The Natural Channel 

of Brandy wine Creek 

Pennsylvania

GEOLOGICAL SURVEY PROFESSIONAL PAPER 271
The Natural Channel 

of Brandy wine Creek 

Pennsylvania

By M. GORDON WOLMAN

GEOLOGICAL SURVEY PROFESSIONAL PAPER 271

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1955
UNITED STATES DEPARTMENT OF THE INTERIOR 

Douglas McKay, Secretary

GEOLOGICAL SURVEY 

W. E. Wrather, Director

For sale by the Superintendent of Documents, U. S. Government Printing Office 

Washington 25, D. C. - Price 55 cents (paper cover)
PREFACE

This report was prepared in the Water Resources Division of the United 

States Geological Survey as part of the investigations of water resource problems conducted by the Technical Coordination Branch, R. W. Davenport, 

chief. The author is greatly indebted to Luna B. Leopold for his help in the 

field and for his many discussions of the manuscript with the author. The advice 

and suggestions of W. B. Langbein have also been invaluable. In addition to 

many colleagues in the Geological Survey to whom the author is indebted for 

aid and counsel, the cooperation of R. W. Struble of the Brandywine Valley 

Association is appreciated. The critical comments of Profs. C. E. Stearns, 

M. P. Billings, J. H. Mackin, and H. A. Thomas, Jr., who kindly read the manuscript, have also been extremely helpful.

M. G. W. 

in
SYMBOLS

a coefficient representing width at unit discharge

A area of cross section of flowing water

b exponent in width-discharge relation

c coefficient representing depth at unit discharge

C Chezy coefficient in r= C\!Rs

d mean depth

/ exponent in depth-discharge relation

g acceleration due to gravity

j exponent in suspended load-discharge relation

k coefficient representing velocity at unit discharge

L suspended-sediment load, weight per unit of time

m exponent in velocity-discharge relation

n Manning roughness factor

	IV

p 

Q 

r

7?

s

t

v

w

y

0

a roughness parameter

coefficient representing load at unit discharge 

water discharge in cubic feet per second (cfs) 

coefficient representing roughness parameter at unit

discharge 

hydraulic radius, approximately equal to mean

depth in wide channels 

slope of water surface in open channel 

coefficient representing slope at unit discharge

width

exponent in roughness-discharge relation

exponent in slope-discharge relation
CONTENTS

Abstract-._ ______________________________ __________

History and perspective.____________________________

The area of study: Brandy wine Creek_ _ _______________

Geologic and geographic setting __________________

Choice of variables._____________________________

Requisites of a measuring reach __________________

Measurements of the stream _____________________

Hydraulic and geologic conditions within selected 

reaches. _________________________________________

The reach at Cornog: an example_________________

Width, depth, and velocity____. __________________

Slope__________________________________ ________

Materials on the bed and in the banks____________

Suspended loacl__ ____________________________

Roughness. _ _---_-_-_---_______________________

Summary and interpretation of at-a-station changes. 

Downstream variation in the river channel..--_________

Downstream variation at low and high flows _ ______

The bankfull stage._____________________________

Page 

1 

1 

3 

3 

6 

6 

9

10

10

11

14

15

19

20

22

24

24

29

Page 

Downstream variation in the river channel—Continued

The headwater tributaries- ______________________ 31

Summary of variations in the downstream direction- 37 

Variability of cross sections in a short reach: the pool

and riffle problem_________________________________ 37

Amount of local variation within a given reach_____ 39

Criteria for distinguishing pools from riffles._______ 40

Equilibrium, grade, and the longitudinal profile_________ 41

Simultaneous adjustments of the variables.________ 41

Comparison of Braiidywiiie Creek with stable irrigation channels_ _ ______________________________ 42

Characteristics of stable channels _________________ 44

The Brandy wine: a channel in equilibrium _________ 45

Equilibrium and the longitudinal profile..________. 45

Summary and conclusions.___________________________ 49

References _________________________________________ 49

Appendix._ ______________________-____--___------_- 51

Index_ __________________________________________ 55

ILLUSTRATIONS

FIGURE 1. Map of drainage basin of Brandy wine Creek, Pennsylvania- ______________________-_---_-----_-----_---- 4

2. Generalized geologic map of the drainage basin of Braiidywiiie Creek, Pennsylvania. ____________________ 5

3. Measuring reach at Cornog_________________________________________________________________________ 7

4. Measuring reach at Seven Springs. ____________________________--_-_--_---_-___------_----_-- 7

5. Measuring reach at South Downiugtown______________________________-_-_-_-----_-_-_---_-_--_------_ 7

6. Measuring reach at Embreeville_ -_________________________________-_-_-_---_-------------_---------- 7

7. Measuring reach at Georgia. ___________________________________---_--------_--__--_--_----_--- 8

8. Measuring reach at Wawaset._________________________________________-_-_----_-_------_-_---------- 8

9. Measuring reach at Lenape__-________________________________-_-_-------------_-----_---_--------_- 8

10. Width, mean depth, mean velocity, water surface slope, and roughness parameter ..' in relation to discharge at

Cornog and Seven Springs.___________________________________________-_____-_-_-----------_------ 11

11. Width, mean depth, mean velocity, water surface slope, and roughness parameter n' in relation to discharge at

South Downingtown and Georgia.____________________________________--_-__------------------_---- 12

12. Width, mean depth, mean velocity, water surface slope, and roughness parameter n' in relation to discharge at

Embreeville and Wawaset____________________________________________--------------_------------- 13

13. Width, mean depth, mean velocity, water surface slope, and roughness parameter n' in relation to discharge at

Lenape_---_________________________________________________________-_-_----_---------_-_---- 14

14. Relation of width, mean depth, and mean velocity to discharge at Coatesville.__________-___________-_---- 14

15. Relation of width, mean depth, and mean velocity to discharge at Chadds Forcl______ ___________________ 15

16. Profiles of cross sections of Braiidywiiie Creek at the principal measuring sectioiis__________________________ 16

17. Cumulative frequency curves showing size of material on the bed of Braiidywiiie Creek______ ______________ 17

18. Examples of bed material___________________________ __________________-_-_-----_-------__-----_- 17

19. Cumulative frequency curves showing size of material in the banks of Braiidywiiie Creek____ ______________ 18

23. Typical bank and flood plain.___________________________________________---------_-------_--_--_- 18

21. Sediment-rating curve for Braiidywiiie Creek at Wilmingtoii, Del____________________-_-_-----------_---- 20

22. Suspended load in relation to discharge at the principal measuring sections__-_---_-_________-_-_-----_---- 20

23. Relation of roughness parameter n' to mean depth at Cornog, Seven Springs, Embreeville, and Lenape_______ 21

24. Relation of roughness parameter n' to size of material 011 the bed__________________---------_____________ 22

:v
VI CONTENTS

Page 

FIGURE 25. Width, depth, velocity, water surface slope, and roughness parameter n' in relation to discharge as discharge

increases downstream.___________________________________________________________________________ 25

26. Hydraulic geometry of Brandywine Creek based on at-a-station and downstream curves. ___________________ 27

27. Relation of suspended load to width-depth ratio_______________________________________________________ 28

28. Example of method used to determine bankfull stage, South Downingtown_______________________________ 30

29. Width, mean depth, mean velocity, and water surface slope in relation to discharge in the downstream direction

at the bankf ull stage.____________________________________________________________________________ 31

30. Determination of the recurrence interval of the bankf ull stage at the principal measuring stations_____ _______ 32

31. Relation of drainage area to recurrence interval of the bankf ull stage, ____________________________________ 33

32. Typical small tributary___________________________________________________________________________ 33

33. Extrapolation of individual measurements to obtain discharges equaled or exceeded 50 and 2 percent of the time_

34. Relation of width, mean depth, and mean velocity to discharge in the downstream direction (50 percent duration) ___________________________________________________________________________________________ 35

35. Relation of width, mean depth, and mean velocity to discharge in the downstream direction (2 percent deration)- 36

36. Relation of water surface slope to size of bed material, mean velocity, mean bed velocity, and to the roughness

parameter «'__________________________________________________-_____________________________-___ 38

37. Effect of local variations in cross section on downstream changes in width, depth, and velocity. _____________ 30

38. Relation of width to depth in canals and rivers and in Brandywine Creek,________________________________ 42

39. Longitudinal profile of Brandywine Creek___________________________________________________________ 46

TABLES

TABLE 1. Summary of data showing rates of change of width, depth, velocity, slope, roughness, and suspended load with

discharge at the principal stations on Brandywine Creek_______________________________________________ 14

2. Size of material on the bed of Brandywine Creek at the principal measuring sections.-___________--__---_-__- 18

3. Location and description of samples from the bank of the channel of Brandywine Greek...____________________ 19

4. Summary of changes in width, depth, velocity, slope, and roughness in the downstream direction for discharge of

different durations, Brandywine Greek.______________________________________________________________ 26

5. Formulas for stable channels in alluvial material for comparison with similar formulas for Brandywine Creek,

Pennsylvania.__-___-___-_-______________________-__-_________-_-_-_______-_-__-----__-______---_- 44
THE NATURAL CHANNEL OF BRANDYWINE GREEK, PENNSYLVANIA

ByrM. GORDON WOLMAN

ABSTRACT

This study of the channel of Brandy wine Creek, Pennsylvania, 

consists of three parts. The first is an analysis of the changes 

which take place in the width, depth, velocity, slope of the water 

surface, suspended load, and roughness factor with changing 

discharge below the bankfull stage at each of several widely 

separated cross sections of the channel. Expressed as functions 

of the discharge, it is found that the variables behave systematically. In every section studied, as the discharge increases, the 

velocity increases to about the 0.6 power, depth to the 0.4, and 

load to the 2.0 power of the discharge. The roughness decreases 

to the 0.2 power of the discharge. The relative magnitudes and 

the direction of these variations are similar to those which have 

been observed in other rivers in the United States, primarily in 

the West. Some modifications of the hypotheses applicable to 

the western rivers are probably required because on Brandywine 

Creek the difference between the materials on the bed and in 

the banks is considerably greater than it is on most of the western 

rivers studied.

In the second part of the paper the progressive changes of the 

same variables in the downstream direction with increasing discharge at a given frequency are described. Despite the disorderly 

appearance of the stream, it is found that the variables display a 

progressive, orderly change in the downstream direction when 

traced from the headwater tributaries through the trunk stream 

of Brandywine Creek. At a given frequency of flow, width 

increases with discharge to about the 0.5 power. Depth increases 

downstream somewhat less rapidly, while the slope and roughness 

both decrease in the downstream direction. Despite a decrease 

in the size of the material on the bed, both the mean velocity and 

the mean bed velocity increase downstream. The rates of 

change of these variables are in close accord with the changes 

observed on rivers flowing in alluvium and in stable irrigation 

canals. These relationships hold for all flows up to the bankfull 

stage. Analysis of the streamflow records indicates that the 

annual maximum discharge equals or exceeds the bankfull stage 

roughly once every 2 years.

The regularity in the behavior of the variables with changing 

discharges both at-a-station and in the downstream direction 

and the similar rates of change of the variables on Brandywine 

Creek and in stable irrigation canals suggest the existence of a 

quasi-equilibrium in the channel of the creek.

Part three of this study is concerned with this concept of 

equilibrium in streams. By analogy with canals and with several 

rivers in diverse regions of the United States it may be concluded 

that this quasi-equilibrium is closely related to the discharge, 

and to the concentration of the suspended load. The shape and 

longitudinal profile of the channel are determined by these two 

independent factors which operate within the limits set by the 

local geology. The latter determines the initial size, shape, and 

resistance of the material provided to the channel.

The existence of a quasi-equilibrium among the variables 

studied suggests that most reaches on Brandywine Creek are at 

grade. This is true if the term "grade," when applied to natural 

rivers, is synonymous with quasi-equilibrium. The adjustability 

of the variables in the channel rather than the stability of any 

particular shape or longitudinal profile of the channel is emphasized when the word "equilibrium" is applied to Brandywine 

Creek.

HISTORY AND PERSPECTIVE

Although the laws and literature of ancient Babylonia 

and Egypt contain references to the natural characteristics of river channels, no systematic development of a 

general principle can be discerned before the 19th 

century. From the early 19th century on, hoAvever, 

the history of speculation on the natural characteristics 

of rivers illustrates the gradual and systematic refinement of a broad generalization. In 1802 Playfair 

observed that the complex drainage network, the proportionality of the rivers to their valleys, and the "nice 

adjustment" between the slopes of the main stream 

and its tributaries made it "infinitely improbable" that 

each of the valleys was not the work of the stream that 

flowed in it. Since the acceptance of Playfair's deduction, elaboration of the generalization has proceeded in 

two ways. First, the concept of the cycle of erosion 

was introduced in part to describe successive stages in 

the erosional history of river valleys. Second, qualitative and, more recently, quantitative inquiries have 

been made into the processes operating through these 

successive stages. This investigation of the Brandywine 

Creek in Pennsylvania falls into the second category.

In the first approach William Morris Davis (1899) 

postulated a cycle of erosion in which he described a 

simplified sequence of stages in the life history of a 

region. Because of the normally slow rate at which 

processes of land sculpture proceed, the actual secuence 

of events in a cycle was inferred by comparing ob^ervations of different rivers and of the same river at different 

reaches. Thus, Davis's system is a qualitative description which may adequately describe a sequence of 

landforms, but it does not necessarily embody the 

physical processes that operate in the development of 

these forms.

In the second approach, fluvia processes, because of

I
THE NATUKAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

their fundamental importance in the cycle of erosion, 

have received special emphasis. In considering the 

behavior of rivers, geologists and engineers have been 

largely concerned with the relationship of a number of 

variables which enter into river processes. Gilbert 

(1877) presented an excellent quantitative picture of 

the mechanics of natural streams in his classic discussion of land sculpture based on his observations on the 

relation among competence, capacity, declivity, velocity 

and quantity of water.

The idea of equilibrium was introduced in studies of 

natural river channels as early as the 16th century 

(Dausse, 1872, p. 340). Later morphologic studies by 

Baulig (1925) and Mackin (1948) and many others 

have been primarily concerned with channel equilibrium. The qualitative analyses of Baulig and Alackin 

on the interaction of several variables which determine 

the size, shape, and profile of a natural channel are an 

application and extension of the kinds of principles 

enunciated earlier by French and Italian engineers 

(Dausse, 1872, p. 340; Baulig, 1925, reprinted 1950, 

p. 43, 50).

In the last, part of the present study the author has 

attempted to relate this idea of equilibrium to the 

observed behavior of a number of variables measured 

on Brandywine Creek, Pennsylvania.

To further our understanding of river morphology, 

qualitative analyses such as those of Baulig and Mackiii 

must be supplemented by quantitative studies for 

several reasons. First, only a quantitative study shows 

the magnitude of the variables involved. Second, if 

their relative magnitudes are known, the choice of 

variables is simpler, as is the recognition of their 

importance in the process being considered. Third, the 

simultaneous action of a number of factors is extremely 

difficult to treat qualitatively. Fourth, quantitative 

methods permit hypotheses to be tested or verified 

either by repetition of the study or by controlled 

experiments.

Engineering problems of water supply and river 

control require measurements of streamflow and sediment content and have led to a large accumulation of 

quantitative data. Such data can be useful in studying 

fluvial processes and the sculpture of the land as 

Leopold and Maddock (1953) demonstrated in thenwork on the hydraulic geometry of river channels. 

Using data obtained from streamflow measurements, 

including measurements of the width, depth, velocity, 

discharge, and sediment load of the stream, they were 

able to describe quantitatively some features of the 

simultaneous adjustments of these variables by which 

an equilibrium may be maintained in natural rivers. 

This study provides the immediate background for the 

present work.

This study of Brandywine Creek is an attempt to 

describe the way in which a particular natural stream 

channel changes in profile and in cross section from its 

headwaters to its mouth. Measurements of the width, 

depth, velocity, slope, suspended load, and size of bed 

material of the stream furnish the basis for a quantitative description of these changes. These measurements 

were specifically designed to test the applicability of the 

methods developed by Leopold and Maddock to a 

stream considerably different from the rivers upon 

which the techniques were developed. The data on 

Brandywine Creek, in addition, provide means for an 

independent test of the validity of some of the generalizations on natural channels. Because the stream is 

small enough to be studied in detail, the author has 

been able to study in particular the extent to which 

variations within individual reaches of the channel 

affect the more general relationships described by 

Leopold and Maddock.

Previous studies utilizing standard stream-gaging 

records have lacked measurements of slope of the river, 

which is not ordinarily measured at gaging stations. 

In understanding fluvial processes, however, this information may be important. On Brandywine Creek 

measurements were made of the slope as well as the 

depth and velocity of the water. From these known 

quantities a hydraulic roughness of the channel was 

computed. Geologic factors such as grain size and 

hardness which interact with the hydraulic variables to 

produce a particular natural channel behave essentially 

as friction or roughness parameters. The correlation 

of the computed roughness with the geological characteristics of the terrain in this study has been limited 

to measurements of the grain size of material on the 

bed of the stream. Unfortunately, in regard to other 

geological variables such as load, this investigation is 

severely limited by the paucity of measurements of 

suspended and bed loads. This deficiency is recognized 

by the author but practical problems made additional 

measurements of the load impossible.

The study is divided into three parts. The first is a 

description of the changes which take place in a given 

reach of the channel as the volume of the f ow changes. 

The second is a discussion of the way in which the 

hydraulic and geologic variables change in the downstream direction at a given flow. The last h an attempt 

to relate these progressive downstream changes to the 

concept of equilibrium in a natural river. Although 

the measurements in parts one and two may have some 

intrinsic value of their own, their primary significance 

here is as a means of illustrating the condition of 

equilibrium. In part one the study of the behavior of 

the variables at a given station is based primarily upon 

data from seven reaches on the Brandywine Creek that
THE AREA OF STUDY

were studied in detail. The changes which take place 

in the downstream direction are based upon comparisons of nine major stations as well as upon a great many 

more cross sections at which measurements were made. 

Such a study of the entire river requires extensive 

coverage of the drainage basin. At the same time, 

however, because it is desirable to obtain data on the 

way in which width, depth, velocity, slope and roughness change with increasing discharge at a given station, 

the choice of the specific locations for study and hence 

the extent of the coverage are partially governed by the 

necessity of having accurate measurements of the 

discharge.

The author is well aware that in a quantitative study 

of fluvial processes too much emphasis can be placed 

upon mathematical relationships to the exclusion of an 

understanding of their physical meaning. Avoidance 

of this danger is made easier by the fact that the data 

on Brandywine Creek rarely permit such an elaborate 

mathematical treatment.

THE AREA OF STUDY: BRANDYWINE CREEK 

GEOLOGIC AND GEOGRAPHIC SETTING

To fulfill the aims of this study the river chosen for 

investigation should meet three principal specifications. 

First, it must traverse a region of heterogeneous rocks 

in order to preclude the possibility that any regularity 

in its behavior could simply be explained as a result of 

especially uniform and thus unrepresentative conditions. 

Second, it must be a single representative river rather 

than individual reaches on a number of streams. 

Third, if one is to test the general applicability of the 

relationships derived by Leopold and Maddock, the 

river must be in a region different in climate and 

physiography from the regions represented by the 

rivers studied by them. These considerations suggested the selection of a river typical of the Middle 

Atlantic region of the eastern United States. Although 

there are no simple fixed characteristics of such streams, 

they are customarily envisioned as streams which flow 

southeast across the Appalachian Valley and Ridge 

province and across the Piedmont. In their lower 

reaches most of them cross the Fall Line, or Fall Zone, 

where the basement rocks have been exposed by erosion 

of the sediments of the Atlantic Coastal Plain. Many 

of the characteristic rivers of this region flow diagonally 

across or at right angles to the alternating bands of 

resistant and weak rocks which strike northeast. They 

do not flow on homogeneous material.

Brandywine Creek in southeastern Pennsylvania fulfills these basic requirements. This creek has a 

drainage area of 314 square miles and an overall 

length of about 45 miles from its headwaters to its

303765—55———2

mouth below Wilmington, Del. (fig. 1). The drainage basin lies almost entirely within the Honeybrook, 

Phoenixville, Coatesville, and West Chester quadrangles in Pennsylvania.

At its highest elevation, approximately 9Cn feet 

above sea level on the southeast slope of Welsh Mountain, Brandywine Creek is perennially fed by springs. 

Most of the major tributaries throughout the drainage 

basin can be traced to their sources in springs that 

reach the surface at the apices of shallow, rmphitheaterlike bowls on the uplands. Flowing through 

an unglaciated area from Welsh Mountain to its 

mouth near the junction of the Christina and Delaware Rivers, the Brandywine crosses a number of 

different types of rock (fig. 2). After traversing a 

rolling upland of metamorphosed sediments and intrusive rocks, the stream successively crosses the 

North Valley Hills, the Chester Valley, and the South 

Valley Hills. The fiist of these is a northeast-striking 

anticline with a resistant quartzite core; the valley 

itself is composed of Ordovician limestone and dolomites. On the southern flank of the valley there is a 

belt of phyllite. The Brandywine then traverses an 

area underlain by schist. Above Chadds Ford the 

course of the stream through the schist is interrupted 

by a belt of gneiss, and about 4 miles above the city 

of Wilmington the stream encounters the Fall Zone in 

an area underlain by gabbro. The Coastal Plain sediments are not reached until the river nears its mouth. 

The passage of the Brandywine through Wilmington 

is obstructed by numerous low dams and the very 

lowest portion at Wilmington Harbor is subject to 

tidal influence. This study is concerned only with 

that part of Brandywine Creek which lies above the 

Fall Line.

Figures 3-9 show the characteristic vegetation of 

the region bordering the river channel. Although trees 

are found adjacent to the channel, most of the flood 

plain of Brandywine Creek is in permanent grass 

pasture. The drainage basin as a whole is farmland, 

about a third of which is covered by woodland. The 

average annual rainfall, which is distributed rather 

evenly throughout the year, is about 47 inches (U. S. 

Dept. of Agriculture, 1941, p. 1087).

In addition to the theoretical requirements which 

Brandywine Creek satisfies, as do many other streams 

in the eastern United States, it also fulfills a number of 

important practical requirements: 1. It has a relatively 

small drainage area. 2. The creek itself is srnrll and 

shallow enough at most stages for measurements of 

flow to be made by wading. 3. A permanent recording stream gage has been in operation since 1932 at 

Chadds Ford. This relatively long and continuous 

record of discharge is required in comparing flows of
THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

PRINCIPAL MEASURING STATIONS

A 

Established for this report

Location of discharge measurement points

WILMINGTCN

FIGTJBE 1.—Map of the drainage basin of Brandywine Creek, Pennsylvania, showing major tributaries, measuring stations established for this study, ga?ing stations of the 

U. S. Geological Survey, and locations of discharge measurements on tributaries. Inset map shows location of Brandywine Creek in southeastern Pennsylvania.
THE AREA OF STUDY

psrs^

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•xxxxxxx\\\ ^xvx *x \vywv\x xx xxyT^*"^*

»wxx XXXXXX\^X>f<\^ViU<'.Hi|ivX^XNX; /r'' *•

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r^x x \XX><^AX f^xAxxxxxxx- x/ f \ /^-^v

Schist

Gneiss

WILMINGTON

10 Miles

FIGURE 2.—Generalized geologic map of drainage basin of Brandywine Creek, Pennsylvania, illustrating the diverse lithology (after Stose and Ljungstedt,
6 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

different durations or frequencies. 4. There is a sediment sampling station 011 Brandywine Creek at Wilmiiigton, Del. Such sampling stations are rare in 

Eastern United States. 5. The area is readily traversed by automobile, a fact which makes it possible 

to obtain measurements of flow in periods of high water.

CHOICE OF VARIABLES

111 an investigation of the river channel both the 

hydraulic and the geologic factors that determine the 

shape and longitudinal profile of the channel must be 

considered. A study of the geological aspects alone, 

such as the distribution of hard and soft rocks, or the 

grain size, shape, and sorting of bed material in the 

channel is insufficient. Although such studies describe 

the physical features of the channel, they shed very 

little light on the mechanism by which those features 

are produced. Similarly, a purely theoretical study of 

the hydraulics of an open channel cannot take into 

account the variety of modifications of any hypothesis 

which the geological setting requires. In the present 

work an attempt to bridge this gap was made by 

measuring some of the hydraulic variables in the field.

As one might suppose, it would be impossible to consider every variable. However, a theoretical minimum 

number of variables must be considered. The methods 

of the present investigation and the reasons that specific 

data were collected are best explained in terms of the 

requirements of the hydraulic study and the practical 

problems involved in measuring the necessary variables 

in the field.

This study deals with the flow of water in open 

channels. Such flow can be described in simple form 

by the familiar Chezy formula,

where v equals the mean velocity; R the hydraulic 

radius, defined as the area of the wetted cross section 

divided by the wetted perimeter; s, the slope of the 

energy gradient, in steady uniform flow equal to the 

slope of the water surface; and C, a coefficient of resistance. Because the coefficient C is not a constant, in 

the numerical evaluation of the three major variables 

C must also be evaluated. The Manning formula is 

one among many developed to overcome this difficulty 

(Ganguillet and Kutter, 1893; Houk, 1918; Schnackenburg, 1951). In this formula

v= 1.486 (2)

where n is a measure of the boundary roughness of the 

bed and banks of the channel. From these basic 

equations it is clear that at least four major variables

must be measured or computed to provide the bare 

essentials in any analysis of the flow of water in a given 

reach of the channel.

Standard stream-gaging measurements (Corbett and 

others, 1945) include data on the width, area of cross 

section, depth, and velocity of the flow. The discharge 

is computed from the measurements of tJh? area and 

velocity. The mean depth as used here is the quotient 

of the cross-sectional area of the flowing water divided 

by the top width of the water surface. Mean velocity 

is equal to the discharge divided by the area of the cross 

section. For relatively wide sections the mean depth 

is approximately equal to the hydraulic radius. On 

Brandywine Creek, more than 160 discharge measurements have been made. These include measurements 

of the fundamental parameters of width, depth, and 

velocity at approximately 50 different locations along 

the creek.

The measurement of another principal variable, the 

slope of the water surface, is much more difficult, however, and is not a customary stream-gaging procedure. 

In order to measure this variable on Brandywine Creek, 

seven stations (fig. 1) were established on the main 

stream. At these sections not only the width, depth, 

and velocity were measured at different discharges, 

but also the slope of water surface.

REQUISITES OF A MEASURING REACH

The discharge at any given point is a function of the 

contributing drainage area. Comparison of the measured variables in the downstream direction with respect 

to discharge therefore requires a range of discharges 

and hence a choice of measuring reaches representing 

a range of sizes of contributing drainage areas. The 

seven major measuring reaches chosen for detailed 

study (fig. 1) have contributing drainage areas ranging 

from 25 square miles at Cornog to 259 square miles at 

Leiiape (app. A). Their distribution spans a relatively 

large part of the total length of the stream.

Within a selected region the choice of the specific 

measuring reach, or length of channel, is more difficult. 

The necessity for accurate discharge measurements 

presents many criteria which must be met or compromised in the selection of each reach. On Brandywine Creek, these criteria include: 1. A natural cross 

section of the channel which could be waded under 

normal conditions of flow. 2. A straight r^ach of the 

channel which would permit a reasonably accurate 

determination of the slope of the water surface. 3. The 

presence of a bridge at or very near the natural cross 

section from which discharge measurements could be 

made at times of high water when wading was impossible. 4. A reach in which the velocity was g~eat enough
THE AREA OF STUDY

FIGURE 3.—Measuring reach at Cornog. A, Upstream view; flow shown is equaled or exceeded about 30 percent of the time. B, Downstream view; flow shown is equaled 

or exceeded less than 1 percent of the time. Note large boulders not submerged until relatively high flow is attained.

FIGURE 4.—Measuring reach at Seven Springs looking downstream from bridge. A, Low flow that is equaled or exceeded about 17 percent of the time and that is similar 

to flow used in comparison of downstream changes in figure 25. JB, High flow that is equaled or exceeded less than 1 percent of the time.

FIGITRE 5.—Measuring reach at South Downingtown. View upstream at low flow FIGURE 6.—Measuring reach at Embreeville at discharge equaled or exceeded about 

from lower end of reach. At right is characteristic flood plain. 20 percent of the time. Downstream view.
THE NATURAL CHANNEL OF BRANDYWTNE CREEK, PENNSYLVANIA

FIGURE 7.—Measuring reach at Georgia looking downstream from bridge. Shows approximate range measured at the principal stations- A, Low flow; B, high flow.

FIGURE 8.—Measuring reach at Wawaset looking downstream from bridge at flow 

equaled or exceeded 15 percent of the time.

FIGUBE 9.—Measuring reach at Lenape. A, View downstream at low flow when mean depth is about 1 foot deeper than lowest depth shown in figure 13. B. View downstream from bridge at high flow. Tree surrounded by water is near car in A. Note broad flood plain.
THE AREA OF STUDY 9

to permit an accurate determination of the discharge 

by means of the standard current meter. 5. A reach 

which was relatively accessible by automobile in rainy 

weather to allow the stream gager to visit as many stations as possible during a single rainstorm. 6. A 

reach at which a reasonable correlation between discharge and elevation of the water surface could be 

obtained.

It was impossible to satisfy all of these requirements 

at each of the gaging locations. Because only natural 

cross sections are required for this study, sensitive 

controls are lacking at many reaches. At Cornog, for 

example, the stage-discharge relation is poor or only 

fair by gaging-station standards. The gaging section 

at Georgia was chosen because of its drainage area, 

accessibility, and proximity to the forks of the river. 

The stage-discharge relation at Georgia is excellent 

because of the presence below the gage of a gravel 

riffle which acts as a control. Measurements of low 

discharges at this natural section however, were 

hampered by the low velocity of flow in the pool. Also, 

conditions in the pool may be unrepresentative of the 

reach as a whole. Photographs (figs. 3-9) of the river 

channel at the principal measuring sections show how 

diverse the individual stations are. How much of this 

diversity is related to the upstream or downstream 

position of the gaging locations along the longitudinal 

profile, and how much is simply a function of the local 

variations at the specific section at which the measurements were taken are discussed later under "Variability 

of the cross section in a short reach." The original 

spacing, on the basis of drainage areas, determined the 

position of the gaging sections in relation to the longitudinal profile. The specific criteria previously listed 

fixed the exact location of the measuring reach within 

the prescribed segment of the profile. The distinction 

between the degree of variation of the channel within a 

local area and the total variation in the downstream 

direction is an important aspect of the present investigation.

Despite the criteria which had to be met in their 

selection, the principal measuring stations should not be 

considered atypical of the Brandywine as a whole. 

Possibly somewhat straighter than most reaches, they 

are otherwise indistinguishable from adjacent parts of 

the stream.

MEASUREMENTS OF THE STREAM

The measurement of the discharge and of the slope of 

the water surface at each of the principal measuring 

sections consisted of the following steps: 1. The staff 

gage nearest to the measuring section was read. 2. A 

measurement of the discharge was obtained by use of a

current meter to measure the velocity of flow. Measurements were generally made by wading and were always 

made at precisely the same cross section at each station. 

3. The height of the water surface at each end of the 

reach was determined from the gages at the upstream 

and at the downstream ends of the reach. 4. At high 

flows when wading was impossible, measurements of 

the discharge were made from bridges nearby (figs. 7-9). 

At these times the elevation of the water surface at the 

regular wading section was marked and the area of this 

cross section measured after the water had receded. 

The slope of the water surface was computed by dividing the difference in elevation between upstream and 

downstream gages by the measured distance between 

the two gages. In addition to these standard observations, measurements of the peak stage at each of the 

seven stations were obtained from peak-stage recorders.

By using the measurements of width, mean depth, 

mean velocity, and slope of the water surface, a roughness parameter n' was computed from a modified 

Manning equation. The quantity n' computed here is 

not equivalent to Manning's n. This distinction is 

discussed later. For each discharge measurement, 

then, four variables have been measured arid one 

computed. The results of these measuremerts are 

listed in appendix A.

Accurate measurements of the water surface slope are 

extremely difficult to make (Houk, 1918, p. 78). First, 

the difference in elevation may be very small, and 

second, the water surface slope often changes within 

very short distances along the thalweg. Although no 

high order of precision is claimed for the measurements 

of slope listed in appendix A, they are reasonable and 

consistent and provide a satisfactory description of the 

range and relative magnitudes of the slopes concerned 

in this study. The values of n' have additioral importance because there are few rivers on which measurements have been obtained that permit comparisons to 

be made of the changes taking place in any roughness 

parameter in the downstream direction.

On occasions when current-meter measurements 

could not be made, weighted floats were used to measure 

the velocity of the flowing water. Correction factors 

for the velocity obtained when the floats wer? used 

were later derived by comparing velocities of the floats 

with the mean velocity determined by use of the current 

meter. Measurements in which floats were us°,d are 

given a different symbol in figures 10-13.

In addition to the seven measuring sections, established for this study, two river gaging stations have 

been operated by the U. S. Geological Survey on 

Brandywine Creek at Coatesville and at Chadds Ford. 

The recording gage at Chadds Ford gives a continuous
10 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

record of the height of the water surface. At Coatesville, from 1943 to 1951, when the station was discontinued, the height of the water on a graduated staff 

gage was read periodically by an observer. At both 

stations rating curves have been established and cover 

a range of river stages.

On Brandywine Creek, analyses have also been made 

of the bed and bank materials. Some measurements 

of the suspended-sediment load have also been made 

which, when supplemented by information obtained 

from other rivers, may be used in the analysis.

HYDRAULIC AND GEOLOGIC CONDITIONS WITHIN 

SELECTED REACHES

Before the behavior of the hydraulic parameters at 

each of the principal stations is described, one example 

will be presented to show the characteristic at-a-station 

changes of all of the variables. Because the behavior 

of the variables with changing discharge at Cornog is 

typical of their behavior at the other stations, it will 

be used as the illustrative example.

The hydraulic variables are treated in accordance 

with the methods developed by Leopold and Maddock 

(1952). At each station the width, depth, velocity, 

slope, and roughness parameter are plotted against discharge on log-log paper. Each variable is thus expressed as a function of discharge. When expressed in 

this way the data at a given station are remarkably 

consistent.

THE BEACH AT CORNOG: AN EXAMPLE

Figure 10 shows the changes in the variables which 

take place at Cornog as the discharge increases. On 

log-log paper, the slope of the straight lines drawn 

through the plotted points is the value of the exponent 

in the equation expressing the relationship between 

the two variables. Thus b, f, m, z, and y (fig. 10) are 

the exponents in the following equations:

d=cQf 

v=kQm

(3)

(4)

(5)

(6)

(7)

In these equations Q is the discharge in cubic feet per 

second; w is the width, in feet; d, the depth, in feet; 

11, the velocity, in feet per second; s, the slope of the 

water surface; and n', a roughness factor; a, c, k, t, and 

r are constants. At Cornog, values measured from the 

graphs of figure 10 are 6 = .04, /=.40, m = . 52, 2=0.0, 

and y= — .23. At the point where Q=l, the width, 

depth, velocity, slope, and roughness equal the constants 

a, c, k, t, and r respectively. In the cross section at

Cornog, the relationship of each variable to discharge is 

given by these equations:

.0032, that is, slope remains 

very nearly constant

and 6+/+m=0.96

It has been shown (Leopold and Maddock, 1953, p. 8) 

that because

(3)

(4)

(5)

and w=aQ* 

d=cQf

v=kQm

then Q=aQ»XcQfXkQ 

whence b -\-f-\-m = 1

The experimental result at Cornog is in close accord 

with the theoretical expression.

The slope of the water surface in the measuring reach 

remains practically constant as the discharge increases. 

Similarly, the width changes only very slightly as the 

discharge increases at Cornog. Both the depth and 

the velocity, however, increase rapidly with increasing 

discharge at a given cross section, while at the same 

time the roughness decreases. As the discharge at 

Cornog rose from 5 to 830 cfs, an increase of 166 times, 

the depth increased approximately 7-fold and the velocity increased 30-fold. At the same time the roughness 

factor was reduced by only half. These figures indicate 

that these hydraulic variables change littJe compared 

with the discharge and the load, both of which are 

extremely variable at any given section, as will be 

shown.

The relative magnitude and the direction in which the 

changes in the variables at Cornog take place are characteristic of the major cross sections studied on Brandywine Creek (figs. 10-13). Very little scatter appears on 

any of these graphs. In contrast, the curves drawn foi 

CoatesviHe (fig. 14), and Chadds Ford (fig. 15), based on 

stream measurements made in the vicinity of the gaging 

stations, show wide scatter. This scatter is characteristic of most of the curves drawn by Leopold and Maddock (1953) from similar data and has raised some 

doubt as to the validity of the relationships expressed 

by their curves. However, these relationships appear 

to be justified if the methods of measuring discharge 

are taken into account.

At the author's seven measuring sections, all discharge measurements were related to a single cross section. At Chadds Ford, Coatesville, and other gaging 

stations, discharge measurements may be made at
K 'oo

•&

- 50 

_cT 

;g

^ 20

5

HYDRAULIC AND GEOLOGIC CONDITIONS WITHIN SELECTED REACHES

CORNOG SEVEN SPRINGS

11

Q.

o; 

Q

0.5

0.2

CD ___

5 0.5 

> 0.3

50

20

5

-•— v ——— -

.-.-.- U- I.L _ l=f|

—

0.5

o>

Q. 

_O

CO

.005

.002

o> 

I

.02

100 1,000 10 

Discharge, in cubic feet per second

1,000

FIGURE 10.—Relation of width, mean depth, mean velocity, water surface slope, roughness parameter n' to discharge at Cornog and Seven Springs. Slopes of curves designated b, /, m, !, and y eive values of esponents in equations relating variable to discharge. Symbol • denotes velocity measurements made by current meter. Symbol 

X indicates velocity measured by floats.

various points in the vicinity of the station. A discharge measurement made at low flow in a narrow cross 

section has a velocity that may far exceed that of a 

much higher flow measured at a wider cross section. 

A study of the discharge records and of cross sections 

of the channel at Chadds Ford and Coatesville shows 

no significant difference between these and the seven 

other sections studied in detail in this investigation. 

The scatter on at-a-station curves is virtually eliminated 

when all of the measurements are made at the same cross 

section. This explanation of the cause of scatter is 

not new, but it is useful to anyone interested in interpreting the data in this or other studies that make use

of regular discharge measurements taken at gaging 

stations.

WIDTH, DEPTH, AND VELOCITY

The curves relating width to discharge (figs. 10-13) 

indicate that at most locations on Brandywine Creek 

the width remains remarkably constant as the discharge increases. With the exception of Chadds Ford 

and Coatesville the value 6 (table 1) is less than .1 at 

each of the sections. These low values of 6 reflect the 

rectangularity of the cross sections of the Brandywine 

(fig. 16).

The mean 6 value obtained by Leopold and Maddock
12

.g 100

_cT 

S 50

Q. 

<U

Q

10

5

THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

SOUTH DOWNINGTOWN GEORGIA

T3

O 

O

Q.

_g 

CO

x: 

en

o

CT

100

50

10

5

I —

10 10

5

0.5

.001

.0005

.(0

.05

'—x— •«

.01

.UUI

.0005

.0001

-•——

ri

i

i

<^^^ * ——_—

10 100 1,000 10,000

O.I 

.05

.02

10 100 1,000

Discharge, in cubic feet per second

FIGUBE 11.—Relation of width, mean depth, mean velocity, water surface slope, and roughness parameter n' to discharge at South Downingtown and Georgia. Symbol i

denotes velocity measured by current meter. Symbol X denotes velocity measured by floats.

(1952, p. 9) was .26. There is a strong suggestion that 

the higher b values result from the fact that the measurements are not all taken at the same cross section. It 

is also true, however, that the western streams toward 

which the Leopold and Maddock data are heavily 

weighted, often have less rectangular or more ellipsoidal 

channels. Some regional factors therefore may influence the rate of change of width with discharge.

The rates of change of depth and of velocity with 

increasing discharge at the principal measuring sections 

are shown in figures 10-13 and in table 1, where / is the 

rate of change of depth and m the rate of change of 

velocity. The curves indicate that both of these 

variables increase at all of the cross sections. In addition, the points on the graphs show that the two 

variables are closely related. Although at Georgia
HYDRAULIC AND GEOLOGIC CONDITIONS WITHIN SELECTED REACHES 13

the velocity increases rapidly at the expense of depth, 

at all of the other stations the rates of change of depth 

and velocity are similar to those described at Cornog. 

With the exception of Lenape where the rates of change 

of the two are equal, the / and m values (table 1 ) show 

that the rate of increase of velocity slightly exceeds 

the rate of increase of depth at the stations on Brandywine Creek.

At each station the range of discharges represent 

stages from very near the bed to within a foot of the 

bankfull stage. There is a suggestion in some of the 

data (figs. 10-13) that the depth-discharge and velocitydischarge curves may actually plot as curved rather 

than straight lines on log-log paper. Such a relationEMBREEVILLE

100

1-50

10

5

"5. 

&

10

5

(U 

Q.

.001

.0005

O.I

.05

.02 —

ship of the at-a-station curves is not uncommon. At 

present, however, the Brandy wine data are not sufficient 

to justify drawing a curve through the points, and a 

straight line is used because it simplifies the general 

analysis.

The mean / value of .41 at the seven measuring 

stations established on Brandywine Creek for this study 

is practically the same as the mean of .40 found by 

Leopold and Maddock (1953, p. 9). The mean m of 

.55 from the Brandywine data, however, is considerably 

greater than .34 value found in their general irtudy. 

These differences may or may not be greater than can be 

accounted for by the scatter alone.

WAWASET

100

50

10

5

10

5

.001

oons

-,*«

— • — •*-• ~^

" ~^?,

"""•x ——

—

O.I

.05

.02

100 1,000 10,000 10

Discharge, in cubic feet per second

100 1,000

FIGUKK 12.—Relation of width, mean depth, mean velocity, water surface slope, and roughness parameter n' to discharge at Embreeville and Wawaset. Symbol • denotes

velocity measured by current meter. Symbol X denotes velocity measured by floats.
14 THE NATURAL CHANNEL OP BRANDYWINE CREEK, PENNSYLVANIA

TABLE 1.—Summary of data showing rates of change of width, depth, velocity, slope, roughness, and suspended load in relation to discharge

at the principal stations on Brandywine Creek

Station

Chadds Ford _ __________________________________

b

. 04

. 05

. 04

1 . 00

1 . 00

. 05

. 08

. 04

. 16

. 22

/

. 40

. 45'

. 36

. 32

.42

. 42

.46

. 41

. 45

. 30

m

. 52

. 48

. 61

. 69

. 59

. 53

. 46

. 55

. 42

. 48

b+/+m

0. 96

0. 98

1. 01

1. 01

1.01

1. 00

1.00

1. 00

1. 03

1. 00

z

1 . 00

1 . 00

-. 03

1 . 03

. 19

2 . 15

. 00

. 05

y

-. 23

-. 22

-. 40

1 -. 10

-. 22

2 -. 15

— . 20 ^u

3-0 20

3

1. 53

1. 41

2. 65

1. 70

1. 30

2. 25

1. 88

2. 37

THE HYDRAULIC FACTORS ARE EXPRESSED BY THESE FORMULAS:

v=kQ L (suspended load) = pQ>

i For high discharges only. 2 por ] ow discharges only. 3 Georgia not included.

200

.£ 100

LENAPE

,.-»—**——

-E 10

D. O005 

£

non?

— _i

=

-- ~\

=

x x —

«

.02

—

~ 1

i 

1 III

*-.

vffifTT

—

IITI

FIGURE 13. 

roughness 

measured

10 100 1,000 10,000 

Discharge, in cubic feet per second

-Relation of width, mean depth, mean velocity, water surface slope, and 

parameter «' to discharge at Lenape. Symbol • denotes velocity 

by current meter. Symbol X denotes velocity measured by floats.

SLOPE

The reaches over which the slope was measured 

range in length from 2,040 feet at South Dowiiington 

to 328 feet at Seven Springs (app. A). These reaches 

are the straightest ones which could be found in the 

vicinity of the principal sections. Because of the 

variety of factors which influence the measurement of 

slope in the field, the reader should refer to the photoCOATESVILLE

100

100 1,000

Discharge, in cubic feetper second

FIGURE 14.—Relation of width, mean depth, and mean velocity to discharge at 

Coatesville. Data from stream-gaging records.
HYDRAULIC AND GEOLOGIC CONDITIONS WITHIN SELECTED REACHES 15

CHADDS FORD

100

Q. "53

J!±

-§ 100 1,000

Discharge, in cubic feet per second

IQOOO

15.—Relation of width, mean depth, and mean velocity to discharge at 

O'hadds Ford. Data from stream-gaging records.

graphs (figs. 3-9) and to the data on local variability 

given later in the discussion of pools and riffles. Certain 

general relationships common to all of the reaches and 

significant peculiarities of individual reaches may be 

appropriately considered here.

The criteria used in selecting the specific reaches at 

which the measurements of the water surface were 

made have already been set forth. To obtain comparable conditions at, each of the stations, an attempt 

was made to select reaches which were, to the eye at 

least, neither pools nor riffles. Specific attempts to 

set up objective criteria by which pools and riffles 

could be distinguished are discussed later.

Generally, in a pool having a section control, as the 

discharge and stage increase, the slope of the water surface increases; while in a riffle the slope decreases as the 

stage increases until the section control has been 

drowned out. The curves of slope plotted against discharge (figs. 10-13) show that reasonable uniformity 

was obtained in the selection of the individual reaches. 

At Lenape the slope remained relatively constant 

throughout the measured range of discharges. A small

increase in the slope of the water surface with increasing 

discharge was recorded at Seven Springs and at Cornog; 

at Embreeville the slope of the water surface increased 

relatively rapidly with increasing discharge. The 

marked increase in the slope of the water surface at 

Embreeville is due in part at least to the presence of a 

minor riffle about 50 feet below the gage at the upstream 

end of the reach. At low flow this riffle causes a break 

or step in the water surface between the two gages. 

After this riffle is drowned out at a discharge of about 

150 cfs, the rate of change of slope becomes les^.

The curve for Wawaset in figure 12 shows that the 

water surface slope increases in the lower part of the 

curve as the discharge increases, but at the highest discharge the slope is markedly reduced. This and the 

fact that the stage increases very rapidly for small 

changes in discharge at this point suggest that backwater conditions from the bridge piling and debrs may 

be causing the water elevation to be unduly high at 

the downstream gage (Corbett and others, 1945 p. 132; 

Ramser, 1929, p. 26). The backwater which produces 

this reduction in the slope of the water surface apparently does not become effective except at higher 

discharges.

At South Downington the decrease in the slope of the 

water surface with increasing discharge may be related 

to some expansion in the area of the cross section of the 

reach and to inflow at the lower end of the reach during 

the high flow.

Generally, in the reaches studied on the Brandy wine, 

as the discharge increases the slope of the water eurface 

approaches the longitudinal profile of the bed of the 

stream as determined from topographic maps. The 

change in slope with changing discharge at-a-station is 

actually small. This approach toward an approximation of the longitudinal profile of the stream largely represents a Siiioothing out of the minor variations represented by the pools and riffles. In this sense, the longitudinal profile is associated with the higher flows 

(Dausse, 1872, p. 320). This relationship, however, 

represents only one stage of a continuous series of 

adjustments. These studies indicate that the rdjustment of the channel to a group of hydraulic variables is 

not restricted to the higher flows, but it is an adjustment 

which prevails over a great range of flows.

MATERIALS ON THE BED AND IN THE BANKS

Before a discussion of the computation and significance of the roughness parameter, some information on 

the physical characteristics of the materials which make 

up the channel should be presented. These features are 

essentially geologic variables, or variables wholly or 

partially independent of the channel itself.
16 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

RIGHT 

BANK

LEFT 

BANK

COATESVLLE

CORNOG

1 SEVEN 

SPRINGS

EMBREEVILLE

WAWASET

VERTICAL SCALE 

rr 0 FEET

-10

-15

-20

SOUTH 

DOWNINGTOWN

GEORGIA

CHADDS FORD

FIGURE 16.—Profiles of cross sections of Brandywine Creek at the principal measuring sections. Dashed line shows bankfull stage at eact station.

The method used in this report to describe the bed 

material is far from satisfactory. Sampling and describing materials from the bottom of a stream are 

difficult, and the relationship between them and the 

physical significance of the results has been indefinite. 

Consequently, in selecting the method, much emphasis 

was given to the practical problem of sampling from a 

gravel bed beneath several feet of flowing water.

Had the material been randomly distributed over the 

bed and had the range of sizes been more restricted, 

mechanical analyses of large samples weighing several 

hundred or several thousand pounds would probably 

have provided the best quantitative description of the 

bed material. For several reasons this method does 

not appear to be appropriate to Brandywine Creek. 

First, because an unrepresentative sample is a distinct 

possibility in a nonrandom distribution typical of a river 

reach or cross section, an elaborate analysis of a huge 

sample from a given location might itself be a detailed 

but still unrepresentative sample. Second, it is important to know the resistance due to friction produced by 

the various sizes of material. Therefore, a description 

of the distribution and the cross-sectional areas of the

materials which obstruct the flow is probably more 

significant than is a description based simply upon a 

sample dug out of the bed. Besides there theoretical 

considerations, it would be almost a practical impossibility to obtain a large sample.

The problem of description is somewhat analogous to 

that of analyzing the size of particles in thin sections. 

In microscopic work, a single linear transect is subject 

to error, but the use of a number of linear transects can 

be shown to be a valid procedure for sampling. However, because the number of transects required for 

validity is a function of the variability of the mateiial, 

it is not easy to determine what that number should be 

(Postel and Lufkin, 1942). The following method that 

was used in sampling and describing the sediments of 

Brandywine Creek is a compromise between theory 

and practice.

At each of the principal cross sections, a transect of 

the bed material was made. Every stone greater than 

2 inches in median diameter was measured and counted. 

Measurements were made of the lengths of three 

mutually perpendicular axes on each of the pebbles. 

Four samples of the fraction of the bed material less
HYDRAULIC AND GEOLOGIC CONDITIONS WITHIN SELECTED REACHES 17

than 2 inches in diameter were taken from each cross 

section. These were sieved and a composite cumulative 

curve of the grain size distribution for this "fine" 

fraction was drawn. The percent of each grade size 

used in the composite curve is the mean value of the 

percents from the four separate analyses. The "pebble 

counts" and mechanical analyses were combined as 

follows: The percent of the total width of the bed 

covered by each of the various sizes 2 inches or larger 

was computed by dividing the total width by the sum 

of the median diameters of each of the pebbles within 

the given grade size. The sum of these percents 

represents the part of the bed covered by material 2 

inches or greater. It was assumed that the remaining 

part of the bed was described by the composite curves 

from the sieve analyses. The percents of the various 

grain sizes less than 2 inches as determined by the sieve 

analyses were converted to percents of the total width 

of the bed according to the formula :

where W equals the total width of the bed, L the width 

covered by material greater than 2 inches, and p the 

percent of the sieved sample in a given grain size. The 

curves in figure 17 show the percent of the width of the 

bed covered by each of the grade sizes. It is important 

to note that statistical parameters (Inman, 1952) derived from these curves cannot be compared with similar 

measures computed from cumulative frequency curves 

based on mechanical analyses.

All of the materials sampled (figs. 17, 18, and table 2) 

are gravels according to the Wentworth scale. At 

Georgia where the curves indicate that the bed material 

is the finest, median diameter is 0.73 inch. The

100

75

CORNOG 

SEVEN 

SPRINGS

SOUTH __ _ 

DOWNINGTOWN

GEORGIA ——• ——

-0*100

75

COATESV1LLE

EMBREEVILLE

WAWASET

LENAPE

CHADDS FORD

0.005 0.01 0.05 0.1 0.5

Diameter, in inches

FIGTJBE 17.—Cumulative frequency curves showing the size of material on the bed 

of Brandywine Creek at the principal measuring sections.

coarsest material is at Cornog where the median is 3.8 

inches; it is composed primarily of cobbles and boulders. 

The bank material of Brandywine Creek presents a 

strong contrast to the bed material everywhere in the 

watershed (figs. 19 and 20). The four curves marked 

Taylor Run (table 3) describe four samples taken from 

a vertical section of the flood plain of the Brandywine 

about 500 yards above Georgia at the point where 

Taylor Run traverses the flood plain before emptying 

into East Branch, Brandywine Creek. These curves,

FIGTTRE 18.—A, Buck Run at Pomeroy. Median diameter of bed material is 6 inches. B, East Branch near Reeds Road. Cobbles 3 to 5 inches in diameter have

been moved by bankfull discharge, which is about 800 cfs.
18 THE NATUEAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

100 -

100

-^ 75

:§ 25

100

75

50

25

DEPTH OF SAMPLE BELOW 

FLOOD PLAIN, IN FEET

LENAPE • 

BROOKFIELD •

•4.1 

2.0

0.005 0.01 0.05 O.I 0.5 1.0 

Diameter, in millimeters

FIGURE 19.—Cumulative frequency curves, based on combined mechanical and 

pipette analyses, showing the size of material in the banks of Brandywine Creek at 

six widely separated localities.

as well as the pairs of curves for Seven Springs, South 

Downingtown, and Embreeville, indicate that regardless of the elevation above the present stream bed, the 

bank material is vastly different from the bed material, 

and that bank material is much the same throughout 

the drainage basin. The variation within a single 

vertical section, such as at Taylor Run, Seven Springs, 

South Downingtown, or Embreeville, is as great as the 

variation among the different cross sections from a

downstream section at Brookfield below Chadds Ford 

upstream to Seven Springs.

TABLE 2.—Size of material on the bed of Brandywine Creek at the 

principal measuring sections

Percent of 

bed covered 

by material 

^>2 inches 

in median 

diameter

77

Station 

Cornog-_______________

Seven Springs._________

South Downingtown___. 

Georgia. _____ __________

Embreeville_ ___________

Wawaset_____________

Lenape.__ _ ____________

Coatesville_ _ ________

Chadds Ford__________.

Median 

diameter 

in inches

3.8 

2.8 

4. 1 

. 7 

' . 9

1. 6 

. 9

2. 4 

1. 1

64

57

11

21

33

7

55

23

At only a few places has it been possible to obtain 

information on the nature and thickness of the coarse 

gravel fill that makes up the gravel bed of Brandywine 

Creek. In excavations for a new bridge at Shaws 

Bridge on the East Branch, 500 yards above the fork of 

the Brandywine, the fill is 10 to 15 feet thick and rests 

upon bedrock. The fill is a mixture of sand and %-inch 

gravel and shows no evidence of stratification or lenses. 

The particles are subangular to subrounded and include 

a variety of rock types. These characteristics indicate 

that this is an alluvial deposit. Records of an excavation for a pipeline indicate that the flood plain at 

Chadds Ford below the railroad bridge is underlain by 

at least 8 to 10 feet of sand and gravel. Bedrock was 

not found in the excavation. No stratification within 

the fill was reported. Near Lyndell and near Dorian 

on the East Branch and near Icedale on West Branch,

FIGURE 20.—Typical bank and flood plain showing uniform silt above gravel, Buck 

Run at horseshoe bend near Ercildoun.
HYDRAULIC AND GEOLOGIC CONDITIONS WITHIN SELECTED REACHES 19

Brandy wine Creek, records of shallow holes beneath the 

flood-plain deposit indicate the presence of coarse 

gravel to depths of 3 or 4 feet. It is possible that in 

some instances weathered bedrock has not been distinguished from, gravel. Present information on the 

depth of the gravel fill throughout the valley is relatively 

scant. It is important to note in considering the 

development of the Branclywine profile that the creek 

does flow upon the gravel fill throughout most of its 

course. There are isolated points, however, at which 

the gravel cover is thin or nonexistent over bedrock 

outcrops.

TABLE 3. — Location and description of samples from the bank of 

the channel of Brandywine Creek

Trask

Station 

Brookfield ______

Lenape_ _ _______

Taylor Run_ _ ___

* Do.________

Embreeville_

South Downingtown_ _ _ - _ 

Do_________________

Seven Springs ___________

Depth below

surface of the

flood plain,

in feet

6.0 

4. 1 

3. 4 

2. 7 

1. 6 

. 7 

4.4 

2. 8

2. 1

3. 0 

1. 3 

5. 1

Median

diameter,

in millimeters

0. 016

. 092

. 072

.11

. 019

.021

.17

. 054

.027

. 035

. 070

.21

coeffi_ll nt _

/Qs 

\ Qi

2. 3G 

2. 30 

2. 72 

3.41 

2. 19 

2. 73 

2.97 

2. 26 

4. 11 

3. 93 

3. 89 

2 79

There is not enough information about Brandywine 

Creek to permit an adequate discussion of the origin 

of the flood plain. It is impossible to say whether the 

coarse material in the bed represents bedload and the 

fine material in the banks represents suspended load 

(Mackin, 1937, p. 828; Happ, 1952, p. 3), or whether 

the coarse material represents the deposit of a forerunner of Brandywine Creek (Fisk, 1951) which existed 

under a different climatic or hydro-logic regime. During 

relatively high flows (those occurring possibly 2 to 4 

times a year) Brandywine Creek does manage to rework 

or at least to toss about these older, coarser materials 

(fig. 185). There is little evidence of deposition of fine 

overbank deposits during floods except in isolated locations and in extremely thin layers. What the cumulative effect of these irregular thin deposits would be is 

speculative. On the other hand, if the formation of 

the flood plain resulted primarily from the accretion of 

material on the insides of bends and meanders as these 

migrate laterally, one might expect the flood plain to be 

composed of alternating silt, sand, and gravel lenses 

similar to those found in the present channel. Thus 

far, the data reveal no such changes in grain size (fig. 

19). Before any answer to these questions can be

given, much additional work is required to establish 

criteria which can be used to determine from any 

deposit its mode of deposition. The writer's present 

tentative conclusion is that the uniform frequercy of 

overbank flooding throughout the drainage basin and 

the apparently small amount of overbank deposition, 

as well as the uniform channel characteristics, indicate 

that the flood plain (deposit and surface) is closely 

related to the present channel and hydrologic regime.

SUSPENDED LOAD

Leopold and Maddock (1953) have shown thr.t the 

quantity and concentration of the suspended lor.d are 

closely related to the shape of a given chamie1 . In 

addition Vanoni (1941, p. 618) and others (Schnackenberg, 1951, p. 378) have shown that the value of a 

roughness factor may be influenced by the sediment 

concentration. Vanoni suggests that a high sediment 

load may reduce the energy loss in friction by dampening the turbulent eddies. These relationships have 

been observed primarily in streams carrying large concentrations of suspended material. Unfortunately, 

little data on the suspended load in rivers in the eastern 

United States is available for comparison with rivers in 

other regions. The data on Brandywine Creek are 

presented here in order to show at least the magnitudes 

of the suspended loads and, perhaps, the kinds of 

changes which take place in the suspended load with 

changing discharge. Although the data on load elsewhere in the Brandywine Creek valley are meager the 

sediment rating curve for the Brandywine at Wilmingtoii, Del., is well defined.

At Wilmington the suspended load increases rs the 

2.37 power of the discharge (fig. 21). This is the value 

of j in the equation:

L=pQ? (9)

where L is suspended load in tons per day, Q the discharge in cubic feet per second, and p is a constant. 

The highest load shown on the graph represents a 

concentration of 2,400 parts per million at a discharge 

of about 16,000 cfs.

The results of measurements of the suspended load 

at seven of the principal measuring sections are shown 

in figure 22. Although the few known values for each 

station do not justify drawing a curve through the 

points, at each it is clear that the load increases rapidly 

with at-a-station discharge. In an ideal examp^ for 

which curves could be drawn for each station, the 

curves would presumably be approximately parallel, 

the most upstream cross section being on the left and 

to its right the stations successively downstream from it. 

The proximity of the stations to one another on Brandy-
20 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

100,000 —

10,000 —

Q> __

•-. 1,000 —

1,000 

Dischorge, in cubic feet per second

10,000

FIGURE 21.—Sediment rating curve for Brandywine Creek at Wilmington, Del. 

Data collected by TJ. S. Geological Survey.

wine Creek and the normal scatter of points on the 

sediment rating curve make it impossible to demonstrate such a relationship here. There is apparently a 

tendency, however, for the points from the upstream 

cross sections to lie successively to the left of the 

straight line representing the sediment rating curve for 

the Brandywine at Wilmington.

Consideration will be given later to the relationship 

between suspended load and channel shape in the parts 

of this report which are concerned with the changes of 

the variables in the downstream direction.

ROUGHNESS

Thus far it has been shown that orderly changes take 

place in the width, depth, velocity, and slope at-astation as the discharge increases. Similarly, it has 

been shown that at every station on Brandywine Creek, 

the materials on the bed and in the banks are distinctly 

different. The changes in a roughness parameter which 

take place at-a-station will now be considered. In a

sense, these changes in roughness reflect the interaction 

of the hydraulic and geologic variables. Although 

difficult to explain, this interaction is fundamental to 

the behavior of all of the variables.

The roughness factor nf used here represents a coefficient in the equation

v= 1.486 (10)

where v is the mean velocity, d the mean depth, and s 

the slope of the water surface. The use of a modified 

Manning equation in the present study is recommended 

by its simplicity and its applicability to a wide range of 

channels. The roughness parameter nf represents some 

measure of the resistance to flow in the channel. It 

differs from the Manning n for two reasons: the mean 

depth rather than the hydraulic radius is used in the 

computation of n', and the slope represents the slope 

of the water surface and not the energy grade line.

100,000 pr

10,000 —

^ o> 1,000,_—

"8

SEVEN SPRINGS

SOUTH DOWNINGTOWN

GEORGIA

EMBREEVILLE 

+ WAWASET 

O LENAPE

100 1,000 

Discharge, in cubic feet per second

10,000

FEGUBE 22.—Suspended load at the principal measuring sections at different discharges. Solid line shows sediment-rating curve for Braniywine Creek at 'Wilmington, Del.
HYDRAULIC AND GEOLOGIC CONDITIONS WITHIN SELECTED REACHES 21

The distinction between n in equation (2) and n' in 

equation (10) is important. In order to determine the 

value of the Manning roughness parameter n, it is 

necessary to obtain as closely as possible a value of the 

slope of the energy grade line; that is, the rate of dissipation of the energy in the flowing water. In addition 

to a precise measurement of slope, this requires an 

evaluation of the changes in velocity head from the 

upstream to the downstream end of the reach. The 

slope of the water surface under some conditions may 

closely approximate this energy slope. The nature of 

the field conditions in the present investigations, however, did not permit the computation of the energy 

slope. Therefore, the values of n' computed for the 

stations on the Brandywiiie cannot be substituted for 

n in the Manning equation when the latter is used in 

the indirect determination of velocity in similar channels elsewhere. That this caution is warranted can 

clearly be seen from the unusually low values of roughness n' shown for the higher stages 011 the Brandywine 

(figs. 10-13).

This study deals only with the relationship between 

the roughness factor and the size of the material in the 

bed of the stream. Actually the roughness n' represents far more than this: n' is related not only to the 

bed and bank materials but also to changes in channel 

shape, the size and distribution of vegetation, bed and 

bank configuration, the amount of sediment, and other 

factors. Unfortunately, it is impossible here to evaluate the separate effects of each of these factors. Consideration of this roughness factor, however, does give 

some way of making the transition between the purely 

physical characteristics of the channel and the effects 

which such characteristics have on the flow.

At all the cross sections studied, the roughness n' decreased with increasing discharge (table 1). This relationship is shown in figures 10-13 and is also illustrated 

by figure 23 which shows graphs of depth plotted against 

roughness (n'} for four stations 011 the creek. Smooth 

curves have been drawn by eye through the plotted 

points. In each of the examples the roughness decreased 

with depth; that is, the roughness decreased with increasing discharge. In the cross sections on Brandywine Creek n' varies approximately as depth to the 

minus 0.6 power. This decrease in n' with depth may 

be associated to some degree with the increasing influence of the relatively fine and possibly smoother bank 

materials, although vegetation on the banks will usually counteract this effect. The decrease in roughness 

at higher flows may also be associated with better alinement of the channel with the banks and increasing 

uniformity of the channel cross sections. Last, and 

probably most important, the decrease in the roughness

factor n' may also reflect a decrease in the friction loss 

accompanying submergence of protuberances on the 

bed. Such a result is suggested by experiments on 

pipes and fixed open channels in which the Chezy C 

friction factor increases as the ratio of the depth of 

flow to the size of protuberances increases (Powell, 

1950, p. 577; Kobinson and Albertson, 1952, p. £82).

Although a number of other factors also affect the 

value of the roughness parameter, the plot of nf at a 

constant depth of 1 foot against grain size in figure 24 

for the major stations on the Brandywine indicates 

that n' is directly related to the size of the material on 

the bed. The small range of grain sizes 011 the Brandywine and the normally narrow range of n' do not permit 

a completely satisfactory verification of this relationship. The results, nevertheless, are in accord with 

those observed by Strickler and others (Lane and Carlson, 1953).

The reader may well wonder at this emphssis on 

roughness. There are two principal reasons: First, if n' 

is to be used in the analysis of river processes because of 

its "simplicity," it is important to have some understanding of its physical meaning. Second, in a sense n' 

is an independent variable in the channel regime. It 

may be argued that because n' is simply computed from 

measurements of width, depth, velocity, and slope, it 

cannot be used as an independent parameter with which 

to explain the reactions of the other variables. Regardless of the method of calculation, the roughness n' is, 

in part at least, a measure of the physical "roughness" 

of the channel. Some such roughness factor is essential 

if we are to relate independent factors such as grain 

size and sediment to the hydraulic variables wHch in 

concert with these independent factors determine the 

form and gradient of the channel.

CORNOG

1

" 1

1

~ °\

1

1

V

SEVEN SPRINGS

1

x

"l

1

1

0

1

}

EMBREEVIULE

1

f

I 1* 

1

1

1

1

LEN/iPE

1

1 

1

bo

1

1

1

2 O6 10 02 .06 10 02 .06 .10 .02 .0£ .K

x - Float ve 

o - Standard

Roughness, /?' OClty ^ ^ 

or current

meter measurement

FIGURE 23.—Kelation of roughness parameter n' to mean depth at Cormg, Seven 

Springs, Embreeville, and Lenape.
22 THE NATURAL CHANNEL OP BRANDYWINE CREEK, PENNSYLVANIA

.10

.05

o tr

i.3

,01 1.0

Median diameter,in inches

FIGITRE 24.—Relation of roughness parameter n' to median size of bed material; n' computed for a constant depth of 1 foot.

10

The problem of choosing the independent variables 

is a complicated one. Where simultaneous interaction 

of at least eight designated variables is involved, no 

simple rules can be laid down by which the independent 

variables can be separated from the dependent ones. 

If the process of development of the river channel and 

profile is a hydrodynamic one, the geologically independent factors must eventually be considered from the 

standpoint of their behavior and not solely by correlating their effects. Thus, although grain size, one of 

many geological factors, may be correlated with slope, 

analysis shows that the hydraulic behavior of "grain 

size" is a frictional factor, and as such, it is one of 

several related factors which exercise a frictional resistance. A roughness n' , although perhaps not the most 

significant measure of energy and friction losses, is, at 

present, one which can be obtained under field conditions. As such, it is a useful tool which can at least be 

applied qualitatively in our effort to find the mechanism 

by which the river channel and its profile develop.

SUMMARY AND INTERPRETATION OF AT-A-STATION 

CHANGES

At any given reach the shape and profile of the 

channel appear to be related to the interaction of a 

minimum of seven variables: discharge, width, depth, 

velocity, water surface slope, roughness (as a measure of 

frictional resistance), and load. At each cross section 

studied on Brandywine Creek an increase in discharge is

associated with the following behavior of the other 

variables: (1) the width increases slowly or remains 

relatively constant, (2) both the depth and velocity 

increase, the velocity as a rule increasing more rapidly, 

(3) the roughness n' decreases, (4) the suspended load 

increases rapidly, and (5) the slope of the water surface 

remains practically constant.

It could be argued that the hydraulic relationships 

observed at the cross sections on the Brmdywine are 

simply the changes which would occur in any fixed 

channel or pipe as the discharge flowing through the 

cross section increased. In a natural channel, however, 

the stream determines its own cross section. This cross 

section is not the only one which would permit the 

passage of given amounts of water. A given flow could 

pass at different velocities through a variety of cross 

sections representing an infinite number of combinations of widths and depths. Despite the possibility that 

various combinations of width, depth, and velocity 

would be capable of discharging the water r,nd sediment, 

natural rivers with self-formed channels have characteristic combinations of these variables. One such 

combination is observed in Brandywine Creek. The 

point of interest is that in a specific geologic and climatic 

setting the river has built for itself a wide, flat channel 

with steep banks of fine material. In some manner this 

particular channel was determined by the kinds of 

hydraulic relationships which have been noted.

If it is assumed that a given drainage basin is under-
HYDRAULIC AND GEOLOGIC CONDITIONS WITHIN SELECTED BEACHES 23

lain by a particular type of rock, the shape of a cross 

section of the stream at any point would presumably 

be determined by the combined effect of the discharge, 

suspended load, and roughness (the hydraulic mechanism through which the material expresses its effect). 

The type of bedrock exercises an influence both within 

the channel where it affects the roughness of the bed 

and bank and within the source area from which the 

suspended load is derived. In a basin in nearly homogeneous rock a hypothetical channel shape imposed by 

the variables above would be modified where the stream 

encountered a particularly resistant rock. The latter 

may cause the width of the channel to expand at the 

expense of the depth. Nevertheless, the combined 

effect of the discharge, suspended load, and roughness 

at that point is to produce a particular channel shape. 

Although possibly a dominant factor at a cross section, 

the resistant bed is not the sole factor determining the 

shape of the channel. That this is true is indicated 

by the fact that the channel is not inordinately wide 

where it passes over the resistant bed material. This 

"constructive" as opposed to "erosive" aspect of channel formation is illustrated by the narrow benches built 

adjacent to the bedrock walls at South Downingtown 

and at Embreeville—benches which are constructed to 

the height of the present flood plain (see fig. 16). 

There is an infinite variety of factors which may modify the results or interrupt the processes described in 

this idealized concept of a self-formed channel. Nevertheless, it is probable that these chance irregularities 

simply obscure but do not suspend the governing 

principles.

With additional information on the rate of change 

of suspended load at-a-station, Leopold and Maddock 

(1953, p. 28) were able to demonstrate that "for a 

given width, and at a given discharge, an increase in 

suspended sediment requires an increase in velocity 

at the expense of depth." Further, they suggest 

(p. 43) that

The characteristically rapid increase of suspended load with 

discharge at-a-station requires for transportation of that load, 

a relatively rapid increase of velocity in comparison with depth

m 

or a relatively large value of -~ • For the cross sections studied

2

this was equal to 0.85. This value, being larger than 7; requires

S'/2

that — increases with increasing discharge. This is accom77-

plished primarily by the decrease of roughness n, resulting from 

the increase of concentration of suspended load with discharge.

The average, rates of change with discharge of the 

width, depth, and velocity at the stations studied on 

the Brandy wine are:

b = .04 

/=.41 

ra=.55

.7 = 2.2

m 

Y=1.34

Based on figure 18 (p. 25) of Leopold and Maddocl", with 

6=.04 and -j=1.34, j (the rate of increase of suspended

load with discharge) should equal approximately 3.2. 

The measured value at Wilmington of 2.37 is therefore

17} apparently too low to account for the high -j ratio.

Aside from the fact that the relationships given by 

Leopold and Maddock are not meant to be exact, 

several qualitative explanations can be advanced which 

may help to account for the differences between the 

behavior of Brandywine Creek and the western rivers. 

First, the decrease in roughness with increasing discharge on the Brandywine appears to be related to 

several factors that include a decrease in the effective 

roughness of the material on the bed as it becomes 

submerged, as well as a decrease in channel irregularities. The effect of increasing sediment concentration 

is probably secondary. Second, the disparity in bed 

and bank materials may be a cause of the large width 

at low flows and hence also of the very low rate of 

change of width with discharge at-a-station. Where 

the materials are more nearly alike, the increase in 

width may be greater. In turn, a more rapid increase

7YL in width at the same -j ratio would require a lower

value of j according to figure 18 in Leopold and Maddock. These suggestions do not explain the operation 

of the phenomena. Nevertheless, they do point out 

some additional factors which would be considered in 

refining the relationship between suspended load and 

channel shape.

Although the data on the Brandywine show that 

the behavior of the variables at a number of different 

locations is very similar, this picture of the process is 

greatly oversimplified. The adjustments whicb take 

place in the channel are simultaneous and not sequential. Our understanding of the processes may be 

enhanced by this piecemeal approach, but we should 

recognize that much of the sequence is artificisl. In 

the absence of experiments in which specific variables 

are controlled and individual ones allowed to vary, 

it is almost impossible to isolate and consider each 

factor separately.
24 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

DOWNSTREAM VARIATION IN THE RIVER CHANNEL

Although the details of the mechanism by which 

a river channel reach is developed and maintained 

are not understood, the data in the first part of this 

study present a picture, at least, of the variables 

involved and the way in which they behave. The 

curves (figs. 10-15) showing the relation of the variables 

to discharge at a given station permit a discussion of 

the changes which take place in a downstream direction in these variables at comparable flows. Again, 

there is no assurance that the variables chosen for 

analysis are physically the most important ones. 

It is probable, however, that an analysis of these 

will demonstrate some significant physical phenomena. 

They may also enable identification of the processes 

by which the measured effects are brought about. A 

voluminous geologic literature deals with this question 

of the way in which a river changes its size and shape 

in flowing from headwaters to mouth. Most of this 

literature has dealt with the longitudinal profile, or 

the way in which the slope of the bed of the river 

changes in the downstream direction. The purpose 

here is to describe the changes in the measured variables 

and to relate whenever possible these changes to changes 

in the longitudinal profile. The ultimate and as yet 

unrealized aim is not solely to demonstrate "relationships"; rather the aim is to distinguish causes from 

results in the observed processes.

Most of the channel characteristics which are being 

compared at various points along the river are closely 

related to the amount of water contributed by the 

drainage basins above the cross sections. In this 

study the hydraulic characteristics of the separate 

reaches will be compared when each is experiencing 

the same duration or frequency of flow. The terms 

"duration" and "frequency" apply respectively to the 

percent of time during which a given flow is equaled or 

exceeded, and to the frequency with which a given flow 

recurs. The distinction between the two terms is 

based upon the way in which each is computed (see 

Linsley, Kohler, and Paulhus, 1949, and other textbooks on hydrology for standard methods of computations; and Leopold and Maddock, 1953).

In the drainage basin of Brandywine Creek the discharge at a given point is directly related to the contributing area at that point. Because the drainage 

area increases downstream, the discharge at any given 

frequency of flow likewise increases downstream. In 

this study, curves showing the hydraulic variables at 

different stations plotted against discharge for a given 

frequency of flow (figs. 25, 26, 29, 34, and 35) describe 

the changes which take place in these variables in the 

downstream direction.

For the downstream comparison of the vddth, depth, 

velocity, slope, roughness, and load, below the bankfull 

stage (fig. 25), a discharge of a constant duration will 

be used. Synthetic duration curves have been drawn 

for each of the seven principal measuring stations 

established in this study on Brandywine Creek. The 

method used hi constructing these curves is based on a 

comparison of miscellaneous discharge measurements 

made at a measuring station with the daily mean flows 

at a permanent recording gage. In this study measurements made at the seven principal stations were compared with the daily mean flows on the same day at the 

Chadds Ford gaging station.

The downstream changes of the variables will be 

discussed in three parts. First, the wry in which 

width, depth, velocity, roughness, and the slope of the 

water surface change between the principal stations, 

including Coatesville and Chadds Ford, will be shown. 

These comparisons will be made for three different 

durations of flow below the bankfull stage. Second, a 

comparison of the downstream changes in the variables 

at the bankfull stage will be made for these same 

stations. Some consideration will be given here to the 

question of the frequency with which the bankfull stage 

is attained at upstream and downstream locations in 

the drainage basin. Separate consideration is given to 

the bankfull stage because of its relation to the flood 

plain and because a different method was ired to determine the bankfull discharge. Third, comparisons of 

the width, depth, and velocity, using all of the measurements covering the entire basin will be made. In 

these comparisons only two durations will be shown. 

Because there is usually only 1 and at the most 3 

measurements at each of these stations covering the 

headwater tributaries, to obtain values fcr the width, 

depth, and velocity at 2 discharges of specific durations, 

it was necessary to extrapolate from the discharge which 

had actually been measured. The methods used in obtaining the discharges for the given durations, as well 

as the values of the variables, will be given in a later 

part.

DOWNSTREAM VARIATION AT LOW AND FIGH FLOWS

In figure 25 the width, depth, velocity, roughness, 

and slope are plotted against the discharge which is 

equaled or exceeded at each station 50 percent of the 

time. For each station the discharge representing a 

duration of 50 percent was determined from the synthetic duration curve for that station. The values of 

the width, depth, velocity, roughness, and slope of the 

water surface at this discharge were read from the at-astation curves shown in figures 10-13. Because the 

values of the variables were read from smooth straight-
DOWNSTREAM VARIATION IN THE RIVER CHANNEL 25

3'°°

,.-50 —

20

3

5

.5

.5

.003

.001

.•-*-

-CO

a:

100

DISCHARGE EQUALED OR

EXCEEDED 50 PERCENT

OF THE TIME

1,000 100 500 1,000

DISCHARGE EQUALED OR 

EXCEEDED 15 PERCENT

OF THE TIME 

Discharge, in cubic feet per second

500 1,000

DISCHARGE EQUALED OR

EXCEEDED 2 PERCENT

OF THE TIME

5,000

FIGURE 25.—Relation of width, depth, velocity, water surface slope, and roughness parameter n' to discharge at the principal stations at flows equaled or exceeded 50, 1 fi, and

2 percent of the time.

line curves, equation (10) may not be verified exactly at 

each of the points on the downstream curves. Each 

of the variables was plotted against discharge and a 

straight line was fitted by eye to the plotted points. 

The slopes of these lines are designated b, f, and m for 

width, depth, and velocity, respectively, as they were 

for at-a-station curves. Similarly, the letters y and z 

are used to denote the slope of the curves of the roughness factor against discharge and of slope of the water 

surface against discharge, respectively. No data were 

available for the slopes of the water surface at Coatesville and at Chadds Ford. The slope of the bed as

determined from the profile of the bed elevation was 

used in the computation of nr and in the plot of slope 

against discharge. No great discrepancy between these 

values and the values of nr and s for the other stations, 

however, is noticeable in the curves showing the changes 

of the variables in the downstream direction. At 

Chadds Ford several measurements of the slope of the 

water surface at low flows indicate that the slope of the 

bed used in the downstream curves is somewhat greater 

than the slope of the water surface at Chadds Ford at 

low flow. The result may be z values that are somewhat too low; that is, the slope of the curve of s plotted
26 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

against Q may tend to be too flat. This effect, however, 

is small.

The other curves in figure 25 were constructed in the 

same way as was the curve for the discharge experienced 

50 percent of the time. A discharge equaled or exceeded 2 percent of the time is a relatively high flow. 

Although there are no pictures of the river at precisely 

this discharge, the high flows shown in figures 4-9 

represent flows equaled or exceeded less than 1 percent 

of the time. The flow at Cornog (fig. 3) is somewhat 

higher and represents a duration of about 0.01 percent, 

or less than 1 day per year.

The values of the exponents in the equations expressing each of the variables in terms of discharge are given 

in table 4. Only those values related to curves drawn 

from data for the main stations at durations equaled or 

exceeded 50, 15, and 2 percent of the time will be 

considered now. At every stage the actual values of 

b, f, and m closely approximate the equation of continuity in which b-\-fJrm = l. Width, depth, and 

velocity all increase as roughness and slope decrease 

downstream on the Brandywine at all frequencies of 

flow (fig. 25). At each successively higher flow—that 

is, decreasing frequency or duration (table 4)—the rate 

of change of depth downstream remains constant, but 

the rate of change of width increases, and the rates of 

change of velocity and roughness decrease. From a 

qualitative analysis of these changes some understanding 

of the physical processes which control them can be 

gained.

TABLE 4.—Summary of changes in width, depth, velocity, slope, 

and roughness in the doivnstream direction for discharge of 

different durations, Brandywine Creek

Duration (percent

of time flow is 

equaled or exceeded)

Rate of change -with discharge of:

Width 

b

Depth Velocity Slope 

f m z

Roughness 

y

BankfulL

PKINCIPAL STATIONS

.34 .45 .32

.38 .42 .32

.45 .43 .17

.42 .45 .05

-. 80 -. 40

-. 92 -.51

-.97 -.32

-1.07 '-. 28

PRINCIPAL STATIONS AND HEADWATER TRIBUTARIES

50______-_____- .57 .40 .03 _--__. ..

2__-______-_-__ .58 .40 .02 ______ __

i Computed from equation (]0).

At no stage does the mean velocity decrease downstream. During periods of low flow the mean velocity 

increases downstream, and at higher flows the velocity 

apparently remains constant downstream. Values of 

m in table 4 decrease with increasing flow. This decrease is probably associated with changes in the rate 

of decrease of roughness rather than with changes in

the rate at which slope decreases. It is logical to 

believe that as the change of roughness vdth respect to 

changing discharge becomes less—that is, as the difference between the roughness upstream and the roughness downstream is reduced—the difference between 

the velocity upstream and the velocity downstream 

will be reduced.

It has already been shown in figure 24 that the 

roughness factor decreases downstream in company 

with the decrease in grain size. The curves of n' 

plotted against discharge (fig. 25) show that the rate 

of change of n' in the downstream direction decreases 

for three different durations of flow. Several suggestions can be made concerning the deceasing rate of 

change of nr at high flows: 1. The range of n', as computed in this study, is greater at an upstream station 

than it is at a downstream one. This may result from 

the fact that the lowest flow used in this downstream 

comparison represents a much shallower depth at the 

upstream stations than it does at the downstream ones. 

2. An increase in the concentration of the suspended 

load downstream at higher flows would tend to reduce 

n' slightly in the downstream direction. 3. The absence of sufficient fine material on the b Q,d of Brandywine Creek precludes the possibility that dune formation in the downstream reaches plays an important 

part in increasing n' at high flows and thus flattening 

the curve of n'.

The slope-discharge curve is steeper for high than 

for low flows (table 4). This possibly car be attributed 

to the fact that the relation of discharge to drainage 

area appears to be different for various durations of 

flow.

These observations on the changes of the variables 

at-a-station and in the downstream direction on the 

Brandywine Creek are combined in figure 26. The 

relationships described in the figure we^e collectively 

termed the "hydraulic geometry" of the stream channel 

by Leopold and Maddock (1953). In figure 26a the 

numbered solid lines show the actual changes of the 

width, depth, velocity, roughness, parameter, and slope at 

each of the seven major stations established on Brandywine Creek for this study. These curves are the same 

as the at-a-station curves shown in figures 10-13. The 

family of curves representing all of the stations indicates 

the uniformity of the behavior of the variables at the 

different cross sections. Two points are ehown on lines 

1 and 7 in each of the groups. The points marked A are 

the values of the variables at Cornog at a low flow: 

specifically, at a discharge equaled or exceeded 50 

percent of the time. Those designated B are the values 

of the variables at Cornog at a high discharge, equaled 

or exceeded 2 percent of the time. A' and B' are the 

values of the variables at Lenape for the low and high
DOWNSTREAM VARIATION IN THE RIVER CHANNEL 27

^100

c !

a.

O)

Q

0.5

8.

.0

CO

.001

.0005

8. 

_s 

en

8o> 

cr>

8 

cr

.05

o

cn

50 100 500 1,000 

Discharge,in cubic feet per second

EXPLANATION

CHANGE DOWNSTREAM FOR DISCHARGE 

OF GIVEN DURATION

CHANGE AT-A- STATION FOR DISCHARGE 

OF DIFFERENT DURATION

FIGURE 26.—a. At-a-station curves of width, depth, velocity, slope, and roughness parameter for seven principal stations on Brandywine Creek, and downstream variation 

in these variables between Cornog and Lenape for flows equaled or exceeded 50 and 2 percent of the time. Numbers refer to stations as follows: 1. Cornog, 2. Seven 

Springs, 3. South Downington, 4. Georgia, 5. Embreeville, 6. Wawaset, 7. Lenape. l>. Schematic diagram showing hydraulic geometry of Brandywine Creek: at-astation and downstream changes in width, depth, velocity, slope, roughness parameter, and suspended load.

303765—55———3
28 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

flows respectively. The dotted lines A-A' and B-B' 

therefore show the rates of change of the width, depth, 

velocit}-, slope, and roughness n', in the downstream 

direction for durations of 50 and 2 percent, as shown in 

the figure. They are near replicas of the downstream 

curves shown in figure 25. The slopes of the lines are 

somewhat different since the lines connect 2 points only 

and do not represent the average lines drawn through 

all 7 points. Figure 26a summarizing both the at-astation and the downstream changes in the variables 

with changing discharge shows the following results: 

First, whereas width and slope change little at-astation, width increases and slope decreases very markedly in. the downstream direction. Second, depth 

increases at-a-station at nearly the same rate as it does 

in the downstream direction. Third, roughness decreases at-a-station somewhat more rapidly than it does 

downstream. And last, velocity increases slightly or 

remains constant in the downstream direction, while 

at-a-station velocity increases rapidly with increasing 

discharge. These are the fundamental changes which 

take place in the measured variables.

These same relative rates of change of the variables 

at-a-station and in the downstream direction are shown 

diagrammatically in figure 266. Although the diagram 

closely resembles the actual results shown beside it in 

figure 26a, they are not identical. The slopes of the 

curves represent the mean at-a-station and downstream 

change in the variables. Also, a diagram has been 

added, based on the information available, showing the 

changes in the suspended load with increasing discharge 

at-a-station. This diagram is a replica of figure 22. 

Figure 266 therefore shows the complete hydraulic 

geometry of the Brandywine Creek.

A comparison of the hydraulic geometry of the 

Brandywine Creek with that of the Bighorn River in 

Wyoming shown by Leopold and Maddock (1953, 

fig. 12, basin A) reveals some similarities in the behavior 

of the variables. The relative rates of change of the 

variables at-a-station and in the downstream direction 

are the same in the two rivers. Why these parallels 

exist is not clear. The similarities however do suggest 

two conclusions: First, the rates of change of the variables are not specifically related to regional differences. 

Second, the processes which are manifested in these 

changes of the variables are similar in different regions.

The hydraulic geometry of the Brandywine Creek 

illustrates a generalization noted by Leopold and 

Maddock (1953, p. 27). At a constant discharge, 

marked Q' on the diagram in figure 266, the conditions which prevail at an upstream station are represented by the intersection of the 2 percent curve with 

the vertical line for the given discharge. The conditions at the downstream station are given by the intersection of this line with the downstream curve for the 

50 percent duration. At a constant discharge the width 

of the downstream cross section is greater than the 

width of an upstream section. The depth is very 

nearly the same at both, while the velocity is higher at 

the upstream section when it is experiencing the same 

discharge as a downstream section. When both the 

upstream and the downstream cross sections are 

experiencing the same discharge, the concentration of 

the suspended load is apparently greater at the upstream 

station.

Figure 27 constitutes an attempt to elaborate this 

relationship between suspended load and channel shape. 

The diagram generalizes some complex relationships 

and at the present time is meant to be suggestive 

rather than conclusive.

In figure 27 the width-to-depth ratio Ii8,s been plotted 

against the suspended load from data collected on the 

Brandywine. The numbers beside each point are the 

discharges, in cubic feet per second, which occurred 

with the given load and width-to-depth ratio. The 

two lines marked 50 percent and 2 percent are mean 

lines drawn through points showing the relation of 

discharge to the width-to-depth ratio as discharge 

increased in the downstream direction for those two 

durations of flow. The diagonal lines are isopleths of 

discharge. The numbers beside the points indicate 

that the discharge in general increase? from left to 

right. The spacing of the isopleths is only approximate. 

It is based upon the numbers shown 011 the figure and 

upon the known decrease in the width-to-depth ratio 

with increasing discharge at each station. This 

decrease in the width-to-depth ratio at a particular 

station, for example, would be given by a diagonal line 

from E on the 50 percent curve to H on the 2 percent 

curve. The slopes of the isopleths of discharge are 

based upon the points shown and upon the spacing 

determined above. They are also only approximately

10 100

Suspended load, in tons per day

1,000

Fir.UBE 27.—Relation of suspended load to width-depth ratio. Comparison of 

channel shapes at constant discharge with different su^tjended loads.
DOWNSTREAM VARIATION IN THE RIVER CHANNEL 29

correct. Assuming that figure 27 presents a fair picture 

of the available data for the principal stations on 

Brandywine Creek, one can make several observations 

concerning the behavior of suspended load on the 

Brandywine.

At the principal stations on Brandywine Creek the 

width-to-depth ratio remains relatively constant with 

increasing discharge in the downstream direction. This 

relationship is shown by the horizontal lines labeled 50 

and 2 percent. (Some increase in this ratio is indicated 

by the data from the headwater tributaries.) The 

downward trend from left to right of the plotted points 

indicates that small loads are associated with low discharges and high width-to-depth ratio, and large loads 

with high discharges and a low width-to-depth ratio. 

Such an association might be expected simply because 

at any cross section an increase in discharge is accompanied by an increase in suspended load and by a 

decrease in the width-to-depth ratio.

For any given discharge the graph also indicates that 

a narrower cross section may be associated with a 

greater suspended load. For example, a flow of 200 

cfs represents a flow equaled or exceeded 2 percent of 

the time at an upstream station (G), while the same 

flow downstream (at E) is equaled or exceeded 50 

percent of the time. When the station at G is experiencing a flow of 200 cfs the width-to-depth ratio is 

about 28 and the suspended load about 65 tons a day. 

When the downstream station at E is experiencing the 

same flow, the width-to-depth ratio is about 60 and 

the suspended load 35 tons a day. A similar general 

relationship can be shown for points H (upstream) 

and F (downstream) when each is experiencing a flow 

of 1,000 cfs. These comparisons suggest that at the 

principal sections on the Brandywine a narrower 

section seems to be associated with a greater concentration of suspended load, as well as with a higher 

velocity (fig. 266).

At a constant discharge the width-to-depth ratio 

might have remained constant between the two stations. 

Because the upstream channel has a smaller area of 

cross section, for a given discharge the velocity would 

be higher. In this instance the larger suspended load 

might be associated with an increase in velocity alone, 

the channel shape being unchanged. For the width-todepth ratio to remain the same at a given discharge, 

however, would require the isopleths of discharge in 

figure 27 to be horizontal. The points on the figure 

indicate that the isopleths of discharge cannot be drawn 

horizontally. Thus, not only in the average data from 

western rivers (Leopold and Maddock, 1953, p. 24), 

but in a single eastern stream the increased suspended 

load apparently is accommodated not only by an increased velocity but also by a change in channel shape.

THE BANKFULL STAGE

The bankfull stage is very close to the stage at which 

overbank flooding occurs. For this study the br.nkfull 

stage has been defined as that stage at a giver cross 

section at which, in a plot of the width-to-deptb ratio 

against stage, the curve breaks sharply and the width 

becomes exceedingly large. Thus the bankfull stage at 

South Downingtown in the example in figure 28 is at 6 

feet. Because the author was unable to make any discharge measurement in the field at this bankfull stage, 

the discharge at each station was determined by extrapolating a curve of discharge against the width-to-depth 

ratio. Because measurements had been made very 

near the bankfull stage at most of the principal sections, 

the extrapolations required were not large. At Coatesville and at Chadds Ford, the width, depth, and velocity 

at the bankfull discharge were read from the smooth 

curves (figs. 14 and 15). At the seven stations established for this study, the values of the width and mean 

depth at the bankfull stage were determined from the 

cross sections (figure 16). The mean velocity r,t the

bankfull stage was then

Figure 29 shows a plot of each of the variables against 

discharge in the downstream direction at the bankfull 

stage. The values of the slopes of the lines are listed 

in table 4. For the graph of slope plotted against 

discharge, the slope used is the slope of the b<?d at 

Chadds Ford and at Coatesville; at each of the other 

seven stations it is the measured slope of the highest 

flow. After the slopes of the curves of width, depth, 

velocity, and water surface slope in relation to discharge 

had been determined, the relation of the roughness nr 

to discharge was computed as follows:

From the equation

V= 1-486-:— nf (10)

substituting equations (4), (5), and (6) for d, v, and s, 

respectively, and letting

n'=rQy (7)

then

and, substituting from table 4,

y=-.2S

A comparison of these curves for the bankfull stage 

with those in figure 25 shows that the trends in the
30 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

2.000

1,000

O

La 

o

500

100

Width-to-depth ratio

FIGURE 28.—Example of method used to determine bankfull stage at each of the cross 

sections. Width-to-depth ratio plotted against discharge and stage at South 

Downingtown.

rates of change of the variables which took place at 

lower and more frequent flows are continued to the 

bankfull stage. The rate of change of width in relation 

to discharge continues to increase. The mean velocity 

at this higher stage becomes very nearly constant 

downstream, and the depth and slope continue to increase downstream at the same rates as they did during; 

the lower flows. The rate of change of roughness is 

somewhat less than the rate during the more frequent 

lower flows.

The bankfull discharge was computed without regard 

to frequency in order to find out whether the baiikful] 

stage represented a particular frequency of flow at both 

upstream and downstream stations. It was felt that 

the available data might be of some use ir approaching 

the question of the significance of the flood plain and 

the frequency with which it is flooded. The frequency 

which the bankfull discharge represents at each of the 

nine stations was computed as follows:

Records were available at Coatesville and Chadds 

Ford from which curves showing the relation between 

discharge and recurrence interval could be drawn. The 

annual-flood method was used in analyzing the 8-year 

record at Coatesville and the 19-year record at Chadds 

Ford; this means that only the highest instantaneous 

peak recorded in a year was used in plotting the high 

flows. (See Jarvis and others, 1936, for standard 

methods of determining flood frequencies.)

At Cornog, Seven Springs, South D owningtown, 

Georgia, Embreeville, Wawaset, and Lenape, where no 

records existed, curves were drawn for each station by 

plotting the discharge per square mile at the station 

against the discharge per square mile at Chadds Ford. 

Figure 30, which uses the South Downingtown station 

as an example, illustrates how this was done.

The computed discharge at the bankfull stage at 

South Downingtown is 1,600 cfs, or 18.6 cfs per square 

mile. Figure 30 shows that this discharge corresponds 

to a discharge of 28 cfs per square mile at Chadds Ford 

(see arrows). The Figure indicates that this flow has 

a recurrence interval of 2.7 years at Chadds Ford; that 

is, the annual maximum flow will equal or exceed 1,600 

cfs, the discharge at bankfull stage, on the average of 

once every 2.7 years.

In constructing the curve in part a, measured lowflow discharges per square mile of drainage area at 

South Downingtown were plotted against the daily 

mean flow at Chadds Ford; at high flows, discharge 

values at peak stages, obtained from a crest-stage 

recorder at South Downingtown, were plotted against 

corresponding peaks obtained from the continuous rec-
DOWNSTREAM VARIATION IN THE RIVER CHANNEL 31

ord at Cliadds Ford. This same method was used in 

calculating the recurrence interval of bankfull stage at 

Cornog, Seven Springs, Georgia, Embreeville, Wawaset, 

and Lenape.

The method is obviously not exact, because the 

assumption of similar frequencies being experienced 

upstream and downstream is not strictly true for infrequent flash flows. Too, the extrapolation required to 

obtain the discharge at the bankfull stage introduces 

the possibility of error. Nevertheless, the following 

results may be of some significance.

Figure 31 is a graph of drainage area plotted against 

the recurrence interval of the bankfull stage for nine 

locations on Brandywine Creek. The graph shows 

that at most places the annual maximum discharge 

will equal or exceed the bankfull stage on the average 

of every 1.2 to 2.7 years. The average recurrence 

interval is 2.2 years. The curve also suggests that the 

bankfull stage may be reached somewhat more often 

downstream than it is upstream.

The overbank stage was not used here because of 

the difficulty in obtaining a consistent definition of the 

overbank stage, as well as the problem of obtaining 

discharge measurements at overbank stage. A close 

examination of most of the cross sections (fig. 16) shows 

that near the overbank stage the channel begins to 

flare out and thus departs from the more rectangular 

shape prevailing below this stage. Because of this and 

because, as already shown, the increase in velocity is 

not constant, the increment of discharge within this 

flaring cross section between the bankfull stage and 

the overbank stage is difficult to calculate. This 

change in the shape of the channel may itself be related 

to the same variables, such as the load, width, and 

depth, which determine the height and structure of 

the flood plain.

Because the difference in discharge between the 

bankfull stage and the overbank stage is small, a 

tentative conclusion concerning the frequency of flooding of the flood plain may be made. The flood plain 

is attained or just over-topped by a discharge corresponding to the annual flood that is equaled or exceeded once very 2.2 years. When expressed in terms 

of a partial duration series—that is, a series of floods 

above a given base—the bankfull stage can be expected 

to recur about once every 2 years (Langbein, 1949, 

p. 880). Although there is a faint suggestion that the 

frequency with which overbank flooding takes place 

decreases upstream, the frequency with which overbank flooding takes place both upstream and downstream ranges between narrow limits. The fact that 

flooding of the flood plain occurs periodically indicates 

that the flood-plain height is closely related to the 

regime of the stream which traverses the flood plain.

200

100

50

10

Q. 

0) 

O

c.

c. o •— o

~ 0>

>> en

3 ——

—

— o

— o

—

o o —

0 ————

o o o _

o

i.u^u 10,000

Discharge, in cubic feet per second 

(Bankfull stage)

Fir.rRE 29.—Variations of width, mean depth, mean velocity, and slope of the water 

surface with discharge in the downstream direction at the bankfull stage.

If the periodicity of flooding actually proves to be 

uniform throughout a particular region, a corollary conclusion might be that a stream does not gradually 

entrench itself below its flood plain until over-bank 

flooding becomes impossible. A given flood plain will 

cease to be flooded only as a result of fundamental 

alterations in the regime of the stream such as climatic 

or tectonic changes might induce.

THE HEADWATER TRIBUTARIES

Thus far only the data from nine cross sections on 

the Brandywine Creek ranging in drainage area from
32 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

9|!uu gjonbs jgd puoogs J9d 

Ojqno Uj'pjoj sppoqop 9BjDqos|Q

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DOWNSTREAM VARIATION IN THE RIVER CHANNEL 33

**-€/>"»- 

0 L Q)

0 w

"5 « -D =

> >» o 3

5.E^

•~ o> ~o

<u R" 3

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Drainage area, in square miles

FIGURE 31.—Relation of drainage area to recurrence interval of the bankfull stage at 

the principal stations on Brandywine Creek.

25 square miles at Cornog to 287 square miles at 

Chadds Ford have been considered. Measurements of 

the discharge have been made at upstream locations on 

tributaries having drainage areas as small as 0.1 square 

mile (fig. 32) and downstream at places having drainage 

areas as much as 298 square miles (fig. 1). With the 

exception of the major cross sections and of five locations on Buck Run at which three measurements at 

different discharges were made, a single measurement 

of the discharge has been made at each of the other 

points. These measurements of the discharge, width, 

depth, and velocity are tabulated in appendixes B 

and C.

In order to compare the changes in the width, depth, 

and velocity which take place at two durations of flow 

as the discharge increases in the downstream direction, 

it was necessary to determine two discharges at each 

point: One was the flow at the section which is equaled 

or exceeded 50 percent of the time; and the other, the 

discharge at the same point which is equaled or exceeded 

2 percent of the time. These durations were arbitrarily 

selected as representating low and moderately high 

flows on Brandywine Creek. For this purpose graphs 

of discharge plotted against drainage area for the nine 

stations were made. On this graph (fig. 33) discharges 

at two separate durations of flow plot as two nearly 

parallel lines. At each cross section of the stream where 

a measurement of the discharge was made, the contributing drainage area was calculated by outlining it on 

a topographic map and measuring the area with a 

planimeter. The discharges corresponding to this 

drainage area for durations of 50 and 2 percent were 

read from the curves in figure 33. It is possible that 

for small drainage areas the extrapolation of the relationship of discharge to drainage area as a straight line 

is not accurate. Because no other method is available 

and because positive data are lacking which indicate 

that a curve rather than a straight line should be

drawn, the method may be adopted for present purposes.

The required discharges for each station having been 

determined, it was necessary to extrapolate the measured widths and depths to the specified discharges. 

The width was obtained by extrapolating from the 

measured width plotted against discharge, using a 

straight line having a slope of .04. This is the mean b 

value obtained from all of the at-a-station curves of 

width and discharge. Similarly, the depth was extrapolated from the measured point along a straight line 

whose slope was .41 or equal to the mean/ value of the 

at-a-station curves. The following example illustrates 

the method used in determining the discharges of the 

given durations and the corresponding width, depth, 

and velocity.

A measurement of the discharge was made near 

Marlboro, Pa., on Pocopson Creek, at which point the 

discharge was 1.45 cfs, the width 7 feet, and the depth 

0.35 foot. In figure 33 the width and depth are plotted 

against this discharge. At this point on Pocopson 

Creek the contributing drainage area is 4.07 square 

miles. Figure 33 indicates that for a drainage area of 

4.07 square miles the discharge equaled or exceeded 50 

percent of the time is 3.3 cfs, and the discharge equaled 

or exceeded 2 percent of the time is 18.0 cfs (see arrows). Extrapolating from the known points along the 

lines with slopes equal to &=.04 for the width and /= 

.41 for the depth, one finds that for a discharge of 3.3 

cfs the width is 7.2 feet and the depth 0.49 foot. Thus

the velocity is —-v=0.94 foot per second. Similarly, at

a discharge of 18.0 cfs the width is 7.7 feet, the depth is 

0.99 foot, and the velocity is 2.36 feet per second. These

FIGURE 32.—Typical small tributary at low flow. Illustrates smallest channels 

included in comparison of downstream changes in channel characteristics shown in 

figures 34 and 35.
34 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

.0 10 100 

Discharge, in cubic feet per second

1,000 5,000

FIGURE 33.—Graphs showing method of extrapolation used to obtain discharges equaled or exceeded 50 percent and 2 percent of the time and corresponding values of the width, 

depth, and velocity at individual measuring sections. Method illustrated by extrapolation of measurement made on Pocopson Creek below Marlboro.

are the values of the variables at this point which are 

subsequently plotted on the downstream curves in 

figures 34 and 35. Some justification for the use of 

this approximate method is provided by the consistency of the at-a-station curves and by the behavior of 

the downstream curves of width, depth, and velocity. 

Figure 34 shows width, mean depth, and mean velocity plotted against discharge in the downstream direction for a duration of flow equaled or exceeded 50 percent of the time. Figure 35 shows these same curves 

at a duration of flow equaled or exceeded 2 percent of 

the time. The points from the nine principal stations 

have been differentiated from the others in order to 

emphasize the difference between the curves based on 

main-station data alone, and those based on data from 

the widespread stations shown in figure 1. These
VARIABILITY OF CROSS SELECTIONS IN A SHORT REACH 35

5

o 

o 

o>

o>

Q.

S i

c 

>^

'6 0.5 

_o

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no

——

———————————— 4

<

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•• * IT <

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'L *

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T

•

—

T PRINCIPAL MEASURING STATION?

SINGLE MEASUREMENTS

AT SAMPLING LOCAIONS

10 100 

Discharge, in cubic feet per second 

(Equaled or exceeded 50 percent of the time)

1,000

FIGURE 34. 

30

—Kelation of width 

3765—55———4

mean depth, and mean velocity to discharge in the downstream direction from headwater tributaries to Wilmington, Del., for flow equaled or

exceeded 50 percent of the time.
36 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

5 

o

<S)

"53 

& 1

£=

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T PRINCIPAL MEASURING SECTIONS

SINGLE MEASUREMENTS 

* AT SAMPLING LOCATIONS

10 100

Discharge, in cubic feet per second 

(Equaled or exceeded 2 percent of the time)

1000

FIGURE 35.—Relation of width, mean depth, and mean velocity to discharge in the downstream direction from headwater tributaries to Wilmington, Del., for flow equaled or

exceeded 2 percent of the time.
VARIABILITY OF CROSS SELECTIONS IN A SHORT REACH 37

differences are also shown by a comparison of the b, f, 

and m values listed in table 4.

Data from the entire watershed (figs. 34 and 35) 

indicate that changes in the variables in the downstream 

direction are the same at high flows as they are at low 

flows. The figures also show, however, that the rates 

of change of the variables are somewhat different when 

smaller drainage areas are included on the graph. Thus, 

the value of 6 —.57 in figure 34 indicates that the rate 

of increase of width downstream with increasing discharge, all stations considered, is greater than is ths 

rate of increase of width between Cornog and Chadds 

Ford. An/ value of .40 shows, however, that the depth 

increases downstream over the entire drainage area at 

approximately the same rate as it does between Cornog 

and Chadds Ford. In consonance with the increased 

rate of change of width downstream, the rate at which 

the mean velocity increases is reduced. The mean 

velocity hence remains practically constant downstream.

The scatter of the points on these downstream curves 

of width, depth, and velocity will be considered later. 

Despite the extrapolations and the amount of scatter, 

particularly on the graph of velocity against discharge 

in the downstream direction, the general direction and 

relative amounts of the changes can be seen.

SUMMARY OF VARIATIONS IN THE DOWNSTREAM 

DIRECTION

At all stages of flow, the width and depth increase as 

the discharge increases in the downstream direction. 

Also, the velocity increases or remains constant. At 

all frequencies or stages of flow, the slope of the water 

surface decreases in the downstream direction.

The physical factors that control these downstream 

changes are the same as those that fix the behavior of 

the geologic and hydraulic variables at-a-station. The 

discharge and the suspended-sediment load are functions 

of the climate and of the drainage area contributing to 

the stream. There is some evidence which suggests 

that these independent factors play a part in determining the ratio of width to depth in the channel. As the 

contributing drainage area increases downstream, there 

is a change in the proportion of suspended material to 

water in the channel. The rate of increase of width in 

the downstream direction may be closely related to 

changes which take place in the concentration of the 

suspended load (fig. 27). Concomitaiitly with these 

changes in the discharge and suspended load the material on the bed of the stream is being abraded, sorted, 

and removed. At any cross section the composition, 

size, and shape of the bed material are originally independent of the channel itself. Once within the channel, 

however, the material is altered by the processes taking

place within the channel. Thus the hardness, initial 

shape, and sorting of the bed material are independent; 

yet in any given reach the actual properties of the 

material are the initial ones which have been modified 

both in place and in movement by the flowing water.

At any reach, therefore, the channel is given a specific 

volume of water, a specific suspended load, and a local 

type of rock modified by other types of rock brought 

in from upstream. The tendency on the Brandy wine 

is for the aggregate size of the bed material to decrease 

downstream. This can be seen in figure 36 011 the 

graph showing slope plotted against size of bed material. 

The data on figure 36 represent slope, bed material 

size, and other characteristics of all the principal measuring stations except Georgia, which was eliminated 

because computations of the roughness parameter were 

meaningless owing to the small values of water surface 

slope associated with the pool reach where the measurements were made.

Local influences such as the slabby material at South 

Dowiiingtown may modify this tendency for b°d material to decrease in size downstream. When the discharge, channel shape, and channel roughness are fixed, 

the depth and velocity of the water are determined. 

Assuming complete adjustment where the flow is 

steady and uniform, the slope would be adjusted in 

accordance with the given velocity, depth, and roughness. Despite the correlation between grain size and 

the slope, both of which decrease downstream (see fig. 

36), neither the mean velocity nor the computed velocity 

at the surface of the bed decreases in the downstream 

direction. Therefore, it cannot be assumed that because slope and grain size are correlative, the slope is 

simply adjusted to the velocity required to c£,rry the 

smaller particles.

It must be emphasized that the classification of a 

variable as being independent rests upon certain rational 

assumptions. The data for the at-a-station and downstream changes simply suggest a number of correlations 

among the variables. Which of them are tl a independent ones cannot be determined from the data alone.

VARIABILITY OF CROSS SECTIONS IN A SHORT REACH: 

THE POOL AND RIFFLE PROBLEM

Anyone who has looked at a natural strearr knows 

that there is rarely a reach along it which is truly 

uniform. Even where the stream looks straight, it 

almost always possesses minute distortions. The 

channel is alternately wide and narrow, straight and 

winding, deep and shallow. These deviations within 

a given reach will here be called local variations. 

While all cross sections are experiencing the same 

discharge, the areas of these cross sections are different
38 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

i. i

^*\ — ^«

_

•

^

•

t

V. —

— •

^^^« 

• \.

4

=

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—

1

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^-^

^•-•^

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———— • ————————

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->*

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•

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100 5021. 05 ID.052O.I 

by of covered Percent material-, median bedvelocity, in BedMan Velocity foot aboveO.I atRoughness 1 n

material 21 inches than greater diameter, in second bed, in feetper 

diameter median in

roughm and be, theo velocity velocity, material bed, to downstream) of near the sizmean to (which decreases slope on Relation ofsurface 36.— water IRE of exceeded percent tequaled 50 duration of flow all the atndstations, study gaging or this 2 established for stations a ofthe measuring n'. Data 6represent jarameter 

lUDaiJSUMOQ
VARIABILITY OF CROSS SELECTIONS IN A SHORT REACH 39

and hence their mean velocities are also different. 

We have seen above that the scatter of points on the 

curves of width, depth, and velocity plotted against 

discharge at Chadds Ford and at Coatesville results 

from the measurements not always having been made 

at the same cross section. The scatter is therefore 

closely related to the choice of the measuring section. 

In making comparisons of changes in the downstream 

direction it is important to be consistent in this choice. 

Theoretically, one should compare pools with pools, 

riffles with riffles, or "in-betweens" with "in-betweens." 

Thus, the problem of scatter is partly a problem of 

pools and riffles. These in turn become partly a 

problem of bends and meanders.

The following brief discussion of the importance of 

the local variations in the river channel consists of 

two parts. The first is an analysis of the magnitude 

of the local variations in width, depth, and velocity compared to the magnitude of the total change of these 

variables in the downstream direction. The second 

part concerns criteria which could be used to assure a 

consistent choice of cross sections.

AMOUNT OF LOCAL VARIATION WITHIN A GIVEN

BEACH

In addition to the measurement of the cross-sectional 

areas of certain stations where discharge was determined, the areas of the flowing water at 10 other cross 

sections near each station were also measured. These 

stations are: Cornog, Seven Springs, and South Dowiiingtown on the East Branch, Brandy wine Creek; 

Embreeville and Wawaset on the West Branch; and 

5 locations on Buck Run. At Embreeville and Wawaset the 10 cross sections were at 200-foot intervals, 5 

above and 5 below the discharge-measuring section. 

At each of the other locations the interval between cross 

sections was 100 feet. Thus, for each discharge 

measurement in a reach, there are 11 measured values 

of width, depth, and velocity.

Based on these measurements, figure 37 shows the 

effect of local variations in cross-sectional area on the 

downstream curves of width, depth, and velocity. 

It also indicates, within each reach, the range of values 

of the variables, as well as the value of each variable 

at the regular discharge-measuring section. The 

method of computation is now described.

Because the measurements in the field were made at 

different frequencies of flow, it was necessary to reduce 

both the discharge and some measure of the range of 

the variables to a constant frequency. A discharge 

having a frequency of 50 percent was chosen. The 

range delineated by the vertical lines oil figure 37 was 

computed as follows.

Measuring section no. 

I 2 345 678 9 10

100

50

.c

-C~

10

f

0.5

&'

•5;

10 50 100 

Discharge, in cubic feet per second

FIGUBE 37.—Effect of local variations in cross section on downstream changes in 

width, depth, and velocity.

(NO. OF MEASURING SECTION)

1. Buck Run at Stottsville. 2. Buck Run near Newlin. 3. Buck Run near 

Ercildoun. 4. Buck Run above Doe Run. 5. Brandywine Creek at Cornog. 6. 

Buck Run at mouth. 7. Brandywine Creek at Seven Springs. 8. Buck Run 1 

mile above mouth. 9. Brandywine Creek at South Downingtown. 10. Brandywine Creek at Wawaset. 11. Brandywine Creek at Embreeville.

A standard deviation from the mean of the 11 values 

of the width, depth, and velocity at each cross section 

was estimated. The standard deviation for each 

variable was expressed as a percent of the width, 

depth, or velocity at the cross section at which the 

discharge was made. The length of the vertical lines 

both above and below the mean is equal to this standard 

deviation expressed in percent multiplied by the width, 

depth, or velocity prevailing at the median-dp.y dis-
40 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

charge. The method of computation for the width 

alone is as follows:

sw= estimated standard deviation of the 11 measurements of width from the mean 

w= measured width

-^ (100)=p=stanclarcl deviation, in percent

w5o= width at a discharge equaled or exceeded 50 percent of the time at a regular measuring section

w5oQ>) = standard deviation of width at discharge equaled

or exceeded 50 percent of the time 

p) = plotting position of solid circles in figure 37

In using this method to reduce the discharge to a constant frequency, it must be assumed that the irregularities in the channel are the same at high and at low 

flow. Within the range of discharges involved in this 

correction, the assumption is probably true. Regardless of the merits of the assumption, however, it would 

be extremely difficult to obtain the actual measurements in the field at a time when each cross section 

was experiencing the same frequency of flow. Some 

such method of making the correction therefore is 

required.

The scatter of the points on the curves of width, 

depth, and velocity plotted against discharge in the 

downstream direction is increased by the addition of 

the local irregularities. Nevertheless, the trends of the 

variables in the downstream direction are similar to 

those shown above on figures 25, 34, and 35. The 

width increases downstream more rapidly than does 

the depth, and the velocity increases slightly or remains 

constant in the downstream direction. These data 

indicate that the apparent rates of change of the variables in the downstream direction can be influenced by 

the choice of the measuring section within each reach. 

They also indicate, however, that their effect is to 

modify but not to mask the changes observed above.

Unfortunately, because the widths, depths, and velocities in a natural channel do not have a normal, random 

distribution, no simple statistical computation procedure can be used to test the reliability of the 11 

measurements. Similarly, because there is a real 

variation in natural-channel cross sections an infinite 

number of measurements would not reduce the error 

to zero. This fact was demonstrated in part by comparing the scatter of points on downstream curves in 

which the means of the 11 measured cross sections at 

each reach were used with the scatter on similar curves 

based upon data from the single discharge-measuring 

cross sections. By this method it was possible to bring 

a few of the points into better alinement. For example, the cross sections regularly measured at Wawaset

and at South Downingtown are somewhat narrower 

than the mean for those reaches. Similarly, the discharge-measuring section at Embreeville is wider than 

the average for the reach. The improvement—that is, 

reduction in scatter—brought about by using the mean 

cross sections in the downstream curves, was not great, 

however.

CRITERIA FOR DISTINGUISHING POOLS FROM 

RIFFLES

Straub (1942, p. 616), Fargue (1908), andmany others 

have pointed out that pools and riffles in r,n alluvial 

channel are associated with the bends and crossovers 

of meanders. Bends and pools and riffles are characteristics of almost all natural channels. True meanders 

are rare on the Brandywine Creek, but where the channel is not tightly confined by sheer rock walls, truly 

straight reaches are equally rare. Detailed mapping of 

apparently straight reaches on Buck Run and on Valley 

Creek indicated that the thalweg of the fow winds 

back and forth within the channel. Because pools and 

riffles, whether at bends or elsewhere, are natural features of any channel, any comparisons which are made 

between the shapes and gradients of the channel at 

different points must take into account systematic 

variations related to pools and riffles.

Analysis of the scatter on the at-a-station curves for 

Chadds Ford and Coatesville indicates that the changes 

of the variables in the downstream direction would 

probably show fewer irregularities if a ratioral method 

of differentiating pools from riffles had been devised.

v2 

By use of a plot of -r against discharge, because pools

are distinguished by low velocities and large depths, an 

arbitrary separation of pools, riffles, and "normal" 

cross sections could be made. Such a division, however, is arbitrary because of the continuous range

v2 

of values of -r between the obvious riffles at the maxiv2 

mum values of -j and the obvious pools at the minimum

v2 

values of -r- Using only "normal" cross sections in

plotting the downstream curves of width, depth, and 

velocity, it was possible to eliminate some extremely 

aberrant points such as Georgia (fig. 25). There is a 

limit, however, to the amount of scatter which can be 

removed from the downstream curves in this way. 

Aside from the arbitrary nature of the selection, the 

method does not take into account the changes in 

width which accompany changes in velocity and depth. 

Many of the cross sections shown in figures 34 and 35 

may have relatively low depths, but because of the 

greater width of the cross section, the velocity is not
EQUILIBRIUM, GRADE, AND THE LONGITUDINAL PROFILE 41

exaggerated. The measurements made thus far show 

no correlation of width with either pools or riffles. 

Although one might expect the pools to be somewhat 

wider, the coarser material on the bed of the riffles 

may result in widening at the riffles at the expense of 

the more erodible banks. The mutual adjustment of 

the three variables is likely to make any simple differentiation of pools and riffles unsatisfactory.

Besides the effects of pools and riffles on the cross 

section, innumerable other factors produce irregularities 

in natural channels. Trees, brush, weeds, plants in general, single stray boulders or erratics, and all works of 

man modify the channel. Their effect cannot be eliminated and will undoubtedly be present in the kinds of 

downstream comparisons which are made in this study. 

In addition, at least in figures 34 and 35, the necessity 

of using extrapolated data may add to the amount of 

the scatter.

From this analysis of channel irregularities it can be 

seen that on a stream such as the Brandywine, and 

probably on most natural channels, it is impossible to 

eliminate all of the scatter of the points on the curves 

of the width, depth, and velocity plotted against discharge in the downstream direction. In homogeneous 

material where the stream is able to adjust all of the 

variables with equal ease, it is conceivable that the 

scatter would be reduced to a minimum. Nevertheless, where bends are allowed to form and pools and 

riffles exist, no random choice of sections will provide 

perfect curves of the change in width, depth, and 

velocity in the downstream direction. As the following 

pages will show, the variability of the downstream 

curves is closely related to the ideas of regime, equilibrium, and grade, as these terms are used by geologists 

and engineers.

EQUILIBRIUM, GRADE, AND THE LONGITUDINAL 

PROFILE

The analysis of the interrelationships of the width, 

depth, mean velocity, roughness, and slope, both at-astation and in the downstream direction on Brandywine Creek, indicates that within a single small drainage basin the behavior of these variables conforms to 

the generalizations derived by Leopold and Haddock 

(1953) from the study of a number of larger rivers. 

Most of those rivers, to the eye at least, are very different from Brandywine Creek; yet between those larger 

rivers and Brandywine Creek there is a striking similarity in the relationships among the variables. This 

similarity and the fact that changes of several variables 

take place harmoniously and simultaneously suggest 

that a kind of equilibrium controls the development of 

the river channel and its longitudinal profile. Consideration of the Brandywine may permit one to extend 

the understanding of equilibrium along the lines suggested by Leopold and Maddock.

The geologist's concept of a graded river also involves 

a consideration of the phenomena that tend to produce 

and maintain equilibrium. The observations on 

Brandywine Creek suggest to the author certain rela-, 

tionships between the geologist's concept of grade in 

rivers and the type of quasi-equilibrium which exists on 

the Brandywine. This discussion includes several 

parts. First, it is argued that the cross section r,nd the 

gradient of Brandywine Creek represent the spontaneous 

or nearly spontaneous adjustment of a number of variables 

to external controls; second, that these adjustments 

suggest at least a phase of an equilibrium process1 which 

is not related to the length of time during which a given 

set of conditions exists; third, there is an analogy 

between behavior of self-adjusted irrigation canals and 

Brandywine Creek which indicates that given a sufficient period during which conditions remain unchanged, 

a river channel will tend to reach a steady state; 

fourth, the characteristics of this stable state or dynamic, equilibrium can be gleaned from the definitions 

used by engineers and geologists to describe channels in 

equilibrium; fifth, the geologist's concept of grade has 

overemphasized the importance of a uniform concave 

longitudinal profile as evidence of the existence of 

channel equilibria. While the measurements on the 

Brandywine do not provide ample evidence with which 

to support each of these points, the creek does provide 

a starting point from which to consider these broader 

problems.

SIMULTANEOUS ADJUSTMENTS OF THE VARIABLES

It has been shown that on Brandywine Creek, as in 

other rivers, the suspended load is related to the channel 

shape. On the Colorado River near Grand Canyon, 

Ariz., on the Rio Grande at Bernalillo, N. Me?*., and 

on the San Juan River near Bluff, Utah, LeopoM and 

Maddock (1953, p. 30) described the sequence of changes 

in the width, depth, velocity, and suspended load which 

take place during the passage of individual floods. 

These analyses indicate a direct correlation between 

changes in the suspended load and changes in the 

width, depth, and velocity of the water in the cl annel. 

The measurements on Brandywine Creek show that during the passage of a flood the concentration of suspended 

material changes and the values of the other variables 

also change. Despite the fact that the specific rates of 

change of these variables differed from place to place, the 

relative magnitudes of the changes in the variables and 

their direction were the same at all of the stations. It is 

clear that despite these changes the channel shape is
42 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

much the same before and after the flood. In Brandywine Creek the changes in width, depth, and velocity 

accompanying changes in discharge are not associated 

with scour and fill of the channel bed to nearly the same 

extent as in the large rivers in the West referred to above.

Because of the relatively small amout of scour and 

fill observed in Braiidywine Creek the observed changes 

in suspended load in a given reach must be attributed 

primarily to changes in the suspended load supplied 

to the reach from upstream. The load, therefore, 

constitutes as external control. As the observed changes 

in width, depth, and velocity with discharge are similar 

to the changes observed in the large rivers, they also 

appear to represent on the Brandywine rapid simultaneous adjustments to changes in external controls.

Several tentative conclusions are suggested by these 

observations. First, factors which control the shape 

and profile of the channel are the same everywhere 

along the channel. Second, the data on Brandywine 

Creek and analogies drawn from data on some other 

rivers indicate that the channel and the flood plain 

associated with it are the result of the simultaneous 

interaction of a number of variables. Third, presumably this quasi-equilibrium prevailed during the 

development of the channel and the flood plain (the 

walls of the channel are the flood plain). The changes 

in the variables which occur at-a-station with increasing 

discharge in the present channel presumably continue 

to reproduce the pattern of changes which led to the 

development of the channel. This adjustability, which 

has been demonstrated both in the large rivers and in 

Brandywine Creek, is one aspect but not the only aspect 

of stream equilibrium.

COMPARISON OF BRANDYWINE CREEK WITH STABLE 

IRRIGATION CHANNELS

In addition to the orderly changes of the variables 

which take place with changing discharge at a given 

cross section, it has been shown that these same 

variables change in an orderly progression downstream 

on the Brandywine Creek. These changes in turn must 

be related to the succession of changes which take 

place in the discharge, sediment load, and material 

on the bed. Despite the scatter of points, the consistency of these downstream changes as illustrated in 

figures 25, 34 and 35 is such as to suggest that a quasiequilibrium prevails in the downstream direction. This 

is to be expected because the discharge and sediment 

load, as well as the volume, density, and size of the 

bed material at any reach, are intimately related to the 

reaches above and below it. Regardless of local

50 100 

Width, in feet

500 1000

FIGURE 38.—Relations of width to depth in canals and rivers in alluvial material 

compared to relation of width to depth in Brandywine Creek.

influences, certain variables, such as the amount of 

water and sediment, maintain their continuity downstream. As Gilbert (1877) termed it, "There is interdependence throughout the system." As the following 

comparison of Brandywine Creek with unlined irrigation 

canals will show, if the controlling variables remain 

constant over a long period of time stability might be 

attained throughout the entire river system.

The similarity between these downstream changes in 

a natural river channel and those which take place in 

stable irrigation canals has already been described in 

detail (Leopold and Maddock, 1953, p. 43). For comparison with the Brandywine data, table 5 shows several formulas derived primarily from canals and rivers 

in India. Additional comparative data are provided 

by figure 38 which shows the relationship between 

width and depth in a number of different rivers and 

canals (Lane, 1937, p. 130). The curve shown for 

Brandywine Creek in figure 38 was computed using 

values of 6 = .57 and /=.40. These are the rates of 

change of the width and depth with disci-arge, respectively, and are derived from the curves shown in figure 

34 in which all sections were used in making the down-
EQUILIBRIUM, GRADE, AND THE LONGITUDINAL PROFILE 43

stream comparisons. The slopes of the lines in figure 

38 were derived from the equations as follows:

w=aQt (3)

d=cQf (4) 

Then

and

c ~ d=—f- wb

Substituting the values from figure 34, (/=.089w7 .

Figure 38 includes three separate curves derived by 

Lacey for canals in India in which the bed material is 

of three different sizes. These lines are nearly parallel 

to the width-depth curve for Brandywine Creek. A 

comparison of these curves with the one shown for the 

Brandywine suggests that the coarser material on the 

Brandywine is responsible for its lower intercept at a 

given width. Additional data, however, do not support 

this conclusion. The curves of width plotted against 

depth for the five rivers shown in figure 38 were derived from the data presented by Leopold and Maddock 

(1953, p. 15). The material on the bed of the KansasRepublican River is of the order of 0.00025 inch. On 

the Tombigbee River the grain size is also small, while 

on the Yellowstone-Bighorn-Wind River system Jjie 

material ranges from gravel in the upstream areas to 

silt downstream. As figure 38 shows, the width-depth 

curves of the Kansas-Republican and the Tombigbee 

Rivers are nearly parallel to the curve of Brandywine 

Creek. The intercept of Brandywine Creek falls between the two. As the grain size on the KansasRepublican River is 1,000-fold smaller than on Brandywine Creek, it might be expected that its intercept 

would be higher. Instead, it has the lowest intercept. 

This specific illustration, combined with the random 

distribution of all of the intercepts shown in figure 29, 

demonstrates that grain size alone does not fix the 

position of the curve. Presumably the rate of change 

of roughness in the downstream direction, the relative 

erodibility of the bed banks, and the nature and amount 

of the suspended load, also help to fix the position of 

the width-depth curve.

The rivers and canals compared in the figure have 

vastly different topographic, lithologic, and climatic 

environments. The striking feature of figure 38 then 

is the close agreement shown between the behavior of 

the width and depth in rivers and in canals. This 

agreement indicates that the processes which determine the shapes of natural alluvial channels are much

the same regardless of their geographic location. These 

same relationships are illustrated by the formulas in 

table 5.

It is interesting to note (table 5) the similarity of 

the rate of change of width with discharge in the 

downstream direction. In the Indian examples 6 = .5, 

while Schoklitsch (1937, p. 151) gives a value of .6 

for natural rivers. These values are very close to the 

value & = .57 found applicable to all stations on the 

Brandywine. Although the hydraulic radius increases 

to the one-third power of the discharge alone—that is, 

y=.33? in the Punjab Research Institute equation, the 

Lacey equations involving the velocity, depth, and 

slope include in addition to the discharge a silt load 

factor or a grain size factor. Nevertheless, the cl aiiges 

of these variables with discharge are in general of the 

same magnitude as those experienced on the Brandywine. The difference in the actual values may be 

related to the concentration of the suspended loe.cl and 

to the relative erodibility of the bed and bank, factors 

which have not been sufficiently studied on the Brandywine to allow detailed comparisons to be made with 

the Indian data.

Although the data for Brandywine Creek indicate 

that the channel of the creek behaves very much like a 

"stable" irrigation canal, the amount of scatter of 

points on the curves is greater for the Brandywine. 

The multiple causes of this scatter have already been 

discussed. It is not due solely to departure from 

so-called equilibrium, although the deviation of some 

few points may be attributed to this cause.

Unlike the Brandywine, the canals are all compared 

at a single discharge, generally a so-called "dominant 

discharge" (Blench, 1951a, 6.9). Thus a family of widthdischarge curves represents (ibid., 5.30) the relation of 

width to discharge at one frequency for channels f owing 

in a number of different materials. A similar family of 

curves, as in figures 25, 34, and 35 for the Brandywine 

Creek, represent different frequencies of flow in the 

same materials. The concept of "dominant discharge" 

may be applicable to natural streams, such as the 

Brandywine. (See Mackin, 1948, p. 477, for geologic 

observations on the relative significance of various 

stages of flow.) The adjustments which have been 

observed at all stages of flow, however, indicate that 

the quasi-equilibrium prevails at more than one 

frequency of flow. A complete comparison of the 

canals with the Brandywine from the standpoint of 

scatter would require a family of curves to be drawn 

for each frequency of flow and for each type of rock 

through which the Brandywine flows. This is an 

impossible task. Unless it were possible first to eliminate the predictable or regular causes of the scatter 

on the downstream curves, such as the pools and
4 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

TABLE 5.—Formulas for stable channels in alluvial material for comparison with similar formulas for Brandywine Creek, Pennsylvania

Author Location or other data Formula 1

Lindley (1919)__ ___________________ India, canals________ ________ P=1.9840-506

Lacey (1939) _______________________ India, Punjab______________ V= 1.1 38R -4995

Do__________________________ India, Madras_______________ r=0.790.R-508

Do____________________________ India, Rivers_____-_-_--___-_ V

Do_____.______________________ India, Punjab___-_____-_____- P

Do__________________________ India, Jamrao__. _____________ P=2.50(?*

Do____________________________ Rivers and torrents, India_____ P=2.63#*

Do______._____________________ India, canals ________________ T r=1.155/*/2*

Remarks

Finer silt than above item.

Only five points on curve.

Do.

Punjab Research Institute (see Lacey, 

1939, p. 24)

r

____do_____________ _________ S-0.000542^Q*

India, ______________________ R=QA7QQ*

m -8

____do._____________________ 5=0.00209 "^

Woods, F. W. (see Lane, 1937, p. 127) _ _ 5 —~ d

Molesworth and Yenidunia (see Lane, ____do_-__-_____________-_-. d =(90605 + .0725)5^

1937, p. 127) 

Schoklitsch (1937)_______-___--_____ Rivers__._________________ W=kQ-»

Wolman____________________ _______ Brandywine Creek___.________ W=aQ-57

Do__________________________ ____do_ _____________________ W=aQM

Do_____ _______________________ ____do__._________________ d=.089Tr-7

Do______-__________--_________ ____do_______________ V=/td-»

1 The following use of symbols applies in this report to this table only:

a = a coefficient.

.6=bed width, in feet.

d=mean depth for wide channels, in feet.

/=a silt factor.

fc = a coefficient.

m = diameter of silt particle, in millimeters.

Tributaries included. 

Average betveen principal

stations.

Tributaries included. 

Do.

riffles, the amount of scatter would not in itself be a 

significant measure of departure from equilibrium. 

The significant observation at this point is the close 

parallelism between data on Brandywine Creek and 

the canals.

CHARACTERISTICS OF STABLE CHANNELS

These, stable channels in alluvial material (Lane, 

1937), or regime channels as they have often been 

called, have been studied in considerable detail. Some 

understanding of the kind of equilibrium which prevails 

in these channels, and by analogy the kind of equilibrium 

which is suggested by the downstream curves on the 

Brandywine Creek can be gained by reviewing a number of definitions of regime and its synonyms.

According to Lacey (1939, p. 1),

Regime flow in silt transporting channels excavated in alluvium connotes physical stability, a balance between silting and 

scouring, and a dynamic equilibrium in the forces generating 

and maintaining the channel cross-section and gradient.

He adds (p. 4),

It is evident that Lindley's concept of regime which postulates 

a delicate and precise dynamic balance in all the dimensions of 

an open alluvial channel can very seldom be perfectly achieved. 

Effectively there are too many variables, and in practice not all 

are equally free to vary.

P=wetted perimeter in feet.

Q= discharge, in cubic feet per second.

R = hydraulic radius, in feet; approximately equal to d.

S=slope, in feet per foot.

y=mean velocity, in feet per second.

TF= width, in feet; approximately equal to P and B.

For regime to be established the fundamental requirements 

are that the discharge should be constant, the channel flowing 

uniformly in unlimited incoherent alluvium of the same character 

as that transferred, and the silt grade and silt charge a constant.

To rivers and torrents Lacey (p. 12) appUed the term 

"quasi-regime."

Blench (1951a, 4.1) states that,

A channel is said to be in regime when it is capable of adjusting 

its cross-sectional form and/or longitudinal slope by means of 

alterations that the flow can impose on the solid, and exhibits 

an average equilibrium between the actions and counteractions 

there.

He states elsewhere (1951b, p. 2) that,

* * * [if] a "specific gage" record shows a trend in time the 

river is not in regime but [its] behavior is consistent with classification as a "regime type." If the record shows no trend, the 

river is practically certain to be in regime during the period for 

all factors provided it is of regime type * * *. Regime implies 

a time scale.

Several other words have also been used to convey 

much the same meaning as regime. Cdom (1950, 

p. 4) points out that,

When the determining factors (slope, load, discharge, character 

of materials on the bed) have remained within reasonable limits 

for a considerable period, over-all cutting or filling of the stream 

bed ceases and the stream is said to be "poised."
EQUILIBIRTJM, GRADE, AND THE LONGITUDINAL PROFILE 45

Bryan (1922, p. 89) defined the "regimen of a stream"

as,

The system or order characteristic of a stream—in other words, 

its habits with respect to velocity and volume, form of and 

changes in channel capacity to transport sediments, amount 

of material supplied for transportation, etc. The term is also 

applied to a stream which has reached an equilibrium between 

•corrasion and deposition or, in other words, to a graded stream.

Kesseli (1941, p. 568) however, quotes Surell to 

the effect that "regimen" is a final period in the development of a river "during which the water overflows and 

returns into an invariable bed."

Mackin (1948, p. 484) uses a number of apparently 

synonymous terms in discussing the engineering literature concerned with transportation by running water. 

Thus he states that the "* * * concept of the 'adjusted' 

or 'stable/ or 'regime' (in the sense of equilibrium) 

cross-sectional form" is emphasized in that literature. 

Very much earlier, however, Davis (1902, 1909 ed. 

p. 391) suggested that, "* * * 'regimen' may better 

be used as meaning the rule of river action under 

which the balanced condition is developed and maintained * * * ," rather than as the condition obtained.

Webster's dictionary (1941, p. 2,097) indicates that 

there is a distinction between regime and regimen in 

rivers as some of these definitions suggest. Presumably 

regimen originally implied stability or regularity and 

related to a particular state of a river system, whereas 

regime meant simply the prevailing mode or condition 

of a river. Bryan's dual use of the term, however, 

seems to weaken this distinction. In the engineering 

use of regime to describe a particular type of canal or 

river, regime is virtually synonymous with regimen. In 

the present paper, where regime is used it is intended to 

apply to a particular state or condition of the river and 

hence is used synonymously with most of the definitions 

of regimen given here. The comparisons to be made 

with canal data unfortunately require this usage.

Despite the wealth of terms, these collected definitions have several fundamental points in common. 

First, regimen, regime, equilibrium, and poised, as used 

in these definitions, all imply a certain amount of 

stability in time. Second, they imply flexibility in the 

adjustments made by the channel itself. Third, there 

are no simple fixed criteria by which we can tell whether 

a given stream is or is not "in regime," or is "poised," 

or "exhibits an average equilibrium." Fourth, the 

definitions do not imply that the adjustments of slope 

are more important than the adjustments in the cross 

section, or vice versa. It is also important to note that 

the "ideal" or perfect equilibrium is seldom attained 

because of many irregularities present in natural 

materials.

THE BBANDYWINE: A CHANNEL IN EQUILIBRIUM

On Brandywiiie Creek it has been seen that rapid 

adjustments take place in the geologic and hydraulic 

variables within a given cross section. The various 

channel shapes and the similarity in the behavior of the 

variables in the channels suggest that the channel has 

been free to adjust itself. These adjustments have 

apparently been made in both the form of the cross 

section and its longitudinal profile. The data in 

figures 25, 34, 35, and 38 and in table 5 indicate thatthrough these adjustments the Brandywine has developed a channel which is very similar to the chmnels 

found in regime canals. This particular channel shape 

and profile apparently represent a stable configuration. 

Unless the measurements on the Brandywine were repeated over a period of years, one could not say with 

assurance that the channel is truly stable. By analogy 

with the canals, however, and bearing in mind that 

there are no simple fixed criteria for determining 

whether a given stream is "in equilibrium," it is suggested that Brandywine Creek falls within the broad 

definitions of regime or quasi-equilibrium.

EQUILIBRIUM AND THE LONGITUDINAL PROFILE

The word "equilibrium" has also been much used by 

the geologists to describe a particular condition attained 

by natural rivers during the "normal cycle of erosion." 

In contrast to the emphasis given to adjustments in 

channel shape by the engineers, the geologic literature 

has emphasized the "equilibrium" slope almost to the 

exclusion of the cross-sectional form. Mackin (1948), 

Rubey (1952) and Leopold and Maddock (1953) have 

all emphasized the need for consideration of both cross 

section and profile in defining equilibrium. Brmdywine Creek fits several definitions of equilibrium or 

quasi-equilibrium. In the minds of many geologists a 

condition of equilibrium is associated with a smooth 

concave-upward longitudinal profile. The conditions 

of quasi-equilibrium on the Brandywine, however, are 

not accompanied by a regular longitudinal profile 

(fig. 39). Because the geological emphasis on the 

profile stems from the introduction of the word "grade," 

it is of some interest at the outset to trace the history 

and meaning of this term.

Gilbert (1877, p. 114) in his original discussion of 

the mutual adjustment of velocity, discharge, dedivity, 

and load in natural river channels referred to the attainment of an "equilibrium of action." Similarly, Davis 

(1902, p. 389) spoke of the "balanced condition." To 

express the equilibrium or balanced conditions Davis 

(p. 395) adopted the word "grade." He pointed out 

that,

River action in such a graded reach may justly be said to be
46 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

o

E H

- -= I
EQUILIBRIUM, GRADE, AND THE LONGITUDINAL PROFILE 47

organized, inasmuch as a change in form or action at any one 

point involves a change at every other point.

Davis's words such as "balanced," "adjusted," and 

"organized" permitted a wide latitude of interpretation. From his own and other geologists' writings, 

however, it was clear that "grade" was very nearly, 

if not exactly, synonymous with "graded slope" (p. 

399). Davis also spoke of "graded reaches" and of 

"graded river systems." Presumably, a graded river 

is a graded river system; that is, one which is graded 

from headwaters to mouth.

The original association of "equilibrium," "grade," 

and "graded slope," has led to the emphasis which has 

often been given to the correlation of a longitudinal 

profile that is concave upward with the idea of a river 

at grade (Cotton, 1941, p. 73; von Engeln, 1942, p. 

133). Both Baulig (1925) and Mackin (1948) have 

demonstrated, however, that the term should be applied not to a stream with a particular profile but to 

a stream in equilibrium. Baulig referred to the "profile of equilibrium" and pointed out that the profile 

of equilibrium may well depart from a regular parabolic curve. Mackin indicated that grade was a condition of equilibrium maintained primarily by adjustments in slope but added that these adjustments 

did not necessarily produce a uniform concave profile.

Although Rubey (1933, 1952) stressed the importance 

of considering both the cross-sectional form and the 

slope of the channel, the geologic literature has continued to emphasize the profile. Mackin (1948, p. 487), 

after considering the importance of changes in channel 

section, suggested that,

* * * while adjustments in section may or may not accommodate the effect of a change in control, slope is always modified, 

by the stream itself, in such a manner as to absorb the effect of 

the stress * * *. In the case of grade it means simply that the 

stream normally reacts to a change in controls calling for an 

increase or decrease in energy required for transportation by 

increasing or decreasing the total energy through modification 

of slope rather than by effecting economies in the energy dissipated in friction.

There is another reason besides Mackin's rational one 

for emphasizing slope in the maintenance of equilibrium 

in rivers. In the comparison of past streams, inferred 

from the reconstruction of the profiles of rivers from 

terrace remnants, with modern streams, and in the 

interpretation of geologic history from such comparisons, the longitudinal profile is the only tool available 

to the geologist.

The downstream curves on Brandywine Creek and 

the canal and river data shown in figure 38 suggest 

that the adjustment of the channel shape may be as 

significant as the adjustment of the longitudinal profile. 

There is no way in which one could predict that the

effect of a change in the independent controls would 

be better absorbed by a change in slope rather than by 

a change in the form of the cross section. This observation does not alter Mackin's (1948, p. 463) point that 

"grade is a condition of equilibrium in streams." It 

suggests, however, that slope is not necessarily the 

primary mechanism by which such an equilibrium is 

maintained. Leopold and Maddock (1953, p. 39) have 

also shown that a change in the governing factors may 

result in changes in channel characteristics without a 

change in river slope. Similarly, Odom (1950) points 

out that, because of the complexity of the relationship 

among discharge, bedload, and suspended load, the 

reduction in discharge of the Mississippi River below 

the Atchafalaya cutoff has not resulted in changir^ the 

slope of the Mississippi River below the cutoff. It is 

not yet possible to say with any degree of confidence 

what proportion of the adjustment to a change in 

external control will be taken up by slope as compared 

with cross-sectional form. Undoubtedly the characteristics of the country rock through which the stream 

is flowing have an effect upon the relative ease with 

which such adjustments can be made.

Prom these considerations one must conclude as 

Mackin did (1948, p. 507) that if we are going tc continue on the present path, the meaning of grade should 

be synonvmous with equilibrium. If grade is to be 

used synonymously with equilibrium, the emphasis on 

the longitudinal profile results in confusion because the 

existence of an equilibrium is not solely dependent 

upon slope. Further, the definition of grade whether 

it is applied to reaches or to river systems, should 

show no preference for adjustments in slope to adjustments in cross-sectional form.

As figure 39 shows, the profile of Brandywine Creek 

and several of its tributaries is broadly concave upward. 

Actually the overall profile consists of segments which 

are concave to the sky and separated by knickpoints. 

Analysis of the changes of the variables in the downstream direction suggests that at those cross sections 

in which measurements have been made the channel is 

in a state of equilibrium or quasi-equilibrium. Thus, 

by definition, throughout those sections the strearr is at 

grade if grade is used synonymously with quasiequilibrium. Figures 34 and 35 (see fig. 1 for location of stations) show then that a large proportion 

of the mainstream and its tributaries is at grade. Few 

measurements have been made at the knickpoint?, but 

one measurement at Rocks, a bedrock outcrop on 

Broad Run is significant. At this point (app. C; figs. 

34 and 39) ^=0.9 cfs and w=20 feet. A low flood 

plain at the measuring section composed of fine material which the stream is obviously capable of moving
48 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

indicates that the channel is adjustable. In addition, 

the velocity is consistent with that at points above and 

below Rocks on Broad Run. The velocity is not 

accelerating through the knickpoint reach. A consistent velocity is maintained through this reach of 

steeper slope by an increase in width and concurrent 

decrease in depth. Thus, the adjustability of the 

section and the continuity of the velocity indicate that 

the presence of bedrock at this point does not preclude 

the possibility of the stream being in quasi-equilibrium 

and, therefore, at grade.

The same case can be made for the knickpoint which 

occurs above Pomeroy on Buck Run. Here the stream 

passes through resistant sandstones where the channel 

is somewhat wider and the velocity somewhat greater 

than upstream or downstream. The reach is not a 

waterfall and possesses a flood plain throughout much 

of its length. Again, the adjustability of the cross 

section is evidence that a quasi-equilibrium exists at 

this section. Elsewhere in the basin, because of the 

presence of manmade dams, it has been impossible to 

study in detail the sections at which the knickpoints 

appear on the longitudinal profile. At Wagontown on 

West Branch Brandywine Creek, however, the knickpoint appears to be due to the presence of resistant 

amphibolite. Several miles below this point, about 1 

mile above the city'of Coatesville, Pa., resistant sandstones occur along the banks of the Brandywine and 

in its bed. There is no evidence to suggest that at 

either of these cross sections the channel cross section 

is adjustable.

The final form of the graded longitudinal profile may 

indeed be a smooth parabolic curve such as the curve 

derived by Unwin (1892, p. 503). Unwin showed that 

if the discharge increased downstream, and the width-todepth ratio and the mean velocity remained constant, 

the longitudinal profile would be a smooth parabolic 

curve. Leopold and Madclock (1953, p. 48) also 

showed that the concavity of the smooth hypothetical 

profile was closely related to the rate of change of roughness in the downstream direction. Because of the fact 

that contributions of discharge and load to a natural 

river by its tributaries may be highly erratic, an absolutely smooth curve is not likely to conform precisely 

to the actual longitudinal profile of a natural river 

(Davis, 1902, p. 400).

Many studies have demonstrated the close relationship which exists between the slope of the profile of a 

river and the size of the material in the bed (Slmlits, 

1941; Pettijohn, 1949, references in chapter on transportation). As Mackin (1949, p. 483) pointed out, 

however, the causal relationship often expressed 

between large grain size and steep gradient is undoubtedly an over simplification.

On Brandywine Creek there is a direct relationship 

between grain size and slope (fig. 36). It is clear from 

the figure, however, that specific velocities "required"" 

for the transport of materials of different sizes are not 

responsible for the correlation of slope and grain size. 

The mean velocity remains constant despite changes hi 

slope and grain size. Not only does the mean velocity 

remain constant, but the computed mean bed velocity 

(O'Brien, 1937; Leopold, 1953) increases downstream. 

Neither mean velocity nor mean bed verity may bethe factor effective in moving materials. As the shear 

decreases in the downstream direction, it is possible 

that this factor may be more closely related to the 

decrease in caliber of material and decline in slope in 

the downstream direction than is the velocity. The 

general term "velocity," in any event, dees not appear 

to be an appropriate parameter with which to explainthe change which takes place in a river channel in the 

downstream direction.

With the exception of a few reaches in which the 

channel is flowing wholly within bedrock, Brandywine 

Creek is a graded stream. That is, it is a stream which 

has attained and maintains an equilibrium or quasiequilibrium through the mutual adjustment of the discharge, load, bed material, slope, width, depth, and 

velocity. The adjustment may actually involve combinations of these variables or adjustments of other 

variables of a similar kind. In addition, the mutual 

adjustments of the variables take place whether the 

stream is at grade, aggrading, or degrading. Hence 

the processes of aggradation and degradation are not 

recognizable at any given time by maladjustments in 

the variables. Save over a long period of time during 

which finite changes may be detected in the shape or 

profile of a given reach, the differences between the 

river at grade, degrading, and aggrading may be indistinguishable. This dilemma forces consideration of 

another aspect of grade and equilibrium.

Strictly speaking, equilibrium is a stable condition. 

Running water carrying material in suspension and on 

the bed of a channel can erode the bed and by removing 

material continue to help erode its basin. Thus, in 

keeping with Davis's original conception, and Mackin's 

observation (1948, p. 500), the attainment of grade 

does not preclude denudation of the land. Mackin 

(1948) has emphasized the perfection of the dynamic 

equilibrium in a truly graded reach of a, channel. He 

distinguished between a graded reach and one which is 

either slowly aggrading or degrading. This distinction 

emphasizes the aspect of stability of grade or equilibrium. Because true stability is difficult to reconcile 

with continued degradation of the land, Mackin uses 

"over a period of years" to define the length of time 

implied in the word stable. On the other hand, as the
SUMMAKY AND CONCLUSIONS 49

time span is reduced, attention is focused upon the 

"equilibrium" processes of adjustment within the 

channel and less upon the stability of a particular shape 

and profile. The processes of adjustment are the 

same whether the stream is at grade, degrading, or 

aggrading. During each period, in accordance with 

the requirements of an equilibrium process, the variables 

may change so as to counteract the effect of a change in 

the external control. The concept of equilibrium 

implies both stability and the ability to make adjustments.

SUMMARY AND CONCLUSIONS

From this study of Brandywiiie Creek in Pennsylvania and from the previous discussion of the concept of 

grade, the following general conclusions may be drawn:

1. So far as it has been possible to measure them, the 

behavior of the variables both at-a-station and in the 

downstream direction on Brandywiiie Creek conforms 

to the generalizations derived by Leopold and Maddock 

(1953) from a larger number of rivers in the western 

United States.

2. In many respects the channel of Brandywiiie Creek 

also bears a remarkable resemblance to regime canals.

3. When casually observed in the field, the channel of 

Brandywine Creek is a picture of disorder. To the 

contrary, however, measurements of the discharge, 

width, depth, velocity, slope, and suspended-sediment 

load, and the computation of a roughness factor indicate 

that the cross-sectional form and the gradient of the 

channel actually approach a dynamic equilibrium.

4. The actual physical mechanisms by which adjustments in the variables are made are not known. The 

gradient and shape of the channel at any given reach 

are assumed to be primarily a function of the discharge 

and load supplied to the reach, and of the material 

supplied to the bed of the stream both within the reach 

and from upstream.

5. Throughout most of its length Brandywine Creek 

is a stream at grade, if grade is used synonymously 

with equilibrium. Like any other natural channel, 

however, neither the adjustability nor the stability of 

any parts of the channel conforms to the rigorous requirements of a stable equilibrium. Hence, quasi- or 

semi-equilibrium more nearly describes these conditions.

6. If grade is used synonymously with quasi-equilibrium, the data 011 Brandywine Creek confirm the 

observations of Baulig (1925), Mackiii (1948), and 

others that neither a reach at grade nor a graded river 

system necessarily has a uniform longitudinal profile 

which is concave to the sky.

7. A decrease in slope in the downstream direction is 

associated with a decrease in size of the bed material. 

No specific functional relationship has been found to

explain this association. The problem of cans? and 

effect is complicated by the fact that the frictioiial 

resistance of the bed material is not a function of grain 

size alone, but is also related to the shape, distribution, 

form, and packing of the particles 011 the bed.

8. The problem of grade, or equilibrium, in natural 

river channels is complicated by the number of variables involved and by the fact that the reactions are 

simultaneous. Quantitative, as well as qualitative, 

reasoning forces one to reject the over-simplified concepts often used to explain the morphology of rivers. 

Unhappily at the present time, an exposition such as 

this illustrates the complexity of the processes without 

offering a comparable or more nearly complete explanation. Because of the complex interrelationships of the 

variables, however, no explanation is likely to be complete that rests solely upon an analysis of a simple 

sequence of events.

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50 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

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APPENDIX

APPENDIX A.—Measurements of discharge and hydraulic variables at the seven, principal stations established for this study on Brandywine

Creek

Date Discharge 

Q (cfs)

Width 

»' (feet)

Mean

depth d 

(feet)

Mean 

velocity 

v (fps)"

Area of

cross section A 

(square

feet)

Slope of 

water 

surface s

Roughness 

parameter n' Remarks

Cornog

Drainage area: 25.7 square miles. Length of reach over which slope was measured: 645 feet.

Aug. 1, 1951_____

Aug. 6, 1951

Aug. 11, 1951____

Sept, 7, 1951____

Sept, 10, 195l____

Apr. 27, 1952____

Apr. 28, 1952, __

Mav 27, 1952____

July 8, 1952

July 9, 1952, _____

Julv 11, 1952____

10. 3

8. 6

12. 9

14. 7

5. 7

304

830

73. 4

19. 4

322. 4

35. 0

23. 5

41. 0

41. 0

41. 5

39. 5

49. 6

55. 8

44. 0

43.0

47. 7

42. 5

0. 82

. 41

.48

. 52

. 34

1. 56

2. 44

. 90

. 57

2. 09

. 79

0. 53

Kfl 

. O\J

. 64

. 69

. 44

3. 92

6. 10

1. 86

.79

3. 22

1. 04

19. 3

17! o

19. 6

21. 4

13. 6

1 77. 6

1 136. 0

39. 5

24. 4

100. 0

35. 0

0. 0033

. 0033

. 0033

. 0033

. 0033

. 0034

.0037

. 0035

. 0034

. 0033

. 0034

0. 140

. 094

. 082

. 080

. 094

. 030

. 027

. 044

.075

. 043

.071

Not at regular cross section.

Velocity measured by float

and corrected to mean

velocity. Do."

Seven Springs

Drainage area: 54.2 square miles. Length of reach over which slope was measured: 328 feet.

July 27, 1951 ____ 

Aug. 3, 1951_____

Aug. 20, 1951____ 

Sept. 10, 1951___ 

Apr. 27, 1952..__

Apr. 28, 1952__._ 

June 2, 1952__.__

July 9, 1952_____

July 11, 1952___ 

Aug. 7, 1952____

20. 7 

20. 8

53. 8 

13. 8 

472

930 

212. 7

411. 0

152. 6 

100. 8

57. 0 

58. 0

58. 0 

56. 5 

66. 6

70. 2 

62. 8

67. 6

62.0 

58. 5

0. 64 

. 64

. 92 

. 53 

2. 47

3. 73 

1. 62

2. 60

1.41 

1. 18

0. 57 

. 56

1. 01 

. 46 

2. 87

3. 56 

2. 09

2. 33

1.75 

1. 46

36.4 

37. 0

53. 4 

30. 1 

164. 8

261. 5 

101. 9

176. 2

87.2 

69. 2

0. 0025 

. 0026

. 0024 

. 0025 

. 0019

. 0028 

. 0027

. 0029

. 0026 

. 0025

0. 091 

. 093

. 067 

. 105 

. 041

. 053 

.051

. 065

. 055 

. 057

Velocity measured by float 

and corrected to mean 

velocity. 

Do. *

South Downingtown

Drainage area: 86.0 square miles. Length of reach over which slope was measured: 2,026 feet.

July 27, 1951____ 

Aug. 3, 1951.____

Aug. 13, 1951____ 

Aug. 20, 1951 _ 

Sept, 8, 1951____ 

Apr. 27, 1952____

Apr. 28, 1952____ 

July 7, 1952_____

July 8, 1952_____

Julv 9, 1952_____

July 10, 1952____

36. 1 

33. 7

52. 0 

154. 4 

29. o 

550

1, 280 

69. 5

105. 5

809. 0

978. 0

53. 0 

50. 0

51. 0 

53. 0 

50. 5 

54. 5

61. 5 

50. 0

56. 7

57. 2

1. 10 

1. 07

1. 20 

1. 78 

1. 05

2. 85

4. 39 

1. 30

1.47

3. 06

3. 64

0. 62 

. 63

. 85 

1. 63 

. 56 

3. 55

4. 74 

1. 07

1. 27

4. 67

4. 69

58. 2 

53. 6

61. 3 

94. 6 

53. 2 

1 155. 0

1 270. 0 

64. 8

83. 3

173. 5

208. 2

0. 00073 

. 00086

. 00086 

. 00086 

. 00086 

. 00071

. 00066

. 00083

. 00074

. 00071

0. 067 

. 072

. 058 

. 039 

. 080 

. 022

. 022

. 044

. 018

. 020

Velocity measured by float 

corrected to mean velocity. 

Do.

51
52 THE NATUBAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

APPENDIX A.—Measurements of discharge and hydraulic variables at the seven principal stations established for this study on Brandyurine

Creek—Continued

Date Discharge 

Q (cfs)

Width

iv (feet)

Mean 

depth d

(feet)

Mean 

velocity 

v (fps)'

Area of

cross section A 

(square

feet)

Slope of 

water 

surface s

Roughness 

parameter n' Remarks

Georgia

Drainage area: 118 square miles. Length of reach over which slope was measured: 489 feet.

July 24, 1951____

July 27, 1951__^_

July 31, 1951__^_

Aug. 3, 195U.__^

Aug. 20, 1951_.__

Apr. 27, 1952____

Apr. 28, 1952. ___

July 9, 1952____.

Aug. 8, 1952. ____

67. 3

57. 9

61. 8

54. 4

182.2

877

2,430

670. 0

924. 0

88. 5

89. 0

89. 0

89. 0

91.0

93. 0

95.0

92.0

Q4 7

1. 80

1. 79

1.78

1. 69

2. 26

3. 82

5. 74

3. 43

3 QQ

0.42

. 36

.39

. 36

0.89

2. 47

4.46

2. 13

2. 45

159.6

159. 5

158. 7

150. 8

205.5

1 355. 0

1 545. 0

315. 1

376.9

0. 00007

0. 0006

.0008

. 0005

.0005

0. 035

.025

Slope too low to measure

with gage? used. 

Do.

Do.

Do.

Velocity measured by float

and corrected to mean 

velocity. 

Do.

Embreeville

Drainage area: 117 square miles. Length of reach over which slope was measured: 850 feet.

July 30, 1951____ 

Aug. 3, 1951_____

Aug. 14, 1951_ __ 

Sept. 7, 1951___.. 

Sept. 10, 1951___ 

Sept, 15, 1951__. 

Apr. 28, 1952- ___

Mav 26, 1952____ 

June 6, 1952__

Aug. 7, 1952 ____ 

Sept. 19, 1952... 

Dec. 16, 1952_.__

73. 8 

57.0

81. 6 

79. 7 

51. 9 

160. 5 

2,340

1,075 

319. 2

259. 8 

540.0 

191.7

76. 5 

76. 0

77. 5 

78.0 

78.0 

79.0 

88. 0

84.0 

81.0

81.0 

81. 0 

80.0

1. 15 

1. 11

1. 21 

1. 18 

1.05 

1. 49 

5. 26

3. 64 

1.98

1.87 

2.31 

1. 62

0.84 

6Q

. 87 

. 86 

.63 

1. 36 

5.05

3.98 

2.00

1.71 

2. 91 

1. 47

88.0 

84.0

93.5 

92. 1 

82. 2 

118. 4 

463. 1

305. 7 

160. 4

151.9 

186. 8 

130.0

0. 00043 

. 00036

. 00042 

. 00035 

. 00031 

. 00043 

. 00069

. 00060 

. 00059

. 00060 

. 00061 

. 00070

0.040 

.044

.038 

.036 

.043 

.030 

. 024

.028 

.028

.032 

. 022 

.037

Velocity measured by float 

and corrected to mean 

velocity. 

Do.

Wawaset

Drainage area: 134 square miles. Length of reach over which slope was measured: 403 feet.

July 24, 1951_._ 

July 30, 1951____ 

Aug. 3, 1951_-___

Aug. 20, 1951_-_. 

Aug. 27, 1951____ 

April 27, 1952 _

Apr. 28, 1952. ___ 

Mav 26, 1952_ __

July 10, 1952_

Aug. 7, 1952____

Sept, 19, 1952_

98.5 

80. 2 

68. 2

312.0 

66. 4 

1, 117

2,400 

1,750

1, 036. 0

293. 9

623. 0

60.0 

61.0 

60.0

64. 0 

60. 5 

67.0

75.5 

69.0

71. 6

64. 5

67. 5

1. 23 

1. 10 

1.10

2.05 

1.08 

3.42

4.93 

4.02

3. 12

2.02

2. 84

1.34 

1. 20 

1.04

2. 38 

1.02 

4. 80

6. 46 

6. 33

4. 64

2. 26

3. 24

73. 6

67. 0 

66. 0

131. 1 

65. 1 

1 229. 0

1 372. 0 

1 277. 0

223. 5

130.0

192. 4

0. 00098 

. 00097 

. 00090

. 00120 

. 00085

. 00062

0.040 

.041 

.046

. 038 

.045

. 017

Velocity measured by float

and corrected to mean 

velocity. Do. ' 

Do.

Area from stage-area relationship.
APPENDIX 53

APPENDIX A.—Measurements of discharge and hydraulic variables at the seven principal stations established for this study on Brandywine

Creek—Continued

Date Discharge 

Q (cfs)

Width 

w (feet)

Mean

depth d 

(feet)

Mean 

velocity

v (fps)

Area of 

cross section A 

(square 

feet)

Slope of 

water 

surface s

Roughness 

parameter n' Remarks

Lenape

Drainage area: 259 square miles. Length of reach over which slope was measured: 578 feet.

July 21, 1951____ 

July 31, 1951____ 

Aug. 7, 1951

Aug. 13, 1951_.__ 

Aug. 20, 1951. __ 

Aug. 27, 1951____ 

Sept. 8, 1951____ 

Sept, 15, 1951. __ 

Apr. 27, 1952____

Apr. 28, 1952_ _ 

May 26, 1952 ___ 

Julv 10, 1952____ 

Aug. 8, 1952_____

218.4 

124. 9 

128.7 

191.5 

646. 7 

115. 8 

107. 2 

385. 7 

1,410

2,420 

2, 190 

1, 310. 0 

864. 3

98. 0 

93.0 

93.0 

97.0 

101.0 

92.0 

91.0 

100.0 

109. 0

121.0 

125.0 

113.0 

103. 4

1.47 

1. 34 

1. 28 

1. 47 

2. 78 

1. 17 

1. 14 

1. 90 

4. 17

4. 42 

4.52 

4. 14 

3.02

1. 51 

. 94 

1.08 

1.35 

2. 30 

1.08 

1. 04 

2.00 

3. 10

4. 52 

3. 87 

2. 80 

2. 75

144. 4 

124. 5 

119.0 

142. 6 

281.0 

107. 6 

103. 7 

190. 1 

1 455. 0

i 535. 0 

i 565. 0 

468. 2 

312. 9

0. 00075

. 00078 

. 00074 

. 00076 

. 00074 

. 00076 

. 00076 

. 00032

. 00028

0.034

.045 

.039 

.035 

.042 

.043 

.031

Weeds in measuring section.

Velocity measured by float 

and corrected to mean 

velocity. 

Do/ 

Do.

APPENDIX B.—Discharge measurements at nine points on Buck Run.

Station

Scrogie

Route 222 and Route 30_ _ _____________

Near Sadsburyyille

Stottsville ___*_____ - --__--_____-________

Newlin _ _

Horseshoe bend near Ercildoun _

Forks_ ________________________________

Road fork___

Mouth _

Drainage 

area

(square 

miles )

10. 6

2. 7

4. 5

6. 9

12.4

22. 6

25.0

48. 1

48. 8

Date

June 6, 1952

Aug. 25, 1951

June 6, 1952

July 2, 1952

Aug. 25, 1951

Aug. 25, 1951

June 6, 1952

July 2, 1952

Aug. 24, 1951

July 2, 1952

Aug. 24, 1951

June 6, 1952

Julv 1, 1952

Aug. 24, 1951

Julv 1, 1952

Aug. 24, 1951

June 6, 1952

June 30, 1952

Aug. 22, 1951

June 6, 1952

June 30, 1952

Discharge 

Q

(cfs)

1. 11

1. 70

5. 01

2. 44

2. 56

4. 81

15.4

8. 26

8. 19

15. 3

15. 6

71. 8

31. 9

18. 7

37. 1

34.2

142. 3

77. 6

42. 8

153. 2

80. 6

Width

w 

(feet)

6. 0

8.5

11. 5

11. 5

9.0

12. 5

13. 3

13. 5

22. 0

22. 0

24. 5

22. 5

20. 0

28. 0

31. 0

51.5

52. 0

52.0

46. 5

48. 0

46. 5

Mean 

depth 

d 

(feet)

0. 26

. 76

. 58

. 54

. 88

. 53

. 72

. 63

.59

. 99

. 65

1. 36

1. 07

. 72

. 99

. 68

1. 24

1.06

. 78

1. 24

1. 03

Mean 

velocity

(fps)

0. 71

. 26

. 75

. 39

.32

. 73

1. 61

. 98

. 63

. 70

. 98

2. 35

1. 49

. 43

1. 21

. 98

2. 20

1. 38

1. 19

2. 58

1. 69

Area of

cross 

section 

A

(square

feet)

1. 5:

6. 4(

6. 6J

6. 2(

7. 91

6. 5r

9. 6i

8. 4J

12. 9

21. 8

15. 9

30. 5

21. 4

20. 1

30. 6

35.0

64. 7

56. 3

36. 0

59. 3

47. 6
54 THE NATURAL CHANNEL OF BRANDYWINE CREEK, PENNSYLVANIA

APPENDIX C.—Discharge measurements at miscellantous stations on Brandywine Creek and its tributaries

Stream and location

West. Branch: 

Brandamore _

Rocklyn

BirdeU______. .__._... _____ _ ___

Siousca

Modena

Northbrook

Cedar Knoll

East. Branch: 

Honeybrook. _ _

Lewis Mills

Do——— _ _. __.

Gleiimore

Reeds RoacL _ ______

Main stream: 

Brookfield _ _ _ . .

Do_ — __________ ._ _

Do_ — __________ . _

Below Pocopson

Above Wllmington _

Smiths Bridge _ . . _ _ _

Doe Run: 

Near Cochranville

Below Cochranville _

Above Springdell _

Springdell. __

Taylor Run: 

Mouth _

Mile above mouthTributary near Route 322

Above West Chester,

Mouth __ _

Vallev Creek: 

Below Route 322__ ________ ____

Mile above Route 322 _ _ _

Smiths Bridge _-_

Pennsylvania R. R

Glad Acres

Whiteford___ _ __

Exton.

Ship Rd., north tributary

Ship Rd., south tributaryGlen Loch

Broad Run: 

Below "Rocks".

Headwater

"Marsh" _ __ __ _ _ _

Buck Hill Farm. ___

"Rocks" ._

Below "Rocks"

Near Romansville

Below Route 162 ___

Edwards Farm_

Above mouth _ _

Do— __ __ -_

Rock Run, }i mile above mouth _ Marsh Creek, near Milford

Indian Run, mouthBrinton Run, mouth

Radley Run, mouth

Plum Run, mouth

Culbertson Run, mouth

Pocopson Creek:

Mouth _ _ __

1.5 miles above mouth- __

2 miles west of Lenape _ _

Tributary at Pocopson Home

Near Pocopson Home,

Near Marlboro_

Headwater

Pocopson Home _

Drainage 

area 

(square 

miles)

20. 6

5. 8

18. 5

35. 2

57. 3

124. 4

25. 0

3. 2

8. 0

8.0

16. 5

31. 8

292. 6

292. 6

292. 6

267. 5

312. 0

298. 0

. 3

2. 0

6. 4

11. 4

5. 6

4. 1

1. 7

r. p.

20. 0

19. 6

1 K. 0

14. 1

11. 3

10. 0

5. 7

. 74

2. 3

1 9

1. 26

. 15

KO

1 03

1.17

1. 26

Q nK

A JO

5. 47

5. 77

c 77

8 n

18. 1

6. 3

1. 3

4. 1

3. 5

4.4

Q *>

8. 6

7. 9

2.0

5. 2

4. 1

1. 3

7.3

Date

July 30, 1951

Aug. 2, 1951

____-do-_--_-

_____do-_----

-___do-_----

____-do------

Aug. 6, 1951

Aug. 6, 1951

Aug. 1, 1951

Aug. 9, 1951

Aug. 6, 1951

Aug. 9, 1951

July 28, 1951

Aug. 4, 1951

Aug. 13, 1951

Aug. 8, 1951

Aug. 30, 1951

Aug. 8, 1951

May 29, 1952

_____do—----

_ _ _do^

_____do------

Aug. 21, 1951

__ _.do------ -

_ -do------

_____do— _---

Julv 25, 1951

Aug. 14, 1951

... _do-_-.--

_-__do------

__-_-do— ---

Aug. 15, 1951

____-do-_ — --

____do_ _____

— __do—-- — -

Julv 8, 1952

June 3, 1952

June 5, 1952

June 3, 1952

___ _do—-_ — _

____-do— — -_

_____do-__-_-

Tn T»P ^ 1 Q^9

____-do— __--

_ — -do-_- — .

Aug. 22, 1952

Aug. 7, 1951

Aug. 9, 1951

Aug. 22, 1952

_-__ ~_do____ — -

_ — _do— - —

____do___-__^

_.__-do- -----

Aug. 8, 1951

Aug. 17, 1951

____ _do—— —

__ __do__ —— -

____-do_______

_____do_______

_____do— __--_

____ _do— - —

Discharge 

Q

(cfs)

3.96

2. 55

6. 88

15. 6

30.4

66. 6

8. 90

1. 10

4. 78

3. 26

5. 68

13. 9

152. 0

123. 6

226. 0

146. 0

132. 0

160. 6

1. 82

9. 23

19. 7

24 2

5. 12

5. 33

i •}_

1. 30

4.07

8. 64

9. 04

6. 14

6. 07

5. 28

4. 37

1. 94

. 25

. 24

. 12

13. 3

. 16

2. 48

4. 47

7 on

8. 92

22. 3

01 n

30. 8

oq 7

3. 22

2. 04

5. 72

i QC.

1. 05

2. 02

2. 10

0 Cl

5. 20

7. 16

2. 75

1. 18

2. 24

1. 45

. 65

2. 75

Width 

HI 

(feet)

25.0

13. 0

14. 0

65. 0

35. 0

48. 0

28. 0

8. 0

21. 0

25.0

33. 0

55. 0

145. 0

142. 5

144. 0

103.0

120. 0

121. 0

4. 7

7. 8

12. 0

16. 0

15. 5

10. 5

6. 5

7. 5

20. 0

12. 5

23. 5

18. 0

19.0

16. 0

17. 0

9. 5

4. 5

3. 5

3. 5

8. 0

3. 5

3. 5

5. 2

19. 5

8. 0

7. 0

15. 5

12. 7

14. 8

8. 0

22. 0

32.0

8.0

3. 5

5. 2

7. 0

9. 5

23.0

18. 0

7.0

6. 5

10. 0

7.0

5.0

7.0

Mean 

depth 

d

(feet)

0. 92

. 75

. 53

. 55

1. 20

1. 97

. 76

. 40

. 38

. 65

. 94

1. 11

2. 40

1. 97

2. 49

2. 98

3. 44

1. 94

. 46

.72

. 83

1. 05

. 38

. 56

. 30

. 19

. 43

. 41

. 30

.43

. 84

. 43

. 23

. 44

. 57

. 10

. 06

. 69

. 17

. 36

. 55

.43

. 68

. 83

1. 12

1. 30

1. 11

. 46

. 46

.73

. 28

. 34

. 57

. 21

. 36

. 76

. 28

. 37

. 20

. 42

.35

. 16

. 37

Mean 

velocity

(fps)

0. 17

. 26

. 92

. 44

. 72

. 71

. 42

. 34

. 59

. 20

. 18

. 23

. 44

.44

. 63

. 48

. 32

. 69

. 83

1. 65

1. 97

1. 45

. 87

. 90

. 67

.90

.48

1. 72

1. 30

. 80

. 38

. 77

1. 13

. 46

. 96

. 67

. 60

2. 40

.27

1. 96

1. 54

. 93

1. 64

3 81

2 34

1. 85

2. 05

. 88

. 20

. 24

. 82

. 89

. 68

1. 46

. 80

. 30

1. 41

1. 06

. 92

. 53

. 59

. 83

1.06

Area of

cross section A 

(square 

feet)

23. 0

9. 81

7.45-

35. 6

42. 0

94. 3

21. 4

3. 20

8. 05-

16. 3

31. 2

61.0

348. 1

281. 0

359. 4

307.4

412. 2

234. 2

2. 18

5. 58

9. 97

23. 2

5. 87

5. 30

1. 96

1.44

8. 50

5. 08

6. 93

7. 70

15. 95

6. 85

3. 86

4. 23

2. 56

. 36

. 20

5. 50

. 59

1. 26

2. 90

8.36

5.44

5. 84

17.4

16. 6

16. 4

3. 66

10. 2

23. 5

2. 27

1. 18

2. 96

1. 44

3. 39

17. 50

5.07

2. 92

1. 28

4. 24

2. 44

. 78

2. 61
INDEX

Page 

Area of cross section, measurements at principal stations. _ 51-53

Bank material, location and description of sizes.-----___ 18, 19

Bankfull stage, defined- _____________________________ 29

determination of mean depth,____________________ 29

determination of width._________________________ 29

method of determining at cross sections. ___________ 30

method of determining recurrence interval _________ 32

relation of drainage area to recurrence interval _____ 33

Bed material, cumulative frequency curves showing size

at measuring sections_____--___________-_ 17

method of sampling and describing________________ 16-17

table of sizes at measuring stations..._____________ 18

Blench, Thomas, quoted.____________________________ 44

Brandywine Creek, compared with stable channels______ 42-44

conclusions drawn ______________________________ 49

equilibrium of__________________________________ 45-49

formulas for comparison with stable channels_______ 44

geographic and geologic setting ___________________ 3-6

index map showing location. ___________________ 4

longitudinal profile______________________________ 46

why chosen for study.__________________________ 3

Bryan, Kirk, quoted._______________________________ 44

Channel shape and profile, relation of variables to, at-astation________________________________ 22-23

relation of variables to, in downstream direction,

general discussion_______________________ 24

Chezy formula _____________________________________ 6

Cycle of river erosion, early references________________ 1

Davis, W. M., quoted_______________________________ 45

Discharge, measurements at miscellaneous stations______ 54

measurements at nine points on Buck Run_________ 53

measurements at principal stations,_______________ 51-53

method of determining for given duration___ _______ 34

method of measurement. _______________________ 9

relation of width-to-depth ratio, downstream.______ 29

Dominant discharge_________________________________ 43

Drainage areas _____________________________________ 51-54

Equilibrium, evolution and definition of term___________ 45, 47

Exponents, values in equations relating variables to

discharge at principal measuring stations __ 14 

values of 6, /, and m compared with Leopold and

Maddock data____._____________________ 11-13

Fall Line (Zone)___ 

Flood plain, origin.

Gaging stations.____________________________________

Grade. See Equilibrium

Hydraulic factors, formulas for expressing._____________

Hydraulic geometry, Brandywine Creek at-a-station and 

downstream direction ___________________

Hydraulic geometry, defined.________________________

3

19

9-10

14

27

26

Lacey, Gerald, quoted__________________________.

Leopold, L. B., and Maddock, Thomas, Jr., quoted.

Mackin, J. H., quoted_________________-________

Manning formula. ____________________________

Mean bed velocity.____________________________

Mean depth, defined.__________________________

effect of variations within reach _____________

measurements at principal stations.__________

relation to discharge at given flow durations, __

26

relation to width in various channels _ Mean velocity, defined _________________

measurements at principal stations. __

Measuring reach, Cornog_______________

criteria for selecting._______________

Embreeville_ _ _____________________

Georgia___________________________

Leriape. _____________-_-_-_-_-_-__

Seven Springs_____________________

South Downingtown_______________

Wawaset _________________________

Measuring stations, location on map____.

Morphology, river, equilibrium in________

Odom, L. M., quoted. 

Overbank stage______

Page 

44 

23

45,47 

6

38,48 

6

39

51-53 

25,

35,36 

42

6 

51-53

7 

6,9

7

8

7

7

7

8

4

2

44

31

Plan of study------------------------------- ----- 2-3

Pools and riffles, criteria for distinguishing _____________ 40

effect on curves of variables._____________________ 40-41

Purpose of study __________________________________ 2

Reach, length over which slope was measured,_________ 51-53

method of computing variation within_____________ 39-40

Regime of a stream, defined. ________________________ 45

Regimen of a stream, defined____ ____________________ 45

Roughness, changes at-a-station ______________________ 20-22

factor ri defined._______________________________ 20

distinction between factors n and .1'_______________ 21

measurements at principal stations_____________ __ 51-53

relation of n' to mean depth at four stations. ______ 21

relation to discharge at given flow durations.______ 25, 26

Scour and fill_____________________________________ 42

Slope of water surface, effect of reach on_______________ 14-15

measurements at principal stations________________ 51-53

method of measurement _________________________ 9

relation to discharge at given flow durations--_____ 25, 26

relation to other variables________-_______________ 38

Stable channels, characteristics___-_____-_____-_______ 44-45

formulas for comparing with Brandywine Creek _ _ _ _ 44

Suspended load, at principal measuring sections-_______ 20

relation to changing discharge.___________________ 19-20

relation to channel shape ________________________ 28

relation to width_______________________ ________ 29

sediment rating curve, Wilmington________________ 20

Symbols used in this report._________________________ iv

55
56 INDEX

Page 

Variables, changes in the downstream direction.._______ 24, 37

choice of_______________________________________ 6

dependent and independent ___________________ 22, 23, 27

relation to discharge at Chadds Ford_______

relation to discharge at Coatesville___ ______

relation to discharge at Cornog____________

relation to discharge at Embreeville______-_

relation to discharge at Georgia_-----_--_^ _

relation to discharge at bankfull stage______

relation to discharge at Lenape____________

relation to discharge at Seven Springs ______

relation to discharge at South Downingtown.

15

14

11

13

12

31

14

11

12

Page

Variables—Continued

relation to discharge at Wawaset_________________ 13

simultaneous adjustment ________________________ 41-42

summary of at-a-station interaction ______________ 22-23

values at Cornog.______________________________ 10

Velocity, effect of variations within reach._____________ 39

relation to discharge at given flow durations._ 25, 26, 35, 36

Width, effect of variations within reach._______________ 39

measurements at principal stations ________________ 51-53

relation to depth in various channels ______________ 42

relation to discharge at given flow durations._ 25, 26, 35, c6

o