Scour and Fill in 

Sand-Bed Streams

GEOLOGICAL SURVEY PROFESSIONAL PAPER 462-D
Scour and Fill in 

Sand-Bed Streams

By BRUCE R. COLBY

SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 462-D

UNITED STATES GOVERNMENT PRINTING OFFICE. WASHINGTON : 1964
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

First printing 1964 

Second printing 1968
CONTENTS

Definitions___ _ --_-----_--_-________________________

Abstract. __--------_-______-______..._____________

Introduction-----------_______________-..__________

Determinations of scour and fill______________________

Photography---------_-____________..____._____

Cross-section soundings________________________

Stage-discharge relations_______________________

Sonic soundings._______________________________

Probing__ _ __________________________________

Plastic ribbons.________________________________

Log chains.____________________________________

Interpreting observations of scour and fill________

General principles of scour and fill____________________

Quantity of sands being transported._____________

Some previous statements or uses of the principles __ 

Discharge of sands as a function of flow characteristics. _ ______________________________________

The simple continuity equation ___________________

Page

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Applications of the principles to assumed channel reaches. D9

Channel downstream from a reservoir.____________ 10

Straight, uniform channel.......----------------- 10

Uniform slope and roughness..____--___--____ 12

Changing slope or roughness.--------.------- 12

Constriction in a straight, generally uniform channel- 14

Winding streams._______________-__-----------_ 21

Scour and fill in natural streams.__-____-___-_-___-_-_ 21

A generally stable sand-bed stream___.._---_-____ 21

Narrow section of a natural sand-bed stream_______ 24

A pool-and-riffle stream.________________________ 25

Streamflow, sediment concentration, and changes

in bed elevation________________________ 25

Application of principles of scour and fill_____ 30

Conclusions. ______________________--_--_----------- 31

References-.-..-.----------_----__----------------- 32

ILLUSTRATIONS

Page 

FIGURE 1. Schematic diagram of instantaneous bed positions at six points in a stream cross section___-_______--__-- D4

2. Approximate volume of moving sands per unit area of streambed for 0.30-millimeter median diameter of bed

material and 60° F water temperature.______________________________________-------------------- 7

3. Approximate discharge of sands for 0.30-millimeter median diameter of bed material and 60° F water temperature...-_____________________________________________________-_-__--__------------_---- 8

4. Properties of assumed cross section--_-__-----______--_-___--_--_-_-------------------------------- 9

5. Streamflow and discharge of sands at the downstream end of an assumed 1,000-foot channel reach immediately downstream from a dam that traps all sands in the flow-------------------------------------- 11

6, 7. Streamflow, discharge of sands, and accumulated sands in an assumed 5,000-foot reach 

6. Straight, uniform channel__________-_-________________--_-__-___--_-___-------------------- 13

7. Straight channel that is uniform in bed material and cross section but has a gradually decreasing slope

toward the downstream end_______-___________________--__-___--____-_-------------------- 15

8. Depths of computed scour at narrow section of an assumed 1,000-foot reach of straight channel-___________ 19

9. Schematic profile of bed position at and near a narrow section of an assumed straight sand-bed channel after

a short period of constant flow------------------------_---_---------_-------------------------- 19

10-13. Changes in average gage height of the streambed of the Elkhorn River near Waterloo, Nebr. 

10. During a period of high flow in 1951----------------------------------------------- ------- 22

11. During a period of high flow in 1952---______-___--__________---__-___-_-,------------------- 22

12. May 18-July 19, 1951_____ _._____ ______ __-_ _ _ __ - - - 23

13. April 1-October 23, 1962..._________________________________________________ _ _ _ _ _ _ 24

14. Comparison of scour and fill computed from changes in bed position at two cableway sections on the Elkhorn

River near Waterloo, Nebr-------__-_-_---____-----__-__----_-----_---------------------------- 25

15. Streamflow and changes in average bed position of Missouri River near Yankton, S. Dak., March-September

1941 ----__---_____-___._______________________________-___________-___---------_-____--_ 25

16. Streamflow, measured sediment concentrations, and changes of average bed position at Lees Ferry and

Grand Canyon, Ariz., April 1-August 31, 1955----------------------------------------------- -- 26

17, 18. Changes of average bed position at Lees Ferry and near Grand Canyon, Ariz. 

17. October-December 1957.--._________________________________________--------- 27

18. April-July 1959______________________________________________________------_____ 28

19. Variable relation of concentration of measured sands to flow of the Colorado River near Grand Canyon,

____-__. _-___-_;______-_-_---_...._--_.__.___________________________ _ __ _ _ _ _ _ 30

in
IV CONTENTS

TABLES

TABLE 1. Differences between two types of concentration 

of transported sands at a vertical___________

2. Volumes of sand removed from a 1,000-foot 

reach of channel downstream from a dam at 

different constant rates of steamflow through 

the standard cross section ________________

3. Fill in an assumed 5,000-foot reach, of straight 

channel of the standard cross section if velocities at a given depth, are 20 percent lower at 

the downstream end of the reach than at the 

upstream end_ _ _____ _____

4. Computed initial rates of scour between cross 

sections A and B at different flows for which 

the depths of flow above the flat beds are assumed to be equal at the two sections-_______

Page 

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10

14

16

Page

TABLE 5. Average scour at section B after five constant flows have established equilibrium bed 

positions _ _______ __ D17

6. Computed scour at cross section where bottom 

width is 66 feet in channel whose uncontracted bottom width is 84 feet __ 20

7. Computed average scour or fill, by months, during 1955 water year in 88-mile reach from 

Lees Ferry to Grand Canyon, Ariz___ 30

DEFINITIONS

Antidunes. Roughly symmetrical sand waves that are about in phase with similar water waves. Definition includes as

antidunes roughly symmetrical sand waves that may exist below quiet and symmetrical standing waves on water

surface. 

Bars. Extensive raised areas or ridges, larger than dunes in same flow, on bed of stream whose bottom shifts readily, at

least at high flow. 

Bed elevation. Vertical position of surface of streambed above arbitrary datum. For flow over sand bed, streambed is

lowest position at which sand grains are generally moving at a particular instant. Perhaps sand grains shift somewhat,

but without moving far, below streambed. 

Bed material. Sediment of particle sizes found in appreciable quantity in streambed. Term is based on particle size rather

than on mode of transportation or location in cross section. 

Dunes. Somewhat irregularly spaced mounds of loose sand whose upstream faces are usually gently inclined as compared

with their much steeper downstream faces. 

Fill. Net deposition of sediment on stream boundary. Fill may be measured in volume of sediment added to a channel

reach, in average depth of sediment accumulation over an area, in average change of depth at a cross section, or in

change of depth at a point.

Fine sediment. Term used in this paper for sediment finer than 0.062 mm. 

Flat bed. Term applied in this paper to general configuration of beds of assumed channels. Does not imply absence of dunes,

antidunes, or bars. 

Measured concentration of sediment or of sands. Concentration based on depth-integrated sediment samples which generally

are obtained by sampling flow from water surface to within 0.3 or 0.4 ft of streambed. 

Measured discharge of sediment or discharge of sands. Discharge that is usually determined directly from streamflow and

depth-integrated sediment samples. Depth integrated samples obtained by sampling flow from water surface to within

0.3 to 0.4 ft of streambed. 

Plane bed. Bed that is free of dunes and antidunes and, consequently, has low resistance to flow. Usually is smoother and

firmer than bed of sand-bed streams of other configurations. 

Pool-and-riffle stream. Stream characterized by alternate pools and riffles. 

Sand. Sediment particles that have diameters between 0.062 and 2.0 mm. 

Sand-bed stream. Reach of stream whose bed consists almost entirely of cohesionless shifting sand and whose channel is

not confined by walls of rough rock or other material that would cause unusually high turbulence throughout much

of flow.

Scour. Net removal of sediment from stream boundary by action of fluid flow. Scour may be measured in volume of sediment removed from a channel reach, in average depth of sediment removal from an area, in average change of depth at

a cross section, or in change of depth at a point. 

Unmeasured discharge of sands. Difference between total discharge of sands and measured discharge of sands at a cross

section.
SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

SCOUR AND FILL IN SAND-BED STREAMS

By BRUCE R. COLBY

ABSTRACT

The surface of a sand bed of a stream rises when more sand 

is deposited on it than is removed from it and falls when more 

sand is removed from it than is deposited on it. The average 

elevation of the surface is constant whenever the volume of 

sand being deposited equals the volume of sand being removed. 

Hence, two basic principles generally are required to explain 

or to understand scour and fill in sand-bed channels. One 

principle is continuity of sediment discharge along a stream 

reach that is, for a sand-bed stream, a balance between the 

volume of sand-size particles deposited on and removed from 

a sand bed. The other principle is that a relation exists between discharge of sands and characteristics of flow and available sands. If no such relation were to exist, the difference between the discharges of sands into and out of a channel reach 

would be wholly indeterminate except by measurement.

Although the basic principles of scour and fill are not new, 

they are frequently disregarded or misunderstood, partly because many observations of scour and fill in streams have been 

misinterpreted. These misinterpretations are due largely to the 

common assumptions that scour or fill at a point is representative, both in quantity and in timing, of average scour or fill at 

a cross section and that average scour or fill at a cross section 

is representative of scour or fill in a long reach of channel.

The principles of scour and fill when applied to assumed 

channels define general patterns of streambed changes for certain types of channel. These principles indicate fairly definite 

limits on the rate of net scour or net fill over long reaches of 

channel.

Observed changes in bed elevation, defined by streamflow 

measurements on the Elkhorn River at Waterloo, Nebr., the 

Missouri River near Yankton, S. Dak., and the Colorado River 

at Lees Ferry and near Grand Canyon, Ariz., show, respectively, 

the different general patterns of scour and fill in a stream that 

has a generally stable sand bed, in a stream at a laterally constricted cross section, and in a pool-and-riffle stream. These 

patterns are explainable in terms of the basic principles of scour 

and fill, although scour and fill in a pool-and-riffle stream may 

be difficult to compute theoretically.

Conclusions indicated by the study, but not universally accepted, include the following:

1. At any instant, the quantity of moving sand, even in deep and 

swift flows, is usually equivalent to an average depth of deposit of only a few hundredths of a foot on the streambed.

2. In a straight sand-bed channel that has uniform cross section, slope, roughness, and bed material, the streambed does 

not either scour or fill appreciably during the passage of a 

flood.

3. In a straight channel having uniform cross section but varying slope or roughness along the channel, fill during a 

rising stage is usually followed by fill on the falling stage, 

and scour during a rising stage is usually followed by 

scour on the falling stage.

4. If a change in roughness or in streamfiow requires a change 

in the depth of flow, the depth generally adjusts through 

a change in elevation of the water surface and not through 

a change in elevation of the streambed.

INTRODUCTION

Sand on the bed of a stream usually shifts readily, 

and unless the velocity of flow near the bed is very low, 

some sand grains are moving at every instant. If the 

velocity near a sand bed is high, many grains move 

from a particular area during each short period of time 

and are replaced by many other grains. Because of the 

continual shifting of individual grains, the average 

elevation of the sand bed of a stream frequently is assumed to be very unstable.

The assumption of instability seems to be supported 

by observations that are commonly made by stream 

gagers, among others, who, when wading on a sand bed 

where velocities are high, feel the sand wash out around 

and under their feet. Also, cross-sectional areas of a 

series of streamflow measurements that are made while 

a flood wave was passing a measuring section generally 

show scour during high and rising stages and fill during 

falling and low stages. When the series of measurements is plotted against gage height, the slope of the 

average relation may indicate that the flow increased 

more rapidly with stage than it would have for a channel having rigid boundaries. Thus, on the basis of general experience, many stream gagers and hydraulic 

engineers assume that the average elevation of a streambed fluctuates widely from .time to time, especially during periods of high and rapidly changing flow. The 

limitations imposed on changes of the average elevation 

of the bed by continuity of quantity of the bed material 

along the stream are frequently overlooked.

The purpose of this paper is to show in reasonably 

simple and practical terms the general principles that

Dl
D2 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

govern scour and fill in sand-bed streams at cross sections that are free from obstructions such as piers or 

pilings. These principles are not new, but they are 

sometimes disregarded or misapplied. In particular, 

the application of the principle of continuity is 

stressed.

Determinations of scour and fill and possible misinterpretations of them are discussed first; then the principles of scour and fill are explained; and, finally, the 

principles are applied to some assumed channel reaches 

and are illustrated by the behavior of a few types of 

natural streams.

Some unpublished information on concentrations and 

particle sizes of sediment and on elevations of the 

streambed of the Colorado Kiver at Lees Ferry and 

near Grand Canyon, Ariz., was furnished by J. M. 

Stow, district chemist, Albuquerque, N. Mex.

DETERMINATIONS OF SCOUR AND FILL

Several direct or indirect methods for determining 

scour and fill are sometimes used in flumes or in natural 

streams. A few of these methods are mentioned briefly, 

and some possibly misleading interpretations of results 

of the methods are pointed out.

PHOTOGRAPHY

The general shape and movements of sand-bed features that are formed by flows in a flume can often be 

best shown by photographs. For example, the topography of a sand bed can be shown by a photograph 

taken after the water has been drained from the flume, 

profiles of dunes or antidunes can be photographed 

through observation ports of flumes during runs, and 

the changing features of a sand bed can often be well 

shown by moving pictures that are taken while water 

is flowing in a flume. Ordinarily, however, photographic observations cannot be made satisfactorily of 

the sand beds in field streams.

CROSS-SECTION SOUNDINGS

Soundings by either mechanical or sonic methods at 

several or many verticals of a cross section provide a 

direct measure of the average position of the streambed at a particular time. Thus, cross-section soundings 

made periodically furnish reasonably good determinations, by difference, of the average scour or fill at a 

cross section. However, even in a fairly uniform channel, the changes in average bed elevation may differ 

from one cross section to another. Much present information on scour and fill in stream channels is based 

on cross-section soundings at bridges or cable ways 

where streamflow measurements are made, but, unfortunately, streamflow measurements are usually made

at narrower-than-average channel sections of sand-bed 

streams. As will be shown later, in the sections entitled 

"Constriction in a straight, generally uniform channel" 

and "Narrow section of a natural sand-bed stream," the 

changes in average bed elevation at narrow cross sections are likely to be much greater than those at nearby 

cross sections of the same stream. Hence, although 

scour and fill can generally be determined with reasonable accuracy at streamflow-measuring sections, the interpretation of the determined scour and fill may be, 

and often is, misleading.

Even though streamflow measurements are made occasionally at wide cross sections, scour and fill information based on soundings for these measurements can 

seldom be interpreted correctly unless amounts of scour 

and fill are determined for the whole width between 

banks rather than for only the width of the water 

surface. Water-surface width may vary greatly with 

time and, for some wide sections of sand-bed streams, 

may even be less at high flows than at some much lower 

flows. Therefore, scour and fill generally should be 

determined for shifting cross sections by defining periodically the position of the surfaces of temporarily unsubmerged parts of the cross section at the same times 

that the position of the submerged streambed is defined. 

Of course, determinations for the whole width between 

banks are made regularly at few, if any, streamflowmeasuring sections of wide sand-bed streams.

STAGE-DISCHARGE RELATIONS

Major changes in the bed configuration of sand-bed 

streams may cause large changes in resistance to flow 

(Vanoni and Brooks, 1957; Brooks, 1958; Liu and 

Hwang, 1961; Simons, Richardson, and Haushild, 

1962). If a sand bed is in the form of dunes at low 

flow and in the form of a plane or antidune bed at high 

flow, the stage-discharge relation may show a large 

increase in flow for a small increase in stage, according 

to Einstein and Chien (in Brooks, 1958, p. 556-557, 

figs. 12, 14), a two-part stage-discharge relationship 

(Dawdy, 1961), or even a discontinuity (Colby, 1960). 

This large increase in flow for a small increase in stage 

has sometimes erroneously been assumed to be an indirect measure of scour of the streambed rather than 

a measure of decrease in resistance to flow.

SONIC SOUNDINGS

Sonic sounders, although they have not yet been used 

extensively by the U.S. Geological Survey, would provide much more dependable information on scour and 

fill at a point in many streams than is commonly obtained by using most other methods for determining 

stream-bed position. These instruments can be used to
SCOUR AND FILL IN SAND-BED STREAMS D3

record not only the time and the elevation of the bed 

when it is at its lowest position at a point but also to 

define a graph of bed position at a point against time. 

Sonic sounders also can be used to establish profiles or 

cross sections of the streambed. Measurements made 

using the sounders should prove helpful in interpreting 

correctly observations of scour and fill made using other 

less informative procedures.

PROBING

The depth of maximum scour during a stream rise 

sometimes can be determined at points in a cross section 

or a stream reach by probing with steel rods or soil 

tubes after the flow has receded. Unless the properties 

of the recently deposited fill differ appreciably from 

those of the material that was not moved during the 

rise, the maximum depth of scour may not be discernible at some points. Even though the maximum depth 

of scour at each of several points in the cross section 

can be determined satisfactorily, it rarely occurs simultaneously at all the points. Hence, the interpretation 

of the maximum depth of scour as determined by 

probing and of the observed fill above the maximum 

depth of scour may be questionable, as is the interpretation of other determinations of maximum scour at a 

point at some unknown time during a flood.

PLASTIC RIBBONS

Maximum scour at points in an ephemeral stream 

can be determined after a flood by observation of plastic 

ribbons that had been inserted vertically into the bed 

oj a stream before the flood (Essex, Terry, and Culbertson, written communication, 1954). If enough ribbons 

at a cross section can be dug out after the flood, the 

points at which the ribbons bend from vertical to horizontal will define the maximum depth of scour across 

the section. However, the ribbons may be difficult to 

recover after a flood, and the time of maximum scour at 

each point is unknown.

LOO CHAINS

Light log chains have been used in some northern 

Mississippi streams by the Agricultural Research Service, U.S. Department of Agriculture, to show maximum 

depths of scour. Each chain was attached to a ring 

that was slipped over the top of a steel angle that had 

been driven into the streambed. A pin kept the ring 

from coming off the top of the angle, but the ring 

could slide down the angle easily if scour removed the 

sand below the ring and the chain. The chain was initially laid at or just below the surface of the bed and 

was extended horizontally downstream from the angle. 

The position of the chain usually could be determined

easily at low flow or no flow whenever knowledge of the 

depth of maximum scour prior to a selected time was 

needed. Probably the observed positions of a chain indicated the maximum depth of scour fairly accurately, 

although the steel angle and the chain may have affected 

the scour a little and the chain may have tended to settle 

slightly into a loose sand bed during periods of flow. 

Of course, the exact time of the maximum scour was 

not known, and the maximum scour at all points along 

a chain may not have occurred at the same time. Also, 

the maximum scour at two or more chains at the same 

cross section may have occurred at different times.

INTERPRETING OBSERVATIONS OF SCOUR AND FILL

Three general problems are involved in interpreting 

some observations of scour and fill. The first problem 

is whether or not an average position of the streambed 

at a cross section can satisfactorily be based on a few 

observations at the section. The second problem is 

whether or not an average bed position at a cross section can be determined from observations of maximum 

instantaneous depths of scour. The third problem is 

whether or not average or typical changes in elevation 

of the sand bed in a stream reach can be determined 

from observed changes in average position of the 

streambed at one or a few cross sections. These problems prevent the behavior of shifting streambeds from 

being easily determined and understood from the usual 

observations available for natural streams.

A few observations are inadequate to determine the 

average position of a shifting streambed at cross sections where the bed position fluctuates widely. Such 

sections include those on the San Juan River near Bluff, 

Utah, on the Colorado River at Grand Canyon, Ariz., 

on the Rio Grande near Bernalillo, N. Mex. (Leopold 

and Maddock, 1953, figs. 22, 24, 26), and on the 

Elkhorn River near Waterloo, Nebr. (Beckman and 

Furness, 1962, figs. 8, 9). For beds as unstable at individual points as those at the above sections, many 

measurements obviously are required to define accurately the bed position or changes in the position even 

for a single cross section. Stream gagers usually make 

soundings at 20 or more points in most cross sections 

to be reasonably sure of defining satisfactorily the cross 

section and the velocity distribution. Changes in bed 

elevation probably should be based also on measurements at about 20 or more points for most cross sections. If only a few soundings at a cross section are 

available for a given time, the average bed position 

based on these soundings should be considered to be 

inexact unless a study has been made to find if fewer 

than 20 points are likely to define accurately a bed position for the section.
D4 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

The second problem, that of determining an average 

bed position at a cross section from observations of 

maximum instantaneous depths of scour, is usually difficult and frequently cannot be solved satisfactorily. 

Suppose that the minimum elevations at 20 points in a 

cross section of a straight, uniform reach were determined after a flood from the positions of plastic ribbons 

or log chains that had been installed before the flood. 

At comparatively low flows at the beginning of the 

flood, the assumed bed was covered with sand dunes; 

at high flows the bed was a combination of plane bed 

near the banks and antidune bed along the middle of 

the channel. A schematic diagram (fig. 1) for 6 of the 

20 points gives a general idea of the difference between 

the scour that is indicated by the ribbons or chains and 

the actual lowest average position of the bed at the 

cross section. As will be shown later, the average

position of the streambed in a straight, uniform channel that has uniform bed material usually remains 

nearly constant during the passage of a flood. This 

stability of the average bed position was assumed at 

each point when the schematic diagram was prepared. 

Even though the average position of the streambed 

remains almost constant during the passage of a flood, 

the maximum scour at each point is roughly 1 foot in 

figure 1. Thus, an entirely fictitious average scour at 

sometime during the passage of the flood and an equally 

fictitious average fill during the recession of the flood 

(a fairly uniform and even bed with ripples usually 

is left when the flow ceases or becomes very low in a 

straight, uniform channel) seem to be indicated by the 

observed minimum elevations of the streambed. Also, 

if the average position of the streambed is virtually 

constant at a cross section, the stability of the average

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TIME

FIGURE 1. Schematic diagram of instantaneous bed positions at six points In a stream cross section. Time of

changes not to scale.
SCOUR AND FILL IN SAND-BED STREAMS D5

position might not be shown when the bed was rough 

unless observations were made at many points in the 

cross section (Beckman and Furness, 1962, p. 12 and 

figs. 8, 9). For the assumed fairly stable time-averaged 

elevation at each point, the maximum point scour does 

not occur at the highest flow. In fact, the least amount 

of point scour would be indicated at the peak flow at 

points where the highest flow was over a plane bed. 

According to the schematic diagram (fig. 1), the maximum point scour does not occur at the peak flow even 

at a point where the highest flow is over antidunes; it 

occurs when the bed is the roughest, which is usually 

when the bed has the largest dunes. In general, the 

largest dunes are formed at about the deepest flows 

at which dunes exist. Dunes may exist at the peak 

flow if the velocity is slow enough in relation to the 

depth, bed material, water temperature, and concentration of fine sediment; but for many sand-bed streams, 

at least part of the cross section will have a plane or an 

antidune bed at the highest flows.

If the assumed cross section had been at a narrow 

segment of a straight channel, the average bed position 

at the cross section generally would have been lower at 

high flows than at low flows. Hence, the observed maximum scour at individual points might have occurred 

when the bed was plane, if a plane bed existed at the 

highest flows at the point, or when antidunes were 

forming and breaking, if antidunes existed at the 'highest flows. However, the problem is further complicated because the average maximum scour for a narrow 

section usually occurs after the peak flow.

Sometimes an adjustment might be applied to approximate the average position of the streambed from 

observations of instantaneous maximum scour at a 

point. If the streambed at a point was known to be at 

its lowest position when the bed was covered with dunes, 

a very rough adjustment to the observed maximum 

scour might be made by adding an approximation of 

half of the probable dune height to the observed lowest 

position at the point. If the bed at a point was known 

to be plane throughout the passage of the flood (an 

unusual condition), the observed maximum instantaneous scour may require no adjustment. If the bed 

position at the point was known to be lowest when the 

flow was over antidunes, the observed position at maximum scour might be increased by approximately half of 

the probable maximum amplitude of the antidunes on 

the bed. Obviously, adjustments such as these, if they 

can be made at all, are usually inaccurate; but they may 

at least indicate the probable inaccuracies in the maximum observed point scour during a flood.

The third problem, that of determining average or 

typical changes in bed elevation for reaches of a stream

from changes at a cross section, requires a consideration 

of the general principles of scour and fill and the application of these principles to different types of channel. 

This problem will be considered at length in the following sections. In general, it will be shown that alternate 

scour and fill at a cross section usually indicate that the 

observed scour and fill are not representative of a long 

reach of channel even though based on accurate soundings at many points in the cross section. Therefore, 

alternate scour and fill at a cross section may lead to an 

incorrect assumption of changes in average bed position 

in a reach of channel in much the same way that alternate scour and fill at a point may incorrectly be interpreted to mean that the average position of the bed 

changed at the cross section.

Also, some observed changes in the average bed elevation at a cross section may not be representative or 

significant even at the cross section. A log or a pile of 

driftwood at a section or immediately upstream from 

if may affect the cross-sectional area at least occasionally. A field stream may shift its alinement somewhat from time to time, and a change in alinement may 

be associated with changes in the shape and size of the 

cross section at a given flow. Sometimes a narrow and 

deep channel may form, at least temporarily, in a sandbed stream (Lane and Borland, 1954, p. 1088-1089). 

Sandbars may shift on the streambed and cause somewhat unrepresentative short-time changes in cross-sectional area. Therefore, some observed variations in 

average bed position at a cross section may be due to 

chance rather than be directly related to changes in flow 

or in the available sediment.

GENERAL PRINCIPLES OF SCOUR AND FILL

Two simple principles can be used to explain scour 

and fill in sand-bed streams. One principle is that a 

relation exists between the discharge of bed material, 

which is sediment of sizes found in appreciable quantity 

in the streambed, and the characteristics of the flow 

and available sediment. Without such a relation, the 

discharge of sands into or out of a reach of sand-bed 

channel would be wholly indeterminate except by measurement. The other principle is continuity of the sediment discharge along a stream reach. Each principle 

is obviously applicable even though some details of its 

use may be obscure or questionable.

Simple logic indicates that the discharge of sediment 

of particle sizes that are readily available in the streambed must, for a steady state, be limited by the characteristics of the flow and sediment. If the sediment 

discharge were not so limited, the available sediment 

would not remain in the bed. Also, the reproducibility 

of sediment discharges in flume investigations at steady
D6 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

flow suggests a relation of sediment discharge to the 

characteristics of the flow and sediment.

Although steady flow is rare in natural streams, an 

approximate relation of discharge of sands in sand-bed 

streams to mean velocity, width, depth, median particle 

size of the bed sands, and viscosity or apparent viscosity 

exists in natural streams (Colby, 1964). If the effects 

of all the characteristics of the flow and sediment could 

be accurately evaluated and if all experimental error 

could be removed, a good relationship could certainly 

be defined between the discharge of sands and these 

characteristics for naturally changing rates of flow as 

well as for constant flows.

Continuity as applied to scour or fill of a channel 

reach requires that the difference between the discharge 

of sediment into and out of the reach be balanced by 

a gain or loss of accumulated sediment within the reach. 

If fine sediment accumulates within a reach during a 

period of time, the inflow of fine sediment must have 

exceeded the outflow of fine sediment during the time. 

However, in general, the silt and clay that enter a sandbed reach also leave the reach; so the principle of continuity need be applied to the sands only. Hence, the 

concentrations of fine sediment in sand-bed streams are 

usually not directly ^elated to scour and fill. Of course, 

inflow or outflow of sands or of sediment of all sizes 

may occur either at the ends of the reach or through 

tributaries or diversions along the reach. Also, a change 

in the quantity of sediment in a stream reach may result 

in a change either in the average elevation of the streambed or in the quantity of sediment that is moving above 

the streambed. (The weight per unit volume of the 

sediment in the streambed is assumed to remain unchanged.)

The principles of scour and fill are more easily stated 

and applied if the quantity of sand being transported 

can be shown to be negligible. Persons aware of the low 

concentrations of sands in natural streams may already 

realize that the amount of sands being transported can 

usually be disregarded in the application of the principle of continuity. Others may want some rough

approximation of the amount of sands being moved at 

different combinations of depth and velocity.

QUANTITY OF SANDS BEING TBANSPOBTED

The depth of all sands in transit, if instantaneously 

deposited, could be computed from the product of a 

concentration by weight; the depth of flow; and the 

ratio, by weight, of a cubic foot of water-sediment mixture to a cubic foot of deposited sands. Unfortunately, 

the type of concentration cr that is needed for such a 

computation is seldom available, because it is the ratio, 

by weight, of the sediment to the water-sediment mixture above a particular area of the bed. In contrast, the 

concentration cz that is usually determined from sediment samples is a weight ratio of the discharged sediment to the discharged water-sediment mixture. In 

terms of sampling at a vertical, the first concentration 

(d) would be obtained if an equal volume of watersediment mixture could be collected from each small 

equal increment of depth, including the bed layer, and 

if all the equal volumes could be mixed and sampled. 

The second concentration (<?2 ) would be obtained if the 

volume from each small equal increment of depth could 

be collected in proportion to the flow velocity through 

the increment and if all the volumes could be mixed 

and sampled. Because the concentration of sands is 

greatest where the velocity is slowest in the vertical, 

the first concentration is larger than the second.

For sediment as coarse as sands, the difference between the two types of concentration is large and variable (table 1). Six computations of the ratio of <?i : c2 

ranged from 1.2 to 2.6 and averaged 1.7. Of course, 

the computations on which information in table 1 is 

based require extensions, usually inexact, of velocity 

and concentration of sands from points of observation 

in the flow to the streambed. However, the inaccuracy 

is not objectionable for estimating the depth of all sands 

in transit, if instantaneously deposited from a flow, because the maximum indicated depth of deposit for the 

flows and concentrations in table 1 is less than 0.01 foot.

TABLE 1. Differences between two types of concentration of transported sands at a vertical

Selected vertical

Assumed flow _____________________________________

Barton and Lin (1955), run 22-. __.-__-_-__.-_ ___

Mississippi River at St. Louis, Mo_. ________ ________

Do__________________ __________________________

Do._________________________________. _________

Do______________________. _____________________

Date

June 7, 1954 

April 24, 1956 

May 5, 1956 

May 13, 1959

Median size 

of bed material (mm)

0.25 

. 18 

.3 

.3 

.3 

.45

Flow

Depth (ft)

5.00 

.76 

33.5 

27.0 

32.8 

36.8

Velocity 

(fps)

3.5 

3.2

5.8 

4.0 

4. 1 

5.0

Concentration of sands 

(ppm)

Cl

1,330 

2,870 

292 

344 

161 

159

C»

980 

1,090 

242 

185 

97 

100

Ratio 

ci:ci

1.4 

2.6 

1.2 

1.9 

1.7 

1.6
SCOUR AND FILL IN SAND-BED STREAMS D7

Approximate depths of average deposits (fig. 2) that 

might result from the instantaneous settling of moving 

sands in a reach can be based on a graph by Colby (1964, 

fig. 21). The graph shows approximate cz concentrations of sands for a wide range of mean velocities and 

depths, a median grain-size diameter of 0.30 millimeter 

for the bed sands, and a water temperature of 60°F. 

These c2 concentrations were determined from an anall.O

O.I

0.01

0.001

0.0001

0.00001

O.I 1.0 10 

DEPTH OF FLOW, IN FEET

100

FIGURE 2. Approximate volume of moving sands per unit area of 

streambed for 0.30-millimeter median diameter of bed material and 

60° F water temperature. Assumed weight per unit volume of 

sands is 85 Ib per cu ft.

ysis of many available records of the discharge of sands 

in flumes and in field streams. They should be increased 

somewhat (Colby, 1964, fig. 24) for high concentrations of fine sediment in the flow. The c2 concentrations from the graph were doubled (table 1) to obtain 

approximate GI concentrations. A weight of 85 Ib per 

cu ft (pounds per cubic foot) was assumed for the deposited sands; and the approximate volume of moving 

sand per unit area of the streambed was computed from 

the product of the GI concentration, the depth of flow,

and the ratio of 62.4 Ib per cu ft to 85 Ib per cu ft. The 

indicated depths of average deposits (fig. 2) are only a 

few hundredths of a foot, even for deep and fast flows; 

for many flows the indicated depth is less than 0.01 foot. 

These amounts of moving sand are so small that they 

can usually be disregarded in studies of scour and fill.

SOME PREVIOUS STATEMENTS OR USES OF THE 

PRINCIPLES

Straub (1934) computed scour for constant flow at a 

laterally contracted cross section from his form of the 

Du Boys equation and from the principle of continuity 

of the bedload discharge. He concluded that observed 

amounts of scour at narrower-than-average cross sections of a stream probably were not representative of 

long reaches of a river.

In 1948, the scour and fill in the Missouri River at 

bridge cross sections where streamflow measurements 

had been made were reported to be greater than the 

scour and fill away from the bridges (B. R. Colby and 

R. E. Oltman, written communication). The conclusion was based on continuity of the sediment discharge 

and on relatively smaller fluctuations in the stage-discharge relation than in the average elevation of the 

streambed at the bridge sections.

In 1952, Billy Mitchell and A. J. Harrison, in separately discussing a paper by Leopold and Haddock 

(1952, p. 172, 173, 176), noted the significance of continuity of the sediment sizes in the streambed. Unfortunately, Mr. Harrison's brief and excellent statement of the principles of scour and fill apparently received little attention, perhaps because it differed from 

the ideas presented by the authors of the paper.

E. M. Laursen (1952) stated well the fundamental 

principles of scour and fill. He applied the principles 

more fully to scour by jets and to scour around bridge 

piers and abutments than to general scour or fill in a 

channel reach.

Lane and Borland (1954) studied scour and fill, particularly in the middle Rio Grande, and concluded that 

the streambed did not generally degrade appreciably 

during floods although scour was pronounced at some 

narrow sections. In spite of this excellent paper by 

Lane and Borland and the statements of principles by 

Straub, Mitchell, and Harrison, the inherent stability 

of the average elevation of sand beds, especially in reasonably uniform channels, is still largely unrecognized.

In 1959, John Fisher Kennedy (oral communication) 

pointed out the stringent limitations imposed on scour 

and fill in a uniform channel by the principle of continuity and the principle of a relation between the discharge of bed material and the characteristics of the 

flow and available sediment. His reasoning was almost
D8 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

exactly the same as that previously used by the present 

writer in unpublished reports.

In 1960, the stability of a sand bed during rises was 

emphasized in a study of the stage-discharge relationship for Pigeon Roost Creek (Colby, 1960, p. 13-15). 

The reasons given for the stability of the average elevation of the sand bed had been arrived at independently, 

but they closely paralleled those given years earlier by 

Mitchell and Harrison (in Leopold and Maddock, 1952, 

p. 176).

The concepts and principles applied by the preceding 

workers were generally consistent, but they differ from 

the idea (Leopold and Maddock, 1952, 1953; Brooks, 

1958, p. 553-555) that an imposed sediment load affects 

the resistance to flow and hence the depth of flow for 

a given rate of flow. According to this idea, scour or 

fill results from an attempt of the stream to adjust its 

depth of flow to a changing sediment discharge. Information on large amounts of scour and fill at the 

measuring section on the Colorado River near Grand 

Canyon, Ariz., was given in support of this idea. On 

the other hand, Einstein and Chien (in Brooks, 1958, 

p. 558) stated that the cited information did not support 

the idea that changes in the elevation of the streambed 

at the measuring section were due to changing resistance 

to flow. The scour and fill and the effect of changing 

roughness on depth at this measuring section will be 

examined in more detail under the heading "A pool-andriffle stream."

DISCHARGE OF SANDS AS A FUNCTION OF FLOW 

CHARACTERISTICS

The discharge of sands can be roughly computed for 

different flows and bed-material sizes according to the 

procedure suggested by Einstein (1950) or according 

to any other of several equations or procedures that are 

available for determining the discharge of sands in a 

sand-bed stream. In this paper, approximate discharges of sands in sand-bed streams are based on relations that were obtained from an analysis of the total 

discharge of sands in field and laboratory streams 

(Colby, 1964). The relations do not apply unless the 

streambed consists almost entirely of nearly cohesionless 

sand and unless the banks are not excessively rough as 

rock walls might be. The relation for low concentrations of fine sediment, a water temperature of 60° F, and 

a median particle size of bed sands of 0.30 millimeter 

is given in figure 3. For the stated conditions, the discharge of sands per foot of width is determined according to figure 3 by mean velocity and depth of flow. 

(Relations for other median particle sizes and approximate adjustments for changes in water temperature and 

for the effects of high concentrations of fine sediment 

are also given in Colby, 1964.)

5000

1000

10 1.0 10 

MEAN VELOCITY, IN FEET PER SECOND

FIGURE 3. Approximate discharge of sands for 0.30-millhneter median 

diameter of bed material and 60° P water temperature. Assumed 

weight per unit volume of sands is 85 Ib per cu ft.

THE SIMPLE CONTINUITY EQUATION

The continuity equation for a reach of channel below 

the highest water-surface position that is to be considered can be written

or

in which

A8=Q,'-Q.' (1)

Q,' is the quantity of sediment that enters the stream

reach during a period, 

Q," is the quantity of sediment that leaves the reach

during the period, and 

AS is the quantity of sediment that accumulates (or

erodes if Afif is negative) during the period within

the reach and below the highest position of the

water surface.

The equation is obviously true for any time period, 

whether the sediment is transported in the main 

channel, in tributaries, or in diversions or whether the 

sediment enters or leaves through the action of wind 

or human agencies.

The three quantities of equation 1 usually can be used 

more conveniently for computations of scour and fill
SCOUR AND FILL IN SAND-BED STREAMS D9

if they are expressed in terms of equivalent volumes 

of deposit rather than in terms of weight. The equivalent volume of deposit is the weight of sediment 

divided by a known or assumed weight per unit volume 

of deposited sediment. The weight per unit volume 

can generally be approximated accurately enough for 

sands, but it is questionable for silt and clay or for 

mixtures of clay, silt, and sand.

Assume that the average elevation of the streambed 

would remain constant if no net weight of sediment 

were added nor taken away during the period to which 

equation 1 is applied and that the amount of sands 

being transported at any instant is so small that 

changes in the amount during a computation period 

can be disregarded. Equation 1 for computing the 

depth Z>, in feet, of average fill on the sand bed of a 

stream reach during a period then becomes

Q.'-Q."

(2)

in which A is the area, in square feet, of the bed of the 

channel reach if the sediment quantities are in cubic 

feet for the period. The quantity of silt and clay being 

transported has not been shown to be negligible, and the 

weight per unit volume of fine sediment may be questionable ; therefore, equation 2 should be limited to flows 

over sand beds. However, Qs' and QB " can generally, 

but not always, be volumes either of sediment of all 

sizes or of only sands, because the difference will usually 

be almost entirely a difference in volume of sands.

APPLICATIONS OF THE PRINCIPLES TO ASSUMED 

CHANNEL REACHES

Although the general principles are broadly applicable, discussions of scour and fill in assumed channel 

reaches will, for convenience, be limited to reaches 

without diversions or tributary inflows of water or sediment. Also, only flows within the channel banks will be 

considered. Because changes in the amount of sands 

being transported must be small (fig. 2), they usually 

will be disregarded.

Two extremes must be avoided if the principles and 

applications of scour and fill in the assumed channels 

are to be reasonably well understood. One extreme is a 

tendency to become enmeshed in questionable details 

and minor effects so that general and significant relations are obscured; the other is a tendency to oversimplify and hence to eliminate some significant effects.

The following assumptions, unless otherwise noted, 

will apply for all the computations in the assumed 

channels:

1. The bed material is cohesionless sand whose median diameter is 0.30 millimeter, and the sand is uniform along the 

reach.

2. The water temperature is 60° F.

3. The combined concentrations of silt and clay do not exceed 

10,000 ppm (parts per million).

4. The standard cross section has a flat bottom 84 feet wide and 

45° bank slopes (fig. 4). (The term "flat" applies to 

general configuration and, as used in this report, does not 

imply an absence of dunes and antidunes.)

5. The stage-discharge relation (stage is measured above the 

flat bed) for a straight reach is assumed to be that shown 

in figure 4.

6. For this stage-discharge relation, the velocity-discharge relation must also be that shown in figure 4 for the standard 

cross section.

7. The discharges of sands for different combinations of depth 

and mean velocity in a straight or nearly straight channel 

are assumed to be as defined by figure 3.

1000 2000 3000 

FLOW, IN CUBIC FEET PER SECOND

FIGURE 4. Properties of assumed cross section.
D10 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

CHANNEL DOWNSTREAM: FROM A RESERVOIR

If all the sediment is removed from a flow in a reservoir, equation 1 for a reach of channel immediately 

downstream from the reservoir reduces to

AS=-Q/', (3)

and the quantity of scour in the reach is known if the 

sediment outflow Qa" is known or can be computed 

for the downstream end of the reach. The conversion 

of weight to volume might be based on the weight per 

unit volume of deposited sands (a weight that can be 

reasonably accurately estimated) if the discharged sediment is entirely sands. If the discharged sediment contains appreciable quantities of silt and clay and the size 

distribution is approximately that of the material in 

place before it was scoured out, the weight per unit 

volume of the material before it was scoured away can 

be used to convert from weight to volume. Even though 

the quantity of sediment discharged from the reach has 

not been measured, the discharge of sands from the 

reach can be computed from such a relationship as that 

in figure 3. However, figure 3 is based on an assumed 

weight of 85 Ib per cu ft for sands and on the assumption that the flow has had time to pick up its equilibrium load of sands. If a cubic foot of material before 

it was removed by scour contained, for example, only 55 

pounds of sand and no great amount of selective sorting 

of particle sizes took place within the reach, the volumes 

from figure 3 should be increased in the ratio of 85:55 

for computing quantities of scour.

Sometimes the amount of sediment released from a 

reservoir is known, and then this amount of sediment 

inflow to the reach can be subtracted from a measured 

or computed outflow of sediment from the reach to compute scour.

For a computation of the volume of scour, assume that 

the standard cross section exists in a fairly straight and 

uniform reach of channel at a point 1,000 feet downstream from a reservoir where all the sand is removed 

from the streamflow. If this distance is sufficiently 

long to permit the streamflow to pick up its equilibrium 

load of sands, the discharges of sands at the downstream 

end of the reach should, according to figure 3, be those 

given in table 2 for different constant rates of flow. If 

all the material scoured from the channel was sand and 

the weight per unit volume of sand in place was 85 Ib 

per cu ft, the volumes in column 6 of table 2 would indicate the volume of scour directly. Perhaps only 60 percent of the eroded material was sands, and the weight 

per unit volume of the material in place before it was 

removed by the flow was 80 Ib per cu ft. Then if the 

amount of selective sorting within the reach was not 

appreciable, the volumes of scour for this material could

be computed from the volumes in column 6 of table 2 

by multiplying by 85/48 or 1.77.

TABLE 2. Volumes of sand removed from a 1,000-foot reach of 

channel downstream from a dam at different constant rates of 

streamflow through the standard cross section

Streamflow (cfs)

(1)

100 

500 

1,000 

2,000 

3,000

Deptb of 

flow above 

the flat bed 

(ft)

(2)

0.07 

2.68 

4.10 

6.20 

8.00

Mean 

velocity 

Ifps)

(3)

1.21 

2.15 

2.76 

3.58 

4.08

Width at 

water 

surface 

(ft)

(4)

86.0 

80.4 

92.2 

96.4 

100.0

Discharge of sands (cu ft per hr)

Per foot of 

width 

(assumed 

SSlbper 

cuft)

(5)

0.11 

3.7 

11.5 

31.5 

52

For tbe total width

Assumed 85 

Ib per cu ft

(6)

9.5 

330 

1,060 

3,030 

5,200

Assumed 48 

Ib per cu ft

(7)

17 

580 

1,880 

5,360 

0,200

The computed amounts of scour are for a 1,000-foot 

reach, but they would be very unequally distributed 

along the reach. Most of the scour probably would take 

place within the first 100 or 200 feet below the dam.

If the hydrograph of figure 5 represents flow through 

the standard cross section 1,000 feet downstream from 

the dam, the velocities and depths can be determined 

from the curves of figure 4 for many times during the 

period of flow. These velocities and depths and figure 

3 can then be used to compute the discharges of sands 

at frequent enough times to determine a graph (fig. 5) 

of discharge of sands during the period of flow. A 

cumulative curve of the volume and average depth of 

sands removed from the reach (fig. 5) can then be drawn 

from this graph. Of course, the volumes indicated by 

the curve are for 85 pounds of sands per cubic foot and 

should be increased if less than this weight of sands 

is removed for each cubic foot of sediment that is scoured 

from the reach. The average depth of sands removed 

is based on the 1,000-foot length of reach and on the 

maximum water-surface width of 100 feet.

STRAIGHT, UNIFORM CHANNEL

The reason for constancy of the average elevation of 

the sand bed of a uniform channel should be clear from 

the principles of scour and fill. (The channel is assumed to be uniform in slope, bed material, and resistance to flow at a given depth and velocity as well as 

uniform in cross section.) For such a channel, the relation between a particular flow and the depth and mean 

velocity of that flow is the same at each end of a reach. 

If the reach is not excessively long nor the volume of 

flow very small in comparison to the size of the channel, 

the hydrograph of flow at the downstream end of the 

reach will lag somewhat but will otherwise differ only 

a little from the hydrograph of flow at the upper end 

of the reach. Hence, the graph of discharge of sands 

from the reach will also lag a little as compared with
HOUR; PER FEET CUBIC IN SANDS, OF DISCHARGE 

SECOND PER FEET CUBIC IN FLOW, 

39\7d3AV Hld30 dO NOI131d30 SONVS Nl ' Nl i33d 

tta

Noii3id3Q do SQNVS NI Ni'HOV3a oiano i33d 

anoos QNV TIM NI aaa-aNvs 
D12 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

the graph of discharge of sands into the reach, but these 

graphs will be otherwise about the same. Therefore, 

the discharges of sands into and out of the reach must 

be about equal during a runoff event, and the average 

elevation of the sand bed in the uniform channel should 

be about the same after a flood as before the flood. If 

an allowance for the lag is made in the hydrographs 

of flow and in the graphs of discharge of sands, the 

inflows and outflows of sands must be about equal during intervals of time during a flood and the average 

elevation of the sand bed must remain about constant 

within the reach throughout the flood.

Perhaps an even simpler concept of the reason for 

constancy of the average elevation of a sand bed in a 

uniform channel can be based on consideration of volumes of water within short segments of length as these 

segments of flow pass along the channel. If the mean 

velocity and average depth of each segment of flow 

do not change as the flow moves through a reach of 

channel, the sand-transporting ability of each segment 

does not change either, and the segments of flow neither 

pick up nor drop appreciable net amounts of sands. 

Hence, the average elevation of the sand bed does not 

change as the flow passes over the bed.

UNIFORM SLOPE AND ROUGHNESS 8

For illustrative computations, assume a straight reach 

of uniform channel that has at all sections the stagedischarge relation (fig. 4) for the standard cross section 

and the relation of streamflow to discharge of sands 

that is shown in figure 3. Constant flows of 100; 500; 

1,000; 2,000; and 3,000 cfs (cubic feet per second) 

would be accompanied by discharges of sands of about 

9.5; 330; 1,060; 3,030; and 5,200 cubic feet of sands 

per hour (table 2, column 6) at each end of the reach. 

Because the discharges of sands into and from the reach 

are equal for any constant rate of flow, the sand bed on 

the average neither scours nor fills within the reach.

If the hydrograph of streamflow and the graph of 

discharge of sands (fig. 5) are assumed to apply at the 

upstream end of a 5,000-foot reach of the same assumed 

straight, uniform channel, the hydrograph of outflow 

at the downstream end of the reach would be about as 

shown in figure 6. The relation between flow and discharge of sands (fig. 5) can then be used with the hydrograph of outflow to compute a graph of discharge of 

sands from the reach. Differences between the discharges of sands into and out of the reach were accumulated by short intervals of time to define the accumulation-of-sands curve (fig. 6). According to equation 1, 

this curve represents the accumulation of sands within 

the 5,000-foot reach from the beginning of the increased 

flow to any time covered by the curve. This accumulation may be either within the flow or on the streambed.

If the accumulation of sands within the reach were 

distributed evenly over the area that is submerged at 

peak flow, 1,000 cubic feet of sands would be equivalent 

to an average depth of accumulation, including the accumulation in the flow, of 0.002 foot (fig. 6). If the accumulation of sands within the reach were distributed 

evenly over the flat part of the streambed, an area 84 feet 

by 5,000, the computed average depths of cumulative 

deposit would be increased in the ratio of 100: 84. Even 

though the 84-foot width is used, the maximum accumulation of sands represents an average depth of less than 

0.01 foot during the total time shown in the assumed 

hydrograph of flow. The accumulation of sands is 

greatest when the outflow of sands becomes equal to 

the inflow of sands.

CHANGING SLOPE OR ROUGHNESS

For the next computations, assume that the reach is 

the same but that the slope of the channel gradually 

decreases through the downstream part of the reach. 

The assumed change in slope begins far enough downstream so that the stage-discharge relation is not 

affected at the upstream end of the 5,000-foot reach, 

but the mean velocity at a given depth is exactly 20 

percent less at the downstream end of the reach than at 

the upstream cross section. This assumed velocity reduction is significantly less than a 20-percent velocity 

reduction at a particular flow. (See fig. 4.)

Volumes and depths of fill in the assumed reach can 

be computed readily. Discharges of sands into the 

reach at constant rates of flow of 100; 500; 1,000; 2,000; 

and 3,000 cfs can be taken from column 6 of table 2. 

The discharges of sands from the reach can be computed 

from the properties of the flow at the downstream end 

of the reach. A curve of relation between flow and 

depth at the downstream end of the reach was drawn in 

figure 4 by reducing the flows for the upstream end by 

20 percent for each of enough stages to define the stagedischarge curve for the downstream end of the reach. 

The stages directly determine the cross-sectional areas 

from the standard cross section (fig. 4) and hence the 

mean velocities for different flows at the downstream 

end of the reach. The mean velocities and the depths of 

flow can then be used to determine the discharges of 

sands per foot of width from figure 3. These discharges of sands multiplied by the water-surface widths 

are the total discharges of sands from the reach. At 

each of the five constant rates of flow, the computed 

differences between discharges of sands into and out of 

the reach gives, according to equation 1, the rate of 

accumulation of sands (expressed in terms of volume) 

within the 5,000-foot reach. (See table 3.) The volume 

of sand accumulation can also be expressed as an average depth of fill on the total submerged area, which is
SCOUR AND FILL IN SAND-BED STREAMS D13

ACCUMULATION OF SANDS IN REACH.IN CUBIC FEET

AVERAGE DEPTH OF ACCUMULATION OF SANDS IN REACH,IN FEET

-anon

ONOD3S U3d

i33d

133d DI9HD Nl MOld

NI 'SQNVS do 39dVHDsio
D14 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

TABLE 3. Fill in an assumed 5,000-foot reach of straight channel of the standard cross secMon if velocities at a given depth 

are 20 percent lower at the downstream end of the reach than 

at the upstream end

Streamflow (cfs)

100 

500 

1,000 

2,000 

3,000

Discharge of sands (cu ft per hr)

Upstream 

end 

(table 2)

9.5 

330 

1,060 

3,030 

5,200

Downstream 

end

1.5 

150 

600 

1,830 

3,350

Difference

8.0 

180 

460 

1,200 

1,850

Average depth of fill (ft per hr)

On horizontal 

submerged area

0.00002 

.00040 

.00099 

.0025 

.0037

On flat-bed area 

(420,000 sq ft)

0.00002 

.00043 

.0011 

.0029 

.0044

the product of the length of the reach and the average 

of the surface widths at the ends of the reach, or on 

the area of the flat bed of the reach, which is 5,000 feet 

by 84 feet. Of course, the volume of fill is greater for 

the larger constant flows, but a smaller percentage of 

the incoming sands remains in the reach at the larger 

flows.

The fill in the same assumed channel reach of 5,000 

feet was computed for the inflow that is represented 

by the hydrograph of figure 5. The computed approximate hydrograph of flow at the downstream end of the 

reach, where the slope is flatter, is shown in figure 7. 

The depths and velocities at many times during the 

flow were computed for the downstream cross section, 

and these depths and velocities were then used with 

figure 3 to compute a graph of discharge of sands at the 

downstream end of the reach through the period of flow. 

The curve of volume of sand accumulation (fig. 7) within the reach was determined from the differences between the inflow and the outflow of sands during short 

increments of time. This cumulative curve is also 

shown as depth of average deposit on figure 7. Similar 

sediment accumulation would be caused if the velocity 

at the downstream end of the reach was reduced 20 percent for a given depth of flow by increased resistance 

such as might result from growth of small willows in 

the channel. In a straight, uniform channel, a change 

of slope or of roughness that causes fill while the flow 

is decreasing can also be expected to cause fill while 

the flow is increasing.

Of course, large amounts of fill over a long period of 

time are inconsistent with the maintenance of a uniform 

channel. A uniform channel was assumed to simplify 

the study of the principles of scour and fill as much as 

possible.

Obviously, an assumed increase in slope or decrease 

of roughness toward the downstream end of a 5,000- 

foot reach of uniform channel of the standard cross section would result in computed scour during periods of 

both increasing and decreasing flow. Also, the amount 

of the scour in the reach would, except for the usually 

small effect of the change in the hydrograph of flow as

the water passes through the reach, be about the same 

as the amount of fill shown in figure 7, if the hydrographs of flow and the graphs of discharge of sands 

were assumed to apply at the opposite ends of the 

reach that is, if the 20-percent lower velocities for a 

given depth existed at the upstream end of the reach 

rather than at the downstream end.

CONSTRICTION IN A STRAIGHT, GENERALLY 

UNIFORM CHANNEL

Consider, next, the scour and fill at a narrower-thanaverage cross section in a long straight reach that has 

the standard cross section (fig. 4) except at the contraction. At a cross section A in the uniform channel upstream from the narrow section, the relations of depth 

and mean velocity to flow are assumed to be those shown 

in figure 4, and the discharges of sands are assumed 

to be given by the curves in figure 3. Also, these relations and figure 3 are assumed to apply at a cross section 

C that is downstream from the narrow section and 

1,000 feet downstream from section A. A suitable distance downstream from section A, the width of the 

channel slowly and smoothly decreases to an assumed 

bottom width of either 75 or 66 feet (9 or 18 ft less than 

the bottom width at section A). The bottom widths of 

75 or 66 feet are assumed to be constant for a short 

distance before the channel gradually widens to the 

standard cross section and bottom width of 84 feet. The 

channel banks are assumed to have a 45° slope throughout the reach. Some computations of scour and fill will 

be based on a cross section B that is 200 feet downstream 

from section A and in the narrow section. Because the 

transitions are slow and smooth and because section B 

is in a short reach of straight channel, the relationship 

of mean velocity to turbulence at section B will be assumed to be normal for a sand-bed reach at the computed depths and velocities. Hence, figure 3 will be 

assumed to apply at section B as well as at sections A

andC.

If sections A and C are far enough upstream and 

downstream from the narrow reach, the net scour and 

fill between these sections would be the same as for a 

straight, uniform 1,000-foot reach. At constant flow, 

neither net scour nor net fill would occur in the reach. 

If the flow indicated by the hydrograph of figure 5 

passes through the reach, the net accumulation of sands 

(expressed in terms of volume) within the reach would 

be about one-fifth as much as that for the 5,000-foot 

reach (fig. 6), but the average net depth of accumulation 

of sands in the reach would be about the same as that for 

the 5,000-foot reach. The average elevation of the 

streambed in the 1,000-foot reach would be about the 

same after the flow has passed through the reach as 

before the flow arrived.
SCOUR AND FILL IN SAND-BED STREAMS D15

ACCUMULATION OF SANDS IN REACH,IN CUBIC FEET

AVERAGE DEPTH OF ACCUMULATION OF SANDS IN REACH,IN FEET

3g0) ?

!*

QNOD3S 83d 133J DIQflO Nl

; ynoH H3d 133J oiano NI 'SQNVS jo 39a\/H3sio
D16 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

Two types of information were computed for five 

different rates of constant flow. One type is the initial 

rate of volume of net scour in the reach between sections 

A and B. This initial rate of net scour is numerically 

equal to the initial volume rate of net fill in the reach 

between sections B and C. The initial rate of scour or 

of fill, respectively, means the difference between the 

discharges of sands from and into or into and from the 

reach, provided that the discharges of sands are based 

on positions of the streambed before these positions are 

changed by the flow. Thus, if a streambed is generally 

flat (although ripples, dunes, bars, or antidunes may 

cause local irregularities) before flow begins, the initial 

rate of scour in a reach between a wide and a narrow 

cross section is the difference between the discharges of 

sands at the narrow section and at the wide section before the streambed scoured noticeably at the narrow 

section. The significance of the initial rate of scour or 

fill is that it represents a maximum rate of removal or 

accumulation of sands in the reach for a particular rate 

of streamflow. The second type of information is the 

approximate depth of average scour to be expected at 

section B after flow has continued at the constant rates 

until an equilibrium cross section or bed position has 

been reached at section B.

The computed initial rates of scour in the 200-foot 

reach between section A and section B (table 4) are 

based on the assumptions that the sand that is removed 

from the channel bed weighs 85 Ib per cu ft and that 

the depths of flow at sections A and B are equal at the 

beginning of each flow at a constant rate. The steps in 

the computation of the initial scour are shown in table

4 and are rather obvious. The discharges of sands 

(table 2, column 6) at the constant rates of flow had 

already been computed for sections such as A and C. 

The depths of flow (table 2, column 2) and the geometry 

of section B determine the surface widths and the crosssectional areas of section B. The streamflow divided by 

the area gives the mean velocity, and the mean velocity 

and the depth define the discharges of sands per foot 

of width in figure 3. The rate of initial scour in the 

reach between section A and B is the difference in discharges of sands at the ends of the reach. The net 

average depth of scour per hour is obtained by dividing 

the volume of scour by the product of the length of the 

reach and an effective bottom width of the reach. The 

effective bottom widths were assumed to be 78 and 72 

feet for the two different reductions in bottom width. 

Of course, the scour at section B probably would be at 

least twice the net average depth of scour and may well 

be somewhat more than twice the average.

A precise effective width of channel subject to scour, 

fortunately, is not necessary for computing approximate 

average depths of scour in the reach for different flows 

and reductions in channel width. In the computations 

for table 4, only the flat bed of the channel was assumed 

to be subject to scour. For a rough computation, both 

the depth of scour and the reduction in channel width 

can be assumed to vary directly with distance from section A. Under these assumptions, the effective width 

of flat bed subject to scour in the reach would be Wi~ 

2(TF1-TT2 )/3 or 78 and 72 feet, respectively, for the 

reductions of 9 and 18 feet in channel width. (Wi and 

W2 are the bottom widths at sections A and B.) The

TABLE 4. Computed initial rates of scour between cross sections A and B at different flows for which the depths of flow above the flat R

beds are assumed to be equal at the two sections

Streamflow (cfs)

Section A

Discharge of 

sands (cu ft 

perhr)

Section B, bed not scoured by flow

Depth of flow 

above the flat bed 

(ft)

Area (sq ft)

Mean 

velocity (fps) Surface width (ft)

Discharge of 

sands (cu ft 

perhr)

Scour between sections or difference In 

discharges of sands (rounded)

Cu ft per hr

Ft of net average 

scour per hr

Bottom width 9 ft less at section B than at section A

100

500

1,000

2,000

3,000

9.5

330

1,060

3,030

5,200

0.97

2.68

4. 10

6. 20

8.00

73.7

208

325

504

664

1. 36

2. 40

3.08

3.97

4. 52

77.0

80.4

83. 2

87.4

91.0

20

506

1,500

3,760

6,280

10

180

440

730

1,080

0 0006

012

028

047

069

Bottom width 18 ft less at section B than at section A

100

500

1,000

2,000

3,000

9.5

330

1,060

3,030

5,200

0.97

2.68

4. 10

6. 20

8.00

64.9

184

288

448

592

1. 54

2.72

3.47

4. 46

5.07

68.0

71.4

74. 2

78.4

82.0

41

770

2, 150

4,940

7,870

31

440

1,090

1,910

2,670

0 002

031

076

133

185
SCOUR AND FILL IN SAND-BED STREAMS D17

effective width can be multiplied by the length of the 

reach and by the average depth of scour to obtain the 

volume of scour under the stated assumptions. This 

effective width is less than the average bottom width because the depth of scour increases as the width decreases. 

Of course, the net average depth of scour would be a 

little less if the width of water surface had been used 

rather than the width of the flat bed.

The 18-foot decrease of bottom width, which represents roughly a 20-percent decrease in area, produced 

initial rates of scour between sections A and B that were 

about 2.5-3.0 times the rates of scour that were caused 

by the 9-foot decrease in bottom width, which represents 

roughly a 10-percent decrease in area. (See table 4.)

Of course, the initial rates of.scour would begin to 

decrease immediately because the scour would reduce the 

mean velocity at section B and hence decrease the difference in discharges of sands between sections B and 

A. The equilibrium bed positions at section B, the 

positions for which the discharges of sands at sections 

A and B are equal, were computed for the same decreases in bottom width and the same five constant flows 

that were used in the computation (table 4) of initial 

rates of scour.

The computation of average depth of scour at equilibrium bed position at section B is indicated in table 

5 and begins with the assumption that the discharges 

of sands are to be the same at sections A and B. The 

discharge of sands at each constant rate of flow is taken 

from column 6 of table 2 and is divided by the surface 

width at section B (the water-surface slope is assumed 

to be the same as though the width were not contracted) 

to obtain the equilibrium discharges of sands per foot of

width at section B. The difference in discharges of 

sands per foot of width at sections B and A requires an 

increase in mean velocity from section A to section B, an 

increase that can be approximated from figure 3, and 

the mean velocity at section B is computed from that at 

section A (table 2, column 3) by adding this increase. 

The flow is divided by the mean velocity to obtain the 

cross-sectional area necessary to discharge the flow. If 

the banks are considered to remain stable at section B 

and all scour is assumed to occur at the flat bed, the 

average depth of flow above the flat bed can be computed from the geometry of section B. The difference 

between this depth of flow and the depth of flow above 

the flat bed at section A (table 2, column 4) is the indicated average depth of scour of the flat bed at section B.

The computations of average depth of scour at 

equilibrium indicate at least three significant points. 

One is that at the narrow section the depths of scour, 

like the initial rates of scour between sections, increase 

fairly rapidly as the flow increases. A second point is 

that at a given flow the depth of scour is about directly 

proportional to the reduction in width. A third point 

is that the length of the reach does not influence the 

computations, and hence the equilibrium depth of average scour at the narrow section is independent of the 

length of reach between the sections unless the reach 

is so short and the change in cross section and streamlines so abrupt that the depths and mean velocities 

cannot be satisfactorily determined or that the relations 

of figure 3 do not apply.

The standard hydrograph of flow and the graph of 

discharge of sands (fig. 5) were assumed at section A 

and also at section C except for a few minutes lag

TABLE 5. Average scour at section B after five constant flows have established equilibrium bed positions

Streamflow 

(cfs)

Section A

Discharge of 

sands (cu ft 

per hr)

Section B

Surface width 

(ft)

Discharge of 

sands (cu ft per 

hr per ft of 

width)

Difference between sections A 

andB

Discharge of 

sands (cu ft per 

hr per ft of 

width)

Mean velocity 

(fps)

Section B

Mean velocity 

(fps)

Cross sectional 

area (sq ft)

Depth of flow 

above the flat bed 

(ft)

Average scour 

of flat bed 

(ft)

Bottom width 9 ft less at section A than at section B

100

500

1,000

2,000

3,000

9.5

330

1,060

3,030

5,200

77.0

80.4

83. 2

87.4

91.0

0. 123

4. 10

12. 7

34.6

57. 2

0.013

.40

1. 2

3. 1

5. 2

0 01

05

06

09

11

1.22

2.20

2. 82

3.67

4. 19

82

227

355

545

717

1.08

2.93

4.50

6. 75

8. 70

0 11

25

40

55

70

Bottom width 18 ft less at section A than at section B

100

500

1,000

2,000

3,000

9.5

330

1,060

3,030

5,200

68.0

71.4

74. 2

78.4

82.0

0. 14

4. 60

14. 3

38.6

63.4

0.03

.9

2.8

7. 1

11.4

0 03

10

14

19

25

1.24

2. 25

2.90

3.77

4. 33

81

222

345

530

693

1. 20

3.25

4.99

7.46

9.56

0. 2

.5

. 8

1. 2

1. 5
D18 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

for the passage of the flow through the 1,000-foot reach; 

scour was computed at section B for reductions in 

bottom width of 9 and 18 feet. At the beginning of 

the standard hydrograph, the depth of the low flow 

was assumed to be the same at all three cross sections. 

Computations were made for 20-minute intervals before 

the peak discharge at section A and for 30-minute or 

longer intervals after the peak. No time lag was applied for the 200-foot reach between sections A and B, 

and scour was assumed to be limited to the flat bed. The 

average depth of scour in the reach was based on effective bottom widths of 72 and 78 feet for the two different reductions in bottom width, and the depth of scour 

at section B was assumed to be double the average 

depth of scour for the reach.

The same type of computation was made for each 

successive time interval. The depths and streamflows at 

section B at the beginning and at the end of the first 

time interval and the channel geometry were used to 

compute the cross-sectional areas and the velocities at 

section B at the beginning and end of the interval. 

From the depths, velocities, and widths at section B, 

the discharges of sands at the beginning and end of the 

time interval were computed from figure 3. The difference between the discharges of sands from and into the 

reach during the interval was used to compute the 

volume of scour and the average depth of scour in the 

200-foot reach. This average depth was then doubled 

to obtain an assumed, but perhaps somewhat too small, 

depth of scour at section B. Next, a similar computation was made for the second time interval except that 

the discharge of sands at section B at the beginning of 

this time interval did not have to be computed unless the 

computed scour at the end of the first interval was large 

enough to require a recomputation of the discharge of 

sands. The same procedure was followed for each successive interval except that as the scour became appreciable, the amount of scour at the end of the interval at 

section B was estimated in advance. A recomputation 

was necessary only if the computed scour differed significantly from the estimated scour. The details of the 

computations for the reduction of 18 feet in bottom 

width are shown in table 6.

The cumulative depths of computed scour at section 

B (fig. 8), although inexact, indicate qualitatively the 

general behavior of scour and fill at a contracted section 

during the passage of an assumed variable flow. The 

average elevation of the bed at the narrow section is 

lowest an appreciable time after the peak of the flow 

when the discharge of sands at the narrow section has 

decreased until it exactly equals the discharge of sands 

upstream from the narrow section. The depths of 

scour are roughly in proportion to the percentage reductions of bottom width. Also, much of the scour is not

replaced by fill during the recession of this particular 

hydrograph of flow. In general, the recession side of a 

hydrograph of flow is long; and in many sand-bed 

streams, even low flows transport appreciable quantities 

of sands; hence, soon after a flood has passed, the streambed at a narrow section may have filled as much as it 

scoured during the earlier part of the flood. Narrow 

cross sections of a sand-bed stream are likely to be characterized by alternate scour and fill with the passage 

of each flood, but the actual amounts of scour and fill 

depend partly on the form and size of the hydrograph 

of flow and partly on the initial position of the streambed.

The maximum computed depth of scour at the narrow section is roughly 40 percent of the equilibrium 

depth of scour for the peak flow of 3,000 cfs (table 5). 

The figure of 40 percent, as well as the amount of fill 

after the flow has passed and the time between the peak 

of the flow and the maximum depth of scour, is largely 

a function of the shape, size, and duration of the hydrograph of flow and of the graph of sands.

Although little scour or fill occurs at representative 

cross sections such as A and C of the assumed uniform 

channel, a pattern of alternate scour and fill is characteristic at a narrow section. (Of course, the scour and 

fill that are discussed here are based on an average bed 

elevation across a channel; such very local irregularities 

as sand dunes are assumed to be smoothed out in the 

computation of the average bed elevation at a cross section.) Because most information on scour or fill has 

been obtained at streamflow measuring sections, which 

are usually at narrower-than-average parts of the channel or at structures that constrict the flow laterally, the 

untypical alternation of scour and fill is often thought 

to be representative for sand-bed streams (Lane and 

Borland, 1954). On the other hand, a sand bed whose 

average elevation is stable during periods when the flow 

changes greatly indicates uniformity of width of flow 

and of available bed sand with distance along the 

channel.

The principles of scour and fill apply, of course, to 

the reach between sections B and C. If this 800-foot 

reach is long enough for normal flow and discharge of 

sands to become reestablished in the straight, uniform 

channel, the discharge of sands at section C differs 

from that at section A only by a few minutes lag. 

Slight adjustment can be made for this lag, or it can 

be disregarded. In simple terms and with reasonable 

accuracy, any net volumes of sands that were removed 

from the reach between sections A and B can be considered to be balanced by equal volumes of net deposit 

in the reach between sections B and C. This broad relation exists in spite of the facts that the bed at section
SCOUR AND FILL IN SAND-BED STREAMS D19

6.01 6000

i.O

o 

tr 

cr

cc. 

^ 

o 

<_> 

to

O.I

0.01

Scour at 66-foot bottom width

of 75-foot bottom width

4 6 8 10 12 14 16 18 

TIME, IN HOURS, FROM BEGINNING OF INCREASED FLOW

FIGURE 8. Depths of computed scour at narrow section of an assumed 1,000-foot reach of

straight channel.

PROFILE

FIGURE 9. Schematic profile of bed position at and near a narrow section of an assumed straight sand-bed channel after a short

period of constant flow.
TABLE 6. Computed scour at cross section where bottom width is 66 feet in channel whose uncontracted bottom wid

Column 1: Time from beginning of increased flow.

Column 2 : Flow from hydrograph on figure 5.

Column 3: Depth, or stage, above flat bed from figure 4.

Column 4 : Mean velocity from figure 4.

Column 5: Discharge of sands per foot of width from figure 3.

Column 6: Surface width of wide section is bottom width of 84 feet plus twice the

depth.

Column 7: Discharge of sands per foot of width times the surface width. 

Column 8: Surface width of narrow section is bottom width of 66 feet plus twice the

depth at the wide section or the surface width at the wide section minus 18 feet. 

Column 9: Bottom width of 66 feet plus the depth at the wide section. 

Column 10: The average width from column 9 minus the computed scour from column

22. For a first trial, the computed scour is estimated. 

Column 11: The same depth as given in column 3 for the wide section. 

Column 12: The depth from column 11 plus the computed scour from column 22. 

Column 13: The product of the. average width from column 10 and the depth from

column 12. 

Column 14 : The velocity is obtained by dividing the flow from column 2 by the area

from column 13.

Column 15 : The discharge of sands per foot of width f

column 12 and the mean velocity from column 14. 

Column 16 : The discharge of sands from column 15 

from column 8.

Column 17: The difference in discharge of sands from c

Column 18: Graphically estimated difference in discha

the times given in column 1. (Figures used only for r

Column 19: Volume removed from reach during period 

column 1. Volume is equal to the product of the le

average difference in discharge of sands in cubic fee

columns 17 and 18.

Column 20 : Cumulative volume of sands removed from re

Column 21: Average depth of scour in reach is compu

from column 20 divided by the product of the 200-f

estimated average bottom width of 72 feet. 

Column 22: The average depth of scour in the reach, fr

This assumes a straight-line variation of the scour fr

end of the reach to a maximum at the narrow section.

Time 

(hr)

(1)

0:00 

0:20 

0:40 

1:00 

1:20 

1:40 

2:00 

2:20 

2:40 

3:00 

3:30 

4:00 

4:30 

5:00 

5:30 

6:00 

7:00 

8:00 

9:00 

10:00 

11:00 

12:00 

14:00 

16:00 

18:00 

20:00 

24:00 

30:00 

36:00 

42:00

Upstream section, 84-foot bottom width

3sy

£ 

o

&

(2)

10 

20 

44 

100 

210 

460 

1,000 

2,000 

2,700 

3,000 

2,800 

2,230 

1,700 

1,280 

960 

740 

470 

320 

230 

175 

140 

115 

84 

65 

52 

43 

31 

20 

13 

10

"5

<c

o> 

£> 

O 

.C

OJJ?

5^ 

%M

Q 

(3)

0.22 

.35 

.58 

.97 

1.56 

2.53 

4.10 

6.20 

7.50 

8.00 

7.70 

6.70 

5.70 

4.80 

4.00 

3.40 

2.60 

2.00 

1.63 

1.36 

1.20 

1.06 

.86 

.74 

.64 

.56 

.46 

.35 

.26 

.22

& 

o

>

CH W

3 OI* 

(4)

0.54 

.68 

.90 

1.21 

1.59 

2,10 

2.76 

3.58 

3.93 

4.08 

4.00 

3.70 

3.40 

3.00 

2.75 

2.50 

2.12 

1.85 

1.63 

1.50 

1.37 

1.28 

1.14 

1.03 

.96 

.90 

.80 

.68 

.58 

.54

 §5

§ 0

2*2

O O fc.

O> -U X 

6JJ 0 ft 

03 °**

f&

Q 

(5)

0.11 

.72 

3.3 

11.5 

31.5 

47 

52 

48 

37 

27 

16.5 

11.5 

7.6 

3.4 

1.7 

.86 

.50 

.27 

.17 

.07 

.02

5-O 

'%

lH

M

(6)

84.4 

84.7

85.2

85.9 

87.1 

89.1 

92.2 

96.4 

99.0 

100.0 

99.4 

97.4 

95.4 

93.6 

92.0 

90.8 

89.2 

88.0 

87.3 

86.7 

86.4 

86.1 

85.7 

85.5 

85.3 

85.1 

84.9

84.7 

84.5 

84.4

I § "fs

0 w fc,

g§&

lls

Wrfl U g«~

(7)

9.5 

63 

294 

1,060 

3,030 

4,630 

5,200 

4,770 

3,600 

2,570 

1,550 

1,060 

690 

304 

150 

75 

43+ 

23 

15- 

6.0 

1.7

Downstream section, 66-foot bottom width

,g 

 o 

%

I 

1 

(8)

66.4

66.7 

67.2 

67.9 

69.1 

71.1 

74.2 

78.4 

81.0 

82.0 

81.4 

79.4 

77.4 

75.6 

74.0 

72.8 

71.2 

70.0 

69.3 

68.7 

68.4 

68.1 

67.7 

67.5 

67.3 

67.1 

66.9 

66.7 

66.5 

66.4

Average 

width

fH

O

o^ 

oS

2

(9)

66.2

66.4 

66.6 

67.0 

67.6 

68.5 

70.1 

72.2 

73.5 

74.0 

73.7 

72.7 

71.7 

70.8 

70.0 

69.4 

68.6 

68.0 

67.6 

67.4 

67.2 

67.1 

66.9 

66.7 

66.6 

66.6 

66.5 

66.4 

66.3 

66.2

ts

sS

t»

(10)

66.4 

66.6 

67.0 

67.6 

68.5 

70.1 

72.1 

73.3 

73.7 

73.3 

72.2 

71.1 

70.2 

69.4 

68.8 

68.0 

67.4 

67.0 

66.8 

66.6 

66.5 

66.3 

66.1 

66.0 

66.0 

65.9 

65.8 

65.7 

65.6

Depth 

above fiat 

bed

(_

la 

oS-

"A

(11)

0.22

.35 

.58 

.97 

1.56 

2.53 

4.10 

6.20 

7.50 

8.00 

7.70 

6.70 

5.70 

4.80 

4.00 

3.40 

2.60 

2.00 

1.63 

1.36 

1.20 

1.06 

.86 

.74 

.64 

.56 

.46 

.35 

.26 

.22

 3

o ^2- 

CO

(12)

0.35 

.58 

.97 

1.56 

2.54 

4.14 

6.31 

7.71 

8.31 

8.13 

7.23 

6.29 

5.43 

4.64 

4.04 

3.24 

2.64 

2.26 

1.99 

1.82 

1.68 

1.48 

1.36 

1.26 

1.18 

1.08 

.97 

.88 

.84

ts 

Itf

<a^n

°c 

S'-s

«!<

(13)

14.6

23.2 

38.6 

65.0 

105 

174 

290 

456 

565 

612 

596 

521 

448 

381 

322 

278 

220 

178 

151 

134 

121 

112 

98 

90 

83 

77.8 

71.2 

63.8 

57.8 

55.1

£ 

o

>

§1

i~

(14)

0.68

.86 

1.14 

1.54 

2.00 

2.64 

3.45 

4.38 

4.78 

4.90 

4.70 

4.28 

3.80 

3.36 

2.98 

2.66 

2.14 

1.80 

1.52 

1.31 

1.16 

1.03 

.86 

.72 

.63 

.55 

.44 

.31 

.22 

.18

 35 

§S_ w £ £ 

"3 "5 t.

So £

111

Q 

(15)

0.07 

.60 

2.55 

9.5 

28 

65 

83 

88 

79 

60 

41 

25 

15.7 

9.7 

3.7 

1.45 

.57 

.21 

.08 

.022

"° 0

11? 

l-a;

HI

5

(16)

5 

41 

176 

675 

2,080 

5,100 

6,720 

7,210 

6,430 

4,760 

3,170 

1,890 

1,160 

706 

264 

102 

40 

14 

5 

2

Difference in discharges of sands 

between sections

g 

1

 gS 

H+>*s 

 <

(17)

5 

31 

113 

381 

1,020 

2,070 

2,090 

2,010 

1,660 

1,160 

600 

340 

100 

16 

-40 

-48 

-35 

-29 

-18 

-13 

-6 

-1.7

gfe

*5

Is

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till

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(18)

12 

70 

200 

610 

1,500 

2,080 

2,050 

1,860 

1,410 

840

3

 O 

O

I

bfl

 c~

7 «* 

Q 

(19)

0.4 

5 

24 

74 

218 

507 

693 

683 

924 

705 

430 

235 

110 

29 

-12 

-44 

-42 

-32 

-24 

-16 

-19 

-8

£

_5

0>

>

3s"3 

6 

a 

0

(20)

0.4 

5 

29 

103 

321 

828 

1,521 

2,204 

3,128 

3,833 

4,263 

4.498 

4,608 

4,637 

4,625 

4,581 

4,539 

4,507 

4,483 

4,467 

4,448 

4,440 

4,440 

4,440 

4,440 

4,440 

4,440 

4,440

Note. The computed scour from column 22 was first needed in the computations for 

columns 10 and 12. At that point in the computations, the scour is estimated and the 

computations of the computed scour are completed. If the estimated scour and the

computed scour do not agree very closely, the computatio

are repeated with either the first computed scour or a

first computed scour.
SCOUR AND FILL IN SAND-BED STREAMS D21

B generally scours when the reach from A to B scours 

and the bed at section C generally shows little net scour 

or net fill. Therefore, the volume of fill in the reach 

between sections B and C must roughly balance both 

the volume of scour in the upstream end of the reach 

between sections B and C and the volume of scour in 

the reach between sections A and B. If the reach downstream from the narrow section is longer than the reach 

upstream from the section, the net average depths of 

fill in the downstream reach must be less than the net 

average depths of scour in the upstream reach. The 

bed profile for a constant flow must have the general 

shape indicated in figure 9. Obviously, the reach between sections B and C must be long, or the flow and 

discharge of sands at its downstream end may be 

affected appreciably by the contraction and expansion 

of the flow.

WINDING STREAMS

The discharge of sands at bends of a sand-bed stream 

may be significantly different from that indicated by 

figure 3 because the turbulence for a given depth, mean 

velocity, water temperature, and bed material may be 

different from the turbulence in a straight reach of 

channel. Although a relation may exist between the 

discharge of sands at a bend and the characteristics of 

flow and sediment, the relation would be complex and 

has not yet been defined. Hence, one of the principles of 

scour and fill cannot now be used at a stream bend; but 

the principle of continuity does, of course, apply. Also, 

both principles can be applied at reasonably straight 

reaches of channel between bends provided that these 

reaches are long enough for the flow to entrain about its 

equilibrium discharge of sands. Thus, in some ways 

a channel reach containing a bend is like a straight 

channel reach that contains a narrow section.

The amount of scour and fill at a bend may be difficult 

to determine, but the net volume of scour or fill within 

a reach that includes the bend can be computed if the 

difference in sediment discharges at the ends of the 

reach can be measured or computed. A general restriction on the scour of sands from a long reach of winding 

channel is, of course, imposed by the total discharge of 

sands at any cross section. Thus, the total discharge of 

sands should be about 13,000 cubic feet (fig. 5) or about 

130 cubic feet per foot of surface width at maximum 

stage for the standard cross section in a fairly straight 

reach of channel (0.30-mm median diameter of the bed 

sediment) and for the hydrograph of flow and the discharge of sands that are given in figure 5. In a reach 

of channel 1 mile long that has no inflow of sands, the 

indicated average depth of net scour during the passage 

of the flow would be only 0.02-0.03 foot. For five constant rates of flow, the discharges of sand per hour

per foot of width of the standard cross section would 

be approximately the same as those given in table 2, 

column 5. At the highest of the constant flows, 3,000 

cfs, the indicated net scour would be about 0.01 foot per 

hour for a reach 1 mile long having no inflow of sand. 

Volumes of discharge of sand, in cubic feet per hour per 

foot of width, are given in figure 3 for many combinations of depth and mean velocity in sand-bed channels 

and for water temperatures of 60° F, bed sediment of 

0.30-millimeter median diameter, and low concentrations of fine sediment. These volumes of sand can be 

converted to net scour, in feet per hour, of the sand bed 

of uniform-width channels by subtracting the inflow of 

sands per foot of width and dividing the difference by 

the length of the channel reach, in feet. Of course, if 

a channel reach cannot be considered to have about a 

uniform width, the volumes of sand per foot of width 

per hour at different flows can be multiplied by the 

channel width at the downstream end of the reach, any 

inflow of sands per hour can be subtracted, and the difference can be divided by the area of the bed of the 

reach to obtain a net depth of average scour, in feet per 

hour. Obviously, large depths of scour or fill in a winding channel like those in a straight channel usually are 

not typical of long reaches of streams.

SCOUR AND FILL IN NATURAL STREAMS

Most natural streams differ considerably from the 

assumed channels for which computations of scour and 

fill have been made for this report. The comparatively 

few streams that have continuous sand beds usually have 

somewhat variable widths, uneven lateral distributions 

of depth and velocity, and some change in particle size 

with distance along the channel. Many natural streams 

are pool-and-riffle streams and have mean velocities, 

widths, depths, and particle sizes that change widely and 

rapidly with distance along the channel. Although the 

pools may have sand beds much or all of the time, the 

pool-and-riffle streams as a whole are not really sandbed streams.

The differences in pattern of scour and fill will be 

shown for only three natural streams.

A GENERALLY STABLE SAND-BED STREAM

Changes of bed position in a generally stable sandbed stream have been defined by many streamflow measurements or cross-section soundings under two cableways over the Elkhorn River near Waterloo, Nebr. 

(Beckman and Furness, 1962, table 2). The general 

stability of the sand bed implies few, if any, large 

changes in bed elevation for a stream reach of appreciable length. This stability exists in spite of frequent 

and rapid changes in bed elevation at individual points 

and even appreciable changes in average bed elevation
D22 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

at entire cross sections (Beckman and Furness, 1962, 

p. 12). The average bed positions at the two cableways, 

which are about 1,670 feet apart, were usually determined at about the same time. The width of the water 

surface at the upstream cableway varied, partly with 

stage and partly with time, from 224 to 283 feet over 

a period of 2 or 3 years. Usually the width was at least 

245 feet; therefore, this width was used in computing 

changes in the bed elevation. Changes in bed elevation 

outside of this width were disregarded. The width of 

the water surface at the downstream cableway was 

usually 285-290 feet, although some widths were a little 

outside of this range. Because the range in width was 

small, the changes in bed elevation at the downstream 

cableway were based on the reported area and width at 

the time of each measurement. The width of the 

channel at the upper cableway was probably significantly less than the average width for the reach of 

channel, and the width of channel at the lower cableway was probably about average for the reach near the 

cableways.

The cableways are in a reach of channel whose bed 

is composed almost entirely of shifting sand that had 

a median diameter of about 0.24 millimeter in 1952 when 

many of the observations were made. The bed shifts 

readily. Driftwood occasionally lodges on or near the 

cableways and somewhat distorts the streambed and the 

pattern of the flow. Also, scour and fill at the upstream 

end of the reach may be affected somewhat by the shifting shape and elevation of a large sand bar a few 

hundred feet upstream from the upper cableway and 

by sometimes active erosion of the right bank about 

opposite the bar. Computed average changes of 0.1 or 

0.2 foot in elevation of the wide and shifting streambed 

at either cableway section may be practically meaningless as measures of scour or fill in the reach.

As Beckman and Furness (1962, p. 12) concluded 

earlier, no clearcut relationship is apparent between 

flow or changes in flow and net scour or fill (figs. 10-13). 

Of course, the position of the streambed generally 

changes faster at high flow than at low flow, but the 

direction of the change is not consistent from one cableway section to the other. The change in average bed 

elevation at the downstream cableway was determined 

from consecutive cross-section soundings for 34 times 

when the difference in flow between the times of the 

soundings was 5 percent or more. Of the 11 times when 

the flow increased 5 percent or more, the computed bed 

elevation increased 3 times at the upstream cableway 

and 8 times at the downstream cableway. If only 

changes greater than 0.20 foot are considered, the computed bed elevation increased one time out of eight at 

the upstream cableway and five times out of five at the 

downstream cableway. Of the 23 times when the flow

decreased 5 percent or more, the computed bed elevation 

increased 16 times at the upstream cableway and 9 times 

at the downstream cableway. If only changes greater 

than 0.20 foot are considered, the computed bed elevation for decreases of flow increased 12 times out of 15 

at the upstream cableway and 3 times out of 7 at the 

downstream cableway. Thus, whether all changes of 

the bed elevation or only those greater than 0.20 foot 

are considered, the tendency is for the slightly narrower 

upstream cross section to scour on increasing flow and 

to fill on decreasing flow, as narrow sections are genUJ 20,000

10,000

_ -2

So, -3 

0= O

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+

IB 19 20

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21

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22 23

+

24 

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29

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ns

4

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III

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IT 10

FIGURE 10. Changes in average gage height of ttie streambed of 

the Elkhorn River near Waterloo, Nebr., during a period of 

high flow in 1951.

-

26 27 

1

26 

JAf

29 

ICH

30 31

++

1

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2

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ew jy

FIGURE 11. Changes in average gage height of the streambed of the 

Elkhorn River near Waterloo, Nebr., during a period of high flow 

in 1952.
SCOUR AND FILL IN SAND-BED STREAMS D23

erally likely to do. The wider downstream cross section shows a slight tendency to fill on increasing flow 

and to scour on decreasing flow, but this tendency may 

not be significant.

The lack of agreement between the computed net 

scour and net fill at the two cableways is clear from a 

graph of change in bed elevation at one cableway (fig. 

14) against change in bed elevation at the other. Of 42 

comparisons, the computed change in average bed position at the downstream cableway was in the same direction as that at the upper cableway 22 times, and once 

a large fill at the upper cableway was accompanied by 

no change at the lower cableway. A reasonable conclusion from figure 14 seems to be that the two cross sections do not show any significant tendency to scour 

or to fill at the same time. Hence, measured scour or 

fill at either cross section is unlikely to represent average scour or fill along a channel reach near the cableways. Also, the lack of consistency indicates that net 

scour or fill is not governed directly by changes either 

in flow or in sediment concentration, because the rate 

of flow and the concentration of sediment are nearly 

the same at the two cross sections. The upstream cross 

section does show somewhat greater amounts of net 

scour and fill, as a narrower section is likely to, than 

the downstream section. At the downstream section, 

changes of the average bed position seem to be nearly 

random, as they should be for a uniform sand-bed

Q 20,000 

o

£ 10,000

a:

LU 

Q_

CD

:Z> 

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± 1000

O ~

CO

- 4

APR

 Sect 

+ Sect

.% 

; \ .

 

%

MAY

ion at upstrean 

ion at downstre

;* ;

JUNE

i cableway 

am cableway-

+ 

+

JULY

FIGURE 12. Changes in average gage height of the streambed of the Blkhorn River near Waterloo, Nebr.,

May 18-July 19, 1951.
D24 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

0

OD

-2

-3

-4

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% + + * +

APR

 

h + + ^ *

MAY

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-T +

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AUG

 

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mstream (

SEPT

, 

+

ileway - 

;ableway

OCT

FIGURE 13. Changes in average gage height of the streambed of the Elkhorn River near Waterloo, Nebr., April 1

October 23, 1952.

channel, except, of course, that the bed position changes 

faster at high flows than at low flows.

The general behavior of the streambed at the downstream cableway on the Elkhorn River near Waterloo 

is probably similar to that of the Colorado River at 

the site of Imperial Dam before the dam was constructed. At that site, too, measurements at a cableway 

at a laterally unconstricted section showed fluctuations 

of the streambed but no appreciable net scour or fill even 

during a high flow (Lane and Borland, 1954, p. 1075).

NAEEOW SECTION OF A NATUEAL SAND-BED STEEAM

Streamflow measurements made from a bridge over 

the Missouri River near Yankton, S. Dak., during 1941 

show scour and fill (B. R. Colby and R. E. Oltman, written communication) that may occur at a cross section 

where the flow of a sand-bed stream is laterally constricted (fig. 15). Measurements were not made at or 

near all peak flows, but the available measurements 

show clearly that the streambed usually was lower than 

its average position whenever the flow was high. After 

the high flow ended, the streambed filled again to about 

the original position. The alternation of scour and fill 

is typical behavior at a narrow cross section of a sandbed stream. A somewhat similar pattern of scour and 

fill was shown by the computations for the narrow section in the assumed uniform channel. The amounts of 

net scour or fill at this narrow cross section are not, 

however, representative for the reaches upstream and 

downstream from the measuring section.
SCOUR AND FILL IN SAND-BED STREAMS D25

-pl.Sft

M

-1.0 ft

= --0.5ft

Scour of downstreom coblewoy

1.0 ft 0.5ft

Fill at downstream cableway

0.5ft l.O ft

--0.5ft

--I.O ft

-1- I 5 ft

FIGURE 14. Comparison of scour and fill computed 

from changes in bed position at two cableway sections on the Elkhorn River near Waterloo, Nebr.

A POOL-ANIX-BIFFLE STREAM

Although pool-and-riffle streams are not strictly sandbed streams, they are discussed here because data from 

such streams have sometimes been used as indications 

of the scour and fill to be expected at sections of sandbed streams.

A pool-and-riffle stream is one in which the watersurface slope alternates between long and comparatively 

level reaches (the pools) and short and comparatively 

steep segments (the riffles). Such a stream may have 

a sand bed throughout most pools, but its riffles normally 

have beds of material that is coarser than sand. The 

presence of the coarser bed material indicates that the 

quantity of available sand at the riffles is less than the 

stream can transport at the riffles. The mean velocity 

in a pool-and-riffle stream varies widely from one cross 

section to another along the channel, whereas continuity 

of the discharge of bed material and the abundant availability of sands prevent large changes in mean velocity 

from one section to another along a sand-bed stream.

STREAMFLOW, SEDIMENT CONCENTRATION, AND 

CHANGES IN BED ELEVATION

Leopold and Haddock (1952, 1953) and Brooks 

(1958) attempted to explain scour and fill at the streamflow-measuring section on the Colorado River near 

Grand Canyon, Ariz., on the basis of an assumed tendency for the depth of flow to adjust itself to changes

< O 100

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FIGURE 15. Streamflow and changes in average bed position of the Missouri River near Yankton, S. Dak., 

March-September 1941. After Colby and Oilman (written communication, 1948).

in resistance to flow, changes that resulted from, or at 

least accompanied, changes in measured sediment concentration. These writers assumed that the pattern of 

scour and fill at the streamflow-measuring section of 

this pool-and-riffle stream was typical of a long reach 

of channel. Brooks' assumption (1958, p. 545) of constant water-surface slope implies a representative relation for a long reach. Actually, the slope in a pool may 

vary widely if the bed elevation changes appreciably. 

The assumptions seem to be questionable for at least 

four reasons, one of which is that the measuring section 

is at a cross section that is probably narrower than 

average for the pool, and hence untypically large 

amounts of alternate scour and fill are to be expected. 

Also, Einstein and Chien (in Brooks, 1958, p. 558) 

pointed out that according to the reasoning of Lane 

and Borland (1954), a change in bed elevation at this 

measuring section was "* * * a transient change of the 

local river bed during a flood." They further stated 

that "It should not be interpreted as an adjustment of 

river regimen in response to the change of independent 

variables imposed on the channel by the watershed."

A second reason for questioning the assumptions is 

that scour and fill may be inconsistent at two cross 

sections even though the flows and the suspended-sediment concentrations are about the same at both sections. 

The inconsistency of scour and fill at the two cableways 

on the Elkhorn River near Waterloo, Nebr., was clearly 

shown in figure 14. Similar inconsistency of scour and 

fill also exists between the measuring sections on the 

Colorado River at Lees Ferry and near Grand Canyon, 

Ariz., for periods of high flow in 1955, 1957, and 1959 

(figs. 16-18). Not only are the patterns of scour and 

fill dissimilar from one measuring section to the other
D26 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

50,000

<X 

Q_

- 10,000

Si

o oc

i. i LU

or Q. 1000

100

500,000

_ o

LiJ 

CO

or

LU 

Q.

-Sediment concentration 

at Lees Ferry

100,000

Sediment concentration 

near Grand CanyonFlow near Grand Canyon

-1

EXPLANATION 

Average gage height of bedA, at Lees Ferry 

B, near Grand Canyon

Cumulative computed monthly change in average bed position from

Grand Canyon to Lees FerryC, based on difference between discharge of sands at Lees Ferry and

Grand Canyon

D, based on discharge of sands at Grand Canyon (assumed no inflow

of sand to reach)

FIGURE 16. Streamflow, measured sediment concentrations, and changes of average bed position at Lees Ferry and near

Grand Canyon, Arlz., April 1-August 31, 1955.
SCOUR AND FILL IN SAND-BED STREAMS

50,000 | 500,000

D27

or «*

O_

- 10,000 -

o

UJ

od

^3 

CO

1000 -

- 10

FIGUEH 17. Cnanges of average bed position at Lees Ferry and near Grand Canyon, Ariz., OctoberDecember 1&57. Concentration and flow near Grand Canyon.
D28 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

- 10,000

LU O

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CO CO

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100

100,000

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1000

10,000 fl- --fl ^ 

T J

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LlJ

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APR

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d C.anyon

i

MAY

/ PPS FPT rv

++ + +

+

+n

9 

4

JUNE

+

+ + +

 ,

JULY

FIGURE 18. Changes of average bed position at Lees Ferry and near Grand Canyon, Ariz., April-July 

1959. Concentration and flow near Grand Canyon.

but also from one period of high flow to another at 

either measuring section, particularly the measuring 

section near Grand Canyon. The flow and the measured concentration of sediment are given (fig. 16) for 

both sections for the high flow in 1955, when they probably differed more than in 1957 and 1959. Obviously, 

neither the flow nor the measured concentration is a 

satisfactory guide to scour or fill at some cross sections.

A third reason for questioning the assumptions is that 

changes in resistance to flow in a pool usually do not 

cause large changes in depth of flow at a measuring 

section that is only a comparatively short distance upstream from a riffle. Suppose that the energy loss is 

0.20 foot between a measuring section and the beginning 

of a fairly stable riffle downstream. This riffle is assumed to determine a stage-discharge relation that is
SCOUR AND FILL IN SAND-BED STREAMS D29

independent of changes in roughness in the upstream 

pool. Now, if at a particular streamflow the Chezy C 

for this reach of channel between the measuring section 

and the beginning of the riffle was to be decreased 50 

percent or to be doubled, the energy loss through the 

reach would be about 0.80 or 0.05 foot, respectively. 

The indicated change in depth at the measuring section 

would be about 0.60 or 0.15 foot. Even for the large 

assumed changes in resistance to flow through the reach, 

only those rather small changes in depths would be 

required. Experimentally, the small effect of changes 

in resistance to flow on depth at the Grand Canyon measuring section is shown by the agreement (Leopold and 

Haddock, 1953, fig. 23) between the stage-discharge 

relations for the same flows at different times regardless 

of whether the stage was rising or falling. (The gage 

is at the measuring section.)

The change in depth because of a given change in 

resistance depends oh distance upstream from the riffle. 

If a pool extended far enough upstream along a uniform channel, the depth effect at the head of the pool 

would approach the depth effect in a long uniform channel. In such a channel, a change in resistance to flow 

along the channel results in a change in depth because 

the slope of the energy gradient in the uniform channel 

(uniform resistance is, of course, assumed along the 

channel) must be nearly constant at all steady flows. 

If the channel is wide and the width is constant as the 

depth changes in the channel, a decrease of 50 percent 

in Chezy C requires an increase in depth of about 60 

percent. This adjustment of depth in a uniform sandbed channel is the sort of effect that Brooks (1958) 

found in his flume investigations.

A fourth reason for questioning the assumptions, 

rather an obvious one even without the preceding discussion, is that any adjustment required in depth of flow by 

changes in roughness, flow, or energy gradient will generally appear as a difference in elevation or slope of the 

water surface rather than in elevation of the streambed. 

In other words, if the water surface is not somehow 

fixed in position, it can adjust readily in response to 

changes in flow, roughness, or energy gradient; but the 

streambed adjusts only as sediment accumulates or 

erodes. Thus, a useful general guide is that changes in 

bed elevation should be explained on the basis of the 

principles of scour and fill whereas changes in depth 

of flow that result mainly from changes in water-surface 

elevation should be explained in terms of changes in 

rate of flow, energy gradient, or resistance to flow.

Depth adjustments resulting from changes in watersurface elevation may be correctly called adjustments in 

cross; section of flow but not adjustments of the stream 

channel. Also, in a channel of uniform roughness and 

cross section, the adjustment in the water surface, at

least at constant flow, will usually be an adjustment in 

elevation rather than in water-surface slope. On the 

other hand, in a pool upstream from a riffle that controls 

the stage-discharge relation, the adjustment in the water 

surface will usually be mainly in slope rather than in 

elevation.

Briefly summarized, scour and fill at the streamflowmeasuring section on the Colorado River near Grand 

Canyon represent local changes that are untypical of 

long reaches of channel; neither concentration nor flow 

is generally a satisfactory guide to scour and fill at the 

Grand Canyon or Lees Ferry sections; changes in resistance in a pool cause little depth change immediately 

upstream from a riffle that controls the stage-discharge 

relationship; and when adjustment of depth is required 

by change in flow, roughness, or energy gradient, the 

adjustment is usually in water-surface elevation rather 

than in bed elevation.

The variability of the relation between discharge of 

bed material and flow in sand-bed streams should be 

noted before any attempt is made to apply the principles 

of scour and fill to a pool-and-riffle stream such as the 

Colorado River between Lees Ferry and Grand Canyon. 

If neither the banks of a pool nor the riffle at the 

downstream end of the pool shifts appreciably, any 

major change in the average elevation of the streambed of the pool must cause an appreciable change in 

the mean velocity through the pool for a given rate 

of flow. For an abundant supply of bed material of a 

given particle size, even a small change in mean velocity usually causes a significant change in discharge 

of sands (fig. 3). Hence, the relation between discharge 

of bed material and flow usually changes, sometimes 

widely, if the mean velocity at a particular flow is 

changed because of scour or fill in a pool. Furthermore, the scour that might reduce the mean velocity at a 

particular flow is usually accompanied by an increase 

in particle size of the bed material In the pool and, 

perhaps, by a decrease in the proportion of the streambed that is covered with sand. Thus, when the elevation of the streambed is low in a pool, three factors 

the velocity decrease, the particle-size increase, and the 

decrease of sands on the bed tend to decrease the discharge of sands for a given flow. Of course, factors 

other than these also affect somewhat the relation 

between measured discharge of sands and flow.

The large overall effect of the difference in elevation 

of the streambed at the measuring section near Grand 

Canyon and in upstream measuring sections probably 

caused the large difference (figs. 16,17) in measured discharge of sands between 1955 and the fall of 1957 for 

given flows (fig. 19). The long-continued high flows 

during the summer of 1957 scoured the bed of the pool 

at the measuring section near Grand Canyon and prob-
D30 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

10,000

1000

100

10

2000 10,000 

FLOW, IN CUBIC FEET PER SECOND

100,000

FIGURE 19. Variable relation of concentration of measured 

sands to flow of the Colorado River near Grand Canyon, 

Ariz.

ably also scoured the beds of many pools upstream. 

These pools could not refill except as sediment, presumably nearly all sands, was trapped in them. Also, flows 

high enough to bring in appreciable quantities of sand 

may also be high enough to carry much of the incoming

sand through the pools. Hence, during periods of high 

flow in the fall of 1957 and the summer of 1959, the 

d<ieply scoured pool near Grand Canyon filled very little 

even though the flows and the sediment concentrations 

were moderately high (figs. 17,18).

APPLICATION OP PBINCIPUBS OP SCOUR AND FILL

The principle of a relation between discharge of 

sands and characteristics of the flow and available sands 

applies to a pool-and-riffie stream, but the relation 

existing at a particular time may be impossible to determine. In such a stream, the quantity and particle size 

of the available sediment generally change with time, 

as the particle size of the bed material did in the Colorado River near Grand Canyon (U.S. Geological Survey, 1961, p. 168) in 1956, and with distance along the 

stream. Because of changing configuration within a 

short distance and perhaps because of unusually rough 

walls, the mean velocity is not so good a measure of 

discharge of sands in a pool-and-riffie stream as it is in 

a sand-bed stream where turbulence and mean velocity 

are closely related for a particular roughness and depth 

of flow. Hence, an equilibrium relation such as that 

given between discharge of sands and parameters of 

flow in figure 3 for sand-bed streams can seldom be 

established for permanent use at a section of a pooland-riffie stream. In general, neither theoretical nor 

empirical relations can be used to compute the discharge 

of sands at many cross sections of pool-and-riffie 

streams; the discharge of sands at these sections often 

must be based at least in part on sediment sampling of 

the suspended sands.

In a pool-and-riffie stream the principle of continuity 

applies directly, the same as it does in sand-bed streams. 

This statement does not mean that the bed-material discharges are about the same from one cross section to

TABLE 7. Computed average scour or fill by months during 1955 water year in 88-mile reach from Lees Ferry to Grand Canyon, Ariz.

[Assumed average width of reach, 400 ft; assumed weight of sands, 85 Ib per cu ft]

Month 

(1)

October. __----__ ____________.--___-_-_-_

November.. ________ _____________________

December _________-___-_-_-_-------_-------

January. ______________-___----_------------

February. ______--_________----------_------

April --_- __.__----- _ .

A^Q-tT

June. ___________--___---- _______--__-----_

July_-__._ --- --- - --- ---

August _ _____________________-_----_-----.

September..- ______________---_-_-__--------

Computed total 

discharge of sands 

near Grand Canyon 

(1,000 tons)

(2)

370

22

9

4

6

790

1,470

13, 500

9,340

500

660

33

Average scour in 

reach, based on 

column 2 (ft)

(3)

-0.05

-.003

-.001

-.0005

-. 0008 -.10

-.19

-1.71

-1. 18 -.06

-.08

-.004

-3.38

Computed total 

discharge of sands 

at Lees Ferry (1,000 

tons)

(4)

700

240

150

110

98

1,040

1,220

4,700

4,810

940

690

120

Difference in discharge of sands 

from columns 4 and 

2 (1,000 tons)

(5)

+330

+ 220

+ 140

+ 110

+92

+ 250 -250

-8,800

-4, 530

+440

+30

+90

Average scour ( ) 

or fill (+) in reach, 

based on column 5 

(ft)

(6)

+ 0.04

+.03

+.02

+ .01

+.01

+.03

-.03

-1. 11 -.57

+.06

+.004

+.01

-1.50
SCOUR AND FILL IN SAND-BED STREAMS D31

another along the stream but only that equation 1 

applies.

Continuity of bed-material discharge limits the average scour in the 88-mile reach of the Colorado River 

between the measuring sections at Lees Ferry and near 

Grand Canyon, Ariz. Daily discharges of sands were 

computed roughly for the 1955 water year from records 

of flow, measured concentration of suspended sediment, 

and size distributions of the measured suspended sediment for Lees Ferry and for Grand Canyon (U.S. Geological Survey, 1959, p. 135-137,153-156). These daily 

discharges were added by months and were increased by 

20 percent for unmeasured discharge of sands. The 

estimate of 20 percent was an average of seven compu- , 

tations of unmeasured sediment discharge for the measuring section at Grand Canyon. The seven computed 

unmeasured sediment discharges varied from 12 to 28 

percent of the measured discharge of sands without any 

definite trend of the percentage with flow. If only the 

discharge of sands near Grand Canyon is considered 

that is, no inflow of sands to the reach is assumed the 

computed monthly scour (table 7, column 3) shows an 

average depth for the 88-mile reach of less than 0.01 

foot for each of 5 months of the 1955 water year. The 

computed average depth of scour is 1.71 feet during 

May and 1.18 feet during June and totals 3.38 feet for 

the water year. If scour and fill in the reach are based 

on the difference in discharges of sands at the ends of 

the reach (table 7, column 6), average fill of 0.06 foot 

or less is shown for each of the first 6 and the last 3 

months of the 1955 water year. Computed average 

scour totals 1.71 feet for the combined months of April, 

May, and June; and the net average scour for the water 

year is 1.50 feet. Of course, computed amounts of fill 

would be increased and computed amounts of scour 

would be decreased if the discharge of sands into the 

reach from the Paria and the Little Colorado Rivers 

and from other tributaries was known and could be included in the computations. Cumulative scour from 

April 1 to July 31 is shown in figure 16 and is based 

on the computed figures of scour from columns 3 and 6 

of table 7.

The same sort of limitation that was imposed on scour 

and fill by continuity of bed-material discharge during 

the 1955 water year also applies in general. If the total 

discharge of sands into the 88-mile reach were known 

for any month or year, a limit could be established for 

the volume of sands that might be deposited within the 

reach for the month or year. If both the total discharge of sands into and out of the reach were known 

for any period of appreciable length, the volume of 

scour or fill could be reasonably closely determined for 

the period. Usually, such a determination would be 

more accurate than an estimate of volume of scour for

a reach from observations of change in bed elevation at 

one or two cross sections along the reach.

CONCLUSIONS

Observations of scour and fill in field streams that 

have sand beds are frequently misinterpreted for several 

reasons. The position of the streambed may be defined 

at too few points to determine a satisfactory average 

elevation at a cross section or at too few cross sections 

to establish an accurate average bed elevation for a 

channel reach. Many available observations at points 

represent approximately the lowest positions of the bed 

at some unknown time during a flood. This time may 

have been when dunes were largest and the flow was 

fairly low. Also, when the bed was lowest at one point, 

it may have been much higher at several nearby points. 

Thus, many observations of scour and of subsequent fill 

tend to represent maximum amounts at points rather 

than average amounts for a cross section or an area. 

Also, changes in average bed elevation at narrow cross 

sections are almost always greater than changes at representative cross sections of channel reaches, and many 

streamflow-measuring sections are narrower than the 

average width of the nearby channel.

The sand being transported even in deep and swift 

flows would amount to an average depth of deposit of 

only a few hundredths of a foot if it could all be instantaneously deposited on the streambed.

Two principles are helpful in understanding scour 

and fill in sand-bed streams. One is the principle of 

continuity of volume of the bed material along a stream 

reach. This principle indicates that changes in elevation of an area of sand bed result from a difference in 

the rates at which sand reaches and leaves the area. 

The second principle is that a relation exists between 

the discharge of sands and the characteristics of flow 

and available sands.

These two principles indicate that the average bed 

elevation in a reach of uniform sand-bed channel is 

usually fairly stable during the passage of a flood 

because the discharge of sands into the upper end of 

the reach tends to be about balanced by the discharge of 

sands from the lower end of the reach.

A straight channel of uniform cross section may 

either scour or fill during rising flow because of a 

change in bed slope or in resistance to flow along the 

channel. In such a channel, scour while flow is increasing is usually followed by scour while the flow is decreasing, and fill while flow is increasing also is usually 

followed by fill while flow is decreasing.

Because the characteristics, especially mean velocity, 

of a flow are different at a narrow cross section in a 

sand-bed channel than at more representative cross 

sections, the amounts of scour and fill at a narrow cross
D32 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

section are unrepresentative of a channel reach. The 

amounts usually are larger than the average for the 

reach; to some extent, scour tends to accompany high or 

increasing flows whereas fill tends to accompany low 

or falling flows.

In spite of usually small and somewhat random 

changes in the characteristics of the streambed, the sand 

bed of the Elkhorn River near Waterloo, Nebr., showed 

general stability on the basis of many determinations 

of bed elevation at two cableways. At the upstream 

cableway, where the width was a little less than at the 

downstream cableway, the streambed seemed to have a 

slight tendency to lower during increasing flow and to 

rise during decreasing flow; but the streambed showed 

no significant tendency to change in the same direction 

at the downstream cableway as at the upstream cableway.

Streamflow measurements on the Missouri River near 

Yankton, S. Dak., showed that the streambed at the laterally constricted measuring section usually was much 

lower at high flow than at low flow, a pattern of bed 

change that characterizes many narrow cross sections in 

sand-bed streams.

Although the Colorado River is a pool-and-riffle 

stream rather than a sand-bed stream, many of its pools 

have sand beds. Measurements during three periods 

of high flow showed that the pattern of scour and fill 

as related to flow at Lees Ferry was different from that 

near Grand Canyon, Ariz., and that the pattern was 

inconsistent from one period to another near Grand 

Canyon. Changes in bed elevation at the measuring 

section near Grand Canyon and in upstream pools 

changed the relations of discharge of sands to flow and 

to measured sediment concentration.

Neither flow nor measured sediment concentration 

showed any clearcut relation to the diverse patterns of 

scour and fill for the Elkhorn River near Waterloo, a 

sand-bed stream, or for the Colorado River, a pool-andriffle stream, at Lees Ferry or near Grand Canyon. Because bed changes at one cross section were inconsistent 

with those at another section on the same stream, observed scour and fill at a cross section are, at least sometimes, poor indications of scour and fill in a channel 

reach.

When a change in depth of flow is required because 

the resistance, slope, or streamflow changes, the water 

surface rather than the streambed adjusts to the new 

depth of flow. A useful general guide is that changes 

in bed elevation should be explained on the basis of the 

principles of scour and fill whereas changes in depth of

flow that result mainly from changes in water-surface 

elevation should be explained in terms of changes in 

rate of flow, energy gradient, or resistance to flow.

In a pool, changes in depth of flow caused by changes 

in resistance to flow in the pool are usually small and 

decrease with nearness to a downstream riffle.

REFERENCES

Barton, J. R., and Lin, Pin-Nam, 1955, A study of the sediment

transport in alluvial channels: Colorado State Univ., Fort

Collins, Colo., Dept. Civil Eng. pub. 55JRB2, 43 p., 21 figs. 

Beckman, E. W., and Furness, L. W., 1962, Flow characteristics,

Elkhorn River, Waterloo, Nebraska: U.S. Geol. Survey

Water-Supply Paper 1498-B, 34 p. 

Brooks, N. H., 1958, Mechanics of streams with movable beds

of fine sand: Am. Soc. Civil Engineers Trans., v. 123, p.

526-594. 

Colby, B. R., 1960, Discontinuous rating curves for Pigeon Roost

and Cuffawa Creeks in northern Mississippi: U.S. Dept.

Agriculture, Agr. Research Service pub. AKS41-36, 31 p.

 1964, Sand discharge and mean-velocity relationships in 

sandbed streams: U.S. Geol. Survey Prof. Paper 462-A (in 

press).

Dawdy, D. R., 1961, Depth-discharge relations of alluvial 

streams discontinuous rating curves: U.S. Geol. Survey 

Water-Supply Paper 1498-C, 16 p.

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Lane, E. W., and Borland, W. M., 1954, River-bed scour during 

floods: Am. Soc. Civil Engineers Trans., v. 119, p. 1069-1080.

Laursen, E. M., 1952, Observations on the nature of scour: 

Iowa State Univ., Proc. 5th Hydraulic Conf., Studies in 

Eng., Bull. 34, p. 179-197 [1953].

Leopold, L. B., and Maddock, Thomas, Jr., 1952, Relation of 

suspended-sediment concentration to channel scour and fill: 

Iowa State Univ., Proc. 5th Hydraulic Conf., Studies in 

Eng., Bull. 34, p. 159-178 [1953].

 1953, The hydraulic geometry of stream channels and 

some physiographic implications: U.S. Geol. Survey Prof. 

Paper 252, 57 p.

Liu, Hsin-Kuan, and Hwang, Shoi-Yean, 1961, Discharge formula for straight alluvial channels: Am. Soc. Civil Engineers Trans., v. 125, p. 1787-1857.

Simons, D. B., Richardson, E. V., and Haushild, W. L., 1962. 

Depth-discharge relations in alluvial channels: Am. Soe 

Civil Engineers Proc., v. 88, HY 5, p. 57-72.

Straub, L. G., 1934, Effect of channel-contraction works upon 

regimen of movable bed-streams: Am. Geophys. .Union 

Trans., pt. 2, p. 454-463.

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United States, 1955: U.S. Geol. Survey Water-Supply 

Paper 1403, pt. 9-14.

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1957: U.S. Geol. Survey Water-Supply Paper 1523, pt. 9-14.

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the roughness and suspended load of alluvial streams: 

Pasadena, Calif., California Inst. Tech. Sedimentation Lab. 

Rept. E-68,121 p.

U.S. GOVERNMENT PRINTING OFFICE : 1968 O 298-946