Introduction to

Clastic Sedimentology

(Notes for a University level, second year, half-credit course in clastic sedimentology)

by

R.J. CheelDepartment of Earth Sciences

Brock UniversitySt. Catharines, Ontario, Canada L2S 3A1

rcheel@brocku.cahttp://www.brocku.ca/sedimentology

© 2005 R.J. Cheel

Acknowledgements

The author thanks the following publishers for granting permission to reproduce figures for which theyhold copyright:

Prentice Hall Inc., Englewood Cliffs, New Jersey, figures 2-2, 2-19, 2-14, 2-21,4-17, 4-20, 5-15, 5-17, 5-5;SEPM (Society for Sedimentary Geology), Tulsa, Oklahoma, figures 2-15, 2-16, 2-17, 5-6, 5-12, 5-13, 5-14,6-7, 6-8, 6-9, 6-10, 6-12, 6-17, 6-18; Academic Press, New York, figure 2-31A; Springer-Verlag, New York,figures 2-6, 2-39, 3-1, and table 3-5; The University of Chicago Press, figure 2-20; Chapman and Hall, U.K.,figures 2-4 and 5-24. All permissions were granted free of charge.

Mike Lozon (Department of Earth Sciences, Brock University) is thanked for preparing several of the figures.

Table of Contents

Chapter 1. Introduction 5

Why study clastic sedimentology? 5About this book 6Comprehensive sedimentology textbooks 7

Chapter 2. Grain Texture 8

Introduction 8Grain Size 8

Volume 8Linear dimensions 9

Direct measurement 9Sieving 10

Settling velocity 10Stoke's Law of Settling 11

Grade scales 14Displaying grain size data 16Describing grain size distributions 18

Median 18Mean 19Sorting coefficient 19Skewness 20Kurtosis 20

Paleoenvironmental implications 20Why measure grain size? 22

Grain Shape 22Roundness 22

Wadell 22Dobkins and Folk 22Power's visual comparison chart 22

Sphericity 23Wadell 23Sneed and Folk 24Riley 24

Clast form 24Significance of grain shape 25

Source rock 25Transport 25

Porosity and Permeability 28Porosity 28

Controls on porosity 28Packing density 28Grain size 29Sorting 30Post-burial processes 30

Compaction 30Cementation 30Clay formation 30Solution 31

Permeability (Darcy's Law) 31Controls on permeability 34

1

Porosity and packing 34Grain size and sorting 34Post-burial processes 35

Directional variation in permeability 36Grain Orientation 36

Introduction 36Measuring grain orientation 36Types of grain fabric 37

Isotropic fabric 37Anisotropic fabric 37

a(t)b(i) 38a(p)a(i) 38Complex anisotropic fabric 38

The problem with measuring grain orientation on thin sections 39Displaying directional data 39Statistical treatment of directional data 42Interpretation of grain orientation 44

Chapter 3. Classification of terrigenous clastic rocks 47

A fundamental classification of sediments 47Terrigenous clastic sediments 49

Classification of sandstone 49Basis of the classification 49Genetic implications 52Level of classification 54

Classification of rudite 54Classification of lutite 54

Chapter 4. Unidirectional fluid flow and sediment transport 55

Introduction 55Unidirectional fluid flows 55

Flow between two plates 55Fluid gravity flows 56

Classification of fluid gravity flows 58Shear stress and velocity in turbulent flows 59Structure of turbulent flows 61

Organized structure of turbulent flows 63Sediment Transport 66

Modes of transport 66Quantitative interpretation of grain size curves 67

Threshold of movement 67Threshold of grain suspension 71

Chapter 5. Bedforms and stratification under unidirectional flows 74

Introduction 74Bedforms under unidirectional flows 74

Terminology 74The sequence of bedforms 75Bedform stability fields 85

Cross-stratification formed by bedforms under unidirectional flows 88Terminology 88

2

Origin of cross-stratification 90Cross-stratifcation produced by asymmetrical bedforms 95

Ripple cross-lamination 96Cross-stratification formed by dunes 97

Upper plane bed horizontal lamination 99In-phase wave stratification 101

Chapter 6. Flow, bedforms and stratification under oscillatoryand combined flows 102

Introduction 102Characteristics of gravity waves 102Bedforms and stratification under purely oscillatory currents 105

Initiation of motion 105Bedforms under waves 106Stratification formed by oscillatory currents 109

Bedforms and stratification under combined flows 113The enigma of hummocky cross-stratification 114

HCS - description and associations 116Characteristics of HCS 116

Grain size 116Morphology and geometry 116

HCS associations 118Discrete HCS sandstone 118Amalgamated HCS sandstone 119

The HCS debate 119Experimental evidence 121Evidence based on grain fabric 121

The origin of HCS? 122Conclusion 124

Appendix 1. 125

Appendix 2. 127

References 129

3

Metamorphic27.4%

Igneous64.7%

Sed

imen

tary

7.9%

Relative abundance of rocks inthe earth's crust

Igneous & Metamorphic

34%

Shale35%

Sandstone14.5%

Limestone16.5%

Relative abundance of rocksat the earth's surface

Figure 1-1. Relative abundance of rock types in the Earth's crust and at the Earth's surface.

5

CHAPTER 1. INTRODUCTION TO CLASTIC SEDIMENTOLOGY

Clastic sedimentology is the branch of geology that studies sediment and sedimentary rocks that are madeup of particles that are the solid products of weathering at or near the Earth’s surface. Thus, clastic sedimentologyis concerned with gravel, sand and mud and the rocks that form by the induration (formation into rocks) of theseparticulate materials (rudites, sandstones and shales; see Chapter 3). The goal of this book is to introduce theterminology and fundamental concepts that are necessary for the description and interpretation of sediment andsedimentary rocks.

WHY STUDY CLASTIC SEDIMENTOLOGY?

There are at least two good reasons for studying clastic sedimentology. The first is because humans, andmost other species on the planet, interact with the Earth largely at its surface. Sedimentary rocks make up only 7.9%of the total crust of the Earth which is dominated by igneous and metamorphic rocks (Fig. 1-1). However, the surfaceof the Earth is dominated by sedimentary rocks and almost 50% of that surface is made up of clastic sedimentaryrocks (predominantly shale and sandstone). Humans are not uniformly distributed over the Earth’s surface andif we were to further consider the proportion of the human population that lives immediately on clastic sedimentswe would find that almost all of us interact with the Earth’s surface through a cover of clastic sediments and/orsedimentary rocks. We interact with this sedimentary surface in a variety of ways. We grow food within it and raiselivestock on it so that it is crucial to global food requirements. We build our homes on it and take water and otherresources from it. At the same time we hide our garbage in it and we modify its physical and chemical propertiesin such a way as to render it unsuitable for many of our needs. Thus, given our uses and abuses of the Earth's surfaceit is particularly important that we understand the various properties of sediments and have systematic methodsof describing these properties.

A second reason for studying clastic, and all other, sedimentary rocks is because they preserve the recordof changing environmental conditions at or near the Earth’s surface over almost the whole of geologic time. Allsediment and sedimentary rocks were deposited at the Earth’s surface, either in the oceans or on the continents.As such, these deposits were influenced by processes that were acting on the Earth’s surface, in their environmentsof deposition. A large part of clastic sedimentology is devoted to the development of criteria for the recognitionof the action of various processes on sediments in their environment of deposition. By developing tools for therecognition of the signature of these processes in a sediment we can unravel the history of environmental changethat is preserved in the stratigraphic sequence of rocks that has been laid down over geologic time.

About these notes

These notes were first compiled in 1992 from the author's lecture notes that were prepared over six yearsof teaching Clastic Sedimentology at the Junior Undergraduate level, first at Brandon University and later at BrockUniversity, and are meant to provide an inexpensive "text" to support this half-credit second year course.Sedimentologists will notice that these notes contain material similar to that in some of the "classic" sedimentologytextbooks but also includes much of my own bias and understanding. I was fortunate enough to have been (andwill likely always be) a student of Gerry Middleton at McMaster University and a significant proportion of Chapters2, 3 and 4 are derived from an understanding (or perhaps misunderstanding in some cases) of the great experiencesthat I have had with one of the father's of modern sedimentology. At McMaster I was also provided the opportunityto learn much from Roger Walker, another Canadian sedimentologist of significant stature. Despite theseopportunities, any shortcomings within these notes reflect my own limitations and are certainly despite the inputof these two great educators.

These notes do not aim to cover all of the important aspects of clastic sedimentology but only those that the authorhas decided to stress in this one semester, introductory course. A list of more comprehensive texts is given at theend of this introductory chapter. Some of the examples and figures in these notes were modified from these textsand they will provide a more detailed treatment of several topics that are covered in this course. These notes areexpected to evolve with time as new sections are added and old ones dropped, although sections that are removedfrom the course will remain in the notes, provided that the cost to students remains reasonable. In an earlier addtionof these notes I added several colour plate. Unfortunately these plates resulted in a doubling of the cost of thenotes. For that reason I have not included the plates here.

In 2005 I began to put Power Point lecture presentations that I created for the course onto the World WideWEB. I subsequently added the full course notes to the Web in the hope that they would provide a resource forstudents at other Universities. These notes, and the Power Point presentations are available free of charge to anyonewho wants to download them; they are available at www.brocku.ca/sedimentology. In some cases the notes maybe downloaded and provided to students by a third party. My condition for such downloading and reproductionis that under no circumstances will a profit be added to the cost of the notes to students, other than a charge ofup to 20% of reproduction costs for distribution by a University book store or similar service that prints the notesfor sale to students. There will be no charge that incurs a profit to any other person or group, including myself.The reason for these notes is to provide an inexpensive but useful resource for students. If your school would liketo reproduce these notes for sale to students I ask that I be approached for permission to do so. Please direct suchrequests to me at rcheel@brocku.ca with the subject line "SedNotes Request" and include the per unit reproductioncosts and the proposed sale price. I will respond to such requests promptly.

The notes begin with a section on the most fundamental properties of clastic sediments, those propertiesthat collectively make up the "texture" of a sediment. The next section reviews the criteria for classifying sedimentand sedimentary rocks, criteria that are commonly based on the texture of the rocks. The remainder of the notesfocus on the behaviour of sediments in response to processes in subaqueous settings, the most common settingsin which sediments are deposited. This begins with an examination of some of the important characteristics ofunidirectional fluid motion (like the currents in a river) and the manner in which sediment is moved by such fluidmotion. This is followed by a section that examines the bulk response of a sediment to unidirectional fluid motion(i.e., bedforms) and the criteria for interpreting hydraulic condition on the basis of the internal structure of sedimentsand sedimentary rocks (i.e., internal stratification and cross-stratification). The final chapter briefly considers allof these aspects of fluid motion and sediment response but for currents that reverse in direction over relatively shortperiods (seconds to tens of seconds) and are generated by water surface waves that are common in many marineand lacustrine environments.

6

Comprehensive sedimentology textbooks

ALLEN, J.R.L., 1985, Principles of physical sedimentology. George Allen and Unwin, Boston, 272 p.

ALLEN, J.R.L., 1985, Experiments in physical sedimentology . George Allen and Unwin, Boston, 63 p.

ALLEN, J.R.L., 1982, Sedimentary structures: their character and physical basis (Two volumes) Elsevier, New York,593 p. (v. 1) and 663 p. (v. 2).

ALLEN, J.R.L., 1970, Physical processes of sedimentation. An introduction. George Allen and Unwin, London,248 p.

BLATT, H., MIDDLETON, G.V., AND MURRAY, R., 1980, Origin of sedimentary rocks. (Second Edition) Prentice-Hall,New Jersey, 782 p.

Boggs Jr., Sam, 2001, Principles of Sedimentology and Stratigraphy (3rd Edition). Prentice Hall, New Jersey, 770p.

DAVIS, R.A., 1983, Depositional systems: a genetic approach to sedimentary geology. Prentice-Hall, Toronto, 669p.

HARMS, J.C., SOUTHARD, J.B., AND WALKER, R.G., 1982, Structures and sequences in clastic rocks. Society ofEconomic Mineralogists and Paleotologists, Short Course Number 9, 249 p.

HSÜ, K.J, 1989, Physical Principles of Sedimentology: a readable textbook for beginners and experts. Springer-Verlag, New York, 233 p.

LEEDER, M.R., 1982, Sedimentology: process and product. George Allen and Unwin, London, 344 p.

MIDDLETON, G.V. AND SOUTHARD, J.B., 1984, Mechanics of sediment movement. (Second Edition) Society ofEconomic Mineralogists and Paleontologists, Short Course Number 3, 394 p.

PETTIJOHN, F.J., 1975, Sedimentary rocks. (3rd Edition). Harper and Row, New York, 628 p.

PETTIJOHN, F.J., Potter, P.E., and Siever, R., 1973, Sand and sandstone. Springer-Verlag, New York, 618 p.

POTTER, P.E., MAYNARD, J.B., AND PRYOR, W.A., 1980, Sedimentology of shale. Springer-Verlag, New York, 306 p.

REINECK, H. -E. AND SINGH, I.B., 1980, Depositional sedimentary environments, with reference to terrigenousclastics. (Second Edition). Springer-Verlag, New York, 549 p.

SELLEY, R.C., 1982, An introduction to sedimentology. Academic Press, New York, 417 p.

7

8

CHAPTER 2. GRAIN TEXTURE

INTRODUCTION

All clastic sediment is made up of discrete particles termed grains or clasts. Thus, any description of a clasticsediment must describe the particles, including their individual and bulk properties. Such properties collectivelydefine the texture of a sediment or sedimentary rock and include individual properties such as grain size and shapeand bulk properties such as grain size distribution, fabric (orientation and packing of particles), porosity andpermeability. These properties are important for any complete description of a sediment. In addition, because theseproperties are governed by processes that act at the time of deposition and, in some cases, after burial, they mightprovide insight into the history of a sediment. Furthermore, many of these properties will govern how a sedimentwill behave when used in a particular way, for example, as a final resting place for garbage. This chapter focuseson the concepts and terminology used to describe sediment texture and shows how texture may be used in makingbasic interpretations about the history of a sediment.

GRAIN SIZE

One might expect that the description of the size of particles that make up a clastic sediment would be thesimplest property to describe. However, most sediment is composed of particles with a variety of irregular shapesand may extend over a range of sizes. Therefore, the characterization of the size of particles within a sediment maynot be so straight forward.

Consider an irregularly-shaped pebble that is large enough to cover the palm of your hand. How might youtell someone else precisely how big this pebble is without actually showing it to them? You might decide that thepebble is “moderately large” and describe it as such but this is a rather ambiguous expression that could be easilymisunderstood by the other person. In this case we need some way of measuring the size of the pebble and thenwe need some consistent terminology to describe the size; a terminology that everyone else will use so that theywill know what is meant by an expression such as “moderately large”.

Grain Volume

There are several methods that will provide a quantitative measurement to describe the size of a particle. Theeasiest method would be to physically measure some linear dimension of a particle. This is simple enough if theparticle is a perfect sphere, it will have only one linear dimension: its diameter. However, natural particles arecommonly not very spherical and one linear dimension may not adequately describe their size. A measure of sizethat can be determined while neglecting the shape is the volume (V) of a particle. A simple way to determine thevolume of a particle is to determine its mass (m) and calculate volume from the relationship:

m = Vρ Eq. 2-1

where r is the density of the particle. The mass can be determined by weighing the particle. We might assume areasonable density of the particle, say that of quartz (2650 kgm-3), and solve for V. However, the actual density ofthe material comprising the particle may vary significantly from the assumed value.

The volume of the particle may also be determined directly by measuring the volume of fluid displaced bythe particle when it is immersed in the fluid within a graduated cylinder or beaker. This method does not requirean assumption of the density of the particle but may derive error if the particle is made up of a porous material andall of the pores do not become saturated with the fluid. We can often neglect this source of error.

In the case of particles that are perfect spheres of non-porous solid we can calculate the volume of the particlewith diameter d from:

Vd

=π 3

6Eq. 2-2

9

Linear Dimensions

Direct Measurement

As noted above, the easiest way the determine and express the size of a particle is by describing its lineardimensions. Because a sedimentary grain is rarely a perfect sphere we normally treat each grain as if it was a triaxialellipsoid (Fig. 2-1) and its linear dimensions are described in terms of the lengths of the three principle axes: the longaxis (a-axis or d

L), the intermediate axis (b-axis or d

I), and the short axis (c-axis or d

S)

To measure these lengths on a particle follow the steps below and refer to figure 2-2.

1. Determine the plane of maximum projection of the particle. This is an imaginary plane passing through the particlesuch that the intersection of the particle and the plane produces the largest possible surface area (i.e., the maximumprojection area).

2. Establish the maximum tangent rectangle for the maximum projection area. This is a rectangle that, when placedaround the maximum projection area, results in the maximum possible tangential contact with the outline of theparticle.

3. Measure the length of the a- and b-axes such that the a-axis length is the length of the long side of the maximumtangent rectangle and the b-axis length is the length of the short side of the maximum tangent rectangle.Note that the plane of maximum projection is the a-b plane of the particle.

4. Measure the c-axis length as the longest distance through the particle, perpendicular to the a-b plane.

The above procedure is used so that different workers can produce comparable results; it provides a standardfor all workers. For large particles, reportedly as small as 0.25 mm, all three axes may be measured. For larger particles,in excess of a few millimetres, callipers may be used to measure the axes. For smaller grains the axes may be measuredunder a binocular microscope. When a grain is too small to allow it to be rotated to see the short axis the long andthe intermediate axis may be measured. A common practice is to measure only d

L and different particles may be

compared only in terms of their maximum axes lengths. However, this practice may lead to considerable error,especially if the shapes of particles varies considerably.

In many studies of particle size, especially coarse particles, grain size is expressed in terms of the nominaldiameter (d

n): the diameter of a sphere having the same volume as the particle. If the three principle axes lengths

are known this expression of grain size is easily calculated by:

n L I Sd d d d= 3 Eq. 2-3

Nominal diameter is just one of a large number of expressions of linear dimension that have been used to describe

dL

dS

dI

Figure 2-1. Definition of principle axes of a sedimentary particle.

10

Maximum

Projection area

dId

L

dS

Maximum Tangent Rectangle/a-b plane

Figure 2-2. Sketch showing the method for determining the lengths of the principle axes of a particle. After Blatt,Middleton and Murray, 1980.

the size of a particle.

Sieving

Sieving is a method that is widely used to directly measure the sizes of a large number of grains (samplesrange from 40 to 75 grams) and is normally limited to particles in the range from as fine as 0.0625 mm up to as largeas 64 mm. The detailed procedure that should be followed when sieving a sediment sample is outlined in Appendix1. Again, a specific procedure has been derived so that results of different workers will be reliably comparable. Theequipment that is used world-wide for sieving includes a shaker and a series of nested, square holed screens (thescreen with the largest holes on top, down the smallest holes on the bottom). The sediment sample is passed throughthe screens, by shaking, and the weight of sediment that accumulates on each screen is weighed. Thus, theinformation obtained is not the absolute size of each grain but the frequency of grains (by weight), in a sample, thatfall between the range of sizes represented by the square holes in the screen above and in the screen on which thegrains are resting. The actual dimension of the grains that might be considered is "the largest square hole throughwhich the grains will pass". This dimension is approximately the intermediate axis length. The procedure and themeaning of grain size data will become more clear later in this chapter.

Settling Velocity

Another very useful measure of grain size is the settling velocity of a particle. A particle's settling velocity(or fall velocity) is normally defined as the terminal velocity that a particle will reach while falling through a still fluid,normally water. A particle's settling velocity (ω) is related directly to its size (d) and also to its density (ρ

s) but is

also influenced by fluid properties such the fluid density (ρ) and the viscosity (µ); for all fluids these propertiesvary with temperature.

To measure the settling velocity of a particle we may simply, but accurately, record the time that it takes tofall a known distance through a column of still fluid contained within a transparent tube (termed a settling tube).This is relatively simple if we are timing the settling rate of only one or a few grains but becomes difficult if we mustdeal with a sample from a population of grains of non-uniform size (and therefore varying settling velocity). In thiscase we must time the settling rate of large numbers of grains. A variety of types of settling tube have been devisedand these are described in several publications (see Blatt et al. 1980).

When using settling tubes a number of factors that will influence the results must be considered. First, when

11

a particle first begins to fall through the fluid it will accelerate from rest (i.e., zero velocity) to some constant (terminal)velocity. Thus, the settling velocity may be under-estimated if the distance over which the particle passes duringthe phase of acceleration is large in comparison to the total distance over which settling is timed. This can beovercome by using a tube that is long enough so that the distance over which acceleration takes place is relativelysmall. This may be a problem for relatively large particles but for less than 1 mm diameter particles of quartz-densitythe distance required to achieve terminal velocity is less than a centimetre. When dealing with samples of morethat one particle care must be taken that the grains settle independently and do not interact during passage throughthe fluid. Sample sizes will depend on the type of settling tube used and should generally be less than 10 grams.Samples that are too large may cause settling en masse so that the velocity determined will not be the velocity ofindividual grains. Also, large quantities of grains in the tube may interact directly (e.g., collide), altering the settlingrate, or indirectly by causing the fluid itself to move (i.e., cause turbulence) so that the velocity of the grains is notthat in still fluid. A further requirement of apparati for measuring settling velocity is that the tube diameter mustbe at least five time the average diameter of the particles that will pass through it. When a particle passes througha fluid it must "push" the fluid out of its path. At the tube wall the fluid is more difficult to push than it is well awayfrom the wall region due to fluid viscosity. Thus, a small ratio of tube to particle diameter may result in the passageof grains too close to the tube walls where they will be travel slower than they would through fluid that is notsignificantly influenced by wall effect.

Stokes’ Law of Settling

The relationship between grain size (e.g., diameter) and settling velocity is an interesting one that reflectsa variety of principles that will be of interest later in these notes. The relationship is relatively complex due to a numberof factors. Theoretical relationships have been developed and compared to experimental data collected by actuallymeasuring settling velocity in settling tubes. One such theoretical relationship between grain size and settlingvelocity is Stokes' Law of settling.

The derivation of Stokes' Law is simple but instructive. It is based on a simple balance of forces that acton a grain as it is falling at terminal velocity through a still fluid and the knowledge that the velocity will be affectedby grain size (d), grain density (ρ

s), fluid viscosity (µ), fluid density (ρ), and the acceleration due to gravity (g).

Consider the forces acting on the spherical particle falling through a still fluid as shown in figure 2-3. As the particlebegins its passage through the fluid three forces are involved: the gravity force (F

G), which is the weight of the

particle acting to move it downward through the fluid; a buoyant force (FB), acting to move the particle upward

through the fluid; and a drag force (FD) that retards the movement of the particle through the fluid. The gravity

and buoyant forces may easily be quantified. The gravity force is just the weight of the particle and is equal to thevolume of the particle times its density times the acceleration due to gravity:

FB

FG

FDFD

FB: Bouyant Force

FD: Drag Force

FG: Gravity Force

Figure 2-3. Forces acting on a spheri-cal particle falling through a stillfluid.

12

F gdG s=π ρ6

3 Eq. 2-4

FG acts in the downward direction, causing the particle to settle. The buoyant force acts in the opposite direction

and is simply the weight of the fluid that is displaced by the particle (i.e., the weight of fluid with a volume equalto that of the particle):

F gdB =π ρ6

3 Eq. 2-5

Because FG and F

B act in opposition the net force that acts on the particle (FG

' ) is given by:

F F F s g d g d s g dG G B' ( )= − = − = −

π ρ π ρ π ρ ρ6 6 6

3 3 3 Eq. 2-6

Clearly, when the density of the particle exceeds the density of the fluid it will settle through the fluid becauseF

G > F

B . Note that the expression on the right hand side of Eq. 2-6 is termed the submerged weight of a particle.

The drag force exerted on the settling particle is more difficult to determine. This force arises from the resistanceof the fluid to deformation due to the fluid property termed dynamic viscosity and is normally given the symbol m,and is expressed as Nsm-2. Fluids that offer a large resistance to deformation have a high viscosity (e.g. molasses)whereas fluids with low viscosity deform readily. Experiments have shown that the drag on a particle varies withthe velocity at which the particle passes through the fluid. Specifically, experiments have shown that at relativelylow settling velocities the drag force can be calculated from:

F d UD = 3π µ Eq. 2-7Stokes derived his law of settling from the two forces acting on a settling particle: the submerged weight and thedrag force. When a particle is settling at its terminal velocity no net force must be acting on it (i.e., the velocity isconstant, neither accelerating or decelerating, therefore experiencing no net force). Therefore, when a particle

reaches terminal velocity FD and FG

' must be equal in magnitude but act in opposite directions (FD acts upwards

and the submerged weight acts downwards). Thus, when a particle reaches terminal settling velocity:

F F FD G B= − Eq. 2-8Setting this equality and substituting with equations 2-6 and 2-7:

36

3π µ π ρ ρd U gds= −( ) Eq. 2-9

Because the particle is falling at its terminal velocity the velocity term in FD is the settling velocity so that U=w.

Therefore, we can solve Eq. 2-9 for ω:

ω π ρ ρπ µ

ρ ρµ

= − × =−

6

1

3 183

2

( )( )

ssgd

d

gdEq. 2-10

Thus, Stokes' Law of Settling is simply:

ωρ ρ

µ=

−( )s gd 2

18Eq. 2-11

Table 1 shows an example calculation of the settling velocity of a particle using Stokes' Law. Note that whenmaking such a calculation you must state all of the conditions (e.g., sediment and fluid density, fluid temperatureand viscosity), in many cases these conditions must be assumed. The statement of conditions is particularlyimportant because there are severe limitations to the use of Stokes' Law of Settling. These limitations are outlinedin the following paragraphs.

Stokes’ Law is reliable only for grain sizes finer than approximately 0.1 mm. Figure 2-4 shows that Stokes'Law predicts the settling velocity of quartz-density particles quite reliably up to a grain size of approximately 0.1mm, beyond which the theoretical relationship increasingly overestimates the observed settling velocity. Thisoverestimation is due to the fact that Stokes' Law only includes viscous drag on the particle. When larger grains

13

Table 2-1. Example of Stokes' Law of Settling

Consider a spherical quartz particle with a diameter of 0.1 mm, in still, distilled water.

ρs= 2650 kg/m3

ρ= 998.2 kg/m3, density of water at 20°C

µ = 1.005x10-3 N.s/m2, water at 20°C

g = 9.806 m/s2

(note that in the calculation, for consistent units, the diameter of the particle is expressed as 0.0001 m)

By Stokes’ Law the Settling, the terminal fall velocity of this particle is 8.954 x 10-3 m/s (~ 9 mm/s).

settle they travel at relatively high velocities through the fluid and eddies develop in their wake. These eddies impartan additional form of resistance that acts to further reduce their terminal settling velocity. Because Stokes' Lawneglects this additional force of resistance it predicts a larger settling velocity than that determined by actualexperiments (Fig. 2-4). Figure 2-5 shows an additional shortcoming of Stokes' Law when applied to sedimentaryparticles that settled in a fluid for which the temperature is not known. Because of the effect of temperature on fluiddensity and, especially, fluid viscosity, the settling velocity of a particle will vary over almost an order of magnitudethrough the range of temperatures that might naturally occur at the earth's surface. Thus, application of Stokes'Law to predict the settling velocity of a particle in some unknown depositional environment (and unknowntemperature) may lead to considerable error.

A further problem that arises when applying Stokes’ Law to natural sedimentary particles is that it onlyapplies to spherical particles. Grain shape can have a dramatic effect on settling velocity. In a simple case, shapecan effect the orientation of the grain as it settles thus affecting the surface drag that acts on the particle as it passesthrough the fluid. A more dramatic example is that of a platy particle (e.g., a flake of mica) which tends to drift backand forth as it settles rather than falling vertically though the fluid.

Finally, Stokes’ Law applies to particles falling through a still fluid. That is, the fluid is at rest except where

Figure 2-4. Somewhat schematic illustration showing thesettling velocities of spherical, quartz-density particles instill water at 20°C, as predicted by Stokes' Law of Settling,compared to experimentally-determined settling velocitiesof similar particles. After Leeder (1982).

1000

100

10

1

0.10.01 0.1 1.0 10

Quartz sphere diameter (mm)

Fal

l vel

ocity

(m

m s

-1 )

Sto

kes'

Law

Exper

imen

tal curve

14

it is accelerated, due to viscosity, by the passage of the particle itself. Therefore, it is unreasonable to apply Stokes’Law to the settling velocity of particles in a moving, particularly turbulent, fluid (e.g., particles in a river) where fluidforces will act to move the particle up, down and sideways.

Before leaving Stokes' Law of Settling we can return to the discussion of the expression of grain size as somelinear dimension of a particle. Sedimentologists have traditionally preferred to express grain size as some lineardimension and Stokes' Law can also be used to determine such a dimension for comparison of different sedimentaryparticles. This linear dimension of a particle is termed Stokes' Diameter (d

s; also termed settling diameter) of a particle

and is defined as the diameter of a sphere with the same settling velocity as the particle. We can determine theStokes' Diameter of a particle by measuring its settling velocity in still water and by solving for particle diameterin Eq. 2-11. Thus, if the settling velocity of a particle is known, Stokes' Diameter may be calculated from:

dgs

s

=−

18µωρ ρ( ) Eq. 2-12

Grade Scales

In the introduction to this chapter we posed the question of how do we verbally describe the size of a particlein such a manner that would be understood by others. A grade scale provides such a standard for verballyexpressing and quantitatively describing grain size. Any good grade scale should:

(i) Define ranges or classes of grain size (grade is the size of particles between two points on a scale. e.g., “veryfine sand”, is a grade between maximum and minimum size limits),

and

(ii) Proportion the grade limits so that they reflect the significance of the differences between grades. For example,the change in size from 1 mm to 2 mm diameter sand is an increase of 100%, however, the change in size from 10 mmto 11 mm is on the order of 10%. Therefore, a grade scale in which grade limits vary by 1 mm would not be useful.

The most widely-used grade scale is the Udden-Wentworth Grade Scale (Fig. 2-6). Note that most of the grade

100

10

10 20 40 60 80 100

Temperature (˚C)

Sto

kes'

Set

tling

Vel

ocity

(m

m s

-1)

Stokes' settling velocity:0.1 mm quartz grain in still water

Figure 2-5. Effect of fluid temperature on set-tling velocity. Curve shows settling velocitycalculated using Stokes' Law of Settling appliedto a 0.1 mm diameter quartz sphere setting throughstill water.

15

Boulders

Cobbles

Pebbles

Granules

Very Coarse Sand

Coarse Sand

Medium Sand

Fine Sand

Very Fine Sand

Silt

Clay

Size(mm)

256

Size(φ)

-8

64 -6

4 -2

2 -1

1 0

0.5 +1

0.25 +2

0.125 +3

0.0625 +4

0.0039 +8

Gra

vel

San

dM

ud

Dir

ect

Mea

sure

men

t

Sie

vin

g

Set

tlin

g T

ub

e

Bin

ocu

lar

Mic

rosc

op

e

Measurement techniques

Sed

igra

ph

Figure 2-6. The Udden-Wentworth Grade Scale defining the range of sediment sizes per size class and the verbalexpression used to describe each size class. Also shown are the techniques that may be used to measure grain size andthe grain size limits over which they should be used. After Pettijohn, Potter and Siever, 1973.

boundaries increase by a factor of 2, reflecting significant changes in grain size. Also, the scale defines limits forthe verbal expression of grain size. "Very fine sand" is sand which ranges in size from 0.0625 mm to 0.125 mm

Krumbein (1934) introduced a logarithmic transformation of the scale which converts the boundariesbetween grades to whole numbers. This scale is known as the Phi Scale, it’s values being denoted by the Greekletter phi (φ), where:

φ = -log2 d(mm) Eq. 2-13

where d(mm) is just the grain size expressed in millimetres. For example:

-log2 1(mm) = 0 φ

-log2 1/4(mm) = 2 φ

-log2 4(mm) = -2 φ

φ was later redefined:φ = -log

2 d/d

o2-14

where d is the size in millimetres and do is a standard size of 1 mm; division by 1 mm does not alter the value of φ

but makes it dimensionless.

With a hand calculator the conversion from φ to mm and from mm to φ is as follows:

φ → mm d(mm) = 2-φ

mm → φ φ = -( log10

d)/log10

2

Note that it is traditional among sedimentologists to plot grain size on a phi scale with decreasing grain size to theright as shown in figure 2-7.

16

Frequency

increasing grain size decreasing grain size

0 1 2 3 4-1-2-3-4

Grain Size (φ)

Displaying Grain Size Data

Most data describing the grain size distribution of a bulk sample of sediment are in the form given in Table2-2 which shows the frequency distribution (in terms of weight) of sediment per size class (here grain size classesare at 0.5 φ intervals). In the case of data derived from sieving a sediment sample each value in column 2 is the weight,in grams, of sediment accumulated on the sieve with openings indicated by the grain size class (column 1). Tofacilitate comparison of samples of different weight the frequency distribution is more commonly calculated as aweight expressed as a percentage of the total weight of the sample (column 3). Column 4 is the cumulative weight(%) derived by incrementally summing the values in column 3. Several ways in which such data may be plottedto graphically display the grain size distribution of a sample are shown in figure 2-8.

Histograms, such as figure 2-8A, are valuable because they readily show the relative proportions of eachsize class and the modal class of the distribution (the size class with the largest frequency). Also shown in figure2-8A is the frequency curve for the data, a smoothed version of the histogram formed by joining the midpoints ofeach size class on the histogram. Note that there are two vertical scales in this example, one shows the absoluteweight per size class and the other shows the weight expressed as a percentage of the total sample weight per sizeclass. In this case the data are normally distributed and the frequency curve forms a bell-shape which is symmetricalabout its highest point. Figures 2-8B and C are cumulative frequency curves for the data in Table 2-2, column 4.

Figure 2-7. Conventional phi scale showing grain sizeincreasing to the left and decreasing to the right.

Table 2-2. Example of grain size data produced by sieving (note these data are plotted in Figure 2-8).

1. Grain Size 2. Weight 3. Weight 4. Cumulative WeightClass (φ) (grams) (%) (%)

-0.5 0.40 1.3 1.30 1.42 4.6 5.9.5 2.76 8.9 14.81 4.92 15.9 30.71.5 5.96 19.3 502 5.96 19.3 69.32.5 4.92 15.9 85.23 2.76 8.9 94.13.5 1.42 4.6 98.74 0.40 1.3 100.0Total 30.92 100

17

-1 0 1 2 3 40

1

2

3

4

5

6

0

5

10

15

20

Indi

vidu

al W

eigh

t (gr

ams)

Individual Weight (%

)

Histogram

Frequencycurve

Grain size (φ)

A. Histogram and frequency curve

-1 0 1 2 3 4

Grain size (φ)

Cum

ulat

ive

Wei

ght (

%)

0

10

20

30

40

50

60

70

80

90

100

B. Cumulative frequency curveon an arithmetic scale

-1 0 1 2 3 4

Grain size (φ)

0.01

1

99.99

99

50

80

2010

90

Cum

ulat

ive

Wei

ght (

%)

C. Cumulative frequency curve on a probability scale

Figure 2-8. Various ways in which grain size data (Table 2-2) may be graphically displayed.

Figure 2-9. A. Illustration showing how a percentile may be determined from a cumulative frequency curve. B. Schematicillustration showing the cumulative frequency curve of a sediment population composed of three subpopulations. Note thatboth graphs are plotted on probability scales.

-1 0 1 2 3 4

Grain size (φ)

0.01

1

99.99

99

50

80

2010

90

Cum

ulat

ive

Wei

ght (

%)

A. Definition of Percentile

-1 0 1 2 3 4

Grain size (φ)

0.01

1

99.99

99

50

80

2010

90

Cum

ulat

ive

Wei

ght (

%)

B. Grain size population of three normallydistributed subpopulations

φ20 = 0.86 φ

One of the major benefits of plotting grain size data as cumulative frequency curves is that the data form a uniquecurve for each possible grain size distribution. Such curves may be plotted on graphs with arithmetic axes (Fig.2-8B) or more commonly on graphs with a vertical “probability” scale which expands the low and high ends of thescale and compresses the middle of the scale (Fig. 2-8C). One advantage of plotting such data on a probability scale(Fig. 2-8C) is that normally-distributed data plot as a straight line. Moreover, if the sediment sample is made up ofseveral normally-distributed subpopulations its’ cumulative frequency curve, plotted on a probability scale, willform straight-line segments, each corresponding to a subpopulation of the total sediment (Fig. 2-9B). Anotherbenefit of constructing cumulative frequency curves is that percentiles can be taken directly off the graph (Fig.2-9A). The nth percentile (denoted φ

n in this context) is the grain size, in units of φ, which is finer that n % of the

total sample. In figure 2-9A φ20

is shown to be 0.86 φfor the hypothetical sample. Several descriptive measuresof grain size distributions have been based on percentiles taken directly from cumulative frequency curves (seeTable 2-3).

Describing Grain Size Distributions

18

There are several different properties of grain size distributions that may be described both qualitatively andquantitatively. The most common of these are outlined below. Table 2-3 summarizes some widely-used formulaefor computing the descriptive measures of grain size distributions. Note that the graphic method uses percentilestaken directly from cumulative frequency curves (see Fig. 2-10)

Median (Md)

This is the mid-point of the distribution (i.e., the grain size for which 50% of the sample is finer and 50% is coarser).

Mean (M)

The mean is the arithmetic average size of the distribution (Fig. 2-11). For perfectly symmetrical normal

Table 2-3. Descriptive measures of sediment size distribution.

Measure Graphic Method Moment Method(Folk and Ward, 1957) (after Boggs, 1987)

Median (Md) φ50 —

Mean (M)φ φ φ16 50 84

3

+ + fm∑100

Standard Deviation (s)φ φ φ φ84 16 95 5

4 6 6

−+

−.

f m M( )−∑ 2

100

Skewness (Sk)φ φ φ

φ φφ φ φ

φ φ84 16 50

84 16

95 5 50

95 5

2

2

2

2

+ −−

++ −

−( ) ( )

f m M( )−∑ 3

3100σ

Kurtosis (K)φ φφ φ

95 5

75 252 44

−−. ( )

f m M( )−∑ 4

4100σ

where: φn is the nth percentile of the size distribution taken from the cumulative frequency curve;

φ is the weight of sediment per size class as a percentage of the total sample weight;m is the midpoint of the size class.

0.01

0.050.10.20.5

12

5

10

20

3040506070

80

90

95

9899

99.899.9

99.99

-3 -2 -1 0 1 2 3 4

Cu

mu

lati

ve F

req

uen

cy (

%)

Grain Size ( )φ

φ16 φ

50φ84

Folk and Ward Formula for Graphic Mean:

+ + M z =

3

from the cumulative frequency curve:

φ16 = -0.59 φ

φ50 = 0.35 φ

φ84 = 1.27 φ

∴

M z =−0.59 + 0.35 + 1.27

3

Example calculation of grain size statistics by the graphical method

= .34φ16 φ50 φ84

φ

Figure 2-10. Example of calculating the mean of asize distribution by the graphical method.

19

distributions the mean is equal to the median. Note that the true mean cannot be determined from data collectedby sieving but can be approximated by the formulae shown in Table 2-3.

Sorting or Dispersion Coefficient (σσσσσ)

This is the standard deviation of the distribution and reflects the variation in grain sizes that make up asediment. Its’ calculated value relates to the mean of the distribution as illustrated in figure 2-11. This figure showsthat 68% of the distribution (i.e., 68% of a sediment sample, by weight) has a grain size that is within ±1σ of the mean.For example, if the mean of the distribution is 1.45 φ and the sorting coefficient is 0.30 φ, then 68% of the samplelies in the size range from 1.15 φ to 1.75 φ. Therefore, the larger the sorting coefficient the greater the range of grainsizes that make up the sediment. A sediment that consists of only a small range of grain sizes (i.e., σ is small) is saidto be well sorted whereas a sediment made up of a wide range of grain sizes (i.e., σ is large) is said to be poorly sorted.Figure 2-12 very schematically illustrates the appearance of various degrees of sorting. Descriptive grades of sortingare:

0 < σ < 0.35φ very well sorted0.35 < σ < 0.5φ well sorted0.5 < σ < 0.71φ moderately well sorted0.71 < σ < 1.00φ moderately sorted1.00 < σ < 2.00φ poorly sorted2.00 < σ < 4.00φ very poorly sortedσ > 4.00φ extremely poorly sorted

Figure 2-11. Relationship between sorting coefficient(standard deviation) and mean of a normal distribution.After Friedman and Sanders (1978).

Figure 2-12. Schematic illustration of vari-ous degrees of sorting. After Anstey andChase, 1974.

Indi

vidu

al w

eigh

t (%

)

Grain size (φ) finecoarse1σ

68%

1σ M

Very well sorted Well sorted

Moderately sorted Poorly sorted

20

Skewness (Sk)

Skewness is a measure of the symmetry of the grain size distribution about the mean; it has a maximumpossible value of +1 and a minimum possible value of -1. A of skewness that is close to zero indicates that thedistribution is very symmetrical and the mean is equal, or nearly so, to the median and both fall within the modalclass. A positive value of skewness indicates that the distribution has a larger proportion of fine grains than if thedistribution were symmetrical. Conversely, if the value of skewness is negative the distribution is enriched in coarsegrains. Figure 2-13 schematically shows the various “types” of skewness.

Descriptive terms for skewness are: Sk >+0.3 strongly fine skewed+0.1 < Sk < +0.3 fine skewed-0.1 < Sk < +0.1 near symmetrical-0.3 < Sk < -0.1 coarse skewedSk < -0.3 strongly coarse skewed

Kurtosis (K)

Kurtosis is a measure of the degree of “peakedness” of the distribution. Oddly enough, while it is one ofthe common descriptive parameters for grain size distributions it is widely thought to be essentially of no value (seejust about any textbook). None-the-less, it has become somewhat of a tradition. Figure 2-14 schematically showsdifferences between the three types of kurtosis that a distribution might display: leptokurtic (sharp-peaked; K >1), mesokurtic (normal; K = 1), and platykurtic (flat-peaked; K < 1).

Paleoenvironmental Implications of Grain Size

For many years it was thought (and hoped) that the characteristics of grain size distributions were governedlargely by processes within the depositional environment. Therefore, properties like mean size, sorting, skewness,etc., would cumulatively reflect these processes and provide a basis for interpreting ancient depositionalenvironments. This was a powerful reason for sedimentologists to study grain size distributions and many did just

Indi

vidu

al w

eigh

t (%

)Grain size (φ)

finecoarse

Mo = M = Md Sk = 0In

divi

dual

wei

ght (

%)

Grain size (φ)finecoarse

Mo

Md

M

Excess fineparticles

Indi

vidu

al w

eigh

t (%

)

Grain size (φ)finecoarse

Sk > 0

Indi

vidu

al w

eigh

t (%

)Grain size (φ)

finecoarse

Sk < 0MoMd

M

Excess coarseparticles

Figure 2-13. Schematic illustration of the various types of skewness. Note that dashed lines indicate the symmetricaldistribution for comparison with fine and coarse skewed frequency curves. M is mean, Md is median and Mo is mode.AfterFriedman and Sanders (1978).

21

Indi

vidu

al w

eigh

t (%

)

Grain size (φ)finecoarse

Leptokurtic

Mesokurtic

Platykurtic

Figure 2-14. Examples of different types of kurtosis. AfterBlatt, Middleton and Murray (1980).

+2.50

+2.00

+1.50

+1.00

+0.50

0.00

-0.50

-1.00

-1.50

-2.00

-2.50

-3.00

-3.500.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40

STANDARD DEVIATION (sorting)

SK

EW

NE

SS

River sandBeach sand (ocean)Beach sand (lake)

that throughout much of the 20th century. There was limited success. Results of one successful study are shownin figure 2-15 and suggest that beach and river sediment can be distinguished on the basis of a simple plot ofskewness versus standard deviation (sorting coefficient). This figure shows relatively good separation of beachsands (generally coarse skewed and better sorted) and river sands (fine-skewed and somewhat less well sorted).The difference arises because rivers are capable of transporting a relatively wide range of grains sizes, particularlya large proportion of fine sediment that is transported in suspension when the rivers are in flood. The river deposits,therefore, tend to be relatively poorly sorted and enriched in fine grained sand (i.e., fine skewed). In contrast,beaches experience the ongoing swash and backwash of waves running up and down the beach slope. These veryshallow flows tend to segregate the sediment very efficiently, particularly in washing fine-grained material out tothe sea or lake. Thus, beach deposits are rather well-sorted and somewhat enriched in the coarser grain size (i.e.,coarse skewed)

The success of this approach to paleoenvironmental interpretation has been very limited. The main problemis that factors other than processes in the depositional environment have a profound affect on grain sizedistributions. For example, the size distribution of the constituents of the source rock (i.e., the rock that weathersto produce the sediment) may be reflected in the size distribution of the deposited sediment. If a river is transportingsand that has weathered out of a pre-existing rock formed of sand that was laid down millions of years ago on abeach, the river sediment will continue to display a size distribution which is inherited form the beach deposits.

Figure 2-15. Plot of skewness versus sortingcoefficient based on samples of river and beachsands. After Friedman, 1967.

22

Thus, if we plotted that river's sediment on figure 2-15 we would mistakenly interpret the deposit as that of a beach.

Why measure grain size?

There are many reasons to quantitatively describe grain size distributions. Some of these are:

1. Grain size is important to determining the strength of currents that transported the sediment. Therefore, we needa precise measurement of size to quantitatively interpret paleohydraulic conditions.

2. Sorting reflects the ability of the transport mechanism to segregate grains by size.

3. Skewness reflects the ability of the transport mechanism to selectively remove coarse or fine grain sizes.

4. It appears that grain size distributions like that shown in figure 2-9B have very specific interpretations in termsof how the sediment moved while it was in transport. (More on points 1 through 4 later in Chapter 4.)

5. We need basic descriptors of sediment size to allow us to communicate with others.

6. Grain size and various properties of its distribution are important in determining a sediment's porosity andpermeability.

GRAIN SHAPE

Grain shape is another fundamental property of particles and one that may provide important informationabout the history of a sediment. Like grain size, shape may be expressed in several different ways, including:Roundness, a measure of the sharpness of the corners of a grain; Sphericity, a measure of the degree of similaritybetween a grain and a perfect sphere; Form, an expression of the overall appearance of a particle. Methods ofquantitatively describing these three expressions of grain shape are outlined below.

Roundness

The roundness of a particle refers to the degree of rounding (or angularity) of the edges of a particle. Ofthe three shape properties it is the most difficult to quantify.

Wadell’s Roundness (Rw)

Wadell (1932) introduced this method of determining particle roundness; it is the most accurate method butit involves the greatest effort and time. R

w is defined as the ratio of the average radii of curvature of the corners

of a grain (Σr/N; N is the number of corners) to the radius of the largest inscribed circle within the particle (R;Fig. 2-16). The maximum possible value of R

w is 1 when the particle has no measurable corners (i.e., R = r). Thus,

Rr

NRW = ∑ Eq. 2-15

Dobkins and Folk Roundness (RF)

Dobkins and Folk (1970) introduced a less tedious (and less accurate) method of calculating roundness(symbolized as R

F here) that is defined as the ratio of the radius of curvature of the particle's sharpest corner (r)

to the radius of the largest inscribed circle (R). Thus,

Rr

RF = Eq. 2-16

once again, RF approaches a value of 1 for perfectly rounded grains (i.e., no sharp corners).

Powers’ visual comparison chart

The easiest way to determine the roundness of a particle is by visual comparison with standard forms. Powers

23

Very angular Angular Sub-angular Sub-rounded Rounded Well-rounded21 3 4 5 6ρ

Rw .17 .25 .35 .49 .70 1

Figure 2-17. Powers’ (1953) visual comparison chart for grain roundness with appropriate terms for describing shape classesdefined by R

w and ρ.

R

rr

rr

rr

r

r

Grain outline

Figure 2-16 Wadell Roundness (see Eq. 2-15; after Boggs, 1967).

(1953) provided what has become the most widely used chart of grain roundness (Fig. 2-17) and is a basis for termsdescribing roundness. Note that each box in figure 2-17 shows a particular roundness class and the appropriateterm to describe each class. Also shown are values of R

w and ρ (a logarithmic transformation of R

w to whole numbers)

that quantitatively define the limits of each roundness class.

Sphericity

Various measures of the degree to which a particle resembles a sphere (i.e., its sphericity) have been devised.Sphericity not only describes one aspect of the shape of a particle but it may also be useful to understanding otherproperties of the particle such as its settling velocity. Remember, Stoke’s Law of Settling applies accurately onlyto spherical particles (less that 0.1 mm in diameter) and its error increases as the shape of the particle deviates fromthat of a true sphere. Therefore, a measure of particle sphericity provides a means of quantitatively determininghow well Stokes’ Law will predict a particles’ settling velocity. Note that sphericity is normally given the symbol“ψ”, the lower case Greek letter psi.

Wadell’s Sphericity (ψψψψψ)

Wadell (1932) defined sphericity as the ratio of the diameter of a sphere with volume equal to that of theparticle to the diameter of the sphere which will circumscribe the particle. Wadell’s measure of sphericity maytake the form:

ψ =V

VS

C

3 Eq. 2-17

where VS is the volume of the sphere with volume equal to that of the particle and V

C is the volume of the

circumscribing sphere. In this form VS may be determined by measuring the volume of water displaced by the particle

24

and VC may be calculated from the formula for the volume of a sphere taking the maximum axis length of the particle

(dL) to be the diameter of the circumscribing sphere:

V dC L=π6

3 Eq. 2-18

VC may also be calculated for a particle with shaped approximately like a triaxial ellipsoid (Fig. 2-1) by:

V d d dS L I S=π6

Eq. 2-19

Substituting these expressions for Eqs. 2-18 and 2-19 into Eq. 2-17 yields:

ψ =d d

dI s

L2

3 Eq. 2-20

As ψ approaches 1 the shape of the particle approaches that of a perfect sphere. Eq. 2-20 can be used to calculatesphericity on the basis of measurements of the principle axes of a particle.

Sneed and Folk Sphericity (yP)

Sneed and Folk (1958) argued that the volume of a particle is not as important as its maximum projection areain determining its settling velocity because drag forces act on the particle’s surface. Therefore, any measure ofparticle sphericity must be reflect the maximum projection area of a particle. They defined maximum projectionsphericity (ψ

P) as the ratio of the maximum projection area of a sphere with volume equal to that of the particle

to the maximum projection area of the particle. ψP may be calculated from the formula:

ψ PS

L I

d

d d=

23 Eq. 2-21

This is the most widely-used expression of sphericity.

Note that an expression that is very similar to ψP and based on similar reasoning, termed the Corey Shape

Factor (S.F.), is widely by used engineers to describe the overall shape of a sediment particle:

S Fd

d dS

L I

. .= Eq. 2-22

Riley Sphericity (ψR)

The main problem in calculating ψ and ψP is that both require the measurement of d

S. This is not difficult

for gravel-size particles but it becomes very impractical for sand-size sediment. Riley suggested a method ofcalculating sphericity that relies on measurements that can be taken from the two-dimensional view of a sand grainas seen through a microscope (Fig. 2-18). He defined an expression of sphericity as:

ψ Ri

c

d

d= Eq. 2-23

where dc is the diameter of a circle that circumscribes the grain and d

i is the diameter of a circle that inscribes the

grain (Fig. 2-18). Again, as ψR

approaches unity as sphericity increases.

Clast Form

There are two commonly used methods of describing the overall form of a particle, both based on variousratios of d

L, d

I, and d

S. Figure 2-19 shows the so-called Zingg diagram (after Zingg, 1935) that defines four form

fields and provides terms for clast form according to which field the axes ratios of a clast plot. The clast forms definedby the Zingg Diagram are largely independent of sphericity, except for “equant” clasts which tend to have valuesof sphericity near 1.

25

Figure 2-20 shows a second commonly-used scheme for classifying grain form that was proposed by Sneedand Folk (1958). This method classifies form into 10 possible classes and the appropriate terms for each form classis shown in figure 2-20 along with blocks that are drawn in the approximate proportions for each form field. Notealso that maximum projection sphericity may also be taken directly from this diagram and that different form classesmay have the same sphericity (i.e., form and sphericity are independent).

Significance of grain shape

Like grain size, the shape of a particle will provide some basis for making fundamental interpretations aboutthe “history” of a sediment, particularly something of the source rock of the grains and of their transport history.

Source Rock

The lithology of the source rock exerts a strong control on the form of particles, especially gravel-size clasts.Bedded source rocks (e.g., horizontally bedded limestones) tend to produce platy clasts, due to the parallel planesof weakness that the bedding produces. Massive source rocks (e.g., granites that have equal strength in alldirections) tend to produce more equant clasts. The hardness of the source rock will also control the overall shape

didC

Figure 2-18. Illustration showing the measurements required to calculatesphericity by the method proposed by Riley.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

dd

S

I

I

L

dd

Zingg form index

OBLATE(DISK) EQUANT

BLADEDPROLATE(ROLLER)

Figure 2-19. Zingg diagram showing the classifica-tion of grain form. After Zingg (1935) and Blatt,Middleton and Murray, 1980.

26

Figure 2-20. Sneed and Folk (1958) classification ofgrain form.

0.3

0.5

0.7

0.670.33

COMPACT

COMPACT

PLATY COMPACTBLADED

COMPACT

ELONGATED

PLATY BLADED ELONGATED

VERYELONGATED

VERYBLADED

VERYPLATY

Sneed & Folk form index

dL d I

dL ds

ds

dL

0 1

1

0.20.3

0.4

0.7

0.8

0.9

ds

2d

L dI

3

ψp

=

0.5

0.6

of a particle, particularly after prolonged periods of transport (see below).

Transport

While in transport a grain will change in shape due to two major processes that may act in the transportingmedium. Interaction between grains during transport results in a change in shape and a reduction in grain size. Asgrains move along their sharp edges are lost due to chipping and grinding, increasing their roundness and sphericityand producing debris as finer sediment. Even when large clasts are partially buried in sediment beneath a currenttheir exposed surfaces are sand-blasted by particles in transport, thereby increasing their overall roundness.However, grains may be crushed during transport, producing smaller, more angular and less spherical grains.Overall, the most common outcome of transport is an increase in roundness and sphericity.

Figure 2-21 shows a plot of data derived by Humbert (1968) from experiments in which angular pebble-sizeclasts of chert (a very hard rock type) and limestone (relatively soft) were transported over several hundreds ofkilometres in a circular flume. Note that the roundness of the limestone increased more rapidly with transport thatthe chert because the softer limestone clasts are more prone to breakage (which increases rounding) than the harderchert clasts. Also, the rate of increase in rounding with increased transport distance is initially very large butbecomes smaller as the grains became rounder. This reflects the fact that an angular clast, with very sharp corners,easily becomes rounder as the fragile corners are broken off. However, a consequence of this increased roundingis that the remaining corners are more massive and difficult to break off. Therefore, as the degree of roundingincreases the rate of rounding decreases. The change in the sphericity of limestone clasts shown by the top curvein figure 2-21 is at a comparatively low rate throughout the entire distance of transport. This is because the changesthat are required to increase sphericity involve removal of a considerable amount of material from the clasts incontrast to the removal of relatively small sharp corners that results in initially dramatic increases in clast roundness.

Changes in roundness and sphericity with transport of gravel-size clasts occur at much greater rates thanfor sand-size grains (e.g., Kuenen, 1964). Because little quantitative research on changes in grain shape withtransport has been conducted on fine-grained sediment we can infer something of such changes from studies ofthe weight loss of sand with transport (increased rounding and sphericity both require weight loss). It has beenestimated that under water flows (e.g., rivers) the rate of weight loss of quartz grains finer than 2 mm is less than0.1% per 1000 km of transport. The rate increases only by a factor of two for feldspar grains. In all cases the rateof weight loss decreases sharply with decreasing grain size and becomes negligible for sizes finer than 0.05 mm.Such low rates of weight loss in very fine sediment is largely attributed to the fact that the particles spend muchof their time in transport in suspension well above the bed where interaction between particles is less frequent. In

27

air the rate of weight loss increases by a factor of 100 to 1000 times the rate in water. In water the abrasion betweenparticles that is required for such weight loss is inhibited by two factors: (1) the small submerged weight of the grainsgives them only little momentum during collisions, momentum that is required to cause damage; and (2) the viscosityof the water between colliding grains tends to dampen the collisions (further decreasing the transfer of momentum).In air, which has a considerably lower density and viscosity than water, the transfer of momentum between collidinggrains is much more effective and the resulting damage to the particles is more extensive. This (partly) explains thecommon observation that wind-blown (eolian) sands are typically much rounder and have higher values ofsphericity than sands from other environments. In ancient sediment that is clearly not of eolian origin good roundingand high sphericity are commonly interpreted to indicate that the sediment has been multi-cycled (i.e., the productof several cycles of weathering and erosion of pre-existing sedimentary rocks).

Transport may have another effect on the shape by selectively sorting clasts by shape. For example, imaginea cube and a sphere resting on the bed of a river. The sphere need only roll almost effortlessly along the bed inresponse to the fluid drag of the flowing water. In contrast, the cube must pivot over 45°, against its own weight,to roll a distance equal to its own length (if friction between the clast and the bed is large enough to inhibit it fromsliding). Thus, the sphere will more readily move under the river’s current and after a short time it will be separatedfrom the cube by a considerable distance. This processes is termed selective sorting by shape whereby easilytransported round, spherical clasts are removed from a sediment much more readily than their angular counterparts,even if both clast types have essentially the same weight. This mechanism has been suggested by studies of somemodern river gravels where there is an increase in roundness and sphericity in the downstream direction that cannotbe explained solely by abrasion (e.g., Gustavson, 1974).

There has been limited success in distinguishing ancient depositional environments on the basis of clastform. One such study showed that the deposits of gravel beaches and rivers (remember grain size) can bedistinguished on the basis of clast form. When data from these environments are plotted as in figure 2-20 the beachgravels tend to fall towards the bottom of the graph, in and about the very platy and very bladed fields, and theriver gravels tend to fall near the top of the graph, in and about the compact field. The distinction may be attributedto two causes. First, due to selective sorting of bladed clasts on the beach slope where rounded, compact clastswould be easily rolled off the beach by the backwash (water running back off the beach after a wave has rushedup its slope- the uprush is termed swash). This process would tend to enrich the beach with bladed clasts overtime. Second, the swash and backwash act to move larger, less mobile particles up and down the beach slope without

Limestone roundness

Chert roundness

Limestone sphericity

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Rou

ndne

ss o

r S

pher

icity

0 50 100 150 200Distance of transport in a circular flume (Km)

Figure 2-21. Change in rounding (Rw) and sphericity (y

P) with transport distance in a circular flume. Based on experiments

by Humbert (1968). After Blatt, Middleton and Murray, 1980.

28

moving them entirely. This back and forth motion would result in abrasion along the basal plane of the clast andas it is periodically flipped over it would become increasingly flat (i.e., bladed or platy). These two mechanisms areabsent in rivers where currents can more efficiently roll the more compact particles along their beds.

In conclusion, grain shape is an important sediment property that, like grain size, provides hints as to theprocesses that acted to transport particles in ancient depositional environments. It is also an important variablein determining the conditions that will cause a sediment to be transported by a specific mechanisms (see chapter4). However, as with grain size, caution must be used in inferring ancient depositional conditions on the basis ofgrain shape because it is also controlled by factors other than those acting in the depositional environment.

POROSITY AND PERMEABILITY

Porosity, the proportion of void or pore space in a sediment, and permeability, which is related to how wellthe voids are connected, are important properties of any sediment or sedimentary rock. This is particularly truebecause of the possible fluids that might be contained within sedimentary deposits; e.g., water, oil and/or gas,contaminants that humans have added to the earth surface. This section defines and discusses porosity andpermeability and shows how they are related.

Porosity

Any sediment contains a certain proportion of void space; that is, the proportion of the sediment that is notoccupied by particulate solids (i.e., grains). Porosity (P) is defined as the ratio of void volume (V

P) to total rock or

sediment volume (VT), expressed as a percentage of the total sediment volume. Therefore,

PV

VP

T

= ×100 Eq. 2-24

The pore volume is generally difficult to measure whereas the total volume of grains in a sediment (VG) is relatively

easy to determine (by weighing a specimen or by placing it in water and measuring the total displacement). Becausetotal volume is the sum of pore and grain volumes V

P can easily be calculated: V

p = V

T-V

G. Substituting for V

P in

Eq. 2-24, porosity is most commonly defined as:

PV V

VT G

T

=−

×100 Eq. 2-25

Porosity in natural sediment varies from 0 to approximately 70% due to a number of factors.

What Controls Porosity?

Packing Density

There are a variety of ways that grains may be arranged in a sediment and the spacing of the particles, referredto as the packing density, exerts a strong control on porosity. This may be very simply illustrated by the mannerin which perfect spheres of equal size may be packed (Fig. 2-22). This figure shows the two-dimensional view ofspheres resting with cubic and rhombohedral packing. In the case of cubic packing the grains are stacked directlyon top of and beside each other and the pore space is relatively large (48% porosity). In contrast, with rhombohedralpacking each grain rests in the space between the subjacent grains so that the grains fill more of the space and theporosity is lower (26%). These two styles of packing produce the theoretical maximum (cubic) and minimum(rhombohedral) porosities of any sediment composed of perfectly spherical particles of equal size. In nature,particles may be arranged in styles intermediate to these two end-members and their porosities will also be of anintermediate value (i.e., between 26 and 48%). The rate of deposition of a sediment exerts considerable control overgrain packing. Rapidly deposited sands tend to have a more open framework (like cubic packing) whereas slowrates of deposition often lead to tighter packing and therefore lower porosity.

Natural sediment is rarely composed of spheres so that porosity varies over a wider range than predictedabove. Porosity varies particularly with grain shape due to the differences in packing density that may be achievedby non-spherical grains. Figure 2-23 shows how much greater porosity may vary when the particles are non-

29

spherical. In nature, angular grains tend to support a looser packing arrangement (with larger porosity) than roundedgrains.

Grain Size

By itself, grain size has no direct effect on the porosity of sediment. For example, for a sediment composedof equant spheres with cubic packing it can be shown that Eq. 2-25 reduces to

P = − ×( )16

100π

Eq. 2-26

and is, therefore, constant (P = 48%) and clearly independent of grain size.

Indirectly grain size may have some influence on porosity. For example, packing of a sediment tends toincrease with the settling velocity of the particles. When relatively larger particles, with high settling velocities,impact on a substrate they tend to jostle the previously-deposited grains into tighter packing (with lower porosity).

Several studies have shown various relationships between grain size and porosity that could be explainedin terms of other factors. In unconsolidated sands porosity tends to increase as grain size decreases. This is becausethe finer sand tends to be more angular than coarse sand and, therefore, will support a more open packing. Insandstones the opposite trend has been recognized: porosity tends to increase with increasing grain size. Thisoutcome is due to the greater tendency of fine sand to lose volume when compacted upon burial (see below)compared to the lower compaction of coarse sand. Thus, the conflicting apparent relationships between grain sizeand porosity are due to factors other than grain size.

Figure 2-23. Influence of particle shape on packing and porosity.

<50% Porosity 0% Porosity

B. Bivalve shellsA. Porosity of sediment of tabular particles

P>90%

Figure 2-22. End-member styles of packing of spheres of equal diameter and their associated porosities.

Cubic packing Rhombohedral packing

Increasing packing density, decreasing porosity

48% porosity 26% porosity

30

Sorting

The relationship between sorting and porosity is fairly straight-forward: as sorting becomes poorer (i.e.,the standard deviation of the grain size distribution increases) porosity decreases. This is illustrated in figure 2-24 for the case of clay-free sand. The reason for this relationship is obvious: the poorer the sorting the wider rangeof grain sizes within the sediment and the greater likelihood of finer grains filling the void spaces between largergrains. What is perhaps less obvious is the reason for the differences in the curves for fine and coarse sand (Fig.2-24). Note that the minimum porosity for fine sand is greater than the minimum porosity for coarse sand. This isbecause a poorly sorted coarse sand has an abundance of fine material to clog the large pore spaces between thecoarse grains. However, a poorly sorted fine sand which is free of clay (as in the case of the curves shown above)the poor sorting is by virtue of the presence of coarser sands that will not clog the pores of the finer sand. A similardifference in minimum sorting would not be expected if clay were available to clog the pores of the fine sand. Indeed,the presence of clays tends to significantly reduce the porosity of sediment.

Post-burial Processes

Several different processes act in conjunction to alter porosity after a sediment has been buried due tosubsequent sedimentation on the overlying depositional surface. Figure 2-25 shows the typical reduction ofporosity of sands with burial (although the forms of such curves vary) that can be explained by the processessummarized below. Note that some post-burial processes cause an increase in porosity; such porosity is referedto as secondary porosity. The expression Primary granular porosity is often used to distinguish porosity due tothe character of the sediment prior to burial from porosity due to changes after burial.

Compaction ⎯ With burial the weight of overlying sediment may force the sediment into a tighter packing, therebyreducing its porosity. Quartz sand shows little response to compaction; experiments have reduced the porosityof quartz sand by only a few percent due to rearrangement and breakage under stress. The reduction in porositywith compaction increases with increasing proportions of ductile (deformable) particles, especially clays minerals.When first deposited, some muds (composed largely of clays) have a very high porosity (in excess of 70%) butcompaction with burial under a thousand metres of sediment reduces that porosity to 5 to 10%.

Cementation ⎯ The precipitation of cements within a sediment may begin almost immediately following depositionor make take place after millions of years and relatively deep burial. The most common cements are calcite and quartzthat precipitates out of solution from saturated waters between grains and gradually fills in the previously openvoid space, potentially reducing porosity to 0%.

Clay formation ⎯ Chemical alteration of some minerals, particularly feldspars which are a common constituent of

01020304050

0 1 2 3 4

Standard deviation (φ)

Por

osity

(%

)

Decreasing sorting

fine sand

coarse sand

Figure 2-24. Schematic illustration showing the relationship between sorting and porosity in clay-free sands. Based on datafrom Nagtegaal (1978) and Beard and Weyl (1973).

31

clastic sediment, results in the formation of very fine-grained clay minerals that tend to accumulate within the porespaces between primary sediment grains. The formation of clays, therefore, may increase the total volume of therock at the expense of the pore volume, thereby reducing porosity.

Solution ⎯ Waters within and passing through sediment and sedimentary rocks may not precipitate cements. Ifthe waters are undersaturated with respect to the minerals with which they are in contact, these waters may dissolvethe sediment grains, leaving a cavity or vug where there had once been solid rock (note that the vugs can rangefrom microscopic in size to the size of caverns). Such solution increases the total void space at the expense of thetotal rock volume; i.e., it increases the porosity. The additional void space produced by solution is termed secondaryporosity. Carbonate rocks (which are relatively soluble) are particularly prone to the development of secondaryporosity by solution.

Pressure solution is a process that can cause a reduction in porosity following burial when mineral grainsbegin to dissolve at the grain contacts (Fig.2-26). Minerals under stress solution tend to dissolve more readily thanunstressed minerals. Within a sediment the weight of overlying material is transferred between particles at the pointswhere they are in contact. At the contacts the pressures can be immense and solution may occur as long as thereis fluid within the adjacent pore spaces. The tangential contacts between grains become increasingly flat as materialis removed into solution. The boundary between grains that have undergone pressure solution is said to be sutured.As the grains lose material at their contacts the pore space becomes markedly shorter and its volume is reduced.Concurrent with pressure solution insoluble material may accumulate in the pore space along with cements.Combined, the reduction of pore length and the deposition of material in the remaining space can cause a dramaticreduction in pore space. Once again, carbonate rocks are most prone to pressure solution and it has beendocumented to occur at burial depths of only a few tens of metres. Modern carbonate sediment has porositiesranging from 40 to 80% but carbonate rocks rarely have porosities larger than 15%. Much of this reduction canbe attributed to pressure solution. Siliciclastic sedimentary rocks can also undergo pressure solution but this ismuch less effective in reducing porosity than in carbonates.

Fracturing ⎯ Fracturing of any rock, sedimentary or otherwise, will lead to an increase in porosity. Such additionalporosity contributes to the secondary porosity of a rock. Fracture porosity may be especially important in rocksin which primary granular porosity is not well preserved.

Permeability (Darcy's Law)

Permeability (k) is related to the ability of a fluid (liquid or gas) to flow through any porous substance (e.g.,a sediment). It was first defined by the French hydrologist Henri Darcy in 1851. Figure 2-27 very schematicallyillustrates how permeability is defined. Darcy’s Law (Eq. 2-27) is an empirical formula that predicts the rate of flowof fluid through a sediment. It can be used to experimentally determine permeability by measuring the rate of

Figure 2-25. Illustration showing the reduction inporosity of tertiary sands in Louisiana. After Blatt,Middleton and Murray (1980).

0

1

2

3

4

5

6

Dep

th o

f bur

ial (

km)

12 16 20 24 28 32 36 40 44

Porosity (%)

32

discharge of fluid of known viscosity through a specimen under a known pressure gradient and solving for k inEq. 6-3. Note that k is just an empirical constant, that depends on conditions within the rock , but one which hasan important physical interpretation (termed permeability).

Darcy's Law: Q kA p

L=

∆µ Eq. 2-27

The terms are defined in figure 2-27.

For the sake of simplicity we will re-arrange the formula as follows:

Q

Ak

p

L= × ×

1

µ∆

Eq. 2-28

Q/A is equal to the average (apparent) velocity (V, in cm s-1) at which the fluid is being discharged from the sediment.Assuming that there is no compression of the fluid this is also the average velocity at which the fluid will pass throughthe sediment. Eq. 2-28 may be re-written:

V kp

L= × ×

1

µ∆

Eq. 2-29

Beginning with the right-hand-side of this formula, ∆/L is the pressure gradient along the distance of flow and

Decreasing porosity

Porosity reduction by pressure solution

Figure 2-26. Schematic illustration showing the reduction in porosity due to pressure.

Figure 2-27. Schematic illustration defining Darcy’s equation. See text for discussion.

A L

Q

p

µk

p∆

Darcy's Law

Q is the fluid discharge (cm3 s-1);k is permeability (darcies, cm2);A is cross-sectional area (cm2);∆p is the pressure difference (bars, g cm-2);L is distance over which the flow passes (cm);µ is the fluid viscosity (centipoises, g cm-1 s-1 x 10-1).

33

Region of low velocityand high resistance

Figure 2-28. The relationship between the cross-sec-tional area of a pathway and the relative proportion of thatcross-sectional area where fluid velocity is small due toviscous resistance along the wall of the pathway.

Figure 2-29. Definition of tortuosity. See text fordiscussion.

Pathway

L2

L1

Tortruosity = L2/L1

represents the force that is acting to push the fluid through the sediment. As the pressure gradient increases sowill the average velocity. The term 1/µ is a measure of how easily the fluid can flow through the sediment (i.e., asviscosity, a measure of fluid resistance to deformation, increases, 1/µ decreases). Therefore, as viscosity increasesthe velocity must decrease. So the velocity of a fluid passing through a sediment depends on the force appliedto that fluid and on how much the fluid will resist flow through the sediment. But we have not yet considered howthe sediment itself will influence the velocity of the fluid passing though it; that is the role of the permeability termin the Darcy’s Law. Permeability is a measure of how the pathway(s) through the sediment will affect the resistanceof the flow of fluid; that is, it is related to the average diameter and total length of the pathways. Certainly if the“holes” or “pathways” through the sediment are small it will be more difficult to push the fluid through than if theholes are large. This is because the greatest viscous resistance to flow occurs along the walls of the pathway; thesmaller the diameter of the pathway the greater the viscous resistance. This is illustrated in figure 2-28 which showsthat for a pathway with a small cross-sectional area the region close to the wall of the pathway, where flow velocityis low due to viscous resistance, will be a relatively large proportion of the total cross-sectional area of the pathway.Conversely, when the cross-sectional area is large the region of low velocity will be relatively small compared tothe total cross-sectional area of the pathway. Because k has units of cm2 (this is required to make the equationdimensionally correct) it is useful to think of permeability as some measure of the cross-sectional area of thepathways through which the fluid flows. Therefore, all other things being equal, a large pathway with large cross-sectional area (i.e., k is large) will allow fluid to move at a higher average velocity, in response to a given pressure,than will a smaller pathway (k is small) with fluid under the same pressure. Of course, the pathways through asediment are very variable in size so that permeability should be thought of as some average cross-sectional areafor the entire network of pathways through a sediment. This is a simplistic but useful view of permeability. Thetotal viscous resistance will also vary with the total distance that the fluid must pass along the pathways; thus,permeability also varies inversely with the tortuosity of the pathways. In this context tortuosity is a measure ofthe degree of deviation of a pathway from a straight line; the more irregular the pathway the greater the tortuosity(Fig. 2-29). Thus, the larger the tortuosity of the pathways through a rock the lower the permeability. Again, thisis related to the degree of resistance to flow that is due to the total character of the pathway.

The units of permeability are termed darcies (d) and have dimensions of cm2. However, the permeabilityof many rocks is much less than 1 darcy so that it is commonly expressed in terms of millidarcies (md; 1/1000 of adarcy). As originally defined, 1 darcy is the permeability which allows a fluid of one centipoise viscosity to flow

34

Cleangravel

clean sands,mixtures of cleansands and gravels

Very fine sands, silts,mixtures of sand, siltand clay, glacial till,stratified clays, etc.

Unweatheredclays

105 103104 102 10 1 10-1 10-2 10-3 10-4 10-5

Permeability, k (Darcies)

Figure 2-30. Values of permeability of unconsolidated sediments. After Pettijohn, Potter and Siever, 1973.

at one centimetre per second, given a pressure gradient of one atmosphere per centimetre. Figure 2-30 shows therange of permeability of unconsolidated sediment.

Controls on Permeability

It should be obvious from the above discussion that any property of a sediment that controls the size ofthe pore spaces and/or the tortuosity of the pathways will control the permeability. These are much the same asthe controls on porosity, with a few notable exceptions.

Porosity & Packing

Figure 2-31A schematically shows the general relationship between porosity and permeability. Permeabilitytends to increase with increasing intergranular porosity. The reason for this should be made clear by re-examiningfigure 2-22: the tighter the packing (and the lower the intergranular porosity) the smaller the “pathways” throughwhich a fluid will move and the lower the permeability of the sediment. Therefore a sediment of a given size withcubic packing will have a higher permeability than a sediment of the same size with rhombohedral packing. However,many rocks (especially shales and some carbonates, particularly those with secondary porosity formed by solution)may have high porosity but low permeability. This occurs when the pore spaces are not well interconnected andthe “average” pathway size is very small. Conversely, the presence of fractures in rocks may significantly increasethe permeability while increasing the porosity only slightly. A fracture of 0.25 mm width. in a rock will allow thepassage of fluid at a rate equal to that passing through 13.5 metres of unfractured rock with a uniform permeabilityof 100 md. The relatively large permeability along fractures is due to a combination of their size, compared to thepore spaces of many rocks, and also because they are especially well-connected. Thus a rock with very low porositymay have very high permeability if it is extensively fractured.

Grain Size and Sorting

Unlike porosity, permeability varies with the grain size of the sediment. This arises from the fact that, inaddition to packing, the size of the pores between grains is determined by grain size. To illustrate the relationshipbetween grain size and pore area consider a sediment of uniform spheres with cubic packing (see Fig. 2-22). It canbe shown that the average pore area (P

A) is related to sphere diameter (d) by:

PA = 0.74 d2 Eq. 2-30

Figure 2-32 illustrates the graph of the solution to Eq. 2-30 and shows how pore area increases with increasing grainsize. Because the pore spaces form the pathways for fluid flow, the permeability of the sediment varies in a similarmanner: as grain size increases so does the size of the connected pores (although the total pore volume remainsunchanged) so that, all other factors being constant, and permeability also increases. This tendency also is shown

35

Figure 2-31. A. Schematic illustration showing the relationship between porosity and permeability (after Selley, 1982).B. Illustration showing the effect of grain size and sorting on the permeability of clay-free, unconsolidated sands (based ondata from Nagtegaal, 1978 and Beard and Weyl, 1973). C. A graph showing the reduction of permeability with burial basedon data from the Ventura Oil Field, California (based on data from Hsü, 1977).

very

well

wellmoder

ate

poor

ly

very

poor

ly

0 1 2 3Standard deviation (φ)

0.01

0.1

1

10

100

1000

Per

mea

bilit

y (m

d)

Sorting

coarse sand

medium sandfine sand

very fine sand

Primary

interg

ranular p

orosity

Shale, chalk and

vuggy rocks

Fractures

0.1 1.0 10 100 10000

10

20

30

40

Permeability (md)

Por

osity

(%

)

0.1 1 10 100Permeability (md)

1000

0

1

2

3

4

Dep

th o

f bur

ial (

km)

A CB

for natural sands in figure 2-31B.

The sorting of a sediment will have an obvious effect on its permeability. Well-sorted sands will have openpore spaces and, therefore, high permeability. As sorting becomes poorer the finer fraction will tend to clog pores(i.e., reduce their average area) and pathways so as to retard the flow of fluid, thereby reducing permeability. Figure2-31B shows that sorting may cause permeability to vary over several orders of magnitude in sediment of the samemean size.

Post-burial Processes

Like porosity, processes acting after burial of a sediment can have a considerable influence on permeability.Figure 2-31C shows an example of the typical decrease in permeability that occurs with increasing burial depth.

Figure 2-32. Relationship between graindiameter and average pore area forspheres with cubic packing. Curvegraphically illustrates solutions to Eq. 2-30.

.01

1

.1

10

100

1000

.001 .01 .1 1 10 100 1000

Pore area (mm2)

Gra

in s

ize

(mm

)

36

Compaction, cementation, pressure solution, and formation of clay minerals all act to reduce the permeability of arock by reducing the size of pore spaces and by increasing the tortuosity of the pathways. As noted above, however,fracturing can increase permeability immensely. Also, solution along pathways may enhance permeability.

Directional Variation of Permeability

An important attribute of the permeability of many rocks is that it may vary with direction. Such anisotropic(not equal in all directions) permeability is particularly notable in bedded sediment where permeability is generallygreatest along planes parallel (or at a slight angle) to bedding. The presence of bedding usually reflects somevariation in grain size through a package of sedimentary rock. Layers of fine sediment bounding coarser layers willimpede permeability in the direction perpendicular to bedding and fluid will flow most easily through the coarserlayers, parallel to the plane of bedding. Permeability may also be larger in a given direction along the plane parallelto bedding. When elongated grains are abundant in a deposit they commonly have a preferred orientation and fluidsmoving through the sediment will receive least resistance along the direction parallel to the grain axes. Fracturesin rocks commonly develop with a preferred strike direction and permeability will be greatly enhanced in the directionparallel to strike.

GRAIN ORIENTATION

Introduction

In sedimentology the term fabric refers to all aspects of the spatial arrangement of particles in a sedimentand includes both packing (which was dealt with in the previous section) and grain orientation. However, in practice,the term has come to refer principally to the orientation of grains (this practice may have been inherited frommetamorphic geology). Grain orientation is one of the fundamental sediment properties and should be includedin any complete description of an in situ sediment or sedimentary rock. It may influence other properties such aspermeability; therefore, grain orientation may be an important consideration in predicting the direction of movementof contaminant through a sediment.

Grain orientation is a potentially powerful tool for the interpretation of processes that acted on a sedimentat the time of deposition. As we have seen, other sediment properties have been of limited value in interpretingsedimentary processes. Both grain size distribution and grain shape may reflect something of the processes in thedepositional environment but both may also preserve characteristics inherited from the source material thatproduced the sediment. In contrast, grain orientation is determined at the time that the sediment was deposited;it inherits no attribute from its source material. Once a particle has attained a preferred orientation at the time ofdeposition it remains constant unless: (1) it is modified by compaction, a trivial concern in sand-size sediment; (2)it is modified by post-depositional disturbance: soft-sediment deformation or bioturbation (both relatively easilyrecognized); (3) it is modified by tectonic deformation (shearing and folding by tectonic processes). This sectionof these notes describes the nature of grain orientation, how directional data are graphically displayed and treatedstatistically, and how such data may be used to infer processes that acted in ancient depositional environments.

Measuring grain orientation

The orientation of a grain is determined by measuring the directional attributes of the long (a-axis) andintermediate (b-axis) axes of particles. In gravel-size sediment we may use a compass to directly measure the trendof a clasts a-axis and the strike and dip of its a-b plane (Fig.2-33), with respect to the plane of the surface on whichthe particle lies. The angle of dip of the a-b plane is termed the imbrication angle and the direction of dip is termedthe imbrication direction of the clast and such a dipping clast is said to be imbricate. From studies of modernsediment it is well known that particles tend to be imbricate into the flow (i.e., the imbrication direction points inthe up-current direction).

In sand-size sediment the orientations of the apparent axes are measured in thin sections but suchmeasurements are generally limited to grains with apparent length to width ratios of 3:2 (this ensures that the true

37

longest axis is measured; when grains have lower ratios it is difficult to distinguish the length from the width). Notethat for the measured directions of axes to be of any value the orientation of the specimen in its original positionin outcrop must be marked directly on the specimen; aspects of the original orientation include: direction to top,direction to north, the plane of bedding seen in outcrop and the horizontal plane. It is particularly important to notethe orientation of the plane of bedding because grains will have been aligned on their depositional surface, the samesurface that controls the geometry of bedding . The trends of the apparent a-axes are measured on thin sectionscut in the plane parallel to bedding in a specimen of sediment or rock. The imbrication direction is determined fromthin sections cut in the vertical plane (normal to bedding). The determination of imbrication direction must be madeon the basis of two thin sections: one in the vertical plane that is parallel to the mean long axis trend, as measuredon the bedding plane, the other in the vertical plane trending normal to the mean long axis trend. The reasons forthis will become apparent in the following section.

Types of grain fabric

The grain fabric of a sediment, determined by either method described above, may be classified into two broadgroups: isotropic fabric and anisotropic fabric. The latter may be further classified according to the specificdirectional attributes of the particles.

Isotropic fabric

A sediment that displays no preferred alignment of grains is said to have an isotropic fabric (i.e., grainorientation varies uniformly and displays no preferred alignment of particles; syn. disorganized fabric; Fig. 2-34).Such a fabric will appear the same in every plane through a specimen. Any sediment that consists of particles witha high sphericity will appear isotropic for the simple reason that it is not possible to measure axes orientations (Fig.2-34A). In sediment of non-spherical particles (Fig. 2-34B) such fabrics may be primary (i.e., developed at the timeof deposition) or result from reworking of the sediment by burrowing organisms.

Anisotropic fabric

Any sediment that displays a preferred alignment of particles in any direction and in any plane is said tohave an anisotropic fabric (i.e., not equal in all directions). Figure 2-35 shows the two common forms of anisotropicfabric that develop in sand and gravel.

plane of beddingstrike

dip

imbricationangle

a-b plane

ab

Figure 2-33. Illustration showing the directional attributes of gravel size sediment that are commonly measured in outcrop.Note that the a-b plane is the plane passing through the clast such that the surface defined by the intersection of the clastand the plane has the maximum possible area (i.e., the a-b plane is the maximum projection plane) and that the c-axis ofthe particle is orthogonal to the a-b plane. In practice, for such a clast we measure the trend of the a-axis, the strike of thea-b plane and its dip direction, which is at 90° to the strike, and angle; termed the imbrication direction and angle,respectively).

38

a-axis transverse, b-axis imbricate [a(t) b(i)]

The particle shown in figure 2-33 has this orientation. Figure 2-35A also shows this fabric as it would beseen in three orthogonal planes through a sediment or sedimentary rock. From studies of modern sediment, wherecurrent directions are known, we recognize this style of fabric as being characterized by a-axes (seen on the planeof the depositional surface; shown as the bedding plane in Fig. 2-35A and B) that are aligned transverse to the currentdirection. Imbrication of the a-b plane is up-current (i.e., into the flow) and the direction of imbrication is along thetrend of the b-axis. Hence, the shorthand notation: a(t) b(i) - a-axis transverse to flow and b-axis imbricate. Notethat the grains appear horizontal in the vertical plane parallel to a-axes on bedding.

a-axis parallel, a-axis imbricate [a(p) a(i)]

This fabric is shown in figure 2-35B. In this case the a-axis of each grain is aligned parallel to the currentand the a-b plane is imbricate into the flow; the imbrication direction is along the trend of the a-axis. The short-hand notation for this fabric is a(p) a(i): a-axis parallel to flow and a-axis imbricate. The mean trend of grains seenin the vertical plane aligned normal to a-axes is essentially parallel to bedding or nearly horizontal.

Complex anisotropic fabric

Some sediment displays an equal or unequal mixture of a(t) b(i) and a(p) a(i) fabrics. Such sediment has acomplex fabric that may be difficult to distinguish from an isotropic fabric.

Figure 2-34. A. Isotropic fabric in a sediment made up of spherical particles. B. Isotropic fabric due to random alignmentof grains.

Figure 2-35. Schematic illustration showing examples of anisotropic fabric. A. Grain a-axes are aligned transverse to flowand b-axes dip into the flow. B. Grain a-axes are aligned parallel to and dip into the flow. See text for further explanation.

BA

Isotropic fabrics

bedding bedding

FLOW a(p)a(i)B

FLOWa(t)b(i)

A

Anisotropic fabrics

beddingbedding

39

The problem with measuring grain orientation on sections

Take a close look at figure 2-35 (A and B) and consider how the angular relationship between the grain andthe plane of view will influence the apparent axes lengths and orientations. In figure 2-35A, on the plane of bedding,we see the a-axes lengths but the b-axes lengths are shortened by an amount that depends on the angle of imbrication(the steeper the angle the shorter the apparent b-axis; if the grains were vertical we would see the c-axes on thatplane). In the plane that is vertical to bedding and transverse to the trend of a-axes we see the form of the grainthat is defined by the b-c plane. In the vertical plane that trends parallel to the long axes on the bedding-parallelplane we see the form of grains that is defined by the a-c plane (actually the apparent c-axis will exceed the true c-axis length as the angle of imbrication increases; if the grains were vertical we would see the b-axis on this plane).This illustrates the problem that the two dimensional view of a three dimensional object depends on the angularrelationship between the directional attributes of the object and the orientation of the plane of view.

In thin sections cut from rocks this problem is compounded by the fact that we have little control over wherethe plane of view will intersect the grains. The control on apparent axes lengths by the plane of intersection of thesurface with the grain is shown in figure 2-36B and C. Figure 2-36D shows how the angular relationship betweena grain and the plane of section will affect not only its apparent axes lengths but also their orientations. Specifically,if the plane of view is not in and parallel to the a-b plane of the particle we will see a distorted form of the grain. Thus,we have a considerable error in measuring grain orientation in rocks consisting of sand-size sediment. This isminimized if care is taken to prepare the first section in the plane of bedding (a known surface with a predictablerelationship to the orientation of the grains). Also, because the angle at which the grains dip below the beddingsurface is relatively small (averaging up to approximately 25°) the distortion of form and orientation is relatively small.Finally, when collecting such data we must ensure that a large enough number of grains are measured to balanceout such error (i.e., normally in excess of 100 grains must be measured but this number will vary depending on theamount of true variation in grain orientations). It should be clear from this that when such data are described it isvery important to make it clear that data are based on measurements made on apparent axes, not true axes; all suchdata collected from thin sections have this limitation. In all remaining figures that deal with grain orientation in thischapter only data on apparent axes are reported.

Displaying directional data

Grain orientation is just one of a large number of types of directional data that may be gathered by asedimentologist (well see others through the course of these notes). When such data have been compiled theymust be displayed to recognize, at least visually, any significant directional trends that may be interpreted. For

Plane of section

A

D

Plane of sectionC

Plane of sectionB

Figure 2-36. Illustration showing the relationship betweenapparent axes lengths and orientation due to the position of theplane of section. A. A three-dimensional sketch of a particleas viewed at an angle to its a-b plane. B. Shaded area showsthe form and area of the grain at its intersection with a plane(termed the plane of section) that is parallel to the a-b plane,but well above the a-b plane. Note the reduction of axeslengths in the image seen on the plane of section. C. Shadedarea shows a larger form and area than in B as the plane of thesection moves nearer to the a-b plane of the grain. Note thattrue axes lengths will be seen only if the plane of the sectionis in the a-b plane of the particle. D. Shaded area shows theform of a grain in section oblique to the a-b plane of the particle.Note that apparent axes (on the shaded area) are not parallelto the true axes of the grain. Similar distortion arises whengrains have variable orientation with respect to a fixed planeof section.

40

example, if we have a set of measurements of the imbrication direction of a number of particles we will need todetermine whether the data are consistent and what is the mean imbrication direction. The imbrication directionwill indicate the upstream direction with respect to the current that deposited the sediment. If such deposits containfine gold particles we might want to know where the sediment came from so that we might find its source; theimbrication direction will point towards that source. Statistical treatment of directional data will be considered inthe next section.

Sedimentologists display directional data on a form of circular histogram that is commonly referred to as arose diagram. Consider the directional data plotted in the form of an ordinary histogram in figure 2-37A. The numberof measurements (i.e., the frequency) in 30° classes is shown over the range 0 to 360° (a full circle; north is generallytaken to be towards 0°). You can see from this histogram that the modal direction is in the range 120-149° (thesoutheast) and the data varies almost symmetrically about that mode. Inset in figure 2-37A is a rose diagram forthe same data and it shows the same trend. However, it is obviously easier to visualize the directional significanceof the data plotted on the rose diagram. Figure 2-37B shows another set of data plotted on a regular histogram anda rose diagram. In this case we can recognize two prominent modes, one from 0 to 29° and another from 330 to 359°.It is not readily apparent how these two modes are related until we look at the form of the rose diagram (inset in Fig.2-37B). The rose diagram more clearly shows the overall trend of the directional data (in this case pointing towardsthe north-northwest). The visual impact of a rose diagram is important when we compare different data sets andparticularly when we plot such diagrams on maps.

Figure 2-38 illustrates how rose diagrams are constructed using the example of data describing imbricationdirection. The class intervals define the angle of curvature of pie-shaped segments that have a length equal to thenumber (this may be expressed as a percentage) of observations per class (i.e., the frequency per class). In figure2-38 the class interval is 30° and the scale of frequency is shown radiating from the centre of the circle. The centreof a rose diagram is commonly reserved to note the total number of observations used to construct it.

Two forms of rose diagram are shown in figure 2-38 based on the data given in the table. In rose A the distance(L) from the centre of the circle to each value representing the number of observations per class (N) is equal to the

Fre

qu

ency

(Num

ber

of o

bser

vatio

ns p

er c

lass

)

0˚N

90˚E

180˚S

270˚W

360˚N

0

5

10

15

0˚N

90˚E

180˚S

270˚W

360˚N

0

5

10

15

Direction Direction

0 5 10 15

Number of observations

62

N N

0 5 10 15

Number of observations

62

Figure 2-37. Examples of directional data plotted as conventional histograms and as rose diagrams (inset).

41

value itself; that is L = N in whatever units you like; the distance between each value on the frequency scale is equal.Thus, a segment of the rose corresponding to a class containing 5 observations is five times longer (extendingoutward from the origin) than a segment corresponding to a class containing only 1 observation. Rose B showsanother frequency scale that many argue is more appropriate. In the case of Rose B the distance from the centreof the circle to any value is given by L = N0.5. Therefore, the spacing of the frequency increments decreases outwardform the centre of the circle. The reason for this is that in this type of diagram the frequency of observations perclass is not shown in terms of the length of the segments but is shown by the relative area of each segment. Forexample, when such a scale is used the area of a segment of the rose representing 5 observations is equal to fivetimes the area of a segment representing 1 observation. Because rose diagrams are used for their visual impact,some sedimentologists argue that this is a more representative scale than the linear scale. Note the difference inthe form of the two roses in figure 2-38. The equal area scale reduces the visual impact of the difference betweenintervals (compare the relative sizes of the segments corresponding to the 180-209° and 210-239° intervals). All rosediagrams in these notes will be presented with a linear frequency scale.

The form of the rose diagram indicates the directional trends of the data in a visually useful manner. Likeany such data, the distribution may have one or more modes and is termed unimodal if one mode is present, bimodalif two modes are present (bipolar if the modes are at 180° to each other) or polymodal if more than two modes arepresent (Fig. 2-39). Also, the data may vary symmetrically or asymmetrically about the mode(s). The interpretationof the form of a rose will depend on what directional attributes of a sediment have been measured to collect the data.In the case of grain imbrication, the rose points towards the average direction of dip of the a-b planes of the particlesand this direction is at 180° to the current that formed the imbrication. Therefore, the rose shown in figure 2-38 showsa mean imbrication direction to the south-southeast, produced by a current flowing towards the north-northwest.Many other directional attributes point in the direction of the current under which they formed and the roses thatthey produce will point directly in the flow direction.

Many types of directional data are like that shown in figure 2-38 based on grain imbrication; i.e., theimbrication direction is a specific direction determined by the dip of the a-b plane of a particle. However, manyfeatures for which directional data may be collected do not point to a specific direction. Consider the long axisorientation of a particle; it does not point in a given direction but rather lies on a directional trend. For example,an a-axis oriented towards 10° has the same orientation as an a-axis oriented towards 190°. Thus, the a-axis of thegrain is said to trend along 10-190°. Such directional data are said to be bidirectional; pointing in either of twodirections which are at 180° to each other. Figure 2-40 shows how such data are commonly dealt with in theconstruction of rose diagrams. Note that rose diagrams based on bidirectional data are symmetrical with only oneside of the rose representing actual measurements and the other side is just its mirror image to give the impressionof the bidirectional nature of the data. Figure 2-41 shows the forms of rose diagrams produced by measuring grain

0˚

90˚

180˚

270˚

30˚

60˚

120˚

150˚210˚

240˚

300˚

330˚

12

3

45

12

34

50˚

30˚

60˚

90˚

120˚

150˚180˚

210˚

240˚

270˚

300˚

330˚Raw data: 96, 121, 135, 146, 152, 160, 165, 172, 178, 185, 186, 201, 209, 212, 219

Class Interval Frequency

90-119˚ 1120-149˚ 3150-179˚ 5180-209˚ 4210-239˚ 2 Total: 15

A B

15 15

Equal length Equal area

(degrees)

frequency frequencyData recording imbrication direction

Figure 2-38. Illustration showing the construction of a rose diagram. See text for explanation.

42

Figure 2-39. Classification of form of rose diagram. After Pettijohn, Potter and Siever, 1973. See text for explanation.

28

Bimodal

22

Unimodal

N N

44

Polymodal

N

29

Bipolar

Figure 2-40. Illustration showing the significance of a rose diagram basedon bidirectional data. In this case, the rose diagrams represent the long axisorientation of grains seen on a bedding plane; one particle is shown but theroses represent a population of grains with a mean a-axis orientationparallel to that shown on the example particle. Because the a-axis of aparticle (like that shown in the illustration) has a trend but no absolutedirection, either of the top roses may equally well describe its orientation.However, visually these two roses give two diametrically opposing impres-sions of the a-axes they represent because they impart a sense of directionto the data; the left-hand rose suggests that the axes point to the west andthe right-hand rose suggests that the axes point to the east. To counter thisperceived sense of direction it is common practice to construct rosediagrams that are symmetrical, as shown in the lower part of the figure.Note that the lower rose has doubled in size without increasing the numberof measurements on which it is based.

N68 68

68

axis of symmetry

+

=

axes orientation in sediment with the anisotropic fabrics described in the previous section. A word of caution: becareful to note what is represented by any rose diagram. For example, truly directional data, based on structuresproduced by tidal currents that flow in directions that vary by 180° over time, may produce essentially symmetricalroses like those shown in figures 2-40 and 2-41.

Statistical treatment of directional data

Normally, directional data are collected in sets of N observations from some particular deposit or associatedstructure. Like grain size data, a set of directional data will be distributed about some mean direction and will havea range of variation about the mean direction. Unlike grain size data, directional data cannot be treated by regularstatistics because each value in a set is a vector quantity and each vector has two components: each observationin a set of directional data will have a direction (θ) and a magnitude (normally each vector has unit magnitude, i.e,its magnitude equals 1; such a vector is termed a unit vector). The descriptive statistics are similar to those appliedto scalar quantities (like grain size) but their computation is fundamentally different. Without going into the detailsof vector algebra this section provides an outline of the various statistics and their implications.

For any distribution of directional data the mean is the resultant vector (sometimes called the vector mean)formed by summing all of the unit vectors that comprise the data set (e.g., Fig. 2-42). Once again, the resultant vector

includes a direction (θ ) and a magnitude (R):

θ = −tan 1 w

vEq. 2-31

43

Note that θ must be corrected as follows: if w > 0 and v > 0 then θ = Eq. 2-31;

if w > 0 and v < 0 or w < 0 and v < 0 then θ = Eq. 2-31 + 180˚;

if w < 0 and v > 0 then θ = Eq. 2-31 + 360˚

and

R v w= +2 2 Eq. 2-32

where

w ni ii

N or NC

==∑ sinθ

1

Eq. 2-33

and

v ni ii

N or NC

==∑ cosθ

1

Eq. 2-34

and N is the number of observations in the data set. Note that directional data may be treated as either individualmeasurements (i.e., as raw data) or grouped data (i.e., data are expressed as frequency per class interval). In thecase of ungrouped data n

i is equal to 1 (unit magnitude of each vector) and θ

i represents each of the measured

directions (from i = 1 to N) of the N vectors in the data set. When data are grouped the quantities above are: ni is

the number of observations in each class interval (from i = 1 to NC) and θi is the mid-points of each class interval

(from i = 1 to NC) and NC is the number of class intervals (i.e., NC = 360°/CI, where CI is the class interval in degrees).

The magnitude of the resultant vector can also be expressed as a percentage (L) of the total number of

Figure 2-42. Example of the derivation of a resultant vector(syn. vector mean; large arrow) by the summation of unit vectors(small numbered arrows). Each unit vector has the directionshown and a magnitude (r

i) of 1. Use Equations 2-31 to 2-35 to

demonstrate the calculation of the direction and magnitude ofthe resultant vector. Note that the arithmetic mean would be146° and clearly not representative of the three vectors.

Figure 2-41. Rose diagrams representing two forms of anisotropic fabric (see Fig. 2-35 but note that the blocks shown hereare rotated at 180° to the blocks in Fig. 2-35).

A

BC

North

68

bedding

S N

68

bedding

W E

C

B

a(t)b(i)

N

A68

N

A

BC

North

68

bedding

S N

68

bedding

W E

A

C

B

a(p)a(i)68

(i) (ii)

1

+

2

3+

=

θ1

= 346˚

r1

= 1

θ2

= 24˚

r2

= 1

θ3

= 67˚

r3

= 1

θ = 25.5˚

R = 2.52

L = 84%

44

observations (or more specifically, as a percentage of the total length of the unit vectors that comprise the dataset):

LR

N= ×100 Eq. 2-35

The magnitude of the resultant vector, particularly when it is expressed as a L reflects the amount ofdispersion of the data (i.e., the degree of variation of directions in the data set). Another, more important, measureof the dispersion of the distribution of directional data is p (defined as the probability that the data are from apopulation that is uniformly distributed) where:

p e L N= − ×1 0 00012( . ) Eq. 2-36

p ranges in value from 0 to 1 where p = 0 indicates that there is no chance that the data are uniformly distributed(i.e., the data are from a population that has a preferred orientation) and p = 1 indicates that the data are from apopulation that is definitely uniformly distributed (i.e., the population has no preferred orientation).

Table 2-4 provides an example of the treatment of directional data, both as ungrouped and grouped data.With a little experience it is possible to acquire some intuition about these statistical properties on the basis of theform of the rose diagram representing a data set.

Interpretation of grain orientation

In the previous section one interpretive use of imbrication direction has already been established; it pointsin the direction 180° to the direction of the current that acted on the particles. Most workers who have includedgrain fabric in studies of ancient rocks have used their results to reconstruct paleoflow directions. However, thereare several other ways in which fabric may be interpreted. To begin with, the anisotropic fabrics can be interpretedin terms of how the particles moved under a flowing current. The a(t)b(i) fabric is produced by currents that rollthe particles along the bottom. This is the typical fabric of gravel size sediment; because of their large mass it isdifficult to move them in any other way but to roll them along the bottom. In contrast, the a(p)a(i) fabric tends tobe the most common fabric in sands and is thought to develop under relatively high rates of deposition and sedimenttransport and high sediment concentration. Thus, for a given grain size a(p)a(i) fabrics in gravel will be producedunder much more energetic currents than those that produce a(t)b(i) fabric. The classic experimental studies ofgravel orientation by Johansson (1965, 1976) are required reading for anyone concerned with gravel fabrics.Isotropic fabrics may result from rapid deposition of thick slurries which restrict the independent movement of aparticle, hindering the establishment of a preferred orientation.

Figure 2-43 shows an example of a complex anisotropic fabric based on data collected from a sandstone thatwas deposited in a shallow-marine environment. Two major modes, at 90° to each other, are clearly seen on the rosediagram. This fabric was produced by oscillating currents induced by storm waves in a shallow sea and thebimodality is interpreted in terms of a mixture of transport modes at the time of deposition. The NW-SE moderepresents grains with long axes aligned normal to the wave-induced current; these grains were rolled along thebed. The SW-NE mode represents grains that were aligned with their a-axes trending parallel to the current andrepresent sand that was rapidly deposited from suspension. In this particular case, the bimodality of the fabricreflects processes that act under currents produced by waves and may provide a basis for inferring wave-inducedcurrents in other ancient deposits.

In most sands the imbrication angles of grains range from 5 to 20°, although higher angles have been recordedin sands that are very rapidly deposited by sediment gravity flows. Some research has suggested that imbricationangle may also reflect the power of the current that act to align particles. Experiments by Gupta et al. (1987) showeda distinct steepening of imbrication angles with increasingly strong flows. However, recent experiments have shedsome doubt on the relationship between imbrication angle and flow strength (Arnott and Hand, 1989).

Results of one recent study have suggested that grain imbrication may be particularly useful in interpretingthe nature of transporting currents. Specifically, the results suggested that variation in imbrication angle through

45

Table 2-4. Example of the statistical treatment of ungrouped and grouped directional data.

Raw data: 184 187 191 196 198 201 204 205(degrees) 205 207 208 210 212 214 216 222

224

Grouped data:Class Interval Midpoint Frequency

180-189° 184.5° 2190-199° 194.5° 3200-209° 204.5° 6210-219° 214.5° 4220-229° 224.5° 2

Total: 17

Treatment of ungrouped data:

w ni ii

N= ∑ = −

=sin .θ

17 04 v ni i

i

N= ∑ = −

=cos .θ

115 13

θ θ θ=−

−= < < = − + ∴ = + =−tan

.

.. ; . .1 7 04

15 1324 95 180 24 95 180 204 95 w 0 and v 0 then Eq.2 31 o o

R L= − + − = = ×( . ) ( . ) . ;.

15 13 7 04 16 6916 69

172 2 100 = 98.17%

p e= = ×− × × −1 98 17 17 0 0001 82

7 67 10( . . ) .

Treatment of grouped data:

w ni ii

NC= ∑ = + + + + = −

=sin sin . sin . sin . sin . sin . .θ

12 184 5 3 194 5 6 204 5 4 214 5 2 224 5 7 06

o o o o o

v ni ii

NC= ∑ = + + + + = −

=cos cos cos cos cos cos. . . . . .θ

12 184 5 3 194 5 6 204 5 4 214 5 2 224 5 15 08

o o o o o

θ θ θ=−

−= < < = − + ∴ = + =−tan

.

.. ; . .1 7 06

15 0825 09 180 25 09 180 205 09 w 0 and v 0 then Eq.2 31 o o

R L= − + − = = ×( . ) ( . ) . ;.

15 08 7 06 16 6516 65

172 2 100 = 97.95%

p e= = ×− × × −1 97 95 17 0 0001 82

8 25 10( . . ) .

Note the slight discrepancy between calculations based on grouped and ungrouped data due to the useof the class mid-points in the case of grouped data.

46

151

N

10 observations

Apparent a-axes orientations froma shallow marine sandstone

FLOW

Figure 2-43. An example of a complex anisotropic fabric. Rose diagramshows the trend of apparent long axes seen in the plane parallel to beddingfrom the Upper Cretaceous Chungo Member (Wapiabi Formation) in theRocky Mountain Foothills. Note the prominent bimodality of the rose. Seetext for explanation. After Cheel and Leckie (1992).

a sediment may record changes in current strength and direction over time. Figure 2-44 shows data collected bydetermining the mean imbrication angle in thin (0.1 to 0.2 mm) vertically contiguous layers through a sediment. Inthe plots in figure 2-44 each point represents the mean imbrication angle (and direction ) in one such layer. Therose diagrams in figure 2-44 are based on all data in the subjacent plot. Figure 2-44 A shows data from the depositsof a river in which currents flowed consistently in the same direction over the course of deposition of the sediment(such currents are said to be “unidirectional”). Within these river deposits the mean imbrication angle is atapproximately 13°, dipping into the current. There is little systematic variation in imbrication angle through thedeposit, reflecting the constant nature of the flow strength over the period of deposition. For comparison, figure2-44B shows data from the deposits of a shallow marine setting that was influenced by waves, specifically powerfulstorm-generated waves. The currents produced by such waves would flow back and forth over 180° over periodson the order of 10 seconds, and are termed oscillatory or “bidirectional” currents. In contrast to the river-formedfabric, the mean imbrication angle is parallel to bedding, reflecting the alternating direction of the current (i.e., thenumber of grains that are imbricate in one direction are essentially cancelled out by the equal number of grains thatare imbricate in the opposing direction). The plot showing variation in imbrication through the shallow marinesandstone is also strikingly different: there is wide variation in imbrication angle and it appears to vary symmetricallyabout a mean of 0°. Not only is this variation symmetrical, but it is also has a cyclicity that can be proven statistically.The cyclic variation in imbrication angle through the deposit records variation in the magnitude of flow strengthunder waves: flow strength decreases, from a maximum in one direction, to zero and then increases to a maximumin the opposing direction. Thus, detailed studies of the variation in imbrication angle in sediment may provide apowerful tool in distinguishing the products of unidirectional and oscillatory currents.

DIS

TA

NC

E A

BO

VE

DA

TU

M (

MM

)

0

2

1

-90 900

θ341

FLOW

ANGLE (˚)

B. Fabric produced by oscillatorycurrent under waves in an

ancient shallow sea

ANGLE (˚)

-90 0 900

1

2

3

4

θ

186

FLOW

A. Fabric produced by unidirectionalcurrent in an ancient river

B BFigure 2-44. Comparison of the fabric of sandstonesdeposited in fluvial (A) and shallow-marine (B)settings. See text for discussion. After Cheel(1991).

47

CHAPTER 3. CLASSIFICATION OF TERRIGENOUS CLASTIC ROCKS

In nature there is a wide variety of sedimentary rocks and each type differs from all other types in terms of physicalproperties, composition and/or mode of origin. The classification of sedimentary rocks is a necessary exercisethat provides consistent nomenclature to facilitate communication between sedimentologists (i.e., the classificationsets limits to the attributes of any given class) and most classification schemes are based on characteristics thathave some genetic significance. This chapter briefly describes the classification of sedimentary rocks on variousscales and then focuses on a particular class: terrigenous clastic sedimentary rocks.

A FUNDAMENTAL CLASSIFICATION OF SEDIMENT AND SEDIMENTARY ROCKS

Figure 3-1 shows the the relationship between sedimentary rock classificaiton and the origin of the sedimentthat makes up the rocks. All sedimentary rocks are composed of the products of “weathering”, the process thatcauses the physical and/or chemical breakdown of a pre-existing rock (termed a source rock). These “products”include detrital grains (chemically stable grains) and material in solution. Detrital grains are normally dominatedby quartz, with lesser amounts of feldspars, rock fragments, micaceous and clay minerals, insoluble oxides, and asmall proportion (normally less than 1%) of what are termed “heavy minerals” because they have a higher densitythan the quartz and feldspars. The heavy minerals may be relatively non-reactive to chemical weathering but formonly a small proportion of a source rock (e.g., tourmaline and zircon) or they may be less stable minerals that comprisea relatively large proportion of the source rock (e.g., the amphiboles and pyroxenes). Rock fragments (syn. lithicfragments) may include as wide a range of particles as there are source rocks but only fragments composed ofrelatively resistant (physically and/or chemically) minerals withstand transport over great distances. Detrital grainsalso include some micaceous and clay minerals and insoluble oxides that are formed by chemical reactions on thesurfaces of some minerals during chemical weathering. The micaceous minerals produced by weathering arerelatively unstable. However, clay minerals, dominated by kaolinite, illite and montmorillonite, and insoluble oxides,including hematite, bauxite, laterite, and gibbsite, are generally very stable. The exact composition of detrital grainsproduced by weathering will depend on the relative importance of chemical and physical weathering and thecomposition of the source rock.

Sediment formed from the products of weathering are normally deposited following a period of transport to somesite of deposition. The various types of sedimentary rocks may be most fundamentally classified according to thetype of weathering product from which they form: as chemical sediment, composed of material that was transportedin solution and deposited by precipitation from solution, or clastic sediment, that include all of the particulateproducts of weathering (i.e., the detrital grains produced by weathering) that are transported to their site ofdeposition by a variety of physical processes: by running water (rivers, currents in lakes, seas and oceans), glaciers,wind, volcanic eruptions (non-igneous rocks produced by explosions and breakage during lava flow), and gravity(e.g., landslides).

The chemical sediment may be further subdivided according to the specific mode of formation. Sediment thatprecipitates directly from solution is termed orthochemical sediment (e.g., halite, gypsum, some limestone anddolomite) whereas those that are precipitated by organisms, to form their own shell material, are termed biogenicsediments. Biogenic sediment is dominated by calcium carbonate (i.e., they form many limestones or have beendiagenetically altered to dolomite) but also include siliceous sediment (e.g., biogenic chert) composed of theexoskeletons of siliceous-shelled organisms (e.g., diatoms).

Clastic sediment may also be divided into subclasses on the basis of their composition and mode of origin. Themost common is the terrigenous clastic sediment, including all sediment composed of detrital grains (derived fromany source rock) that were transported to their site of deposition. Clastic sediment that is derived from the productsof volcanic eruptions is termed pyroclastic sediment. A third, special type, of clastic sediment that spans betweenclastic and biogenic sediment is the bioclastic sediment that is composed of reworked biogenic sediment (i.e., shellmaterial that is reworked by currents). Each of these subclasses of clastic sediment can be subdivided according

48

Source Rock

Physical and Chemical weathering

TRANSPORTRiversWindGlaciersOceanic currentsVolcanic explosions

solutions solid particles

detrital grainsclayinsoluble oxides

DEPOSITIONPrecipitation Cessation of movement

dire

ct fr

om so

lutio

n as shell material

Orthochemicalsediment

Biogenicsediment

Clastic sediment

Bioclasticsediment

Terrigenousclastic

sediment

Pyroclasticsediment

Chemical Sediment

reworking

Figure 3-1. Illustration showing the relationship between sedimentary rock classification and the origin of the sedimentmaking up the rocks.

49

to a variety of characteristics and the remainder of this chapter will focus on the classification of terrigenous clasticsediment. However, note that many of the criteria for subdividing terrigenous clastic sediment may also be usedto further subdivide pyroclastic and bioclastic sediment.

CLASSIFICATION OF TERRIGENOUS CLASTIC SEDIMENT

Most widely-used classifications of terrigenous clastic sediment or sedimentary rocks are based on thedescriptive properties of a rock (e.g., grain size, grain shape, grain composition). The classifications summarizedhere are largely descriptive but they are based on properties that may have important genetic implications (seebelow).

A descriptive classification of any rock may be made at various levels and precision. The classification ofterrigenous clastic sediment and rocks given in Table 3-1 represents the simplest subdivision and is based solelyon grain size (note that the boundaries between sediment/rock types are from the Udden-Wentworth grade scale).This classification should be considered a “first-order” classification and each class may be further subdivided onthe basis of a variety of characteristics.

CLASSIFICATION OF SANDSTONES

Basis of Classification

Sandstones may be further classified on the basis of the composition of the grains and the proportion of therock that is fine-grained matrix (dominated clay size sediment), as determined by examination of specimens in thinsection. The major components of most sandstones are: quartz (including chert and polycrystalline quartz),feldspars, rock fragments and matrix; most other minerals are not sufficiently stable to survive significant transportand comprise only a small proportion of grains in comparison to the major components, and are neglected in mostclassifications. Note that sediment with the composition described is commonly termed siliciclastic sediment.Several schemes for classifying sandstones have been proposed, based on the relative proportions of the majorcomponents listed above. Figures 3-2 and 3-3 show a classification proposed by Dott (1964), defining thecompositional limits of each subclass of sandstone. Note that in this classification Dott defines matrix as all particlesfiner than 0.03 mm; within the range of clay-size particles. This classification limits the term arenite to rocks withless than 15% matrix while a rock with between 15% and 75% matrix is termed a “graywacke” (also spelled“greywacke” or, in German, “grauwacke”; commonly abbreviated as “wacke”). All sedimentary rocks with morethan 75% matrix are termed mudstones in this scheme. The arenites and graywackes are further subdivided on thebasis of the relative proportions of their major constituents (excluding matrix) by plotting their relative proportionson a ternary diagram. Figure 3-2 is rather schematic so take a close look at figure 3-3 to see the limits assigned toeach subclass of arenite and graywacke. According to figure 3-3A a quartz arenite contains no less than 90% quartzgrains and a subarkose contains between 5 and 25% feldspars, less than 25% rock fragments (but the proportion

Table 3-1. Classification of terrigenous clastic sediment/rocks based on grain size.

Grain size1 Sediment name Rock name Adjectives(mm)

>2 Gravel Rudite cobble, pebble, well-sorted, etc.

0.0625 - 2 Sand Sandstone or arenite coarse, medium, fine, well-sorted, etc.

<0.0625 Mud Mudstone or lutite silt or clay

1For the purposes of this general classification we will assign the rock or sediment name shown if more than50% of the particles are in the size range shown. More detailed classification schemes will limit terms onthe basis of different proportions of sediment within a give size range (see text).

50

100%Quartz

100%

Rock

frag

ments

100%

Feld

spars

55

25

25

Sublitharenite

Subarkose

Arko

se

ArkosicArenite

LithicArenite

50%

Quartz arenite

Quartzwacke

50

LithicGraywacke

FeldspathicGraywacke

Arko

sic w

acke

15%

75%

Percent m

atrix (<

0.03 mm)

MUDSTONES

WACKES

ARENITES

Figure 3-2. Classification of sandstones. After Dott, 1964, as modified by Potter, Pettijohn and Siever, 1972.

A. Total rock

Component Proportion%

Quartz 26Feldspar 20Rock fragments 12Matrix 42 (∴ a graywacke)

Total: 100Total Q, F, and Rf: 58

B. Quartz, feldspars and rock fragments

Component Proportion1

%

Quartz 45Feldspar 34Rock fragments 21

Total: 100

The proportions above plot in the field classifying this rockas a feldspathic graywacke (see Fig. 2B).

1Calculated as the proportion of each component in thetotal rock divided by the total proportion of quartz,feldspars and rock fragments (in this example this total is58).

Table 3-2. Example of the treatment of data collected by determining the proportions of quartz (Q), feldspars (F),rock fragments (Rf) and matrix, as seen in thin section. A. Total composition, including matrix, indicates that therock is defined as a graywacke. B. Proportions of quartz, feldspars, and rock fragments "normalized" to 100% sothat the data may be plotted on a ternary diagram (see Fig. 3-2B).

51

Figure 3-3. Details of the classification of arenites and graywackes as depicted in figure 3-2. Note that the corners of thetriangles represent 100% of the constituent indicated and solid and dashed lines (at 5% intervals) within the ternarydiagrams delineate lines of equal proportion of each component, decreasing to 0% for a given component on the side of thetriangle opposite each corner labelled for that component.

10

20

30

40

60

80

80

90

70

60

50

40

30

20

10

90

80

70

10

20

30

40

50

60

QUARTZ

RO

CK

FRAG

MEN

TS

FELDSPAR

90

50

3see table 2.

QUARTZ

RO

CK

FRAG

MEN

TS

FELDSPAR

QUARTZWACKE

ARKO

SIC

WAC

KE

FELDSPATHICGRAYWACKE

LITHICGRAYWACKE

10

20

30

40

60

70

80

80

90

70

60

50

40

30

20

10

90

80

70

10

20

30

40

50

60

QUARTZ

RO

CK

FRAG

MEN

TSFELD

SPAR

90

50

60% QUARTZ30% FELDSPAR10% ROCK FRAGMENTS

40% QUARTZ20% FELDSPAR40% ROCK FRAGMENTS

QUARTZ

RO

CK

FRAG

MEN

TS

FELDSPAR

QUARTZ ARENITE

SUBLITHARENITESUBARKOSE

ARKO

SE

ARKOSICARENITE

LITHICARENITE

}}

LithicArenite

ArkosicArenite

1

2

1

10

20

30

40

60

70

80

80

90

70

60

50

40

30

20

10

90

80

70

10

20

30

40

50

60

90

50

2

70

3

45% QUARTZ34% FELDSPAR21% ROCK FRAGMENTS

A. Classification of arenites

B. Classification of graywackes

} Feldspathicgraywacke

52

of feldspars always exceeds the proportion of rock fragments) and between 50 and 95% quartz. Figure 3-3A alsoshows the compositions of two rocks and points, based on the relative proportions of their constituents, plottedon the ternary diagram. Note that the proportions plotted on a ternary diagram must be recalculated from the originaldata describing the total composition of the rock so that quartz, feldspars and rock fragments total 100% (i.e., theproportions of quartz, feldspars and rock fragments must be “normalized” to 100%; see table 3-2). This proceduremust be applied to all such data that includes any proportion of matrix (i.e., arenites and graywackes).

Note that clastic sediment may contain detrital grains made up of chemical sedimentary rocks (i.e., they havebeen eroded from a source rock that was a chemical sediment and subsequently transported to the site of depositionof the clastic rock in which they occur). Particles derived from chemical sediment are generally relatively unstable(with obvious exceptions like chert) and do not survive transport to a distant site of deposition and are notconsidered here. However, the classification of terrigenous clastic rocks may be more specific than that shown here.For example, the lithic arenites may be further classified on the basis of the relative proportion of the types of rockfragments (e.g., proportions of sedimentary, metamorphic or igneous rock fragments). The rock names given infigure 3-2 may also be modified to refer to the type of cement; e.g., a calcareous quartz arenite would have a calciumcarbonate cement. Howe in these notes we will limit the level of classification to that shown in figure 3-2.

Genetic implications

Rock names based on the relative proportions of their constituents not only provide us with a basis forsystematic classification but also tell us something about the history of the rock.

Textural maturity refers to the maturity of a rock in terms of its grains size distribution and shape. As a populationof sediment undergoes more and more transport, and/or cycles of erosion-transportation-deposition, it tends tobecome better sorted (sands are said to become “cleaner’ as they lose their silt and clay fractions) and its’ particlesbecome rounder and more spherical in shape (see the section on Grain Shape and consider the generalizations madehere in light of all of the constraints on grain shape). A sedimentary rock is said to be mature if it well-sorted andconsists of rounded clasts. Thus, a quartz arenite, with less than 15% matrix, is texturally more mature than a lithicgraywacke (in terms of sorting and also in terms of grain shape; graywackes commonly have more angular grainsthan arenites). Clearly, the name applied to a terrigenous sedimentary rock reflects is textural maturity and, therefore,has implications related to the distance from the source that the sediment was transported prior to deposition and/or the nature of the source-rock that produced the sediment.

Compositional maturity refers to the relative proportions of stable and unstable grains comprising a sediment(quartz is the most stable component whereas feldspars and rock fragments are less stable). Like textural maturitythe degree of compositional maturity of a rock increases with transport and number of cycles of erosion-transportation-deposition (i.e., as a sediment matures it loses its less stable components and becomes better sorted).The unstable grains are destroyed by a variety of processes during weathering and transport: these processesinclude physical processes (e.g., removal of unstable minerals by breakage) and chemical processes (e.g., solutionor transformation of unstable minerals to produce clay minerals). For example, the average proportion of feldsparsin igneous and metamorphic rocks is approximately 60% whereas the average proportion of feldspars in sandstonesis 12%. The difference is due to the relative ease with which feldspars may be destroyed by abrasion and/or chemicalweathering, in comparison to quartz that dominates most sandstones, and the fact that source rocks commonlyinclude older sandstones that have already been through the geologic cycle (maybe several times). Rock fragmentsare also generally less stable than quartz grains and so that their proportions are smaller in mature sandstones thanin immature sandstones. As such, a quartz arenite is the most compositionally mature clastic sedimentary rock. Theultimate formation of a quartz arenite commonly requires several passes through the geologic cycle. Clearly, texturaland compositional maturity go hand in hand, both depending on many of the same factors.

The composition, and therefore the rock name derived from the above classification, will also reflect somethingof the nature of the source rock and the tectonic setting of the source area (referred to as the provenance of asediment). Taking a very simplistic view, we can think of the feldspars in a sediment as reflecting the contributionfrom a granitic source and the rock fragments as reflecting a volcanic or low-rank metamorphic source (these typicallyfine-grained rocks tend to produce abundant rock fragments rather than individual mineral grains). Thus, we can

53

Latit

ude

(deg

rees

nor

th o

f equ

ator

)

% Feldspar

20

30

40

50

60

0 20 40 60 8010 30 50 70

Figure 3-4. Proportion of feldspars in sands plotted against the latitude at which the sands were collected. Data are fromeastern and southern North America as summarized in Pettijohn, Potter and Siever (1973).

make some broad inferences regarding the nature of the source area of a sediment comprising a sedimentary rock,given its formal name and an understanding of the basis for the name: e.g., an arkose represents a sedimentary rockwith sediment derived from a source area with abundant granitic rocks, a shield area for example. Of course,knowledge of the specific type of feldspar or the specific composition of the rock fragments will tell much more aboutthe source rock and the tectonic setting of the source area.

To summarize the above discussion, the class of terrigenous clastic rock, by virtue of its basis on texture andcomposition, reflects something of: (1) the intensity of weathering that the material experienced (related to the climatead relief of the sources area); (2) the extent of transport that the material has undergone; and (3) the nature of thesource rock (original mineralogy and/or rock type: e.g., igneous, sedimentary or metamorphic) and the tectonicsetting of the source area. To illustrate, consider the data plotted in figure 3-4 which shows a general decrease inthe feldspar content of sands in the southward direction, through eastern and southern North America (these sandswould form arenites, specifically arkosic and quartz arenites, if they were cemented). This southward decreasereflects several factors. First, in the north the source rocks are dominated by rocks of the Canadian Shield that includea variety of feldspathic igneous and metamorphic rocks. Such source rocks provide a local supply of feldspars sothat the sands are relatively rich in that mineral. In contrast, to the south there are fewer igneous and metamorphicsource rocks and sediment is derived, to a greater extent, from weathering of pre-existing sedimentary rocks thathave gone through a least one cycle of weathering and lost a proportion of their feldspars. The second factor isthe difference in the style of weathering in the north and south. In the south, a warmer, moist climate facilitateschemical weathering that readily alters feldspars, producing soluble products and clay minerals. In the north,physical weathering is more important (especially during the Pleistocene glaciation of the region that originallyproduced much of the sand-size sediment in modern rivers of glaciated areas). Thus, the chances of feldsparssurviving weathering are greater in the north. Finally, for the data set described, from north to south, the averagetransport distance from the original source tends to increase. The sands in the north are closer to their richest sourceof feldspars than the sands in the south that include particles that originated on or near the Canadian Shield butwhich have lost much of their feldspar content due to abrasion and further chemical weathering over the greatdistance of transport. These are broad generalizations and the extensive scatter of points in figure 3-4 reflects thecomplex interaction of these and other factors.

As noted earlier, other minerals only rarely make up more than a few percent of terrigenous clastic sediment butthese may be of great interpretive importance. For example, a sandstone may consist of a relatively large portionof detrital carbonate, such as limestone or dolomite particles, derived from a carbonate source rock. However, thesegrains will be destroyed within a short distance of transport from their site of origin. Thus, the presence of detrital

54

carbonate grains in a sediment reflects close proximity to exposed carbonate rocks at the time that the sediment wasdeposited.

Level of classification

How specifically a rock is classified depends on the purpose of the study for which the classification is made.In many cases classification based only on grain size will be adequate (especially if the origin of the sediment particlesis not of interest). However, there are many different types of study that require a more detailed classification. Instudies that aim to delineate the geological history of a region the identification of the various classes shown infigure 3-2 will help with the interpretation of aspects of the nature of the source rock and source area and the extentof transport from the site of weathering. In another situation a sedimentologist may be required to provideinformation to engineers who are planning to excavate or drill through sedimentary rocks. In this case theclassification based on composition will be necessary to determine the cost of the work in terms of time requiredand the type of excavating or drilling apparatus that must be used. Both time and equipment influence the cost ofsuch a project so that a sedimentologist must conduct the necessary petrographic analyses to describe the rockand give it a name (that reflects its’ composition). For example, an arkosic graywacke will contain a smaller proportionof quartz than a quartz arenite. Because the quartz grains, that make up more than 95% of a quartz arenite, are harderthan the matrix and feldspars that make up a relatively large proportion of a feldspathic graywacke, the cost ofexcavating or drilling an arkosic graywacke may be less than for the quartz arenite.

Note on genetic classification of sedimentary rocks

It is worth commenting here that some rock and sediment names that are commonly used are based on the modeof origin of the rock (i.e., based on a genetic classification). The broad classification into clastic and chemicalsediment described at the beginning of this chapter is such a genetic classification. The classification of sandstonesis descriptive but those rocks may also be classified according to their origin at very specific levels. For example,the term “turbidite” is applied to any rock that was deposited from a turbidity current (a type of sediment gravityflow). A turbidite may be composed of carbonate or siliciclastic sediment that may range in grain size from gravelto mud, but will contain a certain arrangement of internal structures and will occur in a particular stratigraphic context.Hence, the term turbidite is largely independent of the fundamental properties of the rock and is defined in termsof the mode of origin of the rock. The term “tillite” is another rock name based on mode of origin: a rock depositedas glacial till. A tillite is typically composed of poorly sorted clasts, ranging from mud to boulders. Therefore, theclassification of a rock as a tillite requires a knowledge of the overall depositional environment that can only comefrom a regional study of the tillite and associated rocks. In contrast, descriptive classifications of rocks may be madeequally well in the field, in the original stratigraphic context, or in hand specimens where the stratigraphic contextmay not be known. In any study of a suite of sedimentary rocks it is usually advisable to classify rocks accordingto their descriptive properties, at least in the beginning, possibly later classifying them on a genetic basis whenthe depositional setting is better understood.

CLASSIFICATION OF RUDITES

Rudites have not been subjected to as much detailed subdivision as the sandstones. However, rudites maybe further classified on the basis of shape, packing and the composition of the lithic fragments that dominate thisclass of terrigenous clastic sediment. Table 3-3 reviews the classification of rudites by summarizing the commonterminology, including a brief description of the distinctive characteristics and possible genetic significance of eachtype of rudite. This classification is largely descriptive but in some cases the basis includes an understanding ofthe genesis of the clasts (e.g., the intraformational and extraformational rudites). While this discussion of theclassification of rudites is limited to the broad generalizations contained in Table 3-3, it is important to realize thatthe concepts of textural and compositional (lithological ) maturity apply to rudites in a manner similar to sandstones.

CLASSIFICATION OF LUTITE (SHALE)

A detailed treatment of the classification of lutite, that is dominated by the fine-grained clay minerals producedby weathering, is beyond the scope and purpose of these notes. The definitions given in Table 3-4 should be learned

55

Table 3-3. Definition of terms used to classify rudites.

Distinguishing Characteristics

A rudite composed predominantly ofrounded clasts.

A rudite composed predominantly ofangular clasts.

A rudite composed of poorly sorted, mudto gravel-size sediment, commonly withangular clasts.

A conglomerate in which all clasts are incontact with other clasts (i.e., the clastssupport each other). Such conglomer-ates may have no matrix between clasts(open framework) or spaces betweenclasts may be filled by a matrix of finersediment (closed framework). See figure3-5.

A conglomerate in which most clasts arenot in contact; i.e., the matrix supportsthe clasts. See figure 3-5.

A conglomerate in which clasts includeseveral different rock types.

A conglomerate in which the clasts aremade up of only one rock type.

A conglomerate in which clasts are de-rived locally from within the deposi-tional basin (e.g., clasts composed oflocal muds torn up by currents; such clastsare commonly termed "rip-up clasts" or"mud clasts").

A conglomerate in which clasts are ex-otic (i.e., derived from outside the depo-sitional basin).

Term

Conglomerate

Breccia

Diamictite

Orthoconglomerate(clast-supported conglomerate)

Paraconglomerate(matrix-supportedconglomerate)

Polymictic conglomerate

Oligomictic conglomerate

Intraformational conglomerate

Extraformational conglomerate

Genetic Significance

Rounded clasts may indicate considerable dis-tance of transport from source. The signifi-cance will vary with the lithology of the clast(i.e., limestone clasts will become round a shortdistance from their source whereas quartzitewill require much greater transport).

Generally indicates that the clasts have nottraveled far from their source or were trans-ported by a non-fluid medium (e.g., gravity orglacial ice).

Commonly refers to sediment deposited fromglaciers or sediment gravity flows, particularlydebris flows.

Clast-supported framework is typical of grav-els deposited from water flows in which gravel-size sediment predominates. Open frameworksuggests an efficient sorting mechanism thatcaused selective removal of finer grained sedi-ment. Closed framework suggests that thetransporting agent was less able to selectivelyremove the finer fractions or was varying incompetence, depositing the framework-fillingsediment well after the gravel-size sedimenthad been deposited.

Typical of the deposits of debris flows or waterflows in which gravel size clasts were notabundant in comparison to the finer grainsizes.

Conglomerates that include clasts from a wide-variety of source rocks, possibly derived overa wide geographical area or a smaller butgeologically complex area.

Suggests that the source area was nearby orsource rock extended over wide geographicarea.

Deposition in an environment where mudsaccumulated. Muds were in very close proxim-ity to the site of deposition as the clasts wouldnot withstand considerable transport.

Clasts derived from a distant source.

Note: in the following the rock names are given for rudites consisting of rounded clasts (conglomerates) but the termconglomerate may be replaced with the term "breccia" if the clasts comprising the rock are angular.

56

Clast-supported(open framework)

Clast-supported(closed framework)

Matrix-supported

ParaconglomerateOrthoconglomerates{ {Figure 3-5. Schematic illustrations of orthoconglomerates and paraconglomerate. Refer to Table 3-3.

in order to begin to understand nomenclature that has developed around this class of terrigenous clastic sedimentand sedimentary rocks. Table 3-5 outlines several descriptive properties of lutite and offers a descriptiveterminology. Table 3-5A summarizes a detailed classification of lutites that is promoted by Potter, Maynard andPryor (1980), based on the composition and bedding characteristics of lutite. Note that the term indurated (Table3-5A) refers to any rock that is hardened by pressure and/or cementation; an indurated sediment is a rock and anon-indurated sediment is an unconsolidated sediment. Table 3-5B summarizes the terms used to describe thelayering (stratification) of lutite and the manner in which a lutite “parts” or breaks along planes that are parallel toprimary bedding. Lutites, in particular, are characterized by their parting which is well-developed due to the parallelalignment of platy minerals along the bedding planes (rendering the bedding planes particularly weak and termed“parting planes”). In Table 3-5B “thickness” refers to the thickness of slabs of lutite that break along parting planes.For those with additional interest the book by Potter, Maynard and Pryor (1980) is an invaluable text on the topicof lutites.

Term

Shale

Lutite

Mud

Si l t

Clay

Fissility

Mudstone

Argillaceous sediment

Argillite

Psammite

Siltstone

Definition

The general term applied to this class of rocks (> 50% of particles are finer than0.0625 mm).

A synonym for "shale".

All sediment finer than 0.0625 mm. More specifically used for sediment in which33-65% of particles are within the clay size range (<0.0039 mm).

A sediment in which >68% of particles fall within the silt size range (0.0625 - 0.0039mm).

All sediment finer than 0.0039 mm.

Refers to the tendency of lutite to break evenly along parting planes. The greaterthe fissility the finer the rock splits; such a rock is said to be "fissile".

A blocky shale, i.e., has only poor fissility and does not split finely (see table 5).

A sediment containing largely clay-size particles (i.e., >50%).

A dense, compact rock (poor fissility) composed of mud-size sediment (low grademetamorphic rock, cleavage not developed)

Normally a fine-grained sandstone but sometimes applied to rocks of predominantlysilt-size sediment.

A rock composed largely of silt size particles (68-100% silt-size)

Table 3-4. Definition of terms used to desribe mudrocks.

57

Table 3-5. A. Classification of lutite. B. Terminology for stratification and parting in lutites. From Potter, Mayardand Pryor (1980).

Percentageclay-sizeconstituents

0 - 32 33 - 65 66 - 100

Fat or slickLoamyGrittyFieldadjective

Bed

s

> 1

0 m

mth

ick BEDDED

SILTBEDDED

MUDBEDDED

CLAYMUDLa

min

ae

< 1

0 m

mth

ick LAMINATED

SILTLAMINATED

MUDLAMINATEDCLAYMUDN

ON

IND

UR

AT

ED

Bed

s

> 1

0 m

mth

ick

Lam

inae

< 1

0 m

mth

ickIN

DU

RA

TE

D

BEDDEDSILTSTONE

MUDSTONE CLAYSTONE

LAMINATEDSILTSTONE

MUDSHALE CLAYSHALE

ME

TA

MO

RP

HO

SE

D

Deg

ree

of m

etam

orph

ism

LOW

HIGH

QUARTZARGILLITE

QUARTZSLATE

ARGILLITE

SLATE

PHYLLITE AND/OR MICA SCHIST

Table 3-5A

Thickness Stratification Parting Composition

Table 3-5B

30 cm

3 cm

10 mm

5 mm

1 mm

0.5 mm

Thin

Verythin

Thick

Medium

Bed

ding

Lam

inat

ion

Thin

Verythin

Slabby

Flaggy

Platy

Fissile

Papery

Incr

easi

ng s

and,

silt

, and

car

bona

te c

onte

nt

Incr

easi

ng c

lay

and

orga

nic

cont

ent

55

CHAPTER 4. FLUID FLOW AND SEDIMENT TRANSPORT

INTRODUCTION

In this chapter we will examine the important characteristics of unidirectional fluid flows and sedimenttransport under such flows. An understanding of the nature of fluid flow is crucial in sedimentology becausethe particles that comprise most sediments and sedimentary rocks were deposited following transport in a fluidmedium (either water or air). The treatment of this subject, below, concentrates on the properties and charac-teristics of fluid flow that are particularly relevant to the interpretation of sediments. Note that the principlesoutlined below are also important to environmental geology because particulate contaminants and solutionsare commonly transported by a fluid media, including surface waters.

In order to understand fluid flows a basic knowledge of Newtonian Mechanics and calculus are neces-sary and the treatment below assumes both. However, because students of GEOL 2P31 have a “mixed”background the derivation of the following relationships will not be stressed on tests and assignments.Instead, you will be expected to understand and be able to apply some of the more important relationships. Asummary of symbols and important relationships are given in Appendix 2. Whenever problems must be solvedin tests and exams a copy of Appendix 2 will be provided.

UNIDIRECTIONAL FLUID FLOWS

Unidirectional flows are characterized by a constant mean flow direction, in contrast to oscillatory flowsthat periodically reverse in direction. This section begins by examining the causes of fluid flow in terms of theforces that act on a fluid. Next, the classification of fluid flow is discussed followed by a detailed descriptionof the “structure” of a particular type of flow termed a “turbulent flow”. Note that the best detailed account ofunidirectional fluid flows is given by Middleton and Southard (1984).

Flow between two parallel plates

To understand how fluid flow takes place imagine the flow of fluid trapped between two parallel plates(Fig. 4-1): a top plate that is moving at some velocity, U, and a bottom plate that is stationary (i.e., its’ velocityis zero). When the top plate just begins to move (in response to some force F; see t

1 in Fig. 4-1) the layer of

fluid in immediate contact with it will be accelerated to exactly the same velocity as the plate itself (i.e., there isno slip between the plate and the fluid; the fluid will have the same velocity as a solid surface with which it isin contact). The fluid is accelerated as the force that is acting on the plate is transferred to the fluid along thecontact between the fluid and the plate. At some time (t

6 in Fig. 4-1) the plate will achieve some terminal

velocity (U) when the force that is driving it is balanced by a force of equal magnitude acting in the oppositedirection. This second force is imparted on the fluid by the stationary plate. Note that both forces acting onthe fluid are shear forces (i.e., they are tangential to the surface of contact with the fluid). Once the top platereaches its terminal velocity the entire column of fluid between the two plates will have reached some terminalvelocity that decreases linearly from a maximum equal to the velocity of the top plate to zero where the fluid isin contact (with no slip) with the lower, stationary plate.

Why does the entire package of fluid go into motion rather than the just the top layer of fluid that is incontact with the moving plate and why does the plate and fluid reach some terminal velocity rather thanaccelerating infinitely? Because of the viscosity of the fluid (dynamic viscosity is given the symbol µ; SIunits of dynamic viscosity are Ns/m2 or kg/ms). Viscosity is the property of a fluid that acts to resistdeformation and it arises because of the attraction between fluid molecules; it can be thought of as a “force”that resists deformation (although it is not a real force) and is sometimes referred to as “fluid friction”. When aforce is applied to a fluid molecule the molecule will accelerate and its’ momentum (its’ mass times its’ velocity)will be increased. Because of viscosity, that molecule will cause adjoining molecules to accelerate as well, thusit must exert a force on those molecules in order to change their momentum. This force between molecules in a

56

fluid can be thought of as a shear force acting along an almost infinite number planes lying parallel to theplates in figure 4-1. Thus, because of the viscosity of the fluid, the shear forces exerted by the plates aretransferred through the fluid and when they balance each other exactly the top plate reaches its terminalvelocity. Also, when terminal velocity is reached the shear forces acting across any plane within the fluidmust be balanced (i.e., of equal magnitude but opposite direction above and below each plane) and equal tothe shear force imparted by the plate. The shear force within a fluid is typically referred to as the shear stressin the fluid (and given the symbol τ and has units of force per unit area: N/m2 or kg/ms2). The shear stressalong any such plane through the fluid is given by:

τ µ=du

dyEq. 4-1

where du/dy is the velocity gradient or the rate of change in velocity in the direction normal to the two plates(note that du is not grain size times velocity). This is equal to the slope of the line defining the velocitybetween plates (Fig. 4-1). In figure 4-1 this line is straight, its’ slope is constant and so is the velocity gradient.Therefore, it should be obvious from equation 4-1 that the shear stress acting though the fluid is the samealong every plane between the two plates. Note that Eq. 4-1 applies only to so-called Newtonian Fluids, fluidsthat deform at constant rate regardless of the applied stress (i.e., µ is constant).

From Eq. 4-1 we can develop a general relationship to predict the velocity at any point between the twoplates. As noted above, the shear stress within the fluid is equal to the shear force (F) applied to the plate inmotion: i.e., τ = F/A, where A is the area over which the force is acting. Substituting into Eq. 4-1 andrearranging the terms:

τµ=

du

dy Eq. 4-2

We can solve for u (velocity at height y from the stationary plate) by integration with respect to y:

udu

dydy dy c y c= = + = +z zτµ τ

µ Eq. 4-3

where c is the constant of integration (the velocity at y = 0). Because we know that there is no slip at theboundary then u = 0 at y = 0, therefore c = 0. Thus, the velocity (u) at any distance (y) above the stationaryplate is given by:

u y=τµ Eq. 4-4

Clearly, the velocity between the two plate increases linearly from 0 against the lower plate (y = 0) to amaximum at the upper plate and the magnitude of velocity varies directly with the magnitude of the appliedforce and inversely with the viscosity of the fluid.

Fluid gravity flows

Flows of water, like that in a river, move down a slope and are driven by gravity: gravity acting on the

U = 0

Moving plate

Stationary plate

t1t2t3t4t5t6

} Shear stress (τ)within the fluid.F

luid

Force

y

UForce

Figure 4-1. Schematic illustration of fluid flow between two parallel, sliding plates. See text for details.

57

fluid causes it to flow down-slope. This situation is illustrated in figure 4-2 for a steady, uniform flow1 down asurface dipping at some angle θ. For the sake of simplicity, this example is for the case of open channel flowwhere every fluid molecule is moving downslope along a straight path that is parallel to the lower, rigid surface(termed the boundary). Note that this is the special case where shear stress is transferred through the fluid byviscous forces only (i.e., this is a laminar flow, see below).

In the example of flow between two plates the force that caused the fluid to flow was the force appliedto the top plate and this force was transferred though the fluid due to its viscosity; only the fluid molecules incontact with the plate directly experienced the driving force. In contrast, when gravity drives a fluid everymolecule “feels” this driving force. The shear stress acting along any plane parallel to the boundary, is equalto the downslope component of the weight of fluid above the plane. Consider the shear stress acting on aplanar surface at some distance y above the bed and overlain by a volume of fluid of unit width and lengthand height equal to D-y (see Fig. 4-2A). The volume of fluid above that surface is (D-y)×1×1 and its weight isρg(D-y)×1×1 (where D= flow depth, ρ = fluid density, and g is the acceleration due to gravity; note thatbecause we are dealing with unit width and length we will express the volume only in terms of D-y). Thedownslope component of the weight of fluid (F

G) in this volume is given by F

G = ρg(D-y) sin θ: it acts in the

downslope direction along the plane that is tangential to the flow boundary. Thus, the shear stress acting onthe bottom of this volume of fluid is also:

τ ρ= −g D y( ) sin 0 Eq. 4-5

This distribution of shear stress through such a flow is shown in figure 4-2B. Note that within fluid gravityflows the shear stress is not uniform, as in the fluid between two plates, but increases linearly from a minimumof zero at the free surface (where y = D) to a maximum at the boundary (Fig. 4-2B).

Figure 4-2. Schematic illustration of steady, uniform laminar flow down an incline due to the force of gravity. A. Ablock of fluid with unit width and length to illustrate the shear stress acting on a plane passing through the fluid. See textfor details. B. The distribution of shear stress and velocity through a steady, uniform laminar flow.

1Note that the term “steady” means that the flow depth and velocity are not changing with time and the term “uniform”means that the flow depth and velocity are not changing along the flow direction.

y τy

τy = FG = ρg(D-y)x1x1 sinθFG

τy = ρg(D-y) sinθ

A

Bu = ρg sinθ

µ (yD - y2

2 )

water surfaceD -

y

D

θ

τo

uθ

58

Note that the shear stress acting on the boundary represents the case where y = 0 (such that D-y = D)and is termed the boundary shear stress (given the symbol τ

o). From Eq. 4-5:

°=τ ρgD sin 0 Eq. 4-6.

This is a particularly important component of fluid shear because it acts on the bottom of a flow where muchsediment is transported (i.e., it is the force per unit area acting on the boundary and is the force that causessediment in contact with the bottom to move).

Given the relationship shown in Eq. 4-1 we see that the velocity gradient cannot be uniform within afluid gravity flow but must vary from a maximum at the boundary to a minimum at the water surface. Combin-ing Eq. 4-1 and Eq. 4-3 we derive:

du

dy

g D y=

−ρ

µ

sin ( )0Eq. 4-7

By integrating Eq. 4-7 with respect to y we can solve for u (the velocity at height y above the bed):

udu

dydy

gD y dy c

gyD

yc= z = −z + = − +

ρ

µ

ρ

µ

sin( )

sin( )

0 0 2

2Eq. 4-8

Given that u = 0 at y = 0 (i.e., no slip along the boundary) we find that the constant of integration (c) is equal tozero, such that:

ug

yDy

= −ρ

µ

sin( )

0 2

2Eq. 4-9

Thus, the velocity of such a flow varies as a parabolic function from 0 at the bed (y = 0) to a maximum at thesurface (y = D). This velocity distribution is shown schematically in comparison to the distribution of shearstress in figure 4-2B.

Classification of fluid gravity flows

In his classic experiments Osborne Reynolds (circa 1883) described that fluid motion could be charac-terized as laminar (i.e., fluid motion follows a linear path that parallels the flow boundaries) or turbulent (i.e.,fluid motion follows a chaotic path that appears to be random and varies in magnitude in 3 dimensions: itincludes downstream, upward and lateral components of motion). Reynolds’ experimental set-up is schematicallyillustrated in figure 4-3. A tank filled with fluid was drained through a transparent tube such that the velocityof fluid flowing through the tube was dictated by the height of fluid in the tank and the tube diameter. Dye

R < 1000

1000 < R < 2000

R > 2000

D

Figure 4-3. Schematic illustration of Reynolds’ experiments onthe nature of fluid flow. See text for a detailed discussion.

59

was injected into the fluid at the entrance of the tube and the behaviour of the flow was visualized by watchingthe behaviour of the dye streak in the tube. For a given fluid and constant tube diameter Reynolds found thatat low velocities the streak of dye followed a linear path through the length of the tube. At high velocities thefluid paths were very irregular and the dye was quickly distributed uniformly through the tube (i.e., no dyestreak persisted through the tube). At intermediate velocities a dye streak persisted but its’ path was ratherirregular and not parallel to the walls of the tube. By conducting the experiments using a variety of fluids (ofdifferent viscosity) and different tube diameters Reynolds found that he could predict whether a flow would belaminar or turbulent (the intermediate flow type is termed “transitional”) by the relationship:

R =ρ

µ

UDEq. 4-10

where U is the mean flow velocity and D can be flow depth in channels that are much wider than they are deepor D can be tube or pipe diameter. Viscosity is commonly expressed as kinematic viscosity (ν, where ν = µ/ρ; SIunits are m2/s) and Eq. 4-7 is commonly written:

R =UD

νEq. 4-11

R is termed the flow Reynolds Number and it is dimensionless (i.e., it has no dimensions). In open channelsflow is laminar when R < 500 and turbulent when R > 2000 and transitional when 500 < R < 2000 (these limits aresomewhat different for flow through tubes or pipes; see Fig. 4-3).

The flow Reynolds’ Number can be thought of as the ratio between inertial forces (flow inducingforces) to viscous forces (flow resisting forces). When viscous forces are large, relative to the inertial forcesthe structure of the flow will be dominated by viscosity (i.e., momentum is transferred by way of viscousattraction between fluid molecules) and the flow is laminar. Flow is turbulent when viscous forces are smallcompared to inertial forces (i.e., deep fast flows) and momentum is transferred by turbulence (see below).

Fluid flows with a free surface (i.e., excluding flows in pipes) can also be classified by the anotherdimensionless number termed the Froude Number (F), where:

F =U

gDEq. 4-12

F can be thought of as ratio of inertial forces acting on the fluid to gravity forces that act on the water surface.When F < 1 a flow is said to be subcritical or tranquil and when F > 1 a flow is said to be supercritical orshooting (a flow for which F = 1 is said to be critical). In practical terms F is significant in two related ways.

First, the term gD is equal to the celerity of waves on a water surface (i.e., the speed at which such waves

propagate over the water surface). As such, if F < 1 then U < gD and water surface waves will propagate

upstream because their celerity is faster than the flow velocity. If F > 1 then U > gD and water surface waves

will be swept downstream. A more important implication of F to our later consideration of bedforms underunidirectional flows is shown in figure 4-4. When F < 1 the water surface may be out-of-phase with a mobilesediment bed whereas when F > 1 the water surface is in-phase with the mobile sediment bed.

Shear stress and velocity distribution in turbulent flows

Most fluid flows that are geologically important are turbulent and their characteristics vary consider-ably from the laminar flows that were described earlier in this section. Among other things, because fluidmotion is so irregular (and three-dimensional) turbulent flows are very difficult to treat mathematically. Thus,this section will consider turbulent flows in a much more qualitative manner.

Figure 4-5 compares the velocity profiles of laminar and turbulent flows. For now, disregard thedifference in turbulent flows over rough and smooth boundaries and focus on comparing the forms of the

60

water surfaceWater surface

F < 1 F > 1

Figure 4-4. Schematic illustration showing the significance of the Froude Number in terms of the phase relationshipbetween the free water surface and a mobile sediment bed.

Figure 4-5. Schematic illustration comparing the distribution of velocity through turbulent and laminar flows. See textfor a detailed discussion.

U U U

D

Water Surface

Laminar Flow Turbulent Flow(smooth boundary)

Turbulent Flow(rough boundary)

curves for laminar and turbulent flows. Note that the lower portions of the curves for turbulent flows are like acompressed version of the curve for laminar flow (i.e., there is an initially rapid increase in velocity away fromthe boundary). However, the remainder of each curve for turbulent flows shows a much lower rate of increasein velocity than does the curve for a laminar flow (i.e., in turbulent flows du/dy in the upper portion of the flowis much more uniform than is the case for laminar flows). These two features reflect the fundamental differ-ences in the manner in which shear stress is distributed though the flows.

In laminar flows the momentum of the fluid was determined by the viscous shear stress acting on thatfluid within the flow. However, as figure 4-6 illustrates, in turbulent flows fluid momentum is also changed aspackages of fluid move up and down throughout the flow (the characteristics of these moving packages will bedescribed below). Low-speed fluid from near the boundary moves up into the region of high speed fluid, atsome distance from the boundary, and the high speed fluid loses momentum. Conversely, high speed fluidfrom the region away from the boundary may move downward and increase the momentum of the fluid near theboundary. This physical movement of fluid through the flow accounts for the more uniform distribution ofvelocity, well above the boundary region. This transfer of momentum differs fundamentally from viscousshear stress but it has the same outcome and is often termed turbulent shear stress or reynolds stress.Viscous shear stress is also important within a turbulent flow, in fact it predominates in the region closest tothe boundary where the velocity gradient is large. Thus, the total shear stress along any plane passingthrough a turbulent flow depends on the viscous and turbulent components of shear stress and takes a formsimilar to Eq. 4-1:

τ η µ= +( )du

dyEq. 4-13

which can also be written:

61

τ η µ= +du

dy

du

dyEq. 4-14

where η (the Greek letter eta) is termed the “coefficient of eddy viscosity”, a measure of the effectiveness withwhich momentum is transferred through the flow by eddies (packages of fluid). Figure 4-6 illustrates the abovediscussion and shows that the upper portions of a turbulent flow are dominated by turbulent shear stresswhile the region nearest the boundary is dominated by viscous shear stress. In fact, the boundary itself is somuch dominated by viscous shear that the boundary shear stress for turbulent flows may be determined bythe same relationship used for laminar flows. (i.e., Eq. 4-6). However, the velocity distribution within aturbulent flow is considerably different from that in a laminar flow (compare curves in Fig. 4-5) and can only bedescribed in terms of experimentally-determined relationships: one for turbulent flow over smooth boundariesand another for rough boundaries (i.e., covered with sediment and/or bedforms composed of sediment). Theformula for predicting flow velocity (u) at some distance (y) from a rough boundary (the most common type ofboundary of concern in sedimentology) is given by:

u

U

y

yo*.

.log= +8 5

2 3

κ Eq. 4-15

where κ is termed “von Karman’s constant” and is equal to 0.4 for most fluids, yo is a measure of the height of

the roughness elements on the boundary (either the grains and/or the bedforms) and U* is termed the shearvelocity of the flow and it is related to the boundary shear stress by:

U o* =

τ

ρEq. 4-16

Shear velocity has the dimensions of velocity and is a convenient way in which to express boundary shearstress. By Eq. 4-15 the velocity is zero at some point just beneath the surface of the rough boundary, incontrast to turbulent flow over smooth boundaries where velocity is zero at the boundary surface (see Fig. 4-5). Note that the mean velocity of a turbulent flow over a rough boundary occurs at y = 0.4D. Thus Eq. 4-15can be used to calculate mean velocity by substituting this value.

Structure of turbulent flows

Turbulent flows can be subdivided into three zones on the basis of the way in which momentum istransferred (Fig. 4-7); these subdivisions are also characterized by difference in the behaviour of the flow. Theviscous sublayer is the zone that extends upwards from the boundary and is dominated by viscous shear(much like a laminar flow). The thickness of the viscous sublayer (δ) is given by:

U

Dτ = η + µ dudy

dudy

}}

turbulentshear stress

viscousshear stress

Turbulencedominated

Viscositydominated

Figure 4-6. Schematic illustration ofthe nature of shear stress in a turbulentflow. See text for details.

62

U

D

viscous sublayer

transition layer

outerlayer

Figure 4-7. Subdivisions of turbulent flows based on the major mecha-nisms of momentum transfer. The outer layer is dominated by turbulentshear stress while the viscous sublayer is dominated by viscous shear.

δν

=12

U*Eq. 4-17

and may range from a fraction of a millimetre to several millimetres in thickness. Many older texts refer to theviscous sublayer as the “laminar sublayer” of a turbulent flow but flow in this zone is not strictly laminarbecause it experiences fluctuations in velocity (in both speed and direction) due to interaction with turbulencefrom higher levels in the flow. The buffer layer is the zone which has characteristics that are intermediatebetween those of the viscous sublayer and the outer layer; it is a region of transition from turbulent shear toviscous shear. The outer layer is the zone where turbulence is dominant, i.e., momentum is transferredpredominantly through turbulent shear stress. This zone extends from the free surface to the buffer layer andis characterized by eddies (packages of rotating fluid to be discussed in detail in a later section).

As noted in the previous section, the velocity profile of a turbulent flow depends on the nature of theboundary (whether it is smooth or rough) and turbulent boundaries may be classified on the basis of therelationship between the thickness of the viscous sublayer and the size of the grains on the boundary. Aboundary is said to be dynamically smooth when the viscous sublayer is thicker than the height of grains onthe bed (i.e, the grains are entirely within the viscous sublayer) and dynamically rough when the grains arehigher than the viscous sublayer (i.e., they protrude out of the viscous sublayer). This is an importantconcept because whether a turbulent boundary is dynamically smooth or rough will influence, among otherthings, the forces that act to move particles on the bed (see below).

Turbulent boundaries may be classified by a form of Reynolds number termed a boundary ReynoldsNumber (R

*, also termed a grain Reynolds number), where:

R**=

U d

νEq. 4-18

where d is the average grain size of the bed material. We can easily determine R* for the condition (between

dynamically smooth and rough boundaries) where a boundary is covered by spherical grains of uniform sizethat extend exactly to the top of the viscous sublayer (i.e., δ = d). Substituting Eq. 4-17 for d in Eq. 4-18:

R** *

*= = × =

U U

U

δ

ν ν

ν1212

Thus, when the height of the grains on the bed is exactly equal to the thickness of the viscous sublayer, R* =

12. In actual fact, partly because natural sediments are not composed of uniform spheres, it has been foundthat boundaries behave as dynamically smooth when R

* < 5 and as dynamically rough when R

* > 70. Turbu-

lent boundaries are said to be transitionally rough when 5 < R* < 70. Figure 4-8 schematically defines turbulent

boundaries in the manner outlined above. Note that on beds of grains much larger than the thickness of theviscous sublayer the sublayer develops over the surface of the large particles.

63

Organized Structure of Turbulent Flows

As described above, turbulent flows are characterized by a chaotic pattern of fluid flow and waterparticles are accelerated and decelerated in all directions due to the transfer of fluid momentum through theouter zone of the flow. However, organized structures within turbulent flows can be recognized on a variety ofscales. Figure 4-9 shows a hypothetical curve depicting the variation in velocity (in the downstream direction)over time in a turbulent flow (i.e, if velocity at some depth were measured instantaneously and repeatedly overtime). Note that the pattern of variation in velocity can be thought of as consisting of two components: aslowly varying component and a relatively rapidly varying component. These two components representdifferent patterns of organized fluid motion that act on different scales along with essentially random fluidmotions. The organized patterns of motion are known to result from “structures” within a turbulent flow thatbehave in a quasi-regular manner.

The outer layer of a turbulent flow is dominated by secondary flows and eddies of various type thatinteract in a complex manner. Secondary flows in the outer layer can be considered as rotating packages offluid that spiral along an axis that is parallel to the mean flow direction; secondary flows impart a cross-channeland vertical component of fluid motion onto the mean, downstream flow. In straight channels two such spiralflows typically develop, side-by-side and counter-rotating. In sinuous or meandering channels the form ofsuch secondary flows varies as shown in figure 4-10; one spiral at the bends of the meander (with surface flowtowards the outside of the bend) and two spirals at the points of inflection between bends. In fact, channelsmeander because of the variation in the distribution in boundary shear stress and sediment transport that iscaused by secondary flows. Boundary shear stress is greatest on the outside of the bend (enhancing erosion)

Figure 4-8. Classification of turbulent boundaries on the basis of the relationship between the thickness of the viscoussublayer and the size of grains on the boundary.

Figure 4-9. Schematic illustration of the variation in downstream velocity at a point in a turbulent flow, measured overtime. Solid line shows the rapidly varying component of turbulence; dashed line shows the slowly varying component ofturbulence. See text for discussion.

U

t

δ

δ > d δ ≈ d δ < d

δδ

R* > 70

Rough5 < R

* < 70

TransitionalR

* < 5

Smooth

64

and smallest on the inside of the bend, enhancing deposition. Deposition is further enhanced on the inside ofthe bed due to the component of flow velocity that acts from the outside towards the inside of the bend,transporting sediment in that direction.

Eddies or vortices in a flow are packages of fluid that rotate about an axis that extends perpendicular tothe direction of mean flow and these eddies travel in the mean direction at a speed equal to 0.8U∞, where U∞ isthe free-surface velocity of the flow (i.e., the downstream velocity that the water surface is moving). They maybe of smaller scale than secondary flows and may be superimposed on secondary flows. Eddies may extendthrough the entire thickness of the outer zone and may have smaller eddies superimposed on them. As eddiesmove in the mean flow direction they result in temporal and spatial variation in boundary shear stress due tothe changes in the rate of shear that they induce in the viscous sublayer. Figure 4-11 shows a particular typeof eddy that does not move along the flow direction but develops in the lee of a negative step on theboundary (such a step might be produced by a bedform; see below). Over such a step the flow is said to“separate”, become detached from the boundary, and becomes attached to the boundary at some point downstream. Upstream of the attachment point, below the step, the flow is directed in the upstream direction andforms a “roller vortex”, or “roller eddy”, that extends across the flow in the lee of the step. Downstream of theattachment point the flow “attaches” to the boundary and behaves essentially identical to flow before the stepon the boundary. The development of such roller eddies in the lee of a step is important to the formation ofmany of the bedforms that will be described and discussed in the following chapters.

AA'

BB'

CC'

C C'

A A'

B B'

flow

Figure 4-10. Secondary, spiral flows superimposed on the mean, downstream flow in a meandering channel. Arrowsindicate the direction of the component of fluid motion due to secondary flows that are superimposed on the mean flow.See text for discussion.

65

A component of the flow structure of the viscous sublayer includes a series of mean-flow-parallel,alternating (across the flow direction) rows of high-speed and low-speed fluid termed streaks (Fig. 4-12A). Inthe absence of sediment in transport by a current the spacing of streaks (λ) is given by:

λ

ν

U* ≈ 100 Eq. 4-19

This spacing increases when sediment is in transport over the bed (Weedman and Slingerland, 1985). Some ofthe small-scale fluctuations in velocity in a turbulent flow are due to a process termed the bursting process, orbursting cycle, that begins in the viscous sublayer due to some instability of streaks. Ideally, the bursting

0 0 0U

0

s

av

Figure 4-11. Development of a roller vortex in (v) the lee of a negative step on a boundary. Curves show the velocitydistribution in the flow near the boundary below. The letter “s” indicates the point where the flow separates from theboundary and the letter “a” indicates the position of the attachment point. See text for a discussion of details.

View looking down onto alternating high-speedand low-speed streaks in the viscous sublayer

viscoussublayer

12

3

4

1 & 2

4 3

1

1

2

3

43

2

3

4

U

U

A. Streaks

B. The bursting process

A burst

A sweep

viscoussublayer

Flowdirection

y

y

Figure 4-12. Organized structure of theviscous sublayer. A. The distribution ofalternating high- and low-speed streaks alongthe boundary (lengths of arrows are pro-portional to fluid speed). B. The burstingprocess or cycle. Left hand sideschematically shows the behaviour of fluidin and near the viscous sublayer (numberedsequentially over time). The right hand sideshows the effect of the movement of fluidon the near-boundary velocity profile (num-bers correspond to events shown on theleft hand side). See text for a detailed dis-cussion.

66

cycle includes two events (Fig. 4-12B): bursts, which involve the ejection of low speed fluid away from theviscous sublayer, out into the outer layer, and sweeps, which involve the injection of high speed fluid from theouter layer, into the viscous sublayer. Note that bursts and sweeps have a significant effect on the localvelocity profile and, therefore, local instantaneous boundary shear stress: high boundary shear stress undersweeps, low boundary shear stress under bursting. Most sweeps involve packages of fluid that had previ-ously been ejected away from the boundary by bursting and many burst are initiated by disturbance of theviscous sublayer by incoming sweeps. However, while bursts and sweeps occur with similar periodicity andfrequency, not every burst will result in a sweep and not every sweep will induce a burst. None-the-less, thebursting process is a particularly important process in generating turbulence and bursts may provide aneffective mechanism of suspending sediment (see below). Finally, note that bursting does not require a well-developed viscous sublayer. Even on boundaries dominated by sediment that is much larger than the viscoussublayer the bursting process is known to occur although its origin is not well understood.

SEDIMENT TRANSPORT UNDER UNIDIRECTIONAL FLOWS

Unidirectional flows have the potential to transport sediment (depending on the “flow strength” andthe size of sediment that is available for transport). Thus, on the basis of preserved sediments or sedimentaryrocks we might be able to make some fundamental interpretations about the paleohydraulics of ancientdepositional environments on the basis of an understanding of the relationship between sediment transportand flow conditions.

Modes of sediment transport

The sediment that is transported by unidirectional currents can be classified into two broad types: washload and bed material load. Wash load is the part of the total sediment “load” that is transported continuouslyin suspension by the current; that is, fine-grained sediment (silt and clay) that is held in the main body of thecurrent and rarely settles to the bed (it makes up generally <1% of the material on the bed). In rivers thiscomponent of the total sediment load is in transport regardless of the rivers’ rate of discharge. In contrast, bedmaterial load is the part of the total sediment load that is in transport only during periods of high discharge(e.g., when a river experiences an annual flood due to runoff of snow meltwater). This material may includesand to boulder size sediment that will only move under the strongest flows. During periods of “normal”discharge this component of the total sediment load is stored in the bed (hence it is called “bed material load”).Because most sand-size sediment that forms sedimentary rocks was laid down during periods of maximumdischarge (e.g., flooding events) it is useful to focus on the hydraulic significance of the bed material load.

Bed material load includes three components: contact load, saltation load and intermittent suspension

contact saltation

suspensive saltation intermittent suspension

Modes of particle movement

A. B.

C. D.

FLOW

Figure 4-13. Modes of transport of the components of bed material load. See text for detailed discussion.

67

0.01

12

510

30

50

70

9095

9899

99.99

-3 -2 -1 0 1 2 3 4Grain Size ( )

Y

Cu

mu

lati

ve F

req

uen

cy (

%)

φ

X

contact

and

salta

tion

inte

rmitt

ent s

uspe

nsio

n

wash

Figure 4-14. Interpretation of a segmented cumulative grain sizefrequency curve in terms of sediment transport sub-populations.

load (Fig. 4-13). Contact load or traction load is the part of the total sediment in transport that moves only incontact with the bed (Fig. 4-13A). This normally includes the largest particles in transport that move by rollingor sliding over the bed. Saltation load includes all sediment that moves only by a series of short “hops” thatfollow an approximate ballistic trajectory (i.e., there is a brief upward motion of the grain, due to lift forcesexerted by the fluid, followed by a period over which the grain returns to the bed while being carried down-stream by the fluid; Fig. 4-13B). In water, a saltating particle will ascend only 2 to 4 grain diameters above thebed but will travel 30 to 50 grain diameters downstream during its return to the bed. Intermittent suspensionload includes all grains that undergo transport while held up by the vertical component of turbulence (i.e., thecomponent of the turbulent velocity of the flow associated with upward movements of fluid and sediment).Figure 4-13D shows the path of a particle in intermittent suspension load. Note that suspension of this part ofthe bed material load occurs only “intermittently”, i.e., under the most powerful flows associated with highdischarge events. Figure 4-13C shows a form of transport that is intermediate between saltation and intermit-tent suspension when the ballistic path of the particle is interrupted by fluid motions (possibly bursting); wedescribe such transport as suspensive saltation.

The grains that make up the transport modes described above differ in terms of size (and density, andto a lesser extent shape) because of the different mechanisms that cause them to be transported. Thus, whenthey comprise the bed material, as they do most of the time, they may be distinguished on the basis of theirgrain size characteristics. In the chapter on grain size distributions it was mentioned that segmented cumula-tive frequency curves, plotted on a probability scale, commonly consist of subpopulations corresponding tospecific transport modes. Figure 4-14 shows the interpretation of these transport modes in terms discussedabove. Most such curves consist of at least three segments corresponding to: contact and saltation loads (thecoarsest sub-population), the intermittent suspension load (the middle sub-population) and the wash load (thefinest sub-population). Two grain sizes, “X” and “Y”, are noted on figure 4-14. “X” corresponds to the largestgrain size that was transported by the flow as contact load and “Y” is the largest grain size that was sus-pended by the flow. Each of these sizes is particularly amenable to quantitative interpretation.

Quantitative interpretation of grain size curves

Threshold of grain movement

The coarsest grain size in the contact load (“X” in Fig. 4-14) is the largest size that could be transportedby the flow, if we assume that there are no limitations on the grain sizes available for transport; any sizecoarser would not be moved by the flow and would be present in the bed material. Thus, we can quantitatively

68

interpret this grain size by asking the general question “What is the critical flow condition that will cause agiven particle to move?”. This question can only be answered by experimentation where flow conditions(velocity, depth, boundary shear stress, etc.) are recorded for the instant a particle just begins to move under acurrent.

Figure 4-15 shows results from classic experiments by Hjulstrom (1939) that provide a possible answerto the above question. The experiments on which this figure is based were conducted in a flume, at a flowdepth of 1 metre, over a wide range of grain sizes. The experimentally-determined curve tells us the velocityrequired to cause the initiation of movement of a particle of a given size. For example, the maximum grain sizethat could be transported by the current that deposited the sediment represented by the cumulative frequencycurve in figure 4-14 was approximately -1.5 φ or approximately 2.8 mm. From figure 4-15 we see that a velocityof approximately 0.20 m/s is required to move such a particle. Therefore, we infer the current that transportedand deposited this sediment was flowing at 0.20 m/s, such that 2.8 mm particles were the coarsest grains thatwould move. Thus, we have made a quantitative interpretation of the nature of the current that deposited thesand with the distribution shown in figure 4-14.

Unfortunately, the simple approach possible with Hjulstrom’s diagram is severely limited. Intuitionshould tell us that the condition for initial motion must depend on the boundary shear stress rather thanvelocity (boundary shear stress is the force that does the work to move sediment). Certainly boundary shearstress is related to velocity (see Eq. 4-15) but it depends on flow depth (Eq. 4-6). In addition, sedimentproperties will influence the condition of initial movement, certainly a particle of a given size with a relativelyhigh density, will offer more resistance to movement than a grain of the same size but lower density.

Figure 4-16 shows the forces that act on a grain resting on the boundary beneath a fluid flow. The mainforce that resists movement is the weight (W) of the grain on the boundary. The fluid forces that contribute tothe movement of the grain are the shear force exerted by the fluid on the grain (the boundary shear stress, τ

o)

and a lift force (L). This lift force is due to variation in pressure around the particle; the pressure exerted on theparticle is inversely related to the velocity of the flow in contact with the particle. Flow is fastest across thetop of the particle (imagine a grain resting in the viscous sublayer of a turbulent flow where the velocity

Figure 4-15. Hjulstrom’s diagram showing the critical velocity required to move sediment of a given grain size. Note thatthe relationship shown is limited to a flow depth of 1 m and that the data for silt and clay size sediment are few in number.After Sundborg (1956).

clay and silt sand gravel

0.00

1

0.00

2

0.00

5

0.01

0.02

0.05 0.1

0.2

0.5

1.0

2.0

5.0

10.0

20.0

50.0

100.

0

200.

0

500.

0

1000

.0

Grain size (mm)

0.01

0.02

0.05

0.10

0.20

0.50

1.0

2.0

5.0

10.0

20.0V

eloc

ity (

m/s

)

Hjulstrom's diagram

unconsolidated mud

erosionconsolidated mud

deposition

69

W

L

το

το

L

net f

luid

forc

e Figure 4-16. Schematic illustration of the forces actingon a particle beneath a flowing fluid. “W” is the weightof the particle and “L” is the lift force acting on theparticle.

gradient is strong), therefore pressure is least over the top, and their is a net pressure force directed upwards.Together, the lift and shear forces act at some angle, upwards from the horizontal, in the direction of flow.When these forces exceed some threshold such that they overcome the weight of the particle, it will movealong the boundary in the flow direction. Note that the larger the magnitude of the lift force, the smaller theboundary shear stress required to move the grain.

Shields’ (1936) approach to the question of the critical condition for the initiation of sediment move-ment considered the forces acting on a particle as outlined above. Based on experiments over a wide range ofgrain sizes and grain densities he constructed the curve shown in figure 4-17. In figure 4-17 the condition forthe initiation of motion is defined in terms of the boundary Reynolds number and the ratio of the boundaryshear stress to the weight of grains per unit area of the bed. This ratio is expressed as:

Figure 4-17. Shields’ diagram for determining the critical boundary shear stress required to move a grain on a bed ofuniform spheres of equal size. See Table 4-1 for an example of the use of Shields’ diagram. After Blatt, Middleton andMurray, 1980.

0.01

0.02

0.03

0.04

0.050.060.070.080.090.1

0.2

0.3

0.4

0.5

0.60.70.80.91.0

0.1 .2 .3 .4 .5 .6 .8 1.0 2 3 4 5 6 8 10 20 30 40 60 80 100 200 300 500 700 1000

2 3 4 5 6 8 10 20 30 40 60 80 100 200 300 500 700 1000

d ν

0.1(ρs - 1)gd ρ

U*dν

τ o(ρ

s -

ρ)gd

Shi

elds

' β

Boundary Reynolds number

Shields' Diagram

70

Table 4-1. Determination of the critical boundary shear stress required for the initiation of motion of a particleusing Shields’ diagram.

In this example we will solve for the critical boundary shear stress for the initiation of motion of thelargest particle in the contact load of the bed material with the grain size distribution shown in Fig. 4-14(i.e., d=1.5φ)

Values needed to calculate to:

d = -1.5φ = 2.8 mm = 0.0028 m (see Fig. 14).ν = 1.1 x 10-6 m2/s (water at 20ºC)ρ

s = 2650 kg/m3 (density of quartz)

ρ = 998.2 kg/m3 (density of water at 20ºC)g = 9.806 m/s2

Step 1. Calculate the value of d s gdν

ρ

ρ0 1 1. ( )− , which, in this case equals 172.

Step 2. Find the value calculated in step 1 on the scale in the middle of Shields’ diagram and follow thediagonal line down to the curve. Read the value of β off the vertical scale on the left.

In this case β = 0.047.

Step 3. We can solve βτ

ρ ρ=

−o

( s )gd for τo by rearranging to form: τ

o = β(ρ

s - ρ)gd

In this case τo = 2.13 N/m2. This is the critical boundary shear stress required to move a 2.8 mm diameter

quartz grain on a bed of uniform spheres under a flow of water at 20ºC.

Figure 4-18. A schematic illustration showing the error in estimating critical boundary shear stress using Shields’ diagramdue to variation in the size of grain relative to the average size of the bed material. Note that τ

c is the actual critical

boundary shear stress for the initiation of motion of a grain and tcs is the critical boundary shear stress predicted fromShields’ diagram; d is the grain size for which τ

c and τ

cs apply and d

m is the mean size of the bed material over which the

grain will move.

τc

ddm

1

τcs (Shield's)

ddm

<1; d < dmτc > τcs:

ddm

=1; d = dmτc = τcs:

τc < τcs:ddm

>1; d > dm

d dm

τo

71

βτ

ρ ρ=

−o

( s )gd Eq. 4-20

where ρs is the density of the grains, ρ is the fluid density, g is the acceleration due to gravity and d is grains

size. This ratio is referred to as Shields’ Beta (βββββ). The curve illustrated in figure 4-17 the shows theexperimentally determined relationship between β and boundary Reynolds Number, on beds of spherical grainsof uniform size. For a grain of given size and density β varies with the boundary shear stress required to movethat grain. Thus, the form of the curve reflects how the boundary shear stress required to move a grain varieswith boundary Reynolds Number due to changes in the magnitude of the lift force acting on a grain. Forexample, β decreases with increasing R

* to a minimum value of 0.032 at R

* = 12 (the theoretical limit for smooth

turbulent boundaries) and then increases to a constant value of 0.06 at R* > 600 (i.e., over beds covered with

very coarse particles). The decrease in β at low boundary Reynolds numbers is due to a correspondingdecrease in the boundary shear stress required to move a particle because the lift force acting on the grainincreases as the velocity gradient increases in the thinning viscous sub-layer (compare Eqs. 4-17 and 4-18).The value of β subsequently increases with increasing R

* because the boundary shear stress required to move

the grain increases as the viscous sublayer is disrupted the lift forces acting on the grain are reduced (becausethe strong velocity gradient in the viscous sublayer is reduced). A final, constant value of β is reached whenR

* exceeds approximately 600 and the lift force acting on the grains no longer varies with increasing R

* and the

critical boundary shear stress required to move a grain varies only with grain size.

Shield’s diagram may be used to determine the boundary shear stress required to move a particle by theprocedure outlined in Table 4-1, applied to the example of the maximum grain size in the bed material repre-sented by the cumulative frequency curve shown in figure 4-14. From the results of the calculations in table 4-1 we may assume that the current that transported the sand represented by figure 4-14 exerted a boundaryshear stress of approximately 2.13 N/m2. If the boundary shear stress were greater, larger sizes (>2.8 mm)would be preserved in the deposit (i.e., we assume that there were no limitations on the grain sizes available fortransport). If the boundary shear stress were below this value, the current could not have moved the 2.8 mmsand and it would not be present in the bed material. However, this approach is limited because it does notconsider all the variables that will determine the critical condition for movement. For example, the Shields’curve does not include the effect of shape and packing (see section on grain shape). More angular particleswill be more difficult to move than the spheres used in the experiments to determine the curve shown in figure4-17. More important, however, the experimental curve is based on beds of uniform grain size. Thus, errors inestimating the critical boundary shear stress for motion over a bed will depend on the size of the particle forwhich to is being determined, relative to the average size of the bed material. Specifically Shields’ curve willunderestimate the critical boundary shear stress to move a grain if the grain if much smaller than the averagesize of the grains that make up the bed on which it rests. This is because a small grain will tend to becometrapped in the space between larger particles on the bed, making it more difficult to move. Conversely, Shields’curve will overestimate the critical boundary shear stress to move a grain that is much larger than the averagesize of the particles that make up the bed on which it rests (Fig. 4-18). This is because a large grain will rollmore easily over a bed of much finer particles than over particles of the same size. This is a major limitation onthe use of Shields’ diagram (see discussion by Komar, 1987).

Threshold of grain suspension

The particles in transport as suspension load are supported (i.e., kept off of the bed) by the verticalcomponent of turbulence. Middleton (1976) argued on theoretical grounds that the critical flow condition atwhich a particle became suspended was when the average component of turbulence that acts in the upwarddirection (we’ll give this the symbol “v”) is exactly equal to the settling velocity (ω) of the particle. Clearly,when the v < ω the particle will eventually settle to the bottom. Thus, Middleton’s criterion for suspension is:a grain will be suspended by a flow when upward velocity of the vertical component of turbulence exceeds its’settling velocity. Unfortunately, the mean upward component of turbulence is difficult to measure. However,experiments have shown that the shear velocity of a flow is essentially equal to the upward component ofturbulence so that Middleton’s criterion for suspension may be stated in a more practical manner as in figure 4-

τo

72

19. Therefore, the largest grain that will be suspended by a flow (i.e., the coarsest grain size in the intermittentsuspension population of a bed material) is one with a settling velocity equal to the shear velocity of the flow.So we have another means of interpreting grain size curves such as figure 4-14 in terms of the hydraulics of thecurrents that transported (and deposited) a sediment. Table 4-2 shows the relationship between the maximumshear velocity of a number of rivers and the settling velocity of the coarsest grain size in the intermittentsuspension load of the bed material of these rivers. The theoretical relationship is respectably close. In thecase of the grain size curve shown in figure 4-14 the largest grain size in the intermittent suspension load is1.3φ (0.41 mm) and the shear velocity of the flow can be estimated by calculating the settling velocity of thisgrain size. We could use Stokes’ Law to calculate ω, by making all of the necessary assumptions, but wecannot overcome the error due to the fact that the grain size exceeds 0.1 mm. As an alternative, there areseveral experimentally-derived relationships to determine the settling velocity of particles beyond Stokes’range, one of these is shown in figure 4-20 (for quartz-density particles in water at 20ºC). Figure 4-20 indicatesthat a quartz-density, spherical grain with a mean diameter of 0.41 mm has a settling velocity of 0.044 m/s. Thissuggests that the flow that transported the bed material had a shear velocity of 0.044 m/s. We can comparethis result with that based on Shields’ diagram by determining the shear velocity produced by to = 2.13 N/m2

(see Eq. 4-16) and we find U* = 0.046 m/s, very close to that predicted by Middleton’s criterion.

Figure 4-20 also shows the critical shear velocity for the initiation of movement of grains on a bed (fromShields’ diagram). Note that below the curve for the initiation of motion grains will not move. Between the twocurves grains will move as part of the contact load and above the curve U

* = ω grains will be in suspension.

An important point that is illustrated by this figure is that very fine sand (less than 0.15 mm) tends to go intosuspension essentially as soon as it begins to move. This has very important implications to a variety ofsedimentary processes.

Figure 4-19. Schematic illustration definingMiddleton’s (1976) criterion for suspension of aparticle. See text for a detailed discussion.

ω

V

Suspension when V ω

Middleton's criterion for suspension:

where V is the upward component of velocity

due to turbulence and ω is the settling velocityof the particle.

Table 4-2. Comparison of measured peak shear velocities of flows in rivers to the settling velocity of thecoarsest particles in the intermittent suspension population of bed samples taken from each river. Data fromMiddleton, 1976.

River U* ωωωωω

(cm/s) (cm/s)

Middle Loup 7 - 9 7 - 9Middle Loup ≈20 ≈20Niobrara 7 - 10 7 - 9Elkhorn 7 - 9 2.5 - 5.0Mississippi (at Omaha) 6.5 - 6.8 2.5 - 5.0Mississippi (at St. Louis) 9 - 11 3 - 12Rio Grande 8 - 12 ≈10

73

grain suspension

grain

trac

tion

U *=

ω

Shields'

no movement

0.01 0.1 1.0 10.00

0.05

0.10U

* or

ω (

m/s

)

Grain diameter (mm)

Figure 4-20. Curves showing thecritical value of U

* for the initiation of

motion (based on Shields’ diagram)and suspension as a function of grainsize. Note that the curves apply toquartz-density grains in water at 20ºC.After Blatt, Middleton and Murray,1980.

74

Chapter 5. Bedforms and stratification under unidirectional flows

INTRODUCTION

When the flow of water (or air) over a bed of non-cohesive sediment is strong enough to move the particlesof the bed material the bed becomes molded into some topographic form with vertical relief ranging from fractionsof millimetres to up to several metres. The three-dimensional geometry of the bed topography is governed by theinteraction of the fluid and the sediment. Experimental studies in flumes and in modern rivers and in intertidal areashave found that their are a few common forms of bed geometry that develop under unidirectional current and thatthe overall geometry depends on the flow conditions and the sediment properties. The overall geometry of sucha bed is referred to as a bed configuration and it is made up of many individual topographic elements termed bedforms (or bedforms).

Over the past century much work has been directed at the description of the characteristics and behaviourof bedforms under unidirectional flows. This extensive research has been conducted by both sedimentologistsand civil engineers for two very different reasons. Civil engineering is concerned with the effects of fluid flow onmobile sediment beds because this broad field includes the practical problems associated with moving water fromone place to another (e.g., for hydroelectric generation, irrigation, or city water supplies). Open canals are oftenexcavated in unconsolidated sediment and the flow of water through the canals may cause sediment to betransported and ultimately results in the formation of bedforms. The most fundamental problem with thedevelopment of bedforms in canals is that they alter the character of the flow and affect fundamental flow propertiessuch as discharge. For example, in chapter 4 Eq. 4-15 shows that the velocity of a turbulent flow varies with theinverse logarithm of the size of the roughness elements on the boundary (y

o; i.e., the greater the boundary roughness

the slower the velocity). Most bedforms comprise major roughness elements that are many times larger than theroughness due to the grains on the bed. Hence, the presence of bedforms will retard fluid flow and their presenceand forms must be predictable in order to design canals on unconsolidated sediments. Sedimentologists havebenefited greatly from the work of engineers in their study of bedforms. However, sedimentologists derive theirinterest from the fact that bedforms are commonly preserved on ancient bedding planes in sedimentary rocks.Bedforms are primary sedimentary structures; structures that form at the time of deposition of the sediment inwhich they occur and they reflect some characteristic(s) of the depositional environment. In addition, bedformsproduce a variety of forms of cross-stratification (also primary sedimentary structures) that are very common in thegeologic record. Because bedforms and their behaviour are governed by fluid processes, they, and theirstratification, provide an unequalled basis for making paleohydraulic interpretations of ancient depositionalenvironments. The work conducted by engineers, that attempted to predict the types of bedforms that developon the basis of the sediment and flow characteristics, has been particularly useful in contributing to our ability tomake paleohydraulic interpretations of bedforms and their associated stratification. The aspects of paleohydraulicsthat can be inferred from the forms of cross-stratification include the relative flow strength, the direction of thecurrent, the type of current (e.g., upper or lower flow regime).

This chapter will focus on the types of bedforms that develop under unidirectional flows and emphasizeaspects of their character and behaviour that are particularly useful in the interpretation of sediments andsedimentary rocks. In addition, the final sections of this chapter will describe the forms of stratification that thevarious bedforms produce.

BEDFORMS UNDER UNIDIRECTIONAL FLOWS

Terminology

As we will see below, there are a variety of bedforms that are most precisely classified on the basis of theirgeometry and relationship to the water surface. However, the broadest classification of unidirectional flow bedformsis based on the flow regime under which the bedforms develop. This concept is widely used by sedimentologistsand was introduced by Simons and Richardson (1961; a pair of civil engineers) who distinguished lower flow regimeand upper flow regime, partly on the basis of the bedforms that are produced under unidirectional flows. Table5-1 summarizes the main criteria for distinguishing these two flow regimes. Note particularly that the relationship

75

Table 5-1. Definition of the flow regime concept of Simons and Richardson (1961).

Flow Regime Bedforms Characteristics

Lower flow regime Lower plane bed, • F < 0.84-1.0*;Ripples, Dunes • low rate of sediment transport,

dominated by contact load;• bedforms out-of-phase with the

water surface.

Upper flow regime Upper plane bed, • F > 0.84 - 1.0*;In-phase waves, • high rates of sediment transport,Chutes and pools high suspended load;

• bedforms in-phase with the watersurface.

*Note that Simons and Richardson (1961) set F < 1.0 for lower flow regime and F > 1.0 for upper flowregime. However, subsequent work indicated that in-phase waves began to develop over the range 0.84< F < 1.0. Because in-phase waves were particularly characteristic of the upper flow regime the limitingvalue of F has been adjusted accordingly here.

between the bedform and the water surface figures prominently in this scheme. The lower flow regime is dominatedby bedforms that are out-of-phase with the water surface and the upper flow regime is dominated by bedforms thatare in-phase with the water surface.

Figure 5-1 defines the general terms that will be used to describe bedforms in the following section. Termsfor asymmetric bedforms (including ripples and dunes; these are out-of-phase with the water surface) are well-established and widely used. The descriptive terms for symmetrical bedforms (the in-phase waves) produced underunidirectional flows are of only limited scope. The in-phase waves have been relatively little studied and a morecomplete descriptive terminology will certainly follow the current peak in interest in this class of bedform.

The sequence of bedforms

By “sequence” of bedforms we refer to the hypothetical, sequential development of different bedforms ona mobile sediment bed with increasing flow strength (e.g., increasing velocity with constant flow depth). Theconcept of bedform sequence is useful in describing bedforms because it provides a basis for a qualitativeappreciation for the relative flow strengths that are required to form them. As we will see later, not all of the bedformsdescribe in this sequence will develop on a bed of any one size of sediment; some bedforms are limited to coarsebed material while others are limited to fine bed material. Thus, we can consider the following to be a “hypothetical”sequence but one that provides valuable insight into the interpretation of bedforms and their stratification. Figure5-2 (A and B) schematically illustrate the variety of bedforms, in sequence, with increasing flow strength. Note thatthe two sequences in figures 5-2 A and B form a continuum, more-or-less.

We can think of the sequence of bedforms in terms of the changes in bed geometry under a flow that is slowlyand incrementally increasing in velocity, at constant depth, over a mobile sediment bed. Each specific bedformrepresents the equilibrium bed state for a constant flow velocity and depth. On beds of relatively coarse sand, justas the flow strength exceeds the threshold required for sediment movement, the first bed configuration will be aflat, planar surface. Such a bed is termed a lower plane bed to distinguish it from another type of plane bed thatdevelops under higher flow strengths. The lower plane bed will only form on beds of sediment coarser than 0.7mm and is characterized by its planar surface and relatively low rates of sediment transport (limited to contact load).The limitation of lower plane beds to relatively coarse sand indicates that this bed configuration will only form underdynamically rough turbulent boundaries (i.e., relatively large boundary Reynolds numbers). Sand that is depositedon a lower plane bed is characterized by relatively low angles of particle imbrication (both up- and downstream-dipping; see Fig. 5-3).

76

On beds of sediment finer than 0.7 mm, just as the current exceeds the flow strength required for sedimentmovement, the bed will be molded into a series of small, asymmetrical bedforms termed ripples (some texts call thesebedforms “small ripples”). Figure 5-4 shows the form of a ripple and the flow pattern that it induces near the boundary.Ripples behave like negative steps on a boundary and result in flow separation at the brink of the ripple andattachment just downstream of the trough where fluctuations in boundary shear stress are particularly large anderosion is intense. Ripples migrate downstream, in the direction that the current is flowing, as sediment is erodedfrom the trough and lower stoss slope and moves up the stoss slope to become temporarily deposited at the crest.As the deposit grows on the crest it eventually becomes unstable and avalanches down the lee slope where it isdeposited. The avalanche deposit is subsequently buried by continued avalanching and the entire ripple formmigrates downstream. In this way the pattern of flow separation, which is governed by the ripple geometry, alsomoves downstream, as does the site of erosion at the point of flow attachment. Over time, as the region of scourmigrates downstream the sediment deposited previously on the lee slope is eroded from the bed and begins its cycleof transport all over again. Hence, migration results from the pattern of spatial variation of erosion and depositionalong the length of the ripple.

Figure 5-1. Definition of terms used to describe asymmetrical and symmetrical bedforms that develop under unidirectionalflows.

Height (H)Stoss

Length (L)or

Spacing (S)

Summit

Crest

Brink

Lee orslipface

Trough

Anatomy of an asymmetrical bedform

FLOW DIRECTION

symbol for water surface

Water surface out-of-phase with bed surface

FLOW DIRECTION

Length (L)or

Spacing (S)

Height (H)

Water surface in-phase with bed surface

stoss slope angle

Anatomy of a symmetrical bedform

lee slope angle

77

Ripples range in length from approximately 0.05 m to about 0.6 m and in height from 0.005 m to just less than0.05 m. The size of a ripple is independent of flow depth but there is a crude correlation between ripple length andgrains size; length (L) increases with grain size (d) such that L≅1000d. The lee slope angle of ripples is close to angleof repose for the sediment, in the range of 25 to 30°. In plan form ripples are highly variable and a terminology hasdeveloped to describe these forms (see Fig. 5-5). The first ripples that form when sediment is moved over a bedare straight-crested (2-dimensional) but these do not appear to be a stable bedform because they rather quicklyevolve into 3-dimensional forms (with irregular crests). Some experimental evidence suggests that the forms shown

LO

WE

R P

LA

NE

BE

DR

IPP

LE

SD

UN

ES

2-di

men

sion

al3-

dim

ensi

onal

INC

RE

AS

ING

FL

OW

ST

RE

NG

TH

SEQUENCE OF LOWER FLOW REGIME BEDFORMS PRODUCEDUNDER UNIDIRECTIONAL FLOWS

FLOW DIRECTION

Figure 5-2A. The sequence of bedforms that develop under lower flow regime conditions (after Simons and Richardson, 1961).Note that washed-out dunes (Fig. 2B) are also lower flow regime bedforms.

78

Figure 5-2B. The sequence of bedforms that develop under upper flow regime conditions (after Cheel 1990a). Note that washed-out dunes are actually lower flow regime bedforms.

Up

per

Pla

ne

Bed

Inp

has

e w

aves

incr

easi

ng

in a

mp

litu

de

and

wav

e-le

ng

th

Direction of bedwave/water surface wave migration.

Water surface

STRATIFICATION

UPPER PLANE BED

HORIZONTAL

LAMINATION

DOWNSTREAM-MIGRATING

INPHASE WAVE

HORIZONTAL

LAMINATION

DOWNSTREAM-MIGRATING

INPHASE WAVE

FORESET

CROSS-LAMINAE

STANDING

INPHASE WAVE

DRAPE

LAMINAE

ANTIDUNE

BACKSET

CROSS-LAMINAE

INC

RE

AS

ING

FL

OW

ST

RE

NG

TH

Was

hed

-ou

t d

un

esFLOW DIRECTION

WASHED-OUT DUNE

HORIZONTAL

LAMINATION

0.84 < F < 1.0

SEQUENCE OF UPPER FLOW REGIME BEDFORMS PRODUCEDUNDER UNIDIRECTIONAL FLOWS

79

FLOW DIRECTION

deposition

by avalanchingerosionerosion

Flow over a ripple

in sequence in figure 5-5 develop with increasing flow strength. Note that ripples form on beds of sediment finerthan 0.7 mm, suggesting that they are typical of dynamically smooth turbulent boundaries. In fact, rippledevelopment seems to require the presence of a well-developed viscous sub-layer. Irregularities on an otherwiseplanar bed over which sediment is transported may protrude through the viscous sub-layer and cause flowseparation, much like that over fully developed ripples (see Fig. 5-4). Once established, the flow character associatedwith separation and attachment will cause erosion just downstream of the irregularity and deposition even furtherdownstream. Thus, an irregularity on the bed that is high enough to protrude through the viscous sublayer willset up a pattern of erosion and deposition that will propagate laterally and downstream, causing ripples to form.

Dunes are the next bedform to develop with increasing flow strength beyond the upper limit of ripples. Theyare similar in form to ripples (i.e., asymmetric bedwaves) but are larger than ripples with lengths ranging from greaterthan 0.75 m to in excess of 100 m and heights ranging from greater than 0.075 m to in excess of 5 m. Dunes tendto be most common on sand beds with a mean size in excess of 0.15 mm. Over the past decade several books andarticles have considered dunes to be just large ripples (and they have been termed “large ripples” or “megaripples”in the literature). However, dunes appear to be distinctly different bedforms and there is not a clear continuum insize from ripples to dunes. For example, when we plot bedform length against height of ripples and dunes we seethat they are related in the same linear fashion but there is a distinct break between the fields defined by ripplesand dunes (Fig. 5-6). This break indicates that asymmetric bedforms with lengths and height over the range of the

Figure 5-3. Comparison of vector mean imbrication angles, from the plane normal to the depositional surface and parallel toflow, of lower plane bed and upper plane bed deposits. After Gupta et al. (1987).

Figure 5-4. Flow separation over thenegative step on a boundary due to aripple.

0 180

ANGLE ( )

90

Upper plane bed

Lower plane bed

Grain Imbrication Angles

FLOW DIRECTION

80

FLOW DIRECTION

Straight Sinuous Caternary Linguoid Lunate

2-dimensional 3-dimensional

Plan-view shapesof ripples

Discriptive termfor crest-line

Increasing flow duration and/or (?) strength

0.01

0.001

0.1

1

10

100

0.01 0.1 1 10 100 1000

Spacing (m)

Hei

ght (

m)

Dunes

Ripples

H = 0.0677 L0.8090

r = 0.98

n = 1491

Figure 5-5. Terms used to describe plan forms of asymmetrical bedforms (ripples and dunes). After Allen, 1968 and Blatt,Middleton and Murray, 1980.

break are largely absent in nature. Thus, dunes appear to be distinctly different bedforms than ripples. In fact, undercertain hydraulic conditions dunes may have ripples superimposed on their stoss sides (Fig. 5-7). In the lee of duneslarge rotating eddies develop (due to the negative step on the bed) and when the upstream velocity of the currentgenerated by the eddy is large it will actually cause the formation of upstream-migrating ripples on the lee slope ofthe dune (termed “regressive” ripples; Fig. 5-7). Unlike ripples, dune size seems to be not related to the grains sizeof the bed material but is related to the flow depth (i.e., mean dune length and mean height increase with mean flowdepth). This relationship between dune size and flow depth suggests that the bedforms result from some interactionbetween large eddies in the flow and the sediment bed. Furthermore, dunes appear to interact with the water surfaceby producing a “boil” or bulge on the water surface just downstream of the trough due to the ejection of eddiesfrom the trough outwards towards the water surface (it has been suggested that these boils are a form of burstingassociated with dunes).

Dunes have been among the most studied bedforms and this has led to considerable confusion of theterminology applied to dunes and to their descriptive characteristics. Finally, in 1989 a symposium was held to arriveat a consensus on many fundamental concerns, beginning with the name to give to these large bedforms (they agreedto call them dunes). Table 5-2 summarizes the main conclusions of this symposium and outlines the descriptivecharacteristics of dunes that are thought to be important (and these are listed in their order of importance). This

Figure 5-6. Schematic illustration of the field (shadedareas) of ripples and dunes as defined in terms of the lengths(L) and heights (H) of these bedforms. Solid line is basedon a linear regression applied to measurements of 1491bedforms. After Ashley, 1990.

81

Ripples

DuneRegressive ripples

FLOW DIRECTION

A compound dune

Figure 5-7. Schematic illustration of a dune with ripples migrating up its stoss side (a compound dune). Note that “regressive”ripples (i.e., upstream-migrating ripples) are also shown in the trough and basal lee slope of the dune.

table provides a basis for a consistent terminology regarding dune size and indicates that their are two fundamentallydifferent types of dunes that can be distinguished in terms of their shape: 2-dimensional dunes and 3-dimensionaldunes.

2-dimensional and 3-dimensional dunes are so different that many workers have suggested that they arefundamentally different bedforms. However, we now believe that they are part of a continuum of dune forms overthe range of flow conditions over which they are stable (the two types of dunes do not form separate fields in plotssuch as figure 5-6). None-the-less, these two types of dunes differ in many ways and produce distinctive stylesof cross-stratification (Fig. 5-2A shows some of the subtle differences between these two types of dunes). 2-dimensional dunes develop under somewhat lower flow strengths than 3-dimensional dunes and they tend to berelatively long and low and straight-crested. The length of the crest is relatively long, in terms of the distance fromthe summit to the brink. The lee slopes of 2-dimensional dunes are approximately at angle of repose (25-30°) andscour in the troughs is not well-developed. The lee face forms a rather straight, planar surface. Sediment transportmay be dominated by contact load and avalanching down the lee slope. Under higher flow velocities 2-dimensionaldunes are replaced by 3-dimensional dunes which range from sinuous to lunate in plan form, are shorter and higher,overall, than 2-dimensional dunes and have shorter crests. They have lee slopes that are less than angle of reposeand their basal lee faces form curved surfaces extending down into deeply scoured troughs. Sediment transportover 3-dimensional dunes includes contact, saltation and intermittent suspension loads.

With increasing flow strength the 3-dimensional dunes become much longer and lower and their lee slopesundergo a reduction in angle. Such dunes are commonly termed “washed-out” dunes because the strong currentseems to wash out the dune form. The heights of such dunes become progressively smaller and their lengths becomelonger as the flow strength increases and they may form very subtle bedforms as long as several metres with heightson the order of a few millimetres (and may not be considered as dunes as described above). They appear to forma very gradual transition with the next bed form with increasing flow strength: the upper plane bed. Note that washedout dunes are shown in figure 5-2B. The reasons for relating this type of dune to plane beds and in-phase waveswill become apparent in the section on stratification.

The development of upper plane bed marks the onset of upper flow regime conditions in the scheme of Simonsand Richardson (1961). However, this is the one upper flow regime bedform that doesn’t really fit into this scheme.While all other upper flow regime bedforms require a free-water surface, upper plane bed does not (unlike in-phasewaves, upper plane bed will form in conduits that are completely full, i.e., have no free water surface). In addition,upper plane beds can develop at Froude numbers significantly less than 0.84 (i.e., as low as F = 0.4 under deep flows

82

Table 5-2. Classification and descriptive characteristics of dunes.

Classification scheme for large-scale, flow transverse bedforms (excluding in-phase waves)recommended by the SEPM Bedforms and Bedding Structures Research Symposium (Austin,Texas, 1987). After Ashley, et al. 1990.

General class: Subaqueous dune. The modifier “subaqueous” should only be used when aclear distinction between eolian and subaqueous dunes in necessary.

First order descriptors (necessary)

Size:

Spacing 0.6-5m 5-10m 10-100m >100m

Height1 0.075-0.4m 0.4-0.75m 0.75-5m >5m

Term small medium large very large

1Based on the relationship: H = 0.0677L0.8098

where H is height, L is spacing

Shape:

2-dimensional. Straight-crested, little or no scour in trough.

3-dimensional. Sinuous to short-crested, deep scour in trough.

Note: no quantitative expression of shape has been agreed upon.

Second order descriptors (important)

Superposition:

Simple. No bedforms superimposed.

Compound. Smaller bedforms superimposed (note size and relative orientation).

Sediment characteristics:

Size

Sorting

Third order descriptors (useful)

Bedform profile: note stoss and lee slope length and angles.

Full-beddedness: fraction of bed covered by bedforms.

Flow structure: time velocity characteristics (e.g., steady flows, tidal flows, etc.)

Relative strengths of operating flows: (e.g., tidal asymmetry)

Dune behaviour-migration history: vertical and horizontal accretion of bed with migration.

83

over very fine sand). Upper plane bed seems to have been placed among the upper flow regime bedforms as a matterof convenience to distinguish this bedform from the lower plane bed. Upper plane bed is most common on bedsof fine sand and under deep flows and is absent on beds of coarse sand and/or under shallow flows.

Upper plane bed (short for “upper flow regime plane bed”; synonyms include “upper stage plane bed”) iswidely defined as a flat, planar bed configuration with no significant relief beyond a few grain diameters. The one,widely acknowledged form of regular relief consists of flow-parallel ridges, a few grain diameters high, that are termedcurrent lineation that are thought to form due to streaks on the boundary (Weedman and Slingerland, 1985). Currentlineations are commonly visible on bedding planes within upper plane bed deposits (and also on the planar stossslopes of some dunes). In addition to streaks, the bursting process is thought by some to be particularly importantto forming horizontal lamination under upper plane bed conditions (more on this below). Several workers haverecently suggested that a variety of very low relief bedforms are actually present on upper plane beds (see Bridgeand Best, 1988, 1990; Paola et al, 1989; Cheel 1990a,b; Best and Bridge, 1992). The question of the existence of atrue plane bed, as defined above, remains and is beyond the scope of these notes.

Sediment transport over upper plane beds is intense as contact, saltation and suspension load and near thebed the concentration of sediment is transport is particularly high. So high, in fact, that a “bedload layer” iscommonly prominent that appears as a sheet of moving sediment that been termed a “rheologic layer” by some,a “traction carpet” by others. Grain imbrication is generally well-developed and relatively steep (up to 25° frombedding, dipping consistently into the current; see Figs. 5-3 and 5-8B) and grain a-axes are aligned parallel to theflow (Fig. 5-8A). Note that a-axes alignment on bedding surfaces is bimodal, as seen in plan view with respect tothe alignment of current lineation and flow direction; i.e., one mode on either side of the trend of the lineation. Thiswell-developed, flow-parallel alignment of grains is responsible for a structure that is seen in consolidatedsandstone termed parting step lineation: a structure that occurs on bedding plane exposures of upper plane beddeposits that reflects the tendency for sedimentary rocks that were deposited under upper plane bed conditionsto break along vertical planes that are parallel to the direction of flow that deposited the sediment. Note that bothcurrent and parting step lineation are useful in determining the “sense” of paleoflow direction. However, becausethey are lineation they are bidirectional and, therefore, cannot be used alone to determine the absolute paleoflowdirection. One common method of determining the absolute paleoflow direction from such lineation is to cut thinsections in the plane perpendicular to bedding and parallel to the lineation. The upstream imbrication direction,seen on that plane, can then be used to infer the absolute flow direction. An easier method of determining absolute

L A

LA

LA

LA

LA

AL

L A

A L

A

L A

A

L

L

A

A = vector mean a-axis orientationL = orientation of current lineation

B = bedding I = vector mean imbrication angle

B

IB

IB

IBI

B

I

B

I

L

0 10 20%

FLOW DIRECTION

FL

OW

DIR

EC

TIO

N

A

B

Figure 5-8. Long axes orientations of sandgrains deposited on an upper plane bed. A.Long axes orientations, measured on a beddingplane, in comparison to the trend of flow-parallel current lineation. B. Apparent longaxes, measured in the plane parallel to flow andperpendicular to bedding. After Allen (1968).

84

FLOW DIRECTION

current lineation

decreasing heavy mineral concentration

Directional significance of heavy mineral shadlows

Figure 5-9. Heavy mineral shadows on a current-lineated bedding surfaces on an upper plane bed. Note that the absolute flowdirection can be determined as being parallel to the lineation, in the direction of decreasing heavy mineral concentration throughthe shadow. After Cheel (1984).

paleoflow direction is possible if heavy minerals are abundant on bedding plane exposures of upper plane beddeposits. On upper plane beds heavy minerals commonly form patches on the active bed that display a downstream-decreasing gradient in the concentration of heavy minerals. Such heavy mineral patches are termed heavy mineralshadows and may be used, in conjunction with parting and/or current lineation, to determine the absolute paleoflowdirection from upper plane bed deposits (Fig. 5-9).

With increasing flow strength beyond the true plane bed very low, asymmetrical bedwaves develop. Thesebedforms reflect the gradual on-set of the formation of in-phase waves that takes place over a range of Froudenumbers from 0.84 to 1.0. Note that on beds of sand coarser than approximately 0.3 mm the transition from dunesto inphase waves may not include a true upper plane bed. Instead, washed-out dunes may pass directly, butgradually, into in-phase waves, with increasing flow strength, and their may be a stage where dune-like forms andin-phase waves co-exist (and this has been reported by Southard and Boguchwal, 1990).

In-phase waves are a class of bedform that are all characterized by their symmetrical form and in-phaserelationship with the water surface (although not all in-phase waves are always symmetrical and/or in-phase withthe water surface all of the time); hence, in-phase waves are generally described in terms of the water surface waveand the corresponding bedwave and in the description that follows when the bed and water surface wave arebehaving in the same fashion these two elements will not be separated. Unlike the other bedforms described above,in-phase waves derive their form and behaviour from the interaction of the water surface and the mobile sedimentbed under supercritical (shooting) flows. Cheel (1990a) reviewed literature that indicated that with increasing flowstrength, over the range of conditions for in-phase waves, a sequence of different forms of in-phase wave developed(see Fig. 5-2B). Starting with upper plane bed, the first form of in-phase wave is a very low (millimetres high; seeFig. 5-10), downstream-migrating form that increases in height and length with increasing flow velocity. Note that

all in-phase waves scale with flow velocity by the relationship UgL2

2=

π (Kennedy, 1963). As flow strength

continues to increase the symmetrical bedwaves cease to move downstream and remain stationary on the bed. Thistype of in-phase wave is termed a “standing” or “stationary” wave. Finally, with a further increase in flow strengththe in-phase waves are characterized by upstream migrating forms that Gilbert (1914) first termed “antidunes” (fortheir upstream migration in contrast to the downstream migration of dunes). Note that all of the forms of in-phasewave may not form on beds of a given sand size or flow depth. Further experimental work is required to establishthe fields of hydraulic stability of this class of bedform.

85

.12

.10

8 10 12 14 16

.130

.126

15 16 17

Upper plane bed

Low in-phase waves

.14

.12

8 10 12 14 16

Post aggradation bed surface

.136

.13215 16 17

Distance from entrance (m)Distance from entrance (m)Vertical exaggeration X40 Vertical exaggeration X50

Hei

ght a

bove

flum

e flo

or (

m)

Post aggradation bed surface

FLOW DIRECTION

Figure 5-10. Profiles of the sand bed on a flume of upper plane bed and very low in-phase waves. Note the vertical scale. Notethat “entrance” refers to the entrance to the flume channel. From Cheel (1990a).

The term antidune is one of the most abused terms used for any bedforms; in many texts it has come to applyto all forms of in-phase waves, a practice that has limited the understanding of these bedforms for almost 30 years.Bed behaviour associated with the antidunes of Gilbert (1914) is very complex; the actual bed state at any one timemay be plane bed, downstream-migrating in-phase waves, stationary waves or true antidunes (i.e., upstreammigrating). The change in state of the bed associated with antidunes is cyclic, and figure 5-11 shows the sequenceof stages of water surface and bed behaviour through a complete cycle of events associated with true antidunes.Note that under flow conditions that produce antidunes all of the stages shown in figure 5-11 may not occur duringany particular cycle. For example, the bed and water surface may develop to the stage where stationary in-phasewaves form and then the waves may subside back to a plane bed. In addition, all waves on the water surface maynot break simultaneously, as shown. In most cases one wave will break and it will disrupt the flow to cause otherwaves to break subsequently. All of the forms of in-phase waves shown in figure 2B are 2-dimensional (i.e., straight-crested). At flow strengths greater than those required to form 2-dimensional antidunes there are a variety of 3-dimensional in-phase waves (short-crested) of which we know virtually nothing.

Bedform stability fields

In the above description of bedforms we developed some vague idea that certain factors will control whichbedforms will develop, depending on flow strength and/or grain size. In fact, many related factors will govern whichbedforms will develop on a mobile sediment bed. These factors include those related to the fluid: flow velocity (U),flow depth (D), flow temperature (which controls fluid viscosity and density), and those related to the sediment:grain size (d), grain density (ρ

s), sediment sorting, and particle shape. The effects of particle shape and size sorting

on bedforms are not well known and will not be considered here. However, largely because of experimental flumeresearch, carried out by John Southard and his students at the Massachusetts Institute of Technology, we nowhave a good picture of the fields of hydraulic stability of bedforms on sand beds representing a wide range of grainsizes. Figures 5-12 and 5-13 are “bedform stability” diagrams and show the range of conditions over which thebedforms are stable on diagrams plotting flow velocity versus grain size (each diagram for a given range of flowdepths) and flow depth versus flow velocity (each diagram for a given range of grain sizes). Note that the variableson each diagram are scaled to represent conditions where the flow temperature is 10°C. Thus, they are limited toflows of that temperature but they are fairly representative of most natural flows.

All of these figures show that with increasing flow velocity the exact sequence of bedforms that will developwill vary with flow depth and the grain size of the bed material. For example, figure 5-12 shows that with increasing

86

FLOW DIRECTION

STAGE 1

STAGE 2

STAGE 3

STAGE 4

STAGE 5

STAGE 6

STAGE 7

STAGE 8

IDEAL BEHAVIOUR OF ANTIDUNES

Figure 5-11. A highly schematic illustration showing a complete cycle of the stages of bed and water surface behaviour associatedwith true antidunes. Note that dashed lines indicate the previous position of bed and/or water surface waves in each stage. Thedevelopment of cross-stratification is also shown (solid lines within the bed). After Udri (1991). Stage 1: under conditionsfavouring antidune development the bed may remain essentially flat for part of the time. Horizontal lamination may form duringthis initial stage. Stages 2 & 3: stationary in-phase waves develop as sinusoidal water surface waves grow in place and forma sinusoidal bed wave, of lower amplitude. In this stage the bed is molded by erosion under high velocity flow under water surfacewave troughs and deposition under the relatively lower flow velocities under the water surface wave crests. In-phase wavedrape laminae may develop during this period of in situ growth of the bedforms. Stage 4: after the water surface wave reachessome critical height and steepness it begins to slowly migrate in the upstream direction. Stage 5: the bedwave slowly respondsby similarly migrating upstream; the bed and water surface are slightly out-of-phase during this stage. Low-angle backset beddingdevelops during this stage. Stage 6: as the water surface wave continues to migrate upstream and become steeper the bedwavedevelops what appears to be an asymmetrical bedform on its upstream side; growth of this bedwave is particularly rapid asthe water surface wave begins to break by collapsing in the upstream direction. Breaking of the water surface wave results inupstream sediment transport and large quantities of sediment are taken into suspension. Relatively steep (>15°) backset beddingmay develop over this stage. Stage 7: following collapse of the water surface wave the water surface becomes flatter and thebedwaves are planed off by the very rapid flow. This bed-planing stage involves erosion from the wave crests and depositionin the troughs in the form of a fast-moving, asymmetrical bedform that migrates downstream. Relatively high angle down-stream-dipping cross-strata may develop as this bedform migrates across the trough of pre-existing in-phase wave. Stage 8: the waterand bed surfaces are planar and the cycle may begin again with deposition of horizontally laminated sand, truncating theunderlying cross-stratification produced by the in-phase waves.

87

Figure 5-12. Velocity/grain size diagrams showing the fields of bedforms stability for two ranges of flow depth (all variablesare scaled to 10°C water temperature). See text for discussion. After Southard and Boguchwal (1990).

0.1

0.2

0.4

0.6

0.8

1.0

2.0

10˚C

-equ

ival

ent m

ean

flow

vel

ocity

(m

/s)

0.1.06 0.2 0.4 0.6 1.0 2.0

10˚C-equivalent sediment size (mm)

F = 1

F = 0.84

D = 1.0 m

No movement

Ripples

Upper plane bed

Dunes

2-D

3-D

Lower plane bed

Abrupt Gradual

Gradual

Gradual

In-phase waves

4.0

?

Abrupt

Transition

0.1

0.2

0.4

0.6

0.8

1.0

2.010

˚C-e

quiv

alen

t mea

n flo

w v

eloc

ity (

m/s

)

0.1.06 0.2 0.4 0.6 1.0 2.0

10˚C-equivalent sediment size (mm)

F = 1F = 0.84

D = 0.30 m

No movement

Ripples

Upper plane bed

Dunes

2-D

3-D

Lower plane bed

Abrupt Gradual

Gradual

Gradual

In-phase waves

4.0

Abrupt

?

Transition

Figure 5-13. Depth/velocity diagrams showing the fieldsof bedforms stability for three ranges of grain size (allvariables are scaled to 10°C water temperature). See textfor discussion. After Southard and Boguchwal (1990).

10˚C

-equ

ival

ent m

ean

flow

dep

th (

m)

.02

.04

.06

.080.1

0.2

0.4

0.6

0.81.0

0.1 0.2 0.4 0.6 0.8 1.0 2.0

10˚C-equivalent mean flow velocity (m/s)

F =

0.84

F =

1.0

10˚C

-equ

ival

ent m

ean

flow

dep

th (

m)

.02

.04

.06

.080.1

0.2

0.4

0.6

0.81.0

0.1 0.2 0.4 0.6 0.8 1.0 2.0

10˚C-equivalent mean flow velocity (m/s)

F =

0.84

F =

1.0

Nomovement

Ripples

Upperplanebed

In-phasewaves

trans

ition

In-phasewaves

trans

ition

d = 0.10 - 0.14 mm

d = 0.40 - 0.60 mm

abru

pt

Nomovement

Ripples

grad

ual U

pper

pla

ne b

ed

grad

ual Dunes2-D 3-D

10˚C

-equ

ival

ent m

ean

flow

dep

th (

m)

.02

.04

.06

.080.1

0.2

0.4

0.6

0.81.0

0.1 0.2 0.4 0.6 0.8 1.0 2.0

10˚C-equivalent mean flow velocity (m/s)

F =

0.84

F =

1.0

d = 1.30 - 1.80 mm

trans

ition

Nomovement

grad

ual

Dunes2-D 3-D

grad

ual

Low

er p

lane

bed

to upper plane bed

In-phasewaves

88

velocity, flow over a bed of 0.1 mm sand will produce the sequence ripples → upper plane bed → in-phase waves,whereas the same flow over a bed of 1 mm sand will produce the sequence lower plane bed → dunes (2-D followedby 3-D) → in-phases waves (with upper plane bed following dunes under deep enough flows). Lower plane bedand dunes are limited to relatively coarse grain sizes whereas ripples and upper plane beds dominate fine grain sizes(particularly for the narrow range of flow depths represented by the figures: 1 to 3 m). Note that the maximum velocityat which dunes and plane beds are stable increases with flow depth and is governed by the flow Froude number.Also, note that the transitions form ripples to dunes and from ripples to plane bed are abrupt whereas all othertransitions form one bedform to another are gradual. The effect of flow depth on the sequence of bedforms thatdevelop with increasing flow velocity is apparent in figure 5-13. For example, the depth/velocity diagram for 0.4mm to 0.60 mm sand shows that at flow depths less than 0.1 m the sequence of bedforms that develops with increasingflow velocity is ripples → dunes (2-D followed by 3-D) → in-phase waves. However, at depths above 0.1 m upperplane bed becomes stable and the sequence of bedforms becomes: ripples → dunes (2-D followed by 3-D) → upperplane bed. → in-phase waves. Figure 5-13 also shows that the range of flow velocities over which ripples are stabledecreases with grain size, presumably because the viscous sublayer is destroyed under increasingly lowervelocities as the boundary roughness increases. Dunes and upper plane bed are both stable over wider rangesof velocities as flow depth increases (again, because their upper limit is governed by the flow Froude number). Incontrast, lower plane bed exists over an increasingly narrower range of velocities as depth increases. Note thatthe in-phase waves are not subdivided in diagrams like these because most experimental studies to date do notextend through the upper flow regime.

Figure 5-14 shows how the height and lengths of dunes varies as a function of flow depth and velocity andgrain size, within the dune stability field. The effect of flow depth on dune size (that was mentioned earlier) is clearlyevident in the depth/velocity diagrams: as flow depth increases dunes become both longer and higher. The velocity/grain size diagrams show that dune length increases with flow velocity but also with decreasing grain size: thelongest dunes develop on beds of the finest sand. In contrast, dune height is only loosely determined by grainsize (fine sand has a greater likelihood of developing higher dunes). For all sand sizes, with increasing flow velocity,dune height first increases and then decreases towards the upper velocity limit of dune stability. This reflects thewashing out of dunes described above.

Figures like those shown here are necessary for the interpretation of paleoflow conditions based on thebedforms that are preserved in ancient sediments. However, they continue to allow only qualitative interpretationsregarding the relative flow strength. A major limitation of data like that shown in figures 5-12 to 5-14 is that theyare based on relatively shallow flows depths, no more than a couple of metres deep whereas natural flows may rangeup to several tens of metres, beyond the range represented by the experimental data. As our understanding of thecontrols on bedform stability improve so will our ability to interpret these structures.

CROSS-STRATIFICATION FORMED BY BEDFORMS UNDER UNIDIRECTIONAL FLOWS

Cross-stratification is a type of primary structure that occurs in a wide variety of forms and develops insediments and sedimentary rocks due to temporal and spatial variation in deposition and erosion on a bed, normallyin association with the migration of bedforms. In this section we will see how the form of cross-stratification maybe used to infer type bedform that produced it and of the bedform behaviour (which may also be used to infersomething of the conditions in the depositional environment). However, first the terminology that has beendeveloped for describing cross-stratification must be introduced.

Terminology

Once again, consistent terminology is required so that sedimentologists can communicate with each other.Agreement on definitions of terms is not always easy (for example a symposium had to be held to come to a consensuson the name “dune” for large, asymmetrical, flow transverse bedforms). However, there is a fairly consistent andsimple set of terms to describe layered sediments; the general terms will be defined here and more specific termswill be introduced in the following section.

Figure 5-15 shows a hypothetical sequence of layered sediments or sedimentary rocks in order to illustrate

89

some of the fundamental aspects of sedimentary layering. The term stratum (plural: strata) is a general term appliedto any layered rock unit that is clearly distinguishable from units above and below due to some discontinuity inrock type (i.e., composition or texture). Any stratum may be simple (i.e., not internally divisible) or complex(composed of a number of distinguishable internal units). A bed is any stratum that is greater than 1 cm thick anda lamination (or lamina, plural is laminae) is any stratum that is less than 1 cm thick. The term layer is reserved fora part of a bed which is bounded by some minor, but distinct, discontinuity in texture or composition. The contactbetween beds is very distinct (e.g., the contact between beds in Fig. 5-15 is reflected by the abrupt vertical changefrom shale to sandstone). The contact between layers is generally much more subtle and may represent an erosionalsurface between packages of similar lithology. Such erosional surfaces between layers are termed amalgamationsurfaces. A division is a layer, or part of a layer, that is characterized by a particular association of primarysedimentary structure. A band is a laterally continuous (on outcrop scale) portion of a layer that is distinguishableon the basis of colour, composition, texture or cementation. A lens is a laterally discontinuous (on outcrop scale)

Figure 5-14. Bedform stability diagrams showing variation in dune spacing and dune height as a function of grain size, flowdepth and flow velocity (all scaled to 10°C water temperature). See text for discussion. After Southard and Boguchwal (1990).

10˚C

-equ

ival

ent m

ean

flow

vel

ocity

(m

/s)

0.2 0.4 0.6 1.0

10˚C-equivalent sediment size (mm)

10˚C

-equ

ival

ent m

ean

flow

dep

th (

m)

.08

0.1

0.2

0.4

10˚C-equivalent mean flow velocity (m/s)

.06

0.4 0.6 0.8 1.0 1.5

0.4

0.6

0.8

1.0

1.5

Ripples

Dunes Upper plane bed& in-phase waves

Upper plane bed& in-phase waves

Dunes

Ripples

Lowerplane bed

10˚C

-equ

ival

ent m

ean

flow

vel

ocity

(m

/s)

0.2 0.4 0.6 1.0

10˚C-equivalent sediment size (mm)

10˚C

-equ

ival

ent m

ean

flow

dep

th (

m)

.08

0.1

0.2

0.4

10˚C-equivalent mean flow velocity (m/s)

.06

0.4 0.6 0.8 1.0 1.5

0.4

0.6

0.8

1.0

1.5

Ripples

DunesUpper plane bed& in-phase waves

Upper plane bed& in-phase waves

Dunes

Ripples

Lowerplane bed

2 m

4 m

1 m

0.5 m

2 m4 m

1 m

12 cm

6 cm

3 cm

6 cm

12 cm

6 cm

3 cm

Dune spacing (length)

Dune height

d10 = 0.30 - 0.40 mm

d10 = 0.30 - 0.40 mm

D = 0.25 - 0.40 m

D = 0.25 - 0.40 m

90

portion of a layer that is distinguishable on the basis of colour, composition, texture or cementation.

All stratification that is inclined due to primary processes (i.e., not tectonic deformation or tilting) is referredto as cross-stratification. Cross-stratification typically includes packages of parallel beds or laminae that arebounded by planar surfaces (bedding planes that are termed bounding surfaces). The upper bounding surface isnormally an erosional surface and truncates underlying internal strata. (Note that this allows us to determine theoriginal direction to top in tectonically deformed sediments.) The strata between bounding surfaces are often termedinternal strata or internal stratification. Internal strata are distinguishable as beds or laminae on the basis of oftensubtle changes in grain size and/or mineralogical composition. A group of similar internal cross-strata, betweenbounding surfaces, is referred to as a cross-strata set; a group of similar sets of cross-strata is referred to as a coset.Any description of cross stratification must include the form of the internal cross-strata (see Fig. 5-16), the thicknessof the sets, the direction of dip of internal strata, the form and geometry of the bounding surfaces and the thicknessof cosets. Figure 5-17 outlines some of the common forms of cross-stratification and provides terms for describingthese structures. Note that the term cross-stratification is a general one; more specific terms are cross-laminae(internal strata are less than 1 cm in thickness) and cross-bedding (internal strata are greater than 1 cm).

Origin of cross-stratification

The occurrence of bedding plane exposure of bedforms is relatively rare in outcrop. We typically can seeonly the two-dimensional view (in vertical section) of the internal structure produced by bedforms as they moveover a bed surface. Because bedforms essentially migrate through each other (see above and further discussion

bed

bed

bed

layer

layer

Division of cross-stratification

Division of horizontal lamination

Lens of pebbles

Amalgamation surface

Cementation band

Band of concretions

Band of pebbles

Figure 5-15. Terms used to describe various types of layering in sediments. Note that the lithology that is black is shale andthe white lithology is sandstone. After Blatt, Middleton and Murray (1980).

91

Major types of internal cross-strata(in cross-section, parallel to dip-direction)

Angular internal cross-strata:

angular contact with top and bottom bounding surfaces.

Tangential internal cross-strata:

bases of cross-strata are in tangentialcontact with bottom bounding surface,angular contact with top bounding surface.

Sigmoidal internal cross-strata:

tops and bases of internal cross-strataare in tangential contact with boundingsurfaces.

Figure 5-16. Forms of internal cross-strata. Note, parallel horizontal lines are bounding surfaces.

Set

Set

Set

Coset

Planar tabular cross-stratification

Set

Set Coset

Planar wedge-shaped cross-stratification

Trough cross-stratification

SetCoset

Set

Figure 5-17. Terminology for cross-stratification. Note that planar tabular cross-stratification is characterized by planar,parallel bounding surfaces, wedge-shaped cross-stratification is characterized by planar but not parallel bounding surfaces, andtrough cross-stratification is characterized by trough- or scoop-shaped bounding surfaces (also called festoon cross-stratification). After Blatt, Middleton, and Murray, 1980.

below) we are usually limited to only partial preservation of that internal structure. The internal structure of thebedform that we normally see is some expression of the lee face and trough of the bedform, and only rarely (foranything but ripples) the stoss slope. The expression of these aspects of bedforms form may be visible as layersdue to subtle variation in grain size and/or mineralogy (seen as variation in colour) on the surfaces preserved withinthe deposits. These layers comprise packages of inclined strata (internal strata paralleling depositional surfaces)within the deposits that may be cut by erosional surfaces (bounding surfaces).

The following discussion is directly relevant to the formation of cross-stratification by asymmetric bedformsbut also illustrates principles that are important to the formation and preservation of forms of stratification producedby other bedforms. Figure 5-18 shows the hypothetical lateral migration of an inclined surface (like the lee slopeof a bedform) due to periodic deposition of sediment on the surface (from times t1 to t

9 forming layers L

1 to L

8). Each

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t1

L1 L2 L3 L4 L5 L6 L7 L8

t2 t3 t4 t5 t6 t7 t8 t9

Position of brink over time

Cross-strata

Formation of internal cross-strata by bedform migration

A BC

A B C

Figure 5-18. Illustration showing the for-mation of internal stratification with bedformmigration by episodic deposition on the leeslope. Insets show internal form of cross-strata formed by avalanching (A; inverselygraded), periodic fallout from suspension(B; normally graded) and deposition of heavymineral-enriched sediment (C). See text forfurther discussion.

layer, termed a foreset layer or foreset, represents a particular depositional event on the inclined surface. Eachindividual layer may be visible due to some mineralogical and/or textural differences between layers that might arisefrom a number of possible sorting mechanisms that act while they are deposited. For example, a particular foresetlayer may preserve variations in grain size that developed as the sediment avalanched down the inclined surface;avalanche deposits tend to be inversely graded (i.e., become coarser within cross-strata, in the direction of migration,normal to the lee face; Fig. 5-18A). Avalanching induces shearing within the sliding sediment and the grains exerta “dispersive pressure” (a force per unit area acting in the direction pointing away from the solid surface that arisesdue to grain collisions) which forces coarse grains upwards relative to fine grains. The largest grains also tend toroll further down the lee slope so that internal strata tend to become finer-grained up slope and sets of such cross-strata are normally graded (i.e., become finer upward) vertically through a cross-strata set. Internal strata formedby avalanching tend to have angular contacts with underlying bounding surfaces (see Fig. 5-16). In contrast toavalanching, sediment that falls periodically and in pulses from suspension tends to be normally graded (i.e.,becomes finer, within cross-strata in the direction of migration, normal to the lee face; Fig. 5-18B) because the largestparticles tend to reach the depositional surface first, followed by increasingly finer grains (i.e., the grains are sortedaccording to their settling velocities; internal strata produced by this mechanism tend to be tangential or sigmoidalin form, see Fig. 5-16). Alternatively, the layers that are deposited on the lee surface of the bedform may reflectvariation in mineralogy; any particular layer may be formed when a heavy mineral accumulation is swept over thebrink of the bedform to deposit as a heavy mineral rich foreset (Fig. 5-18C). In addition, particularly in the case ofripples, micaceous minerals commonly lie more-or-less parallel to the plane of the lee slope and highlight the internalstratification. Note that because internal strata lie on the plane of the lee face of the bedform they dip in the directionof bedform migration and this direction is, on average, also the direction of current flow. Therefore, the dip directionof internal stratification produced by ripples and dunes is a valuable paleocurrent indicator and one that is veryeasy to measure in the field, in contrast to grain orientation. The majority of paleocurrent data are based on thegeometry of cross-stratification.

The above example of deposition on an inclined surface is instructional to visualize how internal stratificationdevelops but it neglects the fact that as bedforms migrate so do the regions of deposition and erosion that theygenerate by their interaction with the flow. Deposition is largely limited to the lee face of the bedform (as outlinedabove) and erosion occurs from the trough and continues along the stoss side of the bedform. As the bedformmigrates by deposition on the lee face it forces the trough, and its region of scour, further downstream, up the stossslope of the next-downstream bedform. Thus, as one ripple migrates it consumes the next-downstream ripple, andso on. Figure 5-19A illustrates the condition where there is no net deposition on a bed during ripple migration; asthe bedform migrates the volume of sediment deposited on the lee face will equal the volume of sediment removedfrom its stoss side. Only if there is net deposition on the bed (i.e., more sediment is added to the bedform than isremoved with erosion) will all or any part of a bedform survive destruction associated with ripple migration. Figure5-19B shows the example where there is sufficient deposition on the bed (i.e., the bed undergoes aggradation whilethe bedform migrates) for preservation of the form of the bedform with downstream migration. In this case, the

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volume of sediment added to the bed over the time for migration by one ripple wavelength (for example, sedimentdeposited by fall-out from suspension) is more than the volume of each bedform so that the complete form of thebedform is preserved. Note that if the volume of sediment added to the bed were less than the volume of the bedformonly partial preservation of the bedform is possible.

The extent to which a bedform is preserved depends on the relationship between the rate at which the bedis aggrading and the rate at which the bedform is migrating. Figure 5-20 shows how various proportions of bedforms(and their associated internal stratification) will be preserved as a function of these two variables. Figure 5-20Ashows that the path of a migrating bedform on a bed that is aggrading may be defined as a vector that is the sumof the net migration and net bed aggradation over some fixed period of time. If the bed is aggrading the vectordescribing this path is inclined upward, in the direction of bedform migration, and the bedform is said to “climb”along this path. The angle of climb (β) is determined by the ratio of the aggradation rate to the migration rate. Notethat if we consider the migration path with a point of origin in the bedform trough then it delineates a line of erosionas the bedform migrates (or plane of erosion if we consider three dimensions); everything above the migration pathis eroded due to scour in the trough and everything below the line of migration is preserved and buried bysubsequent deposition. This plane of erosion results in the bounding surfaces that define cross-strata sets andthe internal strata are formed directly by deposition on the lee slope of the bedform and preserved between thebounding surfaces. Thus, in figure 5-20B, where there is no bed aggradation, the migration path is simply the pathof ripple migration and the line of erosion is horizontal. As in figure 5-19A, all downstream bedforms are planedoff by erosion in the trough of the migrating upstream bedform. In this case the only internal stratification that wouldbe preserved would be within any ripple forms that survived after migration (and presumably the current) hadstopped. In figure 5-20C bedform migration is accompanied by a moderate rate of bed aggradation compared tothe ripple migration rate and the internal deposits of bedforms are partially preserved. However, the path of themigrating ripple trough climbs at an angle that is smaller than the angle of the stoss slope (α) of the bedform. Asthe ripple migrates the plane of erosion passes beneath the stoss slope of the next downstream bedform and thetop portion of the internal strata of the downstream ripple are eroded. Whenever α > β only part of the internal stratawill be preserved; as a approaches equality with β more and more of the internal stratification will be preserved. Figure5-20D shows the situation where the aggradation rate is high, relative to the migration rate, such that the angle ofclimb is larger than the stoss slope angle of the bedform (i.e., α < β) and all of the internal deposits of the bedforms

Figure 5-19. Schematic illustration of the pattern of erosion and deposition with ripple migration. A, with no net depositionor erosion on the bed the volume of sediment eroded from the stoss side of the ripple must equal the volume deposited on thelee slope. Thus, with migration by one ripple wavelength, from time t

1 to t

2 all of the sediment contained within ripples at t

1

will have been eroded and deposited within the ripples at t2. B, with net deposition on the bed no erosion takes place and the

entire ripple form is preserved.

t2

sedimenteroded

sedimentdeposited

t2

t1

A. Migration with no net deposition

B. Migration with net deposition.

Height of bed aggradation

due to net deposition.

t1

94

are preserved as they migrate downstream.

From the above discussion it should be clear that cross-stratification, like that shown in figures 5-16 and5-17, forms in response to bedform migration and the particular style of cross-stratification that is preserved willdepend on: (1) the type of bedform (this controls, among other things the thickness of the cross-strata sets andthe thickness of internal strata); (2) the relative rates of bedform migration and bed aggradation; (3) the nature ofsedimentation on the bedding surfaces. Figure 5-21 illustrates how the various forms of cross-stratification shownin figure 5-16 might develop. Angular internal strata (Fig. 5-21A) form when deposition on the lee face of the bedformis dominated by avalanching (resulting in the angular basal contact of internal strata with the lower bounding

Figure 5-20. Illustration of the effect of the relative aggradation rate and bedform migration rate on the preservation of the internaldeposits of asymmetric bedforms. Note that dashed lines indicate the portions of ripples at t

1 that are eroded with migration

to positions shown for t2. See text for discussion.

t1 t2

path of migrating trough(a plane of erosion)

α (stoss angle) β (angle of climb)

t1

t2

t1

t2

ripple migration

α >> β, β = 0

β > α

ripple migration

bed aggradation

A. Definitions

B. No preservation of ripples with migration and no net deposition.

C. Partial preservation of ripples with migration and relatively little net deposition.

D. Complete preservation of ripples with migration and considerable deposition.

ripple migration

bed aggradation

α > β, β = 0

ripple migration

bed aggradation

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surface) and the top portions of internal strata are removed by erosion because the angle of climb of the bedformis less than its stoss slope angle (resulting in an angular contact with the upper bounding surface). Tangentialinternal strata (Fig. 5-21B) form when deposition on the lee face of the bedform includes an important componentof sediment that is falling out of suspension (much of it having gone into suspension after passing over the brinkof the bedform). The base of the lee face is gently curved and tangential with the lower bounding surface formedby erosion in the downstream trough. The tops of the internal strata have an angular contact with the upperbounding surface where they are erosionally truncated. Sigmoidal internal strata (Fig. 5-21C) form when bedformswith a concave lee slope climb at relatively high angles such that the erosional bounding surface passes close tothe bedform crest. This form of cross-stratification is especially well-developed by bedforms with long crests andrelatively short lee faces (e.g., washed out dunes).

Cross-stratification produced by asymmetrical bedforms

There are two major differences in the forms of cross-stratification produced by ripples and dunes. Becauseof the small size of ripples the internal stratification is typically thin (i.e., they are cross-laminae, < 1 cm in thickness)whereas dunes commonly produce thicker internal strata (commonly cross-bedding, > 1 cm in thickness). Alsobecause of the small size of ripples they tend to form a variety of forms of climbing ripple cross-stratification. Becauseof the relatively small volume of sediment in a ripple the aggradation rates required to cause these bedforms to climbat relatively steep angles are not difficult to achieve in nature. However, large dunes contain such large volumesof sediment that it is only rare that aggradation rates are large enough to cause the bedforms to climb at angles greaterthan their stoss slopes. Ripples and dunes bear the common characteristic that their plan form dictates the overallform of cross-stratification. Straight-crested (2-dimensional bedforms) tend to form planar tabular and planar-wedge-shaped cross-stratification (the planes of erosion associated with trough migration are relatively flat). Incontrast, 3-dimensional forms produce trough cross-stratification because the planes of erosion are highly irregular,

A. Angular lee slope dominated by avalanching with partial preservation of lee slope deposits forming angular internal cross-strata.

B. Concave lee slope with avalanching plus fallout from suspension with partial preservation of lee slope deposits forming tangential internal cross-strata.

C. Concave lee slope with avalanching plus fallout from suspension with complete preservation of lee slope deposits preserving sigmoidal internal cross-strata.

Figure 5-21. Styles of internal stratification shown interms of their mode of formation and style of preserva-tion (see text for discussion).

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like the troughs and crests of the bedforms, and internal strata are best preserved where the bedforms migrate intoa particularly deeply scoured trough.

Ripple cross-lamination

Internal laminae produced by ripples can have a variety of forms: from angular to sigmoidal and planar totrough-shaped in cross-section. The thickness of sets of ripple cross-lamination is limited by the upper limit to rippleheight, approximately 0.05 m. Most ripple cross-lamination is classified according to the angle of climb of thebedforms and the resulting forms of internal laminae. Figure 5-22 shows a variety of ripple cross-lamination types.Figure 5-22A is a form of horizontal lamination that is produced when the angle of climb of the ripple is negligible(i.e., much less that the stoss slope of the bedform). Any deposits that are preserved between bounding surfacesas the ripples migrate over the bed surface are too thin to allow the recognition of internal laminae. With increasingangle of climb beyond some critical angle, that will depend on the sorting and mineralogy of the sediment, thedeposits between bounding surfaces become thick enough to preserve visible internal lamination (Fig. 5-22B).Because the angle of climb is smaller than the stoss slope angle of the bedform (i.e., α > β) the foreset laminae are

Styles of ripple cross-stratification

aggr

adat

ion

migration

climb

Sub

cric

ritic

ally

clim

bing

rip

ple

cros

s-st

ratic

atio

nS

uper

cric

ritic

ally

clim

bing

rip

ple

cros

s-st

ratic

atio

n

α = angle of stoss slope

βα

β = angle of climb

α >> β

α > β

α < β

α << β

A

B

C

D

Figure 5-22. Forms of ripple cross-stratification that develop as a function of the angle of climb of the bedforms. See textfor discussion of details.

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truncated on top and all of the stoss slope deposits have been eroded. Such ripple cross-stratification is often termedsubcritically climbing ripple cross-stratification; subcritical means that α > β. This form of cross-stratificationdevelops under moderately low rates of bed aggradation due to fallout from suspension onto the bed. Withincreasing angle of climb such that α < β the entire ripple form is preserved with ripple migration, including depositsof the stoss slope (Fig. 5-22D; such internal strata are often termed formsets). Such stratification is termedsupercritically climbing ripple cross-stratification. (Note that when α = β the stratification is referred to ascritically climbing ripple cross-stratification and the entire foresets are preserved.) With increasing angle of climbthe thickness of the stoss-slope deposits increases and when the angle of climb reaches 90° the stoss and lee slopedeposits are equal in thickness. The forms of ripple cross-stratification that develop with particularly high anglesof climb indicate very rapid deposition from suspension while ripples were actively (but relatively slowly, migratingon the bed. Note that the forms of ripple cross-lamination shown in figure 5-22 occur in nature as a continuous rangeof forms from 0 ≤ β ≥ 90. Figure 5-23 is a sketch of a common vertical sequence of the forms of ripple cross-stratification and shows the interpretation of this sequence that develops under waning flows, in terms of the rateof ripple migration and the rate of bed aggradation due to fallout from suspension..

The forms of climbing ripple cross-stratification shown in figure 5-22 are often termed ripple-drift cross-lamination. In addition to the terminology described above many workers use the classification scheme of Joplingand Walker (1968). In that scheme subcritically climbing ripple cross-stratification (such as that shown in Fig. 5-22B) is termed Type A ripple-drift cross-lamination; critically to slightly supercritically climbing ripple cross-stratification is termed Type B ripple-drift cross-lamination (e.g., Fig. 5-22C); and highly supercritically climbingripple cross-stratification is termed sinusoidal ripple-drift cross-lamination (Fig. 5-22D).

Cross-stratification formed by dunes

As noted above, dunes produce forms of cross-stratification that are geometrically similar to cross-stratification produced by ripples. Figure 5-24 shows two forms of cross-stratification formed by dunes: planarcrossbedding (both tabular and wedge-shaped) and trough cross-bedding. Both of these are distinguished fromripple cross-stratification by their larger scale and the cross-stratification produced by dunes is commonly termed“large-scale cross-stratification” in contrast to the “small-scale cross-stratification” produced by ripples. Thegeometry of the internal strata will be governed as outline in figure 5-21 and the bounding surfaces depend on theangle of climb of the bedform (although, as noted above, large dunes only rarely climb at high angles). The thicknessof cross-strata sets formed by migrating dunes are generally larger than that produced by ripples (because dunesare higher) but set thickness is also governed by the angle of climb of the bedform. As noted above, the thickness

incr

easi

ng

fal

lou

t fr

om

su

spen

sio

n

dec

reas

ing

rip

ple

mig

rati

on

rat

e

Figure 5-23. A relatively common vertical sequence of climbing ripple cross-stratification that is produced by waning, sedimentladen flows. Note that the angle of climb increases continuously upwards as the current wanes, resulting in a temporally increasingrate of fallout from suspension and a decreasing rate of ripple migration.

98

of internal strata is also greater than those produced by ripples; their is a general tendency for the thickness ofinternal strata to increase with the height of the lee face of the bedform.

Large scale planar cross-stratification is produced by 2-dimensional dunes that are dominated by depositionby avalanching on their lee slopes (i.e., internal strata may be inversely graded). Internal strata typically dip at anglesof approximately 30° and typically range in thickness from a few decimetres up to several metres. In outcrop cautionmust be made in interpreting such cross-strata because the form will vary due to the relationship between theorientation of internal stratification and the orientation of the exposure. For example, in figure 5-24, planar cross-stratification looks like a horizontal stratification when viewed on a vertical section aligned normal to the flowdirection (the direction of bedform migration; note that the internal strata seen in this view need not parallel the lowerbounding surface exactly, as shown). Thus, to avoid mistakenly interpreting such cross-stratification you musttry to view it on at least two vertical planes. Note that in plan view, through a planar cross-strata set, the straight,parallel strikes of the planar internal strata are visible, aligned normal to the direction of bedform migration. Theinternal strata, in plan view, may be traced laterally for up to tens of metres. To determine the paleoflow directionbased on planar cross-stratification, we need to find the strike of internal strata and the dip direction: the paleoflowdirection will be perpendicular to the strike, into the direction of dip.

Large scale trough cross-stratification is produced by 3-dimensional dunes where deposition on the lee faceincludes both contact and suspended loads. Dip angles of internal strata vary from 20 to 30°, generally lower thanthe planar forms, and sets range from a few decimetres to several metres in thickness. The form of such trough cross-stratification varies significantly depending of the view of the exposure. In vertical sections, parallel to the flowdirection, sets may appear very similar to planar cross-stratification. However, in the vertical plane, normal to theflow direction, the diagnostic trough-shape is apparent. In plan view the form the stratification appears as inter-cutting, elongate troughs (defined by the bounding surfaces) and internal strata are curved, into the direction ofbedform migration, where they terminate against their lower bounding surfaces; thus, their concave surfaces facethe direction of migration. Such troughs, in plan view and in vertical section may extend for up to several tens ofmetres.

FLOW DIRECTION

2-dimensional dunes(planar tabular or wedge-shaped

cross-stratification)

FLOW DIRECTION

3-dimensional dunes(trough cross-stratification)

Figure 5-24. Forms of large-scale cross-stratification produced by dunes. See text for details. After Allen, 1970.

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Upper plane bed horizontal lamination

Deposition under upper plane bed conditions produces a variety of forms of horizontal lamination (sometimestermed “parallel lamination”, although parallel laminae need not be horizontal). While one might think that horizontallamination would be the simplest form of stratification to interpret (i.e., deposition on a planar, horizontal surface),this has actually been one of the most controversial styles of lamination. For example, a recent (not soon-to-be-published) manuscript listed no less than 20 hypotheses for the origin of horizontal lamination formed under upperplane bed conditions. Each such hypothesis suggests various sorting mechanisms that segregate sediment onupper plane beds into “packages” of distinct grains size (and/or sorting) or mineralogy that comprise laminae. Itis likely that there are several mechanisms that cause the sorting of the sediment that leads to the formation oflamination on upper plane beds (or nearly plane beds with low bedwaves). The major mechanisms can be listedas: (1) local and small-scale sorting by the bursting process (both bursts and sweeps) on a true plane bed; (2)selective sorting by size, shape and density on a true plane bed; (3) migration of low bedforms over which sedimentsize, shape and density varies regularly. Each of these will be discussed below.

Bridge (1978) was the first to suggest that horizontal lamination (that is, thin, < 1 mm, horizontal lamination)formed in response to the bursting cycle. He postulated that temporally decreasing boundary shear stressassociated with bursting would lead to deposition of increasingly finer sediment over the period of bursting andthis sediment would be preserved as a fining-upwards lamination. However, Cheel and Middleton (1986) showedthat horizontally-laminated sand and sandstone deposited under upper plane bed conditions were not composedof predominantly fining-upward lamination but consisted of a mixture of fining- and coarsening-upwards laminaeof limited lateral extent (several millimetres across the flow direction). Such laminae are shown in figure 5-25A from

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.05 0.10 0.15 0.20 0.25

D(mm)

DIS

TA

NC

E A

BO

VE

DA

TU

M (

mm

)

F.U.

?

C.U.

C.U.

C.U.

F.U.

C.U. ??

C.U. ?

C.U.

Quartz &Feldspar

Garnet

Magnetite

A. Upper plane bed

B. Low in-phase waves

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0.05 0.10 0.15 0.20 0.25 0.30 0.35

F.U.

F.U.

F.U.

F.U.

F.U.

F.U. ?

C.U. ?

F.U.

F.U. ?

F.U. ?

C.U. ?

C.U. ?

D(mm)

DIS

TA

NC

E A

BO

VE

DA

TU

M (

mm

)

Figure 5-25. Internal grading and the distribution of heavy minerals in horizontally-laminated sediments: (A) deposited onplane beds, and (B), deposited on low, downstream-migrating in-phase waves. All points in each plot indicate the mean grainsize in thin, contiguous layers up through the horizontally-laminated deposits. CU indicates coarsening-upwards lamina andFU indicates fining-upwards lamina. Question marks indicate that the textural interpretation is rather uncertain. See text fordetailed discussion. After Cheel, 1990a.

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upper plane bed deposits of a flume (the same textural and mineralogical characteristics were found in ancientsandstones and modern inter-tidal deposits). In addition to these “textural” laminae, they identified “mineralogicallaminae” that were much more extensive than the textural laminae and were made up of very thin sheets of heavyminerals (termed “heavy mineral sheets”) that seemed to be predominantly associated with coarsening-upwardstextural laminae. They attributed the textural laminae to the bursting cycle and the heavy mineral sheets to themigration of heavy mineral accumulations over the active plane bed (e.g., heavy mineral shadows). In their modelfining-upward laminae formed by bursting. They suggested that the ejection of fluid away from the boundary carriedsand upwards and as the burst decayed this sand fell back to the bed, the largest, heaviest grains first, followedby the smaller, lighter grains (i.e., see Stokes’ Law of settling); thus the material deposited on the bed followingbursting would form an upward-fining layer. In their model coarsening-upwards laminae formed as an in-comingsweep locally increased the shear stress acting on the granular boundary and the sediment that was taken intotransport would re-deposit just downstream as the sweep decayed. Shearing within the material in transport wouldcause the grains to experience dispersive pressure which pushes coarse grain upwards relative to the finer grainsin the moving sediment layer on the bed. Deposition of this coarsening-upwards layer of moving sediment wouldform the coarsening-upwards lamination. Heavy mineral sheets represent a lag of fine-grained heavy minerals leftbehind as a heavy mineral accumulation passed over the bed. The fine, high-density grains would easily becomedeposited, particularly in the spaces between immobile quartz grains on the bed. The association of heavy mineralsheets with coarsening-upwards laminae suggests that the heavy mineral accumulations were responsive to thehigh bed shear stress due to sweeps but essentially armoured the bed to the effects of bursting. The formationof heavy mineral sheets, described above, is an example of lamination formation by selective sorting by size anddensity. Extensive horizontal lamination are probably commonly formed by this mechanisms (and low bedforms;see below) whereas the laminae formed by the bursting cycle comprise laterally less extensive lamination in upperplane bed deposits.

The formation of heavy mineral sheets may also represent an example of the formation of horizontallamination by the migration of low bedforms or bedwaves. A heavy mineral accumulation may be thought of as

∆

X

T1

T2

θ

Y

∆

Figure 5-26. Schematic illustration showing the formation of fining-upward laminae with heavy minerals concentrated in thelaminae tops due to migration of low in-phase waves with heavy minerals concentrated on their crests. Upper diagram showsthe migration path of in-phase waves (lines with open arrows) during bed aggradation of ∆Y and bed wave migration of ∆Xfrom time T

1 to T

2. Lower diagram is the enlarged view of boxed area showing the nature of grading and the distribution of heavy

minerals on the in-phase wave and within laminae produced with migration. θ is the angle of climb: θ = tan-1 (∆Y /∆X). Verticalexaggeration approximately X25. From Cheel, 1990a.

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a mineralogical bedform that migrates over the bed. In fact, the formation of a variety of textural laminae that areof much greater extent that the textural laminae formed by bursting must involve the migration of bedforms overan otherwise plane bed. This raises the question, again, of whether a bed with even the lowest bedwaves is reallya plane bed (e.g., see Fig. 5-10). For now, we will extend the definition of upper plane bed to include “true” planebeds and “nominally” plane beds over which low bedforms may migrate. This is probably a reasonable extensionof the definition of plane bed, for the purposes of our discussion of horizontal lamination because low bedwavesare definitely responsible for some forms of this structure.

It has long been known that very low bedforms such as washed-out dunes will produce a horizontallamination if the preserved deposit generally lacks internal stratification (because the lamination is too thin to seethe cross-stratification or because the sediment is too well-sorted to undergo significant segregation by size onthe lee of the bedform). Indeed, as we saw in the section of bedform migration (see Fig. 5-22A) any asymmetricbedforms with a very low angle of climb might produce a form of horizontal lamination. However, there have beenseveral suggestions of upper plane bed horizontal lamination formed by migration of low bedwaves on a nominallyplane bed. For example, Allen (1984a) suggested that the passage of eddies over a bed would produce low-relief,downstream-migrating bedforms that would form extensive horizontal lamination. He described the predictedcharacteristics of such bedwaves in detail but such waves, with those characteristics, have not been identified inexperimental or field studies. However, experimental work by Bridge and Best (1988) and Paola et al. (1989) and Cheel(1990a) have described a variety of low bedwaves that are thought to be responsible for the formation of extensivehorizontal lamination under what are essentially upper flow regime plane bed conditions.

Bridge and Best (1988) described low, asymmetrical bedforms (possibly very washed out dunes) and notedthat they formed extensive laminae in the bed material of their flume. Paola et al. (1989) described very low symmetricalbedwaves (presumably similar to those shown in Fig. 5-10) that migrated downstream. By recording the bedbehaviour with high speed video photography they observed very low angle, parallel lamination (essentiallyhorizontal lamination) formed by a combination of small-scale fluctuations of turbulence (bursting), selectivesorting, and bedwave migration. Cheel (1990a) described the internal grading and distribution of heavy mineralsin low, downstream-migrating symmetrical bedwaves on which grain size became finer towards the tops of thebedwaves and heavy minerals were concentrated in their tops. With migration of such low bedforms predominantlyfining-upwards laminae formed with heavy minerals concentrated in the tops of such laminae (compare with thedistribution of heavy minerals in true plane bed horizontal lamination, Fig. 5-25). Figure 5-26 illustrates the formand origin of such laminae. Coarsening-upward laminae, and some thin fining-upwards laminae, also associatedwith this form of horizontal lamination, were attributed to the action of bursts and sweeps, concurrent with bedformmigration.

As for a conclusion to the origin of horizontal lamination, it is likely that all of the major mechanisms (andsome others not discussed here) will lead to the formation of this structure. Some or all of the three major mechanismslisted above may act together to form horizontal lamination under the same flow conditions and any one depositof upper plane bed, horizontally laminated sand or sandstone, probably preserve laminae formed by at least two,and possibly all three of these mechanisms. The laterally extensive forms of horizontal lamination certainly involvethe migration of coherent sediment structures over an active plane bed (heavy mineral accumulations or full-fledgedbedforms) whereas the more subtle laminae are produced by processes associated with near-boundary turbulence.The key to distinguishing the products of these various mechanisms likely lies in detailed studies of the texturalcharacteristics (including grain size, shape and orientation) within individual laminae.

In-phase waves stratification

Descriptions of in-phase wave stratification have been limited and figure 2B shows rather “ideal” forms of internalstratification associated with the variety of forms of in-phase waves. The section on the formation of cross-stratification described conditions that are also necessary for preservation of in-phase wave stratification (i.e., netbed aggradation is require to form thick sequences of in-phase stratification, in conjunction with sorting by sizeand/or mineralogy that is required to produce visible stratification). However, relatively little is known about the

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specific origin of internal stratification produced by in-phase waves so that the discussion that follows will be largelylimited to its geometry. Note that figure 2B is largely based on descriptions in the literature of such stratification,some of which is reviewed below.

Power (1961) coined the term “backset bed” for upstream-dipping cross-strata (becoming finer-grainedtowards their tops) formed under antidunes (while Power coined this widely-used term the so-called backset bedsthat Power described are probably not formed by antidunes at all!). Middleton (1965) characterised in-phase wavestratification by its association with upper plane bed horizontal lamination, low angle (<10°) cross-laminae dippingboth upstream and downstream, and by the confinement of cross-laminae to symmetrical lenses related to the formof the in-phase wave. Hand, Wessel & Hayes (1969) described in-phase wave stratification in which cross-laminaedip at angles of up to 24°. Based on flume experiments, McBride et al. (1975) documented thin (0.2 to 4 mm thick),laterally extensive, near-horizontal, parallel lamination, characterised by alternating coarse and fine laminae, whichformed by downstream migration of low in-phase waves. Allen (1966) noted that in-phase waves will form lensesof backset cross-strata only if the water waves break and/or migrate upstream, whereas undulating parallel laminaedraped over the symmetrical bed forms are produced if the water surface wave grows in place during net deposition.Furthermore, downstream dipping cross-strata form if the in-phase waves migrate downstream (Allen, 1966; alsosee Allen, 1984b, Fig. 10-21). Barwis & Hayes (1985; p. 908) suggested that the occurrence of low angle truncationsurfaces in massive or horizontally laminated sands may indicate the presence of in-phase waves. They alsoprovided an excellent description of the variability of in-phase wave stratification on a washover fan in a barrierisland complex. They noted that down-fan, in the flow direction, in-phase waves decreased in length (reflectingdecreasing flow velocity) and amplitude and passed into plane bed. The form of the in-phase wave cross-stratawithin the deposits also varied downfan from: (1) lenses of backset cross-laminae, to (2) lenses of laminae subparallelto bounding surfaces, and to (3) lenses of foreset laminae. This sequence was interpreted to reflect downfanvariation in the relationship between the water and bed surfaces from: (1) upstream-migrating in-phase waves, to(2) in-phase waves under stationary water surface waves, to (3) downstream-migrating antidunes. Langford &Bracken (1987) described variation in in-phase wave stratification in a fluvial setting, as lenses of backset and foresetcross-laminae of smaller downstream extent than cross-stratification formed under lower flow regime conditions.

Figure 2B shows the ideal forms of stratification produced by the specific behavioural forms of in-phasewaves that have been recognized and shows a gradual continuum of stratification styles through the transition fromwashed-out dunes to antidunes. Washed-out dunes (climbing at a low angle) and upper plane bed, forminghorizontal lamination, develops as the upper flow regime threshold is exceeded. Low (on the order of millimetre high)downstream-migrating in-phase waves develop next and form horizontal lamination of the characteristics describedhere (termed in-phase wave horizontal lamination; see Fig. 5-26). As the in-phase waves grow in height andwavelength they continue to migrate downstream, producing lenses of downstream-dipping cross-laminae (termedin-phase wave foreset cross-laminae). The absence of visible cross-stratification formed by low in-phase wavesis likely due to the very small thickness of the deposit and to the relatively poor development of sorting and fabricalong the low-angle lee of the bed form. The next bed phase is characterised by standing in-phase waves whichproduce laminae which approximately parallel the bed forms (termed in-phase wave drape-laminae). With the onsetof upstream-migrating and/or breaking wave antidunes, lenses of upstream-dipping cross-laminae form (termedantidune backset cross-laminae). This sequence applies to the 2-dimensional forms of in-phase wave only. The3-dimensional forms are not well known from flume studies and are not included here. Also not included in figure2B are associations of the various forms of in-phase wave stratification, due to temporal variation in the form ofin-phase waves. Given the behaviour of the in-phase waves bedforms shown in figure 5-11 it is certain that suchcomplex associations are very important and may be diagnostic of stratification formed by in-phase waves.

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CHAPTER 6. FLOW, BEDFORMS AND STRATIFICATION UNDER OSCILLATORYAND COMBINED FLOWS

INTRODUCTION

Bedforms and stratification produced by unidirectional flows are well understood after almost a century ofstudy in flumes and natural settings. However, in the geologic record of marine and lacustrine depositionalenvironments other classes of fluid flow are important in producing structures that are preserved and form aninterpretive basis for those deposits. Figure 6-1 summarizes a continuum of flow types, from purely unidirectionalflows to purely oscillatory flows, between which are a range of mixed flows and their components are shown in thelower half of the figure. In this section we will begin by examining the nature of currents, bedforms and stratificationproduced by wind-generated waves on a free water surface (i.e., purely oscillatory flow). This will be followed bya very brief description of the bedforms and stratification that develop when unidirectional and oscillatory currentsare superimposed to produce a general class of flow termed a combined flow. Note that figure 6-1 is a simplificationof possible natural flows. To begin with, the classification shown in figure 6-1 does not consider the angularrelationship between the oscillatory and unidirectional components: the resultant current produced by co-linearflows (i.e., both act in the same direction) will be much less complex than in the case where the oscillatory componentacts at some angle to the unidirectional component. In addition, in natural flows several different oscillatory andunidirectional components may act simultaneously to result in a very complex flow pattern. In nature such patternsexist and experimental work, like that described in the section on bedforms, is limited to the relatively simplisticapproach.

GENERAL CHARACTERISTICS OF GRAVITY WAVES

Waves generated by wind blowing over a water surface are prevalent in most marine and lacustrine settings.Such waves are commonly referred to as gravity waves because gravity is the most important force acting to dampenthese waves (i.e., gravity acts to restore the water surface to a flat surface after a wave has been generated by wind).

Figure 6-1. Flow classes defined in terms of the relative strength of their oscillatory and unidirectional components. Top graphsillustrate the pattern of variation of velocity over time of the net flow and the lower graphs show the oscillatory and unidirectionalcomponents of each flow separately. After Swift et al., 1983.

1050

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Resultant Flow

Flow components

Unidirectionalcomponent

Oscillatory component

Time (seconds)

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/sec

)

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flow

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flow

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flow

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Flow Class:Combined flows

103

Gravity waves are just one particular type of wave that acts in large water bodies such as oceans (Fig. 6-2) and theyaccount for much of the total energy possessed by the worlds oceans.

The majority of gravity waves on a water surface are generated by wind moving over that surface. Whilethe exact mechanisms of wave formation are not well understood the response of the water surface to wind is wellknown. Waves develop as sinusoidal oscillations of the water surface that propagate (i.e., travel) in the directionof the wind. The characteristics of wind-generated gravity waves (e.g., their size) depends on a number of factorsthat include wind speed, duration of wind and the distance over which the wind acts on the water surface (thisdistance is termed fetch). Waves will propagate beyond the region of wave generation and may travel thousandsof kilometres and take days to dissipate. Waves that have left their region of generation are termed swell. Swellwaves are typically very long, relatively low and straight-crested.

Any description of gravity waves must include wave length (L; also referred to as wave spacing), waveheight (H), wave celerity (C; the speed at which the wave moves along the water surface, and wave period (T, whereT = L/C; the time required for one full wave-length to pass a fixed point). All of these wave characteristics aredetermined by the intensity, duration and fetch (the distance over which the wind acts on a water surface) of thewind that generates them. To illustrate figure 6-3 shows the scales of waves produced by various wind speeds actingover seas with unlimited fetch.

Gravity waves are classified by the relationship between their wave length and the depth of water throughwhich they are moving (this affects the form of the waves and waves in deep water undergo a number of changesas they move into shallow water; see below). Deep water waves are waves with lengths that are no more than twotimes the depth (h) of water over which they are moving (i.e., h > L/2; note that this depth is commonly called wavebase). Transitional waves have longer wave lengths relative to water depth, between the limits L/20 < h < L/2. Shallowwater waves are waves have lengths that are at least 20 times the water depth (i.e., h < L/20).

The geologically important result of gravity waves on the water surface is the fluid motion that they generate,motion that may result in sediment transport and the formation of primary sedimentary structures. The nature offluid motion associated with waves is very complex and this section will only discuss this topic in a very superficialmanner and in terms that will be particularly important to later discussion of the sedimentary structures generatedby waves. The best general text that describes gravity waves and their products is that by Komar (1976).

To begin the discussion of fluid motion under waves we will consider so-called deep water waves that donot interact significantly with the bottom (i.e., the sediment substrate at some depth below the water surface isunaffected by the passage of such waves). Figure 6-4A shows the characteristics of a train of deep water waves.Fluid motion beneath such waves is complex and is shown in a simple but instructive manner in Fig. 6-4A. If we

Figure 6-2. Wave classification by wave periodand the mechanisms that generate waves on largewater bodies such as oceans. Each shaded bar showsthe range of wave periods generated by each mecha-nism and the solid black bars indicate the waveperiod that generates the maximum wave energy inthe oceans due to each generating mechanism. Thebar at approximately 10 seconds represents wind-generated waves by average wind conditions on theoceans; the bar representing a period of just over100 seconds is associated with long period and wavelength “swell” generated by storms. The bar asso-ciated with wave periods of approximately 2000seconds (about half an hour) are waves generated byearthquakes (Tsunamis). The two bars on the farright represent, from left to right, tides withsemidaily (or semidiurnal; two tidal cycles per day)and daily (or diurnal; one tidal cycle per day)periods.

10.10.01 10 100 1,000 10,000 100,000 1,000,000

Wave Period (seconds)

wind-generated gravity waves

storms

diurnal semi-diurnalEarthquakes(tsunamis)

Sun/moon(tides)

Mec

han

ism

of

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e g

ener

atio

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104

follow the path of a single water molecule as a wave passes one complete wave-length the path appears circular:motion is in the direction of propagation under the crest and in the opposing direction under the trough and atintermediate angles and directions, through 360°, at other positions beneath the passing wave (Fig. 6-4A). Thiscircular path is referred to as the wave orbital and its diameter is termed the orbital diameter (do). Under deep waterwaves the orbital diameter becomes exponentially smaller with depth until it is of negligible size and the orbitaldiameter is related to depth by:

do He Ly

=2π

Eq. 6-1

where y is a negative number indicating the distance below the water surface and L is the wave length. The waveorbitals persist but become negligible below the depth equal to L/2. The speed of fluid motion about the orbitalsat any time during the passage of a single wave is termed the orbital speed. The orbital speed just as the crest andtrough pass by is termed the maximum orbital speed (Um); Um is equal but acts in opposite directions beneath thecrest and trough, as outlined above. The maximum orbital speed varies with depth by its relationship to the orbitaldiameter, given by:

Udo

Tm =

πEq. 6-2

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ind

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eed

(km

/h)

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0 1 2 3 4 5 6 7 8 9 10111213

Average period (s)

0 20 40 60 80 100120140

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160180200

Figure 6-3. Wave conditions as a function of wind speed for a fully developed sea. Data from Thurman, 1988.

0Time

Vel

ocity

0.5T T

wave crest

wave trough

+um

-um

+

-y

-um

+um

still water level

L

HC

do

h < L/2

B. Transitional waves (L/20 < h < L/2)

do

-um

+um

still water level

L

HC

y = L/2

A. Deep water (h > L/2)

shal

lowin

g

y h

h

Figure 6-4. Characteristics of deep water and transitional waves. See text for discussion.

105

At the water surface the orbital diameter is equal to the wave height (i.e., do = H), such that:

UH

Tm =

πEq. 6-3

As waves propagate into shallow water (at depths less than L/2) they begin to interact with the bottom andundergo several changes and the relationships given above no longer hold. A particularly important changeinvolves the refraction of the wave crests to a position that parallels the isobaths of the bottom (isobaths are linesjoining points of equal depth). Because isobaths approximately parallel the shoreline, refraction also aligns thewaves parallel to the shoreline. In addition, waves become steeper due to an increase in H and the formulae givenabove no longer accurately describe the behaviour of the current that they generate. Steepening increases untilthe deep water wave height becomes 75% of the water depth (i.e., h = 4H/3); at this depth the wave will break.Concurrent with steepening of the wave the wave orbitals become elliptical, and increasingly flatter with depth untilalong the bottom (Fig. 6-4B) the motion of fluid molecules (and any sediment in transport) follows a line that is parallelto the bottom surface. Under transitional waves the rate of decrease in orbital diameter with depth is much less thanthe rate under deep water waves (Fig. 6-4B). Under transitional waves the orbital diameter is given by:

dHT g

ho =

2πEq. 6-4

where g is the acceleration due to gravity. Note that under shallow water waves the orbital diameter does not changewith distance below the water surface but is constant. The maximum orbital speed under transitional waves is givenby the general relationship for shallow water waves :

UH

hghm =

2Eq. 6-5

(note that from here on we will assume that Um is the maximum orbital speed acting on the bottom because that iswhere the sedimentologically important work is done).

BEDFORMS AND STRATIFICATION UNDER PURELY OSCILLATORY CURRENTS

As under unidirectional flows, bedforms begin to develop under oscillatory flows as soon as the flowconditions exceed some threshold for the initiation of movement of sediment. Also, the forms of bedforms, andtheir associated internal stratification, vary with the strength of the current. However, because of the veryfundamental differences between unidirectional and oscillatory flows the properties of the flow that control whensediment will move and what bedform will be stable must also differ.

Initiation of sediment motion under waves

It is the to and fro fluid motion acting on the bottom that may produce bedforms and stratification if thestrength of the oscillatory current exceeds some threshold condition required for the initiation of particle motion.Under unidirectional flows we saw that the critical flow condition for motion of a particle depends on the size anddensity of the particle, the density of the fluid, and the boundary shear stress (that was related to the flow velocityand depth). The condition for the initiation of motion of sediment under oscillatory currents may be similarlydetermined by the fluid and sediment properties but the flow strength is normally represented by the maximum orbitalspeed and the orbital diameter or wave period. Orbital diameter and wave period are important components of flowstrength because they determine the spatial extent over which the maximum orbital speed acts and its duration.Komar and Miller (1973) determined that the critical condition for the movement of sediment under waves can bedefined in a manner that is superficially similar to Shield’s criterion for motion under unidirectional flows, by thegeneral relationship:

ρ

ρ ρ

U

gDC

d

D

nt

s

o2

( )−=FHGIKJ Eq. 6-6

where Ut is the threshold maximum orbital speed required to move sediment; ρ is the fluid density, ρs is the density

106

of the sediment, g is the acceleration due to gravity, and D is the size of the particles. C and n are constants thatare determined by the size of the sediment. For grain sizes finer than 0.5 mm C = 0.21 and n = 0.50. For grain sizescoarser than 0.5 mm C = 1.45 and n = 0.25. Note in Eq. 6-6 that do varies with wave period (see Eq. 6-2 and 6-4) sothat the threshold maximum orbital speed similarly varies with wave period: the longer the wave period the largerthe threshold maximum orbital speed required to move the sediment.

Bedforms under waves

For many years one particular type of bedform has been considered diagnostic of the influence of waveson a sediment substrate: symmetrical ripples (commonly called wave ripples and the modern term is 2-D vortexripples). Today we know that there are actually a variety of bedforms that develop under waves but beforeconsidering these in detail we will briefly focus on the common wave ripple.

So-called “wave ripples” are distinct from the “current ripples” produced by unidirectional flows by theirsymmetrical profile, relatively peaked crest and broad trough, and by their straight to bifurcating crestlines (Fig.6-5). These are a very common bedform in shallow marine sediments and occur extensively on bedding planeexposures. The spacing of such ripples (λ) ranges form centimetres to in excess of a metre and they range in heightfrom millimetres to about a decimetre. The overall size of wave ripples varies directly with grain size and small scalesymmetrical ripples form in fine sand and large scale symmetrical ripples form in gravel. When such ripples formthey are molded on the bed beneath the propagating gravity waves and their crests are aligned parallel to the crestsof the water surface waves (but do not confuse these with in-phase waves of unidirectional flows). An importantimplication of the alignment of ripple crests is that, like the wave crests, such ripples will be aligned parallel to regionalshoreline (a characteristic that was nicely demonstrated by Leckie and Krystinik, 1989). Some symmetrical rippleshave two crests, aligned at approximately 90° to each other, and normally one crest is better-developed (the dominantcrest) than the other. For many years such ripples were termed interference ripples and taken to represent thecondition when symmetrical ripples form under two sets of gravity waves that propagate at right angles to eachother. However, recent, as-yet unpublished, experimental studies have shown that interference patterns of ripplecrests can develop under certain conditions when only one wave train is active. Thus, the standard interpretation

SHORELINE

OSCILLATINGCURRENT

Figure 6-5. Schematic illustration of classical “wave ripples” or 2-D vortex ripples. Note the cross-sectional form and straightto gently bifurcating crestlines. The direction of oscillation and inferred shoreline orientation are shown for comparison. Notethat no scale is shown but ripple spacing may range from centimetres to in excess of a metre, increasing with grain size. Seetext for discussion.

107

of this particular form of symmetrical ripple is now in doubt.

Many basic textbooks that discuss the humble wave ripple point out that the spacing of such ripples isdetermined by the diameter of the circular wave orbital that forms them. In general, the ripple spacing is slightlyless than the orbital diameter acting on the sediment substrate (see Eq. 6-8). Considering the above discussion thisidea makes sense for transitional waves under water depths just less than L/2 when a more-or-less circular orbitalis present. However, it is difficult to conceive of shallow-water waves (as described above) producing such ripples.Figure 6-6 schematically illustrates observations of symmetrical ripple spacing (λ, the orthogonal distance betweencrests) from natural and experimental settings where the orbital diameters of the formative waves are known. Whenwe plot ripple spacing against orbital diameter the data occur in two indistinct groups. Some ripples have spacingsthat are directly related to wave orbital diameter by the relationship:

λ = 0 8. doEq. 6-7

Such wave ripples are termed orbital ripples. However, many wave ripples do not fall on the line defined by Eq.6-7 but deviate from that line because their wave-lengths are much shorter than the orbital diameter and such ripplesare termed anorbital ripples. Ripples with wave-lengths that fall between these two classes are termed suborbitalripples. Note that with increasing grain size the maximum ripple wave-length also increases (i.e., the coarser thesediment the larger the maximum possible ripple wave-length). This suggests that for any grain size there is an upperlimit to the size of ripples that will form. Below the limit, a single orbital ripple will exist for every orbital diameteracting on the sediment surface and beyond that limit several individual anorbital ripples will be stable under arelatively long orbital diameter. The relationship between ripples and wave orbitals is shown schematically in figure6-6.

The above discussion points to the fact that there are a variety of bedforms that develop under oscillatoryflows. Over the past decade, experimental studies have helped describe these bedforms in terms of their morphologyand behaviour and the hydraulic conditions that are necessary to form them. Harms et al. (1982), based largely onexperimental work in John Southard’s labs at M.I.T., described bedforms under oscillatory flows in the manner thatthe unidirectional bedforms are described. In the remainder of this section we will consider the sequence of bedformsthat develop under oscillatory flows much like we did the unidirectional flow bedforms in the previous chapter ofthese notes.

Consider an experiment where we induce oscillatory fluid motion (i.e., back and forth) over a sandy substratein a closed tunnel. There is no free-water surface but the speed of the oscillatory current varies gradually in both

Figure 6-6. Schematic illustration show-ing the relationship between symmetricalripple wave-length and the diameter ofwave orbitals. Data are not shown. Thedark, shaded line (labelled “orbital rip-ples”) is defined by ripples with lengthsequal to 80% of their associated waveorbitals. Solid lines (labelled with grainssizes) indicate the limiting ripple wave-length for the grain size indicated. Theshaded area represents the region ofanorbital ripples. The drawings illustratethe relationship between symmetrical rip-ples and their wave orbital for orbital rip-ples (upper left) and anorbital ripples (lowerright). See text for further discussion.

1 105 100 10001

10

100

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al rip

ples (

λ = 0.

8d o)

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?

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Rip

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spac

ing

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m)

Orbital diameter (do; cm)

108

speed and direction (over 180°) and the maximum velocity, in either direction, is comparable to the maximum orbitalvelocity under waves. The duration of the current in either direction is related to the wave period and the distanceover which the current moves during one oscillation is related to the wave orbital diameter. Like the mental experimentthat we conducted to describe bedforms under unidirectional flows, in this case we will consider the sequence ofbedforms that develops with increasing flow strength (maximum orbital speed) holding all other variables constant.Note that not all of the following bedforms will develop under a given wave period or bed grain size, as we shallsee when we look at a bedform stability diagram for oscillatory flows.

The sequence of bedforms that develops under oscillatory flows is shown schematically in figure 6-7. Justas the current begins to move the sediment (i.e., the maximum orbital speed exceeds the threshold for motion) thefirst bedforms that develop are termed rolling grain ripples. These bedforms are small with lengths less than 10cm and heights on the order of a few millimetres to approximately 1 centimetre. Rolling grain ripples are symmetricalin profile, have low slopes, and are straight-crested. Sediment movement over rolling grain ripples is as traction:grains roll back and forth under the oscillating current. This bedform is thought to be “metastable”, that is, overtime they slowly grow in size (particularly height) and become vortex ripples. Under somewhat larger maximumorbital velocities vortex ripples develop quickly and are stable. Vortex ripples include the common form of waveripple described above in the opening paragraphs of this section on bedforms. As noted above, they are symmetricalin cross-section and have slopes that are steeper than rolling grain ripples. Vortex ripple lengths very fromcentimetres in fine sand to in excess of a metre in gravel and similarly vary in height from approximately a centimetreto a decimetre. Sediment transport over vortex ripples is both in traction and suspension. The first vortex ripplesto develop are straight-crested (2-dimensional) as depicted in figure 6-5. However, as the orbital speed increasesvortex ripples become increasingly 3-dimensional in plan view. The 3-D vortex ripples grow larger (with wave-lengths in excess of 1 m) and form rounded bedforms with hummocks (areas of positive relief) and swales (areasof negative relief) on the bed (see top block diagram in Fig. 6-11). Such large, 3-D vortex ripples are sometimes referredto as “hummocky ripples”. Note that these large ripples may someday become thought of as dunes under oscillatoryflows. With a further increase in maximum orbital speed the 3D-vortex ripples become flatter and somewhat shorter

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REVERSING - CREST(long wave periods)

POSTVORTEX

VORTEX

ROLLING - GRAIN

}FLAT BED

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Figure 6-7. Schematic illustration of the bedforms that develop under oscillatory currents. From the bottom upwards thesequence of bedforms reflects the sequence that develops with increasing maximum orbital velocity (except for the reversingcrest ripples). See text for further discussion. After Harms et al. (1982).

109

to form post-vortex ripples. These bedforms can be thought of as transitional forms with flat bed, an essentiallyflat, featureless bedform that develops under the highest orbital velocities and experiences intense sedimenttransport. The so-called flat bed of oscillatory flows is morphologically similar to upper plane bed of unidirectionalflows (and many authors term the oscillatory bedform plane bed). Another bedform that develops under oscillatorycurrents, but only under waves with relatively long periods, is termed reversing-crest ripples. These bedformsare generally less than 10 cm long and rather low and are somewhat asymmetrical. They derive their name from thefact that they reverse in direction as the current reverses. This characteristic is possible because under waves withlong periods the duration of flow in one direction is sufficiently long to generate a small, asymmetrical ripples (likeripples under unidirectional flows). Thus with each oscillation the current reverses and so does the direction ofmigration of the asymmetric ripples. These particular bedforms illustrate that bedforms under oscillatory flows arevery similar to those under unidirectional flows but the current acting in one direction, under normally short periodwaves, does not persist long enough to generate bedforms like those under unidirectional flows.

Figure 6-8 shows a bedform stability diagram for fine, quartz sand (0.15 to 0.21 mm) in terms of orbital speedand wave period (we are ignoring the effects of temperature). Note that diagrams for other grain sizes will havesimilarly formed fields but at different positions (i.e., for coarse sand the fields shift upwards so that all transitionsoccur at higher maximum orbital velocities). Figure 6-8 shows that the sequence of bedforms with increasingmaximum orbital speed are essentially as outlined above and reversing-crest ripples appear to replace otherbedforms at significantly high wave periods. The range of conditions over which post-vortex ripples are stableextends to lower maximum orbital velocities with increasing period. Figure 6-8 is after Harms et al. (1982) but morerecent experiments by Southard et al. (1990) have documented, in more detail, the development of vortex rippleswith increasing orbital speed.

Stratification formed by oscillatory currents

Like bedforms that develop under unidirectional flows, bedforms under oscillatory currents result in a varietyof forms of cross-stratification that differ in detail due to the geometry and behaviour of the generative bedforms.

1.0

0.8

0.6

0.4

0.2

0.11 2 4 6 8 10 20

PERIOD T (sec)

MA

XIM

UM

OR

BIT

AL

SP

EE

D U

m (

m/s

ec)

FLAT BED(sand movement)

POSTVORTEX RIPPLES

VORTEX RIPPLES( start as rolling - grain ripples )

THREE - DIMENSIONAL ( in part ? )

TWO - DIMENSIONAL

ROLLING - GRAIN RIPPLE (metastable)

NO RIPPLE ACTIVITY

REVERSINGCREST

RIPPLES

??

??

??

??

Figure 6-8. Diagram illustrating the fields of stability of wave-formed bedforms on beds of fine sand (0.15 to 0.21 mm). AfterHarms et al. (1982).

110

To make a broad generalization: the features of wave-formed cross-strata that distinguish them from those formedby unidirectional flows are (1) a wide variation in cross-strata dip directions (although this is not always the case;see below) and (2) the symmetrical, curved bounding surfaces that are sometimes preserved, reflecting thesymmetrical form of the bedform. In addition, as we saw in the section on grain fabric, particle alignment in thedeposits of wave-generated currents is distinctly different from that under unidirectional flows; under purelyoscillatory flows the vector mean dip of particles may be horizontal.

Figure 6-9 depicts the forms of cross-stratification that develop under waves with increasing maximum orbitalvelocity (i.e., under the various bedforms described in the previous section) under conditions of vertical aggradationof a bed. Note that exact forms of stratification will vary, depending on the aggradation rate and the rate of ripplemigration (if any; see below). This description follows that of Harms et al. (1982) and remains somewhat speculativealthough the discussion below includes some observations from recent experiments by Southard et al. (1990) andArnott and Southard (1990).

Rolling grain ripples produce thin, sub-parallel laminae that may or may not display internal cross-lamination.As the bedforms grow to 2-D vortex ripples the internal cross-laminae become prevalent and the form of cross-stratification is complex. Under purely oscillating currents the internal cross-strata produced by 2-D vortex ripplesform sets of alternately dipping laminae bounded by curved erosional surfaces. Such sets are sometimes said tobe “braided” and form a particular type of cross-stratification that is termed chevron cross-stratification. The exactform of the cross-stratification produced by 2-D vortex ripples will depend on the behaviour of the bedform, whichin turn depends on the symmetry of the oscillatory current. The thickness of cross-strata sets depends on the sizeof the ripples that form them; therefore, set thickness will be strongly influenced by the grain size of the sediment(larger bedforms and cross-strata sets are possible in sediment of coarse grain size). The form of cross-stratification

Flat bedNear - horizontal fine laminae

Three - dimensional vortex ripples( flatter)λ dm - m, I 15 - 20Swaley sets of cross laminae withset contacts sloping less than 10˚,hummocks rarely preserved

Three - dimensional vortex ripples( steeper)λ dm - m, I~10Hummocky sets of cross laminae with set contacts sloping as muchas 10 - 15˚

Two - dimensional vortex ripplesλ cm - dm, I= 6 - 10 Small intricately "braided" setsof cross laminae

Rolling grain ripplesλ cm, I largeThin laminae, low angle crosslamination or flat lamination

INC

RE

AS

ING

OR

BIT

AL

VE

LOC

ITY

Figure 6-9. Forms of cross-stratification produced by wave-generated bedforms on slowly aggrading beds. Note that “I” isthe ripple index (ratio of length to height). After Harms et al. (1982).

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shown in figure 6-9 is limited to purely oscillatory flows but in nature it is not unusual for even wave-generatedcurrents in shallow water to have an asymmetry that induces a somewhat stronger current in one direction. Suchasymmetrical currents may cause 2-D vortex ripples to migrate in the direction of the strongest component of theoscillatory current. Like current ripples, the form of cross-stratification that is preserved will vary with the rateaggradation of the bed. Thus, there are a variety of forms of cross-stratification produced by 2-D vortex ripplesthat depend on the combined effects of bed aggradation and the rate of ripple migration. Figure 6-10 shows thecontinuum of forms of cross-stratification that may be produced by 2-D vortex ripples as a function of bedaggradation and ripple migration rates. Note that with ripple migration and no bed aggradation the internal cross-strata dip only in one direction and might appear like cross-strata produced by current ripples under unidirectionalflows. The similarities continue for the migrating ripples as increasing aggradation rates produce climbing formsof ripple cross-stratification. The preservation of symmetrical ripple forms is probably necessary to allow reliabledistinction of cross-stratification produced by waves from current ripple cross-stratification formed under similarconditions of bed aggradation. In addition, even the migrating wave-generated bedforms may preserve local setsof cross-strata that dip in the direction opposing the average direction of ripple migration, the presence of suchsets should suggest wave-generated currents. Only when the ripples do not migrate in a preferred direction willtrue chevron cross-stratification develop in which the proportion of cross-strata dipping in one direction or the otherare approximately equal.

Returning to figure 6-9, with increasing maximum orbital velocity, as the bedforms grow and become morerounded and three-dimensional, the forms of cross-stratification change to mimic the morphology of the bedforms.As depicted in figure 6-9 there are two forms of stratification produced by 3-D vortex ripples. The first to develop,

NOMIGRATION,

NOAGGRADATION

AGGRADATIONRATE

aggradation, no migration

migration + aggradationlarge angle of climb

migration + aggradationsmall angle of climb

MIGRATIONRATE

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Figure 6-10. Schematic illustration showing the idealized forms of cross-stratification produced by 2-D vortex ripples as afunction of their rate of migration and the rate of bed aggradation. After Harms et al. (1982).

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with increasing orbital velocity above that which 2-D vortex ripples are stable, is characterized by sets of bothconcave- and convex-up internal strata (termed swaley strata and hummocky strata, respectively) bounded bysimilarly shaped bounding surfaces. This form of cross-stratification is termed hummocky cross-stratification(HCS), consistently one of the most enigmatic primary sedimentary structures over the past two decades (see thefinal section of this chapter for a closer look at this form of cross-stratification and the debate that it has generated).Figure 6-11 shows a large, hummocky, 3-D vortex rippled bed surface and the form of HCS that the bedform is thoughtto produce. Internal lamination in HCS dip at angles up to 15° and are isotropic (i.e., dip with equal frequency inall directions). Based on the descriptions of the bedforms that are believed to generate HCS, the spacing ofhummocks should vary from approximately a decimetre to in excess of a metre. With increasing maximum orbitalvelocity figure 6-9 suggests that the next form of stratification is similar to HCS except for two distinctivecharacteristics: internal laminae dip at shallower angles (not exceeding 10°) and only swaley (concave) laminae arepreserved. This form of cross-stratification is termed swaley cross-stratification (SCS). The different form of SCS,compared to HCS, may be attributed to the lower relief of the bedforms and the greater scour of the bed (planingoff hummocks) under the greater maximum orbital velocities. With increasing maximum orbital velocity, throughthe field of post-vortex ripple stability, the internal strata and their bounding surfaces must become increasinglyflat and produce a form of horizontal lamination by deposition on a wave-generated flat bed. This form of horizontallamination has not been extensively studied and descriptions of such lamination does not provide a basis fordistinguishing it from upper plane bed horizontal lamination of unidirectional flows. Intuition suggests that internalfabric may provide a diagnostic criterion for the identification the formative processes responsible for this structure.Note that the internal structure of reversing crest-ripples is not well known. Presumably, with bed aggradation, theywill form laterally and vertically alternating (or at least varying) sets of cross-strata that dip in opposing directions.

Figure 6-11. Schematic illustrations of hummocky cross-stratification. A. A block diagram showing the form of thethree-dimensional bedform that is thought to produce HCS (3-D vortex ripples). B. The details of the internal structureof HCS. See text for details. From Cheel and Leckie (1992).

First-order surface

Second-order surface

Third-order surface

HummockSwale

Sole marks

cm -

dm

cm -

dm

A

B

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Figure 6-12. Plan view of ripples produced on beds of 0.28 mm sand by current, waves and combined flows. Note that netflow is from right to left and the boxes are 75 cm across. After Harms (1969).

The asymmetrical ripple form, if preserved, should distinguish this form of such cross-stratification from thatproduced by 2-D vortex ripples.

BEDFORMS AND STRATIFICATION UNDER COMBINED FLOWS

This brief section relies heavily on the results of recent work on the bedforms and stratification that developunder combined oscillatory and unidirectional flows. The flows considered are the result of the superposition ofwave-generated currents (see above) on simple unidirectional currents to produce temporal variation in flowstrength (e.g., velocity) as shown for combined flows in figure 6-1. Because this section is brief, the descriptionof both bedforms and their stratification will be combined.

Figure 6-12 shows the differences in plan form of wave ripples (2-D vortex ripples; formed under purelyoscillatory flows), current ripples (produced under purely unidirectional flows) and combined flow ripples. Notethat the major differences are the smaller spacing, straighter crests of wave ripples, and symmetrical profile of waveripples, compared to combined flow and current ripples, and the changes are gradual from one ripple type to theother. Internal stratification may reflect the differences in ripple behaviour: current ripples may have betterdeveloped unimodal dip directions of internal cross-strata. However, as noted above, true wave ripples may alsomigrate in one direction to produce similarly unimodal-dipping internal cross-strata.

Figure 6-13 is based on experimental work by Arnott and Southard (1989) who used a combined flow waveduct to simulate waves with an 8.5 second period acting on a bed of very fine sand. The apparatus was similar tothat used to simulate purely oscillatory flows but included a recirculating pump to induce a unidirectional currentwithin the duct. The graph shown in figure 6-13 is one of many such graphs that could be constructed for combinedflow, each representing a narrow range of grain size and wave periods. The sequence of bedforms along the verticalaxis is the sequence described for purely oscillatory flows (note that they did not go to high enough maximum orbitalvelocities to produce a wave-formed flat bed). Note also, that this figure suggests that there is a gradual transitionfrom 2-D vortex ripples to 3-D vortex ripples and current ripples. The large 3-D vortex ripples have been termedhummocky ripples in an attempt to emphasize that these bedforms likely produce the hummocky cross-stratificationthat is common in the geologic record of shallow marine sediments. These hummocky ripples appear to be stable

CurrentRipples

current - dominated wave - dominated

Combined - FlowRipples

OscillationRipples

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only under purely oscillatory flows or under combined flows with only a very weak unidirectional component. Whenthe velocity of the unidirectional component of a combined flow is only a few percent of the oscillatory velocitythe hummocky bedforms become asymmetric in profile and migrate in the direction of the unidirectional component,producing low angle inclined cross-strata that are isotropic (dip in one direction in contrast to the anisotropic dipof hummocky cross-strata). With a further increase in the strength of the unidirectional component the bedformsbecome steeper and more asymmetric and produce high angle cross-stratification. Note that these large bedforms,formed under combined flows are termed dunes in figure 6-13 (they were termed “large ripples” in the original paperby Arnott and Southard, 1989) and that they form under much lower unidirectional velocities than under purelyunidirectional flows because of the superimposed oscillatory current. Indeed, combined flows appear to producelarge, dune-like bedforms that would not develop under purely unidirectional flows over fine sand. In addition, thesuperposition of an oscillatory component reduces the threshold unidirectional flow strength required to produceupper plane bed (or flat bed).

Figure 6-14 is similar to figure 6-13 but includes a wider range of orbital and unidirectional flow velocitiesand figure 6-15 schematically shows the forms of cross-stratification produced by the various bedforms. Figure6-15 indicates that there is a more-or-less gradual change in the styles of cross-stratification produced by oscillatoryand combined flows but that the overall geometry of the stratification should provide a basis for distinction of thegenerating flow types. However, this work is currently in its infancy and will require further experimentation andobservations from the ancient record before we have a good foundation for interpreting the formative processesthat produce these primary structures.

THE ENIGMA OF HUMMOCKY CROSS-STRATIFICATION

Note: Earlier in this chapter we introduced a form of cross-stratification termed hummocky cross-stratification (HCS) andattributed its formation to the presence of 3-D vortex ripples on an aggrading bed. This view of HCS is one of several that appearin the literature and the debate on the origin of this primary sedimentary structure is ongoing. This section aims to focus onHCS and to show that its interpretation is not so straight-forward.

Hummocky cross-stratification became popular during the late 1970’s and early 1980’s as its widespreadrecognition followed description by Harms et al. (1975), although it had been earlier reported under different names(e.g., truncated wave-ripple laminae, Campbell, 1966; crazy bedding, Howard, 1971; truncated megaripples, Howard,1972). The presence of HCS has since become a prime criterion for the recognition of ancient shallow-marine stormdeposits; however, its reliability as an unequivocal criterion for this environment is now less certain. Despite thefact that HCS is widely accepted to be the product of waves, the structure continues to be the focus of ongoingdebate regarding its mode of formation and, by implication, its specific paleohydraulic interpretation. HCS in marinedeposits has been variously attributed to formation by oscillatory flows produced by waves, combined oscillatoryand unidirectional flows, and purely unidirectional flows. This diversity of hypotheses for HCS formation is justified

no movement?

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0 .04 .08 .12 .16 .20 .24

10 cm scale insketches Figure 6-13. Very schematic illustration show-

ing the bedforms and their internal stratifica-tion produced in a duct by combined flows withan oscillatory component with a period of 8.5seconds acting on a bed of very fine sand. Notethat structures along the vertical axis are formedunder purely oscillatory flows and structuresalong the horizontal axis are formed underpurely unidirectional flows. After Arnott andSouthard (1989) and Duke, Arnott and Cheel(1991).

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NoMovement

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tren

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nent

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symmetrical dunes (HCS)

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Weakly asymmetricaldunes (anisotropic HCS?)

weakly asymmetrical ripples(small-scale anisotropic HCS?)

Plane bed

Plane bedCurrent ripple cross-lamination

strongly asymmetrical ripples(small-scale trough cross-stratification)

strongly asymmetrical dunes(large-scale trough cross-stratification)

asymmetrical 3-D ripples

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Oscillatory flow dominant

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Max

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peed

(cm

/s)

Figure 6-14. Extended bedform stability diagram for combined flows with an 8.5 second period oscillatory component actingon a bed of fine sand. Compare with figure 6-15 for the forms of cross-stratification produced under the conditions shown.After Myrow and Southard (1991).

Figure 6-15. Highly schematicillustration showing the forms ofcross-stratification produced bycombined flows with an 8.5 sec-ond period oscillatory componentacting on a be dof fine sand. Com-pare with figure 6-15. Based onMyrow and Southard (1991).

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because the basis for its interpretation is not as sound as that for many other sedimentary structures that are readilyvisible in modern settings or produced in laboratories. For example, our knowledge of the relationship betweenpaleohydraulics and stratification formed under unidirectional flows is quite advanced because we can easilydissect recognizable bedforms developed under known hydraulic conditions in rivers, intertidal areas and flumes,to precisely document the geometry of their internal structure. In contrast, HCS is found largely in ancient sedimentsand sedimentary rocks where paleohydraulic conditions must be inferred from associated deposits and structures.Thus, HCS has been largely interpreted on the basis of inference rather than direct observation of the relationshipbetween hydraulic processes and the form of the structure. The argument over the origin of HCS has been furthercomplicated by the recognition of similar forms of stratification in settings in which wind-generated water surfacewaves were an unlikely mechanism in its formation. The growing range of physical settings in which HCS-likestratification may have formed may justify the suggestion (Allen and Pound, 1985) that HCS has become just a“bucket term” that embodies similar stratification styles that may be generated by a variety of processes orcombinations of processes.

This section of these notes aims to describe HCS, in detail, and indicate that there is no consensus at thistime as to the exact origin of this form of cross-stratification although recent data points to an origin of HCS underpurely or strongly oscillatory dominant flows.

HCS — description and associations

Because of the paucity of modern and experimental examples of HCS, its paleohydraulic interpretation haslargely been based on its preserved characteristics in the geological record. The following description will reviewthe basis for the recognition of HCS, including its common stratigraphic and sedimentologic associations, andstress characteristics that have led to the various ideas on HCS formation.

Characteristics of HCS

Grain Size

The grain size of sediment in which HCS occurs varies from coarse silt to fine sand (Dott and Bourgeois,1982; Brenchley, 1985; Swift et al., 1987). HCS in coarser sediment is relatively rare but has been reported. Brenchleyand Newall (1982) described HCS in sandstones with mean grain sizes ranging from 0.7 - 1.1 mm (i.e., up to coarsesand). Walker et al. (1983) noted that gravel may comprise beds displaying HCS. However, in cases where gravel-size sediment is present in HCS, the gravel normally makes up a small proportion of the total grain size distributionand is found largely as lag-deposits on surfaces within or at the base of beds displaying HCS. Therefore, theconsensus seems to be that HCS is most common in very fine to fine-grained sand with the frequency of itsoccurrence decreasing dramatically with increasing grain size.

Morphology and geometry

Harms et al. (1975, p. 87) provided an early description of HCS (Fig. 6-11A) that has remained fundamentalover the years (cf. Harms et al., 1982). They pointed to four essential characteristics of individual cross-strata setsas being: (1) low-angle (generally less than 10° but up to 15°), erosional bounding surfaces; (2) internal laminaethat are approximately parallel to the lower bounding surface; (3) individual internal laminae that vary systematicallyin thickness laterally and their angle of dip diminishes regularly; (4) dip directions of internal laminae and scouredsurfaces are scattered (i.e., dipping with equal frequency in all directions). They also postulated that thestratification was due to deposition on a scoured bed surface characterized by low hummocks (bed highs) andswales (bed lows) with a spacing of one to a few metres and with a total relief of between 10 and 50 cm. Hence,the form of the internal stratification was one of convex-upward hummocky laminae and concave-upward swaleylaminae, essentially draped over the hummock and swale topography of the basal scoured surface (Fig. 6-11B).

Following Campbell (1966), Dott and Bourgeois (1982) employed a hierarchy of surfaces (Fig. 6-11B) toprovide a careful description of HCS based on their observations. Here we employ their descriptive terms but detailsalso come from other sources (Bourgeois, 1980; Hunter and Clifton, 1982; Brenchley, 1985).

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First-order surfaces are surfaces of lithic change in discrete HCS beds (discussed below) and may boundHCS cosets or beds containing a sequence of various structures. The basal surface is commonly nearly flat anderosional, although local relief, up to several tens of centimetres may occur due to the presence of tool marks(scratches, grooves, prod-marks), gutter casts and/or rare flutes. This surface may be mantled with a lag of coarsedebris of inorganic (intraclasts and/or extraclasts) or biogenic origin (e.g., shell or bone material). In some instances,the upper surface is deeply-scoured with a hummocky appearance whereas in many other cases this surface isrippled. Cosets of HCS range from decimetres to several metres in thickness, although the thickest cosets mayactually consist of several amalgamated beds.

Second-order surfaces are erosional surfaces within HCS cosets and are normally responsible for the formof the stratification. They cut third-order surfaces but are contained by the first-order surfaces and therefore boundHCS sets (Fig. 6-11B). These surfaces commonly form the distinctly “hummocky” surfaces in HCS that arecharacterized by laterally alternating synforms (swales) and antiforms (hummocks), although the antiforms aregenerally less common than the synforms. The relief on second-order surfaces ranges from several centimetresto approximately 50 cm (the same relief reported for hummocks and swales) with rare, extreme dip angles of up to35°. HCS sets range from several centimetres up to two metres in thickness although the latter extremes are probablythe result of the amalgamation of beds. The erosional character of these surfaces may be obvious where relativelysharp, angular discordances occur between second and third-order surfaces or may be subtle where third-ordersurfaces are nearly tangential with second-order surfaces. The angular relationship commonly varies laterally,giving second-order surfaces the appearance of changing from discordant to concordant surfaces along anindividual bed. The visibility of second-order surfaces is due largely to their angular relationship with underlyingstrata. A change in grain size is typically not evident across second-order surfaces although they may be mantledby dispersed shale and/or siderite rip-up clasts, pebbles and shell debris. Such surfaces, exhumed in outcrop,typically display the hummock and swale topography described above. Most commonly, the plan form ofhummocks and swales is circular, although elongate forms have also been reported (e.g., Handford, 1986). Exhumedsecond-order surfaces may display forms of parting lineation including parting-step lineation (McBride and Yeakle,1963) and current lineation (Allen, 1964).

Third-order surfaces bound individual laminae within HCS sets and account for many of the diagnosticcharacteristics of this structure although their visibility may depend on such fortuitous factors such as the degreeof weathering of the outcrop and cementation. Third-order surfaces are nearly concordant with underlying second-order surfaces which normally determine their overall geometry of internal laminae. Angles of dip are typicallyhighest directly above erosional second-order surfaces but decrease upwards. Third-order surfaces are commonlymantled by mica, clay or comminuted plant debris (in many post-Silurian examples). Laminae defined by third-ordersurfaces tend to pinch and swell laterally and are most commonly thickest within swales, thinning upwards overhummocks. Individual laminae may or may not display internal grading, depending largely on the sorting of sand-size particles; well-sorted sand may not have a sufficiently wide range of grain sizes available to develop visiblegrading. For example, Bourgeois’ (1980) observations of the Upper Cretaceous Cape Sebastian Sandstone ofsouthwest Oregon reported centimetre-scale internal laminae in which grading was not detectable. However,Hunter and Clifton (1982), also working on the Cape Sebastian Sandstone, noted that under certain conditions,light/dark couplets that comprise the internal laminae bore characteristics that suggested that they were normallygraded. Like second-order surfaces, exhumed third-order surfaces may display various forms of parting lineation.

The scour and drape form of HCS is the most common variety of this structure although other forms havealso been recognized, including vertical accretionary forms and migrating forms (Fig. 6-16). Several workers (e.g.,Hunter and Clifton, 1982; Bourgeois, 1983; Brenchley, 1985; Allen and Underhill, 1989) described verticalaccretionary HCS in which internal laminae thickened over hummocks rather than swales. As such, the hummocksappeared to have grown by accretion rather than formed by erosion of second-order surfaces. However, this“accretionary” HCS was thought to be relatively rare. A variant of this form of HCS displays internal laminae thatparallel the hummock and swale morphology of second order surfaces (e.g., Allen and Underhill, 1989). Anothertype of HCS, described by Nöttvedt and Kreisa (1987) and Arnott and Southard (1990), is characterized by low-angle cross-strata sets (generally >5 cm thick) filling shallow scours (swales) and which have a preferred, unimodaldip direction; hummocks are generally rare to absent. This latter structure has been termed “low angle trough cross-

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stratification” by Nöttvedt and Kreisa (1989) and “anisotropic” HCS by Arnott and Southard (1990) in contrastto the more common isotropic HCS.

The adherence by later workers to the essentials of the definition of HCS given by Harms et al., 1975) abovehas been a matter for some debate. For example, Brenchley (1985) reported slopes on hummocky surfaces up to35° and spacings as small as the spacing of wave ripples (i.e., centimetres). These deviations from the originaldefinition have led some to suggest that these smaller forms (commonly termed micro-HCS) are not true HCS asHarms et al. (1975) had defined it. For example, Duke (1987, p. 345) argued that HCS-like stratification with hummockspacings below the “1 m lower limit assigned to HCS by Harms et al. (1975)” is not true HCS. However, the firstexperimentally-produced analogs of HCS identified by Harms et al. (1982) had spacings on the order of a coupleof decimetres (and Harms et al., 1982, point out the discrepancy with HCS observed in the field). The scale of HCSmay or may not be a limiting factor in defining HCS but may represent a natural variation in scale that reflects thebreadth of conditions over which this structure may form. Sherman and Greenwood (1989; p. 985) emphaticallystate that “there is no apparent physical rationale for the 1 m lower limit on hummocky cross-stratificationwavelength” and Campbell (1966) suggested that hummocks may occur with wavelengths of 0.1 to 10 m.Alternatively, many of the examples of HCS that differ significantly from the form defined above may representsimilar stratification styles that formed by different processes. The breadth of variation in form of HCS that hasgrown in the literature may be partly responsible for the similar proliferation of ideas regarding its mode of formationthat is discussed below.

HCS associations

With the onset of widespread recognition of HCS, several studies reported its occurrence in particularassociations with other structures (e.g., Hamblin and Walker, 1979; Dott and Bourgeois, 1982; Brenchley, 1985).The most common occurrences of HCS in the ancient record can be classified into two such associations: (1) discretesandstone beds interbedded with mudstones (commonly termed HCS storm beds), and (2) amalgamated sandstones;however, specific associations vary widely in nature (cf. Dott and Bourgeois, 1983).

Discrete HCS sandstones

Dott and Bourgeois (1982) were the first to propose an “ideal sequence” or model showing structures that

Forms of HCS in shallow-marine sandstones

Sco

ur

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An

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CS

Figure 6-16. Schematic illustration showing the various forms of hummocky cross-stratification. Note that the unidirec-tional component of the flow forming anisotropic HCS is from right to left. See text for discussion.

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are preferentially associated in outcrop within sandstone beds containing HCS; an ideal sequence that began alineage of such sequences for HCS sandstones. This sequence consisted of sharp-based sandstone interbeddedwith bioturbated mudstone; the basal-scour surface includes sole marks and is mantled by a lag of coarse debrisoverlain by an interval of hummocky cross-stratification passing upward into flat lamination and ultimately to cross-laminae associated with symmetrical ripple forms that cap the sandstones. A similar sequence of associations wasproposed by Walker et al. (1983) which differed in detail from that proposed by Dott and Bourgeois (1982; 1983)by the occurrence of a basal parallel (horizontal to sub-horizontal) laminated interval directly overlying the basalerosional surface. The evolution of the model continued as more observations were made and data collected. Forexample, figure 6-17 (Leckie and Krystinik, 1989) shows a recent version of the early ideal sequences and containsconsiderably more information and more variability than the earlier sequences; the new information includes theoccurrence of parting lineation on surfaces within HCS, paleocurrent relationships and a range of ripple typescapping the beds, from purely wave-formed ripples through to purely current ripples. The trend of the partinglineation is generally orthogonal to wave-ripple crests and sub-parallel to sole marks at the bases of hummockybeds (e.g., Brenchley, 1985; Leckie and Krystinik, 1989). Rarely, a polymodal trend of parting lineation has beenobserved on second or third-order surfaces (D.A. Leckie and L.F. Krystinik, unpublished observations). Cappingwave-ripples are typically straight crested with bifurcating patterns, although irregular forms are not uncommon,including: polygonal, ladderback and box patterns (all forms of interference ripples). In addition, Leckie andKrystinik (1989) include directional relationships between structures in the HCS beds with regional shoreline andpaleoslope (Fig. 6-17). Specifically, they showed that directional structures such as sole marks and parting lineationindicate paleoflows directed offshore, orthogonal to regional shoreline-trend indicators. Similarly, capping rippleshave crests aligned approximately parallel to regional shoreline and the internal cross-laminae, when present,indicate migration offshore. Such relationships had been suggested earlier on the basis of local studies (e.g.,Hamblin and Walker, 1979; Brenchley 1985, Rosenthal and Walker, 1987) but data provided by Leckie and Krystinik(1989) suggested that the directional associations may be the norm for discrete HCS sandstones.

Amalgamated HCS Sandstones.

This association of HCS is characterized by thick (up to several tens of metres) sandstones and differs fromthe other association by the lack of mudstones (except as local lenses) and the absence of a preferred sequenceof structures. Amalgamated HCS commonly occur above the discrete HCS beds in regressive shorelinesuccessions (e.g., Hamblin and Walker, 1979; Leckie and Walker, 1982) and is representative of sedimentation inthe lower shoreface. First-order surfaces may be recognized within amalgamated sandstone beds by the presenceof a lag or where they overlie intensely bioturbated horizons, discontinuous mudstone beds or concentrations ofmica and fine plant debris.

Swaley cross-stratification, as originally defined by Leckie and Walker (1982), is not the amalgamated HCSas described here, although there is a growing tendency amongst some authors to state this. For example, Dottand Bourgeois (1983), McCrory and Walker (1986) and Plint and Walker (1987) suggest that the swaley cross-stratification is typical of amalgamated sandstone beds. Duke (1985, p. 171), however, specifically stated thatswaley cross-stratified sandstones do not show evidence of amalgamation. In a vertical, progradational successiondiscrete HCS is overlain by amalgamated HCS which, in turn, is overlain by SCS.

The HCS Debate

When Harms et al. (1975) first introduced HCS they suggested that the structure was the product of theinteraction of waves with a sandy substrate during powerful storms (i.e., the waves that produced HCS wereparticularly large and powerful). Emplacment of sand by storms is widely accepted because discrete HCS bed,encased in shale, must represent extreme sediment transport events that introduced sand into a marine setting thatreceived only deposition of mud during periods of “normal sedimentation”; storms are known to provide such amechanism of sand emplacment. The specific depositional setting of the HCS storm beds is thought to be at depthsabove 200 m (the approximate maximum depth of the continental shelf) but blow the depth where normal, fairweatherwaves will affect a sediment substrate (i.e., below effective fairweather wave base). This setting is suggested bythe normal mud deposition and by the trace fossils and body fossils associated with HCS storm beds. In supportof a storm origin are estimates of the recurrence intervals of emplacement of the sands forming discrete storm beds.

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Table 6-2 shows that independent estimates suggest that HCS storm beds are emplaced only every few thousandyears. Hence the intensity of storms that transported the sand out onto the shelf and formed the sequence ofstructures within the deposits were likely of an intensity not yet recorded. However, as more became known aboutHCS the role of waves in its formation became less certain. Particularly, the presence of unidirectional paleocurrentindicators (such as sole marks on the bases of HCS storm beds), along with the recognition of anisotropic HCS,suggested that unidirectional currents may play a role in forming this structure. The importance of unidirectionalcurrents in the emplacement of HCS storm beds in settings that normally received mud is obvious: the sand istransported offshore, from beach and near-shore environments where it dominates during fairweather conditions,and such transport requires a directed current (i.e., a unidirectional current). Furthermore, such unidirectionalcurrents are well known to modern oceanographers who have measured their intensity during historically moderateto large storms.

In light of the recognition of the importance of unidirectional currents in emplacing HCS storm beds severalnew mechanisms of HCS formation were proposed, stressing the importance of unidirectional currents. Swift etal. (1983) reported HCS on modern shelves in 15 to 40 m water depth. Their examples of HCS collected from wave-modified dunes that formed under combined flows, with a powerful unidirectional component, generated by storms.Box cores taken from the bedforms displayed wedge-shaped sets of heavy mineral-rich laminae in fine to very fine-grained sands. Swift et al. (1983) explained the wedge-shaped sets as curved lamina intersections and likened thestructure to HCS. Hence, the HCS was attributed to combined flows, and specifically to wave modification of whatwould have otherwise been dunes under unidirectional flows. Elsewhere, Greenwood and Sherman (1986)suggested that, in a lacustrine setting, a unidirectional flow (in their case, a longshore current) was crucial to theformation of HCS. They argued that without the combined-flow component, purely oscillatory waves would onlyproduce flat-bed conditions. Similarly, Allen (1985) argued convincingly on theoretical grounds that oscillatorycurrents alone could not form hummocky bedforms of metre-scale wavelength, the development of which requireda unidirectional current.

The above debate progressed on the basis of inference from the ancient record and from observations in

Figure 6-17. Ideal sequence of structures found in HCS storm beds. See text for discussion. From Leckie and Krystinik (1989).

121

the modern , shallow-marine environments where no unequivocal HCS could be identified. Over 1990-91 the resultsof two different approaches to the study of HCS were reported that added more factual knowledge of HCS thatpointed to a return to the original ideas on HCS formation under waves. The new data included experimental studiesand studies of grain fabric in HCS sandstones.

Experimental Evidence

Southard et al. (1990) conducted experiments in a wave duct that produced three-dimensional bedforms(large, 3-D vortex ripples) that behaved in such a way under the purely oscillatory currents that they would producea form of stratification that very closely resembled HCS observed in the ancient record (Fig. 6-18). This experimentalevidence showed that purely oscillatory flows could form HCS but did not rule out the possibility that combinedflows could also produce this structure. However, the combined-flow experiments of Arnott and Southard (1990)appeared to eliminate combined flows as having an important role in forming HCS. In their experiments, symmetricalbedforms could only be produced under purely oscillatory flows or combined flows with a negligible unidirectionalcomponent (Fig. 6-13). As soon as the velocity of the unidirectional component exceeded a few percent of themaximum orbital velocity the bedforms became asymmetrical and the stratification that they produced would appearas anisotropic HCS. These experiments suggested that combined flows with strong unidirectional currents couldnot form HCS, in contrast to the result of Allen’s theoretical analysis and the suggestions of oceanographers.

Evidence based on grain fabric

A detailed study of grain fabric (Fig. 6-19) was reported by Cheel (1991) based on samples oriented withrespect to sole marks on the base of discrete HCS sandstone beds interbedded with shale. This study showedthat particle a-axes, measured in plan view (Fig. 6-19A) varied widely but displayed modes oriented approximatelynormal to sole marks and parallel to the associated ripple crest. This suggests that a-axis alignment of grains inHCS develops by rolling (a-axes transverse to the oscillatory current) and the wide variation in a-axes orientationpoints to deposition under a complex array of surface gravity waves with a mode aligned parallel to the shoreline.In vertical section through HCS sandstones imbrication of grains varied about a mean of 0°, parallel to visiblelamination (Fig. 6-19B). In some cases, this variation in imbrication was markedly cyclic about the 0° mean. Thispattern of imbrication was interpreted in terms of the action of symmetrically oscillating currents during the

Table 6-2. Estimates of recurrence intervals between events (storms?) that emplaced HCS storm beds.

Age Recurrence Interval Author

Kimmeridgian 3,200 - 4,000 years Hamblin and Walker (1979)Walker (1985

Devonian 400 - 2,000 years Goldring and Langenstrassen (1979)

Ordovician 10,000 - 15,000 years Brenchley et al. (1979)

Triassic 2,500 - 5,000 years or Aigner, 19825,000 - 10,000 years

Ordovician 1,200 - 3,100 years Kreisa, 1981

Figure 6-18. HCS simulated from large 3-D vortex ripples. Total length of bed shown in 2.15 m. From Southard et al.(1990).

122

L

θ

0

2

1

DIS

TA

NC

E A

BO

VE

DA

TU

M (

MM

)

-90 90ANGLE ( )

0

= Lθ

FLOW

DIS

TA

NC

E A

BO

VE

DA

TU

M (

MM

)

B

C

341

˚

ANGLE ( )-90 900

2

1

0

˚

566

A REGIONALSHORELINE

C

A

B

PROD MARKS

294

D

Figure 6-19. Grain fabric in a hummocky cross-stratified sandstone. A. Apparent grain long axes as seen in plan view on surfacescut parallel to bedding. B and C. Apparent grain long axes seen in vertical section on surfaces cut perpendicular to beddingand parallel from inferred flow direction. Note that the relative positions of A and B are shown in the block diagram.

formation of isotropic HCS. In contrast, in the basal parallel-laminated interval of an HCS bed, the mean particleimbrication was approximately 13° into the flow direction (based on the sole marks) and varied quasi-cyclically aboutthat mean (Fig. 6-19C). The interpretation of this pattern of variation in fabric suggested that an offshore-directedunidirectional current was active during deposition of the horizontally-laminated portion of HCS storm beds(producing the onshore imbrication). However, when the HCS formed this unidirectional component had eitherstopped or had become too weak to influence grain fabric. Thus, in the HCS storm beds it appeared that HCS formsin response to oscillatory flows, the same conclusion that arose from the experimental evidence.

Origin of HCS?

Given the above interpretation of HCS storm beds we must explain the different forms of HCS that have beenobserved (Fig. 6-20). The HCS storm beds displaying isotropic HCS form as sediment is delivered onto the shelfby offshore-directed combined flows generated by storms; combined flows with powerful oscillatory componentsdue to large gravity waves and offshore (or offshore-oblique) unidirectional components. Such currents are well-known to modern oceanographers, although their directional relationship to the shoreline is more complex thandescribed here (see Duke 1990 for a full discussion). While the unidirectional component of the current is activenot only is the sediment transported offshore but erosion of the substrate occurs, forming the basal solemarksoriented onshore-offshore and any directional solemarks (such as flutes and prod marks) are directed offshore.

123

With the onset of sand deposition conditions are initially on a flat bed (which has a wide stability field undercombined flows; see Fig. 6-14) but this bed state is replaced by large, 3-D vortex ripples as the unidirectional currentwanes and only a powerful oscillatory current continues. Deposition under this oscillatory flow forms isotropicHCS. As the oscillatory flow wanes small, 2-D vortex ripples are formed, capping the sandstone. Under fairweatherconditions, following the storm, muds are deposited, encasing the storm-deposited sandstone bed.

The above scenario accounts for much HCS associated with HCS storm beds but it does not explain thevariety of forms of HCS shown in figure 6-16. Certainly, the isotropic forms are likely formed by oscillatory currents.However, the interaction of these currents with a sandy substrate must differ in detail to produce scour and drapeversus vertical accetionary HCS. In the case of the scour and drape form of HCS the fact that internal laminae drapeand diminish the hummocky relief suggests that a hummocky bedform is not stable under conditions that form thestructure but are only stable during periods of intense flow that causes erosion of the hummocky surface. Sucherosion during a storm might occur due to the temporary development of constructive waves on the water surface.Swift et al. (1983) described the generation of thick clouds of sediment that rose off the bottom during a storm withthe passage of groups of exceptionally high waves formed by constructive interference. The formation of theseclouds must involve the local addition of sediment into suspension by erosion that might form hummocky second-order surfaces. Between periods of wave construction the “normal storm waves” would act as sediment continuedto deposit, causing the temporal variation in flow strength that results in the formation of internal laminae. In thisscenario the hummocky surface is not due to the generation of a stable bedform but is inherited from the form ofan erosional surface, a surface that might mimic the morphology of the stable bedform under the same conditionsbut with net deposition on the bed. Subsequent deposition of sediment onto the hummocky erosional surface actsto bring the topography into equilibrium with the normal storm-wave conditions (and this equilibrium bed appearsto be more-or-less flat). Hence, the currents that produce the hummocky form are associated with erosion, duringthe most intense conditions on the bed.

SUBSTRATE

BED RESPONSE

BED STATE

CURRENT

EROSION

WANING OSCILLATORY FLOWCOMBINED FLOW

FLAT BED

HUMMOCKY BED 2-D WAVERIPPLES

IRREGULAR SCOUR ANISOTROPIC ISOTROPIC

DEPOSITION

NO BED-FORMS

OF

FS

HO

RE

ON

SH

OR

E

0

TIME

SHORE-NORMALFLOW SPEEDNEAR THE BED

INSTANTANEOUSTIME-AVERAGED

cm -

dm

Figure 6-20. Schematic illustration of the temporal variation in currents involved in the formation of HCS storm beds. Seetext for discussion. After Cheel (1991), Duke, Arnott and Cheel (1991) and Cheel and Leckie (1992).

124

Vertical accretion forms (Fig. 6-16), characterized by thickest laminae within hummocks, appears to involvethe growth of a depositional hummocky topography. This type of HCS might represent the product of currents,possibly produced by sustained constructional waves that are less transient than the case of scour and drape.Under conditions of rapid deposition, such sustained currents, capable of building a stationary hummocky bedformthat is in equilibrium with the prolonged, oscillatory currents generated by the constructional waves, would formvertical accretionary sets. Hummocky bedforms that would produce such stratification were produced under purelyoscillatory flows and strongly oscillatory combined flows by Arnott and Southard (1990). In this case, a truebedform is constructed under conditions of net aggradation.

The low angle migratory forms (Fig. 6-16) may represent similar bedforms that form accretionary HCS butwhich migrate over the substrate, preferentially preserving internal laminae that dip in the direction of migration(Nöttvedt and Kreisa, 1982). Arnott and Southard (1990) produced stationary and migrating bedforms experimen-tally in a wave duct. The stationary forms were stable under oscillatory flows and strongly oscillatory-dominantcombined flows. However, migrating forms developed when the velocity of the unidirectional component exceededa few percent of the orbital velocity of the oscillatory component. Hence, anisotropic HCS may be interpreted toindicate the influence of a unidirectional component of a combined flow under conditions that would otherwise formisotropic HCS.

Conclusion

The debate over the origin of HCS continues and has become more perplexing over the same period thatnew evidence for a wave-induced origin. Work by Rust and Gibling (1989) and Prave and Duke (1990) providedexamples of cross-stratification that appears like HCS but was produced by three-dimensional in-phase waves. Nowwe have an HCS-look-alike. The paleoenvironmental interpretation of each structure is radically different; HCSformed by in-phase waves is definitely indicative of unidirectional flows whereas the shallow marine forms of HCSappear to have been largely influenced by waves. The most promising approach to distinguishing the two formsis through detailed studies of grain fabric and this research is on-going and the general debate continues.

125

APPENDIX 1: Standard procedure for sieve analysis of sand

One of the fundamental problems encountered by a sedimentologist is in the description of the size ofparticles which make up sediments and sedimentary rocks. A good description of particle size is important fora number of reasons: (1) description provides a basis for comparison with other deposits, (2) size and/orsorting (in part) will control the porosity and permeability of a rock, and (3) size, sorting, etc., reflect theprocesses that were active in the depositional environment.

This appendix illustrates the standard procedure for determining the size distribution of particles in anunconsolidated sediment by passing them through stacks of nested sieves with square openings of knowndiameter.

Procedures

Note that samples that include a considerable amount of silt and clay size sediment (i.e., <4 φ) shouldnormally be wet-sieved, that is, washed through a 4φ screen to remove all particles finer than 4 φ. The sizedistribution of the material that passes through that screen can be determined by the pipette method or byanalyzing the mixture with a sedigraph.

Step 1. Using a sample splitter obtain approximately 30 to 50 g of sample for sieving. Weigh a beaker on thescale and record that weight on the Sieve Data Sheet. Pour the portion of the sample to be sieved into thebeaker and determine the combined weight of the sediment plus beaker. Record this weight on the Sieve DataSheet. Determine the absolute weight of sediment to be sieved and record this weight on the Data Sheet. Savethe remaining portion of the sample in a labeled sample bag.

Step 2. Select sieves to be used (in this case -1.0φ to 4φ at 0.5φ intervals) and nest them in their proper order,coarsest at the top, pan at the bottom. Hand sieve, for three minutes, a small stack consisting of the coarsestsieves, down to 0φ. The remaining sand should be poured onto the top sieve in the remaining stack for sievingon the sieve shaker.

Step 3. Place the stack of sieves onto the sieve shaker and place the three-armed bracket on the lid of the stack.Lower the straight-arm bracket and make sure that the end pins are penetrating appropriate holes on the frameso that the stack is secure. Set the timer to 15 minutes and turn the power switch to the “on” position.

Step 4. When the shaker has turned itself off empty the contents of each sieve, one at a time, onto a piece ofglazed white paper. Invert the sieve and strike it sharply on the paper to dislodge any sand grains that are lodgedin the mesh. With the soft brush, gently, wipe the bottom of the screen so that all of the relatively loose grainsfall back through the mesh and onto the paper. Never wipe the top of the screen with the brush as this may forcegrains that are too large to pass through the screens, damaging the mesh. Carefully pour all of the sand fromthat size fraction into the appropriate plastic beaker and place the beaker into the beaker holder. If the finestscreen in your stack is not 4φ then pour the contents of the pan into the top screen on a second stack. If thefinest screen is 4φ then pour the contents of the pan into the appropriate plastic beaker.

Step 5. Weigh the sediment in each size fraction (recording the weight on the Sieve Data Sheet) and return thesieved fraction to the original sample bag. (Note that you would normally store the sand from that sievefraction in a labeled plastic bag; e.g., Sample 1, .5φ to 1.0φ .) Repeat the procedure for each of the sievefractions and the contents of the pan.

Step 6. Determine the weight of sediment in each size fraction and the proportion of sample that was lostduring sieving (from the total summed on the sieve data sheet and the total initially poured into the sieves).Calculate the percentage (by weight) for each fraction and the cumulative weight (%) for the sample. Save thisdata for Laboratory two. Hand in your final sieve data sheet with Laboratory 2.

Note: The sample supplied should be almost aggregate-free (i.e., it contains little or no cemented clumps of

126

sand). Normally a “step 7” would involve using a hand lens to briefly examine each of the size fractions todetermine if there are significant quantities of aggregates. The approximate proportion aggregates, as a per-centage of the total number of grains in the fraction would be estimated and entered into an appropriate columnon the data sheet. This information would be used to “correct” the percentages calculated for each sievefraction.

127

APPENDIX 2. FORMULAE FOR TESTS AND EXAMS

Note that a summary of symbols is given at the end of Appendix 2.

1. Flow Reynolds’ Number (R):

2. Froude Number (F):

3. Boundary shear stress (τo): τ

o = ρgDsinθ

4. Shear velocity (U*):

5. Law of the wall for rough boundaries:

Note that for closely packed spheres of uniform size yo=d/30. Mean velocity of a turbulent flow

occurs at a height of 0.4D above the bed.

6. Boundary Reynolds’ Number (R*):

R* is used to classify boundaries of turbulent flows based on the relationship between the rough-

ness of the boundary (grain size of the sediment comprising the bed material) and the thickness ofthe viscous sublayer (δ), where:

When R*< 5 the boundary is smooth (δ > d), when 5 < R

*T < 70 the boundary is transitional, and

when R* > 70 the boundary is rough (δ < d).

128

7. Shields’ criterion for sediment movement:

Calculate:

and find the corresponding value of β on Shields’ diagram (Fig. 4-17, a copy will be supplied fortests), where:

and solve for

8. Middleton’s criterion for suspension: A particle will go into suspension when U* > ω. The

settling velocity of quartz grains with mean size equal to or finer than 0.1 mm may be calculatedusing Stoke’s Law of settling:

For quartz grains coarser than 0.1 mm Fig. 4-20 may be used.

Summary of symbols

D flow depth;d grain size;F flow Froude Number;g acceleration due to gravity;R flow Reynolds’ NumberR

*boundary Reynolds’ Number

θ slope of water surface and boundary for uniform, steady flow;U flow velocity (usually mean flow velocity in the downstream direction);U

*shear velocity;

u mean flow velocity at some height (y) above a boundary;y some height above a boundary;y

othe height of the roughness elements on a boundary;

β Shields’parameter;δ thickness of the viscous sublayer;µ dynamic viscosity;ν kinematic viscosity (ν=µ/ρ);ρ fluid density; Note that in one instance, on page 23, p is a logarithmic transformation of

roundness.ρ

sdensity of a sediment particle;

τo

boundary shear stress;κ von Karman’s constant (normally assume κ=0.4);ω settling velocity.

129

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