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/ Ex Libris (^^9* II "^o^e Vo^w. -^ ? Oil v*\^<l ■ t \ ^ 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 /■ QC / / I I An J f^ 1 / ^^ 1 / ^cT / ^^.^ / y / ^ / n ' . .-■* a;^ U'-^^ ^-..-/^^ 1 1 'f i i .01 .. / J ^ 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 ^ S / ird edi Qua n tile / / / / ' T rst Qua rtile .2 ai 08 .06 .04 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. 100 075 UJ => 2 Li. -.50 r / 'HI iL>7^ \ / \\\.\y^ / \ iy^ ' .^1 1 Jit i 1^' 10 5 3 2 I .5 .2 2 .1 .05 .02 .01 .005 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 30] ^,1^ m ■^ vo ■* 00 (N 0 "oiO <N 00 v3" s ^ > -+ IX p) 0 N 0 q q ^-^ ,^ 0 ,^^ -^ •+ tx ON <o On Os On tx ^ :o 0 2> q 8 q S 'w' •-~ 0 \ vo VO 00 1 lO ro ro 0 IX q q q "^ 0 -^ 0 1- 0 0^ ts^ 2 ^ co 10 VO (N q X: -T3 ^ ^ ^ 0 ^ q^ ^ 00 qV ~ q q 0 10 <N '^1 <^ hH fO 1-^ 1 N "q" < Tl" Tf 0 On ^ ■^ > CO 10 vd ^ II (N l-l 0< "a ^^ , — ^ •^ 2 ^- to tx ON ro V -^ — ro In, VO a, ^ 5 VO m ■^ — ■ ti? t-^ tN. tx 0) as 0) 00 a;^-w s M '5 ^' CB< 0 1 '"' <N fO "C ■-S C 0 'o* ^ a, 0 c cx r; a ct3 bX) bi) be 1 bn c bxi IS G -'t! c/l S •v 3 u c rt 0 D <u f ) n! 1-c Ifl u rt c i ID JJ u c H-. M-( tC c ° ° -g O V. u bjo — •;: w oj rt 4j -g oi ^ ^ S S "C •r> E £ A 3 i> S .2 .2 § o -^ Gi-O -O G > !o II II II II II II CX, -y"Q ^"> ^ .HJ tx ol 00 C X c F . d 3 0 0 0 0 ^ H B G u r; UH (U C bjo 0 (L) c -C 3 M 03 01 •^ -" G C -C o " fe <u t« (/I O <5 OJ >^ <^ ^ g"S S 3 ::; •- 3 "^ J! G -- 3 I § s 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.