Flume Studies of Gravel Bed Surface Response to Flowing
FLUME STUDIES OF GRAVEL BED SURFACE RESPONSE TO FLOWING
WATER
By
JOHN FREDRIC WOLCOTT
B.Sc, B.F.A., University of Washington, 1982
M.Sc, University of British Columbia, 1984
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
in
THE FACULTY OF GRADUATE STUDIES
Department of Geography
We accept this thesis as conforming to
the required standard
THE UNIVERSITY OF BRITISH COLUMBIA
January, 1990
® John Fredric Wolcott, 1990
08 In presenting this thesis in partial fulfilment of the requirements for an advanced
degree at the University of British Columbia, I agree that the Library shall make it
freely available for reference and study. I further agree that permission for extensive
copying of this thesis for scholarly purposes may be granted by the head of my
department or by his or her representatives. It is understood that copying or
publication of this thesis for financial gain shall not be allowed without my written
permission.
Department of Geography
The University of British Columbia Vancouver, Canada
Date Ffthrnary 7.
DE-6 (2/88) ii
ABSTRACT
Almost all sediment transport equations incorporate the Shields parameter, which is a ratio of the total boundary shear stress as a driving force and the particle weight as a resisting force. Shields (1936) equated particle resistance to entrainment with particle weight, which is proportional to particle diameter, or bed texture. The present work analyses the particle resistance term in the Shields parameter.
As the bed material adjusts to a given flow condition, bed stability increases. The arrangement of particles into more stable configurations is here termed geometric structure, and includes the formation of pebble clusters, and imbrication. After an initial surface coarsening, here termed textural structure, particle resistance to movement is a function primarily of geometric structure.
The Shields number for entrainment is thus a measure of particle resistance due to both types of bed structure rather than the conventional notion of particle resistance due to particle weight.
The response of a mobile bed surface composed of < 8 mm diameter gravels to flowing water was explored in a 6 meter by 0.5 meter flume using four different slopes and various water depths. Corrected bed shear stresses varied between 0.05 and 2.79 Pa. Step increases in discharge with a constant slope caused the bed surface to develop a structure which was more stable at the end of a run than at the beginning. Under these conditions, the Shields number for incipient motion was found to vary between 0.001 and 0.066. This variability can be explained by the degree of geometric structure present. Previous studies, including Shields' work (1936), have implicitly included the effects of geometric structure on incipient motion. iii
Surface coarsening develops with very low flows, but subsequent coarsening in higher flows is minor, with less than 5% increase in median diameter following a 50% increase in bed shear stress. Calculations of Manning's n based on depth, slope, and velocity measurements show an increase in flow resistance as structure develops. The development of a coarse surface layer appears to be limited by flow characteristics near the bed which are in turn modified by the development of structure. Measurements of the area occupied by the largest stones show that they do not cover more than 14% of the surface during maximum coarsening. Froude scaling of the flume data indicates that the time necessary for development of maximum strength is on the order of a month for natural rivers under steady flow conditions. This suggests that gravel river beds are rarely in equilibrium with natural flow conditions. iv
TABLE OF CONTENTS
Abstract ii
Table of Contents iv
List of Tables vi
List of Figures vii
Acknowledgements ix
CHAPTER ONE: INTRODUCTION 1
Prologue 1
Field Studies , 2
Flume Experiments 4
Shields' Work 6
Other Flume Studies 9
Objectives of This Study 14
CHAPTER TWO: METHODOLOGY 17
Justification of Approach 17
Models.... 18
A Note on Units 21
Experimental Apparatus 22
Experimental Procedure 28
Summary of Runs 34
CHAPTER THREE: OBSERVATIONS 41
Series One Runs 41
A Qualitative Model 54
CHAPTER FOUR: BED STABILITY AND INCIPIENT MOTION 57
An Overview 57
The Effect of Bed Structure Development 60
Bed Structure and Incipient Motion 63 V
The Shields Parameter Re-Examined 71
Bed Structure and Manning's n 77
Summary 78
CHAPTER FIVE: LAG TIME 80
CHAPTER SIX: LIMITS TO COARSENING 88
CHAPTER SEVEN: CONCLUSIONS 96
REFERENCES 103
APPENDIX 108 VI
LIST OF TABLES
2.1 Velocity Profiles Across the Test Section 25
2.2 Hydraulic Parameters of the Experiments 35
4.1 Measured and Calculated Sediment Transport
Values for Later Runs 64
4.2 Shields Parameter Based on Various D50 68
4.3 The D50 and the Amounts of the +4 mm and +5.66
mm Material Captured in Series 3 Runs 69
5.1 Variations in Transport Rates through Time for
Series 3 Runs 82
5.2 Variations in Transport Rates through Time for
Series 4 Runs 83
5.3 r^ for Linearized Exponential and Power functions of
Sediment Transport through Time 86
6.1 Weights and Percentage of the Total Area of the
+ 5.6 mm Size Class for Several Surface Samples 91
A.l Weights and Areas of Various Size Classes 110
A.2 Ratio of A and B Axes and Areas Occupied by
weight Ill vii
LIST OF FIGURES
1.1 Shields Curve, after Shields (1936) 8
1.2 Illustration of a particle cluster and imbrication 12
2.1 Sketch of flume 23
2.2 Cumulative grain size distribution curves for
sediment used in the experimental runs 29
3.1 Loose, well mixed material (top) and armored, and
settled material after Run 1-3 (bottom) 42
3.2 Surface web of larger stones as seen on bed (top),
and with smaller stones removed for clarity (bottom) 44
3.3 Photograph of incipient web development after Run
4-3 45
3.4 Comparison of bulk sample with surface samples after
0.02 m, 0.03 m, 0.05 m deep flows (Runs 1-1 to
1-4), and thalweg from degrading run (Run 1-5) 49
3.5 Textural differences between channel surface
(above) and bar top surface (below). Flow
from left to right 52
3.6 Comparison of bulk sample, sediment trapped
during degrading run, and surface sample from
thalweg and top of bar 53
4.1 Proposed relation between hydraulic variables
and sediment supply (Parker, 1987, pers. com.) 58
4.2 Cumulative grain size distribution of trapped
material during Series 4 runs 62
4.3 Decline in transport rate through time 66
4.4 Proposed model of bed structure development 73 viii
4.5 Shields parameter based on material moved
compared to Shields parameter based on
subsurface D50 74
4.6 Difference in Shields parameters of Figure 4.5
(which is the excess bed strength due to geometric
structure) compared to Shields parameter based
on subsurface D50 75
5.1 Coarsening of the surface median sizes following
disturbance by three major storms, with discharges
of 0.068, 0.039, and 0.056 m3/s 84
6.1 Comparison of typical surface (top, after 0.02 m
flow depth, Run 1-2) and very well sorted surface
(bottom, after Run 1-5) 95 ix
ACKNOWLEDGEMENTS
Since it is probable that this is the most read page of any thesis, I feel compelled to mention two things, before getting on with the traditional thank yous. First, in my two and half decades of doing research in two countries, which included one province and three states, the most enjoyable experience was completing the M.Sc. The most miserable was dealing with the examining committee during and following the PhD defense, and the administration. The bitterness has not yet dissipated. In fact, it is only through incredible encouragement of a small circle of very good friends that this thesis has continued to the final stage. May it rest in peace on some dark and dusty library shelf. Second, Dr. Humphrey of Cal. Tech. has pointed out the following irony. Since an examiner has asked that the centimeter/gram/second system of units be changed to SI units, this thesis has grown in length and therefore stature simply by the addition of zeros. Think about this when you try and figure out what 0.000158 m^/s means (it is 9.5 liters per minute or 2.5 gallons per minute), or recognize that if a 2.3 mm particle rolls 0.005 m, it has gone 2 grain diameters. To the examiner who edited the original acknowledgements page I apologize. I guess the thesis was dropped on the way to the library and the wrong page inserted. The success of this thesis is the result of generous help from many individuals, a few of whom are mentioned here. First and foremost, I would like to thank Dr. M. Church, one of the best supervisors I have ever met. He supported my initial explorations into this subject in spite of no clear objective on my part, other than curiosity. Later, through suggested readings and provocative questions, he guided me to the point where everything fell into place, and even I could see the way home. Dr. O. Slaymaker of UBC and Dr. T. Hicken of SFU provided moral support and helpful suggestions during Dr. Church's sabbatical leave, and I am very grateful for their encouragement during a discouraging period. Dr. P. Ward of Ward and Associates, Vancouver, was extremely helpful with pump and motor calculations and the design of fluid flow systems. In addition to my committee, I would like to thank Dr. N. Humphrey of Cal. Tech., sans Twit, D. McLean of UBC, and Dr. A. Nowell of the University of Washington for many stimulating discussions and words of encouragement about life in general and my thesis in particular - in other words, for being good friends. C. Souch and S. Grimmond helped at critical times, such as during comps. C. Souch also assisted with flume measurements and other aspects of my program as well as coping with department life. Dr. T. Oke of UBC provided both a data logger and an adventure, the former was essential and the latter was one of the highlights of my graduate career. The kids, Tony Cheong and Stevie Rice, did their best to distract me, and life was more fun after their arrival. All shortcomings of this work are probably their fault. The office staff, especially S. Lapsky, and the technical staff, especially R. Schulte and J. Skapski, provided excellent technical help and more, when needed. And thanks to my parents for their unflinching support; they must have thought my student days would never end. 1
CHAPTER ONE
INTRODUCTION
Prologue
The gravel bed of a stream responds to the water flowing over it.
How it changes, how much it can change, and how long it takes to change are questions addressed in this study. As water flows over .the bed, the least stable particles move into more stable positions. The increase in bed stability is the result of individual particle arrangements which lead to the development of what is here called bed surface structure. Intimately related to the bed response is sediment transport; it both controls in part the extent of the response and is itself a result of the response. Consequently, this study also describes the effects of changes in the bed surface on sediment transport. The approach taken here to observe bed surface evolution and sediment transport is controlled experimentation in a laboratory flume.
Progress toward understanding the processes involved in sediment transport and their effects on gravel bedded streams and rivers has been made by both field studies and flume experiments. Some of this work has been motivated by theoretical considerations of forces acting on bed particles, and some has been motivated by careful observations of actual transport events. The interplay between theory and observation and between field and flume is both complex and necessary. It is complex because the relation between variables in the field, in the flume, and among field, flume and theory, are not well understood. It is necessary because the complexity requires multiple approaches, with advances in one area 2
affecting the research directions of the others. A major problem is
recognizing the important variables. An example of the evolution of
perceived necessary variables is presented in the overview in Chapter Four.
Field Studies
Field studies are the ultimate test of the validity and usefulness of
any theory of river behavior. It is in the field that sediment transport
matters, both in terms of human concerns and in affecting the evolution of
Earth's surface. But no two rivers are exactly alike. This problem is not
unique to fluvial geomorphology; no two biological cells are exactly alike
either, yet molecular biology has made substantial advances (see Piatt,
1964, for a critical discussion of this issue). In spite of the large variability
among rivers, certain general characteristics have emerged. One approach
- has been the regime theory of stable channels which relates channel
stability, and hence sediment transport, to general hydraulic variables such
as discharge and channel slope, width, and depth. Ackers (1988) provides a
current summary for sand bed channels. But little work has been done on
gravel rivers, where the shear stresses rarely exceed 2 or 3 times that
necessary to move the median size stones (Parker, 1980).
Although the regime approach may be useful in some instances, it
does not provide insight into the mechanisms involved in actually moving
sediment. Hysteresis effects between sediment transport and discharge
show that the relation is not simply based on shear stress and grain size
(Nanson, 1974; Klingeman and Emmett, 1982, among others). The same 3
discharge on the rising and falling limbs of a hydrograph move's different
amounts of sediment, as does the same discharge in sequential floods.
Andrews (1983) and Church (1985) report work in which bedload was not
found where the hydraulic conditions predicted it would occur, namely in the
thalweg. Church (1985) discusses many of the factors which influence bedload transport rates besides the hydraulic conditions. A consequence of
all of these additional factors is large spatial variability in sediment
movement; thus sample site location becomes critical (Meade, 1985).
Given the complex nature of mechanisms affecting sediment
transport, two approaches to studying the phenomenon seem most
profitable. One approach is to measure sediment transport over time scales which are larger than the oscillations of the relevant variables. For many of the variables, such as hysteresis effects, unsteady and non uniform flow and sediment supply, and sample site locations, detailed studies with observation times of 1 to 10 or more years may be necessary to reveal consistent
average conditions. Careful surveys such as that carried out on the Fraser
River (McLean, 1990) provide estimates of the amount of material exchanged locally within a 40 kilometer reach over 30 years. From these spatial and temporal perspectives the relations between conditions in
various parts of the river can be studied and sediment exchange rates compared to integrated averages of the conventional hydraulic variables.
Insights into sediment sources, sinks, and transport rates should be forthcoming. This elevates field observations from the level of making
isolated, unrelated, qualitative statements about a specific site to the level
of providing new understanding of the behavior of sediment movement in a
natural reach. From this, the appropriate time and space scales for more 4
detailed analyses of the specific mechanisms can be determined for future studies.
A second approach is to investigate the bed surface-water flow interaction. Aspects of the fluid turbulence and aspects of the bed surface may be studied individually, typically in a laboratory flume, or the combined effects may be studied, either in the field or in a flume. The relation between turbulent fluctuations of marine flows and incipient motion of bed sediments has been studied by Heathershaw (see Thome, Williams and
Heathershaw, 1989) looking at gravels, Soulsby (Soulsby, et al., 1987) looking at sands, and West and Oduyemi (1989) looking at silts. These studies, however, ignore the effect of the bed surface on sediment transport.
Current second and third generation tracer studies, using magnets and radio transmitters installed in individual stones in river beds (Hassan, et al.,
1984; Emmett, et al., 1989) not only monitor the surface and subsurface behavior, but also they support and extend earlier flume studies by Einstein
(1937), Gessler (1970), and Stelczer (1981).
Flume Experiments
Laboratory flumes eliminate many of the problems associated with field studies. Most variables considered to be important can be controlled to a degree that is impossible in the field. Experimental conditions can be duplicated to confirm results, and interdependent variables, such as water depth and slope, can be varied together to produce the same end result, such as total boundary shear stress, to determine the generality of a result. 5
A particular strength of flume experiments is that they permit the study of events which at full scale would pose serious threat to life and equipment. Parker (1980), for example, was able to observe bed surface behavior during conditions equivalent to the highest flows of a river. He observed that the coarse layer on a gravel bed surface remains in place during high flow, a fact which was unexpected at the time. Another advantage of flumes is that if complete scaling is preserved (see Chapter
Six), time can be compressed in the model, which permits observation of events which take months to years to unfold in nature. Schumm, et al.
(1987) have explored this aspect, but only in a very qualitative sense because the experiments they describe did not preserve dynamic or kinematic scaling.
There are, however, several major limitations to flume studies. Due to logistical problems of space, hardware, and funding, most flumes are two- dimensional models. Rivers are definitely three-dimensional, and it appears that this third dimension accounts for much of the discrepancy between
model and prototype. For example, the phenomena reported by Andrews
(1983) and Church (1985) noted above cannot be duplicated in a two- dimensional model (but see Chapter Three for some qualitative observations of three-dimensional flows). Models which attempt to be three-dimensional cannot maintain complete scaling because of size restrictions (see Chapter
Six). At best, they provide qualitative insights; at worst, they give erroneous results. Even using two-dimensional models investigators have difficulty maintaining scaling relations. Flumes, therefore, are very useful tools within their limits, but they cannot give a complete picture. 6
The present work describes a series of flume experiments designed to explore the combined effect of the flow and the bed surface on sediment transport. A flume was chosen for several reasons. The flow can be stopped at any time to permit detailed observations and sampling of the bed. All the sediment leaving the test reach can be collected. It permits greater control over hydraulic variables such as discharge and slope than is possible in the field. Steady, uniform flow conditions can be maintained indefinitely.
Although this study has utilized the experience gathered from previous studies in both its design and execution, the conclusions are unique.
A fuller appreciation of these aspects may be gained by a brief review of some previous studies.
Shields' Work
The forces acting to move a bed particle are due to the pressure differences created by the flow around it. Local accelerations and decelerations caused by constrictions and expansions of the flow, by changes in boundary roughness, by changes in flow directions and by turbulence affect the net force. Motion is resisted by the weight of the stone. Other factors which affect resistance to movement are the size, shape and density of the particle as well as the orientation, degree of exposure, and supporting structures of the particle. Trying to calculate the force balance between driving and resisting forces on every particle on the bed of a three- 7
dimensional study section is futile, given the immense variability in natural bed material and flow conditions over on the bed.
Shields made a fundamental contribution to the study of particle entrainment when he showed that if an averaged driving force and an averaged resisting force are used, the ratio of these two forces remains constant above a certain minimum condition (Shields, 1936). He summarized the driving forces by the total boundary shear stress, which is based on the product of the water depth and water surface slope. He represented the resisting forces by the particle weight of the bed material,
D. This ratio of driving and resisting forces, which he called r* , is known now as the Shields parameter. It figures prominently in almost all bedload equations used today (exceptions include Einstein, 1937; Parker, et al.,
1982; and Lavelle and Mofjeld, 1987).
)D) r* = r0/((7s-7 (1.1)
where Tq is the shear stress, T , at incipient motion,
r = (jRhS) (1.2)
and y is the density of water, RD is the hydraulic radius of the bed corrected for smooth sidewalls (see pages 38- 39 for methods to calculate
RD), S is the water surface slope, 7S is the density of the sediment, and D is the diameter of the bed material.
Figure 1.1, which is taken from a translation of Shields (Shields,
1936), shows the original form of the Shields curve with T* plotted against
Re*,
Re* = V*DA' (1.3) 8
«fe 11 I I I III!
It ~TT^—
• »*»/>c r, • «*' GraaLu.ai Shortening anot
• • - (Vomer) PdSt.pQ.nino of Bo.cC Formation *+ • - • f USW) ft: t 1 *K~\~~ IT <7af Bed ffarcLtioas +Sca/es 4* (ikortBars) i -fa P P
Figure 1.1 Shields curve, after Shields (1936). 9
and
1/2 V* = (gRbS) (1.4)
where g is the acceleration due to gravity and v is the kinematic viscosity
of water. Shields determined that r* had a value of 0.06 for flows with
Re* > 1000. Subsequent investigators have suggested values of T*
varying from 0.0075 to 0.075 (Paintal, 1971). Notable estimates of the
constant limit value include 0.03 (Neill and Yalin, 1969), and 0.047 (Meyer-
.' Peter and Muller, 1948). Yalin (1972) suggested 0.05 above Re* = 70.
The present study may be considered an expansion of Shields' work
because the ratio r* is explored in more detail. The results of this study
show that another factor in addition to particle weight determines particle
resistance to movement, and the range in r* values mentioned above may
be explained by this additional factor.
Other Flume Studies
Shields (1936) studied the concept of a threshold of motion, below
which sediment entrainment and transport did not occur. Both Paintal's
work (1971) and this study show that such a threshold defies definition, yet
the concept may still be useful for many situations. In one sense, this study
is an extension of Paintal's, because it provides the temporal detail missing
in his data, as well as the size distributions of the transported material. For
Paintal's purpose, which was to clarify the concept of a threshold of motion,
these were not relevant. Similarly, it was not the purpose of the present 10
study to investigate the concept of a threshold of motion; Paintal has
already done so.
Harrison's classic study (1950) also is similar to this study, and
several of his results are mentioned in the following chapters. His objectives were to determine the depth of scour in a degrading reach, to predict the grain sizes that limit degradation using Einstein's bedload formula, and to
study the segregation of large particles in the bed. Although his experimental arrangement was different in some respects, for example he did not maintain uniform flow, his measurements of the change in transport through time and the time to achieve negligible transport agree very closely
with those of this study. Furthermore, he addressed an issue of particle
segregation which is ignored in this study, namely the effect of mixtures
with varying proportions of grains too large to be moved by the flow.
This difference in studying the large grains is noteworthy because it
illustrates the different perspectives in the two studies. The present work
assumes that all particles in a reach are mobile, following the concept of a
graded river (Mackin, 1948). Although this is not always the case, it is believed that if non-mobile particles begin to constitute a significant portion
of the surface, the bed slope will increase until they become mobile.
Harrison, on the other hand, believes that there must be a certain
percentage of material too large to be moved. These stones will form a non-
moving layer, which he calls pavement, that protects the underlying smaller
material. Every non-degrading reach must therefore have some particles
which are too large to be moved by the flow. 11
Several of Harrison's qualitative observations agree with those in
Chapter Three. For example, he noted that partial pavement, here considered to be pebble clusters and illustrated in Figure 1.2, is effective in limiting degradation and that complete pavement is not necessary. More importantly, he observed that imbrication, shown in Figure 1.2, is effective in stabilizing the^bed. In cases where he removed imbricated or "shingled" stones for measurement and then replaced them on the bed, degradation immediately accelerated until imbrication was reestablished. In fact, in 2 of his 14 conclusions he discussed particle alignment. The importance of bed structure was implicit in his findings, but he failed to make the point. It is a major theme of this study.
In one sense, Harrison did not miss the importance of particle alignment, for he both photographed and illustrated the phenomenon. The nature of his experiments, however, did not allow him to explore the effects of bed structure in general or the effect of imbrication in particular. First, because his sediment mixtures contained non-moving stones, the effect of structure was confounded by the effect of grain size. Second, the mathematical model he was using did not have a term for bed structure.
This is still the situation today. No bedload transport formula has provision for bed structure.
Parker's original flume study (1980) was a motivating factor during the inception of this thesis. His objective, however, was to clarify the behavior of the surface layer during flood events. Since this layer was identified by grain size, changes in grain size of the surface layer were his primary concern, especially during equilibrium, or sediment feed, conditions. 12
Particle Cluster
Stoss side Obstacle Wake zone accumulation stone
Imbrication
Figure 1.2 Illustration of a particle cluster (top) and imbrication (bottom). 13
It was not his purpose to determine the best index of bed stability. This thesis will demonstrate the relative insensitivity of the grain size of the bed surface to various flow conditions, once a coarse layer has been established.
Parker did do two runs in which sediment feed was halted part way through the run to observe the effect on the bed surface, and he noted the same results as others who have done similar studies, namely the bed surface became coarser. However, he did not pursue aspects of the bed geometry.
In a recent flume study, Wilcock and Southard (1989) investigated the concept of "equal mobility," defined as the occurrence of identical size distributions in transport and subsurface material, in a recirculating flume.
They described the fractional transport rates, bed surface textures, and bed form development during start-up as well as equilibrium conditions for seven different flow strengths. Measurements made at the start and end of each run included surface texture and fractional transport analysis. Their results from the start-up periods show the development of bed structure, but they did not comment upon this, except to note the development of a partial pavement or armor, because it was not part of the intent of their study.
Nor did they measure changes in bed texture or fractional transport rates during the runs. Several of their figures, however, could be substituted for those in Chapter Three, particularly Figures 3.3 and 3.4. This similarity, despite the distinctive behaviors of bed adjustment in sediment recirculating and non recirculating flumes (see Wilcock and Southard, 1989, for an excellent discussion), supports the generality of these results. 14
Objectives of This Study
One objective of this study has been to determine how particles on the surface of a stream bed respond to a shearing flow. A second objective has been to explore the limits to coarsening of a mixed size mobile boundary. A third objective of this study has been to quantify the lag time between a given flow condition and the bed response.
Particles on the bed surface respond to a shearing flow of water by local segregation and by particle alignment. Local segregation occurs both vertically and horizontally. Vertical sorting has been described by others as armor, pavement, or simply surface coarsening (Harrison, 1950; Bray and
Church, 1980; Parker, 1980; Sutherland, 1987). Local horizontal segregation has become known as pebble clusters (Dal Cin, 1968; Laronne and Carson, 1976, Brayshaw, 1985), and particle alignment as imbrication
(see Johnston, 1922). These characteristics of the bed surface response to flowing water are henceforth called bed surface structure, or simply structure.
Structure may be divided into two aspects: textural and geometric.
The textural aspect is the strength of the bed which comes from the weight of material on the surface, which is indexed by size. The conventional
Shields approach to entrainment considers only this part of structure. The geometric aspect of structure, which includes particle alignment and particle arrangement, is not indexed easily. In fact, this study was not able to quantify adequately the extent of geometric structure on a bed surface.
Although it has been observed previously, this aspect of structure has not 15
been studied systematically. Some qualitative work, however, has been done by sedimentologists seeking a basis for paleoflow reconstruction (for example, Church, 1978). A major result of this study has been to quantify the effects of geometric structure.
In the experiments reported here, no sediment coarser than 0.200 mm was recirculated, and no additional sediment was supplied to the flume.
These conditions caused rapid development of bed structure and a minimum of bed forms. With no sediment supplied except that which was eroded from the upstream section, the material trapped came from the bed only.
Analysis of the trap material and the surface texture provide complementary views of the evolution of bed structure. Both show that the ability of the bed material to resist transport is much more a function of the architecture of the surface material than it is of the grain size. This result is the most important contribution of the research. A detailed description of the equipment and procedures used is given in Chapter Two. Observations made during the development of bed structure are recorded in Chapter
Three.
Consequences of the development of structure include an increase in the shear stress necessary to entrain particles. Quantitative analyses of the effects of bed structure and a graphical representation of its development are presented in Chapter Four.
Although imbrication, clusters, and surface coarsening have been described in detail by others, little work has been done on the time necessary for the development of these features. A field study by Gomez 16
(1983) remains one of the few examples. By viewing the flume as a scale model of a generic river, the temporal development of bed structure can be monitored and then related to field conditions. The time necessary for the development of maximum structure and the sensitivity of the results to the displacements of a few stones are discussed in Chapter Five.
Not only does the flow create structure on the bed, but also it may limit the growth of particle clusters and surface coarsening. Morris (1955) describes "skimming flow" which occurs when surface roughness elements are sufficiently close together to cause the flow to skim over the tops of the elements instead of flowing around and between them. Additional laboratory work by Nowell and Church (1979) and Brayshaw, et al. (1983) using fixed geometric objects, such as cubes fastened to the bottom of a flume, support Morris's work. At sufficiently high flows, however, the surface structure is destroyed. Chapter Six contains the results from some additional explorations into this issue. 17
CHAPTER TWO
METHODOLOGY
Justification of Approach
Any study of the evolution of the surface of the earth is constrained by two major factors. First, the time scales at which most processes occur are on the order of hundreds to millions of years, with a few between tens and hundreds of years. Second, high energy events, which may be rare but very effective, pose serious threat to life and equipment. Laboratory models of surface processes allow an investigator to circumvent both obstacles.
To study the development of bed structure a laboratory flume was used to provide steady uniform flow for time periods of days to weeks. One advantage of flume studies is that they allow repetition of a particular set of flow conditions.
Another advantage is that one variable can be changed while others remain the same. The strategy adopted in the present study was to change one variable, water depth, between runs, and to change one variable, slope, between series.
Total boundary shear stress, proportional to the product of the water depth and slope, could be held constant although the water depth and slope could vary to confirm the generality of the results. This permitted exploration of a greater range of conditions, with some overlap of bed shear stresses between runs of different series, at the expense of replication within a series. Series 1 and Series
4 had nearly identical flow conditions, but different aspects of the interaction between flow and sediment were measured. The results of all runs are consistent, and they are similar to those reported in other studies. What this provides is a clarification of an incompletely understood phenomenon. It is gained at the 18 expense of not learning the possible variability for individual runs. However, learning and interpreting the overall pattern of results is the logically prior task.
Models
Representative models are difficult to construct because not all properties of the prototype can be scaled down; viscosity, for example, is the same, usually, in both. There are several approaches to this dilemma, ranging from simply ignoring the problem to recognizing specific constraints on both the size of the model and the generality of the results. Fortunately, one aspect of fluid flow justifies relaxing some of the mutually exclusive requirements under certain conditions. If the model is operated with the appropriate constraints, the model results can be scaled back to the prototype. In this study, the time necessary for the development of bed structure can be scaled to time in the field.
To be representative, models must maintain geometric, kinematic and dynamic similarity. For geometric similarity all length dimensions in the model must be scaled exactly, using the same ratio to the prototype. Model and prototype will then have the same shape. Kinematic similarity requires the ratios of corresponding velocities and accelerations to be the same between model and prototype. Dynamic similarity requires the ratios of corresponding forces to be the same.
The forces acting on a fluid include gravity, viscosity, inertia, pressure, elasticity, surface tension, buoyancy and Coriolis (Vennard and Street, 1982). For models of open water channels, only the first three are important, although if the model is small, surface tension may become important. Dimensionless ratios of 19 the inertial to viscous forces, called the Reynolds number, Re, and inertial to gravitational forces, called the Froude number, Fr, provide a basis for dynamic
scaling.
Re = Vl/j> (2.1)
and
Fr = V/(lg)1/2 (2.2)
where V is the velocity of the flow, 1 is a length scale, v is the kinematic viscosity
of water, and g is the acceleration due to gravity. If the corresponding Reynolds
and Froude numbers are the same in both model and prototype, the model is
considered to be dynamically similar. Models which are geometrically and
dynamically similar are also kinematically similar (Vennard and Street, 1982).
Unfortunately, it is difficult to maintain both Reynolds and Froude scaling
because viscosity tends to be the same in both prototype and model.
Consequently, hydraulic models of the physical environment are used in
three distinct ways. At the most general level, gross characteristics of the
interactions between the variables of interest may be observed. Work by
Schumm et al. (1987) typifies this approach. Geometric, kinematic, and dynamic
similarity are ignored, and the results are expressed in qualitative terms. As a
first approximation to understanding the behavior of a complex system of
variables, there may be some merit in this approach, although different processes
can produce similar results.
A second approach is to model either a specific or generic site by striving
for geometric and some dynamic similarity, and then observing carefully what
occurs. Parker (1980) was thus able to observe the behavior of the coarse surface
layer on a modeled stream bed during floods. 20
The most productive but also the most difficult approach is to achieve complete geometric, kinematic, and dynamic similarity. Here model lengths, velocities, and forces are all scaled from the prototype. Consequently, volumes, velocities, and forces measured in the model relate directly to the those in the prototype.
The difficulty arises in maintaining kinematic similarity. As the length scale changes, so must the viscosity of the fluid acting on the system. Three solutions are to use a different fluid, to use the same fluid at a different temperature, and to use the same fluid at the same temperature but use a scaling factor to adjust. Most hydraulic studies cannot afford the cost, equipment and time necessary to use different fluids. Compared to water at 20 °C, gasoline, for example, is not quite 1/4 as viscous, liquid sodium is 1/4 as viscous, and liquid hydrogen is 1/50 as viscous (Vennard and Street, 1982, p 8). Water at 55 °C has half the viscosity it has at 20 °C, but this means the model length dimensions can be only half as long as the prototype. Most laboratories do not have the room for
1/2 scaling.
The conventional approach (Vennard and Street, 1982) is to use water at ambient temperatures and maintain Froude similarity, which causes the model
Reynolds number to be much less than the prototype, and then correct the results using a scaling factor. Unfortunately, scaling factors are empirically determined since they are the result of all frictional elements in the system. Except for very simple systems, such as hull design studies, the scaling factor is not known
(Vennard and Street, 1982).
Parent (1988) provides an excellent discussion of scaling theory as it relates to flume studies of rivers. He notes that since the gravitational acceleration of 21 water is opposed by the friction of the channel, these are the dominant forces which must be scaled correctly. For a straight reach in a gravel bedded river, the friction is from the bed and bank material, and is found by multiplying the wetted perimeter by a friction factor. The friction factor is independent of viscosity if the flow is hydraulically rough, and therefore the Reynolds number is not critical. As long as Froude scaling is correct, and the flow is hydraulically rough, Reynolds scaling may be relaxed. Fortunately, natural streams and rivers normally are hydraulically rough during transport events. Parent recommends the use of the following formula (from Rouse, 1946) to ensure a hydraulically rough flow and a constant friction factor (ff) in flume studies:
1/2 4Re(ff) ks/4R > 200 (2.3) D50 where Re is the Reynolds number, ks is the roughness height, typically or
DgQ of the bed material, and R is the hydraulic radius.
The deeper flows of Series 2, 3 and 4 in the present study satisfy equation
(2.3). Although the shallower flows do not, there is no major change of behavior between flows which are hydraulically rough and those that are not.
A Note on Units
The scale of the processes studied in the flume and the precision of the measurements were best characterized by the cgs system of centimeters, grams, and seconds, with two exceptions. Particle diameter has been measured traditionally in mm, in part because the screens used to sort material are graded by millimeters and fractions of millimeters. That tradition is continued here to facilitate comparison of the results with other studies. Second, the sediment transport rates were sufficiently low that kilograms per hour were recorded rather 22 than kilograms per second. All measurements have been converted to SI units of meter, kilogram, and seconds, with the two exceptions mentioned previously.
Experimental Apparatus
The flume, illustrated in Figure 2.1, had a rectangular cross section 0.5 meters wide, 0.3 meters deep and 6 meters long between the downstream side of the headbox and the upstream side of the tail gate. Sidewalls were clear plexiglas, 0.0127 m thick and 0.30 m high: The exit portion of the headbox was a honeycomb of 0.025 m square tubing 0.25 m long. Water surface waves created by the exit conditions at the headbox were damped by a 0.07 m thick Styrofoam float 0.6 meters long and 0.01 m less than the width of the flume. The water surface, however, was not unnaturally tranquil. Since the approach adopted in this study was that of averaging conditions and results in the hope of better understanding the natural processes, slight fluctuations in the free surface were desirable.
The first 0.75 meters downstream of the headbox contained a fixed bed made from stones glued to a plywood sheet. The intermediate diameter of these stones was equal to the intermediate diameter of the largest stones of the bed material, but the fixed stones were more angular to facilitate mixing and dispersion of eddies from the headbox. For the remainder of the flume the bed consisted of loose material. The 0.5 meter long working section of the flume, where all flow measurements and bed sampling occurred, was 3.75 meters downstream from the headbox and 1.75 meters upstream from the tailgate. The adjustable tailgate consisted of vertical slats which were closed slightly to offset the drawdown effect as water spilled into the mesh-lined tailbox. 23
Figure 2.1 Sketch of flume. 24
Flow characteristics in the flume have been described by Nowell (1975). To assess the validity of his assumption of uniform flow, he measured crosswise and downstream velocities over a smooth bed. He reports (page 30), "No obvious pattern exists and thus the flow is taken to be suited for the assumptions about its two dimensional nature..." Additional measurements were made during the present study with sediment in place on the bed across the working section to within 0.005 m of the walls. Velocity measurements were obtained with a hot film probe using the same electronics utilized by Nowell. Table 2.1 contains velocity profiles measured at 8 different depths at each of 11 stations across the flume. Each value is the average of four measurements; each measurement is the
60 second average of 10 readings per second. The decrease in the near surface velocity has been reported in other flume studies, including Shields (1936) and
Nowell (1975). One explanation is that the air immediately above the water surface exerts a drag on the surface water. Drag from the side walls may have slowed the surface water since there was nothing above the top layer to overcome the sidewall effect. Another explanation is that since the probe tip was within
0.002 m of the surface, it may have been momentarily exposed during the passage of a water surface perturbation.
Data from Table 2.1 show that the flow was fairly uniform to within 0.03 m of the walls. The variability between repetitions is disconcerting. It suggests that either the flow was changing or the instruments were drifting. A water surface follower connected to a chart recorder showed less than 0.002 m change in water depth in 7 days. One explanation is that the probe tip may have been dirty during some of the measurements, in spite of a standard cleaning procedure which included a rinse in potassium dichromate to remove organic matter. At low velocities, bubbles formed at the probe tip and were removed by a tap on the probe support. Near bed velocities, however, do not always show the same bias 25
TABLE 2.1
Velocity Profiles (m/s) Across Test Section at Water Depth of 0.08 m and Slope of .001.
Height Date, Time, and Distance in meters from left wall above bed Mar 5 Mar 5 Mar 5 Mar 5 Mar 6 Mar 6 Mar 6 1145 1250 1530 1630 1310 2130 1910 (m) 0.005 0.010 0.015 0.030 0.050 0.050 0.100
0.078 0.31 0.15 0.20 0.29 0.33 0.43 0.55 0.060 0.38 0.25 0.40 0.50 0.38 0.51 0.54 0.040 0.36 0.41 0.46 0.49 0.35 0.49 0.49 0.032 0.36 0.40 0.44 0.45 0.33 0.45 0.45
0.020 0.26 0.33 0.40 0.40 0.26 0.38 0.38 0.015 0.26 0.34 0.35 0.33 0.24 0.26 0.34 0.010 0.23 0.28 0.27 0.27 0.18 0.33 0.28 0.005 0.18 0.22 0.17 0.18 0.12 0.18 0.16
(continued)
Height Date, Time, and Distance :i n meters from left wall above bed Mar 7 Mar 11 Mar 6 Mar 7 Mar 11 Mar 11 Mar 11 Mar 11 1055 1600 2010 1010 1515 1430 1350 1300 (m) 0.100 0.100 0.233 0.365 0.365 0.435 0.455 0.460
0.078 0.45 0.56 0.43 0.28 0.24 0.24 0.060 0.44 0.56 0.56 0.56 0.44 0.41 0.33 0.25 0.040 0.40 0.49 0.51 0.52 0.40 0.42 0.33 0.25 0.032 0.38 0.47 0.48 0.49 0.39 0.40 0.34 0.25
0.020 0.33 0.40 0.42 0.45 0.35 0.35 0.31 0.25 0.015 0.29 0.36 0.31 0.42 0.33 0.31 0.30 0.23 0.010 0.27 0.32 0.35 0.36 0.29 0.27 0.23 0.20 0.005 0.19 0.23 0.23 0.27 0.22 0.20 0.18 0.14 26 as the upper flow velocities. More importantly, bed configuration may have changed between measurements. Readings within a few hours of each other, however, appear to be similar.
Of the 4 pairs of repeat measurements, two pairs on March 7 - March 11 show a fairly consistent reduction of about 20%, suggesting equipment problems, but this bias is not constant across the flume. At 0.1 m from the left wall, the low reading was on March 7 and the high reading on March 11. At 0.1 m from the right wall, the low reading was on March 11 and the high reading on March 7
Comparisons between velocities taken 0.10 m from either wall on different days show little variation. Thus, the flow was steady over the span of hours and uniform when averaged over several days.
All water and sediment leaving the flume passed through a 0.200 mm mesh screen, and the water was recirculated. No sediment coarser than 0.200 mm was able to enter the flume headbox. With no sediment supply, the surface developed maximum structure in the shortest time. Other flume studies with no sediment supply include those of Harrison (1950), Gessler (1970), and Paintal
(1971). Studies with sediment feed or sediment recirculating systems, such as those of Parker (1980) and Wilcock and Southard (1988, 1989), have examined bed texture and sediment transport characteristics at equilibrium. In contrast, this study examined the development of bed structure, which implies non- equilibrium conditions.
Water was recirculated by a variable speed electric motor and an axial pump. The pump bearings were water lubricated, which added 0.0000158 m^/s
(9.5 liters per minute) to the circulating water. To adjust for this, an overflow pipe on the tailbox maintained a constant head for the pump. The input of fresh 27 water continually diluted the small concentration of suspended silt and clay, less than 1% of the total bed material, which eventually passed out of the system through the overflow pipe. The fresh water also kept the temperature uniform during a run. Water temperature varied between 7 and 14 degrees centigrade over the course of a year. For any given run the temperature varied less than 2 degrees from start to finish, usually remaining within 0.3 degrees.
Discharge at low flows was calculated from velocity profiles measured with a hot film anemometer as well as by capturing all the water for a set time interval in a bucket which was 0.30 m in diameter and 0.25 m deep and then weighing the water. The maximum decrease in pump head from the interception of the flow by the bucket was 0.015 m, or 2.5%! Agreement between these two techniques was within 10%. The discharge calculated from the velocity measurements was always between the extreme values of the weights of
individual water samples. No trend in the comparisons was observed. At the higher flows, only the velocity profile method was used.
Bed slope was measured by a transit and metric ruler which was read to
the nearest mm and averaged over a horizontal distance of 4 meters. Two
readings were taken before and after each run and were always within 2 mm of each other. Water surface slope was calculated from water depths measured to
the nearest mm at 1/2 meter intervals along the flume. Fluctuations of depth at
a point were 5% or less. The water depth was uniform to within 0.001 m until within 0.50 m of the tailgate, where it began to increase slightly. At the tailgate
it was typically 10% greater than elsewhere in the flume. No noticeable amount of sediment accumulated here. 28
The sediment used was rounded to subrounded natural stream gravel.
Three different mixtures were employed during the research. Mix 1, with a median diameter, or D5Q, of 3.50 mm, was used for the first set of experiments,
Mix 2 with a D50 of 2.30 mm for the second set, and Mix 3 with a D5Q of 2.39 mm for the rest of the experiments. Median diameter changed slightly because new material was purchased after the first set of runs. No stones larger than 8.0 mm were found in ten bulk samples of Mix 1 and five bulk samples of Mix 3, although there were a few stones collected during subsequent sampling which were between 8.0 and 11.2 mm. Mix 2 had approximately 0.1% material coarser than 8.0 mm. Figure 2.2 shows the cumulative grain size distribution curves, which have been truncated at 8 mm. The measured density of the material was
2,810 kg/m^; bulk density was 1,790 kg/m^. Initial volume of sediment in the bed was 0.317 m^, with a depth of 0.13 m and a total mass of 567 kg.
Subsequent settling caused the bed material to lower approximately 0.03 m. A
0.10 m high board across the width of the flume and under the tailgate prevented bed material from eroding below this depth at the tail end of the flume. The maximum sediment transported during a single run was 8.3 kg, or 1.5% of the total bed material.
Experimental Procedure
Material was placed in the flume in layers, each bucketful spread along the length of the flume to minimize local variation in size distributions. Before each set of runs, a composite bulk sample was taken from five locations along the flume bed for analysis. All sieving was done with dry material at 1/2 phi intervals down to 0.063 mm, where applicable. Each sample was oven-dried at
100 °C for 24 hour, then allowed to air dry for 48 hours. The additional air- 29
GRAIN SIZE (mm)
Figure 2.2 Cumulative grain size distribution curves for sediment used in the experimental runs. 30 drying minimized the increase in sample weight during the analysis as the samples absorbed moisture from the air. Surface samples were taken with a 1 to
2 mm thick layer of very soft clay spread over a square piece of 0.0254 m thick foam rubber which was glued to a piece of 0.0127 m thick plywood 0.33 m by
0.33 m with a handle in the center on the side opposite the foam. The surface area was 0.100 m2. The clay used was powdered ball clay, Kentucky OM-4, purchased from a ceramic supplier and mixed with water.
The consistency of the clay was important. Tests showed that stiffer clays pushed the larger material into the bed without extracting the largest stones or the finest material. Any bias of the stiffer clay toward the fines was the result of undersampling the coarsest particles rather than oversampling the fines. If the clay was too soft, water which had been absorbed by the foam during application of the clay was released when the board was pressed onto the surface, and the larger particles were washed off the sampler. The optimum consistency contained between 44% and 48% water by weight and resembled chocolate frosting.
After the foam was coated with clay and pressed onto the bed surface, the sampler was carefully removed, placed in a bucket and washed with warm water to suspend the clay. After decanting the clay and water, the remaining stones were oven dried and sieved at 1/2 phi intervals. The distributions were truncated at the smallest size class that weighed 0.0001 kilogram or more. Data from the surface samples were converted to bulk sample equivalents using the formula of
Kellerhals and Bray (1971):
Wtc = Wts/Dm (2.4)
where Wtc is the converted weight of a particular size fraction, Wtg is the weight
of material on that size sieve, and Dm is the geometric mean diameter of the material on the sieve; 31
Diplas and Sutherland (1988) found that if molten wax was used to sample the surface, the following conversion should be used
0 42 Wtc = Wts/(Dm) - (2.5)
If adhesive tape was used, they found that the Kellerhals and Bray formula should be used.
They also confirmed that grid by number (Kellerhals and Bray, 1971, among others) was equivalent to the bulk sample technique. Since it is difficult, if not impossible, to obtain or even recognize a truly undisturbed surface, the best comparison of conversion techniques is between grid by number and the converted sample. The shortcoming of the grid technique is that it is difficult to sample grains smaller than about 4 mm. To overcome this limitation, a special grid of
0.020 m squares was constructed with 100 intersections. The grid consisted of two tiers of very fine wire to eliminate parallax problems. Tweezers were used to sample grains as small as 0.5 mm.
Five pairs of samples using the clay technique and the grid method taken from the bed surface of the flume showed that the Kellerhals and Bray converted samples were always within 1% of the grid samples for any size class. It appears that the clay technique performs more like the adhesive tape method than the wax method.
At the start of each set of experiments the bed was prepared, leveled, and sampled. The analyzed material was returned to the flume and spread evenly over the surface. Then the entire bed was thoroughly mixed in the flume and leveled again. The surface was gently moistened with mist from a hand sprayer to minimize the effect of surface tension of the wetting front on the smallest particles as water began to flow down the flume. Discharge was slowly increased, 32 flooding the bed from below to decrease entrapped air. Water depth was then set to a predetermined level, and the clock started. To assure that no fines were being stored in the return system, each time the flow was stopped the return line was disconnected and inspected.
For the first series of experiments (runs 1-1 through 1-4), the bed was allowed to equilibrate for 24 hours to a flow 0.016 m deep, and then the flow was stopped. The bed surface was dried with heat lamps and a fan for several days to prevent finer particles from sticking to the bottoms of the larger particles, and the surface was sampled. After analysis the sampled material was returned to the surface area of the flume from where it had been removed, and the start up procedure repeated. Bed material was not remixed, since the subsurface material was still damp, and finer particles stuck to larger particles when the latter were brought to the surface. Water depth was set to 0.010 m. Twenty-four hours later it was increased to 0.020 m for 24 hours and then stopped, and the bed surface was sampled. For the third run, the flow was set at 0.010 m for 24
hours, 0.020 m for 24 hours, and then 0.030 m for 24 hours before sampling the
surface. The fourth run followed the same procedure, with water depth increased to 0.050 m after 24 hours at each of the preceding depths. Thus, each run
repeated the conditions of the previous runs. At the 0.050 m deep flow, the run was stopped after 15 minutes due to the extremely high volume of sediment in transport.
A second experiment, run 1-5, investigated the effects of declining flows to determine if at high transport rates the coarsest fractions might be the last to
come to rest. The tail box was modified to accept large amounts of sediment.
After the bed was mixed and leveled, the discharge was set to produce a shear
stress about 5 times above the threshold for motion. After two hours the water 33 had carved a channel only 2/5 the flume width. Discharge was reduced to give a water depth of 0.035 m, and a Helley-Smith type sediment bag with 0.200 mm mesh was installed at the outlet to trap sediment. The bag was changed after 15 hours, and the second bag was changed after 72 hours. The run was stopped 24 hours later because the third bag had less than 0.005 kg of sediment. A detailed account of observations made during the first two sets of experiments may be found in Chapter Three.
The goal of the initial runs, Series 1, was to assess the change in grain size of the bed surface as the bed adjusted to a given shear stress. Subsequent series provided data on the effect of the bed surface structure on sediment transport and on the length of time necessary for the structure to develop. For Series 3 and 4, surface samples were taken before the first run and after the last run so the flow was not interrupted between changes in flow depth. All sediment leaving the flume was trapped and analyzed. A microbalance was used to weigh the very small amounts of material Captured in the sediment trap to the nearest 10"® kilogram. Sediment size distributions for the trapped material were truncated at
0.180 mm. Sediment bags containing all the material which moved out of the flume were changed at 1, 2, 4, 8, and 16 hour intervals, usually. If sediment was still accumulating at a lower rate than the previous bag, sampling continued until the rate of accumulation stabilized. In several cases this took 5 or more days.
Grain size analysis of each sample showed not only the total amount of material moving, but also the behavior of individual size fractions through time. 34
Summary of Runs
A variety of slopes and depths were used to simulate conditions in natural rivers. The range of slopes studied, 0.001 to 0.01, was chosen because it represents common field conditions for gravel bedded rivers. As noted in Chapter
One, Parker (see Parker, 1980) observed that bed shear stresses in natural gravel bedded streams rarely exceed by a factor of more than 2 or 3 the stress necessary to entrain the median size particles. Consequently, depths which gave bed shear stress of 1, 2, 3, and 5 times that necessary to initiate motion were used to explore the development of the coarse layer at flows similar to a one or two year flood event and at a very high flow.
Table 2.2 lists measured and calculated hydraulic parameters for all runs.
Velocity measurements are 30 minute averages of 1 minute averages of 10 measurements per second. Readings were continued for 30 minutes to ensure no large fluctuations or drift occurred. For flows deeper than 0.005 m, all velocities were within 5% of the 30 minute mean. No drift or trend was observed. This suggests that the flow remained steady. Maximum water depths for each series of runs were the maximum obtainable with the existing flume configuration.
Constraints included pump and motor output and the onset of cavitation in the return line.
An interesting aspect of the velocity data is that the velocity increases much less with increasing slope than predicted by the Manning equation, listed in note 2 of Table 2.2. The values of Manning's n listed in Table 2.2 have been calculated by using the observed velocities. Two patterns emerge. First, within a given slope, n increases or remains the same and then decreases. The decrease occurs at depths greater than about 3 times the dominant roughness height 35
TABLE 2.2
Hydraulic Parameters
Run Bed Water vi Manning's 2 Discharge 3 Mix No. Slope Depth n Q D50 (10"2m) (m/s) (10-3m/s) (mm)
Series 1 1-1 0.0117 1.6 N/A N/A N/A 3.5 1-2 99 2.0 N/A N/A N/A 99 1-3 99 3.0 N/A N/A N/A 99 1-4 99 5.0 N/A N/A N/A 99 1-5 N/A 5-3.5 N/A N/A N/A 99
Series 2 2-1 0.0050 1.0 0.177 0.018 0.82 2.30 2-2 >> 2.0 0.268 0.018 2.49 >> 2-3 >> 3.0 0.458 0.014 6.39 »> 2-4 JJ 6.0 N/A N/A N/A
Series 3 3-1 0.0010 0.5 0.070 0.0130 0.16 2.39 3-2 99 1.0 0.077 0.0185 0.36 99 3-3 99 2.0 0.217 0.0100 2.02 99 3-4 99 4.0 0.340 0.0098 6.32 99 3-5 0.0017 8.0 0.416 0.0151 15.48 79
Series 4 4-1 0.0100 0.5 0.137 0.021 0.32 2.39 4-2 >> 1.0 0.179 0.025 0.83 97 4-3 2.0 0.280 0.025 2.60 99 4-4 >> 3.0 0.520 0.017 7.25 99
N/A = not available
1 V = velocity measured by hot film probe. This is the average of 30 one minute averages at 0.4 times the water depth above the bed located on the center line of the flume.
2 n = (R2^sl/2\/y where R is the hydraulic radius (next page) and S is the slope
= V x water depth x flume width (0.465 m) 36
TABLE 2.2 (continued)
Hydraulic Parameters
6 7 8 9 R4 5 Run Rb Water Water Fr Re We No. Viscosity Temp.
(10"2m) (10"2m) (10-6 m2/s) (°C) Series 1 1-1 1.59 12.8 1-2 1.84 1-3 2.66 1-4 4.12 5.0 1-5 N/A 5-3.5 »
Series 2 2-1 0.96 0.99 1.287 10.6 0.57 1,320 4.22 2-2 1.84 1.95 1.276 10.9 0.61 3,790 19.4 2-3 2.66 2.89 1.219 12.6 0.86 9,990 84.8 2-4 4.77 N/A 1.186 13.6 N/A N/A N/A
Series 3 3-1 0.49 0.49 1.199 13.3 0.32 286 0.330 3-2 0.96 0.97 1.283 10.7 0.25 576 0.799 3-3 1.84 1.87 1.327 9.5 0.51 3,000 12.7 3-4 3.41 3.38 1.327 9.5 0.59 8,740 62.3 3-5 5.95 6.90 1.425 7.2 0.51 7,400 187
Series 4 4-1 0.49 0.50 1.433 7.0 0.62 468 1.26 4-2 0.96 0.99 1.420 7.3 0.57 1,210 4.32 4-3 1.84 1.96 1.404 7.7 0.64 3,670 21.1 4-4 2.66 2.85 1.348 9.0 0.98 10,300 109
4 R is the hydraulic radius, equal to Area/Wetted Perimeter.
5 Rjg is the hydraulic radius of the bed, corrected for smooth side walls.
^ Average of start and finish temperatures
7 Fr is the Froude number, based V, Ru, and g where g is the acceleration due to gravity, taken as 9.81. Fr = V/CgR^)1/2
8 Re is the Reynolds number, based on R, V, and the viscosity of water at the temperature of the run. Re = RV/j>
9 We is the Weber number, based on V, R, p, which is the density of water, and a , which is the surface tension of water. We = pRV^/a 37
TABLE 2.2 (continued)
. Hydraulic Parameters
10 11 12 T 12 Run r Maximum MAX rmin No. Error Factor (Pa) (Pa) (Pa) Series 1 1-1 1-2 1-3 1-4 1-5
Series 2 2-1 0.48 (.005±.15) 0.55 0.41 2-2 0.95 1.10 0.81 2-3 1.37 1.58 1.17 2-4 N/A
Series 3 3-1 0.05 (.025±.55) 0.08 0.02 3-2 0.10 J? 0.16 0.05 3-3 0.18 0.28 0.09 3-4 0.33 0.52 0.16 3-5 1.15 1.81 0.55
Series 4 4-1 0.49 (.00251.10) 0.54 0.44 4-2 0.97 1.07 0.88 4-3 1.92 J> 2.12 1.73 >> 4-4 2.79 3.08 2.52 10 the shear stress, based on the slope, m/s *z, an- d the density of water at the temperature of the run. r = (7RjjS) where y is the weight of water.
11 (Slope error)(Depth error) ± (Slope error + Depth error) where Slope and
Depth errors are the % errors in the original measurements: {S ± (S)(error)}{Rr.
± (Rb)(error)}
*2 T ± T (Maximum Error Factor) Maximum and minimum possible values 38
(Nowell and Church, 1979), here taken as 8 mm. One explanation, found in
Morris (1955), and Nowell and Church (1979), is that at shallow flows, the water flows around the largest stones easily; at about 0.02 m there is maximum eddy interaction from the bed roughness elements; and at deeper flows, the bed surface effect is drowned out.
Second, the pattern between slopes shows that as the slope increases, n increases. If n is considered an index of bed friction, then the bed surface must appear less streamlined to the flow. Two means of increasing bed friction are to increase the size of the roughness elements or to change their spacing. This issue will be discussed in Chapter 4.
The hydraulic radius of the bed, Rb, corrects for the smooth sidewalls. Two similar approaches were used, which gave almost identical results. First,
Einstein's method (see Einstein, 1942) based on the Manning equation was used.
An n value of 0.01 was assumed for that portion of the perimeter occupied by the sidewalls, and this was used to calculate a hydraulic radius, R^, for the sidewalls using the Manning equation,
V = (R2/3Sl/2)/n (2.6) where V = mean velocity, R = hydraulic radius defined as cross sectional area A divided by wetted perimeter P, S = slope, and n = Manning's n. According to the Einstein method, V and S are assumed to be the same for the wall and the rest of the flow. Therefore,
= (Vnw/Sl/2)3/2 (2.7)
where nw = 0.01, the n value for the sidewalls. Since
Rb = Ab/b (2.8) 39
where b is the flume width and Ab is the proportion of the cross sectional area occupied by the bed,
Ab = A - Aw (2.9) where A is the cross sectional area, b-d, d = depth, and
Aw = 2dRw. (2.10)
Rb can be found by substituting equations (2.10) and (2.9) in equation (2.8).
Second, the method of Vanoni and Brooks (1957), which is based on the
Darcy-Weisbach formula, was used. The Darcy-Weisbach friction factor, f, is found by the formula,
f = 8gSR/V2 (2.11) where g = gravitational acceleration. Since S and V are again assumed to be the same for the wall and the bed, equation (2.11) can be rearranged to
Rb = fbv2/(8gS) (2-12) where
fb.= f+(2d/b)(f-fw) (2.13)
and fw is found using the graph in Vanoni and Brooks (1957) or by
2 fw = 8gSRVV (2.14)
where is found by equation (2.7). The values of Rb in Table 2.2 were
calculated by the second method using equation (2.14) for fw.
Froude number, Fr, varied between 0.25 and 0.98, which is comparable to the range of Froude numbers of natural gravel bedded streams. During run 1-4 small standing waves developed indicating that Fr was about 1. Series 3 had values from .32 to .59 whereas Series 2 and 4 had values from .57 to almost 1.
Reynolds number was less than that of natural systems because water was used in the flume, with the same viscosity as water in rivers, as discussed earlier. 40
Although the shallowest flow in Series 4 had a Reynolds number below the turbulent region found in natural streams, its sediment transport did not appear anomalous.
The two shallowest flows in Series 3, however, were unusual in that the shallowest flow transported more sediment than the next two higher flows. One explanation is that the transport rates were so low that the outcome was due to a random large stone moving in the lowest flow but not the next higher flows.
Alternatively, the lowest flow was so shallow and at such a gentle slope that surface tension may have contributed to the movement of material. The Weber number, We, which is a ratio of the inertial forces to the surface tension forces, is less than 1 for the two shallowest flows.
Shear stress was calculated using the water depth and slope. The velocity profile method was not used for several reasons. It is the average shear stress applied to the bed, integrated over a large area, that is of interest here, not the shear stress at any particular point. Variations in particle arrangements on the surface would require many such profiles to obtain a representative value. Also, velocity fluctuations near the bed vary considerably immediately downstream of larger particles, which would require long sampling times to ensure stable readings. 41
CHAPTER THREE
OBSERVATIONS
Series One Runs
Detailed observations were recorded during the Series 1 runs only.
Subsequent series exhibited a subset of shear stresses and bed features of the initial runs. The primary difference between the first series and the others is that
the former included bed surface sampling between flow depth changes, but not
sediment transport analysis, and the latter included continuous sediment
transport analysis but not bed surface sampling between flow depth changes.
Uniform water depth was maintained by use of the tail gate to ensure steady
uniform flow over all but the last 1/2 meter of the flume.
The Series 1 runs had a bed slope of 1.17%. This slope was chosen in part
because it is typical of gravel bedded rivers and in part because if a Shields
parameter of 0.03 for particle motion is assumed (Neill and Yalin, 1969), and the
median diameter of the mixture is 3.5 mm, then the critical water depth for
particle movement is 0.01 m. At a depth of 0.02 m, the shear stress is twice that
for initial motion. At 0.03 m, it is 3 times the predicted value, and at 0.05 m, it
is 5 times.
During the passage of the wetting front, smaller particles, less than 1.4
mm, moved downward into the bed. This movement into the subsurface
continued for several hours. Within the first hour a well developed layer of fines just below the surface grains could be seen through the flume sidewalls. Also the
bed settled about 25% and developed a coarse surface layer. The combined effects
may be seen in Figure 3.1. Here, part of the bed has been left as it was at the 42
Figure 3.1 Loose, well mixed material (top) and armored, and settled material after run 3-5 (bottom). Scale approximately 1:1. 43 end of a series of flows, and part has been well mixed in preparation for the next series. No material has been added, yet the mixed portion is about 0.02-0.03 m higher than the unmixed.
The initial flume settings resulted in a water depth of 0.016 m instead of
0.01 m. Sediment movement was fairly vigorous in a closely observed 0.10 m by
0.10 m area, with a particle displacement every 1-5 seconds. All sizes were observed to move, although smaller particles moved more often. When a large particle moved, several smaller ones were released also. Movement continued fairly steadily during the first hour, even though some parts of the bed did not experience sediment transport.
After 2 hours, there was little transport, but numerous small particles, 0.5 mm to 1 mm in diameter, still vibrated in place. Occasionally a grain would vibrate in place for several minutes, then move downstream 0.003 to 0.005 m, vibrate at the new location for a minute or so, and then be carried back upstream to its original location by an eddy caused by a larger particle immediately upstream. Several examples of imbrication had developed within 15 minutes of the start of the flow.
During the 0.02 m deep flow, isolated clusters consisting of several large stones developed. At the completion of the 0.03 m deep flow, the surface of the bed was covered with an interlocking web of coarse stones and stone clusters.
Virtually every large stone was resting against at least one other large stone, with finer sediment visible in between the lines of this web. This is shown in
Figure 3.2, which is a detailed drawing of the actual bed texture. A photograph of incipient web development from Series 4 is shown in Figure 3.3. Figure 3.2 Surface web of larger stones as seen on bed (top), and with smaller stones removed for clarity (bottom). Actual drawing of bed surface. 45
Figure 3.3 Photograph of incipient web development after run 4-3. Flow from left to right. 46
The web developed as clusters grew linearly in response to increasing shear flow, and, therefore, shear stress. Eventually the clusters were large enough to begin to coalesce. The example in Figure 3.3 is at the incipient stage, whereas
Figure 3.2 shows a more advanced form. Both figures convey the subtlety of the phenomenon. When the force on the bed exceeded the strength of the web, as in run 1-4, the web disintegrated. As soon as a few large stones moved, the addition of underlying sand to the bed load aided the transport of the larger stones by decreasing the local roughness. Typically, the first stones to move when the flow increased were the most recent arrivals at a cluster. The moving stones struck the remaining clusters more vigorously than at lower flows, and the clusters broke up. Although the formation of a web took several days in the flume, its destruction occurred in minutes.
The support or clustering of larger stones on natural river beds has been reported by Brayshaw and others (see Brayshaw, 1985). The features noted in the flume, however, differ in two ways from those previously reported. First, these features formed during increasing flows rather than decreasing flows, which is consistent with Parker's (1980) observation that the coarse layer remains intact during typical runoff events. Second, the morphology was different from the field examples of clusters described by Brayshaw (1985) because the size of the flume material making up the clusters was much more uniform, consisting of the 4 mm and 5.6 mm classes only. Consequently, the distinctive stoss side accumulation, obstacle stone, and finer wake zone illustrated in Figure 1.2 were not present.
Imbrication did occur, but over no more than about 10% of the bed area, as noted by Brayshaw.
An example of a web structure of boulders on an intertidal flat may be seen in Figure 2 in Gilbert and Aitken (1981). They refer to the net-like structure as a 47
"garland." The author was involved in a class exercise in 1982 at The University of British Colombia, Department of Geography, which surveyed stone lines of large boulders in Furry Creek, near Squamish, B. C.
During the 0.03 m deep flow a meander developed within the bed with a spacing between crossings of approximately 6 times the flume width. The bed surface appeared slightly coarser after the 0.03 m deep flow than it did after the
0.02 m deep flow, but analyses of surface samples showed little difference in the
D5Q. An excellent description of particle rearrangement and coarsening during flume studies under similar conditions may be found in Sutherland (1987). He noted that during surface coarsening all particles were rearranged, and that "The resistance to entrainment will vary among particles of any given sieve fraction by virtue of small changes in particle size and density and variations in shape which affect both interlocking and placement of particles.
At each increase in flow depth, all sizes were observed to move initially.
After 2 hours, movement appeared to involve primarily particles smaller than 3 mm. Initial movement following a flow increase resulted in much larger travel distances before particles came to rest than subsequent movement after the first hour. At each flow increase, the water became murky, with visibility less than
0.40 m. This was caused by small amounts of silts and clays going into suspension when larger surface grains moved, exposing finer subsurface material.
The release of sands appeared to assist the larger material in moving farther, much like thousands of miniature ball bearings.
At a flow depth of about 0.05 m, the shear stresses on the bed were sufficiently high to prevent the formation of a coarse layer (Parker and
Klingeman, 1982), and the entire bed was active, almost as a single sheet. 48
Antidunes formed adjacent to the thalweg, moved upstream, and then disappeared within 15 minutes as a flat bed formed. This required erosion of the
antidunes. Consequently, more sediment transport was observed in these areas
than in the thalweg. After 15 minutes the sediment trap was full and the run
was stopped. The bed surface appeared much finer than after the two previous
flows.
Figure 3.4 is a plot of the surface sample data from the 0.02, 0.03, and
0.05 m deep flows as well as the dry bulk sample of the original material.
Noteworthy are the results of the 0.05 m deep flow. Not only is the bed surface
much finer than after the 0.03 m deep flow, but it is finer than the original
mixture. The additional fines most likely had been stored just below the surface.
(Recall that to assure no fines were being stored in the return system, the return
line was disconnected and inspected each time the flow was stopped. No evidence
of sediment storage was found.)
The flume results indicate that the transition from a very coarse surface to
a very fine one is abrupt rather than gradual, suggesting the occurrence of a
threshold condition. The implications are that relatively fine bed surface textures
may be created by very low or very high flow conditions, and a slight increase in
shear stress may change a coarse bed surface to a much finer one. The important
variable is sediment supply. If supply rate increases due to general mobilization
of the bed surface, the bed will become finer. This matter needs careful field
evaluation to determine the validity of the flume results.
Run 1-5 involved investigations of the effects of declining flows. After the
bed was mixed and leveled, the discharge was set to produce a water depth of
0.05 m, which was identical to the preceding run. The sediment trap had been 49
GRAIN SIZE (mm)
Figure 3.4 Comparison of bulk subsurface sample with surface samples after 0.02 m, 0.03 m, 0.05 m deep flows (Runs 1-1 to 1-4), and thalweg from degrading run (Run 1-5). 50 modified to accommodate much larger volumes which allowed the high flows to run for several hours. The initial 15 minutes produced a surface layer similar to the preceding high flow. Shortly, however, pulses of fine sediment and pulses of coarse sediment began to form tongue-like bed forms (that is, very long and narrow low dunes) which migrated down stream. A single 16 mm stone placed in the test section was alternately buried and exposed as the pulses passed.
Gradually, during the next two hours, a channel formed along the left half of the test section. Since there was no systematic change in flow velocities across the cross section initially (see Chapter 2), it is assumed that this was the result of sediment transport only. First a point bar began to develop beginning 1.5 meters upstream of the tail gate and extending about 1 meter downstream. As the bar grew in height, the flow was forced to the left half of the channel. Thus confined, it commenced to lower the bed in this area, which eventually left the bar top about 0.02 m above the water surface. As the water surface lowered, the newly exposed bed material appeared slightly coarser than the previously exposed surface. Thus, there was a textural gradation from the finer bar top to the coarser thalweg. After 2 hours the discharge was reduced to give a water depth of 0.035 m to 0.04 m across the narrow thalweg which now occupied the left 2/5 of the flume. At this time a collection bag was installed at the outfall to capture all bed material larger than 0.177 mm. After 15 hours, 0.0574 kilograms of material had accumulated; the first bag was removed, and a new bag was installed. It was replaced after 72 hours, by which time 0.092 kilograms had been trapped. A third bag was installed, but 24 hours later less than 0.005 kilograms of sediment had been trapped, and the run was halted.
The purpose of run 1-5 was to study the change in surface texture during declining flows. It was not a study of the development of meanders. Although the flow configuration changed from two-dimensional to partially developed three- 51 dimensional, surface samples were taken from well defined homogeneous areas.
This was accomplished by taking longer and narrower samples, 0.05 m by 0.10 m, instead of the usual 0.10 m by 0.10 m . These samples are assumed to represent a range of textures in two-dimensional slices within an evolving three-dimensional flow pattern.
Textural differences between the bar top and the adjacent channel may be seen in Figure 3.5 and Figure 3.6. At the high flow, the coarse surface layer completely disappeared, but it developed again during the declining flows.
Coarsening had already begun before the bar top was exposed because it is coarser than the bed surface of run 1-4 (see Figure 3.4). These observations confirm Parker's (1980) observation that the coarse layer remains in place during typical flood events. They also confirm the opposing notion that the coarse layer is destroyed during high flows, and it redevelops during declining flows. The crucial factor is the size of the flood event.
Figure 3.6 summarizes the surface changes to the two different flow depths and sediment supply conditions during run 1-5. The bar top sample, an area of high sediment supply, and the thalweg sample, an area of low sediment supply, are compared to the original dry bulk mixture and also to the results of the three bag samples of transported material. Although the bar top sample is not as fine as the bed after the 0.05 m deep flow of run 1-4, it is almost identical to the original mixture above Dg5. Below this, the bar top is coarser, presumably a result of the fines being removed as the bar top was gradually becoming exposed.
The trapping of the fines by bag 1 and bag 2 did not reflect the sedimentation process on the bar top, since the bar became exposed before the bags were installed. Figure 3 5 Textural differences between channel surface (above) and surface (below). Flow from left to right, scale approximately 1:1 53
GRAIN SIZE (mm)
Figure 3.6 Comparison of bulk subsurface sample, sediment trapped during degrading run, and surface sample from thalweg and top of bar. 54
The progressive coarsening of the bag samples over a 4 day period is the result of two factors. First, the difference between the contents of bag 1 and bag
2 is primarily the effect of a single large stone in bag 2. Bag 1 had no material larger than 5.65 mm; bag 2 and bag 3 each had one stone between 5.65 mm and
8.0 mm. Second, the amount of material captured was small, a total of 0.1542 kilograms in 110 hours. The difference in texture between the contents of bag 2 and bag 3 is the result of differences in volume of material. Both had a single large stone, but the total material captured by bag 3 weighed 0.0048 kilograms compared to 0.0920 kilograms for bag 2. These individual large stones were 0.3% of the weight of bag 2 material and 6.5% of the weight of bag 3 material. Here the difference is due to a decrease in the amount of fines rather than an increase in the amount of coarse material. This suggests that the fines have been removed and mostly larger stones are available for transport. The small amount of fine material trapped also suggests that part of the surface coarsening occurs because fines are moving into the bed.
A Qualitative Model
The trend of the preceding observations was found in all runs of all series.
This suggests the following evolution of bed surface structure. On any given surface, some grains are more unstable than others. Instability is a function of both the geometrical arrangement with respect to neighboring grains and the local shear stress. Any size grain may be relatively unstable. When water moves over the surface, the least stable particles will move into more stable positions. For small grains, this may be into the bed through pore spaces on the surface, or downstream to a more sheltered position or a more firmly supported one. For larger grains, this may mean a slight realignment to a more stable orientation, or 55 downstream movement to a more firmly supported position or a more sheltered position due to the close proximity of other large grains (see Langbein and
Leopold, 1968, and Brayshaw, et al., 1983, for data on this last effect).
As the flow increases, the stabilization process continues since the increasing stress requires the particles to take up increasingly protected positions to resist movement. At a higher flow, initially less stable grains are moved.
When larger grains move, they expose the smaller material underneath, which may move also. The release of smaller grains, which act like ball bearings, inhibits the formation of new structures involving larger grains which are moving.
The effect of the fines is a function of the volume of fines. If only a few large stones move, releasing a small amount of fines, the effect is negligible, and the bed is observed to coarsen slightly as the flow increases (see Figure 3.4). Further coarsening is minor once it has initially developed, and this led Parker to conclude that the coarse layer remains in place during higher flows (Parker, 1980). As the supply of fines increases, the larger stones which are moving are less likely to stabilize, and this led Parker, et al. (1982) to conclude that the bed surface becomes finer at higher flows (see Figure 3.4).
If the supply of larger stones from upstream decreases, for example due to the development of more stable positions if the flow is only slightly higher, then the larger stones which are in stable positions will remain stable while smaller stones are removed. This is what Wilcock and Southard (1989) observed in their recirculating flume. The reason smaller stones are removed preferentially on a structured surface is the excess stability of the larger stones. A stable structure of large stones may be able to just withstand the imposed stresses, or it may be able to withstand much higher stresses. A group of unsheltered smaller stones, however, will have a much lower limit to the maximum stress they can 56 withstand. As the stress on the bed increases, larger stones will develop more
stable positions. Smaller stones will be either removed completely or incorporated
into the larger stone configurations so the bed surface will not be totally devoid of
finer material.
If no material is supplied to a reach, the bed surface will respond to
increasing flows by coarsening, sooner or later. This temporal variability is an
important aspect. The bed surface may become coarser, or finer, or remain
unchanged immediately after the discharge increases. If the flow is very high, the bed will initially become finer due to the increased amount of material in
transport (see Figure 3.4, 5 cm surface). Ultimately, however, it will become coarser and more stable (see Figure 3.4, Thalweg). The results of the increased
stability are discussed in Chapter Four. 57
CHAPTER FOUR
BED STABILITY AND INCIPIENT MOTION
An Overview-
One line of reasoning about sediment transport begins with Shields' (1936) approach which related the size of material moved by flowing water to two variables, the force applied to the bed by the water and the resistance of the material to move. The latter was based on the weight of the individual stones. If a uniform density is assumed, the weight may "be characterized by the average diameter, which is usually assumed to be the intermediate or B axis diameter.
Shields suggested and others have shown (for example, Neill and Yalin, 1969) that for mixtures of different sizes, the median diameter of the mixture may be used to characterize the bed resistance to movement. Thus, sediment transport is taken to be a function of bed stress and median diameter of the material.
In the late 1960s and early 1970s, Gessler (i.e. Gessler, 1970) showed that grain movement could be described in terms of the probability of movement of individual stones within a mixture. This was a consequence of applying the
Shields approach to each size fraction of the material on the bed in proportion to the area occupied by each size fraction. Thus, the size distribution of the bed material was important also.
Parker (see Parker, et al., 1982) proposed that grain movement was controlled by the incoming sediment, as well as by the shear stress, the median diameter, and the grain size distribution of the bed material. Figure 4.1 (Parker, pers. com., 1987) shows a proposed relation among incoming sediment supply, Qs,
median diameter of the bed surface material, Ds5Q, slope (of both bed and water 58
Figure 4.1 Proposed relation between hydrauhc variables and sediment supply (Parker, 1987, pers. com.). 59 surface), and water discharge, Q. Here bed stress may be considered to be a function of discharge.
A limitation of Figure 4.1 is that it is intended for two-dimensional gravel systems with fixed sidewalls. Bank composition, sinuosity, channel bars, and bed configuration are not considered. Thus, in addition to flow conditions and sediment conditions, which are included in the figure, rivers also have channel conditions. If the channel conditions are relatively stable, however, then Figure
4.1 could apply in the field (see Lisle and Madej, 1989). As a guide to thinking about sediment transport, Figure 4.1 is useful.
There are several aspects of Figure 4.1 worth emphasizing. If incoming sediment supply increases as discharge increases, the bed surface becomes finer.
This has been confirmed experimentally by Parker (1980), Wilcock and Southard
(1988), the present study (see Figures 3.3 and 3.4), and supported in the field by data of Lisle and Madej (1989). If the sediment supply increase matches the increased ability to move sediment, the slope remains constant. A greater increase in sediment supply requires the bed slope to steepen in order to move the increased input, which occurred in one of Harrison's (1950) runs. It is possible to increase Qs as Q increases and maintain nearly constant D50.
According to Figure 4.1, as sediment supply from upstream decreases, the bed surface should coarsen. Indeed, Figure 3.4, which compares the initial surface and subsequent surface median diameters, shows that the bed surface becomes coarser. If the bed shear stress is increased by almost 50%, however, the change in median diameter is only about 3%, as shown in Figure 3.4. Figures from Harrison (1950), Gessler (1970), and Wilcock and Southard (1989) show from their flume work similar small changes in median diameters with increasing 60 shear stresses. The ability of bed particles to resist motion is the result of more than just surface coarsening.
Hidden in Figure 4.1 are the effects of grain size distribution and bed
structure. Both control the degree of surface coarsening as well as sediment
output. The effects of size distribution have been discussed by Gessler (1970) and
Parker, et al. (1982). The effects of geometric structure form the basis for the
remainder of this and following chapters.
The Effect of Bed Structure Development
Direct measurements of bed structure are difficult because it is a complex
phenomenon. It includes particle size, particle alignment, particle exposure, and
proximity of supporting grains. Attempts by others to assess grain stability
include measuring the pivot angle of grains (Miller and Byrne, 1966; Komar and
Li, see references in Komar and Li, 1988; and Wiberg and Smith, 1987). Another
approach is to attach string or rope to individual rocks and measure the force
needed to move the stone out of its original position. The angle of pull will affect
the results, and there is little guidance to determine the proper orientation of the
pull or the place of attachment for the string or rope. These were considered and
abandoned in the present study because they do not duplicate all the factors
involved in grain movement. An attempt to quantify the surface structure was
made by making measurements on parallel transects oriented in a downstream
direction. The number of large stones was recorded, as well as the number of
clusters and the number of stones resting against other stones. The high degree
of subjectivity in the two latter approaches can be appreciated by studying Figure 61
3.2. The "number-of-large-stones" approach was also ineffective in discriminating the degree of geometric structure.
An indirect measure of surface structure on a bed may be inferred by analyzing the material captured downstream of a reach. In the present study, since no material was fed into the system, all material trapped is from the flume bed. Although vertical segregation occurred in part from the infiltration of finer surface particles into the subsurface, as noted in Chapter 3, the distribution of the trapped sediments show which sizes were moving horizontally at what flow conditions. During the later flows of a given set of runs, the pore spaces may have become sufficiently blocked (see Beschta and Jackson, 1979) that all fines moved out of the system rather than some moving into the bed.
Figure 4.2 is a series of cumulative plots of the size distribution of all the material trapped during each of the discharges for the Series 4 runs. Coarsening of transported sediment as discharge increases shows that the finer material is winnowed first, as the bed adjusts to higher flows. Comparison with Figure 3.4 confirms the initial coarsening of the bed between the freshly prepared surface and the surface after a 0.02 m deep flow, as well as the small change in texture between the 0.02 m and 0.03 m deep flows. There is little difference in Figure
4.2 between the material in transport at the higher flows, just as there is little difference in the bed surfaces. Water depth and slope were the same for the runs in Figures 3.4 and 4.2, although the mixture was slightly coarser in the runs in
Figure 3.4 (see Table 2.2). Information not conveyed by Figure 4.2 is the amount of material transported. The 0.5 cm deep flow moved 1 gram of sediment out of the flume, and the 1 cm deep flow moved 4 grams out of the flume. Initial coarsening of the surface probably occurs more by the downward movement of fines into the bed, as noted in Chapter 3. Although the distributions are similar 62
GRAIN SIZE (mm)
Figure 4.2 Cumulative grain size distribution of trapped material during Series 4 Runs. 63 for the 0.02 m and 0.03 m deep flows, the former moved 0.037 kg of material whereas the latter moved 8.328 kg. This large difference in transport with little difference in texture indicates either remarkable sensitivity to hydraulic constraints, or that something other than grain size controls what moves.
Bed Structure and Incipient Motion
Shields (1936) determined the threshold of motion by extrapolating back from higher transport rates to a zero transport rate. The concept of a threshold of motion has been challenged by Paintal (1971) and by Lavelle and Mofjeld (1987).
Work by Paintal (1971) has shown that transport can occur at very low shear stresses if sampling is continued long enough (see Neill and Yalin, 1969, for a discussion of sampling time). Table 4.1 compares initial and final bedload transport rates over a range of Shields parameters covering one and a half orders of magnitude. Although Lavelle and Mofjeld are correct in claiming that a threshold below which no motion occurs is not clear-cut, for many applications the very low transport rates at the lower shear stresses can be ignored and the concept of a threshold retained, if modified to mean threshold above some value considered unimportant for the problem at hand. Paintal observed that below r*
= 0.05, sediment transport increased in proportion to r* raised to the 16tn power, but above 0.05 it increased as r* raised to the 2.5 power. The converse of this statement may clarify the point. Above T* = 0.05, Qs decreases relatively slowly as T* decreases. Below T* = 0.05, Qs decreases very rapidly as T* decreases, in fact, as T* raised to the 16tn power. He therefore suggested that 0.05 be considered the limit for meaningful transport. 64
TABLE 4.1
Calculated Flow Parameters and Measured Sediment Transport Values for Series
3 and Series 4 Runs
Run r* 1 Re* 2 Initial Final Bedload Bedload No. Bedload Bedload Median Maximum Rate Rate Size Size (lO-Skg/h) (10"3kg/h) (mm) (mm)
3-1 0.001 13.8 3.03 0.10 0.45 4.00 3-2 0.002 18.2 1.80 0.06 0.37 2.83 3-3 0.004 24.4 very low very low 3-4 0.008 32.8 3.73 0.07 4.11 8.00 3-5 0.027 56.9 579. 0.82 1.25 8.00
4-1 0.012 36.9 0.37 0.007 0.40 2.00 4-2 0.023 52.5 0.29 0.022 0.54 2.83 4-3 0.045 74.6 11.3 0.062 1.17 8.00 4-4 0.066 93.7 2206.5 1.47 1.15 8.00
1 3 Calculated as RbS/C where Rb and S are from Table 2.2. C = 1.81 kg/m (submerged relative density) times 2.39 x 10"3m (D50 of mixture)
2 1/2 3 Calculated as V*D5Q/I> where V* = (9.81RbS) , D50 = 2.39 x 10' m, and
Rb, S, and v are from Table 2.2. 65
Figure 4.3 shows the decline in sediment transport with time for Series 3 and Series 4 runs. A comparison of initial transport rates in runs 4-3 and 4-4
(see Table 4.1) could lead one to conclude that somewhere within the range
0.045 <, T* < 0.066 sediment really begins to move, but even then the bed was able to stabilize eventually with no measurable change in slope. An investigator observing the bed near the end of run 4-4 who was unfamiliar with the flow history would conclude that a Shields number greater than 0.066 was necessary to entrain significant numbers of particles.
A major finding of this thesis is that the bed surface material becomes more resistant to transport by a means other than increasing particle size, and, hence, particle weight. Figure 4.3 shows the progressive decline of transport through time, which signifies increased resistance to motion. Figure 3.4 shows that this increase in resistance is not due to an increase in the size of stones present on the bed, once the initial coarsening is completed. Figure 4.2 shows that the size of material in transport also shows little change after the unstable, smaller particles are removed. The resistance term in the Shields parameter, therefore, includes more than just particle weight.
There are three key points to Shields' (1936) work. First, by extrapolating back to a zero transport rate, he assumed a threshold which may or may not be satisfactory for a particular problem. The transport rate at T* = 0.06 may be sufficiently low, or extremely high, depending upon the condition of the bed or the nature of the problem. Furthermore, the material that Shields used was described as angular, which would increase further the value of r* . A bed surface composed of subrounded or rounded stones is likely to begin to move at lower shear stresses than his angular material. 66
lOOOOOq Legend
3-1
10000-3 • ^2
CO O • 3-5 1000 • 4-1
A 4r_2_ O) O 4-3.
a V 4-4
o Q. in c a
o L. o <>
I I I I I ll| 1 1 I I M ll| 1 1 I I I I 111 i—i 1111111 1—i i 111111— 1000 10000 100000 10 100 Time (hours)
Figure 4.3 Decline in transport rate through time. 67
Second, since his mixtures were more or less uniform, the D5Q of the bed material was the same as that of the transported material. With mixtures, the appropriate D50 to use to calculate T* is that of the transported material. In the field situation this is usually not available so the subsurface bed material is used, and it is assumed that all sizes move at the highest flows. Table 4.2 shows values of r* based on the DgQ of transported material as well as those based on the largest stone moved. The spread in the Shields parameter is not much less with either of the two transport sizes. Field investigators who do not have access to transport data may find some relief in this observation, but there still remains the problem of an order of magnitude variation in the parameter, regardless of the grain size used. It is not due simply to the wrong grain size being used.
Although the D50 of the material in transport is the appropriate grain size to use to calculate r* , it has not been used in Table 4.1 because few previous studies have done so, therefore it is difficult to compare results if a different D5Q is used, and also because the transport data are subject to undetermined
variations. Table 4.3 shows how sensitive the D5Q is to a single large stone which happens to fall into the trap. This is most noticeable in run 3-4. At the lower transport rates, where a single large stone will affect both the size distribution
and the total amount of material accumulated, there will be variation from one run to the next. The answer to the problem of variability is repetition. For logistical reasons, this issue was not pursued here. Instead, only a single run was conducted at each flow condition to permit the investigation of many different flow conditions. (Shields (1936) does not mention any repetitions of his runs either.)
The third key point is that his experiments were done by sequentially increasing discharge at a given slope (Shields, 1936, p 61), as were the runs in 68
TABLE 4.2
Shields Parameter Based on Various D50 3 Run ?50, Pmax r* 1 r*rmed ^ T*Tmax (mm) (mm) 3-1 0.45 4.00 .001 .006 .001 3-2 0.37 2.83 .002 .015 .002 3-3 very low transport rates 3-4 4.11 8.00 .008 .005 .003 3-5 1.25 8.00 .027 .052 .009
4-1 0.40 2.00 .012 .069 .014 4-2 0.54 2.83 .023 .101 .019 4-3 1.17 8.00 .045 .092 .013 4-4 1.15 8.00 .066 .137 .020
1 From Table 4.1
2 Based on median size of transported material andr from Table 2.2
3 Based on maximum size of transported material andr from Table 2.2 69
TABLE 4.3
The D50 and the of the +4 mm and +5. mm Material Captured in
Series 3 Runs
Run Bag + 4 mm +5.66 mm Total # of Stones D5ft (mm) (10"3kg) (10-8kg) 10"3kg) & size (mm)
3-1 A 0.328 3.03 B 0.379 3.63 C 0.476 3.41 D 0.503 6.96 E 0.469 9.70 F 0.463 2.42
3-2 A 0.316 1.89 B 0.337 0.26 C 0.382 0.32 D 0.403 0.51 E 0.441 1.37 F 0.401 2.51
3-4 A . 4.35 1.60 0.64 3.73 2@ + 5.66 8@ + 4.00 B 0.58 0.43 C 0.85 0.18 0.65 l@ + 4.00 D 4.20 0.40 0.29 1.25 l@ + 5.66 2@ + 4.00 E 0.72 0.11 0.51 l@ + 4.00 F 4.38 1.48 2.18 8@ + 4.00
3-5 A 3.34 174 45.1 579 B 1.06 2.09 516 4@ + 4.00 C 1.02 6.15 2.81 416 2@ + 5.66 ll@ + 4.00 D 1.14 14.2 1.81 446 2@ + 5.66 E 1.09 8.94 457 16@ + 4.00 F 1.13 15.2 0.72 512 l@ + 5.66 G 1.18 21.3 4.78 889 2@ + 5.66 H 1.39 3.35 112 70 the present study. Neither Johnson's (1943) summary of flume studies nor
Shields' (1936) paper mention sediment feed in the experimental set-up that
Shields used. Since he was investigating incipient motion of bed material, it is possible that Shields did not use a sediment feed system or a sediment recirculating system. On the other hand, if he did not, it is not clear how he could have obtained equilibrium conditions. Although Shields (1936) does not provide much information about his experimental procedures, he does state that for each new hydraulic condition, "stability conditions were waited for." (Shields, 1936, p
61) prior to making measurements. Since he waited for stability conditions, he automatically included the effect of any bed structure. In fact, his procedures would develop extensive structure. The particle resistance to motion he was measuring was due not only to particle weight, which he recognized, but also to particle arrangement. As the size of material increases, the potential for structural features to develop increases because larger material may be less well sorted than finer material. In natural sediments, this is almost always the case.
Also, sands tend to be more equant than larger stones, hence structure is less likely to develop as extensively. The minimum T* that Shields measured, 0.03, occurred for sands. This does not mean that there was no structure present on those beds, only that it was less developed than on beds with coarser material.
The increase in T* for material finer than sand is not due to structure.
Fenton and Abbott (1976) also investigated variations in T* . They explored the effect of particle protrusion on entrainment. By moving a loose particle to different elevations relative to a fixed bed, they were able to obtain values of r* from 0.01 to 0.35. Their explanation of the dip in the Shields curve was that sand size particles tend to be equant with some surface particles sitting atop the rest, causing the uppermost particles on the bed to protrude more, and therefore be entrained more easily. With coarser particles, the chance of a 71 particle being less exposed increases because, they reason, larger particles can be packed down more easily by the investigator in an experimental situation so few particles will be sitting above the rest.
Protrusion may be considered an aspect of bed structure, but there is more to bed stability than just protrusion. The changes in r* in Table 4.2 occurred with no interference with grain placement by the investigator, nor was there much change in the surface texture once a coarse layer had been established. The variations in amount of exposure of grains on the bed between runs was minimal, yet T* varies by over an order of magnitude.
The Shields Parameter Re-Examined
The two variables displayed on a Shields diagram are the ratio of the applied force to the particle weight, called r* (see equation 1.1), along the ordinate, and the ratio of near bed velocity and particle diameter to fluid viscosity, called Re* (see equation 1.3), along the abscissa, as shown in Figure
1.1. Beyond Re* ='70 (Yalin, 1972) or 1000 (Shields, 1936), r* appears to be a D50 constant; T0 increases as increases. Below this value, the stress necessary to entrain particles changes more quickly than D50, which means that something other than particle weight is responsible for the changing resistance. Below Re*
= 2, the increase in particle resistance with decreasing grain size is due to the well known cohesive strength of silts and clays. Between Re* = 10 and 70 (or
1000) the extra strength is proposed here to be due to the increasingly effective development of geometric structure as particle size increases from sands to gravels. The value of r* when Re* becomes irrelevant has not been unequivocally established, as noted in the introduction to this chapter. 72
Consequently, developing geometric structure may affect a greater portion of the
Shields diagram than the region between Re* = 10 and 70.
At very high flows, however, bed structure is destroyed, the surface
becomes finer, and rQ decreases. Figure 4.4 shows a proposed relation between
bed resistance to transport and shear stress. Implicit in Figure 4.4 is time for
structure to develop at a given stress. The sudden decline is based on
observations of this study, run 1-4, and Wilcock and Southard, (1989). Data from
both studies suggest that whenr* based on the D5Q of the subsurface material
is greater than about 0.12, bed structure will not develop.
Data from Table 4.2 can be plotted to show graphically the development of
geometric structure, r* based on the subsurface D50, T*sub» assumes no
geometric structure, r* based on the D5Q of the material in motion, T*trans >
gives the actual critical Shields number. A plot ofr*trans against ^*sub
would plot as a 1:1 line if bed resistance were based on weight only. Figure 4.5
shows that the bed is able to withstand greater stress than that based on weight
alone because all points are above the 1:1 line, except run 3-4. Table 4.3 lists the
total amounts and number of large stones present for this run, as well as the rest
of Series 3 and all of Series 4. The small amount of material trapped in run 3-4
combined with the few large stones transported explains the anomalous behavior
of this run. The excess bed strength reflected in Figure 4.5 may be seen more
T* b clearly if the textural strength component is removed by subtracting su from
T and T* *trans plotting the difference against r*sut). When trans equals r*sub ' tne result is zero and the point will fall on the abscissa. Figure 4.6 is in
this form, with run 3-4 removed. Error bars are based on error estimates of
depth and slope used to calculate shear stress, reported in Table 2.2. 73
Figure 4.4 Proposed model of bed structure development. Figure 4.5 Shields parameter based on material moved compared to Shields parameter based on subsurface D5Q. Error bars show maximum range. 75
In
— — — 0.0001 | i 1—i—i i i i 11 1 1—i—i i i i 11 i 1—i i i i i 11 1 1—i i i i i 11 0.0001 0.001 0.01 0.1 1 T*sub
Figure 4.6 Difference in Shields parameters of Figure 4.5 (which is the excess bed strength due to geometric structure after the strength due to particle weight has been removed) compared to Shields parameter based on subsurface D50. Error bars show maximum range. 76
The values ofr* in Table 4.2 are based on the presence of material in the sediment trap which signifies particle movement. At any given depth, the initial transport rate is relatively high, which shows thatr^ , the bed shear stress, is
greater than TQ . Final bedload rate is relatively low, showing that TB is almost
SOT equal T0 . Since depth, slope, and roughness are constant, rb is constant Q must be increasing during a run as structure develops. If D50 is not changing
and T0 is increasing, then T* is also increasing.
The increasing values of T* with subsequent increasing flows show the sequential development of bed structure. If the surface texture is not changing much (see Figure 3.4), then the increased strength must come from geometric structure.
One approach to quantifying the effect of bed structure is to consider the minimum values of r* to represent a bed with no geometric structure. Here only particle weight resists movement. The fully exposed grain in the study by
Fenton and Abbott (1976) is such a case. Any increase in this value must be due to the development of geometric structure. The range ofr* in Table 4.1 covers most of the range of published values for T* , but these values may be extended in both directions by more carefully controlled experiments. The published values of T* may be considered to be determined almost totally by geometric structure.
This suggests that different amounts of bed structure were present in the various published studies. This can explain the scatter found when several studies are plotted on a Shields diagram. 77
Bed Structure and Manning's n
In Table 2.2, the values of Manning's n based on velocity, depth and slope increased for a given depth of water as the slope increased in 11 cases out of 12.
In the one case when it did not, Run 3-2, the value is 3% larger than Run 2-1, which had the steeper slope. A comparison of the 0.02 m deep flows shows that a tenfold increase in slope (Runs 3-3 and 4-3) caused a 250% increase in n. A fivefold increase (Runs 2-2 and 3-3) caused a 180% increase, and a doubling of the slope (Runs 2-2 and 4-3) caused a 138% increase. This depth is the most extreme case because, as shown by Nowell and Church (1979), maximum wake interaction, and therefore flow resistance, occurs when flow depth is about 3 times the height of the roughness elements, taken here as 8 mm.
Two means of increasing flow resistance are to increase grain diameter or to change the arrangement of the largest elements. Unfortunately the experimental design of Series 3 and Series 4 did not permit sampling the bed surface to assess the surface texture. The behavior of the bed surface during
Series 1 and the size of material in transport in Series 4 both suggest that little change in surface texture occurred. Transport rates for Series 3 at the 0.02 m deep flow were too low to permit analysis.
By itself, the increase in Manning's n in this study does not demand the development of geometric structure. The effects of the errors associated with the depth and slope measurements used to calculate n are less than with the shear stress calculations because each term is raised to a fractional power. Manning's n, however, is an index for the interaction between the flow accelerated by gravity and slowed by friction. The relative roughness of the bed material is only one aspect. In terms of bed friction, the arrangement of structural elements may 78 be the most important feature (Morris, 1955; Nowell and Church, 1979). These results, although not definitive, are consistent with the change toward increased roughness of the bed.
Summary
The most significant results of the present work are contained in the preceding three sections. First, Figure 4.3 shows that sediment transport decreases with time if no sediment is fed into the upstream reach. Figures 3.4 and 4.2 show that after an initial coarsening occurs, little change in texture develops. Figures 3.2 and 3.3 suggest that the increase in bed stability develops because geometric structure occurs over the bed surface. These results are purely qualitative and require only that the slope and water depth remain constant.
Second, calculations of Shields parameters based on material in transport show that the force necessary to entrain material increases as the bed adjusts to a given flow condition, or series of flows. Although the grain diameter increases slightly as the flow increases, it does not completely account for the increase in shear stress necessary to entrain similar size material. The extra particle resistance must come from some other characteristic of the bed surface, namely geometric structure. By removing the effect of increasing grain size, the amount of extra resistance provided by the bed can be seen. Figures 4.4, 4.5, and 4.6 develop the evidence which shows the importance of geometric structure in bed stability, based on the material in transport.
Third, supporting evidence is provided by the behavior of Manning's n, which increases as slope increases, suggesting that bed friction may be increasing. 79
Although the summary character of Manning's n precludes unequivocal identification of the development of geometric structure as the cause, it is consistent with the preceding results. These three lines of evidence show that
geometric structure plays a major role in sediment transport in mixed gravel
beds.
A serious consequence of this observation is that paleohydrology studies
which contain a Shields approach to water depth based on grain size may be
suspect unless the degree of bed structure can be determined. Another
implication is that under certain conditions the bed is able to withstand much
higher shear stresses with much less erosion. The justification for this statement
is the increased values of r* over time. Slowly increasing flows allow the
particles to find the most stable positions within the shortest distances. This
permits maximum development of geometric structure. Anyone concerned with
controlling bed scour may find this useful. The effectiveness of rip-rap can be
increased if it is allowed sufficient time to adjust to flow conditions. The length of
time necessary for the bed to adjust to a given flow condition can be inferred from
the time needed for the. bed to stabilize in the flume studies through scaling. 80
CHAPTER FIVE
LAG TIME
Since the Froude numbers are the same in both the model and the prototype, the relation between model time and prototype time may be found by substituting length per time, L/T, for velocity in the Froude ratio, Fr.
2 Fr = Vm /(Lmg) = Vp2/(Lpg). (5.1)
Since in general
2 2 2 Vx = Lx /Tx (5.2) substituting (6.3) into (6.2) gives
2 2 Lm/Tm = Lp/Tp (5.3) which can be rearranged as
2 Tm/Tp = (Lm/Lp)l/ . (5.4)
where Vm, Vp = model and prototype velocity
Lm, Lp = model and prototype length
Tm, Tp = model and prototype time
g = acceleration due to gravity
The length of interest here is water depth. Average model velocity at maximum water depth of 0.08 m was 0.55 meters per second. Gravel bedded rivers with slopes of 1% typically have velocities between 1.5 and 2 meters per second. The model velocity therefore represents rivers with three to four times that velocity.
This means model depth was 1/9 to 1/16 river depth. Thus, 0.08 m deep flows in the model were equal to 0.72 to 1.28 m deep flows in rivers, and the equivalent median sediment diameter was 21 to 38 mm. Substituting length ratios of 1/9 or
1/16 into equation (5.4) gives a time ratio of 1/3 or 1/4. Events which occur in the model will take 3 to 4 times longer in the prototype. 81
Figure 4.3 shows the amount of sediment transported and the elapsed time, which are also listed in Table 5.1 and Table 5.2. Sediment transport reached very low levels after 2 to 8 days, depending upon flow conditions. With a scaling factor of 4, the prototype bed would take up to 32 days to stabilize to a flow with a shear stress necessary to entrain the median size particles. Parker (see Parker,
1980) suggests that the shear stresses of the mean annual flood are only 2 to 3 times this level.
Gomez (1983) measured the time necessary for the reestablishment of the coarse surface layer following several flood events in a small stream in southern
England. Figure 5.1 is a plot of his data. Under field conditions, the surface layer took 1 to 3 months to develop. Presumably the flow was not steady during the entire period. Hence, the 2 to 4 weeks obtained from this study do not contradict the field results.
If the bed surface takes a month or more to adjust to flow conditions, then it is rarely in equilibrium with the flow. As a result of the lag time, there will be moments during the falling stages of the hydrograph when the bed is in equilibrium, but it will be less stable than it could be. For maximum stability, a sustained flow lasting a month or more is probably necessary.
The shape of the curves in Figure 4.3 suggests a model for the development of bed stability. One model is that sediment transport is a function of the amount of material available. Structure develops upstream and then proceeds downstream as more of the bed becomes stable and less material is supplied downstream. This implies an exponentially decreasing relation between sediment transport and the amount of material available for transport. It means that the time for structure to develop is dependent upon the length of the flume. 82
TABLE 5.1
Variations in Transport Rates through Time for Series 3 Runs
Run Water Time Rate of Total No. Depth Interval Accumulation Time (10-2m) (hours) (KHkg/m-h) (hours)
3-1 0.5 1 6.06 1 2 3.64 3 4 1.90 7 10.5 1.32 17.5 24 0.88 41.5 24 0.20 65.5
3-2 1 1 3.78 1 2 0.26 3 4 0.16 7 8 0.12 15 16 0.18 31 42 0.12 73
3-4 4 1 7.46 1 2 0.44 3 4 0.32 7 8 0.32 15 16 0.06 31 31 0.14 63
3-5 8 1 1158.8 1 2 516.4 3 4 208.2 7 8 111.4 15 16 57.2 31 32 32.0 63 128 14.0 191 120 1.88 311 360 1.64 671
* TABLE 5.2
Variations in Transport Rates through Time for Series 4 Runs
Run Water Time Rate of Total No. Depth Interval Accumulation Time 3 (10_2m) (hours) (10" kg/m-h) (hours)
4-1 ,0.5 1 0.732 1 2 0.066 3 4 0.054 7 8 0.038 15 16 0.020 31 32 0.014 63
4-2 1 1 0.580 1 2 0.234 3 4 0.520 7 8 0.090 15 16 0.090 31 63.5 0.044 94.5
4-3 2 1 22.70 1 2 3.82 3 4 1.63 7 8 0.818 15 16 0.666 31 32 0.392 63 64 0.124 127
4-4 3 1 4413.0 1 2 1995.6 3 4 384.6 7 8 163.2 15 16 101.4 31 32 43.4 63 64 17.6 127 120 7.00 247 144 2.94 391 84
-5 1 1 1 1 1 1 1 1 1 1 • 1 • i i i i • -4 - Site 3 1 -3 - ~^s>^>-°-° Ds50 "~ -2
f< . ' ' • " P -t—, si-
... 1 i l i 1 i i i i i 1 ..... J FMAMJ JASONDJ FMAMJ J
J 1978 Q=0.068 j I 1979 Q= 0.039 - Q= 0.056
Figure 5.1 Coarsening of the surface median sizes following disturbance by three major storms, with discharges of 0.068, 0.039, and 0.056 m3/s. Open circles are surface samples; closed circles are subsurface samples, Sheepstor Beck, southern England. (After Church, 1987; data from Gomez, 1983) 85
Table 5.3 lists the r2 values for each of the data sets of the Series 3 and 4 studies which have been fitted to a linearized exponential relation and a power relation. The better fit of the power relation is seen in the higher r2 values. The better fit of the power relation means that sediment transport is a function of time only. An important implication of this is that the amount of material in the flume is not important. The results would be the same regardless of the length of the flume, assuming steady, uniform flow. For this to be true, structure must develop simultaneously on the surface over all parts of the flume, rather than progress from one end of the flume to the other.
The coefficients and exponents listed in Table 5.3 may be used to calculate the rate of accumulation at any time interval by using the power relation
y = a(T)b (5.5) where y is the rate of accumulation in kg/mh, and T is time in hours. To calculate the length of time necessary to reduce the rate of accumulation to a given fraction, say 1% of the initial value, a ratio of equations of the form (5.5) gives
T = (0.01)(1/b) (5.6)
where T = T2/Tx, T2 > Th Tx = 1 hour, and b may be between -0.7 and -1.2.
If b = -0.7, then it would take 30 days in the flume to reduce the rate to 1% of the initial rate, or 3 to 4 months in the field. If b = 1.2, then it would take 2 days in the flume, or 8 days in the field.
The sensitivity of time to small changes in the exponent in equation (5.6) precludes a more precise estimation of the time necessary for the development of bed structure. Since the probability of any stone on the bed to move is initially random, the development of structure will not proceed exactly the same way each time. As noted previously, the variability in the points in Figure 4.3 is not 86
TABLE 5.3
r2 for Linearized Exponential and Power Functions of Sediment- Transport
through Time
Run Exp. Power Power Power No. r2^ r2 coefficient exponent
3-1 0.90 0.88 3.61 -0.70 3-2 0.23 0.64 0.62 -0.67 3-3 0.38 0.38 0.71 -0.71
3 4 0.64 0.61 279 -0.69 " 92 3-4 0.80 0.97 794 -1.03
4-1 0.53 0.87 0.18 -0.86 4-2 0.72 0.79 0.32 -0.57 4-3 0.66 0.96 7.40 -0.96 4-4 0.71 0.99 2629 -1.21 based on 191 total hours based on 671 total hours 87
known, and so confidence limits cannot be assigned to the exponents in Table 5.3.
At low transport rates, a single large stone can affect the results. Repeated runs
at low rates may show different slopes when plotted on Figure 4.3. At higher
rates, the large number of stones moving will mask the behavior of an individual
particle so variation should be much less. The consistent trend in the data of
Figure 4.3, however, suggests that the range of values for the exponents in Table
5.3 may be reasonable approximations.
If bed structure is such a major factor in particle movement, why have
numerous investigators measured approximately the same values for r* (within
one and a half orders of magnitude)? There are two aspects to the answer. First,
for flume studies, most investigators follow similar procedures which create
similar amounts of structure. They wait for "equilibrium conditions" before
beginning their measurements. Studies which explicitly avoid waiting for
equilibrium, such as Paintal (1971) and this study, report a much larger spread in
T* . Second, for field studies, natural conditions may constrain both the range of
structural development and the accuracy of the measurements, particularly
bedload measurements. Constraints on the development of structure are
suggested in the next chapter. 88
CHAPTER SIX
LIMITS TO COARSENING
Is there a limit to the amount of coarsening of the bed surface, or will finer material continually be removed as shear stress increases until a uniform bee] of the coarsest material exists? Engineers involved in the prediction of the depth of scour following a change in flow conditions have failed to assess adequately the interaction between the flow and the bed material. A typical example in the literature is Livesey's report of the Fort Randall Dam (Livesey, 1965) where scour was 3.5 ft instead of the predicted 15 ft. Livesey attributes the error to two causes: initial samples to determine grain size of the bed material were not representative, and the assumption that large stones must form a contiguous layer to prevent erosion was false. The observations presented in this thesis permit a speculation on this.
According to Morris (1955), at high concentrations of the dominant roughness elements the flow appears to skim over the tops of the obstructions, and at low concentrations the flow passes easily around the particles. In both cases flow resistance is minimal. At some intermediate concentration, interference of the flow by obstructing particles reaches a maximum, and flow resistance is maximized. The basis of Morris' concept is wake interaction. At low concentrations, wakes generated by the roughness elements dissipate before encountering other elements. At high concentrations the elements are too close to permit the full development of a wake. Morris predicted that maximum resistance from a single size roughness element would occur when the downstream
spacing equaled the resistance height. If cubic objects are used, maximum
resistance should occur when they occupy 50% of the surface area. 89
Suppose that most particles on the bed move when maximum wake generation occurs. This will create the most extreme velocity fluctuations near the bed and should give the greatest transient lift and drag forces on individual particles. If this occurs when the roughness elements occupy 50% of the area, then maximum entrainment should occur at this density of roughness elements.
It is assumed in this study that a shearing flow will cause the development of both textural and geometric structure so long as most particles are not moving.
As textural structure develops, the concentration of coarse particles increases, which increases the size of eddies generated near the bed. These in turn increase the probability of particle entrainment, including the coarse particles. If the largest particles are able to withstand the maximum stress most of the time, they will continue to be concentrated on the surface. When their density is high enough, the flow will skim over the bed which is now stable.
On the other hand, if the largest particles are not able to withstand the maximum shear stress most of the time, then the bed will not stabilize beyond some density below 50%. Small flows on an unstable bed will initially cause textural structure to develop (see Figure 3.4), but since greater concentrations of coarse particles will increase the velocity fluctuations at the bed, and, hence, entrainment, geometric structure must occur to stabilize the bed. Since there is no change in the number of coarse particles with the development of geometric structure, wake generation at the bed is not increased. If this reasoning is correct, there should be some maximum density of coarse material, which is below
50% on a stable bed if all particles can be moved by the flow. Once maximum structure has developed, any increase in bed stress will destroy the surface structure. 90
Nowell and Church (1979) used a flume with a 0.072 m deep flow and a depth-limited boundary layer to approximate a natural channel. They investigated the effect of the concentration of uniform size rectangular roughness elements fixed to the bed on flow resistance. Their data show that maximum flow resistance occurred when the bed roughness elements occupied approximately 8%
- 12% of the area of the bed. The data from the Nowell and Church (1979) study are based on a single size of roughness elements on a fixed bed, and on a single flow depth and slope. The maximum concentration they used was 12%.
The relations between flow conditions, energy dissipation, and roughness spacing are not explored in any detail in this study. Rather, the results of particle concentrations are presented for comparison with the proceeding theoretical and experimental work to stimulate further research and to suggest a possible approach. The numbers, however, may be useful to anyone involved in calculating scour depths in mixed size gravels.
A two step procedure was used. First, an empirical relation was developed between stone weight and stone area for particles retained on each of seven sieves
(at 1/2 phi spacing) between 0.71 mm and 5.66 mm. All particles passed through an 8 mm sieve. Second, the surface texture was sampled after each of several runs with different water depths. From the weight of the individual size fractions the area occupied by that fraction could be calculated. The actual procedure used is described in the Appendix.
With the area per weight information it is possible to calculate the area occupied by certain size classes if the weight is known. Table 6.1 contains the area occupied by the largest class and its percentage of the area sampled for each of several surface samples from Series One. Two percentage figures are given in 91
TABLE 6.1
Weight and Percentage of the Total Area of the +5.6 mm Size Class for Several
Surface Samples From Series One
Sample Weight Area Percent
Name (10"3kg) (10"3m2) (%)
A2cmSURF 62.1 7.268 7-10
A3cmSURF 68.3 7.994 8-10
A5cmSURF 45.2 5.290 5-6
THALWEG 1 109.0 12.757 13-14 92 the table for each sample. The smaller one is based on the area of the sampler
and therefore included voids between stones. The larger one is based on the sum of the areas occupied by each of the size classes, which assumes no voids. Since the sampling technique did not recover every stone exposed on the surface, the area based on the area of the sampler is over estimated. Therefore, the area based on no voids probably is more accurate, and the larger percent in Table 6.1 probably is a truer estimate. These results suggest that about 15% may be a limiting value to the concentration of the coarsest fraction on the bed, if all sizes are mobile. The maximum depth of scour of the thalweg in run 1-5 was 22 mm.
If none of the largest stones had been removed, this should have increased the concentration of the +5.6 mm size class from 9% of the total area, based on a conversion of the bulk sample to a surface sample, to 35%. The actual increase was to 14% in the thalweg.
There are several examples, however, which show that in some cases other factors are involved because the concentration of the largest stones is greater than
15% of the surface. In the field, such conditions may be found at lake outlets and along cobble beaches. In both cases there may be almost no finer material exposed. Two flume studies suggest possible explanations for these situations.
Work on degrading flows by Harrison (1950) shows concentrations of non- moving particles to be 17.1%, 42.7%, and 36.0% for his three experiments. Only the first, however, approached uniform flow. In the second case, the bottom slope was over three times steeper than the water slope, and in the third case, the
water slope was twice as steep as the bed slope.
An additional complication in Harrison's study is that the flume he used
was non-tilting. Bed slope was achieved by the water eroding bed material at the
downstream end and then gradually forming a uniform slope as the knickpoint 93 migrated upstream. Harrison had anticipated that degradation would occur until the surface was covered with non-moving stones. During all degrading runs, he found "non-moving" stones in the sediment traps. Harrison offers two explanations.
...There was a possibility that the bed shear was large enough locally at the downstream end of the flume to move the previously "non-moving" particles, and this may have happened to some degree; but observations seemed to point to a general movement of the larger particles along the entire length of the bed. This is evidenced by the fact that there was very definitely a movement of large particles from the upstream end of the bed which caused excessive degradation there. There must be some type of movement in a degrading bed characteristic of particles which would be non-moving if the bed were in equilibrium. Let us consider what must happen to a "non-moving" particle in a bed that is degrading, and perhaps some conclusions can be inferred as to the nature -of this movement due to degradation. Consider a particle which the flow cannot move on a bed that is not degrading. This we call a "non-moving" particle. This particle covers a certain area of the bed, and no transport can occur from this covered area. If now the bed begins to degrade, the bed material on all sides of the "non- moving" particle will scour away. It is certain that the non-moving particle will not remain at its original elevation supported by the prism of bed material beneath the area it covers; it will move downward to the general elevation of the surrounding bed. Now this particle cannot move vertically downward because of the bed material underneath which the particle itself had protected from the flow; the "non-moving" particle must move along the bed as it moves downward. Since the forces acting on this particle are in the direction of the flow, it is assumed that the motion along the bed is downstream. Thus it is reasoned there must be a resultant motion downstream and downward among the "non-moving" particles in a degrading bed. (Harrison, 1950, pp 159-160)
Harrison himself considers his experimental set-up atypical of natural rivers.
Although he does not explain why "...it is certain that a 'non-moving' particle...will move downward to the general elevation of the surrounding bed.," the results of 39 years of additional research suggest increased particle protrusion
(Fenton and Abbott, 1976) is the reason.
One of Harrison's undocumented conclusions is that the more nearly mobile the largest stones on a bed are, the more complete the coverage necessary to prevent degradation. Thus one explanation for a uniformly coarse bed is degradation in material with large stones only slightly larger than that which the 94 stream can entrain. This may occur at some lake outlets. It does not explain, however, uniformly coarse beach cobbles which are not degrading.
The second flume study which offers another explanation for this feature is the present study. During run 1-5, the degrading run, a large scour hole developed just downstream of the fixed bed. Off to one side, however, a portion of the bed remained in place. After the run was stopped, it was discovered that this area was uniformly coarse and very tightly packed. Figure 6.1 compares this uniformly coarse area with the typical bed surface after a run. There was little degradation of the surface, and the largest stones were removed from other areas of the bed. A possible cause is that due to turbulence and/or intense turbulent fluctuations, smaller stones were removed and the larger ones vibrated into a tightly packed configuration. Since only one small area of the bed was affected, an area 0.15 m by 0.20 m, and since it occurred only once in the course of the experiments, several factors may be involved. Abrasion of the smaller particles, however, is not likely. Whether the same factors are responsible for coarse cobble beaches (Bishop and Hughes, 1989) is not known, but vibration and intense turbulent fluctuations are present on cobble beaches. Under many flume conditions, however, there appears to be an upper limit to the amount of coarsening that occurs. If all sizes are mobile, that limit may be about 15%. 95
Figure 6.1 Comparison of typical surface (top, after 0.02 m flow depth, Run 1-2) and very well sorted surface (bottom, after Run 1-5). 96
CHAPTER SEVEN
CONCLUSIONS
One objective of this study has been to determine how particles on the surface of a stream bed respond to a shearing flow. Water flowing over a loose boundary affects the bed material by vertical and horizontal segregation and by particle alignment. Vertical sorting, or surface coarsening, called here textural structure, does not completely describe the stabilization processes. Both this study, with no sediment feed, and work by Wilcock and Southard (1989), with sediment feed, show that the median diameter of the surface material may hardly change in response to increases in the applied shear stress of up to 50%, after initial surface coarsening has occurred. Figures 3.4 and 4.2 show both this initial coarsening and then the very small change in D5Q with increasing discharges.
Yet, without sediment feed, the bed surface continues to adjust so that further particle entrainment declines dramatically, as shown in Figure 4.3. The ability of the bed surface to resist erosion has been shown in this study to involve more than just surface coarsening, or particle texture. The extra strength comes from particle alignment and arrangement, here termed geometric structure and shown in Figures 3.2 and 3.4.
Another approach which shows the effect of geometric structure is to use the Shields parameter as a measure of particle resistance to entrainment. Using the D50 of material in transport shows the force needed to entrain the particles in motion. As the bed surface becomes more resistant to erosion, the force needed to move particles must increase. Although some strength comes from increasing particle diameter, size alone does not totally explain the additional resistance.
This is shown in Figures 4.4, 4.5, and 4.6. 97
It appears that Manning's n may be influenced by the arrangement of particles on the bed surface. For a given depth, an increase in slope causes the bed to develop more structure to resist erosion. As the geometric structure becomes more clearly organized, flow resistance increases, as shown by an increase in Manning's n (see Table 2.2).
Previous studies of bed stability and particle entrainments, both field and flume, have measured the combined effect of textural and geometric structure. A major difficulty with geometric structure is the lack of a suitable direct measure of its extent, due to its complexity (see Figure 3.2). This study has not been able to solve this problem.
The general concept of geometric structure , however, may be visualized in terms of a group of uniform bricks. If these are arranged on a surface in a haphazard fashion, the ability of any particular brick to resist a shear force may be low. On the other hand, if the bricks are all arranged in certain tight geometric patterns, the stability of each brick will be increased beyond that of any isolated brick. The size of each brick has not changed; only their arrangement has changed. Measuring the DgQ of the bricks in each arrangement would not describe adequately its stability. Here stability is mainly a function of geometry.
A measure that precisely describes the ability of each arrangement of bricks to resist a shear force is the shear stress necessary to move the bricks. If different groups of bricks, each of a different but uniform size, had exactly the same geometric arrangement, then the shear stress necessary to entrain a brick would be a function of size only. Natural sediments are rarely brick-like, nor are they uniformly arranged. Consequently, the shear stress necessary to entrain particles is a measure of both the resistance due to texture and the resistance due 98 to geometry. The Shields parameter really measures both textural and geometric structure.
One consequence of the variation in bed stability is that there is more than an order of magnitude of spread in the values reported for the Shields parameter for initiation of motion. Values as low as 0.001 can be obtained if sufficiently long sampling times are used or if sufficiently exposed particles are studied.
Gradual increases in discharge which allow the bed to reach equilibrium, or stability, with the flow give values of 0.065.
Another consequence of bed structure is the decrease in sediment transport through time if no sediment is supplied beyond that which is eroded from the upstream reach. A second objective of this study has been to quantify the lag time between a given flow condition and the bed response. The geometric and dynamic scaling ratios used here suggest that a typical bed may take from a week to a month or more to fully stabilize under ideal conditions of steady, uniform flow, depending upon the strength of the current. This implies that natural beds never adjust to the highest flows, and indeed only rarely may be in equilibrium with the flow. Correlations between bed surface texture and flow conditions are thus complex, and must include the effects of previous events. More importantly, correlations between sediment transport and flow parameters are poor in the field because equilibrium conditions rarely exist.
Since the stability of an individual grain is variable, sediment entrainment is a statistical event. The development of structure on a surface is the result of many individual particle movements. These have been averaged in the present study by capturing all particles transported out of the flume for a given time interval. Exactly when a particular particle moved or how far it travelled was 99 not determined. Variability between individual particles has been ignored in order that the overall development of structure might be studied. It is the effect • of the sum of individual movements which is important in the study of sediment transport with natural sediments.
Not only is the movement of an individual particle a statistical phenomenon, but the consequent arrangements of groups of particles are variable also. That is, the degree of structure present on two surfaces with the same initial arrangement and subject to the same flow conditions may vary.
Repetitions of the experiments described in this study should be done to determine the variability between runs. The lack of replications in this study limits the generalizations which can be made.
Decrease in rate of sediment accumulation appears to be a power function of time rather than an exponential function. This means that the phenomenon is independent of the amount of material available for transport, and is dependent only on time. If the decrease is due to the development of structure, as proposed here, then structure must evolve simultaneously over all parts of the bed.
A related aspect of Harrison's work (1950) which was not explored here was the effect of different size distributions, especially the effect of stones not capable of being transported by the flow. His work suggests that as the proportion of non-movable stones increases, the development of bed structure is accelerated.
Bed structure is not only created by flowing water, but it may also be limited by the fluid forces. Beyond some density, the roughness elements, be they clusters or simply larger stones, may cause the flow to skim over the tops of the 100 elements rather than to flow around them. If this occurs, then energy dissipation will shift from near the bed to the flow above the bed, which may decrease the entraining forces applied to the particles on the bed. This then should limit the degree of structure that can develop on the bed. Work by Nowell and Church
(1979) suggests that this occurs when fixed geometric elements, such as cubes glued to the bottom of the flume, reach a concentration between 8% and 12% of the total surface area. They, however, investigated only a single discharge, using a constant slope and depth, and a maximum density of 12%.
A third objective of this study has been to explore the limits to coarsening of a mixed size mobile boundary. Results of this study, using multiple depths, slopes, and discharges, and self adjusting roughness elements, show values between 10% and 15%, the latter occurring at the higher flows. Very high flows destroy bed structure. Work by Wilcock and Southard (1989) and from this study shows that structure does not develop if the dimensionless shear stress, r*, exceeds 0.12.
The most important contribution of this study is the identification of the significance of geometric structure in the conventional Shields approach. All sediment transport models using empirical results have implicitly included the effects of bed structure. If the geometric structure of a bed differs from the bed conditions on which the model is based, the results will not follow the model predictions even though the textures may be the same.
Despite the consistent trend in results, this work is exploratory. It has described novel and unexpected results, but the measurements are not definitive.
There remains sufficient scope to quantify the effects of geometric structure. 101
Additional practical contributions include the following:
1) an estimation of the maximum r* a bed can withstand. Under certain conditions, the bed was able to resist movement when T* = 0.065. More careful experiments may be able to push this value higher,
2) an estimation ofr* that will prevent the formation of bed structure.
Both this study and Wilcock and Southard (1989) show that the surface structure is destroyed when T* =0.12. What happens between 0.065 and 0.12 is not known, but presumably these two values converge, and
3) the length of time necessary for a bed to adjust to a higher flow can be up to a month or more.
Implications of the results of this study include the following:
1) previous studies which measured entrainment conditions on movable beds have included the effect of geometric structure as well as particle size,
2) the bed surface may rarely be in equilibrium with the flow, and hence correlations between flow conditions and bed texture or sediment response will be poor. Given the variability of particle movement and structure development, a statistical approach to sediment transport seems more profitable than a deterministic approach,
3) bed stability can be increased if the bed is given time to adjust to lower flows before the discharge is increased. Two examples where this may be useful are in the timing of rip-rap emplacement to control scour, and the timing and size of reservoir releases, and
4) sedimentological studies must include an assessment of geometric structure before paleo flow conditions based on the sediments can be determined. 102
The conventional approach to sediment transport problems involves the use of values as high as 0.06 for the Shields parameter. This study has shown that such a value is reasonable if sufficient time at lower flows is available. Smaller values are more conservative, and armoring will develop at values less than 0.06.
Much above this value, however, the bed surface will be mobilized completely. If incoming sediment supply is not high, scouring will occur. Also, Shields' value of
0.06 included the effects of structural strengthening, which took time to develop, and it was based on the D50 of the mixture, not the surface material.
Furthermore, the value of 0.06 does not guarantee that no sediment will move. It means simply that eventually the transport rate will be very small. 103
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APPENDIX
DETERMINATION OF AREA BY SIZE FRACTION BASED ON WEIGHT BY
SIZE FRACTION
The relation between stone weights and stone areas is based on ten sets of measurements of 100 stones, or 1000 stones for each of the seven size classes, except the 5.66 mm class which had eight sets. For each set the weight was measured to the nearest 10"^ gram. The area was measured by scanning each set of 100 grains with a video camera. A comparison of the number of pixels occupied by the stones with the number of pixels occupied by objects of a similar size but known area gave the areas of each of the sets. Known areas were provided by thirty steel ball bearings of each of the following sizes: 1/32, 1/16,
3/32, 1/8, 3/16, 7/32, and 1/4 of an inch (0.79 mm to 6.35 mm).
Two possible sources of error in measuring area with a video camera are errors due to the system and errors due to edge pixels. The system error was assessed by measuring known areas several times. Sets of five values for each of the camera settings were always within 1%. Edge pixel error was the result of error in determining when a pixel on the edge of the shape should be counted as part of the object and when it should not. By using a set threshold of brightness for pixels to be counted as part of the area of the object, this error was consistent, that is, a bias. This bias, however, was a function of size since the ratio of edge pixels to interior pixels increases as the size decreases. A plot of the number of pixels against area showed that the area was over estimated by 5% at the 1/32" size and decreased to near zero above the 3/32" size. A curve was fitted to the data and all measurements corrected to correspond with the calibrated areas.
The remaining pixel error was due to the difference in shape between the stones and the ball bearings, since any shape that was not circular would have more 109 edge pixels than the ball bearings. The stones, however, were sorted by sieve size, or intermediate diameter, so any variation in shape from a circle caused an increase in total area as well as an increase in edge pixels. Several calculations of various shapes with a fixed intermediate or B axis showed that the ratio of edge pixels to total pixels remained nearly constant. The final results are presented in
Table A. 1.
It was assumed that the grains were oriented with their smallest axis, the
C axis, in the vertical direction. If this is the case, then the area is a function of the product of the A and B axes. To see if there was much variation in shape between the different size classes, the shape was assumed to be rectangular, and the length of the A axis computed by dividing the area by sieve size. A comparison of the ratios of A to B then revealed any anomalies. Table A. 2 shows the results of these calculations as well as the area occupied by a gram of each of the size classes. no
TABLE A.l
Weights and Areas of Various Size Classes
Size Wt/100 stones Std. Dev. Area/stone Std. Dev.
(mm) (10-3kg) (10-3kg) (10"6m2) (mm2)
+ 0.71 0.1204 0.0175 1.09 0.12
+ 1.00 0.3825 0.0432 2.36 0.19
+ 1.41 1.2327 0.4470 5.31 1.40
+ 2.00 3.0475 0.3941 9.41 1.00
+ 2.83 9.1584 1.4708 19.02 2.67
+ 4.00 19.8786 3.1907 30.77 3.04
+ 5.66 40.2050 1.6588 47.04 1.97 Ill
TABLE A. 2
Ratio of A and B Axes and Area Occupied per Kilogram
B axis A axis A/B Area/Wt Std. Dev.
2 3 6 2 (r3 (mm) (mm) (I0_6m /10- kg)(l0- m /1 kg)
0.71 1.51 2.16 906.94 38.02
1.00 2.36 2.36 618.61 24.24
1.41 3.77 2.67 441.32 32.65
2.00 4.57 2.29 300.90 14.74
2.83 6.72 2.37 209.37 9.52
4.00 7.69 1.92 156.08 3.34
5.66 8.30 1.46 117.04 3.34