENTS  OF  STATISTICAL     ANALYSIS 

INTRODUCTION 

A  THOROUGH  laboratory  study  of  sediments  includes  quantitative  data 
on  the  sizes,  the  shapes,  the  mineral  composition,  the  surface  textures, 
and  perhaps  the  orientation  of  the  grains.  These  fundamental  data  are 
related  to  the  physical  and  chemical  factors  in  the  environment  of  depo- 
sition. To  relate  characteristics  with  environment  one  may  investigate 
the  areal  variation  of  the  sediment,  which  implies  the  comparison  of  one 
sample  with  the  next.  This  comparison  is  most  conveniently  accom- 
pHshed  by  means  of  statistical  analysis. 

The  word  statistics  is  defined^  as  "The  science  of  the  collection  and 
classification  of  facts  on  the  basis  of  relative  number  or  occurrence  as  a 
ground  for  induction  i_sysLematic  compilation  of  instances  for^e  in- 
ference of  general^ruths."  This  definition  shows  that  the  study  of  sedi- 
ments is  largely  statistical  in  nature.  Sedimentary-  petrologists  are  inter- 
ested in  the  collection  and  classification  of  sedimentan,-  data  as  a  basis 
for  inferences  about  sediments.  Mechanical  analysis  is  concerned  with 
the  ranges  of  diameters  present,  and  the  relative  abundance  of  particles 
in  each  diameter  range.  This  is  clearly  a  statistical  operation.  The  analy- 
sis of  particle  shape,  of  mineral  content,  and  of  particle  orientation  are 
all  concerned  with  the  collection  of  facts  in  terms  of  the  number  of 
occurrences  of  any  particular  attribute. 

Statistical  technique  may  be  divided  into  several  operations.  The  first 

/  /  operation  is  the  collection  and  classification  of   data.   In  sedimentary 

terms  this  refers  to  mechanical  analysis,  mineral  analysis,  and  so  on. 

,  ^    The  second  step  may  be  the  presentation  of  the  data  in  the  Jonn  of 
tables^ind^graphs.  Finally,  the  data  themselves  may_be^  jnalyzed  sta- 

,  ^    tistically,  and  from  the  values  obtained,  inferences  may  be  drawn  about 
the  sediment. 

Several  approaches  to  statistical  analysis  are  possible,  depending  on 
the  nature  of  the  data.  A  statistical  series  involving  magnitude  (as  size 

1  Webster's  Xcu'  Inteniatio7ial  Dictionary  (Springfield,  Mass..  1926). 

212 


ELEMENTS  OF  STATISTICS  213 

of  grains,  or  percentage  of  heavy  minerals),  is  called  a  frequency  dis- 
tribution. If  geographic  location  is  involved  (as  in  the  comparison  of 
samples  over  the  areal  extent  of  a  formation),  the  statistical  series  is 
called  a  spatial  distribution.  If  time  is  an  imjxjrtant  factor  (as  in  the 
changes  of  sedimentary  characteristics  as  time  goes  on),  the  statistical 
series  is  called  a  time  series.  Each  of  these  cases  is  of  importance  in 
the  study  of  sediments. 


THE     CONCEPT     OF     A     FREQUENCY     DISTRIBUTION 

The  discussion  of  frequency  distributions  will  be  confined  to  size 
frequency  distributions  (mechanical  analysis  data),  although  it  should 
he.  borne  in  mind  that  the  same  principles  may  apply  to  the  study  of 
mineral  distribution  or  the  shape  distribution  of  particles  in  a  sediment. 
In  all  size  frequency  distributions  there  are  two  principal  variables, 
"size"  and  frequency.  The  frequency  distribution  itself  is  simply  the 
arrangement  of  the  numerical  data  according  to  size.  Size  is  considered 
to  be  the  independent  variable,  and  frequency  the  dej^endent  variable. 
This  choice  means  that  frequency  is  a  function  of  size,  expressed  as 
y^^f{x),  where  y  is  the  frequency  and  f{x)  is  some  function  of  size.  S.  «»-v(.> 
By  convention  any  graph  of  the  frequency  distril)Ution  is  drawn  with  ^ 

size  (diameters  in  millimeters  or  any  numbers  representing  size,  such 
as  the  logarithms  of  the  diameters)  plotted  along  the  horizontal  A"-axis, 
and  frequency  (percentage  by  weight  or  by  number  or  any  other  sym- 
bol representing  frequency)   plotted  along  the  vertical  ^/-axis.  '^V 

Frequency  distributions  may  be  of  two  types.  A  discrete  series  is  one<:  1 
in  which  the  independei]t  jyariable  increases  by  finite  increments,  with  ^    ^ 

no  gradations  between.  A  pile  of  coins,  made  of  pennies,  nickels,  dimes,  **^'*t 
and  quarters,  if  assembled  into  a  frequency  distribution  by  counting  the 
number  of  each  coin  present,  constitutes  such  a  discrete  scries.  In  discrete 
series  each  value  of  the  variable  (in  coins  \f,  50,  lo^,  etc.)  is  a  separate 
group  of  items,  so  that  drawing  a  smooth  curve  through  the  data  is 
quite  erroneous.  The  second  type  of  frequency  distribution  is  the  con- 
tinuous distribution,  in  which  the  indet:)endent  variable  increases  by 
infinitesimals  along  its  range  of  values.  That  is,  if  the  individuals  were 
arranged  side  by  side,  there  would  be  complete  gradation  from  one  to 
the  next.  The  heights  of  men  form  such  a  series.  It  is  obvious  also 
that,  with  few  if  any  exceptions,  sediments  fall  within  the  class  of  con- 
tinuous distributions.  Within  a  single  sediment  there  is  a  continuous 
range  of  sizes  from  largest  to  smallest. 


214 


SEDIMENTARY  PETROGRAPHY 


In  continuous  data  there  is  no  inherent  grouping.  Whatever  classes  of 
size  are  erected  are  perfectly  arbitrary,  a  point  which  is  emphasized 
in  Chapter  4  in  the  discussion  of  grade  scales.  However,  some  sort  of 
grouping  is  necessary  in  analyzing  the  data,  so  that  frequency  may  be 
expressed  as  the  amount  ofrnaterial  within^selected  intervals  along  the 
size  scale.  Mechanical  analysis  is  the  operation  of  determining  this 
abundance  or  frequency  within  chosen  size  classes  or  grades. 

HISTOGRAMS     AS     STATISTICAL     DEVICES 


100 

50 
25 

/^ 

/ 

/ 

/ 

0 

Perhaps  the  most  common  graphic  device  used  in  presenting  frequency  data 
of  sediments  is  the  histogram,  described  in  Chapter  7.  This  simple  frequency 
diagram  is  readily  understood  and  has  a  uni- 
versal appeal  because  of  its  clarity  and  sim- 
plicity. From  the  histogram  itself  much  may 
be  learned.  In  the  first  place,  one  may  see  that 
there  is  a  particular  class  which  has  the  great- 
est frequency  of  individuals  within  it,  and 
that  the  frequency  decreases  on  either  side. 
The  class  of  greatest  frequency  is  called  the 
jiwdal  c/a.yj7aii^frbiTrthe  extent  to  which  It 
towers  above  its  neighbors  one  may  note 
whether  it  is  a  conspicuous  modal  group  or 
not.  Likewise,  from  the  rectangles  stretching 
away  on  either  side  of  the  modal  class  one 
may  see  the  range  of  size  in  the  population. 
From  the  extent  of  the  spread  one  may 
roughly  note  whether  the  tendency  is  for  the 
individuals  to  cluster  about  the  most  preva- 
lent size  or  to  spread  widely  on  either  side. 
Finally,  one  may  note  whether  or  not  the  dis- 
tribution of  individuals  on  either  side  of  the 
modal  class  is  symmetrical  or  not. 

Among  the  earliest  workers  to  use  histo- 
grams in  the  study  of  sediments  was  Udden.^ 
He  observed  that  the  histograms  of  sedi- 
ments varied  considerably,  according  to  the 
type  of  sediment  involved.  That  is,  dune  or 
beach  sands  have  well-defined  central  groups, 
whereas  histograms  of  such  sediments  as 
glacial  till  were  wide-spread  and  irregular.  Udden  defined  the  modal  class 
as  the  "maximum  grade"  and  contrasted  this  maximum  with  the  material 
on  either  side,  which  he  designated  as  the  "coarse  and  fine  admixtures." 
In  this  manner  Udden  obtained  a  sorting  factor  which  was  used  in  his 
geologic  reasoning  about  the  sediments. 

ij.  A.  Udden,  The  mechanical  composition  of  wind  deposits,  Augustaiia  Library 
Publications,  no.  i,  1898. 


/^ 

1 

-t 

\-^ 

/ 

1 

SIEVE  MESHES 

Fig.  ioi. — Two  histograms 
from  the  same  cumulative 
curve.  The  units  on  the  .r-axis 
represent  the  logarithms  of 
the  sieve  sizes. 


ELEMENTS  OF  STATISTICS  215 

Unfortunately,  the  histogram  is  influenced  by  the  class  intervals  used  in  the 
analysis,  and  its  shape  varies  according  to  the  particular  class  limits  chosen. 
Figure  loi  shows  the  same  continuous  frequency  distribution  represented 
as  a  cumulative  curve,  analyzed  according  to  two  different  grade  scales,  and 
it  may  be  noted  that  the  histograms  are  not  at  all  alike.  One  of  the  diagrams 
is  symmetrical  and  one  is  definitely  unsymmetrical.  It  would  appear  from 
this  diat  the  particular  form  assumed  by  a  histogram  is  accidental  and  de- 
pends on  the  nature  of  the  classes  used  in  the  analysis.  If  one  therefore  bases 
conclusions  on  a  given  histogram,  he  may  never  be  quite  certain  that  his 
conclusions  are  correct.  This  applies  especially  to  situations  in  which  only  a 
few  classes  are  used.  The  difficulty  widi  histograms  is  due  to  the  fact  that 
the  diagrams  attempt  to  illustrate  a  continuous  frequency  distribution  as 
though  it  were  made  of  discrete  classes.  Consequently,  the  diagram  may  not 
furnish  much  visual  information  about  the  frequency  distribution  considered 
as  a  continuous  variation  of  size.  It  is  because  of  this  that  statisticians  rec- 
ommend the  use  of  smooth  curves  to  represent  continuous  distributions. 

If  one  wishes  to  generalize  from  a  picture  of  the  frequency  distribution,  it 
is  much  safer  to  use  the  unique  frequency  curve,  because  within  small  experi- 
mental limits  the  essential  shape  of  the  distribution  will  be  brought  out  by 
the  continuous  curve.  A  graphic  method  of  obtaining  the  frequency  curve  is 
given  in  Chapter  7. 


CUMULATIVE     CURVES     AS     STATISTICAL     DEVICES 

The  difficulty  attendant  upon  the  variation  of  histograms  has  resulted  in 
the  wide  adoption  by  sedimentary  pctrologists  of  the  cumulative  curve.  Ex- 
perience has  shown  that  whereas  histograms  vary  depending  upon  the  class 
interval  used,  the  cumulative  curve  remains  fairly  constant  regardless  of 
the  particular  class  limits  used.  Within  the  limits  of  errors  of  smoothing,  the 
cumulative  curve  is  a  more  reliable  index  of  the  nature  of  the  continuous 
distribution  than  the  histogram. 

It  was  shown  in  Chapter  7  that  the  frequency  curve  is  the  limit  approached 
by  the  histogram  as  the  class  intervals  decrease  to  zero  and  die  number  of 
individuals  increases  without  bound.  By  the  calculus  it  is  possible  to  show 
that  every  continuous  curve  has  associated  with  it  an  integral  ciu've  and  a 
derivative  curve.  The  integral  curve  is  such  that  its  ordinate  at  any  point 
represents  the  area  under  the  given  curve  up  to  that  point,  whereas  the 
derivative  curve  is  such  diat  its  ordinate  at  any  point  represents  the  slope  of 
the  given  curve  at  that  point.  The  relation  is  such  that  if  one  curve  is  the 
integral  of  a  second  curve,  the  second  curve  is  itself  the  derivative  of  the 
first  curve. 

It  is  a  widely  recognized  fact  that  the  cunnilative  curve  is  the  integral  of 
its  corresponding  frequency  curve,  and  consequently  that  the  frequency  curve 
is  the  derivative  of  its  cumulative  curve.  This  relationship  may  be  demon- 
strated as  follows :  When  die  cumulative  curve  is  prepared  from  the  histo- 
gram, the  percentage  of  material  in  each  class  is  summed  to  obtain  the 
successive  ordinates  of  die  cumulative  curve.  Thus,  inasmuch  as  the  histogram 
classes  represent  areas,  the  ordinates  of  the  cumulative  curve  are  linear  rep- 


2l6 


SEDIMENTARY  PETROGRAPHY 


resentations  of  the  area  under  the  histogram  up  to  that  point  Further,  the 
total  area  under  the  histogram  is  the  sum  of  the  areas  in  the  successive  blocks. 
This  may  be  expressed  in  the  shorthand  of  mathematics  as  follows.  Let  2!  be 
the  summation  s>-mbol,  let  f^,  /s, . .  •  /n  be  the  frequencies  in  each  histogram 
block,  and  let  Ajt  be  the  class  interval,  assumed  constant.  Then,  if  there  are 
n  classes  or  blocks,  the  total  area  under  the  histogram  is 


2  /;£.-r  =  total  area  under  histogram,  where  /;  represents  the  several 
I 
frequencies.  However,  as  the  classes  become  smaller  and  smaller,  or  in  other 
words  as  Ax  approaches  zero,  the  limit  of  this  sum  is  the  integral  of  the 
function  taken  over  the  range  involved.  In  mathematical  notation  this  is 


lim 
Ax-»o 


TA^ 


/ 


f-dr 


where  a  and  b  are  the  limits  of  the  range  of  sizes  in  the  distribution.  This 
mathematical  relation,  often  called  the  fundamental  theorem  of  the  integral 
calculus,  is  proved  in  all  standard  texts  on  the  subject 


100 

i 

/ '  '    .==^ 

/     Xf 

50 

//^          /              ' 

25 

0  . 

-« 

^    lyii 

1 

I       J> 


.01       .005 


J         .05 
DiAMcTER  IN  MM. 
Fig.  102. — Elxamples  of  cumulative  cunes.  A,  beach  sand ;  B,  glacial  till ;  C,  loess. 

The  fact  that  the  cumulative  cur\e  is  the  integral  of  the  frequency  curve 
explains  why  ctimulative  curves  are  less  liable  to  fluctuations  than  histo- 
grams. In  cumulating  the  original  frequency  data  on  which  the  histogram  is 
based,  a  process  of  finite  integration  is  performed  which  converts  the  finite 
class  intervals  into  a  continuous  function  when  the  curve  is  smoothed.  The 
histograms,  on  the  other  hand,  are  plotted  as  "raw"  data  and  so  preserve  the 
accidents  of  treating  a  continuous  function  as  a  series  of  discrete  intervals. 

The  oimulative  curve  may  be  used  in  the  same  manner  as  the  histogram 
in  interpreting  the  nature  of  sediments,  and  the  fact  that  it  is  less  liable  to 


ELEMENTS  OF  STATISTICS  217 

fluctuations  due  to  accidents  of  the  grade  scale  has  led  various  writers  to 
use  it  to  the  exclusion  of  histograms.  The  most  abundant  grains  are  asso- 
ciated with  the  inflection  point  (page  192)  of  the  cumulative  curve,  and  the 
degree  to  which  the  grains  cluster  about  or  spread  away  from  the  modal  group 
may  be  seen  from  the  steepness  of  the  curve.  Irregularities  in  the  smooth 
rise  of  the  curve  indicate  secondary  modal  groups.  Likewise,  the  approximate 
degree  of  sorting  or  sizing  of  the  sediment  may  be  read  from  the  general 
slope  of  the  curve  and  the  range  of  sizes  included  within  it. 

Figure  102  illustrates  several  cumulative  curves  of  sediments.  It  may  be 
seen  at  a  glance  that  curve  A  (a  beach  sand)  is  symmetrical  and  well  sorted, 
curve  B  (a  loess)  is  well  sorted  but  is  not  symmetrical,  and  curve  C  (a 
glacial  till)  is  poorly  sorted.  In  each  case  the  modal  class  is  associated  with 
the  steepest  part  of  the  curve. 


INTRODUCTION     TO     STATISTICAL     MEASURES 

Although  much  can  be  done  by  purely  graphic  methods  in  the  in- 
terpretation of  frequency  curves,  it  is  more  convenient  to  have  the 
characteristics  of  the  curve  expressed  as  numbers.  Statisticians  have 
develoj^ed  analytical  devices  so  that  the  numbers  themselves,  instead  of 
the  pictures  of  the  curves,  may  be  used  in  referring  to  the  distribution. 

Figure  103  shows  six  frequency  curves,  all  drawn  to  the  same  scale. 
The  top  row  of  three  curves  are  all  symmetrical,  but  curve  A  is  less 
peaked  than  B,  and  both  A  and  B  are  less  widely  spread  out  than  C. 
Similarly,  the  curves  in  the  lower  row  are  all  unsymmetrical,  but  curves 
E  and  F,  while  equally  unsymmetrical,  are  inclined  or  skewed  in  opposite 
directions.  Curve  D  shows  an  extreme  degree  of  asymmetry.  In  order 
to  describe  and  compare  this  wide  range  of  curves,  a  number  of  statis- 
tical measures  are  necessary.^ 

Measures  of  the  central  tendency.  Perhaps  the  most  important  measure 
is  a  measure  of  the  central  tendency,  the  value  about  which  all  other 
values  cluster.  In  general  this  value  corresponds  to  the  size  which  is 
most  frequent,  although  in  asymmetrical  curves  this  may  not  be  so.^Such 
measures  of  the  central  tendency  are  called  averages.  They  include  such 
diverse  measures  as  the  aritJimetic  mean  sice,  the  median  size,  the  modal 
size,  the  gcouictrjc  mean  size,  and  others. 

From  a  sedimentary  point  of  view,  the  average  size  of  a  sediment  is 

1  The  reader  is  referred  to  any  standard  textbook  of  statistics  for  more  detailed 
definitions  of  the  terms  used  in  this  section.  Among  elementary,  non-mathematical 
references  may  be  mentioned  F.  C.  Alills,  Statistical  Measures  (New  York.  1924). 
A  more  detailed  discussion  of  theory  may  be  found  in  B.  H.  Camp,  Tlie  Mathe- 
viatical  Part  of  Elementary  Statistics  (New  York,  1931).  A  general  reference  of 
much  value  is  R.  A.  Fisher,  Statistical  Methods  for  Research  Workers  (Edinburgh 
and  London,  1932). 


2l8 


SEDIMENTARY  PETROGRAPHY 


of  interest  because  it  indicates  the  order  of  magnitude  of  the  grains. 
Average  size  is  also  useful  for  comparinc;  sami)les  collected  in  the 
direction  of  transport  as  along  a  beach  or  stream.  Curves  of  the  average 


^SIZE  -*— SIZE 

Fig.  103. — Frequency  curves.  The  abscissae  may  be  diameter  in  millimeters,  logs  of 
the  diameters,  or  any  other  expression  of  "size." 

size  against  distance  may  disclose  some  underlying  law  of  variation.  In 
a  similar  manner,  maps  may  be  prepared  of  the  areal  variation  of  size 
within  a  given  environment,  as  a  basis  for  reasoning  geologically  about 
the  causes  of  the  variation.  The  average  grain  size  is  thus  an  important 


ELEMENTS  OF  STATISTICS  219 

value,  and  the  choice  of  particular  averages  will  receive  detailed  atten- 
tion in  Chapter  9. 

Measures  of  the  degree  of  scatter.  Two  frequency  curves  with  the 
same  average  size  may  have  entirely  different  degrees  of  spread,  such 
as  curves  A  and  C  of  Figure  103,  because  the  average  value  merely  rep- 
resents the  central  point  and  does  not  indicate  the  spread  of  the  data  on 
either  side.  Hence,  a  second  measure  needed  to  describe  the  curves  is 
a  measure  of  the  degree  of  spread  or  degree  of  dispersion  of  the  data 
about  the  central  tendency?  Such  measures  of  spread  mayl)e  the jnean 
deviation,  the  standard  deviation, ^t^uartilj_demation^t\^rajiqe^ and 
so_on.  As  with  measures  of  average  size  there  is  a  choice  of  devices 
available,  and  one  of  the  problems  of  the  practising  statistician  is  to 
determine  the  appropriate  measure  to  use. 

From  a  geological  point  of  view,  the  average  spread  of  the  curve, 
which  means  the  tendency  of  the  grains  to  cluster  about  the  average 
value,  is  another  important  characteristic  of  sediments.  Some  geological 
agents  are  more  selective  in  their  action  than  others,  and  this  may 
manifest  itself  in  the  extent  to  which  the  grains  tend  to  be  selected  or 
"sorted"  according  to  size.  Theoretically,  perhaps,  one  may  expect  a 
perfectly  sorted  sediment  to  consist  of  only  one  size  of  grain,  but  in  any 
natural  situation  there  are  deviations  about  this  size,  due  to  fluctuations 
in  velocity,  shape  and  density  of  the  grains,  and  the  like.  Consequently, 
the  degree  of  spread  may  prove  of  importance  as  a  clue  to  the  nature  of 
the  deposit.  For  example,  it  is  not  known  whether  the  selectivity  of  sedi- 
ments increases  or  decreases  in  the  direction  of  transport ;  what  meager 
evidence  there  is  on  beaches  suggests  that  the  average  spread  may  be 
fairly  constant  over  a  given  stretch  of  beach.  Profiles  of  average  spread 
of  the  curv^es  along  a  traverse  line  or  maps  of  the  average  spread  over  a 
formation  may  furnish  clues  to  variations  in  the  depositing  agent. 

Measures  of  the  degree  of  asymmetry.  The  average  size  and  the 
degree  of  spread  of  two  curves  may  be  the  same,  but  one  may  not  be 
symmetrical.  This  situation  is  illustrated  approximately  by  curves  A 
and  E  of  Figure  103.  Hence  it  is  necessary  to  have  a  measure  of  the 
tendency  of  the  data  to  spread  on  one  side  or  the  other  of  the  average. 
Such  aa\-mmetry  is  called^skewness,  and  various  skewness  measures  are 
available.  Because  skewness  may  be  either  to  the  left  or  to  the  right,  a 
positive  or  negative  sense  is  usually  assigned  to  it.  Thus,  curves  E  and 
F  are  skewed  in  opposite  directions ;  the  choice  of  positive  and  negative 
directions  may  be  conventional.  In  extreme  tj^pes  of  skewness,  such  as 
shown  by  curve  D  of  Figure  103,  additional  measures  may  be  needed  to 


220       SEDIMENTARY  PETROGRAPHY 

describe  the  shape  of  the  curve,  or  mathematical  methods  may  be  used 
to  "symmetrize"  the  curv-e  by  changing  the  independent  variable. 

Skewness  is  an  attribute  of  sediments  about  which  relatively  little  is 
known.  When  sedimentary  curves  are  plotted  with  diameter  in  milli- 
meters as  the  independent  variable,  they  almost  invariably  show  extreme 
types  of  skewness,  but  the  asymmetry  is  reduced  when  logarithms  of 
the  diameters  are  used  as  independent  variable.  For  this  reason  it  is 
often  simpler  mathematically  to  analyze  sedimentary  data  on  a  loga- 
rithmic basis.  The  physical  meaning  of  skewness  is  not  easily  interpreted, 
for  several  reasons.  For  example,  sampling  errors  may  manifest  them- 
selves in  skewness,  either  if  more  than  one  size  frequency  distribution  is 
included  in  the  sample  due  to  improper  selection  of  samples,  or  if  the 
sample  of  a  single  population  is  too  small  to  reflect  the  attributes  of  the 
original  distribution.  Likewise,  skewness  may  result  if  a  symmetrical 
distribution  is  later  acted  upon  by  a  transpxDrting  agent  which  removes 
only  a  portion  of  the  material.  A  sandy  gravel  may  have  some  of  its 
finer  material  removed,  leaving  behind  a  skewed  lag  sediment.  Miss 
Gripenberg  ^  has  suggested  that  skewness  has  a  genetic  significance  in 
some  instances  and  that  a  sediment  deposited  by  a  uniform  current  may 
increase  in  skewness  as  the  material  is  followed  along  in  the  direction 
of  transport. 

Relatively  few  studies  have  been  made  of  the  areal  variation  of  skew- 
ness within  given  deposits,  and  the  data  are  perhaps  too  meager  for 
generalizations.  The  almost  universal  presence  of  skewness  in  sediments, 
especially  in  terms  of  diameter  as  the  independent  variable,  suggests  that 
there  is  a  genetic  relation  between  agent  and  skewness,  as  ]\Iiss  Gripen- 
berg points  out,  and  that  the  skewness  may  vary  areally  in  accordance 
with  definite  laws. 

Measures  of  the  degree  of  peakedncss.  Frequency  curves  which  are 
alike  in  their  degree  either  of  symmetry  or  asymmetry  may  nevertheless 
vary  in  the  degree  to  which  peakedness  is  present.  Curves  A  and  B  of 
Figure  103  illustrate  this  difference.  Curve  B  has  a  more  pronounced 
peak  than  A.  Statistical  measures  designed  to  express  this  attribute  are 
measures  of  kurtosis;  here  also  a  choice  is  available. 

Not  much  is  known  about  the  significance  of  kurtosis  in  sediments. 
It  appears  to  be  related  to  the  selective  action  of  the  geological  agent, 
but  the  sum  total  of  the  factors  entering  into  the  selective  process  are 
not  known.   The  kurtosis  of   a  curve,   and   especially  of   symmetrical 

1  Stina  Gripenberg,  A  study  of  the  sediments  of  the  North  Baltic  and  adjoining 
seas:  Fcnnia,  vol.  60,  no.  3,  1934. 


ELEMENTS  OF  STATISTICS  221 

curves,  has  a  definite  geometrical  significance,  however,  whether  or  not 
it  may  have  a  physical  or  geological  significance.  No  complete  investi- 
gation of  the  areal  variation  of  kurtosis  has  been  made,  and  virtually 
nothing  is  known  of  its  magnitude  or  prevalence  in  sediments. 


ARITHMETIC     AND     LOGARITHMIC     FREQUENCY 
DISTRIBUTIONS 

The  extreme  type  of  skewed  curve  shown  as  D  in  Figure  103  is  com- 
monly encountered  in  sedimentary  practice,  especially  when  the  data  are 
plotted  with  diameter  in  millimeters  as  the  independent  varialile.  When 
the  same  data  are  plotted  with  log 
diameter  as  independent  variable,  the 
curve  becomes  much  more  symmetri- 
cal. The  symmetrizing  influence  of  a  ; 
logarithmic  size  scale  was  mentioned  \ 
above ;  it  will  be  discussed  here  in  ' 
terms  of  the  application  of  statistical 
measures  to  the  data.  From  a  mathe- 
matical point  of  view  it  is  simpler  to 
describe  a  symmetrical  curve  than  one 
which  is  as^•mmetrical.  A  svmmetrical   t 


LOGioOIAMETER 


fr_ec[uency  curve  may  be  completel}-  de-  ^ 
scribed  b\-  two  measures,  ah  average^g 
size  and  the  degree  of  spread  about  ^ 
the  average.  If  the  curve  is  moderately 
skewed,  three  measures  usually  suffice, 
but  for  extreme  skewed  curves  the 
labor  involved  in  computing  the  nec- 
essary number  of  measures  becomes 
quite  tedious.  From  the  point  of  view 
of  convenience  alone,  the  symmetriz- 
ing eflfect  of  logarithmic  plotting  is  ample  justification  for  its  use.  There 
is  another  "justification,"  however;  most  workers  in  sediments  prefer  to 
plot  their  data  on  an  implied  log  scale,  by  drawing  the  classes  equal  in 
width,  to  facilitate  interpretation  of  the  data. 

From  a  strict  statistical  point  of  view,  a  sedimentary  histogram  based 
on  a  geometric  grade  scale,  but  drawn  with  its  blocks  of  equal  width,  is 
no  longer  a  picture  of  the  frequency  distribution  of  diameters.  This 
may  be  more  efifectively  demonstrated  with  frequency  curves,  as  shown 


DIAMETER  IN  MM. 

Fig.  104. — Logarithmic  and  arith- 
metic graphs  of  the  same  frequency 
distribution,  showing  shift  in  posi- 
tion of  central  ordinate. 


222  SEDIMENTARY  PETROGRAPHY 

in  Figure  104,  which  shows  the  same  data  plotted  as  an  arithmetic  and 
a  logarithmic  frequency  curve.  In  the  logarithmic  curve  an  ordinate 
has  been  drawn  in  such  a  manner  that  the  areas  on  both  sides  are  equal. 
This  same  ordinate,  converted  to  its  diameter  equivalent,  is  shown  in 
the  arithmetic  cur\-e :  it  no  longer  divides  the  area  into  halves.^  Inasmuch 
as  histograms  or  frequency  curA-es  are  areal  representations  of  the  fre- 
quency, it  must  be  obvious  that  the  geometrical  interpretations  of  the 
two  diagrams  must  differ.  One  may  say,  for  example,  that  the  distribu- 
tion of  (logarithmic)  individuals  is  symmetrical  about  the  central  or- 
dinate, but  it  is  not  correct  to  say  that  the  distribution  of  sedimentary 
particles,  considered  as  grains  of  a  given  size,  is  equal  and  symmetrical 
about  the  average  value. 

The  distinction  between  two  types  of  frequency  curves,  one  based 
directly  on  diameters  as  the  independent  variable,  and  the  other  based 
on  logs  of  the  diameters,  is  important  in  sedimentary  data.  Numerous 
workers-  have  discussed  the  merits  of  one  or  another  manner  of  plot- 
ting the  data,  and  various  statistical  measures  have  been  proposed  to 
take  cognizance  of  the  shift  in  geometrical  significance  introduced  by  the 
logarithmic  plotting.  Three  broad  t}pes  of  measures  have  been  developed 
as  a  result.  The  first  includes  arithmetic  measures  based  directly  on 
grain  diameters  in  millimeters ;  these  measures  include  a  size  factor. 
The  second  type  of  measure  is  also  based  on  grain  diameters,  but  in- 
volves ratios  between  sizes  to  eliminate  the  size  factor  and  to  emphasize 
the  geometric  nature  of  the  frequency  distribution.  The  third  t}-pe  fore- 
goes the  diameter  distribution  entirely  and  applies  a  series  of  logarithmic 
measures  to  the  logarithmic  frequency  curA'e. 

All  three  kinds  of  measures  are  used  at  present,  but  some  confusion 
has  arisen  due  to  the  use  of  one  kind  of  measure  coupled  with  inferences 
drawn  from  postulates  underlying  another  kind  of  measure.  An  im- 
portant precaution  to  be  used  irr  sedimentary  statistical  practice  is  that 
the  identity  of  the  independent  variable  must  be  known  at  all  times.  It 

1  A  complexity  enters  this  analysis,  due  apparently  to  a  shift  of  the  mode  during 
the  transformation. 

-  The  following  papers  are  among  those  which  bear  on  the  problem :  C.  W.  Cor- 
rens,  Gnmdsatzliches  zur  Darstellung  der  Korngrossenverteilung :  Zentr.  f.  Min., 
Abt.  A.,  pp.  321-331,  1934.  T.  Hatch  and  S.  P.  Choate,  Statistical  description  of  the 
size  properties  of  non-uniform  particulate  substances:  Jour.  Franklin  Inst.,  vol.  207, 
pp.  369-387,  1929.  W.  C.  Krumbein,  Application  of  logarithmic  moments  to  size  fre- 
quencj-  distributions  of  sediments :  Jour.  Scd.  Petrology,  vol.  6,  pp.  35-47,  1936.  P.  D. 
Trask,  Origin  and  Emnronmcnt  of  Source  Sediments  of  Petroleum  (Houston,  Texas, 
1932),  pp.  67  it.  C.  E.  Van  Orstrand,  Note  on  the  representation  of  the  distribution 
of  grains  in  sands :  Researches  in  Sedimentation  in  19.24,  pp.  63-67,  Nat.  Research 
Council,  1925.  C.  K.  Wentworth,  Method  of  computing  mechanical  composition  types 
of  sediments:  Geol.  Soc.  America,  Bulletin,  Vol.  40,  pp.  771-790,  1929. 


ELEMENTS  OF  STATISTICS  223 

is  not  necessary  for  statistical  purix)ses  that  the  independent  variable 
have  any  immediately  comprehensible  significance;  it  is  only  required 
that  a  curve  be  given.  The  identity  of  the  inde^jendent  variable  may, 
however,  be  preserved  by  defining  it  in  terms  of  diameters,  so  that  at 
any  point  in  the  analysis  one  may  convert  his  results  back  to  diameter 
terms  if  he  wishes. 

It  would  appear  at  first  glance  that  the  application  of  logarithmic  methods 
would  complicate  the  essential  simplicity  of  the  sedimentary  picture,  but 
actually  this  is  not  so.  In  order  to  avoid  difficulties,  it  is  only  necessary  to 
set  up  a  mathematical  relationship  such  that  a  new  logarithmic  variable  is 
substituted  for  the  diameters  of  the  grains.  Methods  of  analysis  and  the 
grouping  of  the  data  into  size  classes  are  not  changed;  the  new  variable  is 
used  only  in  the  computation  of  statistical  measures,  and  the  geometrical 
meaning  of  the  measures  is  directly  related  to  the  logarithmic  frequency  dia- 
gram of  the  sediment.  The  phi  and  zeta  scales  mentioned  in  Chapter  4  and 
discussed  more  fully  in  Chapter  9  afford  one  method  of  attacking  this  statisti- 
cal problem. 

gUARTILE     AND     MOMENT     MEASURES 

Coupled  with  the  problem  of  choosing  suitable  independent  variables 
for  sedimentary  data  is  the  choice  of  sets  of  measures  in  terms  of  their 
underlying  mathematical  theory.  In  conventional  statistical  practice  two 
main  types  of  measures  have  been  used. 

Qimrtilc  measures.  If  a  size  frequency  distribution  is  arranged  in 
orderof  magnitude,  with  the  smallest  particle  at  one  end  and  a  continu- 
ous gradation  upward  to  the  largest  particle  at  the  other  end,  it  is  ^x^ssible 
to  choose  certain  particles  as  representing  significant  values.  The  size 
of  the  middlemost  particle  represents  an  average  of  the  group,  and  is 
called  the  median.  To  determine  the~spfeaH  of  the  clistnbirtion  ahoUt 
the  median,  two  other  particles  are  measured.  The  first  is  just  larger 
than  one  fourth  of  the  distribution  (the  first  quartUe),  and^the  second  is 
just  larger  than  three  fourths  of  the  distriljution_(.the^^/i/Vrf  quartile). 
Measures  c)f  spread  arc  based  on  differences  or  ratios  between  the  two 
quartiles,  depending  upon  whether  arithmetic  or  L^cnnictric  measures  are 
to  be  used.  Similarly,  logarithmic  measures^are  based  on  logs  of  the 
quartiles.  For  measuring  the  asymmetry  or  skewness,  a  comparison  is 
made  of  the  median  value  with  an  average  of  the  first  and  third  quartiles, 
either  arithmetically,  geometrically,  or  logarithmically. 

The,  outstanding  feature  pf  quartile  measures  is  that  they  are  con- 
fined to  the  central  half  of  the  frequency  distribution  and  the  values  ob- 
tained are  not  influenced  by  extreme  particles,  either  ven,'  large  or  very 


«^  2^4^"  --^   SEDIMENTARY  PETROGRAPHY 

small.  Furthermore,  quanile  measures  are  very  readily  computed,  and 
most  of  the  data  may  be  obtained  directly  from  the  cumulative  curve  by 
graphic  means.  For  these  reasons  quartile  measures  are  extensively  used 
in  sedimentan.-  data,  and  they  apply  even  to  incomplete  sets  of  data. 
This  is  an  advantage  for  fine-grained  sediments,  where  part  of  the 
material  is  beyond  the  range  of  ordinary  methods  of  mechanical 
analysis. 

^loment  measures.  In  contrast  to  quanile  measures  are  moment 
m^sures.  whiclTaFe'  influenced  by  every  individual  in  the  distribution, 
from  coarsest  to  finest.  Moment  measures  are  much  more^mplex  mathe- 
maticaUylhanquartile  measures,  and  they  involve  rather  tedious  compu- 
tations compared  with  the  quartiles.  Nevertheless,  moment  measures  are 
more  extensively  used  in  conventional  statistical  practice  because  of 
their  greater  sensitivit)-  to  the  influence  of  each  member  of  the  distri- 
bution and  because  of  their  more  unified  mathematical  basis.  A  full 
understanding  of  the  nature  of  moments  cannot  be  had  without  re- 
course to  the  calculus,  but  fortunately  the  computations  may  be  made, 
and  the  geometrical  significance  evaluated,  without  mathematical  knowl- 
edge. 

The_firsLJiiQmenJ_ofa^ frequency  distribution  is  its  center  of  gravity 

'^"'^^'^  '  andis^called  the  adtbmetic  mean.  It  is  a  measure  of  the^^erage  size 

*N«.*.-     of_the_sedinient.  The  second  moment,  or  more  properly  its  square  root, 

4  >^*^*-^^  measures  the  average  spread  of  the  curv^e  and  is  expressed_  as  ^e 

^iX^X^z.^andard  deviation  of  the  distribution.  It  is  analogous  in  physics  totfie 

^  radius  of  g)rati^"  ^^  ^  «;yQfpm    TlipjJTiH^mnment,  or  its  cube  root^^is 

'^"1>I>  a  measure  of  the  skewness  of  the  data.  The  moment  measures  are  thus 

a  set  of  parallel  mea5ures^o~the~quartile  measures,  but  their  geometric 

significance  is  different. 

Mathematically,  the  rth  moment  of  a  distribution  is  defined  as 

-X 
-X 

where  /(x)  is  the  frequency  function  and  X  is  the  total  frequency.  By  set- 
ting r=  I,  2,  3,  ...,  the  successive  moments  result.  The  moments  are  thus 
related  as  the  successive  powers  of  x  into  the  integral.  In  practice,  the  first 
four  moments  are  used;  unfortunately,  there  is  no  physical  analogue  of 
moments  higher  than  the  second,  so  that  the  moments  cannot  conveniently  be 
expressed  in  simple  terms.  In  practice,  where  f(x}  is  unknown,  the  fre- 
quency- in  the  several  classes  of  width  A.r  is  multiplied  by  some  power  of  their 
distance  from  the  origin  and  the  result  is  translated  to  a  value  in  terms  of  the 
first  moment,  as  described  in  Chapter  9. 


*»<«.» 


ELEMENTS  OF  STATISTICS  225 

The  tediousncss  of  computing  moment  measures,  combined  with  their 
mathematical  complexity,  has  miHtated  against  their  extensive  use  in 
sedimentary  petrology.  A  further  difficulty  has  arisen  from  the  fact  that 
conventional  statistics  books  afford  only  methods  of  computation  l)ased 
on  equal  intervals,  whereas  with  sediments  the  data  are  usually  expressed 
in  unequal  grade  sizes.  Finally,  moment  measures  expressed  in  terms 
of  diameters  in  millimeters  often  result  in  complexities  owing  to  the 
extreme  skewness  of  the  data.  Fortunately,  however,  methods  are  avail- 
able for  the  direct  use  of  logarithmic  moments,  which  appear  to  have 
physical  significance  in  sediments  and  which  may  be  converted  to  their 
diameter  equivalents. 

The  fact  that  moments  are  affected  by  the  value  of  every  grain  in  the 
distribution  may  limit  their  application  to  sedimentary  data.  When  anal- 
yses are  so  expressed  that  all  material  finer  (or  coarser)  than  a  given 
grade  is  grouped  into  one  class,  the  values  of  the  higher  moments  are 
distorted.  Improved  techniques  of  analysis,  especially  among  the  fine- 
grained sediments,  may  remedy  this  difficulty,  however. 


THE     QUESTION     OF     FREQUENCY 

There  is  one  conspicuous  manner  in  which  the  statistical  data  of  sedi- 
mentary petrology  differ  from  most  conventional  statistical  data.  Fre- 
quency in  sedimentary  data  is  usually  expressed  by  weight  instead  of 
by  number,  and  it  is  usually  expressed  as  percentage  frequency  rather 
than  absolute  frequency.  No  complete  investigations  of  this  aspect  of 
sedimentary  usage  have  been  made,  and  the  problem  of  weight  vs.  num- 
ber is  still  largely  unsolved.  In  a  given  sample  there  may  be  only  one 
or  two  large  pebbles  to  a  gram,  whereas  there  may  be  literally  millions 
of  small  particles  to  a  gram.  Hence  if  the  grades  are  weighed,  the  result- 
ing frequency  distribution  will  give  greater  significance  to  the  larger 
sizes,  where  a  few  pebbles  will  outweigh  a  tremendous  number  of  fine 
grains.  The  curve,  then,  may  be  inclined  toward  the  coarser  sizes.  If 
the  grades  are  counted,  on  the  other  hand,  the  several  large  pebbles 
would  be  quite  negligible  in  contrast  to  the  millions  of  fine  particles. 

Conventional  statistical  measures  are  defined  in  terms  of  number  fre- 
quency, represented  by  N,  the  total  number  of  individuals.  One  may 
raise  the  question,  however,  whether  it  is  not  possible  to  redefine  exactly 
the  same  types  of  measures  in  terms  of  weight  frequency.  It  is  possible 
to  set  up  statistical  measures  on  a  weight  basis  (or  weight  percentage) 
which  are  directly  applicable  to  conventional  usage  and  may  be  related 


226 


SEDIMENTARY  PETROGRAPHY 


to  probabilities,  areas  under  curves,  and  the  like  equally  as  conveniently 
as  number  measures.  It  will  not  be  true  in  general,  however,  that  there 
is  any  necessary'  simple  relation  between  the  measures  defined  on  a 
weight  basis  and  the  measures  defined  by  number. 

It  would  be  convenient,  however,  to  know  whether  number  or  weight 
is  a  more  important  concept  in  the  interpretation  of  sediments.  As  far 
as  the  writers  are  aware,  no  mathematical  statistician  has  attacked  the 
problem,  and  the  following  discussion  is  to  be  taken  as  a  tentative  quali- 
tative evaluation  of  the  problem.  Part  of  the  discussion  will  be  based 
on  the  only  apparent  research  that  has  been  done  along  these  lines,  and 
part  on  geologic  reasoning.  In  1933  Hatch  ^  showed  that,  if  a  frequency 
distribution  of  grains  is  symmetrical  when  plotted  on  a  logarithmic  basis, 
there  is  a  simple  relation  between  the  weight  frequency  cur^-e  and  the 

number  frequency  curve.  In  such  cases 
only  tivo  parameters  are  involved, 
which  Hatch  defined  as  the  log  geo- 
metric mean  and  the  log  standard 
geometric  deviation.  His  demonstra- 
tion showed  that  the  log  standard 
deviation  remains  constant  when  the 
frequency  is  changed  from  weight  to 
number,  but  the  log  geometric  mean 
diameter  changes  from  one  distribu- 
tion to  the  other.  Thus  two  frequency 
curves  of  the  same  sediment,  s}-m- 
metrical  on  logarithmic  plotting,  one 
based  on  number  frequency  and  the 
other  on  weight  frequency,  have  the  relations  -  shown  in  Figure  105. 
Note  that  the  nimiber  frequency  curve  lies  to  the  right  of  the  weight 
frequency  cur\-e,  which  means  that  the  average  value  has  shifted  toward 
smaller  sizes,  an  expected  result  due  to  the  greater  numerical  signifi- 
cance of  many  smaller  grains  as  opposed  to  a  few  larger  grains. 

When  the  frequency  curve  is  skewed  on  a  logarithmic  basis,  the  simple 
relation  between  weight  and  number  frequency  no  longer  holds,  inasmuch 
as  an  additional  parameter  enters  the  situation.  However,  the  fact  that 

1  T.  Hatch,  Determination  of  "average  particle  size"  from  the  screen-analysis  of 
non-xiniform  particulate  substances :  Jour.  Franklin  lust.,  vol.  215,  pp.  27-37,  1933. 

2  The  relations  between  the  curves  are  determined  graphically  by  the  use  of 
logarithmic  probability  paper  (page  189).  The  necessary  condition  is  that  the  cumu- 
lative data  plot  as  a  straight  line.  The  corresponding  frequency  on  the  alternate 
basis  is  then  foimd  bj'  translating  the  ctirve  parallel  to  itself  in  accordance  with  an 
equation  given  by  Hatch  (loc.  cif.).  The  cur\-es  of  Figure  105  were  obtained  by 
graphic  differentiation  of  the  cumulative  curves. 


--■lit 

l\ 

,.'/\\ 

I  -a  J6      A  2  J4WA6    .04         j02         .01 

DIAMETER    IN   UU. 

Fig.  105. — Weight-percentage  fre- 
quenc}- curve  (A)  and  number-per- 
centage frequency  curs-e  (B)  of 
same  sediment  (a  beach  sand). 


ELEMENTS  OF  STATISTICS  227 

there  is  a  relation  between  the  symmetrical  curves  suggests  that  there 
also  is  a  relation  between  skewed  curves,  however  complex  that  relation 
may  be. 

From  a  geological  point  of  view,  the  problem  of  frequency  may  be 
considered  from  several  angles.  In  terms  of  the  kinetic  energy  of  the 
transporting  medium,  the  work  performed  in  moving  a  pel)ble  varies 
directly  as  its  mass  (weight),  assuming  the  stream  velocity  constant. 
Thus  there  is  a  physical  relation  between  weight  and  energy.  With  a 
given  geometrical  form,  on  the  other  hand,  number  may  be  converted 
into  weight,  and  hence  there  is  also  a  relation  between  number  and 
energy.  Whether  number  or  weight  may  be  taken  as  the  more  significant 
value,  however,  is  not  clear.  If  surface  area  of  a  given  geometrical  form 
is  considered,  it  is  possible  to  relate  surface  area  either  to  weight  or  to 
number  of  particles.  Unfortunately  in  none  of  these  cases  is  the  mathe- 
matical relation  simple,  especially  in  a  distribution  of  diverse  shapes, 
densities,  and  diameters. 

If  convenience  be  taken  as  a  criterion,  there  is  little  doubt  that  in 
the  average  case  weight  is  a  variable  more  readily  determined  than  num- 
ber, especially  among  fine  grains.  With  gravels  it  may  be  more  con- 
venient to  count  the  pebbles ;  likewise  microscopic  methods  usually  in- 
volve counting  the  grains.  As  long  as  current  methods  of  analysis  remain 
in  use,  namely,  sieving  and  sedimentation,  then  weight  will  be  more  con- 
venient to  use  than  number.  However,  if  microscopic  measurements  of 
limited  samples  increase  in  use  as  a  device,  then  number  will  be  more 
convenient  than  weight.  The  number  of  factors  entering  the  problem 
suggests  that  for  immediate  purposes  it  may  be  immaterial  whether  one 
or  the  other  is  used,  as  long  as  measures  based  on  weight  be  not  directly 
confused  or  compared  with  measures  based  on  number.  The  safest  pro- 
cedure to  follow  is  to  indicate  in  the  published  results  the  type  of  fre- 
quency data  involved  and  to  define  the  measures  specifically  in  connection 
with  the  variables  used. 

One  important  point  may  be  stressed  here  :  frequency  is  always  the  de- 
pendent variable,  and  although  the  physical  interpretation  of  the~clata 
may  vary  with  the  manner  of  expressing  frequency,  the  geometrical 
significance  of  the  statistical  measures  is  the  same  regardless  of  the 
particular  choice  of  frequency  used. 


CHAPTER  9 

APPLICATION  OF  STATISTICAL  MEASURES  TO 
SEDIMENTS 

INTRODUCTION 

At  least  three  points  of  view  have  been  manifested  by  sedimentary 
petrologists  in  the  appHcation  of  statistical  measures  to  sedimentary  data. 
One  group  has  developed  measures  designed  to  furnish  a  series  of  num- 
bers for  each  sample,  as  an  aid  in  describing  and  classifying  sediments. 
This  group  has  not  concerned  itself  directly  with  statistical  theory,  on 
the  ground  that  conventional  devices  furnish  too  few  numbers  for  de- 
tailed work. 

A  second  group  has  applied  conventional  statistical  measures  to  sedi- 
ments, so  that  the  relation  of  the  measures  to  the  body  of  statistical 
theory  could  be  known.  The  contention  is  that  unless  the  measures  can 
be  related  to  the  background  of  statistical  theory,  little  more  can  be  done 
than  to  classify  sediments;  that  the  relation  of  the  measures  to  environ- 
mental factors  which  control  the  characteristics  of  sediments  cannot  be 
brought  out  with  arbitrary  measuring  devices. 

A  third  group  includes  those  who  maintain  that  statistical  procedures 
are  essentially  meaningless  as  applied  to  sedimentary  data.  The  errors 
of  sampling  and  analysis  are  so  large,  the  contention  runs,  that  the  data 
have  little  quantitative  significance,  and  no  matter  how  much  statistical 
manipulation  is  involved,  the  final  data  are  no  better  than  the  original. 

It  is  true  that  -statistical  manipulation  cannot  create  data,  but  by 
applying  statistical  reasoning  it  is  possible  to  determine  how  large  the 
errors  are  and  to  devise  corrections  so  that  more  reliable  data  may  be 
obtained.  Furthermore,  statistical  operations  furnish  a  means  of  sum- 
marizing large  amounts  of  information  in  a  convenient  manner,  such 
that  comparisons  and  descriptions  are  greatly  simplified.  Workers  as  a 
whole  are  becoming  increasingly  aware  of  the  advantages  of  a  statistical 
approach,  although  there  is  as  yet  little  tendency  to  standardize  the 
techniques. 

It  is  the  opinion  of  the  authors  that  a  fuller  understanding  of  the 

228 


STATISTICAL  METHODS  22q 

significance  of  statistical  devices  will  do  much  in  clearing  the  present 
confusion  about  means  and  ends.  Toward  that  end,  the  discussion  in  the 
present  chapter  is  concerned  as  much  with  the  geometrical  meaning  of 
the  measures  as  it  is  with  the  mere  mechanical  process  of  arriving  at 
the  numbers. 


QUARTILE      MEASURES 

Quartile  measures  are  perhaps  as  widely  used  as  any  other  device  tor 
describing  and  comparing  sediments.  The  first  use  of  quartile  measures 
in  sedimentary  data  was  by  Trask/  who  introduced  a  set  of  geometric 
quartile  measures  in  1930,  and  discussed  the  theory  of  them  more  fully 
in  1932.  In  1933  Krumbein-  used  conventional  arithmetic  quartile  meas- 
ures for  describing  glacial  tills.  In  1934  Miss  Gripenberg^  demonstrated 
that  Trask's  geometric  measure  of  spread  was  related  to  the  logarithmic 
probability  curve,  and  in  1936  Krumbein*  showed  the  relations  among 
arithmetic,  geometric,  and  logarithmic  quartile  measures  in  terms  of  con- 
ventional statistical  theory. 

The  great  advantage  of  quartile  measures  is  the  ease  with  which  they 
are  determined  from  the  analytical  data.  Three  values  usually  suffice 
for  the  computation  of  the  measures.  These  are  the  median  and  the  first 
and_thijd  quartiles,  each  read  directly  from  the  cumulative  curve. 

Theuicdjan.  Tlie  median  diameter  is  defined  ''  as  the  middlemost  mem- 
ber of  the  distribution ;  it  is  that  diameter  which  is  larger  than  50  per 
cent  of  the  diameters  in  the  distribution,  and  smaller  than  the  other  50 
per  cent.  For  its  graphic  determination,  therefore,  it  is  only  necessary 
to  draw  a  cumulative  curve  of  the  sediment  and  to  read  the  diameter 
value  which  corresponds  to  the  point  where  the  50-per  cent  line  crosses 
the  cumulative  curve,  as  shown  in  Figure  106. 

The  medianhas  the  advantage  that  it  is  not  affected  by  the  extreme 
grains  on  either  end  of  the  distribution,  and  it  is  not  necessary  to  have 
the  complete  analysis  to  determine  it.  AmoQg_lhe— disadvantagesTpf 
the  median  are  that  it  cannot  be  manipulated  algebraically ;  that  is,  the 

1  P.  D.  Trask,  Mechanical  analysis  of  sediments  by  centrifuge :  Econ.  Geology. 
vol.  25,  pp.  581-59Q,  1030.  P.  D.  Trask,  Origin  and  Environment  of  Source  Sedi- 
ments of  Petroleum    (Houston,  Texas,   1932),  PP-  67  fF. 

-  W.  C.  Krumbein,  Lithological  variations  in  glacial  till :  Jour.  Geology,  vol. 
41,  pp.  382-408,  1933. 

3  Stina  Gripenberg,  A  study  of  the  sediments  of  the  North  Baltic  and  adjoining 
seas :  Fcnnia.  vol.  60,  no.  3,  1934,  pp.  214  flf. 

*  W.  C.  Krumbein,  The  use  of  quartile  measures  in  describing  and  comparing 
sediments:  Am.  four.  Sci..  vol.  37,  pp.  98-111.  1936. 

sp.  C.  Mills,  Statistical  Methods  (New  York,  1924),  p.  112. 


230 


SEDIMENTARY  PETROGRAPHY 


100  ■-- 

9 

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Qua 
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/ 

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DIAMETER     IN   MM. 


^^-^^ 


niedians  of^  each  grade  cannot  be  averaged  to  jjive  the  median  of  the 
distribution.  This  disadvantage  is  not  great,  however,  inasmuch  as 
graphic  methods  are  generally  used  in  determining  the  median. 

Quartile  deviation.  The  measure  of  average  spread jwhichjs^commonly 
used  witlitjie  median  is  the  guar  tile  devi_ation.  The  quartiles^  lie^n  either 
side  of  the  median  and  are  the  diameters  which  correspond  to  frequencies 

of  25  and  75  per  cent.  By  convention, 
the   smaller   diametej-  value   iFlaken 
as_the   first   quartile^Qi.    It   is   that 
diameter  which  has  25  per  cent  of  the 
distribution  smaller  than  itself  and  75 
per  cent  larger  than  itself.  It  is  found 
from  the  cumulative  curve  by  reading 
the  diameter  value  which  corresponds 
to  the  point  where  the  75-per  cent 
line  intersects  the  cumulative  curve. 
The    third   quartile,    Q3,    is   that   di- 
ameter which  has  75  per  cent  of  the 
distribution   smaller  than   itself,   and 
25  per  cent  larger  than  itself.   It  is 
found  by  determining  the  diameter  value  corresponding  to  the  intersec- 
tion of  the  25-per  cent  line  and  the  cumulative  curve.  (See  Figure  106.) 
Three  types  of  quartile  deviation  are  used  in  sedimentary  petrology, 
depending  upon  the  particular  features  of  the  curve  which  are  to  be 
emphasized.  They  are  the  arithmetic,  geometric,  and  logarithmic  quartile 
deviations.  For  convenience  these  three  types  of  measures  will  be  dis- 
cussed together,  the  better  to  bring  out  their  relations  to  one  another. 
-   ^^The   simplest   form  of   quartile   deviation  is  the  arithmetic  quartile 
deviation,^  ODa  which  is  a  measure  of  half  the  spread  between  the  two 
quartiles.  The  difference  is  so  chosen  that  positive  values  always  result: 

QDa=(O3-Q0/2 (i) 

The  second_possibiHty  is  a  geometric  quartile  deviatjon^  QDg,  which 
is  based  on  the  ratio  between  the  quartiles,  instead  of  on  their  differences. 
Specifically  it  is  the  square  root  of  the  ratio  of  the  two  quartiles,  so 
chosen  that  the  value  is  always  greater  than  unity : 


Fig.  106. — Method  of  reading  me- 
dian and  quartiles  from  cumulative 
curve. 


QD,  =  VQ3/Q, 


(2) 


Trask  -  introduced  this  measure  as  a  "sorting  coefficient,"  but  owing  to 

1  F.  C.  Mills,  0/-.  cif.  (1924),  p.  158. 

2  P.  D.  Trask,  0/..  cit.  (1932). 


'W^ 


STATISTICAL  METHODS  231 

his  reversal  of  the  usual  definition  of  tlie  quartiles.  his  equation  is 
So  =  \/Oi/Q3.  This  is  identical  with  equation  (2^.  however,  as  long 
as  the  larger  quartile  is  used  as  numerator.  For  convenience,  then,  the 
symbol  So  will  be  used  for  equation  (2),  with  the  understanding  tliat  the 
larger  quartile  is  ahi'ays  in  the  numerator. 

Finally,  there  is  a  log  quartile  deviation,  w^hich  is  equal  to  half  the 
difference  between  the  logs  of  the_quartiles.  This  measure  is  simply  the 
log  (to  any  base)  of  equation  (2)  :  ^"^^^^^T^W^'^'O    ' 

LogOD,  =  logSo=aogQ3-logQO/2 (3!    >^3v^ 

Obviously  equation  (3)  may  be  computed  either  with  the  logs  direct 
or  the  log  of  So  may  be  found  in  logarithmic  tables. 

100 


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i   1^' 

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DIAMETER   IN  MM 

Fig.  10/. — Examples  of  cvimulative  curves.  A.  beach  gravel :  B,  beacli 
glacial  till;  D,  a  Pennsylvanian  underclay.  (After  Krumbein,  1936.) 


:and;  C, 


These  three  measures  each  have  certain  characteristics  which  may  be 
examined  by  means  of  a  few  samples.  Figure  107  shows  the  cumulative 
curves  of  four  sediments ;  a  beach  gravel,  a  beach  sand,  an  underclay. 
and  a  glacial  till.  Table  20  lists  the  median  and  quartiles  of  these  sedi- 
ments as  well  as  the  quartile  measures  defined  by  equations  (i).  (2), 
and  (3),  the  latter  measure  to  the  base  10. 

Inasmuch  as  the  QDa  measures  the  difference  between  the  quartiles,  its 
value  depends  both  on  the  size  of  particles  involved  and  on  the  units  of 
measurement  used.  The  values  for  the  beach  sand  and  the  gravel  illus- 
trate this  size  factor ;  similarly,  if  centimeters  were  used  instead  of  milli- 
meters, the  ODa  for  the  beach  sand  would  be  0.0060  instead  of  0.060. 


232 


SEDIMENTARY  PETROGRAPHY 


Thus  the  arithmetic  quartile  deviation  does  not  directly  compare  the  rela- 
tive spread  of  the  curves,  because  the  size  factor  colors  the  result.  This 
does  not  imply  that  the  QDa  finds  no  use  in  sedimentary  petrology ;  on 
the  contrary,  where  the  size  factor  is  to  be  brought  in,  it  shows  up  most 
clearly  with  this  arithmetic  form  of  the  quartile  deviation. 

Table  20 

Comparison  of  Quartile  Measures  of  Underclay,   Glacial  Till, 
Beach  Sand,  and  Beach  Gravel* 


Sediment 

Median 
{nun.') 

(nun.) 

Q3 

(mill.) 

QD, 

So 

LogioSo 

Underclay    

Glacial  till 

Beach  sand 

Beach  gravel   

0.004 
0.062 
0.3CO 
4.420 

0.002 

O.OIO 

0.240 

3.900 

0.008 
0.290 
0.360 
4.970 

0.003 

0.140 
0.060 
0.535 

2.00 

548 
1.22 
I-I3 

0.301 
0.740 
0.087 
0.054 

*  Data  from  Figure  107. 

The  geometric  quartile  deviation,  So,  being  essentially  a  ratio  between 
the  quartiles,  at  once  eliminates  the  size  factor  and  the  units  of  measure- 
ment. Thus  in  Table  20  the  beach  sand  and  the  beach  gravel  have  very 
similar  values  for  So,  showing  that  the  relative  spread  of  the  curves  is 
very  much  the  same.  This  is  borne  out  by  inspection  of  the  first  two 
curves  in  Figure  107.  Here  the  difference  in  coarseness  has  no  bearing 
on  the  sorting  coefficient,  nor  would  it  make  any  difference  if  centi- 
meters were  used  instead  of  millimeters  as  the  units  of  measurement. 
Thus  in  general.  So  is  a  convenient  measure  to  use  for  describing  the 
spread  of  the  curve,  uninfluenced  by  size  factors. 

On  the  basis  of  nearly  200  analyses,  Trask  found  that  a  value  of 
So  less  than  2.5  indicates  a  well  sorted  sediment,  a  value  of  about  3.0 
a  normally  sorted  sediment,  and  a  value  greater  than  4.5  a  poorly  sorted 
sediment.  These  numbers  do  not,  however,  lend  themselves  directly  to  a 
visualization  of  what  they  signify  in  terms  of  the  actual  spread  of  the 
curve.  That  is,  one  cannot  say  that  a  sediment  with  So  =  3.0  is  twice  as 
widely  dispersed  (i.e.,  half  as  well  sorted)  as  another  sediment  with 
So^  1.5.  This  is  because  the  values  of  So  are  geometric  rather  than 
arithmetic.  However,  it  is  a  simple  matter  to  transform  the  values  of 
So  into  measures  that  may  be  directly  compared  with  each  other. 

It  is  here  that  the  significance  of  the  log  quartile  deviation  becomes 
apparent.  Since  So  increases  geometrically,  the  logs  of  So  will  form  an 
arithmetic  series,  so  that  the  values  of  log  So  may  be  directly  compared 


STATISTICAL  METHODS  2^3 

with  each  other.  The  last  column  of  Table  20  lists  the  logs  of  So  to  the 
base  10.  as  an  illustration.  By  comparing  the  beach  sand  and  the  under- 
clay,  for  example,  we  may  see  that  the  spread  of  the  grains  in  the 
underclay  is  some  3.4  times  as  great  as  that  in  the  sand,  because 
0.301/0.087  =  3.4.  This  is  the  same  as  sa\-ing  that  the  sand  is  3.4  times 
as  well  sorted  as  the  underclay,  but  this  information  cannot  be  read  by 
comparing  the  So  values  directly. 

Although  logs  to  the  base  10  are  most  convenient  to  use  in  ordinary 
cases,  it  is  possible  to  choose  logs  to  such  a  base  that  they  describe  the 
sediments  in  terms  of  some  easily  visualized  characteristic.  For  example, 
if  one  were  able  to  say  that  some  sediment  A  had  two  Went  worth  grades 
between  the  first  and  third  quartiles,  as  against  sediment  B.  which  had 
three  Wentworth  grades  between  the  quartiles,  one  would  have  not  only 
an  easily  visualized  measure,  but  as  well  one  that  would  satisfy  tlie  con- 
dition that  the  relative  spread  of  the  curves  could  be  directly  compared. 

Application  of  the  phi  scale  to  the  qnartUc  dcziation.^  It  is  in  connection 
with  logarithmic  measures  that  the  phi  scale  (see  Chapter  4)  is  most  useful. 
Fisrure  loS  shows  the  ordinary  diameter  scale  above  and  tlie  phi  scale  below. 
Each  Wentwortli  class  limit  is  an  integer,  and  tlie  phi  scale  increases  with 

ZETA      SCALE  ► 


20  109  8    7    6     5      4         3  2  I  :  9  8   7    -6     5     -4         3  2 


Fig.  io8. — Relations  between  logarithmic  grade  scales  and  diameters  in  millimeters. 
The  "zeta  scale"  is  adapted  to  Atterberg  grades,  and  the  "phi  scale"  to  Wentworth 
grades. 

decreasing  grain  sizes.  Since  tlie  phi  inter\-als  are  equal,  ordinarj-  aritlimetic 
graph  paper  may  be  used  in  plotting  cur\-es,  and  the  median  and  quartiles 
may  be  read  ott  in  phi  values  directly,  to  the  nearest  tentli  or  hundreddi,  as 
desired.  Figure  109  illustrates  a  case;  it  is  the  glacial  till  of  Figure  107 
superimposed  on  the  phi  scale.  The  curve  is  in  no  wise  changed;  only  tlie 
independent  variable  has  been  changed,  and  the  graph  paper  is  arithmetic 
instead  of  logarithmic.  The  position  of  the  quartiles  is  conventional  also,  and 
since  phi  increases  to  tlie  right,  Q3  is  greater  than  Q^.  The  tliree  values 
obtained  from  tliis  curs-e,  determined  in  tlie  usual  manner  of  reading  tlie 

^  This  discussion  is  based  largely  on  the  paper  by  Krumbein,  loc.  cit.,  1936. 


234 


SEDIMENTARY  PETROGRAPHY 


median  and  quartiles,  and  expressed  in  (f>  terms,  are  Md^  =  4.00;  Qi^  =  1.80; 
Q30  —  6.70.  Inasmuch  as  the  independent  variable  <P  forms  an  arithmetic 
series,  it  is  possible  to  substitute  these  values  directly  in  equation  ( i )  for 
the  arithmetic  quartile  deviation  in  phi  terms,  calling  the  result  QD^  to 
indicate  that  phi  values  are  used:   QD^  =  (6.70 — 1.80)72  =  2.45. 

The  geometrical  significance  of  this  value  in  terms  of  the  curve  may  readily 
be  seen.  Since  the  phi  values  are  expressed  in  Wentworth  grades  as  units,  the 
difference  between  the  quartiles  indicates  directly  how  many  Wentworth 
grades  lie  between  the  first  and  third  quartile,  and  half  this  value  is  the 
quartile  deviation.  Thus  in  the  glacial  till,  the  first  and  third  quartiles  are 


0^ 


Fig.  109. — Cumulative  curs'e  of  glacial  till  (curve  C  of  Fig.  107),  plotted  with  0 
as  independent  variable.    (After  Krumbein,   1936.)    See  page  237  for  ordinate  A. 


spread  over  a  distance  of  4.9  Wentworth  grades,  and  consequently  the  curve 
is  decidedly  drawn  out.  (This  spread  of  the  curve  may  actually  be  checked 
by  laying  a  ruler  between  the  quartiles  in  Figure  109.) 

The  value  QD0  =  2.45  may  be  converted  to  QDg  by  finding  its  antilog. 
Because  of  the  reversal  of  scale  direction  which  follows  the  use  of  <P,  the  third 
quartile  in  phi  terms  corresponds  to  the  first  quartile  in  diameter  terms  in  its 
position  on  the  scale.  This  means  that  QD^  =  log^,QDg,  rather  than  the 
negative  log,  as  one  may  expect.  To  find  the  antilog  of  QD^j,  accordingly,  one 
may  use  the  relation  log^r,"  =  logifj2  log^H,  where  logi„2  =  0.301.  Substituting 
QDg  for  ti,  and  QD^  for  logoQDg,  there  results  logioQDg  =  0.301  QD^. 
Using  the  value  2.45  for  QD^  one  obtains  logioQDg  =  0.738.  The  antilog  of 
this  is  5.47.  This  is  tlie  value  of  So  shown  for  the  glacial  till  in  Table  20. 

Instead  of  transforming  the  logarithmic  measure  to  its  diameter  equivalent 


■^ 


STATISTICAL  METHODS  2^^ 

by  the  process  outlined,  it  is  convenient  to  prepare  a  graph  which  permits 
a  direct  conversion.  Figure  no  is  such  a  cliart,  showing  the  values  of  So 
from  I  to  10  on  the  vertical  logaritiimic  scale,  and  log.  So  =  QD^  on  the 
horizontal  scale.  Note  tl:at  when  So  equals  2.  4,  8,  QD^  equals  I,  2,  3, 
respectively.  The  latter  values  tell  how  many  Wentwortli  grades  are  involved 
in  half  the  spread  between  the  quartiles,  for  any  given  value  of  So.  Thus 
when  So  =  5.48,  QD^  =  2.45.  which 
means  that  4.9  Wentworth  grades  lie 
between  the  two  quartiles,  and  tlie 
curve  is  drawn  out,  as  curves  of  gla- 
cial till  usually  are.  Since  QD^  is  a 
logarithm,  it  may  be  used  directly  in 
comparing  tlie  relative  spread  of  two 
or  more  curves,  as  noted  above,  and 
furthermore  this  comparison  is  ex- 
pressed directly  in  terms  of  the  num- 
ber of  Wentworth  grades  involved. 
The  value  of  the  phi  notation  in  tliis 
case  is  that  the  "sorting"  values  are 
expressed  in  terms  of  Wentworth 
grades,  which  are  easily  visualized. 
This  result  arises  from  a  deliberate 
choice  of  <P  to  satisfy  these  conditions. 

Quartile  skczcncss.  In  a  perfectly 
symmetrical  curve,  the  median  ex- 
actly coincides  with  the  point  half 

.  ^t        /•     j^         J     ^i_-   J         Fig.   iia — Conversion  chart  for   So 

way    betw^een   the    first    and    third    ^^^  qq^ 

quartiles,  but  if  the  curve  is  skewed, 

the  arithmetic  mean  of  the  quartiles  departs  from  the  median,  and  the 
extent  of  this  departure  may  be  taken  as  a  measure  of  the  skewness. 
Further,  the  direction  of  departure  may  be  included  in  the  measure  by 
assigning  positive  and  negative  senses  to  the  two  possible  directions.  The 
simplest  form  of  this  skewness  measure  is  the  arithmetic  case.  Sk^. 
which  expresses  the  difference  between  the  arithmetic  mean  of  the  qtiar- 
tiles  and  the  median : 


Ska-  [(Q.-^03V'J] 


Md. 


(0,  +  03-2Md) 


(4^ 


This  skewness  measure  may  be  cast  into  a  geometric  form,  which  is 
the  square  root  of  the  ratio  of  the  product  of  the  quartile  to  the  square 
of  the  median : 


Skg=  VQiQs/Md-  (5) 

There  is  an  interestins:  relation  between  this  measure  of  ske\^■ness  and 


236 


SEDIMENTARY  PETROGRAPHY 


the  form  introduced  by  Trask,^  here  referred  to  simply  as  Sk.  In  develop- 
ing his  measure,  Trask  compared  the  ratio  of  the  largest  quartile  and 
the  median  to  the  ratio  of  the  median  and  the  smaller  quartile,  thus : 
Sk  =  03/Md/Md/Oi.  This  simplifies  to  Sk  =  0i03/Md-,  which  is 
obviously  the  square  of  Skg. 

The  relation  between  Trask's  measure  and  Skg  may  be  seen  readily  by 
considering  the  third  possibility,  the  log  geometric  skewness,  log  Skg : 


log  Skg  =3^  (log  Qi-f  log  Q3-2  log  Mdj 


(6) 


This  is  obviously  the  log  of  equation  (5)  to  any  base.  If  one  takes  logs 
of  Trask's  ske\%Tiess  measure,  it  will  be  noted  that  Yz  log  Sk  =  log  Skg. 
Table  21  offers  a  comparison  of  the  skewness  values  of  the  four  sedi- 
ments of  Table  20,  in  terms  of  equations  (4),  (5),  (6).  In  addition, 
the  last  column  includes  a  phi-ske\\Tiess  to  be  introduced  below. 


Comparison  of 

Quartile 

Beach  S 

Table  21 

Skewxess  of  Uxdeeclay,   Glacial  Till, 
axd,  axd  Beach  Gravel  * 

Scd'uncnt 

Sk, 

Sk=(Skg)^  i  2  log,.sk,  =log,,Sk 

Skq0 

Underclav 

Glacial  till   

Beach  sand    

Beach  gravel  .  .  . 

+0.001 

+0.C88 

0.000 

—0.053 

1. 00 
0.74 
0.96 

0.99 

0.000 
—0.130 
— 0.018 

—0.004 

0.000 
+0.250 
+  0.030 
+  0.007 

*Data  from  Figiire  107. 

The  arithmetic  skewness,  Ska,  is  subject  to  the  same  comments  that 
apply  to  the  arithmetic  quartile  deviation,  ODa,  inasmuch  as  the  size 
factor  and  the  units  of  measurement  enter  its  values.  Thus  when  QDa 
is  used  to  bring  out  the  size  factor,  presumably  the  corresponding  Ska  is 
the  best  measure  to  supplement  it.  In  this  connection,  however,  see  the 
comments  under  kurtosis  (page  2"^^^)  concerning  an  arithmetic  measure 
of  skewness  independent  of  size. 

The  geometric  skewness,  or  its  square,  which  is  identical  with  Trask's 
skewness,  eliminates  the  size  factor  and  units  of  measurement  from  the 
resulting  values,  so  that  it  is  a  descriptive  measure  independent  of  these 
two  factors.  When  the  curve  is  syinmetrical.  this  skewness  is  equal  to 
unit>%  but  the  values  obtained  range  from  numbers  less  than  i  to  numbers 
larger.  As  Trask  himself  points  out,  the  significance  of  this  depends  on 


1  P.  D.  Trask,  op.  cit.  (1932). 


STATISTICAL  METHODS 


237 


the  fact  that  numbers  less  than  i  present  a  reciprocal  relation  to  num- 
bers greater  than  i,  so  that  actually  the  spread  is  greater  on  one  side  of 
the  curve  in  the  one  case,  and  on  the  other  side  in  the  other  case.  Thus 
Sk  is  itself  not  an  easily  visualized  measure,  because  reciprocals  are  often 
hard  to  visualize.  For  this  reason  Trask  introduced  logmSk,  which  is 
positive  when  Sk  is  greater  than  unity,  and  negative  when  Sk  is  less 
than  unity. 

Again,  however,  a  measure  based  on  log.Sk  will  yield  easily  visualized 
values,  because  such  a  measure  will  directly  express  the  skewness  in  terms 
of  its  definition,  namely,  the  extent  to  which  the  mean  of  the  quartiles  departs 
from  the  median.  When  the  median  and  quartiles  are  expressed  in  <i>  values. 


-■-    ''s 

': 

r::::::::::-::::::: 

:::: 

mmiiiM 

■^||-:-^;^;::^-;^;--:-:|:i 

;; 

;:    Tt"-    ..:_;:-;:-_-:.:-';.:         ::;.::_:::/     :::; 

:::::::::::::-; 

--:: 

t:--(vi---  +  =--- -r ---;;:--:;;-:-::::: 

::  :  :        ::  :: 

::::::;:: 

K^ 

K^   :'?>r'c.r^i]   j(>>-  C'.!r(Mi   CU.  C.E  "- 

:i:::i:-^;::-5 

:; 

k:>:Hn----$. --■-.?■ 

:    ^-,   :.:       ;::_:::;„__::;::;;.-::-- ^              i 

^/              :      H,  ::::;;:;::::;::::: : 

3c;::        ::::::'  =  ::::::::::::::::::::::::::: 

-  -  -t 

-  _,  f  f|y|||5ffflT^^ 

:  :::::::: 

:::il::;:;;;: 

Hi 

-     - 

p^^"#-::;;";"-^;^;;|;-;;;;^;;; 

Skq^ 


Fig.  III. — Conversion  chart  for  logioSk  and  Skq,. 


equation  (4)  may  be  used.  The  ^-values  for  the  till  in  Figure  109  were,  it 
will  be  recalled,  ]Md,^  =  4.00  ;  Q,^  =  1.80  ;  Q.,,;,  =  6.70.  By  substitution  of  these 
values  in  equation  (4)  there  results  Skq^  =(  1.80  +  6.70  — 2 [4.00])  = 
+  0.25,  where  Skq,;,  is  the  symbol  for  the  phi  quartile  skewness.^ 

Figure  109  illustrates  the  geometrical  meaning  of  Skq^.  The  ordinate  A 
marks  the  position  of  the  arithmetic  mean  of  Q^  and  Q;,  expressed  in  phi 
terms.  Its  value  is  (Qi  +  Qa)/^  =  (1.80  +  6.70) /2  =4.25.  Note  that  4.25 
—  4.00=  +0.25,  the  value  of  Skq,^.  Thus  the  mean  of  the  quartiles  lies  0.25 

1  Elsewhere  the  symbol  Sk .  is  used  to  indicate  the  phi  skewness  based  on  the 
third  moment  of  the  distribution,  hence  Skcj^  is  used  for  the  present  measure. 


2zd>  SEDIMENTARY  PETROGRAPHY 

of  a  Wentworth  grade  to  the  right  of  Md,/,,  and  the  curve  is  skewed  in  the 
direction  of  the  positive  0  axis.  (This  direction  of  skew  is  indicated  by  the 
+  sign;  a  negative  value  would  indicate  that  the  mean  of  the  quartiles  lies 
toward  smaller  values  of  <P\  i.e.,  to  the  left.) 

The  relation  of  Skq,^  to  Trask's  Sk  may  be  found  directly  from  the  relation 
between  Trask's  Sk  and  equation  (6)  above:  Skq^  =  —  logoSkg  =  —  J4 
logoSk  =  Yz  logo  (i/Sk).  In  words,  Skq^  is  one-half  the  log  to  the  base  2  of 
the  reciprocal  of  Trask's  Sk.  This  relation  yields  to  a  conversion  chart,  how- 
ever, if  logioSk  is  used  instead  of  Sk.  Figure  iii  is  such  a  chart,  based  on  the 
equation  Skq0  =  (log^oSk)/ —  0.602,  the  necessary  transformation  equation 
for  the  conversion  of  one  symbol  to  the  other. 

In  Table  21  the  phi  quartile  skewness  of  the  four  sediments  is  included 
in  the  last  column.  These  values  are  directly  interpretable  in  terms  of  Went- 
"A^orth  grades.  Thus,  the  underclay  is  symmetrical  because  Skq^  is  zero;  the 
beach  sand  is  skewed  0.03  to  the  right;  and  the  gravel  is  skewed  in  the  same 
direction,  but  to  a  less  extent.  It  is  to  be  noted  that  the  sign  before  Skq^ 
is  opposite  to  that  before  logmSk,  This  is  a  convention  in  terms  of  the 
change  of  variable,  and  is  consistent  with  the  phi  notation.  The  curve, 
whether  expressed  in  phi  terms  or  in  diameters,  is  of  course  skewed  in  the 
same  direction ;  merely  the  sense  assigned  to  the  direction  is  changed. 

Quartile  kurtosis.  The  degree  of  peakedness  of  a  curve  is  measured  by  its 
kurtosis,  which  may  be  defined  in  various  ways.  Essentially  the  kurtosis  in- 
volves a  comparison  of  the  spread  of  the  central  position  of  the  curve  to  the 
spread  of  the  curve  as  a  whole.  To  obtain  a  measure  of  kurtosis,  one  may 
adopt  Kelley's  equation,^  which  is  the  ratio  of  the  quartile  deviation  to  that 
part  of  the  size  range  which  lies  between  the  lo-per  cent  and  90-per  cent 
lines.  The  latter  values  may  be  referred  to  as  the  tenth  percentile,  Pjo,  and 
the  ninetieth  percentile,  Pgo-  The  arithmetic  quartile  kurtosis  may  then  be 
written  as 

Qa-Qx 

Kqa=p     fp      =(Q3-Qi)/2(P9o-Pio)      ...      (7) 

Equation  (7),  it  will  be  noted,  is  independent  of  size  or  the  units  of  meas- 
urement used,  inasmuch  as  it  represents  a  ratio  of  two  spreads.  In  this  manner 
it  differs  from  the  quartile  deviation  (equation  i),  and  the  quartile  skewness 
(equation  4),  both  of  which  are  influenced  by  size  factors. - 

It  is  not  feasible  to  introduce  a  simple  geometrical  measure  of  kurtosis 
based  on  equation  (7)  ;  however,  the  corresponding  phi  quartile  kurtosis,  Kq0, 
may  be  obtained  merely  by  using  phi  values  in  equation  (7).  The  geometrical 
picture  of  the  measure  is  the  same  as  in  the  aridmietic  case,  except  that  it 
applies  to  the  logarithmic  curve  instead  of  to  the  arithmetic  grain  diameter 
distribution. 

IT.  L.  Kelley,  Statistical  Methods  (London,  1924),  P-  77- 

2  In  conventional  statistical  practice,  especially  in  connection  with  moment  analy- 
sis, both  skewness  and  kurtosis  are  chosen  to  be  independent  of  size.  An  arithmetic 
quartile  skewness  having  this  attribute,  and  expressed  essentially  in  units  of  quar- 
tile deviation,  is  discussed  by  F.  C.  Mills,  op.  cit.   (1924),  p.  167. 


STATISTICAL  METHODS  239 

The  kunosis  as  defined  above,  as  well  as  its  phi  analogue,  yields  values 
which  decrease  with  increasing  peakedness.  in  the  sense  that  as  the  cluster 
of  values  in  the  central  part  of  the  curve  becomes  more  pronounced,  without 
a  corresponding  decrease  in  the  total  spread  of  the  cur\e,  the  ratio  of 
(Qs  — Qi)/-  to  (P90  — Pio)  decreases  also.  If  one  prefers  a  kunosis  value 
which  increases,  tlie  reciprocal  of  equation  (7)  appears  to  be  suitable. 


M  O  M  E  N  T     MEASURES 

Despite  the  wide  usage  of  moments  in  conventional  statistics,  they 
have  found  relatively  Httle  appHcation  in  sedimentar}-  analysis  until 
recently.  The  earliest  applications  of  moments  to  sedimentar}-  data  were 
made  by  \'an  Orstrand^  in  1924;  and  by  Wentworth-  and  Hatch  and 
Choate^  in  1929.  \'an  Orstrand  discussed  the  possibilit)-  of  representing 
frequency  distributions  of  sediments  by  means  of  Pearson  frequency 
functions.  He  computed  the  mode,  the  arithmetic  mean,  standard  de\-i- 
ation,  and  skewness  of  sands,  arguing  in  favor  of  arithmetic  measures 
based  on  equal  class  intervals.  \\'entworth  used  logarithmic  methods  of 
computing  his  moments,  but  referred  to  his  measures  as  though  they 
were  arithmetic  instead  of  logarithmic  in  nature.  Hatch  and  Choate 
detined  their  measures  as  log  geometric  moment  measures  and  confined 
their  theor}-  largely  to  the  moments  of  logaritlimically  S}-mmetrical 
curves  which  could  be  treated  graphically.  In  1936  Krumbein*  devel- 
oped a  series  of  logarithmic  moment  measures  by  means  of  a  logaritlimic 
transformation  equation  and  showed  the  relation  of  his  measures  to  the 
body  of  statistical  theorj-, 

A  consideration  of  moments  as  applied  to  sediments  should  distin- 
guish between  arithmetic  and  logarithmic  measures,  because  the  geomet- 
rical and  physical  significance  is  considerably  different  in  the  two  cases. 
As  in  the  discussion  of  quartile  measures,  the  treatment  of  aritlunetic, 
geometric,  and  logarithmic  measures  will  be  carried  on  simultaneously, 
so  that  similarities  and  differences  may  be  brought  out  as  the  discussion 
proceeds. 

1  C.  E,  \'an  Orstrand,  Note  on  the  representation  of  the  distribution  of  grains  in 
sands :  Researches  in  Sedimentation  in  1924..  Nat.  Research  Council.  1925. 

-  C.  K.  Wentworth.  Method  of  computing  mechanical  composition  tj-pes  in  sedi- 
ments:  Gcol.  Sec.  America.  Bulletin,  vol.  40,  pp.  771-790,  1929. 

3  T.  Hatch  and  S.  Choate.  Statistical  description  of  the  size  properties  of  non- 
uniform particulate  substances:  Jour.  Franklin  Inst.,  vol.  207.  pp.  369-3S7.  1920. 

■*  \V.  C.  Krumbein.  Application  of  logarithmic  moments  to  size  frequency  distri- 
butions of  sediments :  Jour.  SeJ.  Petrology,  vol.  6,  pp.  35-47,  1936. 


240 


SEDIMENTARY  PETROGRAPHY 


Measuhes  of  the  Cextral  Tendency 

The  arithmetic  mean  of  the  diameter  distribution.  The  arithmetic 
mean  *  of  the  diameter  distribution  is  most  convenient!)-  calculated  from 
the  frequenc)'  distribution,  although  it  may  be  determined  graphically.- 
In  computing  the  arithmetic  mean,  a  procedure  such  as  that  shown  in 
Table  22  may  be  used.  The  actual  grades  in  millimeters  are  listed  in  the 
first  column,  and  the  weight  percentage  frequency  is  placed  in  the 
second  column.  The  third  column  has  the  actual  midpoints  (;n)  of  each 
grade,  r^;ardless  of  whether  the  grades  are  equal  or  unequal  in  interval. 
In  the  fourth  column  the  frequencj-  has  been  multiplied  by  the  midpoint, 
(/»«),  and  the  stun  of  the  nimibers  in  this  column  is  43.97.  This  sum, 
divided  by  the  total  frequency  (100 1,  pelds  the  arithmetic  mean  di- 
ameter in  millimeters,  Ma  =  0440  mm.,  approximately. 


COMPUT. 


Table  22 
T  THE  Arithmetic  Mean  Diameter  of  a  Beach   Sand 


Grade  Size 
{mm.) 


Weight  Percentage 

Frequency  (f) 


2 — I   ... 

I— 1/2  . 
1/2^1/4  . 
1/4— 1/8  . 

i^g— I  '16 


0.^ 

2-0 

i.:;o 

5-6 

1-5 

8.40 

1 1.7 

0.75 

8.78 

53-7 

0-375 

20.15 

264 

0.187 

494 

2.1 

0.093 

0.20 

Totals 


43-97 


*  Sample  22  of  F.  J.  Pettijohn  and  J.  D.  Ridge,  A  textural  variation  series  of 
beach  sands  from  Cedar  Pcont,  Ohio :  Jour.  Sed.  Petrology,  vol.  2,  pp.  76-^,  1932. 

The  arithmetic  mean  diameter  in  millimeters  represents  the  diameter- 
value  of  the  center  of  gravity  of  the  frequency  distribution.  The  arith- 
metic mean  is  aflFected  by  everj^  grain  in  the  distribution,  and  in  some 
respects  it  is  therefore  more  tj-pical  of  the  grain  distribution  than  the 
median.  The  arithmetic  mean  may  be  manipulated  algebraically. 

TJie  geometric  mean.^  The  geometric  mean  diameter  of  sediments  has 
not  been  used  extensively  in  sedimentan,-  work,  although  in  significance 
it  appears  to  rank  as  more  important  than  some  other  means  that  have 

1  F.  C.  Mills,  op.  cit.  (1924),  pp.  113  fF. 

2  See  page  255,  mider  Baker's  equivalent  grade. 
»F.  C  Mills,  op.  cit.  (1924),  pp.  135  fiE. 


STATISTICAL  METHODS 


241 


been  used.  The  geometric  mean  diameter  is  defined  as  the  ;fth  root  of  the 
product  of  n  items,  and  its  direct  computation  is  a  tedious  process.  By 
means  of  a  simple  logarithmic  device,  however,  a  fair  approximation  of 
the  geometric  mean  may  be  obtained  from  the  frequency  data.^  The 
method  used  is  similar  to  that  used  in  computing  the  arithmetic  mean, 
except  that  the  logarithms  of  the  midpoints  of  each  grade  are  substituted 
for  the  midpoint  itself.  Table  23  illustrates  the  method.  The  first  column 
lists  the  grade  sizes,  the  second  column  shows  the  weight  percentage 
frequency  (/),  the  third  column  shows  the  midpoint  (///)  of  each  grade, 
and  the  fourth  column  has  the  log  -  of  the  midpoint  to  the  base  10.  The 

Table  23 
Computation  of  the  Geometric  Mean   Diameter  of  a   Beach   Sand* 


Grade  Sice 
(mm.) 

/ 

in 

log  m 

/  log  m 

0.5 
5-6 

11.7 

J6.4 
2.1 

30 
1-5 
0-75 

•375 

■1875 

•0937 

+  .477 
+  .i;6 

-  -125 

-  .426 

•7~7 
-I.ojS 

+  0.238 

+  0.097 

I— I  /2 

-  1.46 

1/2     I  /a 

-22.82 

I/4-I/8    

1/8— 1/16 

—  19.20 

-  2.16 

Totals                              .... 

1 00.0 

-45.40 

*  Tlie  frequency  data  are  the  same  as  in  Table  22. 

last  column  has  the  products  fm.  The  algebraic  sum  of  the  products  is 
—45.40.  This  is  divided  by  the  total  frequency,  100,  to  yield  —0.454, 
which  is  the  log  of  the  geometric  mean.  To  convert  it  to  a  value  which 
may  be  found  in  common  log  tables,  it  is  added  to  10.000 —  10,  yielding 
9.546  —  10,  the  antilog  of  which  is  0.352  mm. 

The  geometric  mean  diameter  is  noticeably  smaller  tiuin  the  arithmetic 
mean  diameter,  as  computed  for  the  same  sediment  in  Table  22.  This 
means  that  the  geometric  mean  lies  to  the  right  of  the  arithmetic  mean 
as  usually  plotted,  in  the  cluster  of  grains  near  the  higher  part  of  the 
frequency  curve.  It  is  thus  associated  with  the  most  abundant  grains  in 


1  For  a  praphic  method  applicable  in  some  cases,  see  page  254. 

-  For  numbers  smaller  than  unity,  the  logarillim  as  obtained  in  ordinary  \og  tables 
must  be  converted  to  its  coloRaritlim.  For  example,  log  0.750=0.875—10.  By  adding 
+9.875  and  — 10.000,  one  obtains  —0.125,  the  value  used  in  the  computations  in 
Table '23. 


242 


SEDIMENTARY  PETROGRAPHY 


an  asymmetrical  distribution.  The  geometric  mean,  like  the  arithmetic 
mean,  is  affected  by  every  grain  in  the  distribution,  but  the  geometric 
mean  is  not  affected  to  the  same  degree.  The  geometric  mean  may  be 
manipulated  algebraically. 

The  logarithmic  mean.  Tlie  use  of  a  logarithmic  mean,  expressed  and  used 
directly  as  a  logarithm,  has  received  relatively  litde  attention  in  sedimentary 
work.  Such  a  mean  is  defined  as  the  arithmetic  mean  of  the  logarithmic  fre- 
quency distribution,  and  it  is  most  conveniendy  computed  by  transforming 
the  grade  scale  in  millimeters  to  the  logarithmic  phi  scale  or  zeta  scale  men- 
tioned in  Chapter  4  and,  in  connection  with  quartile  measures,  in  the  present 
chapter.  The  logarithmic  mean  may  be  computed  in  a  manner  exactly  analo- 
gous to  that  of  the  arithmetic  mean,  using,  however,  the  midpoints  of  the 
logarithmic  grades. 

Table  24  shows  die  limits  of  the  diameter  classes  in  the  first  column.  The 
corresponding  phi  values  from  table  10  (page  84)  are  listed  in  the  second 
column.  Tbe  frequency  (/)  is  shown  in  the  third  column,  and  the  midpoints 
of  the  phi  classes  are  shown  in  the  fourth  column.  The  products  fvi  are  listed 
in  the  fifth  column.  The  algebraic  sum  of  the  numbers  in  the  last  column, 
+  156.15,  divided  by  the  total  frequency  (100),  yields  the  arithmetic  mean 
of  the  phi  distribution,  called  the  "phi  mean,"  M^  =1.651. 

Table  24 
Computation  of  the  Logarithmic  ^^Ieax  of  a  Beach  Sand* 


Grade  Size 
(mm.) 

^ 

/ 

m 

fm 

A — 2 

—2 1 

—  I —      0 

0 —  I 

1 —  2 

2—  3 

3—  4 

0-5 

5-6 

11.7 

53-7 

26.4 

2.1 

-1-5 
-0.5 
+0.5 
+  1-5 
+2.5 
+3-5 

—  0.75 

I-v^ 

I/2-V4    

I/4-I/8    

1/8— 1/16    

-  2.80 

+  5-85 
+80.50 
+66.00 
+  7.35 

Totals    

lOO.O 

+  156.15 

The  frequency  data  are  the  same  as  in  Table  22. 


The  phi  mean  is  die  center  of  gravity  of  the  logarithmic  frequency  curve, 
expressed  with  0  as  the  independent  variable.  It  thus  has  exacdy  the  same 
relation  to  the  logarithmic  curve  as  the  arithmetic  mean  of  the  diameters  has 
to  the  frequency  curve  drawn  with  diameters  in  millimeters  as  the  independent 
variable.  One  should  not  confuse  the  two,  however.  The  phi  mean,  when 
transformed  to  its  diameter  equivalent,  becomes  the  geometric  mean  of  the 


STATISTICAL  METHODS  243 

size  distribution.  In  words  the  phi  mean  is  the  negative  log  to  the  base  2  of 
the  geometric  mean  of  the  grain  diameters.^ 

The  phi  mean  may  be  used  directly  in  describing  sediments,  in  connection 
with  other  logarithmic  measures. 

The  method  used  in  computing  M^  may  also  be  used  as  a  more  accurate 
method  of  finding  the  geometric  mean  of  tlie  diameters.  To  convert  the  phi 
mean  to  its  diameter  equivalent,  the  antilog  of  1.561  must  be  found.  To  con- 
vert any  value  in  the  phi  notation  to  its  corresponding  diameter  equivalent  in 
millimeters,  the  relations  (j>  —  —  logol  and  log^^^  =  logio2  logo?  are  used, 
where  log^o^  =  0.301.  Substituting  —  <P  for  logol,  the  relation  log^,,^  — 
—  0.3010  is  obtained,  and  from  this  equation  the  antilog  of  logj,,!  may  be 
found  in  logarithmic  tables.  In  the  example  given,  M^  =  1.561.  Multiplying 
this  by  —0.301  yields  —0.469.  The  colog  of  — 0.469  is  obtained  by  adding 
this  value  to  10.000—  10,  which  yields  9.531  —  10.  The  antilog  of  the  latter 
expression,  from  any  common  table  of  logs  to  the  base  10,  is  0.340  mm.- 

It  is  more  convenient  to  use  a  graphic  method  for  converting  phi  values 
to  their  diameter  equivalents.  Figure  112  is  such  a  graph,  showing  0  as 
ordinate  and  diameters  in  millimeters  as  abscissae.  The  value  <^  =1.561  is 
chosen  along  the  vertical  scale,  and  where  this  value  intersects  the  diagonal 
line  an  ordinate  is  dropped  to  the  millimeter  scale,  yielding  the  value  0.340  mm. 

TJie  mode.  An  average  which  is  used  rather  frequently  in  conventional 
statistics  is  the  mode.^  The  modal  grain  diameter  may  be  defined  as  that 
diameter  which  is  most  frequent  in  the  distribution.  The  mode,  therefore, 
lies  directly  at  the  peak  of  the  curve,  and  it  may  be  determined  graph- 
ically either  by  locating  the  highest  point  of  the  frequency  curve,  or  by 
finding  the  point  of  inflection  of  the  cumulative  curve.  The  modal 
diameter  of  the  beach  sand  in  Table  22  is  0.300  mm.  The  mode  has 
not  .been  extensively  used  in  sedimentary  work,  but  it  is  an  average 
which  represents  the  most  abundant,  and  therefore  the  most  typical, 
grain  in  the  distribution.  The  mode,  like  the  median,  is  an  average  of 
position,  and  is  not  affected  by  extreme  grain  sizes  in  the  distribution. 

(pi  +  <p.  + +  <Pn 

^  The  arithmetic  mean  of  a  series  of  phi  values  is  M  j,  =  and 

^  n 

by  substituting  —  logs?  for   (p,   one  obtains   M^  =  — — ^I^^^ 2l^ — ^^° 

n 
Such  a  sum  of  logarithms  is  equal  to  the  nth  root  of  the  product  of  the  antilogs, 

which  yields  \/ ^^^.-, ^^.  By  the  definition  of  the  geometric  mean,  however,  this 

last  expression  is  seen  to  be  the  geometric  mean  itself.  Hence  the  arithmetic  me*in 
of  the  phi  distribution  is  a  log  of  the  geometric  mean  of  the  diameters. 

-  The  value  of  the  geometric  mean  obtained  by  the  phi  method  is  about  3  per  cent 
smaller  than  that  found  by  the  earlier  method  of  computing  the  geometric  mean. 
This  difference  depends  upon  the  precise  midpoint  used  in  the  computation.  The  phi 
method  uses  the  midpoint  of  the  logarithmic  classes  directly  and  yields  a  more  rigor- 
ous value.  However,  the  approximation  furnished  by  the  first  method  is  sufficiently 
close  for  most  work. 

3  F.  C.  Mills,  o/".  cit.  (1924),  pp.  124  ff. 


STATISTICAL  METHODS 


245 


The  modal  diameter  cannot  be  manipulated  algebraically,  and  it  has 
its  most  precise  meaning,  perhaps,  in  a  uninifKlal  distribution,  with  a 
sinj,de  peak.  Curves  which  display  several  i)eaks  have  a  corresponding 
number  of  modal  diameters,  but  it  is  not  appropriate  in  such  cases  to 
refer  to  the  mode  unless  one  of  the  peaks  predominates  strikingly  over 


1 

Mode    - 

Median    — 17~;7~ -j 

1 

Arithmetic  Mean 

/ 

\ 

) 

_____-- 

y 

—  DIAMETERS  IN    MM. 

Fig.  113. — Frequency  curve  of  beach  sand  with  ilianiclcr  in  mm.  as  independent 
variable.  The  several  averages  are  from  Table  25. 

the   others.   Frequency  curves   of   glacial   till,   either   on   arithmetic   or 
logarithmic  scales,  commonly  display  several  modes. 

Comparison  of  average  values.  The  variety  of  averages  discussed  in 
preceding  sections  indicates  the  wide  choice  of  measures  available  for 


Table  25 
Comparison  of  Average  Values  Computed  in  Tables  22  to  24 


Average 

Value 

Arithmetic   Mean,   M^ 

Geometric  Mean,  M,, 

Logarithmic  Mean,  M<P 

Median  Diameter,  Md 

Modal   Diameter,  Mo 

0.440  mm. 
0.3 S2  mm. 
1. 56 1* 
0.320  mm. 
0.300  mm. 

*  The  diameter  efjuivalcnt  of  this  value,  0.340  mm.,  is  the  geometric  mean.  Sec 
footnote  2  on  page  243. 

sedimentary  work.^  It  is  instructive  to  compare  the  geometrical  sig- 
nificance of  the  averages  in  the  exami)le  used  for  computation.  The  sev- 
eral values  found  for  the  beach  sand  are  shown  in  Table  25,  which  also 
includes  the  median  and  modal  diameters  for  comparison.  Figure   113 

1  Other  averages,  such  as  tlie  surface  reciprocal  mean,  are  discussed  under  the 
later  heading  "Fineness  factor." 


246 


SEDIMENTARY  PETROGRAPHY 


is  the  frequency  curve  of  this  sediment  drawn  with  diameter  in  milli- 
meters as  independent  variable.  The  diameter  values  of  the  several  av- 
erages are  indicated  by  labeled  ordinates.  The  median,  mode,  and 
geometric  mean  are  clustered  near  the  high  point  of  the  curve,  whereas 
the  arithmetic  mean  is  to  the  left.  As  Table  25  indicates,  the  arithmetic 
mean  is  the  largest  value.  In  all  cases  the  averages  are  drawn  away  from 
the  center  of  the  range  (4  mm.  —  .06  mm.).  Figure  114  shows  the 
same  curs'e  plotted  on  a  logarithmic  scale.  (The  phi  scale  is  indicated 
below  for  convenience.)  The  curve  has  become  much  more  symmetrical, 


DIAMETER  IN  MM. 


i 

i    3       2 

1 

1 

OA     .6 

4 

.2 

0  1    .07 

Mode — ____^ 

Median--Zl3l]~~~~" i 

V 

^«n»«>«.*»;<-     lilT^iC^^-^-                i 

1 

>- 
0 

z 

Arit 

imetic  >i 

can,     X 

\    / 

UJ 

0 

\i 

/ 

UJ 

J 

q: 

X 

u. 

/ 

^i 

Vi 

.301         0.00      -301      -.602     -.903 

^LOG«    DIAMETER 


Fig.  114. — Frequency  data  of  fig.  113  plotted  with  logwdiameter  as  independent 
variable.  Note  shift  in  position  of  average  values.  The  diameter  scale  and  phi  scale 
are  added  for  comparison.  The  figure  has  been  enlarged  to  show  the  relations  clearly. 


and  the  log  of  the  median,  the  log  of  the  mode,  and  the  log  of  the 
geometric  mean  Tthe  phi  mean;  are  essentially  at  the  center  of  the  curve. 
The  log  of  the  arithmetic  mean,  however,  has  been  drawn  relatively  to 
the  left,  and  no  longer  occupies  a  central  position.  In  terms  of  logarith- 
mic frequency  curves  (or  logarithmic  histograms)  the  arithmetic  mean 
is  not  so  truly  a  measure  of  the  central  tendenc\-  as  the  other  averages. 

A  comparison  of  values  such  as  the  foregoing  is  important  in  deciding 
upon  the  appropriate  average  to  use  in  a  given  case.  The  physical  and 
geometrical  significance  of  the  values  changes  when  logarithmic  curves 


STATISTICAL  METHODS 


247 


are  drawn,  and  the  practical  worker  should  understand  the  necessity  of 
making  his  physical  or  geological  interpretation  of  the  curve  conform 
to  its  geometrical  picture. 

Measures  of  Dispersion 

Measures  of  spread  or  dispersion  about  a  central  value  may  be  set  up 
with  respect  to  the  median,  the  arithmetic  mean,  or  any  arbitrary  central 
point  in  the  distribution.  As  in  the  case  of  other  measures  they  may  be 
arithmetic,  geometric,  or  logarithmic  in  nature. 

I\fcan  deviation.  The  mean  deviation,  which  is  used  to  some  extent  in 
conventional  statistics,  is  a  measure  of  the  average  spread  of  the  data 
about  a  mean  value.^  In  this  case  the  mean  value  is  chosen  either  as  the 
arithmetic  mean  or  the  median.  In  the  example  to  be  given  the  arithmetic 
mean  will  be  used.  In  words,  the  mean  deviation  is  i/N  of  the  sum  of 
the  deviations  from  the  mean,  without  regard  to  whether  the  deviations 
are  to  the  right  or  left  of  the  mean.  In  computing  this  measure.  Table 
26  is  set  up.  In  the  first  column  the  grades  in  millimeters  are  shown.  The 
second  column  has  the  percentage  frequency,  (/),  and  the  third  column 
shows  the  mid-point  (m)  of  each  grade.  The  fourth  column  gives  the  ab- 
solute value  of  the  deviation  of  each  mid-point  from  the  arithmetic  mean, 
Ma  =  0.440  mm.  The  fifth  column  shows  the  frequency  multiplied  by 
the  deviation,  and  the  sum  of  the  products  from  this  column,  21.91,  is 
written  below.  By  dividing  this  sum  by  the  total  frequency,  the  mean 
deviation,  (/a  =  0.219,  is  found. 

Table  26 
C0MPUT.A.T10N  OF  THE  Mean  Deviation  of  a  Beach  Sand* 


Grade  Sise 
(mm.) 


Malt 


M.. 


4—2  . . . 
2 — I  . . . 

I — 1/2 
1/2— 1/4 
1/4-1/8 
1/8— 1/16 


0.5 

5-6 
11.7 
53-7 
264 

2.1 


3-0 

1-5 

0-75 

0.375 

0.187 

0.093 


2.56 
1.06 
0.31 
0.07 
0.25 
0.35 


1.28 

5-93 
3.62 

3-75 
6.60 

0.73 


Totals 


:i.9i 


*  The  frequency  data  are  the  same  as  in  table  22. 

fThe  symbol  \m  —  M^|  refers  to  the  absolute  value  of  the  difference,  regardless 
of  sign.  In  this  example,  M^  =  0.440  mm.,  from  Table  22. 

1  F.  C.  Mills,  op.  cit.  (1924),  pp.  149  ff. 


248 


SEDIMENTARY  PETROGRAPHY 


The  mean  deviation  has  been  used  to  only  a  very  Hmited  extent  with 
sedimentary  data,  but  a  modification  of  it  is  used  as  Baker's  grading 
factor  (page  255),  which,  however,  was  introduced  as  an  arbitrary  meas- 
ure and  not  related  to  its  statistical  background  by  any  of  its  users,  as  far 
as  the  authors  are  aware. 

It  is  not  likely  that  the  mean  deviation  will  achieve  the  wide  usage  of 
the  standard  deviation  in  sedimentary  work,  although  logarithmic  meas- 
ures analogous  to  the  arithmetic  case  given  above  may  readily  be  devel- 
oped. 

Arithmetic  standard  deviation.  The  standard  deviation  of  a  distribu- 
tion is  a  measure  of  the  average  spread  of  the  curve  about  its  arith- 
metic mean,  and  it  is  perhaps  the  most  widely  used  measure  of  dispersion 
in  conventional  statistics.^  In  sedimentary  data,  the  arithmetic  standard 
deviation  is  computed  with  respect  to  the  arithmetic  mean  of  the  diameter 
distribution,  i.e.,  the  independent  variable  is  diameter  in  millimeters.  The 
data  are  usually  obtained  in  terms  of  unequal  class  intervals,  which  are 
inconvenient  for  the  computation  of  the  standard  deviation.  The  value 
may  be  found,  however,  as  shown  in  Table  27.  In  the  first  column  are 
given  the  grades  in  millimeters,  in  the  second  column  is  the  frequency 
(/)  in  each  grade.  The  third  column  has  the  mid-point  {m)  of  each 
grade,  and  in  the  fourth  column  is  given  the  deviation  of  this  mid-point 
from  the  arithmetic  mean  of  the  grain  diameters.  This  value  was  found 
to  be  0.440  mm.,  in  Table  23.  The  fifth  column  has  the  deviations 
squared,  and  in  the  sixth  column  the  deviations  squared  are  multiplied  by 
the  frequency  in  each  grade.  The  sum  of  this  column  is  12.75.  Finally, 
the  square  root  of  i/ioo  of  the  summed  value  is  extracted,  yielding  the 


standard  deviation:  Oa  = 


V^ 


2.75  = 


I-I3- 


[OO 

Table  27 
Computation  of  the  Arithmetic  Standard  Deviation  of  a  Beach  Sand  * 


Grade  Sice 
(mm.) 

/ 

m 

in  -  Ma 

(m-M,r- 

f(m-M,y 

0.5 
5-6 
1 1.7 
53-7 
26.4 
2.1 

3-0 

1-5 

0.75 

0.375 

0.187 

0.093 

2.56 

1.06 

0.31 

—0.07 

-0.25 

-0.35 

6.55 

1. 12 

0.09 

0.005 

0.062 

0.122 

3-27 

6.27 

T     1/2        

1.05 

1/2— 1/4   

I /a        1/8             

0.27 
1.64 

1/8— I/I6 

0.25 

Totals 

1 00.0 

12.75 

The  frequency  data  are  the  same  as  in  Table  22. 
1  F.  C.  Mills,  op.  cit.  (1924),  PP-  154  ft- 


STATISTICAL  METHODS 


249 


If  the  frequency  curve  is  perfectly  symmetrical,  the  standard  deviation 
is  a  measure  of  spread  such  that  about  68  per  cent  of  the  distribution 
is  contained  in  the  interval  (Ma  —  0^)  to  (^la  +  Oa).^  If  the  frequency 
curve  is  not  symmetrical,  the  exact  relationship  becomes  less  clear,  and 
the  geometrical  picture  of  the  measure  becomes  clouded,  especially  with 
curves  as  asymmetrical  as  the  average  sediment  plotted  with  diameters 
in  millimeters  as  independent  variable. 

Logarithmic  standard  dcziation.  Wlien  the  logarithmic  frequency  curve  is 
used  for  inferences  about  the  sediment,  a  logarithmic  measure  is  appropriate, 
inasmuch  as  it  may  be  used  directly  in  describing  the  average  spread  of  the 
logarithmic  curve.  A  logarithmic  standard  deviation  may  most  readily  be 
obtained  by  means  of  the  phi  or  the  zeta  scale,  either  of  which  converts  the 
unequal  geometrical  diameter  grades  into  equal  logarithmic  classes.  Hence 
conventional  methods  of  computing  the  standard  deviation  may  be  applied  to 
the  data.  In  Table  28  the  method  of  computation  is  shown.  The  first  column 
contains  the  diameter  limits  of  the  Wentworth  grades,  the  second  column 
has  the  equivalent  phi  classes,  and  the  third  column  shows  the  percentage 
weight  frequency  of  the  data.  In  column  four  an  arbitrary  d  scale  is  chosen 
with  its  zero  value  opposite  the  greatest  frequency.  The  fifth  column  has  the 
products  fd,  which  yield  the  algebraic  total  +  6.2.  This  is  divided  by  100  to 
yield  n^  =  +  .062,  the  first  moment  about  the  d  origin.  Column  six  shows  the 
square  of  the  d  value,  and  column  seven  has  the  products  fd'-.  The  total  of 
this  column  is  +73.4;  it  is  divided  by  the  total  frequency,  100.  to  yield 
»Zo  =  +  0.734,  the  second  moment  about  the  d  origin.  To  convert  the  measure 
to  the  second  moment  about  the  mean,  use  is  made  of  the  standard  equation 
ff  =  V»2  —  («j^)-.  By  substituting  the  corresponding  values  one  obtains: 
00=  ^-734-  (.062)2=  V.73o  =  0.855. 

Table  28 

computatiox  of  the  logarithmic  standard  deviation  of  a 
Beach  Sand* 


The  frequency  data  are  the  same  as  in  Table  22. 
B.  H.  Camp,  op.  cit.   (1931),  pp.  61  flf. 


Grade  Size 
(mm.) 

0 

/ 

d 

fd 

d- 

fd-'     . 

A 2                 

—  I —      0 

1 —  2 

2—  3 

3—  4 

05 

5.6 

11.7 

53-7 

26.4 

2.1 

-3 

—  I 

0 

+  1 

+2 

-  1-5 

—  II. 2 
-11.7 

0 

+26.4 

4.2 

9 
4 

I 
0 

I 
4 

4-5 

2-^-4 

II. 7 

0 

26.4 

8.4 

I-I/2     

1/2— 1/4    

I/4-I/8   

1/8— 1/16  

Totals 

+  6.2 

+  73-4 

^  — 


250  SEDIMENTARY  PETROGR.\PHY 

This  computation  has  been  carried  out  by  the  "short  method  of  computing 
the  standard  deviation"  as  explained  in  every  statistics  textbook.^  The  sig- 
nificance of  o^  in  terms  of  the  logarithmic  frequency  curve  is  shown  in 
Figure  115.  The  horizontal  scale  shows  <i>  as  the  independent  variable,  and  the 
area  under  the  curve  represents  the  total  frequency.  At  the  point  ^  =  1.561  an 
ordinate  has  been  erected  This  is  M^,  the  arithmetic  mean  of  the  phi  dis- 
tribution, and  it  passes  through  the  center  of 
gravit>^  of  the  distribution.  This  is  tlie  central 
value  about  which  the  standard  deviation  is 
computed.  From  the  example,  o^j  was  found 
to  be  0.85,  and  two  additional  ordinates  have 
been  erected  at  M^  +  o^  and  M^  —  o^,  or  at 
the  phi  values  2.41  and  0.71,  respectively. 
Between  these  two  ordinates  lies  the  central 
part  of  the  distribution,  which  in  a  symmetri- 
cal curve  would  include  about  68  per  cent  of 
the  distribution. 

Inasmuch  as  o^  is  expressed  in  phi  units, 
each  of  which  represents  one  Wentworth 
Fig.  IIS— Frequency'  data  of  grade,  the  significance  of  Oa  mav  readilv  be 
^Z^lV^tti't;^  -"■  The  value  o,  =  0.85  n,eans  .ha.  in  .he 
mean,  Mp,  and  ordinates  B  distance  M^  —  o^  to  yi^  + o^  there  are  1.70 
and  C  are  at  a  distance  c<p  Wentworth  grades,  entirely  independently  of 
from  the  mean.  whether   the   sediment   is   coarse   or    fine.    In 

terms  of  "sorting,"  as  the  term  is  commonly 
used,  this  indicates  statistically  that  the  sediment  is  well   sorted. 

Geometric  standard  deviation.  Corresponding  to  o^,  there  is  a  "diameter- 
equivalent"  which  is  a  geometric  measure  of  spread  of  the  curve  with  respect 
to  the  geometric  mean  of  the  grains.  This  value  is  called  o^,  and  is  found  by 
converting  o^  to  its  "diameter-equivalent"  by  means  of  the  chart  of  Figure 
112.= 

Skewxess  and  Kurtosis^ 

The  moment  skewness  of  size  frequency  distributions  is  a  more  difficult 
concept  than  the  average  size  or  the  standard  deviation,  partly  because  of  the 
complexity  of  the  concept  in  terms  of  grain  distribution  curves  and  partly 
because  the  physical  significance  of  skewness  in  sediments  is  not  adequately 
kno\HTi.  Kurtosis,  even  more  than  skewness,  represents  an  essentially  unex- 
plored field  in  sedimentary  analysis.  For  the  sake  of  completeness,  however, 
these  two  statistical  measures  will  be  discussed  briefly,  but  the  discussion  will 
be  confined  to  the  logarithmic  measures,  in  which  the  geometrical  meaning 
can  be  illustrated  more  effectively  than  in  the  arithmetic  case. 

1  The  use  of  the  d  scale  results  in  the  choice  of  an  arbitrary  mid-point  about 
which  all  the  moments  are  computed.  During  the  computations  the  correct  moments 
are  found  in  terms  of  corrections  applied  to  the  arbitrary  origin  by  means  of  the 
equation  for  g.  Thus  one  may  compute  several  moments  in  one  operation,  and  the 
method  is  much  shorter  than  some  others  which  have  been  used. 

-  Details  of  this  measure  are  given  in  W.  C.  Krvmibein,  loc.  cit.,  1936. 

'  B.  H.  Camp,  op.  cit.  (1931),  pp.- 28  ff. 


STATISTICAL  METHODS 


2SI 


Table  29  illustrates  the  metliod  of  computing  the  values  needed  for  skew- 
ness  and  kurtosis.  The  example  is  not  tixe  same  as  that  formerly  used.  Instead, 
another  beach  sand  is  used  which  nearly  approximates  a  normal  9  curve, 
about  which  more  will  be  said  later.  For  completeness  the  entire  set  of 
moments  is  computed  in  a  single  table,  to  illustrate  the  full  sequence  of  steps. 
Table  29  contains  all  needed  data.  The  diameter  classes  are  shown  in  the 
first  column,  the  corresponding  phi  intervals  in  the  second  column,  and  the  fre- 
quency in  the  third.  The  d  scale  is  chosen  with  its  origin  opposite  the  larg- 
est class,  as  before,  and  shown  in  column  four.  In  the  fifth  column  the 
values  fd  are  shown;  the  sixth  and  seventh  colunms  contain  d^  and  fd^ 
respectively.  The  eighth  and  ninth  columns  show  d^  and  fd'^,  and  columns 
ten  and  eleven  show  d*  and  /J*.  The  several  moments  about  the  d  origin, 
found  by  dividing  the  algebraic  totals  of  the  product  columns  by  100,  are 


0.315 
0417 


«i  =  0.441 
The  phi  mean.  M^,  is  computed  by  adding  %  to  the  mid-point  of  the  d 


scale :  M^  =  1.500  +  0.315 
obtained  by  the  equation 


00 


■(«iV 


Similarly,  the  phi  standard  de\-iation  is 
■417-  l.3i5>-  =  0-563 


Table  29 
Computation  of  the  First  Four  Phi  Moments  of  .\  Beach  Sand 


Grade  Size 
(mm.) 

<P 

/ 

d 

fd 

J- 

fd-' 

d^ 

fd- 

d^ 

fd* 

1-1/2  . . 
1/2—1/4  ■ . 
1/4— 1/8  . . 
i/S_i/i6  . 

0— I 
I — 2 
2—3 
3—4 

4-9 
58.9 
36.0 

0.2 

—I 

0 

+  1 

+2 

-  4-9 

0 

+36.0 

+  0.4 

I 
0 
I 
4 

4-9 

0 

36.0 

0.8 

0 

I 
8 

-  4-9 

+36.0 
1.6 

I 
0 

16 

4-9 

0 

36.0 

3^ 

Totals  .  . 

100. 0 

+31-5 

+41.7 

^5^-7 

-^-i 

*  Sample  7a  of  \V.  C.  Krumbein,  The  probable  error  of  sampling  sediments  for 
mechanical  analysis;  Am.  Jour.  Sci.,  vol.  27,  pp.  204-JI4,  1934. 

For  the  skewmess,  the  third  moment  about  tlie  d  origin  must  be  converted 
to  W3,  tlie  third  moment  about  the  mean,  by  using  the  standard  equation  ^ 

WI3  ==  "3  —  3"2'h  "^  ^"i^ 

This  yields  m^  =  .327~3(4I7)  (-315)  +  2(.3i5)'  =  —0.004.  The  third  mo- 
ment, m^,  is  used  in  any  of  several  formulas  for  skewness.  A  common  and 
convenient  one  is  based  on  03  =  mg/o^.  SkewTiess  is  taken  as  Sk  =  03/2.  In 
this  example  00  =  0.563,  so  that  Sk0  =  — .004/0.356  = —0.0 1 1. 

1  B.  H.  Camp,  op.  cit.  (1931),  p.  26. 


252        SEDIMENTARY  PETROGRAPHY 

Kurtosis  requires  that  the  fourth  moment  about  the  mean,  vi^,  be  first  found, 
by  using  the  standard  equation  ^ 

m^  =  n^  —  4"i"3  +  6hi-«2  —  3"i* 
This  yields  ;7u  =  .44i— 4(.3i5)  (327) +6(.3i5)  =  (.4i7) -3(.3i5)*  = +  0.247. 
The  kurtosis  itself  is  computed  in  terms  of  /S^,  where  /3,  =  ju^/o^.  Kurtosis 
may  then  be  defined  at  /S,  —  3.  On  this  basis,  /Sj  =  0.247/0.  loi  =  2.5 ;  K^  = 
2.5  —  3  =  —0.5.  In  statistical  usage  ^3,  is  often  used  as  a  test  for  the  normal 
curve,  because  in  normal  curves  it  is  equal  to  3.  In  the  present  example,  )3, 
is  less  than  3,  and  the  curve  is  designated  as  "platykurtic."  - 

Zcta  moments.  A  set  of  moments  similar  to  those  just  computed  may  be 
used  with  Atterberg's  grade  scale  by  applying  the  zcta  notation  developed  in 
Chapter  4.  Each  of  the  zeta  moments  has  a  simple  relation  to  the  correspond- 
ing phi  moments,  and  may  be  converted  to  its  equivalent  by  means  of  the  fol- 
lowing equations : 

Mr  =  0.301   (M^+i) 

0?  =  0.301    00 

Oaf  =  a^,}, 

y82f  =  )8.0 
It  may  be  noted  that  moments  higher  than  the  second  are  identical  in  both 
notations,  without  the  necessity  of  conversion.   Further  details  of  the  zeta 
notation  are  given  by  Krumbein.^ 

The  Normal  Phi  Curn-e 
The  normal  curve  in  conventional  statistics  is  defined  as  the  function 

y -=t 

where  x  is  any  value  of  the  independent  variable,  M^  is  the  arithmetic  mean 
of  the  x's  and  <r^  is  the  standard  deviation.  The  normal  curve  is  completely 
described  by  two  parameters,  M^  and  <t^.  That  is,  the  third  moment  is  zero, 
and  the  fourth  moment  has  a  value  of  /S,  =  3.  The  normal  curve  is  of  con- 
siderable importance  statistically  because  its  properties  have  been  so  thor- 
oughly studied,  and  tables  have  been  prepared  for  evaluating  the  frequencies 
and  other  characteristics  over  its  entire  range. 

The  tendency  for  sedimentary  curves  to  become  symmetrical  on  a  logarith- 
mic size  scale  suggests  that  logarithmic  (phi)  parameters  be  substituted  for 
the  x-values  in  the  function  above,  to  obtain  a  normal  phi  curve,  analogous 
to  the  conventional  curve.  In  terms  of  the  phi  mean,  ^I^  and  the  phi  standard 
deviation,  0^,  the  function  is 

I        —(<P-yi^)-/2<T^^ 
y  =-e 

where  <t>  is  the  value  of  the  independent  variable  at  any  point.  The  importance 

1  B.  H.  Camp,  op.  cit.  (1931),  p.  26. 
2F.  C.  Mills,  op.  cit.    (1924),  p.  545- 

3  W.  C.  Krumbein,  Korngrosseneinteilungen  und  statistische  Analyse :  Xcucs 
Jahrb.  f.  Min.,  etc.,  Beil.-Bd.  73,  Abt.  A,  pp.  137-150,  1937. 


STATISTICAL  METHODS 


253 


of  this  concept  is  that  among  the  asymmetrical  curves  of  sediments  which 
are  commonly  encountered  when  diameters  are  used  as  the  independent  vari- 
able, some  may  be  approximately  "normalized"  by  a  simple  mathematical 
transformation  equation.  By  means  of  this  normalizing  process  the  curve  may 
be  completely  described  with  tvvo  parameters,  M^  and  0^.  Furthermore,  the 
usual  tables  of  probabilitv*  developed  for  the  conventional  normal  curve  of 
statistics  may  be  directly  applied  to  analysis  of  tlie  phi  curs'e,  with  no  changes 
of  technique  whatever. 

Significance  of  Higher  Moments 

The  normal  phi  curve  affords  a  basis  for  furnishing  the  geometrical  picture 
of  higher  moments.  This  curve,  illustrated  in  Figure  116,  is  symmetrical.  It 
is  possible  to  consider  this  normal  curve  as  the  first  member  of  a  series  of 
successive  derivative  cun-es,^  such  that  each  succeeding  curve  represents  a 
higher  moment.  For  example,  the  third  moment  is  such  a  function  that  a 


0- 
Fig.  116. — Normal  phi  curve, 


Fig.  117. — Graph  of  third  moment. 
(Data  from  Camp,  Appendix,  Table 
III.) 


graph  of  its  effect  on  the  direction  of  skewness  appears  like  that  in  Figure 

117.  If  a  curve  is  skewed,  the  values  of  the  parameters  will  govern  the  exact 
shape  of  the  curve,  but  the  net  eft'ect  will  be  equivalent  to  adding  algebrai- 
cally the  ordinates  of  Figures  116  and  117,  to  obtain  the  curve  of  Figure  118. 
Similarly,  the  fourth  moment  contains  a  function  of  the  type  shown  in  Figure 
119,  and  if  the  ordinates  of  this  curve  are  added  to  Figure  116  or  Figure 

1 18,  the  net  eft'ect  will  be  an  increased  or  decreased  "peakedness,"  depending 
upon  the  exact  values  of  the  fourth  moment. 

In  statistical  practice  it  is  a  common  procedure  to  analyze  curves  statisti- 
cally in  terms  of  the  moments.  The  resulting  function  is  called  a  Gram- 
Charlier  series,  and  the  0  analogue  of  this  series  may  be  called  the  <^-Gram- 
Qiarlier  series.  Details  of  the  conventional  procedures  may  be  found  in  ad- 
vanced statistics  texts.  An  elementary  treatment  is  given  by  Camp.-  The  de- 


1  B.  H.  Camp,  of>.  cit.  (1931),  pp.  225  ff. 
-Op.  cit.  (1931),  Chap.  3. 


254 


SEDIMENTARY  PETROGRAPHY 


Fig.  ii8. — Graph  of  skewed  phi  curve, 
prepared  by  adding  the  ordinates  of 
Figures  Ii6  and  117  algebraically. 


Fig.  119. — Graph  of  fourth  moment. 
(Data  from  Camp,  Appendix,  Table 
IV.) 


termination  of  the  analytical  functions  of  sedimentary  curves  may  in  some 
cases  pave  the  vv^ay  for  a  more  complete  understanding  of  the  environmental 
factors  which  control  the  nature  of  the  frequency  distribution. 

Graphic  Computation  of  Geometric  Mean  and  Geometric 
Standard  Deviation 

For  curves  which  are  symmetrical  when  plotted  on  a  logarithmic  size  scale, 
there  is  a  convenient  graphic  method  of  determining  the  log  geometric  mean 
and  the  log  geometric  standard  deviation.  This  method,  developed  by  Hatch 
and  Choate,^  involves  plotting  the  cumulative  curve  on  logarithmic  probability 
paper.  (In  order  for  the  graphic  method  to  apply,  this  graph  must  be  ap- 
proximately a  straight  line.)  The  diameter-value  corresponding  to  the  50  per 
cent  line  is  the  geometric  mean,  and  the  standard  deviation  is  found  by  com- 
puting the  ratio  between  the  geometric  mean  and  the  diameter-value  corre- 
sponding to  the  15.8  per  cent  line.  This  latter  determination  follows  from  the 
fact  that  in  a  symmetrical  curve  the  interval  between  the  mean  and  o  is 
approximately  34.2  per  cent  of  the  distribution.  Complete  details  may  be 
found  in  the  reference  cited. 


SPECIAL     STATISTICAL      MEASURES 

In  addition  to  standard  statistical  devices,  in  which  the  relation  of 
the  measures  to  statistical  theory  is  known,  a  number  of  special  devices 
have  been  introduced  by  sedimentary  petrologists  to  describe  the  char- 
acteristics of  sediments.  These  empirical  devices  include  an  average  size 
and  one  or  more  measures  of  spread  of  the  data,  some  of  which  are 
related  to  the  skewness  of  the  distribution.  In  some  cases  it  has  been 
possible  to  reconcile  the  measures  with  standard  statistical  devices,  and 

1  Loc.  cit.,  1929. 


STATISTICAL  METHODS  255 

these  relations  will  be  brought  out  in  the  discussion.  In  some  instances 
the  original  authors  have  not  examined  the  exact  geometrical  or  physical 
significance  of  their  measures,  and  apparently  some  followers  of  the 
methods  have  been  content  to  accept  them  without  scrutiny. 

Baker's  equivalent  grade  and  grading  factor.  Perhaps  the  best  known 
of  the  empirical  measures  are  those  introduced  in  1920  by  Baker.^  In 
Baker's  method  two  measures  are  used,  one  representing  an  average 
grain  size  and  the  other  a  measure  essentially  of  the  spread  of  the  data 
about  the  average.  The  method  involved  is  largely  graphic  and  involves 
by  definition  the  use  of  a  cumulative  curve  drawn  on  an  arithmetic  scale 
of  diameters,  as  shown  in  Figure  120.  It  will  be  noted  that  Baker  chose 
frequency  as  his  independent  variable,  presumably  because  his  graphic 
approach  is  more  convenient  on  that  basis.  In  computing  the  equivalenl 
grade,  or  average  size,  the  area  under  the  cumulative  curve  is  determined 
with  a  planimeter,  and  the  area  so  found  is  divided  by  the  length  of  the 
frequency  line  from  o  to  100  per  cent.  The  quotient,  which  is  the  length 
corresponding  to  the  equivalent  grade,  is  then  laid  off  on  the  vertical 
axis,  and  the  diameter  value  at  that  point  is  the  equivalent  grade,  which 
is  indicated  by  a  heavy  horizontal  line  in  Figure  120,  at  the  value  0.430 
mm.  Essentially,  the  equivalent  grade  is  the  mean  ordinate  of  the  curve, 
and  it  may  be  shown  that  it  is  in  fact  the  arithmetic  mean  diameter  of 
the  grain  distribution.^ 

Baker's  grading  factor  is  a  measure  designed  to  indicate  how  nearly 
the  degree  of  grading  approaches  perfection.  A  perfectly  graded  sediment, 
according  to  Baker,  is  one  in  which  all  grains  are  of  the  same  size,  so 
that  there  is  no  deviation  from  the  average  size.  In  that  case  the  com- 
puted value  of  his  grading  factor  would  be  unity. 

Figure  120  also  indicates  the  manner  of  computing  the  grading  factor. 
The  two  shaded  areas,  one  below  the  curve  and  above  the  line  (V,/), 
and  the  other  above  the  curve  and  below  the  line  (Vi,),  are  called  the 
variation  areas  of  the  curve,  and  the  sum  (Va-f  Vb)  is  called  the  total 
variation  area.  These  are  also  found  by  means  of  a  planimeter. 

Baker  defined  his  grading  factor  as 

Total  area  under  the  curve  —  total  variation  area 

G.F.  = 

total  area  under  the  curve  ' 

1  H.  A.  Raker,  On  tlic  investigation  of  the  meclianical  constitution  of  loose  are- 
naceous sediments  by  the  method  of  elutriation,  etc. :  Gcol.  Magazine,  vol.  57,  iip. 
366-370,  1920. 

-  As  far  as  the  authors  are  aware,  Baker  did  not  indicate  that  his  gradin<i  factor 
is  a  graphically  determined  arithmetic  mean.  This  identity  may  be  proved  by  a 
rather  tedious  mathematical  dcmunstration. 


256 


SEDIMENTARY  PETROGRAPHY 


and  in  the  actual  computation,  the  grading  factor  is  found  by  subtracting 
the  total  variation  area  from  the  total  area,  and  dividing  the  difference 
by  the  total  area. 

The  relation  of  the  grading  factor  to  statistical  theory  is  less  direct 
than  that  of  the  equivalent  grade.  It  may  be  shown,  however,  by  a  some- 
what laborious  process,  that  this  measure  is  related  to  the  mean  devi- 
ation of  conventional  statistical  practice.  The  relation  is 


G.F. 


mean  deviation 
arithmetic  mean  size 


1 

A 

^^ 

^^ 

^^^ 

^iiiii^ 

"^ 

CUMULATED  FREQUENCY 


To  obtain  Baker's  equivalent  grade  and  grading  factor  as  he  defined 
them,  it  is  necessary  to  use  an  arith- 
metic scale  for  diameters.  Some  work- 
ers have  drawn  their  cumulative  curves 
on  a  logarithmic  size  scale  and  fol- 
lowed Baker's  graphic  procedure,  call- 
ing the  result  the  equivalent  grade. 
What  is  actually  obtained  in  this  case 
is  the  geometric  mean  diameter  of  the 
grains — an  entirely  different  value.  In 
similar  fashion,  if  the  logarithmic 
cumulative  curve  is  used  for  the  com- 
putation of  the  "grading  factor,"  the 
result  is  not  Baker's  grading  factor, 
but  a  geometric  measure  in  terms  of 
the  logarithmic  frequency  distribu- 
tion. 

Baker's  two  measures  have  been 
widely  used  in  the  examination  of 
sediments,  and  his  equivalent  grade  has  been  chosen  as  the  average  value 
for  other  empirical  statistical  methods  also.  It  is  probably  preferable 
to  indicate  that  the  arithmetic  mean  size  is  used  in  such  instances,  and 
indeed  the  equivalent  grade  may  be  computed  directly  by  the  methods 
shown  in  Table  22. 

Niggli's  statistical  method.  Most  recent  of  the  empirical  statistical 
methods  is  that  introduced  in  1935  by  Niggli.'  Niggli's  method  combined 
Baker's  equivalent  grade  with  various  approximate  quartile  measures 
as  well  as  with  the  maximum  and  minimum  grain  sizes.  The  result  is  a 

1  P.  Niggli,  Die  Charakterisierung  der  klastischen  Sedimente  nach  der  Korn- 
zusammensetzung :  Schweiz.  Min  u.  Pet.  Mitt.,  vol.  15,  pp.  31-38,  I935- 


of 


Fig.  120. — Cumulative  curve 
beach  sand,  showing  Baker's  equiva- 
lent grade  (A),  and  areas  used  in 
computing  grading  factor. 


STATISTICAL  METHODS  2-^,^ 

method  of  characterization  which  Xiggh  did  not  attempt  to  relate  to 
conventional  statistical  practice.  Zingg  ^  applied  the  technique  to  a  num- 
ber of  sediments  and  discussed  the  further  implications  of  a  wide  range 
of  devices  for  describing  sediments.  In  detail,  Xiggli's  method  involves 
the  following  steps : 

Three  fundamental  values  are  chosen,  which  are  called  d^,^,  d,  and 
cf„in.  The  first  of  these  is  the  size  of  tlie  largest  particle  in  the  sediment, 
tlie  second  is  Baker's  equivalent  grade,  and  the  last,  rfmin-  is  tlie  smallest 
grain  in  the  sediment,  usually  taken  as  zero  by  Xiggli.  unless  the  smallest 
size  is  appreciable.  Niggli  points  out  that  in  a  perfectly  symmetrical  cun-e 
(an  arithmetic  size  scale  is  used),  tlie  value  of  d  would  be  exactly  equal  to 

-^^ 5^^  which  is  true  in  an  arithmetic  distribution.  Inasmuch  as  few 

2 

curves  are  s\'mmetrical,  an  important  measure  is  tlius  found  in  tlie  relation 
between  the  value  of  d  and  the  value  -^^ ^^:  hence  a  measure  is  set  up 

from  die  ratio  of  these  values,  and  called  delta:  5=- — ^^^ — .  Xext,  tlie 

total  percentage  of  material  lying  between  d^^^  and  d  is  called  p.  and  a 
second  measure,  pi,  is  defined  from  tliis  as  follows:  .t  = />  50  =  ^Z*  100.  The 
two  measures  delta  and  pi  are  used  as  the  first  characteristic  of  tlie  sediment. 
by  noting  tlie  extent  to  which  they  depart  from  tlie  value  i. 

By  means  of  d  and  />  the  grain  distribution  is  divided  into  two  parts,  one 
fine  and  the  other  coarse.  For  each  of  tliese  portions,  accordingly,  a  similar 
set  of  measurements  is  obtained,  analogous  to  d  and  /•.  For  tlie  fine  fraction 
the  average  value  is  called  d'  and  for  tlie  coarse  d" .  Likewise,  tlie  correspond- 
ing p  values  are  called  />'  and  q  respectively.  Thus  Niggli  obtains  a  series 
of  values  distributed  along  the  curve,  which  serve  to  define  it  by  fixed  values. 
As  Xiggli  points  out,  if  the  curve  were  s>Ttimetrical.  f  would  be  tlie  first 
quartile  and  q  the  third  quartile,  whereas  d  would  be  tlie  median  grain  size. 
Sediments  are  characterized  in  part  by  the  relations  among  d,  d'.  and  d". 
The  greater  tlie  interval  between  d'  and  d".  the  more  widespread  is  the  curve ; 
the  smaller  the  difference  between  d'  and  d",  the  more  nearly  uniform  the 
sediment  is. 

Niggli   also   introduced  a   sorting   index,   uninfluenced  by   size,   which   he 

defined  as  follows:  a=- ,  which  vields  values  approximatelv  equal 

d 

to  unity  for  sediments  tlie  bulk  of  whose  grains  are  well  sorted,  and  values 

greater  than  i  for  poorly  sorted  sediments.  The  sorting  index  is  essentially 

the  ratio  of  the  difference  between  the  average  size  of  the  coarse  and  fine 

portions  of  the  curve  and  tlie  average  grain  size.  The  value  3  was  chosen  as 

a  constant  because,  according  to   Niggli,  tlie   ideal   relation   in  the   Udden 

(Wentworth)  grade  scale  is  (d"  —  d')/d=  li. 

iTh.  Zingg,  Beitrag  zur  Schotteranalyse :  Schzcch.  Miu.  u.  Pet.  Mitt.  vol.  15. 
pp.  39-140,  1935. 


258       SEDIMENTARY  PETROGRAPHY 

Fineness  factor.  In  1902  Purdy^  introduced  a  surface  factor  or  fine- 
ness factor  for  describing  the  texture  of  clay  and  ceramic  materials.  The 
fineness  factor  is  computed  by  multiplying  the  reciprocal  of  the  mid- 
point of  each  grade  size  by  the  weight  percentage  of  material  in  the 
grade,  expressed  as  a  decimal  part  of  the  total  frequency.  The  sum  of 
the  resulting  products  is  the  fineness  factor.  Purdy  based  his  measure 
on  the  assumption  that  the  surface  areas  of  the  two  powders  are  inversely 
proportional  to  their  average  grain  size.  Roller-  examined  Purdy's  factor 
in  terms  of  statistical  theory  and  showed  that  the  factor  is  essentially  the 
reciprocal  surface  mean  diameter  of  the  powder,  providing  the  average 
size  of  each  grade  is  defined  in  terms  of  a  surface  mean  diameter,  da, 
and  a  percentage  weight  of  material,  W,  as  : 


where  the  right-hand  side  of  the  equation  indicates  the  operations  used 
in  computing  Purdy's  factor.  Full  details  of  the  theory  may  be  found 
in  Roller's  paper. 

The  fineness  factor  has  not  been  extensively  used  in  sedimentary 
petrology,  but  in  the  light  of  Roller's  work  it  would  appear  to  offer  an 
excellent  approach  to  the  study  of  properties  of  finer  sediments  in  terms 
of  their  surface  area.  For  the  study  of  pigments,  where  surface  is  per- 
haps the  most  significant  attribute,  the  measure  is  of  considerable  im- 
portance. 

Sorting  indices.  A  number  of  writers  have  introduced  various  meas- 
ures of  the  sorting  of  sediments,  but  apparently  no  one  has  thoroughly 
investigated  the  subject  of  sorting  itself,  to  determine  the  most  suitable 
measure  of  this  attribute.  The  generally  accepted  definition  of  sorting  is 
that  the  more  nearly  a  sediment  approaches  a  single  size  in  its  frequency 
distribution,  the  better  it  is  sorted.  Thus  most  measures  of  sorting  are 
statistical  in  nature  and  measure  essentially  the  spread  of  the  curve. 
This  is  true  of  Baker's  grading  factor  and  of  Xiggli's  index  of  sorting. 
Trask's  measure  of  sorting,  So,  a  geometric  quartile  deviation,  is  also  a 
measure  of  spread.  In  similar  fashion  the  standard  deviation,  either  loga- 
rithmic, geometric,  or  arithmetic,  may  be  used  as  a  measure  of  statistical 
sorting.  There  is  a  field  for  investigation  on  the  physical  significance  of 
sorting,  as  well  as  of  the  possible  influence  of  sorting  on  the  skewness  of 
the  sediments  or  vice  versa. 

1  R.  C.  Purdy,  Qualities  of  clays  suitable  for  making  paving  brick :  III.  State 
Geol.  Survey,  Bull,  g,  pp.  133-278,  1908. 

-  P.  S.  Roller,  Separation  and  size  distribution  of  microscopic  particles :  U.  S. 
Dept.  Commerce,  Bur.  Mines  Teclin.  Paper  490,  1931. 


STATISTICAL  METHODS  259 


CHOICE     OF     STATISTICAL     DEVICES 

The  authors  are  not  prepared  to  commit  themselves  on  any  single 
method  of  statistical  analysis  as  being  the  best ;  the  field  of  statistics  as 
applied  to  sediments  will  require  mathematical  statisticians  to  investigate 
all  the  ramifications  of  the  problem.  For  the  present  the  individual 
worker  must  choose  the  method  which  appears  best  adapted  to  the  end 
he  has  in  view.  In  general,  three  things  may  guide  his  choice:  the  rela- 
tions of  the  measures  to  the  body  of  statistical  theory,  the  relative  sim- 
plicity of  the  mere  mechanical  process  of  arriving  at  the  numbers,  and 
the  simplicity  of  the  geometrical  meaning  of  the  measures.  It  is  largely 
on  the  latter  that  one  bases  his  conclusions,  and  whatever  measures  are 
chosen  should  at  least  be  readily  visualized. 

The  decision  between  arbitrary  methods  of  description  and  conven- 
tional statistical  devices,  related  to  the  background  of  statistical  theory, 
must  depend  upon  the  objects  of  the  study.  If  description  and  classifi- 
cation are  an  end  in  themselves,  any  measures  designed  to  summarize 
the  data  are  adequate.  If  description  and  classification  are  only  a  means 
to  an  end,  on  the  other  hand,  then  the  measures  chosen  should  serve 
other  purposes  as  well.  Every  sedimentary  deposit  has  characteris- 
tics which  depend  upon  the  conditions  of  its  formation,  and  these  char- 
acteristics appear  to  be  most  eflfectively  expressed  in  terms  of 
their  statistical  parameters.  If  the  relation  between  sedimentary 
characteristics  and  environmental  conditions  is  to  be  elucidated,  it 
seems  reasonable  to  suppose  that  the  body  of  theory  behind  con- 
ventional statistical  procedures  will  afford  a  more  direct  relationship 
than  the  use  of  measures  designed  without  regard  to  that  body  of 
theory. 

The  objection  is  occasionally  raised  that  standard  statistical  devices 
yield  too  few  values  for  the  classification  of  sediments.  It  was  partly  this 
reason  that  impelled  Niggli  to  develop  his  measures.  However,  the  ar- 
gument may  be  met  by  proponents  of  the  quartile  measures  by  extending 
the  devices  to  the  deciles  or  intermediate  points.  A  satisfactory  parallel 
of  Niggli's  method  may  be  developed  by  choosing  the  median  and  the 
first  and  third  quartiles  as  fundamental  values,  and  intermediate  deciles 
where  needed.  In  this  manner  as  many  as  ten  values  may  be  had  if 
desired. 

A  choice  between  arithmetic,  geometric,  or  logarithmic  measures  must 
depend  upon  the  type  of  results  which  are  most  immediately  useful  to 
the  investigator.  If  the  influence  of  size  is  to  be  included  in  the  study, 


26o       SEDIMENTARY  PETROGRAPHY 

measures  should  be  chosen  in  which  the  size  factor  is  explicit.  When  size 
is  to  be  eliminated,  geometric  or  logarithmic  measures  may  be  used.  It 
is  important  in  this  connection  that  the  worker  imderstand  the  de- 
pendence or  independence  of  his  measures  on  size,  and  to  that  end  this 
information  has  been  given  in  the  bod)'  of  the  chapter.  A  choice  between 
quartile  or  moment  measures  ma)^  depend  upon  the  anal\l;ical  data  at 
hand.  It  may  be  stated  as  a  general  rule  that  moment  measures  are 
much  more  sensitive  to  "open  ends"  on  the  sedimentarj-  data  than 
quartile  measures  are.  Hence  in  working  with  very  fine-grained  sedi- 
ments, where  present  methods  of  analysis  require  grouping  a  considerable 
amount  of  material  in  the  smallest  class,  the  higher  moments  are  perhaps 
not  very  reliable,  although  the  first  and  second  moments  are  usually  not 
greatly  affected.  Quartile  measures,  on  the  other  hand,  are  usually 
not  affected  at  all  by  open  ends  beyond  the  25  per  cent  hne  in  one 
direction  and  the  75  per  cent  line  in  the  other.  For  partial  analyses,  there- 
fore, the  quartile  measures  are  excellent.  However,  the  disadvantage 
of  quartile  measures  is  often  that  the  behavior  of  the  extreme  parts 
of  the  curve  is  not  at  all  reflected,  and  in  studies  where  departures 
from  the  average  are  to  be  studied  the  quartile  measures  may  be  of 
limited  use. 

The  final  test  of  any  statistical  measure  is  its  mathematical  conven- 
ience, and  this,  combined  with  its  relation  to  the  background  of  statis- 
tical theory,  enables  the  worker  to  gain  the  maximum  value  from  his 
study.  Beyond  standard  devices  for  size  distribution  studies,  there  re- 
mains much  to  be  done  with  statistical  devices  in  mineral  studies  and 
in  connection  with  shape  and  surface  texture.  Likewise  problems  of 
sampling  and  statistical  correlation,  from  the  view^Ktint  of  sedimentary 
data,  have  not  been  investigated  extensively.  The  field  may  therefore  be 
considered  wide  open  for  appropriately  trained  research  men. 

STATISTICAL     CORRELATION 

In  many  scatter  diagrams  the  points  are  dispersed  more  or  less  widely, 
and  a  question  arises  whether  there  is  any  definite  relation  between  the 
two  variables. 

Statisticians  have  de\-eloped  methods  of  testing  data  of  this  nature  by 
means  of  a  coefficient  of  correlation.  This  is  not  to  be  confused  with  the 
term  correlation  as  used  in  a  geological  sense :  statistical  correlation  is  a 
mathematical  procedure  which  yields  a  coefficient  whose  value  extends 
from  —  I  through  zero  to  -f-  i.  If  the  correlation  coefficient  is  equal  to 
-\- 1,  there  is  a  direct  relation  between  the  variables ;  if  its  value  is  —  i, 


STATISTICAL  METHODS 


261 


there  is  an  opposite  (inverse)  relation  between  the  variables;  and  if  the 
coefficient  is  zero,  there  is  in  all  likelihood  no  fixed  relation  between 
them.  For  values  other  than  zero,  but  neither  -|-  i  nor  —  i,  the  signifi- 
cance of  the  correlation  coefficient  depends  partly  on  the  nature  of  the 
data  being  examined. 

Statistical  correlation^  between  two  variables  is  called  simple  correla- 
tion ;  it  may  be  either  linear  or  non-linear.  It  is  also  possible  to  test  the 
relations  among  more  than  two  variables  by  multiple  correlation,  but  the 
methods  become  somewhat  tedious, 
and  the  interested  reader  is  re- 
ferred to  standard  texts  for  meth- 
ods of  computation.  An  example 
of  linear  correlation  will  be  given 
here  to  illustrate  the  method  and 
to  indicate  some  of  the  advantages 
and  disadvantages  of  applying  this 
statistical  technique  to  sedimen- 
tary data. 

A  study  of  beach  pebbles  from 
Little  Sister  Bay,  Wisconsin, 
showed  that  in  general  there  was 
a  relation  between  large  average 
size  and  average  degree  of  roundness.  Eleven  samples  -  were  studied, 
and  by  plotting  average  size  against  average  roundness  the  scatter  dia- 
gram of  Figure  121  was  obtained.  Size  was  chosen  as  the  independent 
variable,  and  roundness  as  the  dependent  variable.  It  will  be  noted  that 
the  points  scatter  too  widely  to  justify  drawing  a  straight  line  through 
them  without  considerable  qualification.  It  is  in  such  cases  that  the  cor- 
relation coefficient  may  shed  light  on  the  problem  of  possible  relations 
between  the  variables. 


^ 

0 

{3 

0 
0 

0 
0 

^0.0 

0 

0 

a40 

GEOMETRIC  MEAN  SIZE  IN  MM. 

Fig.  121. — Scatter  diagram  of  average 
roundness  and  geometric  mean  size  of 
beach  pebbles. 


There  are  several  methods  of  computin.q:  the  coefficient  of  correlation,  de- 
pending- upon  wliether  the  data  are  grouped  or  ungrouped.  In  the  present  case 
they  are  iingrouped.  The  following  method  of  computation  was  chosen  so 
that  eacli  step  in  the  process  would  be  explicit,  especially  the  transformation 
of  variables  that  occurs  during  the  computation  and  final  fitting  of  straight 
lines.  In  Figure  121  the  liorizontal  axis  is  chosen  as  X  and  the  vertical  axis  Y. 
Hence   the   "raw   data"   are   given   in   X   and   Y   units,   the   variables   being 

1  F.  C.  Mills,  ot>.  cit.  (1924),  Chapter  10. 

-  Strictly  speaking,  cloven  samples  are  not  sufficient  for  a  detailed  study  of  cor- 
relation. It  is  preferable  to  use  at  least  twenty-four  samples.  Likewise,  the  direct 
correlation  of  average  values  involves  more  complex  theory,  but  the  example  will  at 
least  indicate  the  method  of  computation. 


262 


SEDIMENTARY  PETROGRAPHY 


X  =  Mg,  the  geometric  mean  size  of  the  pebbles,  and  Y  =  P^v.  the  average 

romidness  of  the  pebbles.  In  computing  the  coefficient  it  is  convenient  to  consider 

the  deviations  from  the  mean  values  of  X  and  Y.  The  steps  are  indicated  in 

Table  30.  The  first  two  columns  list  the  X  and  Y  values,  and  the  averages 

331                               6.86 
obtained  from  each  of  these  columns  is  X„  = ■  =  30  mm.,  Y^ 0.62. 

The  third  and  fourth  columns  represent  the  differences  between  X  and  X„ 
and  Y  and  Y^;  specifically  the  values  are  ;ir  =  X  —  X^^,  and  y  =  Y  —  Y„. 
These  new  variables,  x  and  y,  are  called  the  deviation  values.  In  columns  5 
and  6  the  individual  values  of  x  and  y  are  squared,  and  in  column  7  the 
products  of  X  and  y  are  indicated.  Note  that  these  last  values  may  be  either 
positive  or  negative.  The  figures  in  the  several  columns  are  added  and  the 
sum  indicated  in  the  last  line  of  the  table. 


Table  30 

Computation  of  the  Correlation  Coefficient  of  Average  Roundness 

AND  Geometric  Mean  Size  of  Beach  Pebbles  from  Little  Sister  Bay, 

Wisconsin 


Geometric 

Mean  Sise 

Average 

x-x„ 

Y-Y,, 

(mm.) 
X 

Roundness 
Y 

X 

y 

X- 

y- 

xy 

52 

0.62 

+22 

0 

485 

0.0000 

0.00 

43 

•74 

+  13 

+0.12 

169 

.0144 

+  1.56 

36 

.65 

+  6 

+  .03 

36 

.0009 

4-0.18 

32 

•71 

+  2 

4-  .09 

4 

.0081 

+  0.18 

27 

.68 

-  3 

4-  .06 

9 

.0036 

-0.18 

26 

•59 

-  4 

-  -03 

16 

.0009 

4-0.12 

22 

•49 

-  8 

-  -13 

64 

.0169 

+  1.04 

37 

.67 

+  7 

+  .05 

49 

.0025 

+0.35 

24 

.64 

-  6 

4-  .02 

36 

.0004 

—0.12 

19 

•56 

—  II 

-  .06 

121 

.0036 

4-0.66 

13 

•51 

-17 

4-  .11 

289 

.0121 

4-1.87 

6.86 

1278 

0.0634 

4-5.66 

The  correlation  coefficient,  r. 


is  defined  as  follows^ 
„-     P 


where  p  is  the  product  moment  of  .r  and  3',  and  o-^.  "^y  'ire  respectively  the 
standard  deviations  of  the  x  and  y  values  about  X^  and  Y^.  Inasmuch  as  the 
X  and  y  values  are  expressed  directly  as  deviations  from  Xm  and  Y^,  the 


1  F.  C.  Mills,  op.  cit.  (1924),  pp.  385  ff- 


STATISTICAL  METHODS  263 

three  needed  values   are   readily   found   from   the   values   in   the   table   and 
the  following  relations 

Z(xy)_  +  5.66  _ 


P 


X 


-i^ 


—  \  116=  10.8 


-(y)^ 


*^y  — -1' — ^^^ =  V  0.0057  =  0.07 


■'1  O  ^2 

Hence  r  =  -^—  = —^ =  -ro.6S. 

"x^T       (io.S)(o.07) 

The  correlation  coefficient  thus  has  a  value  between  zero  and  +  i,  indicat- 
ing that  the  expected  relation  is  present,  but  is  by  no  means  perfect.  This 
means,  essentially,  that  size  alone  is  not  the  controlling  factor  in  rouruiness, 
a  conclusion  that  the  geological  evidence  itself  affords.  However,  the  correla- 
tion coefficient  at  least  indicates  that  the  general  relation  between  large  aver- 
age size  and  high  average  roundness  does  hold  on  the  beach  in  question. 

If  one  wishes  to  indicate  the  degree  of  correlation  graphically,  he  may  plot 
lines  of  regression  on  the  scatter  diagram.^  The  lines  of  regression  represent 
tlie  straight  lines  of  approximate  best  fit.  both  of  y  on  x  and  x  on  v.  If  the 
correlation  coefficient  is  equal  to  unit>-,  both  of  these  lines  will  be  identical 
(i.e.,  have  the  same  slope  and  intercepts),  whereas  if  r  =  o  the  indication  is 
diat  no  line  fits  tlie  data  better  than  any  other.  For  values  of  r  between  o  and 
-r  I  the  angle  between  the  lines  is  a  function  of  the  correlation  coefficient. 

The  correlaiion  coefficient  has  not  been  extensively  applied  to  sedi- 
mentary data,  but  it  is  a  common  statistical  device  which  may  well  be 
used  in  appropriate  situations.  Certain  precautions  should  be  followed 
in  drawing  inferences  from  the  correlation  coefficient,  however.  That  is, 
the  correlation  coefficient  is  applicable  directly  only  if  the  attributes 
being  correlated  are  continuous  variables,  expressible  as  numbers  on  a 
continuous  scale.  The  usual  case  involves  n  samples,  each  of  which  has 
two  variables  in  common,  as  in  the  illustration  used.  Appropriate  cases 
would  include  not  only  size  and  shape  attributes,  but  various  mineral  at- 
tributes as  well.  For  example,  it  is  proper  to  use  the  correlation  coeffi- 
cient with  n  samples  of  heavy  minerals  for  the  correlation  of  garnet 
and  hornblende  in  each. 

An  illustration  of  an  unconventional  use  of  the  correlation  coefficient 
is  given  by  a  method  of  mineral  correlation  introduced  by  Drsden.^ 
Instead  of  using  n  samples  of  two  attributes  each,  he  used  2  samples  with 

1  F.  C.  Mills,  op.  cit.  (1924),  pp.  393  ff. 

-  L.  Drvden,  A  statistical  method  for  the  comparison  of  hea\-}'  mineral  suites : 
Ant.  Jour.  Sci.,  vol.  29,  pp.  393-408,  1935. 


264        SEDIMEXTARY  PETROGRAPHY 

;;  attributes  each  (see  page  487).  In  this  case  the  underlying  postulates 
on  which  the  correlation  coefficient  are  based  were  not  satisfied,  because 
the  attributes  used  do  not  form  a  continuous  series.  However,  it  is  per- 
fectly appropriate  to  use  the  process  of  computing  the  correlation  co- 
efficient in  this  case ;  but  perhaps  another  term,  as  "coefficient  of  mineral 
association,"  should  have  been  applied  to  the  result.  The  point  to  this 
discussion  is  that  it  is  suitable  to  make  the  computations  as  Dryden  did, 
providing  no  attempt  is  made  to  fit  r  into  the  extensive  background  of 
theor}'  with  which  the  correlation  coefficient  has  hitherto  been  associated, 
because  it  does  not  satisfy  the  postulate  of  this  existing  theory.  However, 
there  seems  to  be  no  reason  why  a  second  body  of  theory  may  not  be 
developed  based  on  postulates  suggested  by  Dryden's  use  of  r. 

CHI-sou  ARE     TEST 

The  correlation  coefficient,  as  usually  employed  in  statistical  work, 
involves  a  number  of  samples.  In  geological  problems  it  is  often  desirable 
to  "correlate"  two  samples,  to  determine  whether  they  came  from  the 
same  or  different  deposits.  It  is  very  unusual  to  find  two  samples  having 
exactly  the  same  size  frequency  distribution,  or  the  same  percentage  of 
hea\y  minerals,  and  the  question  is  how  much  variation  is  permissible 
without  rendering  invalid  the  inference  that  the  two  samples  are  from 
the  same  parent  deposit. 

Eisenhart^  attacked  this  problem  in  1935  by  means  of  the  chi-square 
test,  which  is  applicable  to  a  number  of  problems  in  sedimentar}'  work. 
The  theory  of  /-  is  beyond  the  scope  of  this  book,  but  the  essential 
features  of  the  test  may  be  described  by  a  simple  example,  as  used  by 
Eisenhart.  Two  samples  of  sediment  have  the  following  numbers  of  lime- 
stone and  shale  pebbles : 

Sample  Limestone  Shale 

1    103  794 

2   109  781 

The  question  is  whether  two  samples  drawn  at  random  from  the  same 
parent  deposit  could  show  such  observed  variations  due  purely  to  chance. 
In  other  words,  what  is  the  probability  that  two  random  samples  would 
show  variations  as  great  or  greater  than  those  observed  ?  The  chi-square 
test,  applied  to  these  data,  shows  that  one  would  obtain  these  or  greater 
variations  in  62  per  cent  of  the  cases.  Thus  there  is  little  risk  in  assuming 
that  the  samples  are  from  the  same  deposit. 

1  C.  Eisenhart,  A  test  for  the  significance  of  lithological  variations :  Jour.  Sed. 
Petrology,  vol.  5,  pp.  137-145,  1935- 


STATISTICAL  METHODS  265 

In  applying  the  chi-square  test,  one  sets  up  a  table  showing  the  samples 
in  vertical  columns,  and  the  attributes  to  be  tested  in  horizontal  rows. 
Each  observed  frequency  is  subtracted  from  an  expected  frequency  (or 
an  "independence  frequency"  if  the  former  is  not  known),  and  the  dif- 
ference is  squared  and  divided  by  the  expected  frequency.  A  series  of 
values  are  obtained,  the  sum  of  which  equals  y\  The  observed  value  of 
yj  is  located  in  a  table  opposite  an  appropriate  number  for  the  "degrees 
of  freedom,"  of  the  table,  and  the  probability  desired  is  found.  Details 
of  the  test  and  an  introduction  to  the  theory  are  given  in  Eisenhart's 
paper.  The  complete  y;  tables  are  to  be  found  in  Fisher's  book ;  ^  a 
partial  table  is  given  by  Camp.^ 

The  chi-square  test  may  also  be  used  in  testing  such  assumptions  as 
were  made  in  connection  with  the  data  of  Table  29,  which  suggested  that 
the  sediment  approximated  a  normal  <^  curve.  By  comparing  the  ob- 
served frequencies  and  the  theoretical  frequencies  of  a  normal  curve 
from  tables  of  probability  integrals,  the  "goodness  of  fit"  of  the  data 
may  be  tested  by  y^. 

The  chi-s(|uare  test  promises  to  be  of  considerable  importance  in  the 
theory  of  sampling  sediments,  but  it  requires  detailed  study  to  deter- 
mine whether  the  conditions  of  sampling  sediments  satisfy,  in  all  cases, 
the  postulates  on  which  the  chi-square  test  is  based. 


THEORY    OF    CONTROL 

The  theory  of  control,  as  developed  by  Shewart,^  affords  a  powerful 
method  for  testing  data  to  determine  whether  observed  variations  are 
due  purely  to  chance  causes  or  whether  they  may  be  attributed  to  assign- 
able causes  of  variation.  The  method  was  used  by  Otto'  in  testing  the 
performance  of  a  Jones  sample  splitter.  His  tests  indicated  that  certain 
subjective  errors,  due  to  differences  in  operators,  were  present.  These 
errors  were  largely  eliminated  by  developing  an  improved  type  of  splitter 
(see  page  45),  in  which  lugs  required  that  all  operations  be  standard- 
ized. 

The  the(M-y  of  control  rests  fundamentally  on  the  fact  that  in  a  nor- 
mal probability  function  99.7  per  cent  of  all  cases  fall  within  the  range 
Ma  -f-  0 ;  in  other  words,  the  chances  are  that  no  more  than  three  out  of  a 

1  R.  A.  Fisher,  op.  cil.  (1932). 

2  B.  H.  Cam]),  of.  cit.  (1931),  p.  265. 

^  W.  A.  Shcwart,  Economic  Control  of  Quality  of  Mamijacturcd  J'roduct  (New 
York,  1931). 

■•  G.  H.  Otto,  The  use  of  statistical  methods  in  effecting  imi)rovcmcnts  on  a  Jones 
sample  splitter:  Jour.  Scd.  Pctrolo<jy,  vol.  7,  pp.  1 10-132,  1937. 


266       SEDLMEXTARY  PETROGRAPHY 

thousand  items  will  depart  from  the  arithmetic  mean  by  more  than  three 
times  the  standard  deviation.  Tests  are  de\-i5ed  to  determine  whether 
this  relation  holds;  if  not,  further  tests  seek  to  assign  causes  to  the 
obser\Ted  departures  from  the  normal  law. 

The  computations  involved  in  appl>nng  the  theory  are  quite  tedious, 
which  is  an  attribute  of  most  detailed  methods  of  analysis.  The  reader 
is  referred  to  Shewart's  book  and  Otto's  paper  for  the  details  of  the 
method. 


THE     PROBABLE     ERROR 

Among  statistical  devices  long  used  in  the  evaluation  of  errors  is 
the  probable  error/  defined  as  that  error  which  will  not  be  exceeded  in 
one  half  of  the  observed  cases.  The  probable  error  bears  a  constant  and 
simple  relation  to  the  standard  deviation,  o.  This  relation  is  expressed  by 
the  equation  P.E.  =  0.67450. 

The  probable  error  was  applied  to  the  problem  of  sampling  sediments 
by  Krumbein,-  who  investigated  the  error  in  terms  of  its  effect  on  the 
median  grain  diameter.^  In  theor\'  the  method  apphed  depends  first  on 
the  fact  that  independent  errors  (sampUng  errors  as  opposed  to  labora- 
ton.-  errors)  are  related  to  the  total  observed  error  E  by  the  expression  * 
E  =  \/(^i)--|-  (^2)",  where  e^  and  ^2  are  the  sampling  and  laboratorj' 
errors  respectively.  A  number  of  samples  are  collected  and  separately 
anal\-zed  to  determine  the  total  error  E.  The  samples  are  then  combined 
into  a  single  composite  which  is  anah'zed  a  number  of  times,  to  obtain 
€2.  This  permits  the  computation  of  e^  from  the  equation  above. 

The  probable  error  of  the  mean,  PEm,  is  defined  as  PE/\/",  where 
PE  is  the  probable  error  of  a  single  observ^ation,  and  n  is  the  number  of 
samples.  This  may  be  expressed  as  PEm/PE=  i/V"-  This  was  the 
equation  used  by  Krumbein.  Some  writers  prefer  to  use  the  standard 
error  of  the  mean,  On.,  defined  as  ^n.  ^  o/\/ii,  where  o  is  the  standard 
deviation.  The  relation  between  o  and  PEm  is  FR^  =  0.67450,  from  the 
definition  of  the  probable  error.  In  general,  one  may  express  the  error  of 

1  F.  C.  Mills,  op.  cit.  (1924),  P-  160. 

2W.  C.  Krumbein,  The  probable  error  of  sampling  sediments  for  mechanical 
analysis:  Am.  Jour.  Set.,  vol.  27,  pp.  204-214,  1934. 

'  It  is  preferable,  perhaps,  to  use  the  arithmetic  mean  in  such  studies,  or  to  ap- 
proach the  problem  logarithmicalh-  in  terms  of  the  phi  mean.  Fortunately  the 
median  deviated  about  its  mean  value  in  a  normal  manner,  so  that  the  method  was 
applicable. 

*  A.  Fisher,  The  Mathematical  theory  of  probabilities  (New  York,  1915),  vol. 
I,  p.  106. 


STATISTICAL  METHODS  267 

the  mean  as  Em,  whereupon  the  relation  is  E,„/E  =  i/V"-  This  function 
is  discussed  in  Chapter  2. 

Further  discussion  of  the  theory  of  probahle  errors  and  details  of  the 
method  for  evaluating  the  error  are  given  in  Krumbein's  paper. 


SUMMARY     OF     STATISTICAL     METHODS 

The  preceding  sections  on  correlation,  the  %-  test,  the  theory  of  con- 
trol, and  the  probable  error  indicate  that  there  is  a  growing  recognition 
of  the  importance  of  statistical  analysis  in  sedimentary  problems.  One 
cannot  ignore  the  contributions  which  such  studies  have  made  and  will 
make  to  a  fuller  understanding  of  the  complex  study  of  sediments.  As 
methods  of  sampling  and  laboratory  analysis  are  improved,  and  as  more 
precise  methods  of  evaluating  errors  are  developed,  the  data  furnished 
by  sedimentary  studies  will  become  more  reliable,  and  consequently  the 
inferences  drawn  from  the  data  may  be  expected  to  be  more  sound.  Mean- 
while, parallel  studies  of  sediment  genesis,  in  terms  of  the  controlling 
environmental  factors,  may  ultimately  lead  to  an  understanding  between 
conditions  of  deposition  and  statistical  parameters,  which  will  pave  the 
way  for  more  quantitative  reconstructions  of  past  environments  in 
historical  geology. 


CHAPTER    10 

ORIENTATION  ANALYSIS  OF  SEDIMENTARY  PARTICLES 

INTRODUCTION 

Under  certain  conditions  of  deposition  sedimentary  particles  may  as- 
sume a  given  orientation  with  respect  to  the  surface  of  deposition.  The 
imbrication  of  stream  pebbles  is  a  common  example,  but  it  is  only  one 
of  a  large  number  of  similar  cases.  Numerous  writers  have  described 
oriented  deposits,  but  comparatively  little  has  been  done  in  a  quantitative 
manner  with  the  large  field  of  study  available  in  the  investigation  of  the 
primary  orientation  of  sedimentary  particles.  By  primary  orientation  is 
meant  the  arrangement  in  space  of  the  component  particles  during 
deposition,  regardless  of  subsequent  changes  in  position. 

The  fertile  fields  of  research  which  have  been  opened  in  the  study  of 
igneous  and  metamorphic  rocks  by  the  techniques  of  petrofabric  analysis 
suggests  that  similar  results  may  accrue  from  a  wider  application  of  like 
methods  to  sediments.^  Among  sedimentary  materials  the  techniques  of 
analysis  may  often  be  more  conveniently  applied  than  to  igneous  or  meta- 
morphic rocks.  In  the  latter  instances  it  is  necessary  to  work  with  ori- 
ented thin  sections,  often  composed  of  small  grains ;  among  sediments 
one  may  study  unconsolidated  gravels,  for  example,  in  which  the  particles 
may  readily  be  examined  individually.  Smaller  particles,  such  as  sand 
grains,  may  of  course  require  special  techniques,  especially  among  un- 
consolidated deposits.  Artificial  induration  with  bakelite  or  similar  ma- 
terial may  preserve  original  relations  among  the  grains. 

Among  orientation  studies  of  large  particles  is  that  of  Wadell,-  who  in- 
vestigated the  orientation  of  pebbles  in  an  esker  and  an  outwash  delta,  to 
determine  whether  eskers  are  necessarily  the  result  of  deposition  in  subglacial 
streams.  His  results  were  extremely  interesting,  inasmuch  as  they  showed  that 
the  long  axes  of  the  pebbles  in  the  esker  gravel  were  in  general  parallel  to 
the  direction  of  dip  of  the  bedding  planes,  whereas  in  the  foreset  beds  of  the 
delta  the  orientation  of  the  long  axes  was  more  or  less  diametrically  opposite 

1  Suggestions  of  the  possibilities  afforded  by  such  studies  were  given  by  E,  B. 
Knopf,  Petrotectonics :  Am.  Jour.  Sci.,  vol.  25,  pp.  433-470,  1933. 

-  H.  Wadell,  Volume,  shape,  and  shape-position  of  rock  fragments  in  open-work 
gravel :  Geografiska  Annalcr,  1936,  pp.  74-92. 

268 


ORIENTATION  ANALYSIS 


269 


to  the  direction  of  dip  of  the  beds.  These  relations  are  shown  in  Figure  122, 
adapted  from  Wadell's  paper.  In  both  deposits  the  dip  of  the  beds  was  about 
the  same  and  the  sizes  of  the  particles  were  of  the  same  order  of  magnitude. 
Winer  ^  had  previously  noted  that  fragments  in  talus  were  arranged  with  their 
long  axes  parallel  to  the  dip  of  the  slope,  and  on  this  basis  Wadell  offered  a 
tentative  conclusion  that  the  accumulation  of  the  pebbles  in  the  esker  was 
essentially  subaerial  in  nature. 

Among  otlier  studies  involving  particle  orientation  may  be  mentioned 
Richter's  studies  of  the  pebbles  in  glacial  till  -  as  a  statistical  device  to  deter- 
mine the  direction  of  ice  movement.  Ricliter  plotted  the  orientations  of  tlie 


30 


Sio 


1                    1 

i 

k 

-Dip  of  bed 

1— 

"~^ 

i 

i 

1 — 

— 

J.. 

30 

Dipof  bed-^ 

20 

> 
1 

J 

■10 

— 1 

> 

0 

000 

ift       0>       <*) 

(vj     <M     n 


ORIENTATION 


o     p     o     o 
n     "^     =     2 

ORIENTATION 


o    o 


Fig. 
the  left 


122. — Histograms  of  pebble  orientation,  after  Wadell,  1936.  The  figure  at 
represents  esker  gravel,  that  at  the  right  is  from  an  outwash  delta. 


long  axes  of  the  pebbles  in  terms  of  compass  direction  and  used  the  orienta- 
tion of  tlie  modal  group  as  an  index  of  direction  of  ice  movement.  The  study 
was  supported  by  actual  examinations  of  the  relation  of  the  long  axes  in  cases 
where  the  direction  of  ice  movement  was  known.  Richter  also  pointed  out  the 
further  implications  of  his  studies  in  terms  of  pebble  orientation  as  a  function 
of  the  nature  of  glacial  movement.  According  to  his  view,  the  arrangement 
of  the  pebbles  argued  for  streamline  motion  of  the  ice. 


1  N.  A.  Miner,  Talus  slopes  of  the  Gaspe  Peninsula :  Science,  vol.  79,  pp.  229-230, 

-  K.  Richter,  Die  Bewegungsrichtung  des  Inlandeises,  rekonstruiert  aus  den  Krit- 
zen  und  Langsachsen  der  Geschiebe :  Zcits.  j.  Gcschiebcjorschung ,  vol.  8,  pp.  62- 
66,  1932. 


2-0       SEDIMENTARY  PETROGRAPHY 

The  orientation  of  roller-shaped  pebbles  on  beaches  was  studied  by  Fraser.^ 
He  found  statistically  that  the  most  common  position  was  for  the  pebbles  to 
lie  with  their  long  axes  parallel  to  the  shore  line.  On  the  average,  only  16.5 
per  cent  were  oriented  with  their  long  axes  more  than  45°  from  parallelism 
to  the  water's  edge.  Fraser  attributed  his  findings  to  the  tendency  for  waves 
to  swing  pebbles  into  that  position  or  for  roller-shaped  pebbles  to  roll  with 
their  long  axes  perpendicular  to  the  direction  of  movement. 

These  examples  serve  to  illustrate  the  types  of  quantitative  data  which 
may  be  obtained  in  orientation  analysis.  The  study  of  particle  positions 
may  properly  supplement  size,  mineral,  and  shape  data  in  the  complete 
study  of  sediments. 

The  study  of  pebble  orientation  is  no  less  a  statistical  operation  than 
mechanical  analysis.  The  orientation  of  a  single  particle  may  be  rela- 
tively meaningless,  but  the  "average"  direction  of  orientation  may  be 
found  from  a  study  of  the  frequency  distribution  of  the  individual  orien- 
tations. Perhaps  the  most  significant  "average"  to  use  in  such  cases  is 
the  mode.  Whether  other  statistical  devices  such  as  measures  of  spread 
or  asymmetry  of  the  distribution  have  significance  in  these  studies  is 
largely  a  matter  for  further  investigation. 


COLLECTION     OF     ORIENTED     SAMPLES 

The  following  details  for  the  collection  of  oriented  sedimentary  par- 
ticles, adapted  from  W'adell,-  apply  to  fairly  large  pebbles  in  uncon- 
solidated deposits. 

The  face  of  the  outcrop  is  cleaned  and  a  rectangular  sampling  area 
enclosing  about  100  pebbles  is  marked  on  the  gravel  face.  Instruments 
required  for  sampling  include  a  Brunton  pocket  transit,  a  soft  pencil, 
two  fine  brushes,  and  cans  of  quick-drj-ing  black  and  red  lacquers.  The 
direction  and  dip  of  the  gravel  bed  are  determined,  and  the  compass 
trend  of  the  face  of  the  outcrop  is  read. 

For  collecting  an  individual  pebble,  the  Brunton  is  held  at  eye-level 
as  a  prismatic  compass,  with  the  mirror  in  position  to  reflect  the  leveling 
bubble  and  the  compass  needle.  The  compass  is  held  in  one  hand  and 
the  pencil  in  the  other.  The  back  edge  of  the  compass  is  set  parallel  to 
the  trend  of  the  gravel  face,  and  the  instrument  is  leveled.  A  vertical 
pencil  line  is  now  drawn  on  the  pebble  parallel  to  the  etched  line  in  the 
Brunton  mirror,  and  a  horizontal  line  is  drawn  parallel  to  the  top  edge 
of  the  mirror.  The  lines  are  checked  by  holding  the  compass  in  both 
hands.  When  they  are  found  correct,  the  horizontal  line  is  drawn  with 
black  enamel  and  the  vertical  line  with  red  enamel.  A  black  dot  is  also 

1  H.  J.  Fraser,  Experimental  study  of  the  porosity  and  permeability  of  clastic 
sediments:  Jour.  Geology,  vol.  43,  pp.  910-1010  (esp.  pp.  978  ff.),  1935. 
-  H.  Wadell,  loc.  cit.,  1936. 


ORIENTATION  ANALYSIS  271 

placed  in  the  lower  right-hand  portion  of  the  pebble  to  indicate  dip-direc- 
tion and  orientation  position. 

After  the  lacquer  has  dried,  the  pebbles  are  individually  removed.  As 
a  general  rule  the  pebbles  may  be  carried  in  an  ordinary  container,  in- 
asmuch as  the  lacquer  will  not  be  affected  by  ordinary  rubbing  of  one 
pebble  against  the  next. 

If  the  sample  to  be  collected  is  sand  or  finer  material,  where  the 
particles  cannot  be  handled  individually,  the  sand  must  be  impregnated 
with  a  binder,  or  a  sample  removed  with  a  device  which  does  not  disturb 
the  relations  of  the  grains  to  one  another.  Sampling  apparatus  similar 
to  that  used  for  porosity  determinations  may  be  suitable  (see  Chapter 
20).  In  all  cases  the  exact  orientation  of  the  sample  as  a  whole  must  be 
recorded.  In  indurated  sediments  orientation  lines  may  be  drawn  directly 
on  the  specimen  to  be  removed  from  the  outcrop. 


LABORATORY     ANALYSIS     OF     PARTICLE 
ORIENTATION 

Methods  of  measuring  and  tabulating  the  data  in  orientation  analysis 
depend  upon  the  nature  of  the  sediments  being  studied.  The  technique 
is  fairly  simple  for  unconsolidated  material  coarse  enough  to  be  handled 
individually ;  for  consolidated  materials  no  simple  procedures  are  known 
to  the  authors.  The  determination  of  primary  particle  orientation,  as 
opposed  to  the  determination  of  crystallographic  axes,  presents  prob- 
lems of  locating  the  long  axis  of  particles  in  thin  section.  The  actual  long 
axis  of  the  particle  may  not  lie  in  the  plane  of  the  thin  section,  and  it 
probably  is  not  sufficient  to  determine  the  apparent  orientation  of  the 
longer  axis  of  the  grain  section.  Methods  of  analysis  analogous  to  those 
used  in  petrof abric  analysis  ^  may  be  developed,  however,  for  grains 
in  which  there  is  a  more  or  less  fixed  relation  between  the  orientation  of 
the  long  diameter  of  the  grain  and  a  crystallographic  axis.  For  example, 
it  is  obvious  that  in  zircon  grains  the  long  diameter  is  parallel  to  the 
c-axis;  for  rounded  grains  the  relations  may  be  less  clear.  Pettijohn  has 
observed  from  his  own  work  that  statistically  the  long  diameters  of 
quartz  grains  tend  to  lie  along  the  c-axis  of  the  original  crystals ;  for 
other  minerals  similar  characteristics  may  be  found. 

Samples  of  pebbles  may  most  conveniently  be  measured  with  an  or- 
dinary two-circle  contact  goniometer.  The  method  is  shown  in  Figure 
123,  adapted  from  Wadell's  paper.  A  vertical  red  line  and  a  horizontal 

1  B.  Sander,  Gcjiigckunde  der  Gcsteine  (Vienna,  1930).  H.  W.  Fairburn,  Struc- 
tural Petrology  (Queen's  University,  Kingston,  Can.,  1937). 


272 


SEDIMENTARY  PETROGRAPHY 


l)lack  line  are  drawn  on  a  plate  of  glass,  which  is  supported  before  the 
goniometer.  The  pebble  is  mounted  on  the  goniometer  with  putty,  in 
such  a  manner  that  the  painted  lines  on  the  pebble  coincide  with  the 
corresponding  reference  lines  on  the  glass.  In  this  manner  the  pebble  is 
in  the  same  position  as  it  occupied  in  the  outcrop.  The  longest  axis  of 
the  pebble  is  then  determined  by  inspection,  and  the  dip  of  this  axis  is 
read  by  rotating  the  horizontal  goniometer  circle  until   the  long  axis 


Fig.  123. — Goniometer  and  glass  plate  used  in  measurement  of  pebble  orienta- 
tion. After  Wadell,  1936. 

coincides  with  the  plane  of  the  vertical  circle,  whereupon  its  dip  is  found 
by  means  of  the  goniometer  ruler.^  Meanwhile,  the  amount  of  horizontal 
rotation  necessary  to  swing  the  pebble  into  position  is  read.  This  angle 
is  a  measure  of  the  deviation  of  the  pebble  from  the  trend  of  the  gravel 
face,  as  represented  by  the  glass  plate,  and  from  this  information  the 
compass  trend  of  the  long  axis  may  readily  be  computed.  The  process 
outlined  yields  two  values  for  each  pebble  which  may  be  tabulated  or 
presented  in  a  graph. 

PRESENTATION     OF     ANALYTICAL     DATA 

The  results  of  an  orientation  analysis  may  be  expressed  as  a  frequency 
distribution  of  the  direction  of  dip  of  the  pebbles  (see  Figure  122),  or 
the  distribution  may  show  the  actual  angle  of  dip  of  the  long  axes.  As  a 


1  It  is  assumed  in  Figure  123  that  the  long  axis  of  the  pebble  is  properly  oriented 
when  the  lines  on  the  glass  and  pebble  coincide.  Usually  the  lines  will  not  coincide 
after  the  pebble  is  rotated. 


ORIENTATION  ANALYSIS  2yi 

first  approximation  the  mode  may  be  taken  as  the  mid-point  of  the  modal 
class,  although  it  may  be  computed,  or  determined  graphically  from  the 
inflection  point  of  a  cumulative  curve  of  the  data. 

In  addition  to  histograms  of  strike  or  dip  direction,  orientation  studies 
may  include  polar  coordinate  diagrams  of  the  dip  and  strike  of  the  peb- 
bles. Such  charts  are  circles  divided  into  degrees  around  the  circumfer- 
ence (for  strike)  and  have  concentric  circles  dividing  their  radii  from 
O  to  90°,  for  angle  of  dip.  A  dot  is  placed  at  the  appropriate  point  for 
each  pebble,  which  results  in  a  circular  scatter  diagram.  The  data  are 
readily  visualized,  and  from  the  concentration  of  the  dots  it  is  possible 
to  estimate  the  modal  trend.  For  a  more  formal  determination  of  the 
mode  the  dots  on  the  diagram  may  be  assembled  into  classes  and  a 
histogram  prepared. 

The  choice  of  the  mode  as  the  significant  average  is  suggested  l)y  its 
nature,  i.  e.,  it  is  a  measure  of  the  most  abundant  individuals  in  the 
population.  If  a  number  of  pebbles  are  being  deposited  under  a  set  of 
controlled  conditions,  it  would  appear  that  most  pebbles  may  tend  to 
conform  to  the  set  conditions,  but  complexities  of  size  and  shape  would 
cause  some  deviations  from  the  fixed  orientation.  The  deviations  them- 
selves may  be  significant,  however,  in  terms  of  their  distribution  on  one 
side  or  the  other  of  the  mode.  Deviations  are  most  conveniently  studied 
in  terms  of  the  arithmetic  mean  of  the  distribution,  by  the  conventional 
methods  of  moment  analysis.  The  standard  deviation  afifords  a  measure 
of  the  average  spread,  and  higher  moments  afford  measures  of  asymmetry 
and  peakedness.  Improper  sampling  methods  may  reflect  themselves  in 
a  skew  distribution,  or  there  may  be  an  actual  genetic  significance  to 
such  measures. 

Several  samples  of  till  pebbles  were  studied  by  Krumbein,^  using  essentially 
the  technique  of  Richter  (page  269),  in  an  attempt  to  apply  conventional 
statistical  methods  to  orientation  data.  One  set  of  pebbles  was  collected  from 
a  till  exposure  overlying  striated  bedrock,  so  that  the  known  direction  of  ice 
movement  could  be  used  as  a  control.  The  striae  varied  from  N  5°  E  to  N 
30°  E,  a  range  of  25°.  The  distribution  of  strikes  of  the  long  axes  of  the  peb- 
bles showed  a  pronounced  mode  at  N  20°  E.  well  within  the  range  of  the  striae. 
The  arithmetic  mean  strike  was  N  8°  E,  also  within  the  range.  The  standard 
deviation  of  strikes  about  the  mean  strike  was  23°.  In  this  sample  either  the 
mode  or  the  mean  could  be  used  as  an  index  of  ice  movement,  despite  the 
difference  of  12°  in  their  trends.  Other  till  samples  showed  a  close  agreeinent 
between  mode  and  mean,  usually  within  5°.  Moreover,  a  composite  of  all 
samples  (400  pebbles)  showed  an  essentially  normal  (or  at  least  symmetrical) 
distribution  about  the  mean,  suggesting  that  the  deviations  are  random. 

1  Unpublished  data  at  the  University  of  Chicago. 


274       SEDIMENTARY  PETROGRAPHY 

Studies  of  serial  sets  of  oriented  pebbles  along  streams  or  other 
traverses  from  source  to  final  disposition  may  shed  important  light  on 
orientation  changes  in  the  direction  of  transport.  These  may  reflect  an 
increasing  control  of  strikes  as  a  function  of  distance  (by  a  decrease 
in  the  standard  deviation),  or  complexities  may  be  introduced  by  changes 
of  size  and  shape.  In  any  event,  there  is  a  fertile  field  for  further 
research,  not  only  on  orientation,  but  also  on  the  relations  between 
orientation,  size,  and  shape.  In  all  such  studies  statistical  analysis  will 
play  an  important  part ;  moreover,  improved  laboratory^  techniques  will 
enlarge  the  scope  of  possible  studies. 

Among  techniques  there  appears  to  be  a  need  for  a  simpler  method  of 
collecting  the  pebbles.  The  authors  have  developed  a  wooden  frame 
measuring  about  5  by  7  in.  (a  small  picture  frame  will  do),  in  which 
two  brass  rods  are  mounted  at  right  angles.  A  spirit  level  is  set  in  the 
frame,  and  by  holding  the  frame  upright,  level,  and  parallel  to  the  face 
of  the  exposure,  lines  may  readily  be  drawn  on  the  pebbles.  In  this 
manner  100  pebbles  may  be  collected  in  an  hour  or  two. 


PART  II 

SHAPE  ANALYSIS,  MINERALOGICAL  ANALYSIS, 

CHEMICAL  ANALYSIS,  AND  MASS 

PROPERTIES 


CHAPTER   11 

SHAPE  AND  ROUNDNESS 

INTRODU  CTI O  N 

The  shape  of  sedimentary  particles,  large  and  small,  is  one  of  the  fun- 
damental properties  of  these  particles  and  is  the  most  recent  to  be  studied 
quantitatively  and  statistically.  Observers  early  noted  the  modification 
of  shape  that  took  place  by  transportation,  and  the  master  experimenter 
Daubree  himself  studied  the  results  of  attrition  of  gravel  in  a  revolving 
cylinder — an  experiment  later  to  be  repeated  by  Wentworth  in  America 
and  Marshall  in  New  Zealand.  Geologists,  moreover,  noted  the  character- 
istic forms  imparted  to  cobbles  and  pebbles  by  ice  action  and  those 
developed  by  wind  abrasion  (Einkanter,  Drcikantcr).  Others  went  so 
far  as  to  generalize  that  marine  and  lacustrine  pebbles  are  round  and 
oval  or  roller-shaped,  but  not  wedge-shaped;  that  fluviatile  pebbles  are 
flat  and  wedge-shaped.^  These  generalizations  and  others  were  based 
on  qualitative  data  only  and  that  none  too  certain.  H.  E.  Gregory,  as  a 
result  of  his  study  of  many  exceptions  to  these  beliefs,  went  so  far  as 
to  say  that  "of  the  many  factors  whose  evaluation  is  essential  in  estab- 
lishing distinctions  between  modes  of  origin  of  conglomerate,  that  of 
shape  of  pebbles  has  perhaps  the  least  significance.  No  constant  diflfer- 
ence  between  the  constituents  of  marine,  lacustrine  and  river  gravel  is 
likely  to  be  established."  -  This,  too,  is  a  generalization  based  on  little  if 
any  quantitative  data. 

The  statement  has  been  made  ^  and  often  repeated  and  also  denied* 
that  wind  is  capable  of  rounding  smaller  grains  than  water.  If  so,  the 
aqueous  or  aeolian  origin  of  an  ancient  sandstone  could  be  determined  by 
noting  the  lower  limit  of  rounding  of  the  grains. 

So  many  geologic  factors  are  involved  in  the  development  of  shape 

1  R.  Hoernes,  Gerolle  und  Geschiebe :  Vcrhandl.  K-K.  Geol.  RciclisanstaU,  no. 
12,  pp.  267-274,  191 1. 

-  H.  E.  Gregory,  Note  on  the  shape  of  pebbles:  Am.  Jour.  Set.,  vol.  39,  pp.  300- 
304,  191S. 

3  Wm.  Mackie,  On  the  laws  that  govern  the  rounding  of  particles  of  sand :  Trans. 
Edinburgh  Geol.  Sac,  vol.  7,  pp.  298-311,  1897.  Victor  Ziegler,  Factors  influencing 
the  rounding  of  sand  grains:  Jour.  Geology,  vol.  19,  pp.  645-654,  191 1. 

*  G.  E.  Anderson,  Experiments  on  the  rate  of  wear  of  sand  grains :  Jour.  Geology, 
vol.  34,  pp.  144-158,  1926. 

277 


278        SEDIMENTARY  PETROGRAPHY 

and  roundness  that  any  criterion  of  origin  based  on  a  single  principle  is 
likely  to  be  unreliable.  Factors  that  control  shape  and  roundness  are: 
(i)  the  original  shape  of  the  fragment,  (2)  the  structure  of  the  frag- 
ment, as  cleavage  or  bedding,  (3)  the  durability  of  the  material,  which 
is  in  turn  a  vector  property  of  the  rock  or  mineral  fragment,  (4)  the 
nature  of  the  geologic  agent,  (5)  the  nature  of  the  action  to  which 
the  fragment  is  subject  and  the  violence  of  that  action  (rigor),  and 
(6)  the  time  or  distance  through  which  the  action  is  extended. 

It  seems  clear  that  if  the  problem  of  development  of  shape  is  to  be 
studied  in  any  other  way  than  casually,  some  rigorous  definition  of  shape 
and  roundness  must  be  established.  Such  qualitative  expressions  as  "an- 
gular," "subangular,"  "subrounded,"  and  "rounded"  are  vague.  No  two 
observers,  moreover,  can  agree  on  the  proper  designation  of  a  given 
sediment.  A  simple  scheme  of  measuring  objectively  and  a  method  of 
expressing  numerically  the  shape  and  the  roundness  of  a  grain  is  neces- 
sary not  only  for  descriptive  purposes,  but  for  the  prosecution  of 
quantitative  studies  of  the  several  factors  involved  in  the  evolution  of 
the  shape  of  a  particle  or  fragment. 

The  shape  of  fragments  and  grains  has  a  bearing  on  several  other 
problems.  It  has  been  generally  assumed — not  entirely  correctly,  as  will 
be  shown  later — that  sand  grains  and  pebbles  become  progressively  more 
round  as  they  are  transported,  so  that  theoretically  the  direction  from 
which  a  sediment  came  could  be  determined  if  a  progressive  increase  in 
roundness,  or  roundness  gradient,  were  detectable. 

A  by-product  of  the  study  of  shape  and  roundness  has  been  the  use 
made  of  these  characteristics  for  correlation  purposes.  Certain  horizons 
have  been  marked  by  specific  degrees  of  roundness  and  sphericity.^ 

Certain  other  properties  of  sediments,  notably  porosity  and  permea- 
bility,^ are  related  to  the  shape  of  the  component  grains  of  the  sediment. 

REVIEW     OF     QUANTITATIVE      METHODS 

Sorby  in  1879  classified  sand  grains  into  five  groups:^ 

I.  Normal  angular  fresh-formed  sand,  as  derived  almost  directly  from 
granitic  or  schistose  rocks. 

1  A.  C.  Trowbridge  and  M.  E.  Mortimore,  Correlation  of  oil  sands  by  sedimen- 
tary analysis  :  Econ.  Geology,  vol.  20,  pp.  409-423,  1925.  Tor.  H.  Hagerman,  Some 
lithological  methods  for  determination  of  stratigraphic  horizons:  World  Petroleum 
Congress,  Proc.  193^  (London),  vol.  i,  pp.  257-259. 

2  H.  J.  Fraser,  Experimental  study  of  the  porosity  and  permeability  of  clastic 
sediments :  Joxir.  Geology,  vol.  43,  pp.  934-938 ;  962-964,  1935. 

3  H.  C.  Sorby,  On  the  structure  and  origin  of  non-calcareous  stratified  rocks: 
Quart.  Jour.  Geol.  Soc.  London,  vol.  36,  Proc,  pp.  46-92,  1880. 


SHAPE  AND  ROUNDNESS  279 

2.  Well-worn  sand  in  rounded  grains,  the  original  angles  being  completely 
lost,  and  the  surface  looking  like  fine  ground  glass. 

3.  Sand  mechanically  broken  into  sharp  angular  chips,  showing  a  glassy 
fracture. 

4.  Sand  having  the  grains  chemically  corroded,  so  as  to  produce  a  peculiar 
texture  of  the  surface,  differing  from  that  of  worn  grains  or  crystals. 

5.  Sand  in  which  the  grains  have  a  perfect  crystalline  outline,  in  some 
cases  undoubtedly  due  to  the  deposition  of  quartz  over  rounded  or  angular 
nuclei  or  ordinary  non-crystalline  sand. 

It  may  be  seen  from  the  above  that  Sorby's  scheme  of  classification 
is  both  descriptive  and  genetic  and  involves  surface  character  as  well  as 
shape. 

Wentworth  seems  to  have  been  the  first  to  develop  a  quantitative  sys- 
tem of  measurement  of  the  shape  of  individual  rock  particles  independent 
of  origin.  Wentworth^  expressed  the  shape  of  pebbles  by  a  roundness 
and  a  flatness  ratio.  The  roundness  ratio  is  r^/R,  where  r-L  is  the  radius 
of  curvature  of  the  sharpest  developed  edge  and  R  is  the  mean  radius  of 
the  pebble. 

The  mean  radius   (one  half  the  mean  diameter)   is  sometimes  difficult  to 

determine.  The  mean  diameter  may  be  the  arithmetic  mean  of  the  principal 

A  +  B  +  C        ,        \    „        ,  ^  ,      , 

diameters,  or ,  where  A,  B,  and  C  are  the  three  major  diameters 

of  the  solid,  the  length,  breadth,  and  thickness,  respectively.  The  geometric 
mean  may  also  be  used.  In  that  case  the  mean  value  is  -^  ABC.  A  major 
difficulty  arises  from  the  fact  that  no  agreement  has  ever  been  reached  in 
defining  the  three  diameters  of  a  non-spherical  object.  Some  workers  require 
that  the  three  diameters  meet  at  right  angles;  others  stipulate  that  the  lines 
of  measurement  must  be  at  right  angles,  but  do  not  require  a  common  point  of 
crossing.  On  pebbles  with  reentrant  angles,  the  terms  length,  breadth,  and 
thickness  become  ambiguous.  W'adell  has  used  the  nominal  diameter,  derived 
from  volume  measurements  of  the  pebbles,  to  avoid  these  confusions. 

The  flatness  ratio  is  expressed  by  n/R  where  r^  is  the  radius  of  curva- 
ture of  the  most  convex  direction  of  the  flattest  developed  face  and  R  is 
the  mean  radius  of  the  pebble.  In  his  study  of  beach  pebbles  Wentworth 
expressed  the  flatness  ratio  as  the  arithmetic  mean  of  the  length  and 

A  -I-  B 
breadth  divided  by  twice  the  thickness,  or  — ^^ — .  The  radii  r^  and  Tz 

were  first  measured  by  a  gage  similar  to  that  used  by  opticians   for 

^  C.  K.  Wentworth,  A  laboratory  and  field  study  of  cobble  abrasion :  Jour.  Geol- 
ogy, vol.  27,  pp.  507-521,  1919:  The  shapes  of  pebbles:  U.  S.  Gcol.  Sun-ex,  Bull. 
730-c,  pp.  91-114,  1922:  The  shapes  of  beach  pebbles:  U.  S.  Gcol.  Sun-ey,  Prof. 
Paper  131-C,  pp.  75-83,  1922. 


( — '"- 

.0 

eSlPo 

^^^L^ 

280       SEDIMENTARY  PETROGRAPHY 

measuring  the  curvature  of  lenses.  Wentworth  later  developed  a  flat-type 
convexity  gage.  This  instrument  consists  of  a  low-angle  measuring 
wedge  sliding  in  a  split  profile  block.  The  profile  block  is  so  constructed 
that  the  radius  of  curvature  of  the  corners  or  edges  of  pebbles  ranging 
from  I  to  100  mm.  may  be  measured  conveniently  and  rapidly  to  within 

2  or  3  per  cent.  The  readings  are 
made  through  a  reading  slot  in  the 
split  block  on  a  scale  which  indi- 
cates the  position  of  the  measuring 
/l^iwt'^wofth^^^^    °^    convexity     ^^^^^^    (pj^^    ^^4).    R   is   computed 

either  as  half  the  arithmetic  mean 
of  length,  breadth,  and  thickness  of  the  pebble,  or  as  half  the  geometric 
mean  of  the  same  dimensions. 

Trowbridge  and  Mortimore^  used  a  visual  method  of  determining 
"roundness"  by  comparison  of  the  material  under  question  with  a  more 
or  less  arbitrary  set  of  standards. 

Lamar  ^  devised  a  mechanical  means  of  determining  the  relative 
"roundness"  or  "angularity"  of  sand  grains  in  the  bulk.  The  method 
consists  of  determining  the  minimum  porosity  of  sand  obtained  by  com- 
pacting. A  cylindrical  metal  tube,  1%  in.  in  diameter,  working  in  two 
guide  sleeves,  was  raised  a  half-inch  from  below  by  a  plunger  operating 
on  an  eccentric  and  allowed  to  drop  about  100  times  a  minute.  The 
cylinder  struck  a  piece  of  felt  so  as  to  produce  a  nearly  "dead"  fall, 
thus  reducing  to  a  minimum  the  amount  of  rebound  imparted  to  the 
sand  in  the  cylinder.  The  machine  was  motor-driven.  The  percentage  of 

porosity  in  the  sand  was  determined  from  the  formula  P  = ^ 

where  C  is  the  volume  of  the  sand  and  voids  measured  in  the  cylinder, 
V  is  the  actual  volume  of  the  sand  grains  determined  by  displacement 
of  water  in  a  graduate,  and  P  is  the  percentage  of  porosity,  with  maxi- 
mum compaction.  The  relative  angularity  for  the  sand  was  determined 
by  dividing  25.95,  the  theoretical  minimum  for  spherical  sand  grains,  by 
the  porosity  of  the  compacted  sample.  The  nearer  the  quotient  to  i.o, 
the  less  angular  the  sand.  In  order  that  the  angularity  be  thus  calculated 
it  is  necessary  to  use  sand  of  one  size  only,  hence  the  sample  to  be 
studied  must  be  screened  and  the  angularity  of  each  screened  separate 
individually  determined. 

^  A.  C.  Trowbridge  and  M.  E.  Mortimore,  Correlation  of  oil  sands  by  sedimen- 
tary analysis:  Ecoii.  Geology,  vol.  20,  pp.  409-423,  1925. 

-  J.  E.  Lamar,  Geology  and  economic  resources  of  the  St.  Peter  sandstone  of  Illi- 
nois:  ///.  Gcol.  Sun'cy,  Bull.  33,  pp.  148-151,  1927. 


SHAPE  AND  ROUNDNESS 


281 


The  first  attempt  to  express  the  "roundness"  of  individual  sand  grains 
was  that  made  by  Pentland  in  1927.^  He  determined  the  percentage  area 
of  the  grain  projection  to  that  of  a  circle  with  diameter  equal  to  the 
longest  diameter  of  the  grain.  The  area  of  the  grain  projection  was 
determined  from  camera  lucida  drawings.  As 
W'adell  -  has  shown,  it  is  possible  to  have  two 
ditterent  plane  figures  of  equal  areas  and 
equal  major  diameters  but  of  distinctly  dif- 
ferent shapes  (Figure  125). 

Cox  also  studied  the  projection  figures  of 
individual  grains.^  He  projected  the  image  of 
the    grains    on   a    screen    and    from   drawinsrs 


' 


Fig.  125. — Figures 
with  same  major  diame- 
ter and  areas  but  with 
different  shapes.  The 
long  diameter  of  the 
grain  "B"  is  not  the  di- 
ameter of  the  circum- 
scribing sphere.  The  dif- 
ference between  the 
method  of  Pentland  and 
that  of  Wadell  is  thus 
demonstrated. 


Fig.  126. — Figures  with  same 
perimeters  and  areas  but  of  dif- 
ferent shapes. 


made  calculated  the  roundness  or  circularit\-  according  to  the  formula 


K 


AX  4-^ 


,  where  A  is  the  area  measured  by  a  planimeter.  and  P  is 


the  perimeter  measured  by  a  map  measurer.  K  is  the  value  of  the  round- 
ness and  depends  on  the  shape  of  the  projection  figure.  For  a  circle  it 


^  A.  Pentland.  A  method  of  measuring  the  angularity  of  sands :  Rcyal  Soc. 
Canada,  Proc.  and  Trans.  (Ser.  3),  vol.  21,  1927,  Appendix  C,  Titles  and  Abstracts, 
p.  xciii. 

2  Hakon  Wadell,  Volume,  Shaf^e  and  Roundness  of  Rock  Particles.  A  dissertation 
submitted  to  the  Faculty  of  the  Division  of  Physical  Sciences  in  candidacy  for  the 
degree  of  Doctor  of  Philosophy.  University  of  Chicago.  June  1932.  MS. 

3  E.  P.  Cox,  A  method  of  assigning  numerical  and  percentage  values  to  the  degree 
of  roundness:  Jour.  Palcon.,  vol.  i,  pp.  179-183.  1027. 


282       SEDIMENTARY  PETROGRAPHY 

is  I.  Wadell  has  also  shown  that  it  is  possible  to  have  two  figures  of 
the  same  perimeter  and  the  same  area  but  of  quite  different  shape 
(Figure  126). 

Tickell  ^  more  recently  used  the  ratio  of  the  area  of  the  projected 
grain  to  the  area  of  the  smallest  circumscribed  circle  to  express  "round- 
ness." This  is  very  nearly  the  same  method  as  that  used  by  Pentland. 
For  pebbles  Tickell  recommends  the  ratio  of  the  volume  of  the  pebble 
to  the  volume  of  the  smallest  enveloping  sphere.  The  volume  of  the 
pebble  would  be  determined  by  weighing  in  air  and  weighing  in  water. 
Tickell  thus  made  an  improvement  over  Pentland  and  Cox,  yet  he  too 
failed  to  differentiate  between  shape  and  roundness. 

Tester,^  in  193 1,  proposed  a  very  different  method  of  expressing 
"roundness."  He  determined  the  ratio  of  the  length  of  the  original  edge 


-.^--i\% 


AB      ■  Zi 

=  75% 


CB-bb'  -  18   _ 
CB  24 


CD-cc    .  19 
CO  24 


79% 


---  67% 


Fig.    127. — Shapometer    of    Tester  Fig.  128. — Example  of  Tester's  method 

and  Bay.  of  roundness  determination. 

or  edges  of  a  grain  to  the  portion  or  portions  worn  away.  Tester  and 
Bay^  devised  a  "shapometer"  to  facilitate  the  measurements  required 
by  his  method  (Figure  127). 

For  example,  in  Figure  128,  the  outline  of  an  abraded  fragment,  the 
remnant  edges  aa',  hh' ,  cc' ,  and  dd'  are  shown  and  are  extended  to  form 
polygon.  A,  B,  C,  and  D.  A,  B,  C,  and  D  are  believed  to  outline  tlie 
original  shape  of  the  block  from  which  the  pebble  was  derived.  The 
length  of  the  lines  representing  the  projected  edges,  as  line  AB,  and  the 
length  of  the  remnant  edges,  as  aa',  are  measured.  The  total  length  of 
AB  is  taken  as  denominator  and  the  part  of  the  line  AB  not  in  contact 

^  Frederick  G.  Tickell,  The  Examination  of  Fragnicntal  Rocks  (Stanford  Uni- 
versity Press,  1931),  pp.  6-7. 

2  A.  C.  Tester,  The  measurement  of  the  shapes  of  rock  particles :  Jour.  Scd. 
Petrology,  vol.  i,  pp.  3-1 1,  1931. 

3  A.  C.  Tester  and  H.  X.  Bay,  The  shapometer :  a  device  for  measuring  the  shapes 
of  pebbles :  Science,  vol.  72,  pp.  565-566,  1931. 


SHAPE  AND  ROUNDNESS  283 

with  the  grain  (or  difference  between  AB  and  aa')  is  the  numerator. 
The  ratio  {AB-aa')/AB  multiplied  by  100  gives  the  percentage  of  the 
original  edge  worn  away.  The  value  for  each  edge  is  separately  deter- 
mined, and  the  average  for  the  whole  pebble  is  then  computed  (see 
Figure  128). 

On  the  basis  of  the  values  obtained,  a  pebble  may  be  classified  in  one 
of  the  five  groups  set  up  by  Tester: 

Percentage  Abraded  Class  Name 

81-100    Rounded 

61-  80    Sub-rounded 

41-  60    Curvilinear 

21-40    Sub-angular 

0-20    Angular 

Much  difficulty  is  found  in  application  of  this  method,  since  it  is  based 
on  the  ratio  of  an  assumed  factor,  the  original  shape  (largely  unknow- 
able), to  a  known  factor,  the  present  shape. 

Wadell  ^  appears  to  be  the  first  to  differentiate  between  shape  (spheric- 
ity) and  roundness  and  to  show  that  these  are  two  independent  variables. 
Wadell  pointed  out  that  roundness  was  a  matter  of  the  sharpness  of  the 
comers  and  edges  of  a  grain,  whereas  shape  has  to  do  with  the  form 
of  the  grain  independently  of  the  sharpness  of  its  edges.  The  several 
geometrical  solids,  for  examples,  cube,  tetrahedron,  dodecahedron,  etc., 
are  clearly  of  diflferent  shapes,  yet  their  respective  edges  or  corners  are 
equally  sharp,  i.e.,  the  radius  of  curvature  of  the  edges  is  o. 

Since  the  sphere  has  the  smallest  surface  area  in  proportion  to  volume 
of  any  solid,  it  has  a  higher  settling  velocity  for  a  given  volume  than 
any  solid  of  any  other  shape.-  Wadell,  therefore,  used  the  sphere  as  a 
standard  of  reference  and  spoke  of  the  "degree  of  sphericity"  as  a 
measure  of  the  approach  of  other  solids  to  the  sphere  in  form.  Hence 
in  the  accumulation  of  sediment  from  suspension  the  degree  of  sphericity 
of  the  component  grains  is  an  important  factor.  So  also  in  the  trans- 
portation of  debris  by  traction  is  the  spherical  form  a  suitable  standard 
of  reference,  since  a  sphere  will  roll  more  easily  than  solids  of  other 
shape.  .\n  expression  which  approximately  reflects  the  behavior  of  a 
particle  in  suspension  is  the  ratio  of  the  surface  area  of  a  sphere  of  the 

1  Hakon  Wadell,  Volume,  shape  and  roundness  of  rock  particles :  Jour.  Geology, 
vol.  40,  pp.  443-451,  1932;  Sphericity  and  roundness  of  rock  particles:  Jour.  Geol- 
ogy, vol.  41,  pp.  310-331,  1933;  Volume,  shape  and  roundness  of  quartz  particles: 
Jour.  Geology,  vol.  43,  pp.  250-280,  1935. 

-  Except  only  some  pear-shaped  solids  with  displaced  centers  of  gravity. 


284       SEDIMENTARY  PETROGRAPHY 

same  volume  as  the  particle  to  the  actual  surface  area  of  the  particle 

expressed  by  the   f ormula-^^  ^  ly,  where  s  is  the  surface  area  of  the 

sphere  of  the  same  volume,  S  is  the  actual  surface  area  of  the  particle, 
and  ip  is  the  true  sphericity.  The  difficulty  of  determining  the  actual 
surface  area  and  volume  of  a  small  grain  led  Wadell  to  adopt  a  working 
formula  similar  to  that  proposed  by  Tickell.  This  gives  a  close  approxi- 
mation of  the  true  sphericity  and  may  be  stated  :-pr-=  4>>  where  dc  is  the 
diameter  of  a  circle  equal  in  area  to  the  area  obtained  by  planimeter 
measurement  of  the  projection  of  the  grain  when  the  grain  rests  on  its 
larger  face,  Dc  is  the  diameter  of  the  smallest  circle  circumscribing  the 
projection,  and  <^  is  the  shape  value  thus  obtained. 

Wadell's  method,  like  that  of  Pentland,  Cox.  and  Tickell,  which  involves 
the  use  of  projection  areas  for  the  study  of  grain  shape,  falls  in  error  in  tlie 
case  of  very  flat  grains.  Such  grains  tend  to  lie  upon  their  flattest  developed 
face.  Under  these  conditions  a  circular  disk  and  a  sphere  give  the  same  pro- 
jection image  and  therefore  the  same  sliape  value.  Some  writers,  therefore 
(Tickell,  for  example),  have  specified  that  the  grain  be  in  random  position 
and  have  taken  precautions  to  insure  such  orientation.  Wadell,  on  the  other 
hand,  specified  that  the  grains  be  oriented  more  or  less  parallel  to  the  largest 
and  intermediate  diameters.  He  found  from  actual  experiment  with  quartz 
particles  that  discrepancies  due  to  this  cause  were  not  great.  Wadell  not  only 
found  the  discrepancies  to  be  small,  but  also  gave  reasons  for  using  the  pro- 
jection area  containing  the  longest  and  intermediate  diameters.  Such  reasons 
were  related  to  the  behavior  of  quartz  particles  to  fracturing,  chipping,  etc. 

For  large  materials,  pebbles,  etc.,  WadelP  developed  a  different  for- 
mula: c/n/Ds  =  ^,  where  da  is  the  true  nominal  diameter  of  the  pebble 
or  the  volume  of  a  sphere  of  the  same  volume  and  Dg  is  the  diameter  of 
the  circumscribing  sphere — usually  the  longest  diameter  of  the  pebble. 

As  stated  above,  roundness  is  a  function  of  sharpness  of  edges ;  hence 
it  is  possible  to  have  solids  with  perfect  roundness  independently  of 
shape.  As  roundness  increases  the  radius  of  curvature  of  the  corners 
increases.  An  object  of  cylindrical  form  with  hemispherical  ends  would 
be  as  perfectly  rounded  as  a  sphere.  Such  an  object  may  eventually  be 
worn  down  to  a  sphere,  but  the  radius  of  curvature  of  its  ends  must, 
during  the  process  of  wear,  remain  always  equal  to  the  radius  of  the 
maximum  inscribed  circle  in  the  longitudinal  section  of  the  solid  (Figure 
129).  Roundness  was  therefore  defined  by  Wadell  as  a  value  computed 

1  Hakon  Wadell,  Shape  determinations  of  large  sedimental  rock-fragments :  Pan- 
American  Geologist,  vol.  61,  pp.  187-220,  1934. 


SHAPE  AND  ROUNDNESS 


28.:; 


from  a  plane  figure,  either  projection  or  cross-section,  in  which  the 
radius  of  the  individual  corners  is  divided  by  the  radius  of  the  maximum 
inscribed  circle.  The  roundness  of  the  individual  corners  thus  obtained 
is  added  up  and  divided  by  the  number  of  comers.  This  result  is  ex- 


r 
25- 


pressed  by  the  formula:  ^^^ 


P,  where  r  is  the  radius  of  cur\-ature 

of  the  corner,  R  is  the  radius  of  the  maximum  inscribed  circle.  N  is  the 
number  of  comers,  and  P  is  the  total  degree  of  roundness.  The  actual 
manipulative  technique  has  been  de- 
tailed by  Wadell  and  is  here  given 
elsewhere  (see  page  298). 


Wadell  also  used,  for  reasons  given 
by  him,  a  formula  which  gave  slightly 
different  roimdness  values  than  the 
one  given  above.  The  same  items  are 
measured  as  before  and  the  degree  of 

roundness  bv  the  expression  :-^; — ^  ,  ^ 
:^    (R/r) 

The    maximum    value    for    roundness 

bv  this  formula  is  also  i. 


Fig.  129. — Wadell's  concept  of  round- 
ness, a,  original  fragment  (with  in- 
scribed sphere)  :  b,  figure  with  maxi- 
mum roundness  (radius  of  curvature  of 
ends  equal  to  radius  of  inscribed  circle)  ; 
c,  figtu-e  with  low  roundness  resulting 
from  radius  curvature  of  the  ends 
greater  than  inscribed  circle. 


In     order     to     obtain     comparable 

values  a  standard  size  was  adopted. 

Large  objects  such  as  boulders  must  be  reduced,  and  small  ones  like  sand 

grains   must   be  magnified   to   approximately   the   same    size.   \\'adeirs 

standard  size  is  7  cm. 

Wadell  has  given  examples  of  grains  differing  from  one  another  in 
roundness  and  sphericity  values  and  introduces  the  term  image  as  a 
"binomial"  expression  of  shape  made  up  of  the  values  for  roundness 
and  sphericity  (see  Figure  130). 

Wadell  has  also  used  the  term  "degree  of  circularity"  defined  as  c/C, 
in  which  c  is  the  circumference  of  a  circle  of  the  same  area  as  the  plane 


.82  .82  ,«3 

Fig.  130A. — Grains  of  same  sphericity  but  differing  roundness  (after  Wadell). 


286       SEDIMENTARY  PETROGRAPHY 

figure  and  C  is  the  actual  circumference  of  the  plane  figure.  This  ex- 
pression is  used  to  describe  the  shape  of  a  plane  figure,  presumably  the 
projection  image  of  a  sedimentar)-  particle. 


Fig.  130B. — Grains  of  about  same  roundness  but  of  differing  sphericities  (after 
Wadell). 

It  is  clear,  then,  from  what  has  been  said  about  the  difference  between 
shape  and  roundness  that  Wentworth  measured  roundness,  whereas  the 
Pentland,  Cox,  and  Tickell  measures  express  the  shape  of  the  grain  pro- 
jections. Wentworth's  "flatness  ratio,"  on  the  other  hand,  involving  the 
ratio  of  the  mean  of  the  length  and  breadth  divided  by  thickness,  is 
more  of  a  shape  expression.  Tester's  method  is  somewhat  akin  to  that 
of  Szadeczky-Kardoss,  to  be  described  later,  and  is  a  type  of  roundness 
measure. 

Wadell  studied  the  same  material  as  was  investigated  by  Lamar,  namely, 
the  St.  Peter  sandstone  at  Ottawa,  Illinois,  and  the  sphericity  values  for 
each  of  the  grades  involved  as  determined  by  Wadell  agree  very  closely  with 
the  "angularity"  values  given  by  Lamar  for  essentially  the  same  grades.  It  is 
evident,  therefore,  that  a  person  can  obtain  an  average  sphericity  value  for  a 
given  grade  by  Lamar's  method.  Lamar's  method,  moreover,  measures  shape 
and  not  "angularity"  or  roundness  in  a  strict  sense. 

Other  workers  have  in  recent  times  studied  quantitatively  the  shape 
of  sedimentary  particles  and  fragments.  Szadeczky-Kardoss  ^  in  1933, 
after  criticizing  the  method  of  expressing  roundness  proposed  by  Went- 
worth and  also  that  of  Cox,  devised  a  new  scheme  of  measurement  and 
presentation  of  data.  The  method  applies  especially  to  grains  of  diameters 
2-100  mm.  The  pebble  to  be  investigated  is  placed  in  an  apparatus  (Fig- 
ure 131)  whereby  the  profile  of  the  fragment  in  one  plane  is  mechani- 
cally traced  without  change  in  size  on  a  sheet  of  paper.  The  outline  thus 
obtained  is  analyzed  and  the  percentage  of  concave,  C,  convex,  V,  and 
plane,  P,  parts  of  the  profile  is  determined.  A  number  of  roundness 
grades  were  defined  as  shown  in  the  table  below : 

1  E.  V.  Szadeczky-Kardoss,  Die  Bestimmung  des  Abrollungsgrades :  Ceniralbl.  f. 
Mill.,  Geol.,  u.  Pal'don.,  Abt.  B,  pp.  389-401,  1933. 


SHAPE  AND  ROUNDNESS 


287 


Table  31 
Szadeczky-Kardoss  Roundness  Classes 


Grade  0   

C  =  100% 

Grade  in           

c>rv  +  P) 

P>  V 

Grade  ib   

V>P 

(V  +  P)>C>V 

rc  +  V)>p 

Grade  2b             

P>(C  + V) 

r^rnrlp    "Jn                          

(C  +  P)>V>C 

PXC  +  V) 

Grade  3b   

('C  + V)>P 

Gradp  /la                    

V>(C  +  P) 

P>C 

C>P 

Grade  5   

V  =  100% 

Fig.  131. — Apparatus  of  v.  Szadeczky-Kardoss  for  tracing  profile  of  pebble. 


288 


SEDIMENTARY  PETROGR.\PHY 


On  a  triangle  diagram  with  C,  V,  and  P  the  three  comers 
(C  +  V  -f  P  =  ioo%  )  were  plotted  the  values  obtained  for  each  pebble. 
Differences  in  roundness  due  to  rock  structure  and  mode  of  origin  of 
the  deposit  are  thus  readily  shown  (Figure  132). 


b/3 


Fig.  132. — Triangle  diagram  of 
roundness  grades  of  v.  Szadeczky- 
Kardoss.  P  =  plane,  V  =  convex, 
and  C  =  concave  parts  of  profile. 
C  +  V  +  P  =  100  per  cent. 


I 

Disc-shaped 
(oblate  spheroid] 

Spherical 

m 

Bladed 
(tnaxia!) 

n 

Rod- like 
(prolate 
spheroid) 

c/b 


i/3 


Fig.  133. — Shape  classes  of  Zingg. 
a,  length ;  b,  breadth ;  and  c,  thickness. 


Zingg  ^  in  a  monographic  study  used  the  Szadeczky-Kardoss  method 
of  expressing  roundness  but  recognized,  following  Wadell,  the  clear  dif- 
ference between  roundness  and  sphericit)-.  Zingg,  however,  measured  the 
three  principal  diameters  of  a  pebble,  a,  b,  and  c.  On  the  basis  of  these 
measurements  he  set  up  four  classes : 


I  b/a>2/z 

II  b/a>2/s 

III  b/a<  2/3 

IV  b/a<2/3 


c/b  <2  3 
c/b  >2  3 
c/b<2  s 
c/b  >  2/3 


where  a  >  &  ><:.  These  are  set  in  a  table  and  given  specific  names  (Fig- 
ure 133)  : 

Hagerman  -  utilized  grain  shape  to  mark  different  stratigraphic  hori- 
zons. He  carefully  split  down  the  sample,  mounted  a  few  hundred  grains 
on  a  microscope  sHde  in  such  manner  that  the  grains  lie  on  their  greatest 
developed  face.  The  length  (/)  and  breadth  {b)  of  each  grain  were 
then  measured  imder  the  microscope  with  micrometer  ocular  or  with 
a  microprojector  and  the  b/l  ratio  computed.  The  values  for  each  grain 

1  Th.  Zingg,  Beitrag  zur  Schotteranalyse :  Schweiz.  Min.  u.  Pet.  Mitt.,  Bd.  15, 
pp.  39-140,  1935- 

2  Tor  H.  Hagerman,  Some  lithological  methods  for  determination  of  stratigraphic 
horizons:  World  Petroleum  Congress,  Proc.  IQSS  (London),  vol.  i,  pp.  257-259. 


SHAPE  AND  ROUNDNESS 


289 


{h/l  ratio  and  /)  determine  a  point  on  the  diagram  (Figure  134).  The 
h/l  ratio  is  diagrammed  as  ordinate  and  /  is  plotted  as  abscissa.  The 
grains  fall  into  four  groups : 


Small,  equiaxial  grains 
Small,  oblong  grains 


Large,  equiaxial  grains 
Large,  oblong  grains 


As  plotting  continues  the  boundaries  of  the  distribution  field  begin  to 
appear.  A  search  for  limit  grains  soon  fixes  the  boundary  of  the  field 
quite  closely.  The  limit  observations  are 
marked  with  heavy  dots  through  which  a 
curve  may  be  drawn.  The  shape  of  the  dis- 
tribution field  is  related  to  sedimentation 
conditions  at  time  of  deposit  (turbulence, 
current  velocity,  etc.)   and  was   found  to  be  269 

ingularity  =  J5Q  =74.7p«r  ce 


\ 


1.0 

F 

. 

\ 

1 

, 

. 

\ 

j- 

\ 

1 

/ 

\. 

; 

'^ 

N  •. 

/ 

•s 

■^ 

^'* 

Fig.  134. — Haserman  plot  of  quartz 
grains  of  a  sandstone.  Limiting  or 
boundary  grains  shown  by  heavy  dots. 


«n|ularily  =353  =  21.9 per  cent 


Fig.  135.  —  Fischer's 
method  of  angularity 
computation.  The  ratio 
of  the  worn  (convex)  to 
unworn  (plane)  portions 
of  the  profile,  measured 
in  terms  of  a  central 
angle,  determines  the 
angularity. 


characteristic  of  certain  stratigraphic  horizons. 

Hagerman  ^  has  more  recently  published  detailed  results  of  his  method 
applied  to  certain  Argentine  formations. 

Fischer,  in  a  recent  paper  on  graywackes,  devised  a  method  of  study- 
ing rounding  of  grains  in  thin  sections.-  There  are  360°  about  any  point. 
Fischer  chose  a  central  point  within  the  sectional  outline  of  the  grain 
and  measured  the  angles  around  this  point  which  were  governed  or 
subtended  by  the  straight  or  non-curved  parts  of  the  profile.  The  ratio 


1  Tor  H.  Hagerman,  Granulometric  studies 
Aiinalcr,  vol.  18,  pp.  125-213,  1936. 

-  Georg  Fischer,    Die   Petrographie   der   Grauwackcn 
Landcsanst.  (Berlin),  Bd.  54,  pp.  322-323,  1933. 


northern  Argentine:  Geagrafiska 
Jahrb.   d.   Prciiss.    Gcol. 


290 


SEDIMENTARY  PETROGRAPHY 


of  the  sum  of  these  angles  to  the  whole  angle  of  360°  gave  the  angular- 
ity value  of  the  grain.  In  Figure  135a.  for  example,  the  angularity  is 
70.5  per  cent,  while  in  Figure  135b  it  is  21.9  per  cent.  Fischer  gives  no 
specific  instructions  for  choosing  the  central  point.  The  authors  suggest 
the  center  of  the  inscribed  circle. 

Recently  Wentworth^  has  fallen  back  on  a  verbal  schedule  for  de- 
scription of  cobble  shapes  in  which  the  shape  is  compared  to  some  well 
known  geometrical  form.  Terms  such  as  prismoidal,  bi pyramidal,  pyra- 
midal, wcdgc-shapcd,  parallel  tabular,  etc.,  are  used.  The  form  of  the 
margin,  as  viewed  from  the  top  was  likewise  described  as  hexagonal, 
pentagonal,  trapezoidal,  oval,  rhombic,  etc.  The  major  diameters  were 
also  measured,  and  ratios  between  these  were  used  in  statistical  study 
of  the  collected  materials. 

SUMMARY 

It  is  clear  from  the  historical  review  just  given  that  there  is  no  ac- 
cepted standard  at  the  moment  for  measuring  the  shape  of  irregular 
solid  particles.  The  studies  reviewed  show  several  lines  of  approach  to 
this  problem: 


Method 

A.    Methods   based  on  measurements  of 
individual  particles  or  fragments 

1.  Visual  comparison  with  arbitrary 
set  of  standards 

2.  Diameter  measurements  :  Ratio  be- 
tween longest  and  intermediate 
and/or  shortest  diameters  or  mean 
of  length  and  breadth  divided  by 
thickness 


Attribute 
Measured 


Auihor 


Roundness  Trowbridge    and 

and/or    shape     Mortimore 


Shape 


Hagerman 

Zingg 

Wentworth 


Measurements    of    projection    or 
sectional  areas : 

a)  ratio  of  plane  to  convex  and/     Roundness 
or  convex  parts  of  profile 

b)  ratio  of  perimeter  to  area  of     Shape 
grain  projection 


Tester 
Szadeczky- 

Kardoss 

Cox 


iC.  K.  Wentworth,   An  analysis  of  the  shapes  of  glacial   cobbles:  Jour.  Sed. 
Petrology,  vol.  6,  pp.  85-96,  1936. 


SHAPE  AND  ROUNDNESS 


291 


B. 


Method 

c)  ratio  of  area  of  projection  to 
smallest  circumscribing  circle 

d)  ratio  of  radius  of  curvature 
of  sharpest  corner  or  mean  radius 
of  curvature  of  all  corners  to  half 
the  diameter  through  tlie  corner  or 
to  radius  of  maximum  inscribed 
circle 

4.  Volume  measurements :  Ratio  of 
diameter  of  sphere  of  same  volume 
to  diameter  of  circumscribing 
sphere :  volume  ratios  of  same 
spheres 

5.  Surface  area  measurements  :  Ratio 
of  surface  area  of  sphere  of  same 
volume  to  actual  surface  area 

Methods   based   on   measurements   on 
the  aggregate 

I.  ^leasurement  of  minimum  porosity 
on  uniformitv  sized  materials 


Attribute 
Measured 

Shape 


Roundnes 


Shape 


Shape 


Author 

Pentland 

Tickell 

W'adell 

\\'entwortli ' 
Wadell 


^^'adell 
Tickell 


Wadell 


Lamar 


Shape 
*  Measurements  made  on  actual  pebble  rather  than  plane  figure. 


PROCEDURES      FOR     ANALYSIS 

Choice  of  metlwd.  The  choice  of  method  depends  on  the  use  to  be 
made  of  tlie  results,  on  the  time  available,  and  on  the  size  of  the  material 
to  be  studied.  Where  the  results  are  to  be  used  tor  correlation  purposes 
or  for  comparison  of  somewhat  similar  materials,  it  is  probable  that  some 
of  the  less  rigorous  and  more  rapid  methods  of  Zingg,  Hagerman,  and 
others  will  suffice.  Should  these  methods  fail  of  the  purpose  intended, 
it  is  possible  that  the  more  strict  methods  of  Wadell  will  succeed.  The 
Wadell  methods  are  very  time-consuming  and  most  likely  will  not  be 
of  value  for  rapid  work  demanded  in  oil-field  laboratories  imtil  they  are 
abridged  and  made  shorter  by  use  of  appropriate  tables  and  computing 
charts  or  alinement  diagrams.  I^Ioreover,  a  method  suitable  for  the  study 
of  sands  may  be  unsuited  or  awkward  when  applied  to  pebbles  and 
cobbles,  and  vice  versa.  W'ent worth's  methods,  for  example,  are  scarcely 
applicable  to  fine  materials,  though  for  many  purposes  they  may  be  sat- 


292 


SEDIMENTARY  PETROGRAPHY 


isfactor}'  for  coarse  sediments.  In  any  event,  the  worker  should  have 
clearly  in  mind  the  distinction  made  by  Wadell  between  shape  and 
roundness.  Whatever  the  method  chosen  it  will  give  in  some  instances 
misleading  or  erroneous  values.  The  worker  should  be  aware  of  these 
limitations  and  interpret  his  data  accordingly.  The  results  obtained  by 
one  method  are  not  usually  convertible  into  values  obtained  by  other 
methods. 

Sphericity.  For  large  fragments  Wadell  devised  a  ver\'  simple  and 
rapid  method.  The  true  sphericity,  ip,  is  de- 
fined by  Wadell  as  s/S,  where  .9  is  the  sur- 
face area  of  a  sphere  of  the  same  volume 
as  the  pebble  and  S  is  the  actual  surface 
area  of  the  solid.  Owing  to  difficulties  of 
measuring  the  surface  area  of  an  irregular 
solid,  Wadell  proposed  a  "practical  method" 
for  actual  analysis.^  In  the  practical  for- 
mula 'V  =  dj,/Ds,  where  d^  is  the  true 
nominal  diameter,  i.e.,  diameter  of  a  sphere 
of  the  same  volume  as  the  pebble,  and  Dj 
the  diameter  of  a  circumscribing  sphere. 
The  value  of  dn  is  computed  from  a  meas- 
f — ;'  ; — z^      urement  of  the  pebble  volume  determined 

p  '  _  ~;  :  '  ,  v^^  by  dropping  the  pebble  in  a  graduated 
cylinder  and  noting  the  volume  of  the  water 
displaced. 


Fig.  136. — Schurecht  over- 
flow volumeter.  For  use  either 
with  water  or  kerosene.  Over- 
flow is  caught  and  measured  in 
a  burette. 


For  small  pebbles  a  small  graduate  (25  or 
30  c.c.)  may  be  used  and  the  volume  deter- 
mined to  within  0.5  c.c.  For  large  pebbles  and 
cobbles  a  cylinder  with  side  spout  for  overflow 
may  be  used  and  the  overflow  caught  and  measured  with  a  small  graduate. 
Schurecht  has  devised  an  overflow  volumeter  (Figure  136)  in  which  a 
burette  is  used  in  place  of  graduates.^  Other  types  of  volumeters  suitable  for 
measurement  of  the  bulk  volume  of  pebbles  and  cobbles  are  described  in 
Chapter  19  in  connection  with  the  measurement  of  the  volume  of  rock  sam- 
ples for  porosity  determination. 

Determination  of  volume  should  be  made  to  0.5  c.c.  on  pebbles  of  10  to 
20  c.c.  in  volume,  to  the  nearest  i  c.c.  on  pebbles  of  20  to  50  c.c.  in  volume, 
1  Hakon  Wadell,  Shape  determinations  of  large  sedimental  rock-fragments :  Pan- 
American  Geologist,  vol.  61,  pp.  187-220,  1934-  A  number  of  typographical  errors 
occur  in  this  paper.  The  integrational  symbol  J*  is  used  throughout  in  place  of  the 
symbol  /,  for  function.  0.7  in  formula  8,  p.  205,  should  be  o.i. 

2H.  G.  Schurecht,  A  direct  reading  overflow  volumeter:  Jour.  Am.  Ceram.  Soc, 
vol.  3.  P-  731.  1920. 


SHAPE  AND  ROUNDNESS 


Vol 
cc 


cm 


293 

cm. 


100 

90  - 

80  - 

70- 

60 

50 

40 

30  ^: 


Fig.  137. — Nomograph  for  computation  of  sphericity  by  Wadell's  method.  For  peb- 
bles, the  volume  value,  determined  by  displacement  method,  is  connected  with  the  D^ 
(maximum  diameter)  value  measured  by  gage.  Point  of  crossing,  of  line  thus  de- 
termined, on  slanting  scale,  gives  sphericity.  For  sand  grains,  the  d^,  value  com- 
puted from  area  measurements  by  means  of  Fig.  139,  is  connected  with  D^  value 
obtained  from  grain  drawing  and  sphericity  read  on  center  scale  as  before. 


294       SEDIMENTARY  PETROGRAPHY 

and  to  the  nearest  2  c.c.  on  pebbles  of  50  to  200  c.c.  in  volume.  This  de- 
gree of  accuracy  will  insure  a  sphericity  value  correct  to  the  second  decimal 
place. 

The  longest  diameter  of  the  fragment  is  assumed  to  be  the  diameter  of 
the  circumscribing  sphere,  Dg.^  This  may  be  measured  by  simple  gauge 
(Figure  Z7)- 

The  computations  on  each  pebble  involve  computing  (/„  from  measured 
volume  and  evaluating  the  ratio  d^/T>^.  The  accompanying  monograph 
simplifies  this  procedure  (Figure  137).  On  this  there  are  three  scales, 
Xo\-d^,  Ds,  and  center  scale.  To  solve  the  above  problem,  locate  the 
volume  of  the  pebble  on  the  left-hand  or  volume  scale  and  the  maximum 
diameter  on  the  D^  scale.  Connect  these  two  points  with  a  straight- 
edge. \\'here  the  straight-edge  intersects  the  center  scale  read  the  value 
of  sphericity. 

Allowing  a  minute  for  measurement  of  maximum  diameter  and  vol- 
ume, and  a  half-minute  for  calculation  with  the  alinement  chart,  it  is 
evident  that  a  gravel  sample  of  about  50-100  pebbles  can  be  readily 
analyzed  and  that  a  suite  of  samples  can  be  studied  in  the  course  of  a 
few  days. 

The  use  of  the  alinement  diagram  does  not  reduce  the  accuracy  of 
the  method,  since  errors  introduced  by  the  displacement  method  of 
volume  and  errors  made  by  measurement  of  the  long  diameter  (to  the 
nearest  o.i  cm.)  are  greater  than  those  involved  in  reading  the  scales.- 

An  example  of  the  use  of  the  method  is  given  below : 

Data  concerning  early  pre-Cambrian  conglomerate  at  locality  M.B.  i,  Manitou 
Lake,  District  of  Kenora,  Ontario. 

Location:  Lahay  Bay.  Manitou  Lake,  District  of  Kenora,  Ontario,  Canada. 
About  50  chains  due  east  of  narrows  between  Lahay  Bay  and  IManitou 
Straits  on  narrow  neck  of  land  separating  Lahay  Bay  from  small  bay 
south  of  the  same. 

Note  on  Exposure:  Vertical  conglomerate  beds,  strike  N  58°  E,  containing 
numerous  pebbles  and  cobbles  and  a  few  boulders  up  to  20  inches  in  di- 
ameter. A  count  in  one  square  yard  disclosed  37  black  cherts  and  iron- 
formation,  36  granites,  60  greenstones  and  metadiorites,  12  felsites  and 
porphyries,   i  vein  quartz.  Bedding  pronounced  with  pea-like  grits  alter- 

1  As  noted  by  Wadell,  this  is  not  always  the  case.  Exceptions  are,  however,  be- 
lieved to  be  too  rare  to  materially  affect  the  analysis. 

-  Wadell  calculated  the  values  obtained  by  this  method  with  those  obtained  by  the 
formula  s/S  for  certain  definite  geometric  forms  and  found  the  values  of  the  prac- 
tical method  here  given  to  be  about  o.io  lower,  on  the  average,  than  the  true  sphe- 
ricit}'.  He  corrected  for  this  difference  by  adding  o.io  to  all  values  less  than  0.80. 
This  correction,  however,  as  noted  by  Wadell,  produced  some  difficulties  of  overlap- 
ping of  values  in  the  0.70-0.90  range. 


SHAPE  AND  ROUNDNESS 


295 


nating  with  pebble  and  cobble  conglomerate.  Pebbles  readily  weather  out 
of  matrix. 


Table  2>^ 


Shape  Analysis:  "«p  =  d^D^ 

Pebble   No. 

Rock  Type 

(cm.) 

Vol. 
ice.) 

W 

Remarks 

1   

2  

3  

4 

5  

6  

7  

8  

9 

Vein  quartz 
Felsite 
Greenstone 
Granite 

Greenstone 

Greenstone 
Felsite 
Felsite 
Granite 

5-31 
4.62 

6.31 
6.42 

5.00 

4.00 
3.80 
3.20 
8.50 

34 
19 
29 
60 

^3 

19 

15 

5 

174 

0.762 
.718 
.604 
•759 

.708 

.823 
.803 
.664 
.815 

Slightly  broken 
Slightly  broken 

A  little  matrix  at- 
tached 

A  little  matrix  at- 
tached 

WadcU's  mctJwd  of  determination  of  volume,  sphericity,  and  round- 
ness of  sedimentary  particles.  Wadell  has  described  in  great  detail  the 
various  steps  involved  in  analysis  of  a  sediment  so  as  to  determine  vol- 
ume, sphericity,  and  roundness  of  quartz  particles  according  to  his  defini- 
tions of  these  properties.^  Since  on  the  whole  the  concepts  of  shape  and 
roundness  presented  by  Wadell  seem  to  the  authors  to  have  the  soundest 
theoretical  basis,  a  brief  outline  of  the  procedure  used  by  Wadell  is  here 
given. 


The  sample  is  split  down  and  screened  into  several  fractions,  based 
on  the  Wentworth-Udden  grade  scale,  which  are  then  weighed.-  Each 
screened  fraction  is  further  split  down  to  an  amount  small  enough  to 
be  spread  over  a  microscopic  slide.  The  grains  on  each  slide  are  counted 
and  also  weighed.  Knowing  the  number  of  grains  and  their  weight  makes 
possible  the  computing  of  the  total  number  of  grains  in  each  sieve  sepa- 
rate, since  the  latter  is  weighed  also.  Thus  the  frequency  of  grains  by 
number  (rather  than  by  weight)  is  known  for  the  sediment.  A  few  drops 
of  clove  oil  (n=  1.560)  serve  as  mounting  medium  for  the  grains. 

The  slide  is  next  placed  under  the  microscope  and,  by  means  of  a 
camera  lucida,  the  outlines  of  each  grain  are  drawn.  It  is  necessary  for 

1  Hakon  Wadell.  Volume,  shape  and  roundness  of  quartz  particles :  Jour.  Geology, 
vol.  43,  pp.  250-280,  1935. 

-  Wadell,  in  fact,  removed  all  minerals  except  quartz  by  bromoform.  Such  pro- 
cedure is  probably  advisable  in  those  sediments  in  which  minerals  other  than  quartz 
are  abundant. 


296 


SEDIMENTARY  PETROGRAPHY 


determination  of  roundness  that  the  average  diameter  of  the  reproduced 
grains  be  about  the  same  size.  Wadell  chose  7  cm.  as  the  "standard  size." 
It  is  evident,  therefore,  that  different  objectives  will  be  needed  for  the 
different  grades  in  order  to  achieve  this  standard.  In  order  to  avoid  dis- 
tortion, especially  with  high-power  objectives,  the  grains  to  be  drawn 
should  be  placed  in  the  center  of  the  field.  About  50  grains  in  each  grade 
size  should  be  drawn. 

The  area  of  each  grain  reproduction  is  then  determined  by  a  polar 
planimeter  (Figure  138).  From  this  value  the  nominal  sectional  diam- 
eter, i.e.,  the  diameter  of  a  circle  with  the  same  area  as  the  projection, 
is  computed.  This  may  be  done  graphically  by  use  of  a  chart  (Figure 


Fig.  138. — Polar  planimeter.  a,  weight  and  pin;  b,  vernier  and  revolving  drum  re- 
cording 0  to  I  sq.  in. ;  c,  revolving  recording  disk,  o  to  10  sq.  in. ;  and  d,  tracing  stylus 
and  thumb-hold. 

139).  The  diameter  of  the  smallest  circumscribing  circle  is  also  measured, 
usually  the  long  diameter  of  the  grain,  and  from  this  and  the  nominal 

sectional  diameter  the  sphericity  is  obtained,  as  per  formula  <^  =  -yY- 

where  d^  is  the  diameter  of  a  circle  equal  in  area  to  the  area  obtained  in 
the  standard  size  when  the  grain  rests  on  one  of  its  larger  faces  and  Dg  is 
the  diameter  of  the  smallest  circle  circumscribing  the  grain  reproduction.^ 
1  Wadell  defines  the  nominal  sectional  diameter  as  that  of  the  non-magnified  grain. 
No  distinction  between  the  non-magnified  grain  and  the  image  as  drawn  is  here 
made,  since  the  sphericity  as  calculated  is  a  ratio  in  which  size  effects  are  canceled 
out. 


SHAPE  AND  ROUNDNESS 


297 


10000 


100 


10  20  30        40       50    60    70  60100 

NOMINAL  SECTIONAL    DIAMETER 

mm. 

Fig.  139. — Chart  for  computing  nominal  sectional  diameter  (of  the  magnified  grain) 
from  measurement  of  the  projection  area. 


298 


SEDIMENTARY  PETROGRAPHY 


While  this  formula  or  definition  of  sphericity  differs  from  that  earlier 
given  by  Wadell  for  large  fragments  (see  page  284),  he  has  shown  that 
since  the  nominal  sectional  diameter  as  above  defined  approximates  quite 
closely  the  true  nominal  diameter,  the  result  obtained  from  grain  pro- 
jections is  quite  comparable  to  that  obtained  on  large  fragments.^  He  has 

shown  also  that  the  values  for  the 
sphericity  thus  obtained  approach  fairly 
closely  the  actual  sphericity  in  which 
surface  areas  are  taken  into  account.^ 
Roundness  is  also  obtained  from  the 
grain  projections  according  to  Wadell's 
formula  for  roundness  of  plane  figures 
(see  page  285).  The  radius  of  curva- 
ture of  each  corner  is  obtained  by  plac- 
ing a  transparent  celluloid  scale  (Fig- 
ure 140)  over  the  image  of  the  grains 
as  drawn  and  magnified  to  the  standard 
size  of  about  70  mm.  This  scale,  on 
which  are  drawn  some  35  concentric 
circles  differing  from  one  another  in 
radius  by  2  mm.,  is  adjusted  so  that 
the  radius  of  curvature  of  each  corner 
may  be  obtained.  The  radius  of  the 
maximum  inscribed  circle  is  likewise  obtained.  The  roundness  is  then 
computed  according  to  Wadell's  formula.  Example  : 


Fig.  140. — Celluloid  circle  scale  of 
Wadell  for  radii  measurements  of 
grain  drawings. 


R  =  19  mm.  =  the  radius  of  the  maximum  inscribed  circle 

r  =  the  radius  of  curvature  of  a  corner.  The  values  for  r  are  shown  in  the 
table  below. 

N  =  io  =  the  number  of  corners,  see  Figure   141B. 

Table  of  values  of  r: 

^1  —  3;  r.,  —  2',  rj  — 10;  r^  — 3;  r-,  —  2;  r,-  7;  r-— 13;  r^  — 4;  ro — 
4;  rio  — 6. 


^''=54. 


=54/10  =  5.4 


5.4/19  =  .28  + 


To  measure  the  roundness  of  pebbles  and  larger  objects  a  different 
scheme  of  obtaining  the  projection  image  is  required.  Hough,^  in  a  study 
of  material  3  mm.  to  100  mm.  in  diameter,  photographed  or  photostated 
the  pebbles,  using  appropriate  lens  combinations  to  achieve  a  standard 
size.  The  pebbles  were  first  separated  into  groups  of  approximately 
uniform  size  and  placed  in  their  most  stable  positions  on  a  black  back- 
ground. A  white  square  bearing  identification  number  and  a  white  cellu- 


1  Hakon  Wadell,  loc.  cit.,  pp.  259-263,  1935. 

2  Idem,  pp.  264-266. 

3  J.  L.  Hough,  personal  communication,  1937. 


SHAPE  AND  ROUNDNESS 


299 


loid  pocket  ruler  were  also  placed  in  the  field.  If  the  images  were  less 
than  standard  size,  further  enlargement  by  projection  printing  is  neces- 
sary. The  authors  have  used  a  simpler  scheme  for  obtaining  projections 
of  pebble  images.  The  pebbles  were  placed  on  a  glass  tracing  table  with 
light  source  beneath.  With  a  "pointolite"  ^  or  similar  illumination  the 
shadow  of  the  pebble  is  sharply  defined  on  a  paper  placed  on  a  glass 
plate  supported  an  appropriate  distance  above  the  pebbles  themselves. 


Fig.  141A.— Wadell 
method  of  determina- 
tion of  shape  and 
roundness  of  sand 
grains.  D^,  diameter  of 
circumscribing  circle ; 
R,  radius  of  inscribed 
circle. 


Fig.  14 1 B.— Wadell  method 
of  determination  of  shape  and 
roundness  of  sand  grains.  D^, 
diameter  of  circumscribing 
circle;  R,  radius  of  inscribed 
circle. 


Short  method  for  determination  of  size  and  shape  of  sand  grains, 
Wadell's  procedure  as  given  above  is  very  time-consuming.  The  follow- 
ing is  suggested  as  a  modification  for  the  worker  who  wants  to  retain 
the  principles  developed  by  Wadell,  but  who,  for  lack  of  time,  cannot 
give  every  sample  exhaustive  treatment. 

The  sample  is  carefully  split  down  with  a  Jones  splitter  to  about 
I  or  2  g.,  which  in  turn  is  split  down  by  some  means — preferably  a 
microsplit — until  some  300  to  500  grains  remain.  These  are  then  mounted 
on  a  glass  slide,  in  piperine  (n=  1.68)  if  possible,  to  bring  the  grains 
into  clear  view.  The  slide  is  then  placed  under  the  microscope  and 
suitably  magnified  so  that  the  outlines  of  the  grains  may  be  drawn  some 
100  times  their  actual  size  with  camera  lucida,  or  they  are  placed  on  a 
microprojector  and  the  images  as  projected  on  a  drawing  board  are 
traced. 

1  "Pointolite"  is  the  trade  name  of  a  bulb  which  emits  rays  from  a  very  small, 
white-hot  bead  of  tungsten.  It  is  essentially  a  point-source  of  illumination  and  gives, 
therefore,  very  sharp  shadows.  Obtainable  from  James  G.  Biddle  Co.,  Philadel- 
phia, Pa. 


300 


SEDIMEXTARY  PETROGR.\PHY 


If  one  wishes  to  know  the  average  size  of  the  grains,  it  is  only  neces- 
sar\',  as  pointed  out  by  Krumbein,*  to  measure  the  maximum  horizontal 
intercepts  of  randomly  oriented  grains.  If, 
however,  one  is  concerned  about  the  sort- 
ing of  the  grains  as  measured  by  the 
standard  deviation,  it  is  better  to  measure 
the  grain  areas,  as  projected,  with  a 
planimeter.  This  is,  of  course,  absolutely 
necessary  if  one  wishes  to  determine  their 
shape  or  sphericitv'.  From  the  measured 
areas  the  nominal  sectional  diameter,  or 
the  diameter  of  a  circle  of  the  same  area, 
is  computed.  The  diameter  of  the  circum- 
scribing circle,  usually  the  maximum 
diameter  of  the  grain,  is  also  measured, 
and  the  sphericit}-  for  each  grain  is  com- 
puted by  taking  the  ratio  of  these  two  measurements  as  stated  above. 
It  is  not  possible  to  measure  roundness  on  the  images  as  drawn  unless 
they  are  all  of  about  the  same  order  of   size  and  preferably  of  the 


Spkericitj 

Fig.  142. — Histogram  of  sphe- 
ricity analysis  of  sample  of  Sl 
Peter  sandstone  based  on  micro- 
scopic  measurements  of  100 
grains. 


VXZA 


i^ 


1/2        1/4         ,/6         ./.. 

Size      mm. 


1/2     ./«     ./a     ./,. 


Fig.  143. — .\,  histogram  of  sieve  analysis  of  St.  Peter  sandstone ;  B,  histogram  of 
same  based  on  microscopic  measurements  of  intermediate  diameter  of  grains ;  C,  his- 
togram of  same  based  on  microscopic  measurements  and  computations  of  nominal 
diameter  ;  100  grains  used  for  B  and  C. 

"standard  size,"  namely,  about  7  cm.  over  all.  If  these  conditions  are 

fulfilled  the  roundness  may  be  measured  as  outlined  in  the  preceding 

section. 

1  W.  C.  Krumbein,  Thin-section  mechanical  analysis  of  indurated  sediments :  Jour. 
Geology,  vol.  43,  pp.  489-496,  1935- 


SHAPE  AND  ROUNDNESS 


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302       SEDIMENTARY  PETROGRAPHY 

The  computations  involved  are  considerable,  and  as  an  illustrative 
example  the  determinations  made  on  a  sample  of  St.  Peter  sand  are  here 
given.  The  example  is  more  elaborate  than  is  required  by  routine 
analysis,  since  the  writers  wish  to  point  out  several  ways  in  which 
the  results  may  be  expressed  and  to  show  differences  in  the  results  when 
the  composition  is  expressed  on  a  weight  basis  from  those  when  it  is 
expressed  on  a  number  basis  and  the  differences  between  an  analysis 
made  by  screening  and  one  made  by  micrometric  methods  on  a  very 
small  sample.  (See  Figures  142  and  143.) 

Sieves  classify  grains  according  to  their  intermediate  diameter.  Hence  in 
order  to  obtain  strictly  comparable  results  by  microscopic  analysis,  it  is  nec- 
essar>-  to  measure  the  shortest  diameter  of  the  projection  area.  Since  grains 
tend  to  lie  on  their  greatest  developed  face,  parallel  to  their  longest  and  inter- 
mediate diameters,  it  is  clear  that  the  shortest  diameter  of  the  projection  is 
really  the  intermediate  diameter  of  the  grain.  The  true  shortest  diameter 
will  be  perpendicular  to  the  projection.  That  this  scheme  of  measurement 
gives  values  more  nearly  identical  with  those  obtained  by  screening  is  seen  by 
inspection  of  Figure  143.  Here  the  data  obtained  by  screening  and  that  ob- 
tained by  microscopic  measurement  and  recomputation  to  a  weight  basis 
may  be  compared.  As  seen  in  143B,  the  data  from  measurement  of  tlie  inter- 
mediate diameter  is  closer  to  that  obtained  by  screening. 


CHAPTER    12 

SURFACE  TEXTURES  OF  SEDHIENTARY  FRAGMENTS 
AND  PARTICLES 

INTRODUCTION 

The  intimate  details  of  the  grain  surface,  independent  of  size,  shape,  or 
mineral  composition,  are  termed  the  surface  texture  of  the  grain.  A 
grain,  for  example,  may  be  polished,  frosted,  or  etched.  A  pebble  may 
be  marked  with  striations  or  by  percussion  marks.  Such  features  are 
here  defined  and  described. 

These  detailed  characters  have  genetic  significance  and  may  be  criteria 
of  value.  The  frosting  on  sand  grains  has,  for  example,  been  said  to 
denote  aeolian  action,^  while  striations  are  most  usually  attributed  to 
glaciation.^  While  these  generalizations  are  open  to  question  and  while 
the  origin  of  many  surface  textures  is  not  at  all  clear,  it  is  evident  that 
as  our  knowledge  increases  these  external  characteristics  of  sedimentary 
particles  and  fragments  deserve  careful  attention. 

Just  as  a  sand  grain  or  pebble  may  inherit  its  shape  from  an  earlier 
deposit  of  different  origin,  so  too  may  a  particle  or  fragment  inherit  the 
surface  markings  that  it  bears.  However,  a  geological  agent  will,  if  time 
be  sufficient,  impose  its  own  unique  character  on  the  particle  or  frag- 
ment. Some  considerable  time  must  elapse  before  size  and  shape  modi- 
fications are  evident,  but  the  surface  characteristics  are  more  readily 
destroyed  or  modified.  It  is  perhaps  the  sensitiveness  of  these  features 
to  change  that  makes  them  all  the  more  important.^  They  record  most 
faithfully  the  effect  of  the  last  cycle  of  transportation. 

1  W.  H.  Shcrzcr.  Criteria  for  the  recognition  of  the  various  types  of  sand  grains : 
Gcol.  Soc.  Am.,  Bull.,  vol.  21,  p.  640,  1910. 

2  C.  K.  Wentworth,  An  analysis  of  the  shapes  of  glacial  cobbles :  Jour.  Scd. 
Petrology,  vol.  0,  p.  85,  1936.  Wentworth  describes  glacial  striations  in  some  detail 
and  points  out  also  the  similarities  between  those  of  glacial  origin  and  those  pro- 
duced by  river  ice.  See :  The  shape  of  glacial  and  ice  jam  cobbles :  ibid.,  vol.  6,  p. 
97,  I93^>- 

3  Wentworth,  for  example,  found  by  experimental  study  that  a  travel  of  but  0.35 
mi.  was  necessary  to  remove  striae  on  hard  limestone  and  greenstone  pebbles.  See: 
The  shapes  of  pebbles,  U.  S.  Gcol.  Survey,  Bull.  730-c,  p.  114,  1922. 

303 


304       SEDIMENTARY  PETROGRAPHY 

But  as  has  been  pointed  out  it  is  not  impossible  to  find  sand  grains  whose 
surfaces  may  be  described  as  smooth,  rough,  glassy,  frosted,  pitted,  and 
stained,  all  in  the  same  sand.  If  the  character  of  the  surface  of  the  grain 
bears  any  relation  to  the  origin  of  the  sand,  it  is  clear  that  the  mixture  of 
the  above  surface  types  in  the  same  sand  indicates  that  the  grains  were  derived 
originally  from  several  different  types  of  deposits  and  that  tliey  have  not  been 
worked  over  sufficiently  to  have  these  inherited  surface  textures  destroyed. 
However,  in  a  great  many  sands  there  is  one  type  of  surface  texture  that 
predominates,  and  it  is  assumed  that  in  such  deposits  an  understanding  of 
the  surface  texture  may  be  extremely  helpful  in  determining  the  conditions 
of  deposition  of  the  sediments. 

The  study  of  these  features  has  lagged  much  behind  that  of  the  study 
of  other  fundamental  properties — size,  shape,  and  mineral  composition. 
Even  now  no  quantitative  method  of  measuring  these  features  is  known. 
Our  discussion  then  is  confined  to  a  brief  definition  of  each  feature  and 
a  purely  descriptive  classification  of  these  surface  characters.^ 

Surface  characters  are  most  conveniently  discussed  with  reference  to 
the  size  of  the  particle  or  fragment  on  which  they  appear.  There  are 
many  features  known  on  cobbles  and  pebbles  which  cannot  or  do  not 
appear  on  sand  grains.  This  is  due  to  the  fact  that  the  pebbles  and  other 
large  fragments  are  generally  rock  fragments — often  of  more  than  one 
mineral — whereas  the  sand  grains  are  largely  single  minerals  and  are 
microscopic  in  size.  Moreover,  the  pebbles  and  cobbles  are  studied  mega- 
scopically,  whereas  the  sand  grains  are  examined  with  the  microscope. 


SURFACE  TEXTURE  OF  LARGE  FRAGMENTS 

(>    2    MM.   diameter) 

The  surface  features  of  large  fragments  fall  into  three  categories, 
namely,  degree  of  smoothness,  degree  of  polish  or  gloss,  and  surface 
markings. 

Polish  or  gloss  has  to  do  with  the  degree  of  luster  of  the  surface. 
This  property  is  primarily  related  to  the  regularity  of  reflection.  Much 
scattering  or  diffusion  of  light  produces  a  dull  surface.  The  presence 
of  high-lights  indicates  a  good  polish.  A  polished  or  glossy  surface  may 
or  may  not  be  smooth,  it  may  be  striated,  grooved,  or  pitted.^ 

1  For  these  definitions  and  classification,  the  writers  are  indebted  to  Miss  Lou 
Williams,  who  prepared  a  summary  of  the  literature  on  this  subject,  while  at  the 
University  of  Chicago,  for  the  Committee  on  Sedimentation. 

2  Miss  Williams  finds,  for  example,  that  a  pebble  with  a  polished  and  apparently 
smooth  surface  reveals  microscopic  striations  under  high  magnification.  In  fact,  the 
presence  of  such  microstriations  may  distinguish  between  polish  induced  by  abrasion, 
which  exhibits  such  striations,  and  gloss  or  chemical  polish,  which  lacks  such  mi- 
crostriations. 


SURFACE  TEXTURES 


305 


Table  34 
Surface  Textures  of  Fragments  over  2  mm.  in  Diameter 


May     be     smooth,     or 

A.  DULL       versus  B.  POLISHED 

scratched,      furrowed 

'             (Gloss) 

and     grooved,     pitted 

or  dented 

Mav  be  dull  or  polished 

C.  SMOOTH    versus  D.  SURFACE 

MARKINGS 

I.  FURROWED  AND 

GROOVED 

2.  SCRATCHED 

3.  RIDGED 

4.  PITTED  OR 

DENTED 

Smoothness  is  the  evenness  of  the  surface.  A  smooth  surface  is  one 
on  which  no  striations,  pits,  ridges,  or  other  features  are  observable. 
A  smooth  surface  may  be  either  polished  or  dull.  The  antithesis  of 
smooth  is  rough.  Roughness  may  be  due  to  pits  or  to  nondescript  small 

irregularities  of  the  surface. 


J 


II 

i 

Subparaliel 

^ 

Random  Grid 

Fig.  144. — Patterns  of  arrange- 
ment of  striations  on  pebble  or 
cobble  face. 


I 


-  + 

ANGLE      or      DEVIATION 

Fig.  145. — Histogram  of  de- 
viations of  striations  from  long 
axis  of  glacial  pebble. 


Much  attention  has  been  given  to  the  nature  of  surface  markings 
and  blemishes  when  these  are  present.  Many  pebbles  are  scratched  or 
striated.  Such  striations  may  occur  in  parallel  or  subparaliel  sets,  grid 
patterns,  or  in  random  or  scattered  fashion  (Figure  144).  The  relation 
of  the  striations  to  the  long  axis  of  the  pebble  may  be  important  and 


3o6       SEDIMENTARY  PETROGRAPHY 

the  deviations  in  direction  from  such  long  axis  may  be  measured  and 
studied  statistically. 

Krumbein/  for  example,  studied  the  striae  on  a  glacial  cobble  and  meas- 
ured their  deviations  from  the  long  axis  of  that  cobble.  The  angle  between 
the  long  axis  and  each  striation  was  determined.  The  data  obtained  were 
then  divided  into  nine  classes  based  on  ten-degree  intervals  and  a  frequency 
curve  was  obtained  (Figure  145;.  Conceivably  such  a  statistical  study  might 
reveal  differences  between  cobbles  striated  by  one  agent  or  another. 

Such  detailed  studies  may  not  always  be  warranted,  but  may,  in  the 
case  of  certain  glacial  deposits,  be  important.  A  distinction  between  ice- 
jam  river  cobbles  and  true  glacial  cobbles  may  be  made  on  the  basis  of 
arrangement  of  striations.-  Some  pebbles  are  furrouced  or  grooved.  The 
opposite  of  grooved  is  ridged. 

Pebble  surfaces  may  also  be  pitted  or  dented.  Pits  vary  much  in  form 
and  size.  Crescentic  impact  scars  or  percussion  marks  are  notable  on 
some  pebbles,  particularly  quartzite  pebbles.  Conceivably  chatter  marks 
may  appear  on  glacial  cobbles.  Indented  pebbles,  pebbles  with  shallow 
oval-shaped  depressions,  due  to  solution  at  point  of  contact  between 
pebbles  under  pressure,  are  also  known. 


SURFACE     TEXTURES     OF     SMALL     FRAGMENTS 
(  <  2      MM.     D  I  A  M  E  T  E  Rj 

The  surface  texture  of  a  sand  grain  may  be  described  as  dull  or 
polished  and  as  smooth  or  rough.  As  suggested  in  the  table  on  page  307, 
various  combinations  of  these  qualities  are  possible. 

The  dullness  or  polish,  as  in  the  case  of  pebbles,  is  a  quality  of  luster. 
A  dull  surface  is  one  lacking  in  brilliance  or  luster,  whereas  a  polished 
surface  is  one  of  high  gloss.  A  grain  surface  may  have  a  low  polish, 
often  pearly,  or  it  may  have  a  high  polish,  vitreous  or  brilliant. 

A  grain  surface  may  be  smooth  or  rough  independent  of  the  luster 
of  the  surface.  A  smooth  surface  lacks  relief  when  seen  under  the 
microscope,^  whereas  a  rough  surface  has  inequalities,  projections  or 
pits.  Where  the  irregularities  are  linear,  the  term  striated  may  be  used. 
\\'hen  they  are  of  geometric  form  and  of  chemical  origin,  the  term 

1  W.  C.  Krumbein,  unpublished  data. 

2  C.  K.  Wentworth,  loc.  cit.,  1936. 

3  A  pebble  may  have  a  smooth  surface  to  the  naked  eye  but  under  the  microscope 
appear  minutely  rough.  As  stated,  the  definitions  for  surface  characters  of  pebbles 
are  given  for  the  unaided  eye,  while  those  for  sand  grains  are  given  as  observed 
under  the  microscope. 


SURFACE  TEXTURES 


307 


etched  is  used.  When  the  irregularities  are  very  minute  a  frosted  surface 
results,  but  where  the  irregularities  are  larger  and  scattered  the  surface 
is  termed  pitted.  Grains  of  some  minerals  are  subject  to  secondary  en- 
largement. The  secondary  growth  is  deposited  from  solutions  in  optical 
and  crystallographic  continuity  with  the  original  material.  Such  grains 
exhibit  microscopically  small  facets. 

Table  35 
Surface  Textures  of  P'ragments  under  2  mm.  in  Diameter 


May  he  smooth  or 
rous:h 


A.  DULL 


B.  POLISHED 


May     be     dull     or 
polished 

C.  SMOOTH 

ver 

;us 

D.  ROUGH 

1.  STRIATED     (usually 
glacial  action) 

2.  FACETED  (second- 
ary growth) 

3.  FROSTED    ("ground- 
glass"  surface) 

4.  ETCHED   (solvent 
action) 

5.  PITTED 

XoTE. — The  term  mat  surface  is  not  well  defined  but  is  perhaps  most  often 
used  in  place  of  frosted  surface,  though  perhaps  the  surface  described  by  the 

former  term  is  one  of  finer  texture. 


It  is  not  within  the  province  of  this  book  to  discuss  the  genetic  sig- 
nificance of  these  characteristics.  The  reader  is  referred  to  the  report  by 
Miss  \\'illiams  for  a  resume  of  current  opinion  on  this  subject.^  It  is 
worth  while  to  recall  that  some  workers  have  given  considerable  weight 
to  surface  texture.  It  is  noteworthy  also  that  some  of  these  textures 
can  form  in  very  different  ways.  Polish  or  gloss,  for  example,  may  be 
produced  by  gentle  attrition  or  wear  or  it  may  be  induced  by  solution 
or,  as  in  the  case  of  the  pebbles,  by  deposition  of  a  vitreous  film,  ex- 
emplified by  desert  varnish.  Likewise  the  frosted  surface  may  be  formed 
by  the  rigorous  action  of  wind,  chemical  etching,  or  incipient  secondary 
enlargement. - 

As  is  evident  from  the  foregoing  summary,  our  knowledge  of  the  sur- 

1  Report  of  the  Committee  on  Sedimentation,  1936-^7,  pp.  1 14-128,  Xat.  Research 
Council,  1937. 

-  R.  Roth,  Evidence  indicating  the  limits  of  Triassic  in  Kansas,  Oklahoma,  and 
Texas.  Jour.  Geology,  vol.  40,  pp.  718-719,  1932. 


3o8        SEDIMENTARY  PETROGRAPHY 

face  characters  of  grain  is  very  incomplete  and  inexact.  Even  our  defi- 
nitions and  our  classification  of  the  surface  textures  are  not  wholly 
satisfactory.  We  have  yet  to  be  content  with  verbal  descriptions,  though, 
as  pointed  out  under  the  discussion  of  striae,  some  attempt  has  been 
made  to  quantify  our  observations  and  to  express  the  results  in  statisti- 
cal terms.  Perhaps  ultimately  the  polish  on  grains  will  also  be  described 
more  exactly — perhaps  even  measured  by  a  photometer. 

The  interpretation  of  these  features  is  even  more  unsatisfactory,  but 
when  we  have  fully  organized  and  adequately  defined  what  we  know 
about  the  surface  characters,  we  may  hope  that  experimental  and  ob- 
servational data  will  accumulate  that  will  make  for  a  better  understand- 
ing as  well. 


CHAPTER    13 

PREPARATION  OF  SAMPLE  FOR  MINERAL  ANALYSIS 

INTRODUCTION 

The  section  of  this  book  which  follows  is  devoted  to  the  techniques  of 
study  which  deal  with  the  mineral  composition  of  the  sediments.  The 
various  steps  involved  are  dependent  on  the  type  of  study  chosen.  When 
the  mineral  grains  are  to  be  studied  (as  in  the  case  of  sub-surface  cor- 
relation), it  is  necessary,  after  collection  of  a  sample,  to  prepare  it  for 
analysis.  Such  preparation  involves  a  disaggregation  or  breaking-down 
of  the  sample  followed  by  a  separation  of  the  sample  into  two  or  more 
fractions  which  are  more  or  less  homogeneous  mineralogically  and 
mounting  of  the  separates  in  a  suitable  way  for  microscopic  study.  The 
separations  are  usually  achieved  by  panning,  or  by  use  of  heavy  liquids, 
the  electromagnet,  or  some  special  method  or  device.  Since  the  average 
sand  is  largely  quartz,  the  separation  methods  are  usually  designed  to 
segregate  the  minor  accessory  minerals  from  the  quartz  (and  occasion- 
ally abundant  feldspar).  The  minor  accessories,  or  so-called  "heavy 
minerals,"  even  though  present  in  very  small  amounts  (tenth  of  i  per 
cent  or  less),  have  proved  of  most  worth  both  in  correlation  and  in 
provenance  studies. 

The  next  step  involves  identification  of  the  minerals.  This  is  best  ac- 
complished optically  by  means  of  the  polarizing  microscope,  though 
microchemical  and  other  methods  are  of  occasional  value.  Following 
identification,  the  mineral  frequencies  are  determined  by  actual  count  or 
by  estimation  and  are  then  recorded  in  tabular  or  graphic  form.  Statis- 
tical analysis  of  the  data  obtained  may  then  be  made  if  desired. 

The  non-clastic  sediments  are  more  generally  studied  in  thin  sections 
in  which  both  the  mineral  components  and  the  textures  may  be  identified. 
Consequently  we  have  given  instruction  in  the  preparation  of  such  thin 
sections  from  both  consolidated  and  feebly  coherent  materials  as  well 
as  methods  of  identification  suitable  to  the  study  of  thin  sections.  Plani- 
metric  analysis  of  the  section  makes  possible  quantitative  results,  if  such 
are  desired. 

309 


3IO        SEDIMENTARY  PETROGRAPHY 

Both  the  thin  section  and  the  mineral  concentrates  of  a  sediment 
should  be  studied  if  full  information  is  required.  The  thin  section  gives 
the  only  method  of  careful  study,  of  textures  and  structure,  whereas  the 
mineral  concentrates  are  most  ideal  for  the  study  of  the  mineral  com- 
position. 

Both  methods  have  their  limitations.  In  the  case  of  exceedingly  coarse 
materials  microscopic  study  is  awkward  and  unnecessary,  and  in  the  case 
of  very  fine  materials  it  is  quite  barren  of  results  owing  to  the  limits  of 
visibility.  Other  methods,  such  as  chemical  analysis,  x-ray  and  spectro- 
scopic investigations,  must  then  be  resorted  to  for  information  concern- 
ing the  mineral  composition  of  the  rock.  In  the  case  of  sedimentary 
materials,  other  than  clastic  rocks,  such  as  the  phosphates,  dolomites, 
gypsum,  and  coal,  the  thin  section  is  the  most  usual  method  of  study, 
though  it  has  been  found  advantageous  to  study  the  detrital  components 
of  these  rocks  which  have  been  isolated  by  some  method  suitable  to  the 
material  in  question.^ 

In  all  cases  an  interpretation  of  the  results  obtained  from  the  mineral 
grains  or  the  thin  section  is  a  matter  of  large  scope  and  beyond  the  pur- 
pose of  this  volume.  For  such  information  the  reader  must  refer  to  the 
voluminous  literature  on  these  aspects  of  sedimentary  petrology  or  to 
such  larger  works  as  the  Treatise  on  Sedimentation,  published  under  the 
auspices  of  the  Committee  on  Sedimentation.  Since  the  industrial  value 
or  use  of  the  various  sedimentary  materials  is  closely  tied  up  with  their 
mineral  composition,  the  study  of  their  composition  becomes  economically 
important.  But  a  discussion  of  this  too  is  beyond  the  scope  of  this  book, 
and  the  reader  is  therefore  referred  to  special  papers  and  works  on  the 
subject. 


preparation    of    sample 
Disaggregation 

Assuming  the  most  usual  case,  namely  a  mineral  grain  study  of  a 
detrital  rock  (supplemented  by  thin  section),  the  authors  will  consider 
the  preparation  of  the  material  for  examination.  Since  this  involves,  as 
pointed  out  above,  some  methods  of  separating  the  whole  sample  into 
fractions  more  or  less  simple  in  composition,  and  this  in  turn  implies 
a  state  of  complete  disaggregation,  it  will  be  necessary  to  give  an  ac- 

1  J.  E.  Lamar,  Sedimentary  analysis  of  the  limestones  of  the  Chester  series : 
Econ.  Geology,  vol.  21,  pp.  578-585,  1926. 


PREPARATION  OF  SAMPLE  311 

count  of  the  treatment  preliminary  to  making  the  mineral  separations  and 
then  discuss  the  separation  methods  themselves. 

Mineral  analyses  and  preparation  of  the  sample  for  analysis  can  be 
intelligently  pursued  only  if  the  general  make-up  of  the  clastic  rocks  be 
kept  in  mind.  The  following  groups  of  materials  may  be  present : 

(After  Holmes) 
I.    Allogenic  coiistititciifs 

1.  Pebbles  or  other  fragments  of  preexisting  rocks 

2.  Composite  grains  consisting  of  more  than  one  mineral 

3.  Simple  grains  or  particles  of  unaltered  minerals  such  as  quartz, 
muscovite,  garnet,  etc. 

II.    Organic  remains 
III.   Authigenic  constituents 

1.  Alteration  products  or  synthetic  recrystallization  products  formed 
in  situ  from  any  of  the  allogenic  constituents:  e.g.,  clay  from  feld- 
spar ;  limonite,  leucoxene,  glauconite,  etc. 

2.  Infiltration  products,  or  materials  introduced  from  external  sources, 
generally  present  as  a  cement 

3.  Kecrystallication  products,  formed  by  the  recrystallization  of  mate- 
rials already  present  in  the  sediment 

Note. — New  materials  produced  during  hydrothermal  alteration  or  contact 
metamorphism  may  also  be  considered  authigenic  constituents. 

A  preliminary  examination  of  the  material  is  advisable  in  order  to 
select  intelligently  the  method  of  treatment  most  likely  to  be  successful. 
Deverin,^  following  Cayeux,-  gives  six  possible  cases  involved  in  dis- 
aggregation problems.  These  are : 

(i)  ^laterial  very  colierent  and  unattacked  by  weak  acid,  for  example 
quartzite. 

(2)  Very  coherent  material  partially  soluble  in  dilute  acid,  for  example 
calcareous  sandstone. 

(3)  Material  slightly  coherent,  unattacked  by  weak  acid,  for  example  clay- 
shale. 

(4)  Material  slighdy  coherent,  but  attacked  by  dilute  acid,  for  example 
chalk. 

(5)  Unconsolidated  materials,  unattacked  by  weak  acid,  for  example  sili- 
ceous sand. 

(6)  Unconsolidated  materials  attacked  by  weak  acid,  for  example  car- 
bonate sand. 

1  L.  Deverin,  L'fitude  lithologique  des  rochcs  sedimcntaires :  Sclnvci::.  Min.  u. 
Pet.  Mitt.,  Bd.  H,  pp.  29-50,  1924. 

-  Lucien  Cayeux,  Introduction  a  I'etude  petrographique  des  roches  sedimentaires : 
Tc.vtc  (Imprimerie  Nationale,  Paris,  1931,  re-impression),  pp.  4-8. 


312       SEDIMENTARY  PETROGRAPHY 

For  (i)  the  thin  section  is  still  the  principal  method  of  study.-  Thin 
sections  are  of  much  value  for  (2),  (3),  and  (4),  but  are  rarely  used 
for  (5)  and  (6).  In  (3)  and  (4)  disintegration  is  accomplished  by  slak- 
ing or  softening  in  water  and  by  rubbing  with  a  stiff  brush.  When  the 
rock  is  attacked  by  weak  acid  (i  HC1:4  H2O),  it  may  be  broken  down 
by  acid  treatment.  It  is  advisable,  however,  to  restrict  the  quantity  and 
strength  of  the  acid  and  the  time  of  treatment  as  far  as  possible,  since  the 
acid  does  in  part  destroy  certain  minerals  either  by  solution  or  by  de- 
composition. Such  loss  may  be  anticipated  by  the  preliminary-  examina- 
tion suggested  above,  and  due  consideration  must  be  given  it.  In  some 
cases  an  alkaline  digest,  either  KOH  or  XaOH  solution,  will  facilitate 
disintegration  of  the  rock.  Goldman^  found  this  to  be  of  use  in  sand- 
stones with  opaline  silica  cement.  The  alkaline  digest  is  of  course  the  only 
one  that  can  be  used  if  calcareous  microfossils  are  to  be  looked  for.  Cer- 
tain siliceous  forms  may,  however,  be  destroyed  by  its  use.  Several  other 
methods  have  been  suggested  for  the  disaggregation  of  rocks  that  do  not 
}-ield  to  the  simpler  methods.  Freezing  and  thawing  have  been  recom- 
mended by  Hanna  and  Church  ^  for  the  purpose  of  disintegrating  shale 
containing  microfossils.  Tolmachoff  •*  soaked  shaly  rocks  in  a  hot  solu- 
tion of  "hypo"  (sodium  hyposulphite).  Since  hypo  is  several  times  less 
soluble  in  cold  water  than  in  hot,  crj-stallization  begins  at  once  as  soon 
as  the  solution  has  been  sufficiently  cooled,  accompanied  by  disintegration 
of  the  shale.  Should  the  liquid  become  supersaturated  and  fail  to  cn-stal- 
lize,  inoculation  with  a  little  of  the  solid  will  start  crystallization  at  once 
and  produce  disruptive  pressures.  Tolmachoft  also  used  sodium  sulphate 
and  ordinary  washing  soda.  The  hypo  was  found  to  be  the  most  satis- 
factory. Saturation  by  heating  the  sample  with  sodium  acetate  ^  has  also 
been  tried,  as  has  saturation  with  sodium  carbonate  followed  by  quench- 
ing with  HCl  with  attendant  evolution  of  CO2,  which  also  has  the  effect 


1  W.  Wetzel,  in  Sedimentpetrographische  Studien,  Neues  Jahrb.  f.  Min.,  B.B.  47, 
PP-  39"92,  1922,  examined  splinters  and  flakes  of  chert  instead  of  thin  sections.  Even 
such  materials  as  the  quartzites  may  be  crushed  and  the  fines  screened  out  and  the 
remainder  treated  in  the  same  manner  as  that  given  for  the  loose  sands.  This  treat- 
ment, however,  is  likely  to  introduce  changes  in  the  mineral  frequencies. 

-  M.  I.  Goldman,  Petrographic  evidence  on  the  origin  of  the  Catahoula  sandstone 
of  Texas:  Am.  Jour.  Sci.  (4),  vol.  39,  pp.  261-287,  1915. 

3  G.  D.  Hanna  and  C.  C.  Church,  Freezing  and  thawing  to  disintegrate  shales : 
Jour.  Palcon.,  vol.  2,  p.  131,  1928. 

*  I.  TolmachoflF,  Crystallization  of  certain  salts  used  for  the  disintegration  of 
shales :  Science,  vol.  76,  pp.  147-148,  1932. 

5  M.  Guinard,  The  disintegration  of  diatomaceous  deposits  :  Jour.  Queckett  Micros. 
Club,  ser.  2,  vol.  3,  p.  188,  1888.  G.  D.  Hanna  and  H.  L.  Driver,  The  study  of  sub- 
surface formations  in  California  oil-field  development:  Summary  of  Operations, 
Calif.  Oil  Fields,  vol.  10,  No.  3,  pp.  5-26,  1924. 


PREPARATION  OF  SAMPLE 


313 


of  breaking  up  the  rock  in  some  cases.^  Most  methods  of  treatment  are 
greatly  promoted  if  the  rock  is  first  crushed.  This  is  usually  accomplished 
with  an  iron  mortar  and  pestle.  A  crushing  action,  rather  than  a  grinding 
or  abrasive  action,  should  be  used,  since  the  latter  gives  an  objectionable 
amount  of  line  dust  and  tends  to  destroy  the  original  form  of  the  grains. 
Occasional  sifting  through  a  sieve  with  openings  of  0.5  mm.  (about  30 
mesh)  and  repeated  crushing  of  the  oversize  is  recommended,  otherwise 
an  undue  amount  of  dust  is  developed  as  a  consequence  of  abrasion. 
Tickell  suggests  that  for  crushing  small  amounts  of  rock  fragments  may 
be  broken  with  a  hammer  and  assayer's  anvil  (3x3x1  in.).-  A  V/i-'m. 
pipe  will  prevent  the  fragments  from  scattering. 
Tickell  ^  also  recommends  the  use  of  a  screen  made 
by  soldering  a  piece  of  wire  mesh  to  the  bottom  of  a 
sheet-iron  cylinder  about  2^  in.  in  diameter  and  2 
in.  long.  If  it  has  been  thoroughly  cleaned  by  brush- 
ing and  jarring  before  use,  the  crushed  sample  can 
be  gently  sifted  through  without  contamination  by 
grains  that  would  otherwise  collect  in  the  meshes.  A 
diamond  mortar  (Figure  146)  may  be  used  where 
small  quantities  or  mineral  grains  are  to  be  crushed. 
More  recently  disaggregation  of  clastic  rocks  by 
means  of  a  pressure  chamber  has  been  tried  and 
found  successful.*  Although  various  solvents  and 
solutions  were  forced  into  rock  specimens  under  high 
pressure,  none  was  found  to  be  as  effective  as  a 
supersaturated  solution  of  sodium  sulphate.  Each 
sample  to  be  treated  is  placed  in  a  beaker,  covered  with  the  solution, 
and  placed  on  a  rack  in  the  pressure  chamber.  The  latter,  see  Figure  9. 
was  constructed  of  an  ordinary  10-in.  steel  casing  some  12  in.  long.  A 
plate  of  ^-in.  steel  was  welded  to  one  end  to  form  a  bottom,  while 
at  the  other  end  a  flange  of  the  same  material,  i^  in.  wide,  was 
welded  to  the  outside  of  the  casing.  A  cover  of  J/^-in.  steel,  14  in.  in 
diameter,  was  bolted  to  the  flange  with  J/2 -in.  bolts.  A  heavy  composition 
boiler  gasket  between  cover  plate  and  flange  prevented  escape  of  any 
vapors.    Holes   in   the   cover   permitted   attachment    of    a    Y^-'m.    stop- 


FiG.  146.— M  o  r- 
t  a  r  for  crushing 
mineral  or  rock 
fragments  (about 
3  in.  high). 


1  Albert  Mann,   Suggestions  for  collecting  and   preparing  diatoms :   Proc.    U.   S. 
Nat.  Mus.,  vol.  Co,  pp.  1-8,  1922. 

2  Frederick  G.  Tickell,   The  Examination  of  P'ragiiiental  Nocks   (Stanford  Uni- 
versity Press,  1931),  p.  35. 

3  Frederick  G.  Tickell,  op.  cit.,  p.  36. 

*  G.  L.  Taylor  and  N.  C.  Georgesen,  Disaggregation  of  clastic  rocks  by  use  of  a 
pressure  chamber :  Jour.  Sed.  Petrology,  vol.  3,  pp.  40-43,  1933. 


314       SEDIMENTARY  PETROGRAPHY 

cock  and  pressure  gauge.  A  pressure  of  350  lbs.  was  developed. 
When  the  sample  has  been  reduced  to  grains  consisting  mainly  of 
single  minerals,  the  fine  dust  can  be  removed  by  screening  through  a 
sieve  with  openings  of  about  0.061  mm.  (about  250  meshj.  The  finer 
material  may  also  be  removed  by  washing  in  water  and  decanting.  In  this 
case  settling  in  a  15-in.  column  of  water  for  5  min.  will  elimmate  every- 
thing below  about  %2  mm-  in  diameter.  Repeated  washings,  of  course, 
are  necessary  to  reduce  the  quantity  of  fine  material  to  a  negligible 
quantity.  It  is  desirable  to  screen  the  dried  residue  in  order  to  divide  it 
into  three  or  four  fractions,  each  quite  uniform  in  size,  which  can  then 
be  mounted  for  study  under  the  microscope. 

Clarification  of  Gr.\ins 

It  is  convenient  for  identification  purposes  to  clear  the  grains  which 
are  coated  with  iron  oxide,  etc.,  or  with  weathering  products  and  to 
dissolve  those  grains  which  have  been  weathered  beyond  recognition, 
particularly  if  the  latter  are  ver>-  numerous.  This  is  accomplished  to  some 
extent  during  treatment  of  the  sample  during  disaggregation.  The  hydro- 
chloric acid  treatment,  for  example,  used  in  the  solution  of  the  carbonate 
cements,  removes  iron-oxide  stains  in  addition  to  removing  coatings  of 
the  cement.^  In  the  case  of  HCl,  however,  certain  detrital  minerals  are 
likely  to  be  partly  or  wholly  dissolved,  and  a  microscopic  check  is  nec- 
essar)'  to  determine  whether  or  not  this  has  been  the  case.  Reed,  however, 
says,-  "In  spite  of  published  assertions  to  the  eflFect  that  this  treatment 
destroys  apatite,  hypersthene,  and  other  minerals  of  a  similar  degree  of 
stability,  in  several  experiments  these  minerals  were  not  visibly  aflPected 
by  boiling  from  as  much  as  an  hour  in  50  per  cent  acid." 

Hydrogen  sulphide  has  also  been  used  to  remove  the  iron  oxide  coat- 
ings." A  water  suspension  of  the  minerals  is  treated  with  hydrogen  sul- 
phide. This  treatment  changes  the  finely  divided  iron  oxide  to  iron 
sulphides  which  dissolve  quickly  in  0.05 X  hydrochloric  acid.  Other  min- 
erals, such  as  silicates  and  apatite,  are  not  appreciably  aflfected,  although 
the  carbonates  would  be. 

Mackie"  made  it  his  procedure  to  examine  each  heavy  mineral  frac- 

1  The  action  of  HCl  in  removal  of  iron-oxide  coatings  is  greatly  accelerated  if  a 
little  stannous  chloride  is  added  to  the  acid. 

-  R.  D.  Reed,  Some  methods  of  heavy  mineral  investigation :  Ecoti.  Geology,  vol. 
19,  pp.  320-337,  1924-  .  .         , 

3  M.  Drosdoff  and  E.  Truog,  A  method  for  removmg  iron  oxide  coatmgs  from 
minerals:  A»i.  Mineralogist,  vol.  20,  pp.  669-673,  1935. 

*  Wm.  Mackie,  Acid  potassium  sulphate  as  a  petrochemical  test  and  solvent : 
Trans.  Edinburgh  Geol.  Soc,  vol.  Ii,  pp.  119-127,  1915-1924. 


PREPARATION  OF  SAMPLE  315 

tion — separated  from  the  light  minerals — after  the  following  treatment: 

(i)  the  fraction  as  originally  separated,  without  treatment 

(2)  hydrochloric  acid  treatment  (heat  with  dilute  HCl) 

(3)  fusion  with  KHSO4  and  solution  in  water 

Mackie  found  that  the  KHSO4  (acid  potassium  sulphate)  fusion  in 
a  platinum  crucible  was  very  effective  in  removing  iron  oxides,  quite 
superior  to  HCl,  and  he  found,  moreover,  that  it  only  slowly  dissolved 
apatite.  It  completely  dissolves  anatase,  chromite,  magnetite,  ilmenite, 
pyrite  and  marcasite. 

The  solvent  action  of  other  acids  or  combinations  of  acids,  such  as 
nitric  acid,  sulphuric  acid,  hydrofluoric  acid,  hydrofluoric  and  sulphuric 
acids,  and  hydrofluoric  and  nitric  acids,  has  been  investigated,^  and  the 
results  can  be  had  by  consulting  the  literature  on  this  subject.  The  reader 
is  also  referred  to  Table  39,  page  355. 

In  some  heavy  mineral  concentrates  certain  minerals  appear  in  such 
quantity  as  to  mask  the  less  frequent  mineral  species  that  may  be  present. 
It  is  then  desirable  to  remove  the  overabundant  mineral,  if  possible,  by 
some  simple  method.  Minerals  that  play  such  a  role  include  pyrite,  gyp- 
sum, anhydrite,  and  barite.  Pyrite  can  be  removed  by  heating  with  15 
per  cent  nitric  acid.  Gypsum  is  usually  not  present  in  the  heavy  residues 
due  to  its  low  specific  gravity,  but  when  present  due  to  abundant  in- 
clusions of  iron  oxide,  it  may,  according  to  Milner,-  be  eliminated  by 
digesting  with  a  strong  ammoniacal  solution  of  ammonium  sulphate. 
Barite  can  be  removed  by  concentrated  sulphuric  acid.  Strong  hot  HCl 
will  dissolve  the  anhydrite.  In  all  cases  it  must  be  remembered  that  some 
of  the  other  minerals  in  the  residue  may  be  adversely  affected  by  the 
solvents  and  may  be  partly  or  wholly  dissolved  (see  section  on  Chemical 
Separation  Methods). 

Bituminous  sands  may  be  clarified  by  treatment  with  a  mixture  of 
petroleum  ether  and  carbon  disulphide  followed  by  a  wash  in  alcohol  or 
with  benzol,  chloroform,  or  ether.  Milner^  gives  a  method  for  the  re- 
moval of  hydrocarbon  materials  which  can  be  used  for  the  quantitative 
determination  of  the  amount  present  in  the  sample  if  desired.  The  ex- 
traction apparatus  is  a  Soxhlet  extractor  (Figure  147)  or  simply  a 
reflux  condenser  attached  to  an  Erlenmeyer  flask,  in  which  the  crushed 

1  C.  L.  Doelter,  Ilaudhuch  dcr  Mincralchcviic  (1914).  A.  A.  Noyes,  Qiialitativc 
Analysis  of  the  Rare  Elements. 

-  H.  B.  Milner,  Sedimentary  Petroc/rapliy,  2nd  ed.  (Thos.  Murby  and  Co.,  Lon- 
don, 1929),  p.  65. 

3  H.  B.  Milner,  of.  cit.,  pp.  70-73. 


3i6 


SEDIMENTARY  PETROGRAPHY 


rW 


sample  is  suspended  in  a  "thimble"  (porous  perforated  porcelain  cup). 
The  whole  apparatus  is  then  placed  on  a  water-bath,  and  after  some 
solvent,  such  as  benzol,  is  placed  in  the  flask  to  a  depth  of  about  an  inch, 
heat  is  applied.  When  clarification  is  complete  the  sample  is  treated  as 
usual.  The  evaporation  of  the  solvent  will  leave  a  hydro- 
carbon residue,  the  amount  of  which  can  thus  be  deter- 
mined.^ 


Special  Preparation  Problems 

The  techniques  just  described  apply  in  the  main  to  the 
arenaceous  sediments.  The  mineralogy  of  these  rocks  has 
been  most  important  in  problems  of  correlation,  and  hence 
these  rocks  have  been  more  widely  studied.  In  some  cases 
it  has  been  found  useful  to  use  some  of  these  techniques 
for  study  of  the  mineralogy  of  other  types  of  sediments. 
This  is  usually  done  when  minor  accessory  minerals  are  to 
be  studied,  and,  as  usual,  this  study  must  be  preceded  by 
some  concentration  of  these  lesser  constituents.  Here 
again  there  is  no  standard  procedure  and  the  problem 
presented  by  each  sediment  must  be  met  in  a  different 
way. 

Fiue-graincd  argillaceous  rocks.  No  special  problems  of 
preparation  are  involved  with  these  materials  that  have  not 
been  treated  at  some  length  in  the  section  dealing  with 
mechanical  analysis  (page  51).  Action  with  acid  is,  how- 
Soxhlet  ex-  ever,  permissible  in  some  cases  where  the  mineralogy  is  to 
tractor.  |^g   studied   without   mechanical   anal}-sis,   though  the   de- 

structive effect  of  the  acid  on  certain  minerals  should  be 
kept  in  mind.  Treatment  with  alkaline  reagents  may  give  better  results 
with  the  argillaceous  materials.  Owing  to  difficulty  of  readily  identifying 
the  minerals  in  the  clays,  it  is  common  practice  to  remove  these  finest 
materials  and  discard  them,  leaving  a  small  residue  of  fine  sand  for 
study  with  the  microscope.  This  is  achieved  by  means  of  a  special  elutria- 
tion  apparatus,  such  as  that  devised  by  Eichenberg-  or  more  simply  by 
elutriation  or  decantation,  the  latter  following  a  5-min.  settling  in  a 
15-in.  column  of  water  which  will  remove  everything  less  than  V32  nim. 
in  diameter,  or  a  like  period  of  settling  in  a  5-in.  column  which  will  per- 


1  See  also  E.  M.  Spieker,  Bituminous  sandstone  near  Vernal,  Utah  :  U.  S.  Gcol. 
Survey,  Bull.  8?2-c,  pp.  77-98,  1930. 

-  W.  Eichenberg,  Ein  Schlammapparat  fiir  Tone :  Ccntralbl.  f.  Min.  Gcol.  u.  Pet., 
Abt.  A,  pp.  221-224,  1932. 


PREPARATION  OF  SAMPLE  317 

mit  washing  out  of  everything  under  ^4  mm.^  The  principal  difficulties 
of  the  study  of  the  frne-grained  sediments  relate  to  the  separation  into 
like  mineral  groups  and  to  identification.  These  problems  are  treated 
elsewhere  (see  section  on  study  of  clay  minerals  and  section  on  heavy 
mineral  separation  in  clays  and  silts). 

Crystalline  rocks.  Owing  to  the  development  of  many  new  minerals  by 
metamorphism,  the  suite  of  minor  accessories  originally  present  loses 
somewhat  of  its  identity.  Nevertheless,  it  is  occasionally  important  to 
investigate  these  minor  constituents,  as  they  may  be  the  sources  of  these 
minerals  in  sediments  of  later  geologic  age  or  may  be  important  in  de- 
termination of  the  igneous  or  sedimentary  origin  of  the  rock.  In  these 
cases  the  rock  is  crushed  to  a  size  about  that  of  the  constituent  minerals, 
and,  following  sieving  out  of  all  the  oversize  and  fine  dust,  then  sub- 
jected to  the  usual  concentration  methods.  For  a  more  detailed  discussion 
of  the  treatment  of  crystalline  rocks  the  reader  is  referred  to  the  liter- 
ature on  the  subject.- 

Calcarcoiis  rocks.  Since  these  are  nearly  all  calcite  or  dolomite,  the 
remaining  constituents  can  be  concentrated  by  crushing  the  rock  and 
leaching  with  acid.  Except  in  cases  of  very  abundant  residue,  it  has  been 
the  practice  to  examine  the  minor  minerals  as  a  whole  rather  than 
separated  fractions.  In  the  more  impure  calcareous  rocks  where  the 
residue  is  large,  mineral  separation  methods  may  be  applied,  the  pro- 
cedure to  be  used  will  depend  on  whether  the  residue  is  mainly  arenaceous 
or  argillaceous.  If  the  residue  is  composed  of  authigenic  minerals,  such 
as  pyrite,  or  chert,  the  separations  made  are  for  convenience  of  study 
and  do  not  have  the  same  significance,  as  for  example,  the  mechanical 
analysis  of  the  arenaceous  residues.  For  details  on  the  method  of  study 
of  insoluble  residues  in  calcareous  rocks  the  reader  is  referred  to  page 
494. 

Coal.  The  minor  mineral  constituents  of  coal  have  rarely  been  in- 
vestigated. To  concentrate  these  minerals  is  a  special  problem.  Two 
methods  have  been  proposed.  The  first  involves  burning  of  the  coal  and 
washing  the  ash  to  eliminate  the  finer  dust.  Such  a  method  is  drastic  in 
its  action.  The  second  method,  the  solution  of  the  organic  matter  with 
caustic  potash,  pyridine,  phenol,  chloroform,  selenium  oxychloride,^  or 

1  Marcellus  H.  Stow,  Washing  sediments  to  obtain  desirable  size  of  grain  for  mi- 
croscopic study:  Am.  Mineralogist,  vol.  i6,  p.  226,  1931. 

-Report  of  the  Committee  on  Accessory  Minerals  for  1936-37  (National  Re- 
search Council,  Division  of  Geology  and  Geography,  Washington,  D.  C). 

■^  E.  Stach,  Kohlenpetrographisches  Praktikiim  ( Gebriider  Borntraeger,  Berlin, 
1928).  Stach  describes  the  chromic  acid  method,  "Schulze's  method"  (potassium 
chlorate  and  nitric  acid),  and  the  diaphonol  method. 


3i8 


SEDIMENTARY  PETROGRAPHY 


sodium  hypochlorite,  is  to  be  preferred  if  it  can  be  made  to  work.  Crush- 
ing before  treatment  with  the  solvent  is  necessary. 


Table  36 

Summary  of  Disaggregation  Procedure  for 
mineralogical  studies 


Condition 

Non-calcarcoiis 

Calcareous 

of  Sample 

Cement 

Cement 

#5                                                   #6 

Unconsolidated 

No  problem  involved  * 

#3                                                      #4 

(a)   Crush    with    fingers,    wood    block,    or    rubber- 

covered  pesde 

Slightly 

(b)   Soak  in  water  and  use  stiff  brush 

coherent  f 

(c)   Digest  in  alkaline  so- 

(c)  Digest   in   weak  acid 

lution  (KOH,  NaOH, 

(HCl) 

or  NH4OH) 

#1 

#2 

(a)  Thin-section 

(a)  Thin  section 

Very  coherent 

(b)  Crush  in  iron  mortar 

(c)  Sodium      thiosulphate 
treatment 

(b)   Digest  in  HCl 

*  It  is  sometimes  desirable  to  remove  carbonate,  which  is  troublesome  in 
bromoform  separations  due  to  nearness  of  specific  gravity  (2.7  —  2.9)  to  that 
of  bromoform   (2.85).  Acid  clarifies  iron  oxide-stained  grains  also. 

f  May  be  thin-sectioned  after  special  treatment  to  indurate  sufficiently  for 
cutting  and  grinding.  Indurate  with  Canada  balsam  or  bakelite. 


CHAPTER    14 

SEPARATION  METHODS 

PRELIMINARY     ENRICHMENT     OF     SAND     IN     HEAVY 
MINERALS 

Owing  to  the  high  cost  of  heavy  Hquids,  some  workers  have  used  other 
methods  of  concentrating  the  hea\y  fraction  of  the  sediment.  Most  often 
these  other  methods  are  but  rough  concentrations  and  are  preUminary 
to  actual  separation  by  other  means.  Such  preliminary  enrichment  re- 
duces the  amount  of  heavy  liquid  required  for  a  separation.  Such 
methods  are  sizing,  panning,  and  vibration. 

It  is  known  from  observation  that  the  heavy  minerals  of  a  sand  are 
largely  concentrated  in  its  finer  grades.  This  is  due  to  the  fact  that  for 
a  certain  dominant  size  of  quartz  and  feldspar  there  is  a  smaller  size 
of  magnetite  and  like  heavy  minerals,  which  are  deposited  together  be- 
cause they  have  what  Schone  called  the  same  hydraulic  value  or  the 
same  settling  rates.  Consequently  by  screening  out  the  coarse,  light 
fractions  the  heavy  minerals  will  be  materially  concentrated. 

Heavy  minerals  can  be  separated  from  the  light  ones  by  repeated  pan- 
ning as  in  panning  for  gold.  The  method  is  used  mostly  for  separating 
small  amounts  of  heavy  accessor}^  minerals  from  a  large  amount  of  rock 
or  sediment  and  works  best  if  there  is  a  large  difference  in  gravity.  A 
conical  pan  is  preferred  by  some  to  the  common  miner's  pan.^  There  are 
rather  serious  objections  to  preliminar}-  panning,  as  pointed  out  by 
Smithson,^  if  exact  frequencies  of  the  minerals  are  to  be  determined. 
The  proportions  are  materially  affected  by  panning. 

Panning  is  carried  out  in  a  shallow  circular  vessel  with  wide  flaring 
rim.  A  common  size  is  some  18  in.  in  diameter  at  the  top  and  8  in.  in 
diameter  at  the  base.  The  depth  is  4  in.  The  pan  is  made  of  thin  black 
sheet  steel.  It  should  be  entirely  free  from  grease. 

^  O.  A.  Derby,  On  the  separation  and  study  of  the  heavy  accessories  of  rocks : 
Proc.  Rochester  Acad.  Sci.,  vol.  i,  pp.  198-206,  1891. 

-  Frank  Smithson,  The  reliability  of  frequency-estimations  of  heavy  mineral  suites : 
Geol.  Magazine,  vol.  67,  pp.  134-136,  1930.  See  also  C.  J.  C.  Ewing,  A  comparison 
of  the  methods  of  heavy  mineral  separation:  Geol.  Magazine,  vol.  68,  pp.  136-140, 
1931. 

319 


320  SEDIMENTARY  PETROGR.\PHY 

The  pan  is  filled  with  the  sediment  to  be  studied  and  set  in  water  a 
few  inches  deeper  than  the  pan.  The  material  is  thoroughly  wetted, 
stirred  up  and  disintegrated.  At  the  same  time  any  large  pebbles  are 
washed  from  the  pan.  From  time  to  time  the  pan  is  sharply  swirled  in  a 
horizontal  plane  until  nothing  is  left  but  clean  sand.  The  pan  is  then 
tipped  gently  forward,  is  held  level  with  the  surface  of  the  water,  and 
is  given  a  circulator)^  motion.  This  alternates  with  stirring  so  that  the 
lighter  materials  are  gradually  spilled  over  the  edge.  Concentration  is 
carried  as  far  as  desired ;  the  first  stages  are  completed  rapidly,  but  the 
later  stages  must  be  done  carefully  to  avoid  loss  of  hea\y  minerals.  A 
further  concentration  is  best  effected  by  means  of  hea\'}-  liquids.  Other- 
wise the  pan  is  so  rocked  (jerk  and  flow)  at  the  end  to  soread  the 
"colors.' 

Salmojraghi^  found  that  when  a  handful  of  drj-  sand  is  agitated  on 
a  sheet  of  paper  there  is  a  marked  enrichment  in  hea\y  minerals  in  that 
part  which  works  to  the  bottom  and  which  may  be  seen  by  inclining  the 
sheet.  He  believed  that  with  three  or  four  such  operations  all  the  com- 
ponents of  a  sand  may  be  determined  with  the  possible  exception  of  the 
extremely  rare  ones.  Salmojraghi,  moreover,  found  that  a  relationship 
could  be  established  between  the  proportions  of  the  minerals  in  the  sand 
enriched  by  dr\-  agitation  and  their  true  proportions  in  the  natural  sand. 
The  authors  believe,  however,  that  while  this  method  is  an  aid  to  study, 
it  can  hardly  take  the  place  of  the  more  complete  and  easy  separation 
with  heavy  liquids. 


S  E  P  A  R  .A.  T  I  O  N      OF      M  I  V  E  R  .\  L  S      OX      B  .A.  S  I  S      OF 
SPECIFIC     GR.\\ITY 

He.\vy  Liquids 

A  heavy  liquid  substance  to  be  satisfactor>'  should  be  (i)  inexpensive, 
(2)  easily  prepared  or  purchased,  C3)  transparent,  (4)  liquid  at  ordi- 
nary- temperatures,  that  is,  having  a  low  melting  point,  (5)  non-corrosive, 
(6)  chemically  inert,  (7)  without  odor,  (8)  fluid,  not  -viscous,  and  (9) 
easily  concentrated  or  diluted. 

Many  liquids  have  been  used  or  investigated,  but  no  one  has  all  these 
desirable  qualities.  Several,  however,  are  decidedly  superior  and  are  to 
be  recommended.  Some  hquids  that  have  imdesirable  features,  notably 
potassium  mercuric  iodide  (Thoulet  solution),  have  been  widely  used. 
Hence  details  concerning  the  properties,  preparation,  and  uses  of  most 

^  F.  Salmojraghi,  Sullo  studio  mineralogico  delle  sabbie  e  sopra  un  modo  di  rap- 
presentarne  i  risultati:  Atti  soc.  ital.  set.  nat.,  voL  43,  pp.  54-89,  1904. 


SEPARATION  METHODS  321 

liquids,  including  those  heretofore  generally  used,  have  been  omitted  and 
only  five  of  the  most  useful  and  satisfactory  fluids  are  described  in 
detail. 

As  Davies  ^  has  suggested,  the  (.K)ublc  noinonclaturo  oi  the  heavy 
liquids,  that  is,  the  use  of  both  the  chemical  name  and  the  name  of  the 
first  user  or  discoverer  of  the  fluid,  is  confusing.  The  usage  is  further 
often  inconsistent.  Consequently  the  liquids  or  solutions  are  here  given 
according  to  their  chemical  name.  The  chemical  name  has  the  double 
advantage  of  giving  the  composition  of  the  liquid  as  well  as  recalling  to 
mind  some  of  the  properties  of  the  substance. 

Bromoform  (tribroin-uivthanc).  Schroeder  van  der  Kolk  usetl  bronio- 
form  in  1895  ^s  a  heavy  liquid  for  the  separation  of  the  minerals  of  a 
sand.-  It  has  since  been  used  rather  widely  by  investigators  antl  is  at 
present  perhaps  the  most  commonly  used  heavy  liciuid. 

Bromoform  (tribrom-methanc),  CIIBr;,,  is  a  halogen  substitution 
product  of  methane.  It  is  a  highly  mobile  liquid  at  ordinary  tcmperatin-es 
and  has  a  specific  gravity  of  2.89  at  10°  C.  Sullivan  also  gives  the  melting 
point  at  9°  C.  and  the  boiling  point  at  151.2°  C.^ 

Bromoform  may  be  more  readily  purchased  than  prei)ared  in  the 
laboratory.  Commercial  bromoform,  however,  is  usually  low  in  gravity, 
often  below  that  of  quartz  (2.66)  due  largely  to  dissolved  alcohol.  For 
work,  therefore,  where  it  is  desired  to  separate  the  quartz  from  the 
heavier  minerals  it  is  necessary  to  remove  the  alcohol  by  the  procedure 
given  below.*  Bromoform  changes  in  specific  gravity  with  temperature, 
about  .0023  per  degree  Centigrade.  Consequently  pure  bromoform  will 
have  a  specific  gravity  of  2.87  at  ordinary  laboratory  temperatures 
(20°  C).  This  is  quite  sufficient  to  affect  a  separation  of  quartz  and 
feldspar  from  the  heavier  minerals.  Since  this  separation  is  usually  all 
that  is  desired,  bromoform  is  likely  to  be  one  of  the  most  useful  heavy 
liquids. 

Bromoform  is  miscible  in  all  proportions  with  carbon  tetrachloride 
(CClj),  benzene  (benzol)  (CoH,,),  alcohol  (CTIbOH),  and  acetone 
(CH3COCH3).  From  carbon  tetrachloride  and  benzene  mixtures  it  may 
be  received  by   fractional  distillation,   using  an   Engler-type  flask  and 

1  G.  M.  Davies,  Nomenclature  of  the  heavy  liquids :  Gcol.  Magazine,  vol,  57,  p. 
287,  1920. 

-  J.  L.  C.  Schroeder  van  der  Kolk,  Beitrag  zur  Kartirung  der  quartaren  Sande : 
Ncucs  Jahrh.  j.  Miit.,  rtr.,  vol.  50,  Bd.  I,  pp.  272-276,  especially  p.  274,  1805. 

■''  John  D.  Sullivan.  Heavy  liquids  for  niineralogical  analyses :  U.  S.  Bureau  Mines. 
Technical  Pafer  .,\V/,  p.  10,  1927. 

4  The  authors,  for  example,  took  commercial  bromoform,  specific  gravity  of  2.(13^. 
and  were  able  with  three  washings  to  raise  the  specific  gravity  to  2.838. 


322       SEDIMENTARY  PETROGRAPHY 

Liebig  condenser.  Benzene  boils  at  80.5°  C.  and  carbon  tetrachloride 
boils  at  76.7°  C.  Since  bromoform  boils  at  151.2°  C.,  the  tetrachloride 
or  benzene  would  come  over  first.  Bromoform  diluted  with  alcohol  (as 
commercial  bromoform  often  is)  or  acetone  could  be  similarly  recov- 
ered, but  a  much  simpler  and  more  satisfactory  method  is  that  suggested 
by  Ross.^  To  the  bromoform-alcohol  or  bromoform-acetone  mixture  is 
added  a  large  volume  of  water  (fifteen  times  as  large)  in  a  two-liter 
bottle.  After  vigorous  shaking  the  heavy  bromoform  phase  separates  out 
with  but  very  little  alcohol  or  acetone  and  the  water  phase  contains  most 
of  the  alcohol  or  acetone.  Most  of  the  water  is  decanted  and  the  process 
repeated  two  or  three  times.  After  the  third  decantation,-  the  bromoform- 
water  mixture  is  poured  into  a  separatory  funnel  and  the  bromoform 
almost  free  of  water  is  drawn  off  and  run  into  a  funnel  fitted  with 
several  thicknesses  of  filter-paper  which  will  absorb  any  dispersed  water. 
The  bromoform  filtered  should  be  clear  and  have  a  specific  gravity  of 
about  2.85  at  room  temperature.  A  little  bromoform  is  lost  due  to  dis- 
persion in  the  decanted  water  but  the  loss  is  not  large  (3  to  6  per  cent). 
Cohee,  using  this  method  of  recovery,  has  described  an  apparatus  set-up.^ 
Another  method  of  recovery  and  of  clarification  in  the  case  of  separa- 
tion of  free  bromine  is  that  of  fractional  crystallization  by  freezing-out 
of  the  bromoform.  If  a  dish  of  bromoform  is  placed  out  of  doors  on  a 
cold  winter  day,  the  bromoform  will  soon  crystallize  out  as  clear,  trans- 
parent platy  crystals.  (Pure  bromoform  freezes  at  9°  C.  but  a  somewhat 
lower  temperature  is  required  for  solutions  containing  alcohol.)  If  the 
residual  liquid  be  strained  off  from  the  crystals  and  the  latter  melted,  it 
will  be  found  that  the  bromoform  will  be  clear  and  of  high  specific  grav- 
ity. A  second  crystallization  may  be  needed  to  completely  clear  the 
substance. 

Sometimes  the  bromoform  becomes  discolored  due  to  petroleum  present  in 
the  sands  undergoing  separation.  Hanna  *  found  fuller's  earth  useful  in  the 

1  Clarence  S.  Ross,  Methods  of  preparation  of  sedimentary  materials  for  study : 
Econ.  Geology,  vol.  21,  pp.  454-468,  1926.  The  method  of  recovery  described  by  Ross 
was  later  redescribed  by  Bracevvell.  See  S.  Bracewell,  Recovery  of  bromoform: 
Geol.  Magazine,  vol.  70,  p.  192,  1933,  and  F.  Smithson,  The  recovery  of  bromoform: 
Gcol.  Magazine,  vol.  71,  p.  240,  1934. 

-  Ross  recommends  shaking  with  water  in  excess  and  allowing  the  material  to 
stand  24  hr.  If  the  volume  of  water  is  greatly  in  excess  and  if  the  shaking  and  de- 
cantation are  repeated  two  or  three  times,  it  will  be  unnecessary  to  allow  the  mix- 
ture to  stand.  By  two  or  three  such  decantations  the  writers  concentrated  bromo- 
form, S.G.  2.590,  to  2.835  with  no  great  loss. 

3  George  V.  Cohee,  Inexpensive  equipment  for  reclaiming  heavy  liquids:  Jour. 
Sed.  Petrology,  vol.  7,  pp.  34-35,  1937- 

4  Marcus  A.  Hanna,  Clarification  of  oil-discolored  bromoform:  Jotir.  Paleon., 
vol.  I,  p.  145,  1927. 


SEPARATION  METHODS  ^^3 

clariticarion  of  tlie  bromofonn  in  such  cases.  The  fuller's  eartli.  however, 
will  not  clarify  bromofonn  discolored  by  free  bromine.  Shaking  witli  XaOH 
or  alcoholic  KOH  will  remove  free  bromine. 

Carbon  tetradiloride  may  be  used  as  diluent  as  indicated  above.  It  must  be 
recovered  by  fractional  distillation,  however,  and  tlierefore  is  less  desirable 
than  acetone  or  alcohol.  It  has  one  advantage,  namely  that,  since  carbon 
tetrachloride  has  a  specific  gravity  of  1.58,  a  larger  volume  of  solution  of 
given  specific  gravity  will  be  obtained  if  it  is  used  instead 'of  acetone,  alcohol, 
or  benzene.  In  case  a  pennanent  series  of  liquids  differing  by  small  inter\-als 
in  specific  gravity  is  desired,  for  t!ie  purpose  of  specific  gravity  detennination, 
perhaps  carbon  tetrachloride  is  tlie  most  economical  diluent.  Sullivan  ^  has 
investigated  tlie  properties  of  tliis  combination  and  tlie  following  table  is 
given  by  him.  Sullivan  used  bromofonn  of  an  original  gravit>'  of  2.61  (com- 
mercial bromofomi)  and  hence  concluded  that  the  solution  had  but  a  re- 
stricted use  in  tlie  separation  of  hea\-y-  minerals  from  gangue  but  that  it  w^as 
satisfactory  for  separating  coal  from  bone. 

Specific  Gravity  of  Mixtures  of  CCU  and  CHBts  at  25'  C. 

Per  Cat  CHBrs  (6v  Vol.)       S.  G.      Per  Cent  CKBt^  (by  Vol.)       S.  G. 

100 2.61  2^ 1 .84 

75 -35  o 

50 2.09 

Bromofonn  is  "commercial" 


i.>S 


The  mineral  grains  collected  on  the  filter-paper  after  gra\'it3-  separa- 
tion may  be  collected  in  a  bottle  la-  ^^ 
beled  '■Bromofomi  washings"  from  29 
which  the  bromofomi  may  be  re-  ^  2* 
covered  when  tlie  quantit}-  becomes  >  ^^ 
sufficiently  large.  The  method  of  g 
recover)^  is  that  of  ^^-ashing  with  ^  ^4 
large  volumes  of  water  as  described  fj  23 
by  Ross.  ^  2: 

XonnaJly     bromofomi     is     used  2' 

without  dilution  for  the  separation  ^^o    a 

of    minerals    with    specific    graA-ity  per  cent  BRcworoRw 

greater  than  2.S5  from  tliose  with  a  Fig.    14S.— Curve    showing    specific 

gravitv  less  than  that  figure.  As  in-    ^'■.^'"^'^'  °*  ''^"''"^  bromoform-acetone 
-.        -  *  mixtures, 

dicated  by  Hanna.-  however,  there 

are  separations  in  which  a  liquid  of  less  densit\-  is  needed.  Bromoform 

with  acetone  will  give  such  a  liquid.  (See  Figure  148.) 

1  John  D.  Sulli\-an.  op.  cit..  p.  11. 

-  Marcus  A.  Hanna.  Separation  of  fossils  and  other  light  materials  by  means  of 
heavy  liquids :  Econ.  Geology,  vol.  22,  pp.  14-17,  1927. 


■    ^                   '    1 

2:c                                       ^ 

y  \ 

/                1 

1 

^ 

/■ 

y 

/ 

/ 

324  SEDIMENTARY  PETROGRAPHY 

Hanna  outlines  the  following  separation :  ^ 

(i)  Bromoform-alcohol  liquid.  S.  G.  2.685.  Quartz,  etc.,  floats,  calcite  and 
aragonite  fossils  sink  as  do  also  the  "hea\->-  minerals." 

(2)  Bromoform-alcohol  liquid.  S.  G.  2.60.  Glauconite,  orthoclase.  glau- 
conite-filled  shells,  and  air-filled  shells  float.  Quartz  sinks. 

(3).  Bromoform-alcohol  liquid.  S.  G.  2.20.  Air-filled  shells  float.  Glau- 
conite, glauconite-fijled  shells,  and  orthoclase  sink. 

Hanna  estimates  that  90  per  cent  or  more  of  the  fossils  are  thus  sepa- 
rated from  the  quartz.  Possibly  a  quantitative  separation  of  quartz  from 
orthoclase  in  arkosic  sands  can  be  made  in  this  way. 

Table  of  Specific  Gr.\vity  Mixtures  of  CHBrj  and  CH3COCH3 
(Bromoform  and  Acetone)  at  20°  C. 

Per  Cent  CHBrg   (&v  Vol.)  S.  G.  Per  Cent  CHBrg   {hy  Vol.)  S.  G. 

100    2.846            85     2.551 

95     2.743           80     2.450 

90    2.652            -^    2.346 

Bromoform  is  not  entirely  free  from  objectionable  features.  These 
are: 

(i)  It  is  decomposed  by  strong  light.  Prevent  by  keeping  in  brown  bottles 
in  the  dark. 

(2)  It  also  deteriorates  with  heat.  Store  in  a  cool  place. 

(3)  It  evaporates  readily,  hence  vessels  containing  bromoform  should  be 
kept  stoppered  or  covered. 

(4)  Cost  is  sufficiently  high,  though  not  excessive,-  to  make  it  profitable 
to  consen-e  the  acetone  washings  and  to  recover  the  bromoform  therefrom. 

(5)  Bromoform  is  subject  to  strong  convection  currents.  Milner  ^  states 
that  this  difficulty  may  be  overcome  by  carrying  out  the  separations  in  a  fume 
cupboard  or  constant  temperature  chamber  or  a  room  free  from  warm  air 
currents. 

(6)  Bromoform  is  toxic.  While  it  is  not  corrosive  nor  are  its  vapors 
strongly  toxic  if  inhaled,  the  fumes  are  mildly  so  if  one  works  with  the 
liquid  in  the  open  laboratory  for  a  long  time  without  interruptions.  If  the 
separations  are  carried  out  in  a  well-ventilated  hood  this  objection  will  be 
overcome. 

Acetone  as  a  diluent  has  some  objectionable  features,  notably  its  rapid 
evaporation,  necessitating  constant  checking  of  the  specific  gravity  of  the 
bromo form-acetone  mixture,  and  also  its  inflammability,  which  makes  it 
impossible  to  use  near  an  open  flame.  The  rapid  evaporation  is  an  ad- 

1  In  this  table  "alcohol"  has  been  substituted  for  "benzol"  of  Hanna. 

2  Bromoform  w-as  obtainable  in  1936  at  about  $2.25  per  pound  (about  157  c.c). 

3  Henry  D.  Milner,  op.  cit.,  p.  40. 


SEPARATION  METHODS  325 

vantage,  however,  in  that  minerals  washed  free  of  bromoform  with 
acetone  dry  ahiiost  instantly  and  are  ready  for  mounting  and  examina- 
tion. Alcohol  has  much  the  same  disadvantages. 

Acetylene  tctrabromide  {tctrabrom-ethane) .  The  most  promising  al- 
ternate to  bromoform  is  apparently  acetylene  tetrabromide.  It  is  rather 
widely  used.  Muthmann  ^  seems  to  have  been  the  first  one  to  use  it  for 
mineral  separations. 

Acetylene  tetrabromide  (tetrabrom-ethane),  CoHoBr^,  is  a  colorless 
mobile  liquid  with  a  specific  gravity  of  2.96  at  20°  C.  It  is  miscible  in 
all  proportions  with  carbon  tetrachloride  or  benzene,  giving  a  range  of 
gravities  from  that  of  the  pure  liquid,  2.96,  to  the  gravity  of  the  respec- 
tive diluents,  1.6  or  0.9.  Either  diluent  may  be  used  for  washing  the 
mineral  grains  free  of  the  tetrabromide,  though  for  rapid  washing  ben- 
zene is  probably  the  better.  The  acetylene  tetrabromide  may  be  recovered 
by  fractional  distillation.  Since  the  compound  is  also  miscible  with  alcohol 
and  insoluble  in  water,  alcohol  may  be  used  as  diluent  and  the  tetra- 
bromide recovered  by  washing  with  water  as  is  the  case  with  bromoform. 
The  alcohol  used,  however,  must  be  absolute  alcohol  otherwise  the 
acetylene  tetrabromide  becomes  clouded  upon  dilution. 

Sullivan,-  who  used  commercial  acetylene  tetrabromide  with  a  specific 
gravity  of  2.89,  gives  the  following  data  on  mixtures  of  the  tetrabromide 
and  carbon  tetrachloride: 

Specific  Gr.wity  of  Mixtures  of  CCI4  and  CoH^Br^  at  25°  C. 

Per  Cent  C,H,Br^  {by  J'ol.)       S.  G.  Per  Cent  CoH.Br^  (by  Vol.)       S.  G. 

100 2.89  25 1.91 

75  2.58  o 1.58 

50 2.24 

Acetylene  tetrabromide  may  be  purchased  (at  about  $1.50  per  pound 
in  lo-lb.  lots)  or  prepared  in  the  laboratory  according  to  the  method 
outlined  by  O'Meara  and  Clemmer.^ 

A  thousand  grams  of  liquid  bromine  is  placed  in  three  or  four  gas- 
washing  bottles,  and  a  small  amount  of  water  is  added  to  prevent  exces- 
sive volatilization  of  the  bromine.  Acetylene  gas,  produced  by  the  action 
of  water  on  calcium  carbide,  is  bubbled  slowly  through  the  series  of 
bottles  until  the  reaction  is  complete.  This  is  indicated  by  a  change  of 

1  \V.  Muthmann,  Uebcr  eine  zur  Trennung  von  Mineralgemischen  geeignete 
schwere  Fliissigkeit :  Zcits.  Kryst.  Min.,  Bd.  30,  pp.  73-74,  1899. 

2  John  D.  Sullivan,  op.  cit.,  pp.  11-12. 

3  R.  G.  O'Meara  and  J.  Bruce  Clemmer,  Methods  of  preparing  and  cleaning  some 
common  heavy  liquids  used  in  ore  testing :  U.  S.  Bureau  Mines,  Rcpt.  of  Jnz'cstiga- 
tions  No.  3S97,  pp.  1-3,  1928. 


326       SEDIMENTARY  PETROGRAPHY 

color,  since  the  acetylene  tetrabromide  is  light  amber.  Excessive  temper- 
ature rise  should  be  prevented.  The  resulting  heavy  solution  is  placed  in 
a  separator}-  funnel  and  agitated  with  a  dilute  caustic  solution  to  remove 
the  uncombined  bromine.  \\"hen  the  water  and  the  acetylene  tetrabromide 
have  separated  into  two  layers,  the  latter  is  drawn  off  and  dehydrated 
with  calcium  chloride.  O'Meara  and  Clemmer  report  a  yield  of  90  to  96 
per  cent. 

Recovery  of  the  acetjlene  tetrabromide  from  the  benzene  or  carbon  tetra- 
chloride may  be  effected  by  fractional  distillation.  Simple  evaporation  will, 
however,  produce  the  same  result.  If  the  washings  are  permitted  to  stand 
open  at  room  temperature,  the  diluent  evaporates  slowly.  When  the  residual 
liquid  is  up  to  the  desired  gravity,  it  is  filtered  and  is  then  ready  for  further 
use.  Gentle  heating  facilitates  the  concentration,  but  rapid  or  excessive  heating 
results  in  partial  decomposition.  If  decomposition  occurs,  as  indicated  by  a 
dark  color,  the  acetylene  tetrabromide  can  be  restored  by  shaking  with  a 
small  amount  of  bromine  in  a  separatory  funnel,  then  adding  a  small  amoimt 
of  soditim  hydro.xide  to  remove  the  excess  bromine.  The  solution  should 
become  light  straw-colored.  Dehydration  with  calcium  chloride  follows.  The 
liquid  is  then  allowed  to  stand  in  an  open  vessel  until  it  loses  the  peculiar  odor 
acquired  during  the  cleaning  process,  when  it  is  ready  for  further  use.  If  the 
liquid  was  ven.-  dark  to  begin  with,  a  repetition  of  the  cleaning  process  may 
be  necessary.  Sulphurous  acid  may  be  substituted  for  the  caustic,  according 
to  O'Meara  and  Clemmer,  and  the  danger  of  formation  of  ethylene  dibromide 
during  the  caustic  treatment  is  thus  eliminated. 

Methylene  iodide.  Brauns  is  credited  with  the  first  use  of  methylene 
iodide  for  mineral  separation.^  O'Meara  and  Clemmer-  give  the  most 
recent  published  statement  of  the  preparation  and  reclamation  of  this 
fluid.  Owing  to  its  high  cost,  the  authors  do  not  recommend  it  for 
hea\-)'  mineral  work.  It  is,  however,  one  of  the  most  useful  immersion 
liquids  for  refractive  index  work,  hence  something  of  its  properties  is 
given  here. 

Pure  methylene  iodide,  CH2I2,  has  a  light  straw  color  and  a  specific 
gravity  of  3.32  at  18°  C.  It  has  an  index  of  1.74.  It  is  miscible  in  all 
proportions  with  benzene  or  carbon  tetrachloride,  from  which  it  may  be 
recovered  by  fractional  distillation  at  reduced  pressure  or  simply  by  al- 
lowing the  more  volatile  lighter  hquid  to  evaporate.  The  products  of 
separation,  therefore,  may  be  washed  with  these  substances  and  the  iodide 
recovered  by  slow  evaporation.  Heating  to  hasten  evaporation  should  be 
avoided,  owing  to  decomposition  of  the  iodide.  Ross^  states  that  the 

1  R.  Brauns,  Ueber  die  Verwendbarkeit  des  Methylenjodids  bei  petrographischen 
und  optischen  Untersuchungen :  Xeues  Jahrb.  f.  Min.,  etc.,  Bd.  2,  pp.  72-78,  1886. 

-  R.  G.  O'Meara  and  J.  Bruce  Clemmer,  op.  cit.,  pp.  3-4. 

3  Clarence  S.  Ross,  Separation  of  sedimentar>^  materials  for  study:  Econ.  Geology, 
voL  2i,  p.  334,  1928. 


SEPARATION  METHODS  327 

methylene  iodide  may  be  diluted  with  acetone  or  alcohol  and  recovered 
by  washing  with  water  in  the  same  manner  as  with  bromoform.  For 
refractive  index  work,  methylene  iodide  is  diluted  with  alpha  mono- 
bromonaphthalene  to  give  a  series  of  liquids  ranging  in  index  from  1.66 
to  1.74.  Methylene  iodide  is  decomposed  by  strong  light  and  should  be 
kept  in  tin-foil  covered  or  brown  bottles.  Mercury  or  copper  foil  may 
be  used  to  restore  normal  color  and  remove  the  free  iodine  liberated,  or 
clarification  may  be  achieved  by  shaking  with  dilute  KOH  or  NaOH 
solution  in  a  separatory  funnel  followed  by  dehydration  of  the  iodide 
with  calcium  chloride. 

Methylene  iodide  cost  $15  per  pound  in  8-lb.  lots  according  to  O'Meara 
and  Clemmer  (1928).  It  may,  however,  be  prepared  in  the  laboratory  by 
the  method  given  by  these  authors. 

Thalloiis  formate.  Clerici  ^  used  thallium  formate  (TICO^H)  for  the 
separation  of  minerals  in  1907.  Since  then  this  substance  has  come  to 
be  more  widely  used,  and  solutions  of  this  salt  together  with  the  double 
thallium  formate-malonate  ("Clerici's  solution")  are  probably  the  most 
satisfactory  substances  known  for  use  where  liquids  of  high  specific 
gravities  are  desired. 

Thallous  formate  (TICO2H)  is  an  organic  salt  very  soluble  in  water. 
Sullivan  -  gives  the  melting  point  as  94°  C.  and  the  specific  gravity  as 
4.95  at  105°  C.  The  aqueous  solutions  have  the  following  gravities: 

Per  Cent  H.O  {by  Weight)  Temperature  S.  G.       Melting  Point 

o  105°  C 4.95 94°  C. 

5  60°  C 4.19 54°  C. 

10  42°  C 3.72 31°  C. 

15  25°  C 3-39 22°  C. 

20  25°  C 3.09 

25  25°  C 2.86 

The  solubility  of  the  salt  in  water  increases  rapidly  with  increasing 
temperature.  The  formate  does  not  decompose  at  the  boiling  point  of 
water  but  does  slowly  decompose  at  higher  temperatures.  The  diluted 
solutions  may  be  concentrated  by  evaporation. 

The  salt  may  be  purchased,  but  is  readily  prepared  from  thallium 
carbonate  by  adding  an  equivalent  weight  of  formic  acid.  Sullivan  ^  gives 
the  procedure  for  preparing  the  formate  from  thallium  sulphate  and 
quotes  the  procedure  given  by  Clerici  for  making  the  material   from 

1  Enrico  Clerici,   Preparazione  di  liquidi  per  la  separazione  dei  minerali :  Atti. 
Rend.  R.  Accad.  Lined.  Roma,  set.  5,  vol.  16,  i  semestre,  pp.  187-195,  1907. 
-  John  D.  Sullivan,  op.  cit.,  pp.  20-21. 
3  John  D.  Sullivan,  op.  cit.,  p.  20. 


328        SEDIMENTARY  PETROGRAPHY 

metallic  thallium.  The  preparation  of  the  salt  is  also  described  by  Vhay 
and  Williamson.^ 

Thallium  formatc-vialoiate  (Clcrici's  solution).  Clerici,-  who  worked 
with  thallium  formate  solutions,  also  discovered  that  solutions  of  the 
double  salt,  thallium  formate-malonate,  were  suitable  for  mineral  sepa- 
rations. Clerici 's  solution  has  also  been  investigated  by  Vassar,^  from 
whose  work  the  following  data  are  taken. 

Clerici  solution  is  a  mixture  of  tliallium  malonate,  CHo(COOTl)2, 
and  thallium  formate,  HCOOTl,  which  at  ordinary  room  temperatures 
has  a  density  of  4.25.  Clerici  is  quoted  as  stating  that  increasing  heat 
will  increase  the  density  of  the  solution  because  of  increased  solubility 
of  the  salts  at  higher  temperatures.  A  concentrated  solution  at  35°  C. 
has  a  specific  gravity  of  4.4  and  at  50°  C.  a  specific  gravity  of  4.65 ;  at 
90-100°  C,  pyrite  floats.  It  is  possible  to  dilute  with  water  in  any  quan- 
tit}'  and  to  reconcentrate.  The  solution  is  more  mobile  than  Thoulet's 
solution  (potassium  mercuric  iodide  j,  is  odorless,  and  has  a  slight  amber 
color.  At  ordinary  temperatures  the  solution  appears  to  be  stable  and 
inert,  but  sulphides  should  not  be  left  too  long  in  hot  solutions. 

The  solution  is  prepared  by  neutralizing  two  equal  parts  of  thallium 
carbonate  with  equivalent  parts  of  formic  acid  and  malonic  acid,  each 
separately,  and  then  mixing,  filtering,  and  evaporating  until  almandite 
floats.  A  weight  of  11 1  g.  of  malonic  acid  dissolved  in  a  httle  water 
will  neutralize  500  g.  of  thallium  carbonate,  and  115  g.  of  55.5  per  cent 
formic  acid  will  neutralize  the  same  amount  of  thallium  carbonate.  One 
kilogram  of  thallium  carbonate  will  make  approximately  300  c.c.  of 
Clerici's  solution.  If  the  solution  is  made  from  the  dry  salts,  thallium 
formate  and  thallium  malonate,  7  g.  of  each  will  dissolve  completely  in 
I  c.c.  of  water,  but  10  g.  of  each  will  leave  a  part  undissolved.  Vassar 
gives  the  method  of  preparing  Clerici's  solution  from  metallic  thallium, 
but  it  is  omitted  here  because  thallium  carbonate  can  be  purchased  and 
the  procedure  for  its  manufacture  from  the  metal  is  quite  laborious. 
Most  recently  Rankama^  has  given  detailed  instructions  for  the  purify- 
ing of  Clerici's  solution,  as  well  as  acetylene  tetrabromide,  by  treatment 
with  bone  charcoal. 

1  J.  A.  Vhay  and  A.  T.  Williamson,  The  preparation  of  thallous  formate :  Atii. 
Mineralogist,  vol.  17,  pp.  560-563,  1933. 

-  Enrico  Clerici,  o/>.  cit.,  pp.  187-195;  Ulteriori  ricerche  sui  liquidi  pesanti  per  la 
separazione  dei  minerali:  Atti.  Rend.  R.  Accad.  Lincei.  Roma.  ser.  5,  vol.  31,  pp. 
116-118,  1922. 

3  Helen  E.  Vassar,  Clerici  solution  for  mineral  separation  by  gravity:  Am.  Min- 
eralogist, vol.  10,  pp.  123-125,  1925. 

*  Kalervo  Rankama,  Purifying  methods  for  the  Clerici  solution  and  for  acetylene 
tetrabromide :  Bull.  Comm.  Gcol.  de  Finlandc,  No.  115,  pp.  65-67,  1936. 


SEPARATION  METHODS 


329 


5 

^     C 
Si 

111 

= 

a| 

c  .1 

C 

S.  G. 
Diluent 

6 

d 

0 
q 

8 

s 

8  08 

< 

III 

w 
1 

1 

A ppr ox. 

Cost 
1,000  g. 

8 

IT) 
#9= 

0 

-J- 
8 

J 

8 

8 

^0 

m 

^0 

°0 

d 

10 

0. 

8 

0) 

8 

1     00. 

++ 
CO 

10 

< 

U 

ffi' 

u 

d' 

u 

H 
0 
0 

Eli 

e 

E 

c 

1>     Ji 
^  i;  0 

< 

1; 

1.1 

<u 

II 

1      OJ 

.ill 

E   S 


0.0 


U 


330        SEDIMENTARY  PETROGRAPHY 

Choice  of  liquid.  An  aqueous  solution  of  thallium  formate  is  probably 
the  most  satisfactory  liquid  for  a  gravity  range  of  2.89  (bromoform)  to 
3.39  at  ordinary'  temperatures  (25°  C.)-  Where  gravities  lower  than  2.89 
are  required,  either  bromoform  or  acetj'lene  tetrabromide  may  be  used; 
where  gravities  higher  than  3.39  are  desired,  an  aqueous  solution  of  the 
formate-malonate  is  more  suitable.  The  formate-malonate  solution,  of 
course,  could  be  used  for  all  ranges  except  that  it  is  more  expensive  than 
the  other  liquids  mentioned.  The  composition,  cost,  and  properties  of 
these  most  useful  liquids  are  summarized  in  Table  37. 

Other  heavy  liquids.  As  pointed  out  at  the  beginning  of  the  section  on 
hea\-y  liquids,  there  are  some  other  liquids  which  may  be  or  have  been  used 
for  mineral  separations  and  specific  gravitj'  work.  Of  the  many  which  have 
been  investigated  or  proposed  but  a  few  ever  achieved  wide  popularity.  As 
noted,  even  these  are  now  largely  replaced  by  the  organic  liquids  described 
above.  Since  a  few  laboratories  still  have  a  supply  of  the  earlier-used  liquids, 
it  seems  advisable  to  indicate  tlie  nature  of  these  liquids  and  to  point  out 
where  instructions  for  their  use  and  recover>'  may  be  obtained. 

Thoulet  solution  (or  Sondstadt's  solution),  potassium  mercuric  iodide,  was 
one  of  the  most  widely  used  heaAy  liquids.^  It  is  an  aqueous  solution  and 
therefore  recovered  b)-  evaporation  and  diluted  with  water.  It  is  very  cor- 
rosive, rather  viscous,  and  reacts  with  certain  minerals  -  and  becomes  dark 
on  the  liberation  of  iodine.  For  these  reasons  its  use  should  be  discouraged. 
It  has  a  specific  gravity'  of  about  3.0-3.2. 

Klein  solution,^  an  aqueous  solution  of  cadmium  borottmgstate,  has  a 
density-  of  3.3-3.4.  This  liquid  is  poisonous,  though  not  corrosive;  it  is  also 
decomposed  by  certain  minerals,  notably  the  carbonates,  and  on  exposure  to 
light  becomes  verj-  dark. 

A  benzol  solution  of  tlie  double  iodide  of  tin  and  arsenic  with  a  densit>'  of 
3.6  was  used  by  Retgers.'*  It  is  verj-  toxic,  readily  decomposed  in  the  presence 
of  water,  and  has  a  dark  red  color.  Retgers  '"  also  used  a  double  salt  of  thallium 
mercuro-nitrate  which  may  be  dissolved  in  water.  Tliis  is  a  quite  fluid  sub- 
stance, is  transparent,  and  does  not  react  with  the  metallic  sulphides  and  is 
therefore  generally  more  satisfactory. 

1  E.  Sondstadt,  Note  on  a  new  method  of  taking  specific  gravities,  adopted  for 
special  cases:  Chemical  News,  vol.  29,  pp.  127-128,  1874.  J.  Thoulet,  Separation  des 
elements  non  ferruginuex  des  roches,  fondee  sur  leur  difference  de  poids  specifique : 
C.  R.  Acad.  Sci.  Paris,  vol.  86,  pp.  454-456,  1878.  V.  Goldschmidt,  Ueber  Verwend- 
barkeit  einer  Kaliumquecksilberjodidlosung  bei  mineral ogischen  und  petrograph- 
ischen  Untersuchungen :  Ncues  Jalirb.  f.  Min.,  etc.,  B.B.  I,  pp.  179-238,  1881. 

2T.  L.  Walker,  Alteration  of  silicates  by  Sondstadt's  solution:  Am.  Mineralogist, 
vol.  7,  pp.  100-102,  1922. 

3  D.  Klein,  Sur  une  solution  de  densite  3.28,  propre  a  I'analyse  immediate  des 
roches :  C.  R.  Acad.  Sci.  Paris,  vol.  93,  pp.  318-321,  1881. 

■*J.  W.  Retgers,  Die  Bestiramung  des  specifischen  Gewichts  von  in  Wasser  16s- 
lichen  Salsen,  III.  Die  Darstellung  neuer  schwerer  Flussigkeiten :  Zeits.  phys. 
Chem.,  Bd.  11,  pp.  328-344,  1893. 

5  J.  W.  Retgers,  V'ersuche  zur  Darstellung  neuer  schwerer  Fliissigkeiten  zur 
Mineraltrennung :  Neues  Jahrb.  f.  Min.,  etc.,  Bd.  II,  pp.  183-195,  1890. 


SEPARATION  METHODS  331 

Rolirbach  ^  used  an  aqueous  solution  of  barium  mercuric  nitrate.  This  solu- 
tion is,  however,  difficult  to  prepare,  very  easily  decomposed,  and  very 
poisonous. 

For  a  brief  review  of  the  large  literature  on  heavy  liquids,  the  reader  is 
referred  to  Sullivan's  paper.-  Sullivan  described  the  properties  of  carbon- 
tetrabromide-carbon  tetrachloride,  stannic  bromide-carbon  tetrachloride,  stan- 
nic chloride,  stannic  iodide,  antimony  tribromide,  antimony  tricldoride,  thal- 
lous  silver  nitrate,  mercurous  nitrate,  thallous  mercurous  nitrate,  mercuric 
chloride-mercuric  iodide-antimony  trichloride,  all  in  addition  to  those  of 
bromoform,  acetylene  tetrabromide,  and  thallous  formate. 

Reasons  for  imperfect  separations  with  heavy  liquids.  Mineral  sepa- 
rations are  often  incomplete  or  imperfect  owning  to  convection  currents 
in  the  separating  fluid,  to  entrapment  of  grains  of  one  density  within  the 
bulk  of  the  fraction  of  opposite  density,  to  inclusions  with  the  mineral 
grain  or  to  attachment  to  other  mineral  grains  of  either  lower  or  higher 
density  or  to  alteration  products  which  cause  density  to  differ  from 
theoretical  value,  to  extreme  fineness  of  size  which  causes  the  mate- 
rial to  "ball-up"  or  settle  with  extreme  slowness,  or  to  smallness  of 
difference  in  density  between  grains  and  liquid.  Careful  control  of 
temperature  will  prevent  convection,  while  repeated  separations  or 
frequent  stirring  will  usually  overcome  the  normal  incomplete  sepa- 
ration. The  settling  rate  of  very  small  grains  and  grains  of  density 
near  that  of  the  fluid  may  be  accelerated  by  use  of  a  centrifuge. 
Separation  of  the  fine  materials  is  promoted  by  the  use  of  a  vacuum 
pump.^ 

Standardization  of  heavy  liquids.  A  simple  and  approximate  method 
of  checking  the  specific  gravity  of  a  liquid  is  to  put  a  small  crystal  of  a 
mineral  of  known  specific  gravity  in  the  liquid  and  determine  whether 
the  grain  floats  or  sinks.  In  this  way  one  can  readily  test  a  liquid  for  the 
separation  of  a  given  mineral,  as  quartz  for  example.  If  it  is  desired  to 
make  a  liquid  of  a  given  specific  gravity,  this  can  be  done  by  diluting  the 
heavy  liquid  with  the  proper  fluid  until  a  mineral  of  the  gravity  desired 
remains  suspended  in  the  solution. 

If  20  c.c.  or  any  other  convenient  volume  of  the  liquid  be  pipetted  into 
a  previously  weighed  container  and  then  weighed,  the  specific  gravity  of 
the  liquid  can  be  calculated  by  dividing  the  weight  found  (in  grams)  by 

1  Carl  Rohrbach,  Ueber  cine  neue  Fliissigkeit  von  hohem  specifischen  Gewicht, 
hohem  Brechungsexponcntcn  und  grosser  Dispersion :  Wildcin.  Ann.,  N .  F.,  20,  pp. 
169-174,  1883. 

-  John  D.  Sullivan,  Heavy  liquids  for  niineralogical  analyses :  U.  S.  Bur.  Mines, 
Technical  Paper  381,  pp.  5-9,  1927. 

3  R.  C.  Emmons,  On  gravity  separation:  Am.  Mineralogist,  vol.  15,  p.  536,  1930. 


2,?>^ 


SEDLMENTARY  PETROGRAPHY 


the  volume  (in  cubic  centimeters).  Goldschmidt,'^  using  this  principle, 
gives  the  details  for  careful  determination  of  the  specific  gravity  of  a 
liquid  which  will  just  suspend  a  mineral 
grain.  His  method  involves  the  accurate 
weighing  of  a  liquid  in  a  25-c.c.  measur- 
ing flask.  The  average  of  three  weighings 
is  assumed  to  be  the  correct  value  and  is 
used  in  calculations.  Greater  accuracy, 
however,  is  achieved  if  some  special  con- 
tainer, such  as  the  Sprengel  tube,-  is  used 
(Figure  149).  In  this  case  the  liquid  is 
sucked  directly  into  the  tube  and  weighed. 
Then,    since   the   weight   of    the   tube   is 

known  and  the  weight  of  the  tube  filled 
Fig.    149. — Sprengel   tube   for         ..       ..    ...    .  ^  1        u       1  i. 

weighing   a   precise   volume   of     ^^''^h    distilled   water   can   also   be   deter- 
liquid.  (After  SoUas.)  mined,  it  follows  that 


where  zv  is  the  weight  of  tube  and  water,  w'  is  the  weight  of  tube  and 
liquid  whose  specific  gravity  is  desired,  and  xv"  is  the  weight  of  tube 
alone. 


The  index  of  refraction  may  be  used  to  check  the  specific  gravity  of  a  fluid 
composed  of  mixtures  of  two  independent  liquids.  To  do  so,  however,  requires 
an  accurate  method  of  measuring  the  index  of  refraction  of  liquids,  preferably 
a  refractometer,  and  this  is  likely  to  be  a  more  circuitous  way  of  obtaining 
specific  gravity  than  some  other  method.  It  is  necessary  also  to  determine  the 
index  of  known  mixtures  of  the  liquids  (with  specific  gravities  of  the  mix- 
tures also  known)  so  that  a  curve  showing  the  relation  between  gravity  and 
index  can  be  constructed.  With  such  a  curve,  however,  it  is  possible  to  de- 
termine the  specific  gravity  of  a  combination  of  the  liquids  with  but  a  single 
drop  of  the  mixture.  Merwin  ^  has  published  the  data  for  a  solution  of  barium- 
mercuric  iodide  (Rohrbach's  solution).  Vassar  •*  has  prepared  a  table  showing 
in  like  manner  the  relation  between  the  index  of  refraction  and  the  specific 
gravity  of  a  solution  of  thallium  formate-malonate  (Clerici's  solution)  : 

1 V.  Goldschmidt,  Ueber  Verwendbarkeit  einer  Kaliumquecksilberjodidlosung 
bein  mineralogischen  und  petrographischen  Untersuchung :  Neiics  Jahrb.  f.  Min., 
etc.,  B.B.  I,  pp.  196-199,  1881. 

-  W.  J.  Sollas,  On  a  modification  of  Sprengel's  apparatus  for  determining  the 
specific  gravity  of  solids :  Proc.  Roy.  Dublin  Soc,  n.s.,  vol.  5,  pp.  623-625,  1886- 
1887. 

3  H.  E.  Merwin,  A  method  of  determining  the  density  of  minerals  by  means  of 
Rohrbach's  solution  having  a  standard  refractive  index:  Am.  Jour.  Sci.  (4)>  vol. 
32,  pp.  425-432,  1911- 

4  Helen  E.  Vassar,  op.  cH.,  p.  125. 


SEPARATION  METHODS 


333 


Indices  of  Refraction 

1.6761    

1.6296   

1.6154   

1-5990   

1.5815    

1-5693    

1-56^0    

1-5515    

1-5363    

1. 5156   

Measurements  at  19.5"  C. 


The  W'estphal  balance  is  perhaps  the  most  accurate  and  most  satis- 
factory- device  of  all,  though  its  use  involves  a  little  more  time  than 
tliat  of  the  hydrometer  described  below. ^  This  apparatus  is  essentially 


Specific  Gravity 

4076 

3695 

3-580 

3434 

3-280 

3-184 

3114 

3-024 

2.884 

2.692 


Fig.  150. — Westphal  balance.  Riders  to  be  suspended  on  beam  are  shown  at  left 
Largest  rider  marks  one  unit  when  placed  at  end  of  beam  and  marks  tenths  when 
placed  in  appropriate  notch.  Next  largest  rider  indicates  hundredths,  and  the  smallest 
rider  marks  thousandths.  Liquid  to  be  measured  is  placed  in  cylinder  at  right 

a  beam  balance  (Figure  150).  From  one  end  of  the  beam  is  suspended, 
by  a  platinum  thread,  a  weighed  sinker,  usually  a  short  glass  thermometer 
tube  (thus  enabling  one  to  make  a  determination  of  the  temperature  of 
the  liquid  in  question  at  the  same  time).  The  balance  is  made  so  that 

1  E.  Cohen,  Ueber  ein  einfache  Methode  das  specifische  Gewichteiner  Kalium- 
quecksilberjodidlosung  zu  bestimmen:  Ncucs  Jahrb.  f.  Min.,  etc.,  Bd.  II,  pp.  87-89, 
1883. 


334 


SEDIMENTARY  PETROGRAPHY 


when  the  sinker  is  immersed  in  water  the  beam  is  horizontal  as  indicated 
by  a  pointer  on  the  left  end.  If  the  liquid  is  heavier  than  water  the  sinker 
rises,  and  to  restore  balance  riders  must  be  hung  upon  the 
right  arm  of  the  balance,  which  is  graduated  into  ten  equal 
parts.  Since  there  are  three  sizes  of  riders,  each  division 
has  a  different  value  for  each  rider.  In  terms  of  specific 
gravity,  each  division  represents  o.i  for  the  largest  or  unit 
rider,  o.oi  for  the  rider  of  intermediate  size,  and  o.ooi  for 
the  smallest  rider.  The  specific  gravity  of  a 
liquid  is,  therefore,  given  by  summing  up 
the  readings  given  by  the  position  of  the 
three  kinds  of  riders  plus  one  (since  the 
beam  alone  is  in  balance  when  the  sinker  is 
immersed  in  water). 

A  simple  and  accurate  method  is  that  in- 
volving a  hydrometer  which  is  especially 
calibrated  for  heavy  fluids^  (Figure  151). 
This  gives  the  specific  gravity  by  direct  read- 
ing, accurately  to  two  decimal  places  (and 
approximate  to  three).  The  principal  objec- 
tion to  such  a  hydrometer  is  that  it  needs  a 
rather  large  quantity  (200  c.c.  or  more)  of 
liquid  to  float  it. 

Tester-  has  devised  a  hydrometer  for 
measuring  the  specific  gravity  of  a  heavy 
liquid.  The  hydrometer  requires  but  5  c.c.  liquid  and  has  a 
range  of  values  from  2.0  to  5.0.  It  is  made  of  glass  tubing 
and  consists  of  a  ball  float  (a)  above  which  is  a  glass  stem 
(b)  graduated  from  10  to  25  g.,  subdivided  into  tenths, 
and  below  which  is  a  liquid  chamber  (c)  (Figure  152).  A 
small  amount  of  mercury,  used  as  a  balancer,  is  sealed  in  a,  ball  float; 
the  ground  glass  stopper  (d)  which  fits  the  liquid  chamber.  stem'-^c^Tiq- 
To  use,  the  liquid  chamber  is  filled  to  a  5  c.c.  mark  and  uid  c ham- 
then  stoppered,  inverted,  and  immersed  in  a  column  of  dis-  ^^^'  ^"^  ^' 
^  ^  .  .ground 

tilled   water.   The  scale  is  read,   and  the  value  noted  is     glass    stop- 
divided  by  five  to  give  the  specific  gravity.  The  instrument     P^^- 
may  be  used  as  a  means  of  measuring  the  specific  gravity  of  a  solid. 


iSi.— 
Heavy- 
liquid  hy- 
drometer 
(Blake). 


1  Such  a  hydrometer,  20  cm.  long,  with  bulb  diameter  of  2  cm.,  and  range  from 
2.6  to  3.0  with  scale  graduated  every  o.oi,  can  be  obtained  from  Messrs.  T.  O. 
Black,  57,  Hatton  Garden,  London,  E.  C.  i. 

2  A.  C.  Tester,  A  convenient  hydrometer  for  determining  the  specific  gravity  of 
heavy  Hquids:  Science,  n.s.,  vol.  72,  pp.  130-131,  1931. 


SEPARATION  METHODS 


335 


Separation  Apparatus  for  use  with  Heavy  Liquids 

Gravity  separations  by  means  of  heavy  liquids  have  been  carried  out 
in  various  kinds  of  apparatus  varying  from  a  simple  evaporating  dish  on 
the  one  hand  to  the  elaborate  Penfield  ap- 
~  paratus   for  use  with   low-melting  solids 

on  the  other.  The  apparatus  should  be  in- 
expensive, not  too  fragile,  and  so  con- 
structed that  little  loss  occurs  with  vola- 
tile liquids.  It  should  permit  the  grains  to 
be  agitated  or  stirred,  should  be  so  con- 
structed that  the  mineral  crops  are  readily 
removed  from  the  apparatus,  and  should 
not  be  so  narrow  that  clogging  occurs 
during  separation.  Moreover,  the  separat- 
ing vessel  should  be  as  broad  as  possible 
so  that  the  floating  minerals  will  have  but 
a  slight  thickness,  should  have  no  abrupt 
reentrants  on  which  the 
grains  may  lodge,  and 
should  have  valves  with 
a  diameter  as  large  as  the 
outlet  tube. 

The  several  types  of 
separating  devices  are 
briefly  described.  Sim- 
plest of  all  is  a  simple 
filtering  funnel,  the  stem 
of  which  is  fitted  with  a 
short  length  of  rubber  tubing  and  pinch-cock  (Figure 
153).  A  watch-glass  may  be  used  to  cover  the  funnel 
to  reduce  evaporation  losses.  Several  workers  have  used 
funnels  of  this  type  but  have  modified  them  by  increas- 
ing the  steepness  of  the  funnel  walls  to  prevent  lodge- 
ment-of  grains  and  have  introduced  stop-cocks  into  the 
stem  ^  (Figure  154).  One  of  the  earliest  devices  was  such  a  funnel,  the 


Fig.     153. — Simple    apparatus 
for     heavy-liquid     separation. 

A,  funnel    with    heavy    liquid; 

B,  rubber  tube  and  pinch-cock; 

C,  funnel  fitted  with  filter-paper ; 

D,  bottle  to  collect  used  heavy 
liquid. 


F  r  G.  IS4-— 
Special  funnel 
for  heavy-liquid 
separation. 


1  Clarence  S.  Ross,  Methods  of  preparation  of  sedimentary  materials  for  study: 
Econ.  Geology,  vol.  21,  pp.  454-468,  1926.  Such  a  funnel  with  steep  walls  and  stop- 
cock is  obtainable  from  Emil  Greiner  of  Fulton  and  Cliff  Streets,  New  York.  N.  Y. 
L.  Van  Werveke,  Ueber  Regeneration  der  Kaliumquecksilberjodidlosung  und  iiber 
einen  einfachen  Apparat  zur  Trennung  mittelst  dieser  Losung :  Ncties  Jalirb.  f.  Min., 
etc.,  Bd.  2,  pp.  86-87,  1883. 


336 


SEDIMENTARY  PETROGRAPHY 


upper  part  of  which  was  cylindrical,  with  a  short  stem,  with  valve,  which 
fitted  by  a  ground  glass  joint  into  a  lower  bottle^  (Figure  155). 

Other  workers  have  used  bulb-  or,  better,  pear-shaped  separatory  fun- 
nels. Earliest  of  these  is  the  Harada  tube  ^  which  was  closed  at  the  upper 
end  by  a  tight-fitting  stopper  and  at  the  lower  end 
by  a  valve,  the  opening  in  which  was  of  the  same 
diameter  as  the  tube  in  which  it  is  located   (Figure 
156).   After  shaking  and  complete   separation,   the 
lower  end  is  placed  in  an  accessory  vessel  and  a  stop- 
cock opened.  Brogger  ^  built  a  similar  tube  but  added 
a  second   large   stop-cock  above   the   first    (Figure 
157).       Separations      are 
rarely  complete  after  one 


Fig.  155.— Apparatus 
of  Church. 


Fig. 
tube. 


156.  —  Harada 


Fig.   157. —  Brogger 
apparatus. 


settling  period,  but  with  the  Brogger  apparatus  it  is  possible  to  make  a 
clean  separation  by  first  closing  the  middle  valve — after  the  preliminary 
settling — and  then  inverting  the  apparatus,  allowing  a  second  separation 
to  take  place  in  both  portions,  then  slowly  returning  the  apparatus  to 
normal  position  and  reopening  the  middle  valve  so  that  the  separated 
portions  of  the  light  and  heavy  minerals  unite. 

Another  device  which  makes  such  repeated  separations  possible  is  that 


1  A.  H.  Church,  A  test  of  specific  gravity:  Mineral.  Mag.,  vol.  i,  pp.  237-238,  pi. 
8,  fig.  7,  1876-1877. 

-  K.  Oebbeke,  Beitrage  zur  Petrographie  der  Philippinen  und  der  Palau-Inseln : 
Ncues  Jahrb.  f.  Min.,  etc.,  B.B.  i,  pp.  45I-50I   (esp.  p.  457),  1881. 

3  W.  C.  Brogger,  Om  en  ny  Konstruktion  af  et  isolations-apparat  for  petro- 
grafiske  undersogelser :  Geol.  Forcn.  i.  Stockholm  Fork.,  vol.  7,  pp.  417-427, 
1884. 


SEPARATION  METHODS 


337 


devised  by  Laspeyres  ^  in  which  two  pear-shaped  vessels  are  connected 
by  a  stop-cock  and  at  each  end  are  closed  with  large  ground  glass  stop- 
pers (Figure  15S).  A  moditication  of  this  device  is  described  by  Hauen- 
schild  -  in  which  the  two  stop-cocks  are  modified  or  converted  into  two 
small  vessels  (Figure  159).  They  are  large  enough  so  that  when  they  are 
detached  tlie  heavy  liquid  and  mineral  grains  can  be  contained  therein. 
Wiilfung^  also  devised  an  apparatus  in  which  repeated  separations 
are  possible.  It  differs  from  the  others  in  that  it  is  a 
linked-shaped  affair  composed  of  two  branches — each  one 
a  curved  tube — which  arc  connected  to  each  other  at  both 


Fig.  1 58. — Laspeyres 
separating  vessel. 


Fig.  150. — Ilauoiis- 
cliild  apparatus  (after 
Cayeux ) . 


Fig.    lOo. — Wiilfuiig 
separation  tulx\ 


ends  by  two  valves  (Figure  160),  and  by  means  of  which  ilio  two  jx-u-ts 
can  be  shut  off.  To  make  repeated  separations  the  valves  are  closed  and 
heavy  liquid  and  minerals  are  introduced,  through  a  glass-stoppered 
opening,  into  one  half  only.  After  partial  separation  the  lower  valve  is 
opened  to  allow  the  heavy  grains  and  some  liquid  to  pass  into  the  other 
half.  The  two  halves  are  then  again  separated  by  closing  the  valves,  and. 
after  some  liquid  is  added  to  both  sides,  a  second  separation  takes  place 
on  each  side.  By  opening  the  lower  valves  and  tilting  the  apparatus,  the 
heavy  fractions  unite,  while  opening  the  upper  valve  permits  the  light 
portions  to  join. 


1  H.  Laspeyres,  Vorrichtung  zur  Scheidung  von  Mineralien  mittelst  sclnvcrer 
Losungen:  Zeits.  f.  Kryst.  sMin.,  vol.  27,  pp.  44-45,  1897. 

-  A.  Hauenschild,  Zcitschr.  f.  Baumatcrialcnkundc ,  Marz,  1898.  Description  and 
figure  in  Cayeux,  L'Etiidc  pctrographiqnc  dcs  rocltcs  scdimciitaircs.  Tcxtc  (1931), 
P-  59- 

3  E.  A.  Wiilfung.  Beitrag  zur  Kenntnis  des  Kryokonit :  Ncucs  Jalirb.  f.  Miii..  etc.. 
B.B.  7,  pp.  164-165,  1891. 


338 


SEDIMENTARY  PETROGRAPHY 


Smeeth^  developed  an  apparatus,  later  modified  by  Diller,-  in  which 
an  upper  pear-shaped  separatory  vessel  could  be  entirely  detached  from 
a  lower  vessel  and  base  of  candlestick  form  (Figure  i6i).  The  junction 
is  a  ground  glass  joint.  The  upper  vessel  could  be  closed  at  its  lower  end 
by  a  glass  stopper  on  the  end  of  a  glass  rod  which  extends  upward  and 
out  through  the  top  opening.  A  ground  glass  cap  in 
turn  closed  the  top  opening.  Luedecke  also  made  a 
separating  device  on  much  the  same  order  ^  (Figure 
162). 

Thoulet.-*  in  1879,  used  a  long,  narrow,  burette- 
like separating  tube,  the  outlet  stem  of  which,  how- 


FiG.  161.— Dil- 
ler modification  of 
S  m  e  e  t  h  appa- 
ratus. 


Fig.   162.- 
ratus. 


-Luedecke  a  p  p  a- 


F  I  G.  163. — Thoulet 
tube  (slightly  modi- 
fied). 


ever,  has  two  valves.  A  parallel  side  tube  of  smaller  diameter  was  joined 
to  the  main  apparatus  between  the  two  valves  (see  Figure  163).  Air 
forced  through  the  side  tube  and  up  into  the  main  separating  vessel  agi- 
tated the  hea\y  liquid.  The  upper  valve  is  then  closed  to  allow  quiet 
separation.  Afterwards  the  valve  is  opened  to  allow  the  heavy  minerals 


1  W.  F.  Smeeth,  An  apparatus  for  separating  the  mineral  constituents  of  rocks : 
Proc.  Roy.  Dublin  Soc,  vol.  6,  pp.  58-60,  1888. 

-  J.  S.  Diller,  The  Smeeth  separating  apparatus :  Science,  n.s.,  vol.  3,  pp.  857-858, 
1896. 

3  O.  Dreibrodt,  Trennungsapparat  nach  Prof.  Dr.  O.  Luedecke:  Centralbl.  f.  Min., 
etc.,  pp.  425-426,  191 1. 

*  J.  Thoulet,  Separation  mecanique  des  divers  elements  mineralogique  des  roches : 
Bull.  Soc.  Min.  France,  vol.  2,  pp.  17-24,  1879. 


SEPARATION  METHODS 


339 


to  drop  into  the  space  between  the  valves,  after  which  it  is  closed  and 
the  lower  valve  is  opened  to  release  the  heavy  crop.  Oebbeke  ^  simplified 
the  Thoulet  apparatus  but  owing  to  its  fragile  nature  and  difficulty  of 
cleaning  it  has  been  but  little  used. 

Eraser-  suggested  the  simplest  of  all  special  devices.  His  apparatus 
is  a  tube,  intermediate  between  a  V-  and  a  U-tube,  which  tapers  from 
one  end  to  the  other  (Figure  164).  The  tube  is  filled  two-thirds  full  of 
heavy  liquid  and  so  held  that  the  wider  limb  is  vertical  and  so  that  the 
liquid  rises  to  the  open  end  of  the  smaller  limb.  The  sand  is  introduced  at 
the  wide  end.  The  light  minerals  thus  accumulate  at  the  top  of  the  liquid 
in  this  wider  portion  while  the  heavy  minerals  settle  to  the  bend  in  the 


Fig.  164. — Fraser  tube.  A,  initial  position;  B,  inverted  position.  Slight  pressure  on 
cork  forces  out  drop  of  heavy  liquid  and  heavy  mineral  onto  slide. 


tube.  The  tube  should  now  be  inverted  in  such  a  way  that  the  heavier 
minerals  fall  into  the  narrow  tube  (B,  Eigure  164),  where  they  collect 
in  the  jet  and  may  be  expelled  on  a  glass  slide  for  observation  by  pushing 
in  the  cork.  The  apparatus  is  easily  made,  without  fragile  or  complicated 
parts,  uses  little  liquid,  and  is  inexpensive.  It  is  probable,  however,  that 
it  is  best  suited  to  rapid  qualitative  work  rather  than  complete  quanti- 
tative study. 

Woodford  ^  suggested  the  use  of  the  Spaeth  sedimentation  glass  for 
heavy  liquid  separation.  This  is  a  conical-shaped  glass  with  wide  base. 
Near  the  base  of  the  cone  is  a  large  stop-cock,  a  part  of  which  is  cut 
away  to  make  a  relatively  large  cup-shaped  hole  of  dimensions  contin- 
uous with  the  upper  part  of  the  vessel.  If  the  stop-cock  is  turned,  its 

^  K.  Oebbeke,  Beitrage  zur  Petrographie  der  Philippinen  und  der  Palau-Inseln : 
Ncucs  Jahrb.  j.  Miii..  etc.,  B.B.  I,  pp.  450-501  (esp.  p.  456),  1881. 

-  F.  J.  Fraser,  A  simple  apparatus  for  heavy  mineral  separation :  Econ.  Geology, 
vol.  23,  pp.  QQ-ioo,  1928. 

3  A.  O.  Woodford,  Metliods  for  heavy  mineral  investigations :  Econ.  Geology, 
vol.  20,  pp.  103-104,  1925. 


340 


SEDIMENTARY  PETROGIL\PHY 


Fig.  I  6  5. — 
Spaeth  sedimen- 
tation glass. 


content  of  hea^^,-  minerals  are  isolated  from  the  glass  above   (Figure 

165)- 

Penfield  ^  designed  an  apparatus  to  be  used  with  low-melting  solids. 
In  its  improved  form  (Figure  166)  it  consists  of  a 
tubular  separating  vessel  fitted,  by  means  of  a  ground 
glass  joint  at  its  lower  end,  to  a  small  hollow  glass  cap. 
A  hollow  stopper  connected  to  a  glass  tube  which  ex- 
tends the  entire  length  of  the  upper  tube  shuts  off  this 
upper  tube  from  the  lower  hollow  cap.  The  whole  ap- 
paratus is  put  in  a  large  test-tube  in  a  beaker  of  hot 
%\-ater.  An  air  stream  through  the  tubular  stopper  agi- 
tates the  melt  and  contained  mineral  grains.  After  sepa- 
ration is  complete  the  stopper  is  inserted  and  isolates  the 
heavier  minerals  in  the  lower  hollow  cap.  from  which 
they  are  then  removed.  The  double  nitrate 
of  silver  and  thallium  is  used  ^\^th  this  de- 
vice. Since,  however,  this  salt  has  a  specific ^^ 
gra\-it\-  of  4.5  at  its  melting  point  of  75°  C,  while  Qerici's 
solution  has  a  gravit>^  of  4.25  at  room  temperature  (20° 
C. ) .  it  seems  to  the  authors  unnecessary'  to  use  the  former 
and  the  complicated  Penfield  separator  except  in  rare  cases. 

Hea\'\--liquid  Fr-^.ctioxatiox  with  Centrifuge 

It  may  be  necessan,-,  in  the  case  of  fine  sands  and  silts, 
to  use  the  centrifuge  to  accelerate  the  separation.  An  inex- 
pensive but  permanent  centrifuge  tube  which  makes  a 
rapid  and  efficient  separation  of  grains  smaller  than  0.5 
mm.  is  described  by  Taylor.-  It  consists  of  a  tube  120  mm. 
long  with  a  cjlindrical  upper  part  having  a  diameter  of  17 
mm.  and  a  similar  lower  part  with  a  diameter  of  9  mm.  A 
cork-tipped  plunger,  slightly  larger  than  the  constricted 
lower  part  of  the  tube,  is  inserted  in  the  upper  wider  part 
to  the  point  of  narrowing,  where  it  effectively  separates  the 
two  parts  of  the  tube.  A  cork  is  pro\nded  for  the  opening 
of  the  centrifuge  tube,  and  this  fits  tightly  on  the  stem  of  the  plunger 
so  that  the  latter  can  be  fixed  at  any  desired  position  (Figure  167).  The 


Fig.  166.— 
Penfield 
separa- 
tion appa- 
ratus  for 
heaivy   melts. 


1  S.  L.  Penfield  and  D.  A  Kreider,  On  the  separation  of  minerals  of  high  specific 
gravity  by  means  of  the  fused  double  nitrate  of  silver  and  thallium :  Am.  Jour.  Set., 
vol.  48,  pp.  143-144.  1894.  S.  L.  Penfield.  On  some  devices  for  the  separation  of  min- 
erals of  high  specific  gravity:  Am.  Jour.  Sci.,  vol.  50.  pp.  446-448.  1895. 

-  G.  L.  Taylor,  A  centrifuge  tube  for  heaw  mineral  separations  :  Jour.  Scd.  Petrol- 
ogy. voL  3,  pp-  45-46,  1933- 


SEPARATION  METHODS 


341 


heavy  minerals  collect  in  the  lower  constricted  portion  during  centrifug- 
ing.  They  are  stoppered  in  the  plunger  while  the  light  minerals  are 
washed  out  of  the  upper  part  of  the  tube. 

Brown  ^  has  also  described  the  use  of  the  centrifuge  in  separation  of 
heavy  minerals.  Brown  used  an  ordinary  centrifuge  tube  and  removed 


Fig.  167. — Taylor 
centrifuge  tube. 


Fig.  168.— Schro- 
der centrifuge  tube. 


Fig.  169. — Kunit/. 
centrifuge  tube. 


the  heavy  minerals  by  means  of  an  ordinary  pipette  iittcd  with  a  rubber 
bulb. 

Miiller  -  and  Schroeder  ^  both  apparently  used  the  centrifuge  in  min- 
eral separations,  Miiller  used  a  modified  sling  tube,  anticipating  in 
principle  that  later  described  by  Taylor,  while  Schroeder  used  a  tube 
with  valve  in  the  middle,  which  upon  closing  separated  the  light  and  the 
heavy  portions  (Figure  168).  Kunitz"*  also  used  a  sling  tube  modified 
with  a  large  stop-cock  at  the  lower  end  for  separation  purposes.  The 
stop-cock,  however,  had  only  a  cup-like  depression  in  which  the  heavier 
constituents  collected.  Turning  the  valve  180°  permitted  the  escape  of 
the  heavy  crop  at  the  lower  end  of  the  tube  (Figure  169). 


1  Irvin  C.  Brown,  A  method  for  the  separation  of  heavy  minerals  of  fine  soil : 
Jour.  Palcon.,  vol.  3,  pp.  412-414,  1929. 

-'  Heinrich  Miiller,  Ncues  Zcntrifugenglas  zum  quantitativen  Trenncn  von  Kor- 
nigcn  und  pulverigcn  Gemengen  verschiedencn  spez.  Gewichts  mit  Hilfe  von 
schwercn  Losungen :  Mitt.  Min.-Gcol.  Staatsinsfitut.  llamburt/,  H.  11,  pp.  1-6,  1929; 
Uber  cin  angeandertes  Zcntrifugenglas  zum  Trennen  nach  dem  spez.  Gewicht: 
Ccntralbl.  f.  Miu..  etc..  Abt.  A,  pp.  QO-91,  1932. 

3  Fritz  Schroeder,  Schcidctrichterzuni  Einsetzen  in  die  Zentrifugc  bcim  Trennen 
von  Mineralgcmischcn  mit  schwercn  Flussigkeiten :  Ccntralbl.  /.  Min.,  etc.,  Abt.  A, 
pp.  38-46,  1930. 

•*  W.  Kunitz,  Fine  Schnellmethode  der  gravimetrischen  Phasscnanalyse  mittels 
der  Zentrifuge:  Ccntralbl.  f.  Min.,  etc.,  Abt.  A,  pp.  225-232,  1931. 


342 


SEDIMENTARY  PETROGRAPHY 


The  centrifuge  effects  a  very  complete  separation  between  minerals 
differing  but  slightly  in  specific  gravity.  Dolomite  and  calcite^  may  be 
quantitatively  separated  by  its  use  as  also  may  anhydrite 
and  rock  salt.- 

None  of  the  various  devices  for  heavy  liquid  separation 
of  minerals  in  the  centrifuge  is  wholly  satisfactory.  The 
stop-cocks  will  not  hold  heavy  liquids  when  subject  to  cen- 
trifugal force.  It  is  necessary,  therefore,  to  employ  only 
those  with  sealed  ends.  With  these,  however,  it  is  difficult 
to  separate  the  light  and  heavy  mineral  crops.  To  overcome 
these  difficulties  Berg  ^  devised  a  pipette  consisting  of  (i) 
a  small  glass  tube  (D)  i6  cm.  long  and  a  2-mm.  bore  but 
tapered  to  a  1^:4 -mm.  opening  at  the  lower  end,  (2)  a  trap 
(C)  which  collects  the  heavy  minerals  carried  through  the 
small  tube  by  the  rising  heavy  liquid,  (3)  a  stop-cock  (B) 
which  permits  drainage  of  the  trapped  minerals  into  a 
filter-paper,  and  (4)  a  rubber  bulb  which  can  be  closed  off 
from  the  rest  of  the  pipette  by  a  stop-cock  (E)  (see  Fig- 
ure 170). 

Berg  used  about  i  g.  of  sediment  in  an  ordinary  15-ml. 
centrifuge  tube  with  about  15  ml.  of  heavy  liquid.  After 
thorough  shaking  to  disperse  the  sediments,  the  material  is 
centrifuged  until  the  liquid  is  clear.  To  remove  the  heavies, 
the  rubber  bulb  is  depressed,  the  bulb  stop-cock  is  closed, 
as  is  also  the  lower  stop-cock,  and  the  small  glass  tube  (D) 
is  lowered  to  the  bottom  of  the  centrifuge  tube.  The  bulb 
stop-cock  is  then  opened  and  the  heavy  liquid  and  the 
heavy  minerals  rise  into  the  trap.  The  bulb  stop-cock  is 
then  closed  and  the  pipette  is  withdrawn  until  it  is  just  out  of  the  liquid, 
when  the  bulb  stop-cock  is  opened  just  enough  to  clear  the  small  pipette 
tube  of  liquid.  The  material  in  the  trap  is  then  released  onto  a  filter- 
paper  by  opening  the  lower  stop-cock.  Or,  if  desired,  the  material  may 
be  released  into  a  second  centrifuge  tube  to  effect  a  second  settling  to 
insure  complete  separation  of  the  heavy  mineral  fraction. 

1  C.  B.  Claypool  and  W.  V.  Howard,  Method  of  examining  calcareous  well  cut- 
tings:  Bull.  A.  A.  P.  G.,  vol.  12,  pp.  1147-1152,  1928. 

-  F.  von  Wolff,  Die  Trennung  fester  Phasen  durch  die  Zentrifuge :  Centralhl.  f. 
Mill.,  etc.,  Abt.  A,  pp.  449-452,  1927.  Carl  W.  Correns,  Ueber  zwei  neue  einfache 
Verfahren  fiir  das  Zentrifugieren  mit  schweren  Losungen:  Centralhl.  f.  Miii.,  etc., 
Abt.  A,  pp.  204-207,  1933. 

3  Ernest  Berg,  A  method  for  the  mineralogical  fractionation  of  sediments  by 
means  of  heavy  liquids  and  the  centrifuge:  Jour.  Sed.  Petrology,  vol.  7,  pp.  Si-54. 
1937- 


SEPARATION  METHODS  343 

Procedure  for  Separ.\tixg  the  Minerals  of  a  Clastic  Sediment 
BY  Means  of  Bromoform 

As  may  be  seen  by  referring  to  the  preceding  section,  there  are  many 
types  of  apparatus  designed  for  separating  the  heavy  minerals  from 
the  light,  using  some  hea\y  liquid.  Nevertlieless,  the  set-up  shown  in 
Figure  153  is  about  as  satisfactory  as  any,  except  for  very  fine  material, 
since  it  is  readily  constructed  from  materials  on  hand  in  any  laboratory'. 

Bromoform  of  known  specific  gravity  is  first  poured  into  the  upper 
funnel,  after  which  a  weighed  amount  of  dry  sediment  (prepared  as 
previously  described)  is  placed  in  it  and  thoroughly  stirred ;  and  the 
heavier  minerals  can  then  be  drawn  oft  by  opening  the  pinch-cock.  If 
the  proportion  of  heavy  minerals  is  small,  as  is  the  case  with  ordinary- 
unconcentrated  sediments,  a  large  amount  of  materials,  say  about  50  g.. 
can  be  added  at  once.  When  the  amount  of  heavier  minerals  is  large,  as 
in  concentrated  sediments  (and  in  many  crushed  igneous  rocks),  the 
mixture  should  be  added  in  several  portions,  the  pinch-cock  being  opened 
each  time,  to  prevent  the  accumulation  of  too  much  material  in  the  neck 
of  the  funnel.  Cover  the  funnel  with  a  watch-glass  to  reduce  evaporation 
losses. 

It  is  convenient  to  have  the  "used"  hea\y-liquid  bottle  at  hand,  pro- 
vided with  a  filtering  funnel  with  filter-paper,  and  to  allow  the  heavy 
minerals  to  drop  on  the  filter-paper.  After  the  heavy  liquid  has  drained 
off,  the  filter-paper  can  be  detached  and  opened,  and  the  hea\y  grains 
washed  off  by  placing  the  paper  face  downwards  in  a  watch-glass  or 
porcelain  dish  containing  alcohol  (or  acetone).  The  washings  thus  ob- 
tained should  be  put  in  the  bottle  marked  "Bromoform  Washings.'' 
(Ultimately,  the  amount  of  hea\T  liquid  in  the  washings  will  become 
considerable  and  it  will  be  recovered.)  After  the  removal  of  the  '■hea^w" 
minerals,  the  bromoform  in  which  the  light  minerals  are  floating  can  be 
run  off  through  the  pinch-cock  and  through  a  clean  filter-paper  into  the 
"used"  heavy-liquid  bottle.  After  replacing  the  "used"'  hea\y-liquid 
bottle  with  that  marked  "Bromoform  Washings,"  alcohol  may  be  used 
to  wash  down  the  light  minerals  remaining  in  the  upper  funnel. 

Weigh  the  "lights"  and  "heavies"  and  put  each  in  a  vial  and  label.  A 
permanent  mount  in  Canada  balsam  should  be  made  of  the  light  minerals 
and  a  mount  of  the  heavy  minerals  should  also  be  made  either  in  Canada 
balsam  or  piperine  (see  instructions  on  same). 

Procedure  for  recovery  of  bromoform.  Add  the  washings  containing 
bromoform  to  a  large  volume  (i  gal.)  of  cold  water  in  a  large  stoppered 
bottle.  Shake  vigorously.  Decant  most  of  the  water.  Repeat  twice  more. 
After  the  last  decantation  pour  the  remaining  water  and  separated 
bromoform  into  a  separatory  funnel.  Draw  the  bromoform  otf  from 
below  and  run  into  a  funnel  fitted  with  several  thicknesses  of  filter-paper. 
Collect  the  bromoform  filtrate.  If  it  is  not  clear  run  it  through  a  second 
funnel  and  filter-paper.  (The  paper  absorbs  any  dispersed  water  and  any 


344  SEDIMENTARY  PETROGR.\PHY 

wax  that  may  be  formed.)  Test  the  bromoform  for  specific  gravity.  Put 
it  in  a  brown  bottle  and  label  "Used  Bromoform." 


5  E  P  .A.  R  .\  T  I  O  X     ON     THE     BASIS     OF      M  .A.  G  X  E  T  I  C 
PERMEABILITY 

Minerals  can  be  classed  as  paramagnetic  when  the  lines  of  magnetic 
force  pass  through  them  more  easily  than  through  air  (air  i),  and 
diamagtietic  when  they  pass  less  readily.  Gammon  paramagnetic  sub- 
stances are:  iron,  nickel,  cobalt,  manganese,  chromium,  cerium,  po- 
tassium, platinum,  and  aluminum.  Common  dimagnetic  substances  are: 
bismuth,  antimony,  zinc,  silver,  copper,  water,  sulphur,  phosphorus, 
boron,  the  halogens,  silica,  etc.  The  grouping  of  the  elements  in  the 
compound  will  influence  its  permeabilit}-.  Both  paramagnetic  and  di- 
magnetic elements  may  be  present,  but  the  compound  will  not  necessarily 
be  either  one  or  the  other.  When  iron  is  present,  the  compound  is  usually 
magnetic.  Ferrous  iron  is  usually  more  effective  than  ferric  iron.  Mag- 
netism of  a  particular  grain  may  be  due  to  the  presence  of  magnetic 
inclusions. 

Because  of  the  xarying  magnetic  permeabilitj-  of  minerals  it  is  possible 
to  make  a  separation  of  one  kind  from  another.  Either  a  permanent 
magnet  or  an  electromagnet  may  be  used.  An  ordinar)-  permanent  magnet 
of  considerable  strength  is  rather  large;  it  requires  remagnetizing  at 
intervals ;  and  it  is  difficult  to  detach  grains  attached  to  it.  On  the  other 
hand,  it  is  relatively  cheap  and  can,  if  fitted  with  special  pole  pieces,  be 
used  for  separating  weakly  magnetic  material.  The  separation  of  the 
weakly  magnetic  materials  is,  however,  better  achie^-ed  by  using  an 
electromagnet,  which,  if  suitably  constructed,  is  much  stronger  than  a 
permanent  magnet  of  the  same  size.  The  electromagnet,  moreover,  exerts 
no  attractive  force  when  the  current  is  shut  off,  and  hence  recovery  of 
the  magnetic  crop  is  rendered  eas)% 

Pennaiient  magnets.  Crook^  recommends  an  instrument  of  U-shape, 
the  limbs  of  which  are  6  in.  long,  made  of  a  steel  bar  about  i  in.  wide 
and  y2  in.  thick.  (See  Figure  i/i.)  Two  adjustable  pole  pieces  of  soft 
iron  fit  against  the  smooth  free  end  of  the  limbs  to  which  they  are  se- 
cured by  binding  screws.  These  pole  pieces  are  slotted  so  that  the  gap 
between  them  can  be  varied.  The  free  ends  of  the  pole  pieces  taper 
gradually  and  are  gently  curved  or  bent  downwards.^ 

1  T.  Crook,  A  simple  form  of  permanent  magnet  suitable  for  the  separation  of 
weakly  magnetic  minerals :  Geol.  Magazine,  vol.  5,  pp.  560-561,  1908. 

2  The  L-shaped  pole  pieces  used  by  some  investigators  are  unsatisfactory  and 
result  in  loss  of  magnetic  intensity.  The  bend  should  be  gradual,  not  right-angled. 


SEPARATION  METHODS 


345 


Smithson  ^  developed  .1  simple  metluxl  for  ohservin.c^  the  mac^netic  properties 
of  single  mineral  gi'^ii"^-  -^  piece  of  cardboard,  about  i-}^  by  i  J/l  in.,  is  bent 
over  at  one  end  to  j?ive  a  scjuare  1J/2  by  1^2  in.  Two  darning  needles  are 
heated  to  redness  and  cooled  slowly.  These  are  then  forced  through  the  fold 
in  the  card  about  ^  in.  apart  and  fixed  to  the  card  with  seccotine  so  that  their 
points  are  close  together.