Storage and 

Flood Routing

Manual of Hydrology: Part 3. Flood-Flow Techniques

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1543-B
Storage and 

Flood Routing

By R. W. CARTER and R. G. GODFREY

Manual of Hydrology: Part 3. Flood-Flow Techniques

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1543-B

Methods and practices of 

the Geological Survey

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1960
UNITED STATES DEPARTMENT OF THE INTERIOR 

FRED A. SEATON, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

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

Washington 25, D.C. - Price 15 cents (paper cover)
CONTENTS

Page

Abstract. _________________________________________________________ 81

Introduction._____________________________________________________ 81

General relation between upstream and downstream hydrographs________ 82

Methods of flood routing-__________________________________________ 85

Basic concepts._______________________________________________ 85

Stage-storage method of flood routing._____________-__--_____---_ 87

Discharge-storage method of flood routing. _______________________ 92

Selection of time interval.__-__-___________-__-__-__--___-__ 94

Computation of storage volume. ____________________________ 94

Evaluation of the constant weighting the inflow and outflow. ___ 96

Evaluation of the slope of the storage-weighted discharge relation. 96

Example of computation procedure._______-__--______-___-__ 97

Determination of storage- ______________________________ 98

Determination of constants. ____________________________ 98

Reservoir-storage method.________________________-___---____.--- 102

Selected references.._______________________________________________ 104

ILLUSTRATIONS

Page 

FIGURE 31. Schematic representation of reservoir-type storage. _________ 83

32. Schematic representation of inflow, outflow, and storage in a

uniformly progressive wave.___________________________ 84

33. Schematic representation of prism and wedge storage. ______ 86

34. Cochrane-Gainesville reach on Tombigbee River.___________ 88

35. Relation of mean gage height to storage in Cochrane-Gainesville reach,__________________________________________ 90

36. Relation of change in storage per foot of gage height to mean

gage height for Cochrane-Gainesville reach_______________ 91

37. Comparison of actual and computed hydrographs for Tombigbee River at Gainesville, Ala__________-___--__--_--_ 91

38. Andalusia-Brooklyn reach on Conecuh River_______________ 97

39. Relation of storage to weighted discharge for different values

of *_____________^__._____________-__-_-_----- 98

40. Comparison of computed and actual hydrograph, Conecuh

River near Brooklyn, Ala___-_-_-_____-_--------------- 100

41. Inflow and outflow hydrographs for reservoir______--------- 104
IV CONTENTS

TABLES

Page 

TABLE 1. Computation of storage for Cochrane-Gainesville reach, Tombigbee River___________________________________________ 89

2. Example of routing for Cochrane-Gainesville reach.__________ 92

3. Computation of storage for Andalusia-Brooklyn reach, Conecuh

River_________________________________________________ 95

4. Computation of weighting factor x for Andalusia-Brooklyn

reach, Conecuh River___________________________________ 99

5. Example of routing for Andalusia-Brooklyn reach, Conecuh

River_._______________________________________________ 101

2 Sf

6. Selected values of-r7 + 0--__---__-_-_-___-------_--_------ 102

t\t

7. Routing through a reservoir.______________________________ 103
MANUAL OF HYDROLOGY: PART 3, FLOOD-FLOW TECHNIQUES

STORAGE AND FLOOD ROUTING

By R. W. CARTER and R. G. GODFREY

ABSTRACT

The basic equations used in flood routing are developed from the law of continuity. In each method the assumptions are discussed to enable the user to select 

an appropriate technique.

In the stage-storage method the storage is related to the mean gage height in 

the reach under consideration. In the discharge-storage method the storage is 

determined, from weighted values of inflow and outflow discharge. In the 

reservoir-storage method the storage is considered as a function of outflow discharge alone.

A detailed example is given for each method to illustrate that particular 

technique.

INTRODUCTION

The storage in the reaches of stream channels is used extensively 

as an index of the timing and shape of flood waves at successive points 

along a river. Among the principal users of the technique are the Corps 

of Engineers who route hypothetical floods through river systems to 

determine the effects from proposed flood-control projects, the U.S. 

Weather Bureau whose forecasts of river stages are based largely on 

flood routing, and the operators of hydroelectric power systems who 

schedule their operations according to the predicted progress of a flood 

wave. The storage index and the techniques of flood routing may 

also be used to advantage in computing and evaluating streamflow 

records.

The primary use of these methods in the Geological Survey is in 

testing and improving the overall consistency of records of discharge 

during major floods in a river basin. The number of direct observations 

of discharge during such flood periods is generally limited by the short 

duration of the flood and the inaccessibility of certain stream sites. 

Through the use of flood-routing techniques, all observations of discharge and other hydrologic events in a river basin may be combined 

and used to evaluate the discharge hydrograph at a single site. For

81
82 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

an example of a study of this type the reader is referred to WaterSupply Paper 1139 (U.S. Geological Survey, 1952).

Storage-routing techniques also are used frequently to determine 

the effect of artificial storage on peak discharges from small drainage 

basins. For economy of operation, gaging stations on many small 

streams are located at the outfall of small detention reservoirs impounded by dams or by highway crossings. The measured peak 

discharges are adjusted for the effects of artificial storage, thereby 

increasing their usefulness in areal studies of the frequency and magnitude of peak flow.

Another use of flood routing in hydrologic studies is in computing 

natural flood hydrographs for long-term gaging stations on streams 

where major storage reservoirs have been constructed. These computations are made to extend the time period for a single condition of 

basin development for use in discharge-frequency studies.

In this report, several different methods of storage routing are 

presented in detailed form. These methods are not new and, in 

general, were not originated in the Geological Survey. They are 

presented for the convenience of engineers engaged in hydrologic 

studies. A previous manual on this subject was prepared by W. B. 

Langbein and R. W. Carter (written communication, 1947). The 

methods and techniques described in this manual are the product of 

development in many engineering offices concerned with floods.

GENERAL RELATION BETWEEN UPSTREAM AND DOWNSTREAM HYDROGRAPHS

Flood waves are subject to two principal kinds of movements 

uniformly progressive flow and reservoir action. A uniformly 

progressive flow designates downstream movement of a flood wave 

without a change in shape, which would occur only under ideal 

conditions in a prismatic channel in which the stage and discharge 

are uniquely defined at all places. Reservoir action refers to the 

modification of a flood wave by reservoir pondage. Flood-wave 

movement in natural-channel systems is probably intermediate 

between the two ideal conditions cited, one or the other predominating 

in a particular place. However, the actual behavior of the wave is 

sometimes obscured by the effects of local tributary inflow.

The attenuation effect of what has been called reservoir-type or 

pondage storage is illustrated by the performance of a detention or 

retarding basin a reservoir impounded by a dam with an outlet at 

stream level. The outlet conduit is designed to permit the normal 

flows of the stream to pass unobstructed, but its capacity is exceeded 

by flood discharges. The impounded water creates a sensibly level 

pool and discharge through the conduit is related solely to the head
STORAGE AND FLOOD ROUTING 83

1000

800

600

UJ rr 

OlU 

Q£ CL

< 

I 

O

400

200

At time TP inflow equals outflow, 

storage is maximum and outflow 

is maximum

TP

Q£

UJ 

CL

fc> 

UJ < 

U.Q

Ofc Suj D°- 

"Q

?§uia 38

Q£ 

O

fc

vSTO RAG E=TOTAL INFLOW MINUS 

TOTAL OUTFLOW

2468 10 12 14

TIME. IN DAYS

FIGURE 31. Schematic representation of reservoir-type storage.

and hence to the storage. Storage in the reservoir will increase so 

long as inflow exceeds outflow, and is still increasing when peak inflow 

occurs. Figure 31 shows that storage and outflow reach a maximum 

(TP) when inflow equals outflow, but this equality cannot occur 

until after inflow has receded from its peak. Therefore, reservoir or 

pondage storage always attenuates peak discharge.

The storage generated in a reach of channel by a uniformly progressive wave, illustrated in figure 32, is completely different. The 

storage prism varies in shape and in volume. River stages and storage 

in a reach are conditioned by the inflow and the outflow rates. Hence,
84 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

in the course of a flood that is controlled only by channel action, the 

storage increases rapidly in relation to outflow. Conversely, the 

storage decreases rapidly when the flood recedes. Storage is increasing when peak inflow occurs, and it continues to increase until outflow 

rises to equal the receding inflow. At this time the flood crest occupies a central position in the reach. At the instant of the maximum 

storage, the net change in storage equals zero and inflow equals 

outflow; the upstream stages are decreasing, the downstream stages 

are increasing and volume of storage in the reach at that instant is 

constant. After this point is passed, outflow continues to increase,

1000

LJ

OL 1200

LU <

u. o

CO £ 800 

OQ

400

-STORAGE=TOTAL INFLOW 

TOTAL OUTFLOW

MINUS

01 3 5 7 9 

TIME, IN DAYS

FIGURE 32. Schematic representation of inflow, outflow, and storage in a uniformly progressive wave.
STORAGE AND FLOOD ROUTING 85

partly at the expense of storage in the reach, until the crest passes 

the lower end.

Figures 31 and 32 show the difference in the behavior of storage 

controlled by a fixed outlet dam and by channel action. The inflow 

hydrographs, total storage, and lag between centers of mass of inflow 

and outflow, are the same in both illustrations. In reservoir-type 

storage, as in figure 31, the peak outflow is much less than the peak 

inflow, whereas in the uniformly progressive wave, as in figure 32, 

there is no reduction in peak discharge.

River channel systems are commonly characterized by alternations 

of "pools" and "rapids" and of "narrows" and "intervals" which in 

the aggregate have much the same attenuating effect as reservoir 

storage. However, where the cross section and bottom slope are 

virtually uniform, little or no attenuation due to storage can be 

expected. The action of storage must not be confused with the 

attenuating effect of the desynchronized timing of tributary and 

mainstream flood peaks.

METHODS OF FLOOD ROUTING 

BASIC CONCEPTS

Many different methods are used to route flood waves through 

river reaches. All these methods are based on the law of continuity 

the volume of water that is discharged from a reach during any 

interval must equal the volume of inflow during the interval plus or 

minus any increment in stored water during the period. In equation 

form it becomes:

o=T-~ (i)

At

where 0=mean outflow during routing period At

/ mean inflow during routing period At 

A$=net change in storage during routing period At

Equation (1) is general. A modification frequently used is:

, . ,2v 

^

where 0, I, S and At are as before, and the subscripts identify the 

beginning and ending of routing period At. The assumption that 

mean discharge is equal to the simple arithmetic average of the flows 

at the end points of the interval can be justified if the period is equal 

to, or less than, the time of travel through the reach and no abrupt 

changes in flow occur during the routing period.
86 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

Two problems arise in using the storage equation in a particular 

reach. The first is to express the storage volume in terms of some 

practical index and the second is to solve equation (2) for 02 . The 

equation can be readily solved by graphical or analytical methods. 

The exact nature of the solution depends on the method used to 

determine the storage in the reach.

Expressing the storage volume in terms of some practical index 

is much more difficult, but it is extremely important because the

.Water surface

3456 

i

SUCCESSIVE POSITIONS OF A WAVE IN 

ITS TRAVEL THROUGH A SINGLE REACH

FIGURE 33. Schematic representation of prism and wedge storage.

accuracy of flood routing depends on the closeness with which the 

storage volume can be approximated. Assume that the reach in 

figure 33 represents successive positions of a wave traveling through 

a single reach. One element of storage commonly called "prism 

storage" may be easily defined in terms of stage or discharge at the 

downstream station. The simpler routing methods attempt to use 

this approximation alone, neglecting a second important volume of 

"wedge storage" which is superimposed on the prism. Wedge storage 

is a secondary approximation, but it is much more difficult to define 

in terms of a simple index because of the variety of shapes of the 

wedges. However, a routing method that ignores wedge storage is 

subject to serious error except in most unusual conditions.
STORAGE AND FLOOD ROUTING 87

The storage in a reach may be determined by direct measurement 

from detailed maps or channel cross sections, as if computing reservoir or earthwork volumes. This approach is laborious and data are 

rarely adequate. Except where stage records are available during 

the flood to be routed, it would be necessary to compute several 

backwater curves defining the water-surface profile under various 

conditions of unsteady flow.

The method most often used is to compute the volume of storage 

by comparing the actual flood hydrographs. The volume of water in 

storage at any instant exclusive of intervening inflow between the 

stations is equal to the accumulated difference between inflow and 

outflow.

STAGE-STORAGE METHOD OF FLOOD ROUTING

The simplest method of flood routing defines the storage in terms 

of the mean gage height in the reach. Thus a gage-height record for 

both ends of the reach must be available if the flood is to be routed. 

The necessary stage-storage relation is, generally defined on the basis 

of past flood-discharge records; although in certain reaches it may 

be defined from topographic data. Areal photographs of rivers in 

flood are also valuable in defining this relation. Because storage is 

directly related to stage, A Ah may be substituted in equation (1) 

for AS

0=I-A t (3)

where A= average area of water surface in the reach during time At 

and Ah= average change in water surface elevation in the reach in 

time At.

In equation (3), the value of / is obtained from the given inflow

hydrograph and from the value which is obtained from the stage

L\t>

record. The water-surface area at a given stage is the slope of the 

stage-storage curve and may be easily computed from a stage-storage 

table, since it is the "first difference." Hence the slope or first difference is a function of the mean stage in the reach. As mentioned 

previously, areal photographs made during record floods can be used 

to construct or extend a stage-water surface relation. The outflow 

may be computed from the mean stage, the rate of change of stage, 

and the inflow.

As an example, the method is applied to the Cochrane-Gainesville 

reach on the Tombigbee Kiver in Alabama. This reach, as shown
88 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

Gainesville

FIOTTRE 34. Cochrane-Gainesville reach on Tombigbee River.

in figure 34, is about 25 miles long and the flow from the main tributaries entering the reach is measured. Records lor the flood of July 

1940 were used to define the stage-storage relation. The small amount 

of unmeasured inflow was estimated by increasing the flow of the 

measured tributaries by the drainage-area ratio. In table 1, columns 

2, 3, and 6 show the several components of inflow, and column 7 

shows the total inflow. Listed in columns 8 and 9 are the accumulated inflow and the accumulated outflow. The difference in accumulated inflow and accumulated outflow, which is the total storage 

in the reach at the end of the selected time interval, is shown in 

column 10. The storage is represented by water within the channel, 

on the river flood plain, in riverbank voids, and in contiguous swamps 

which are affected by river levels.

The mean stage shown in column 11 of table 1 is the mean of the 

midnight gage heights at each end of the reach. Intermediate stage 

records in the reach may be included in this average if available. 

The mean stage and storage in the reach at the time of the mean are 

plotted in figure 35.
TABLE 1. Computation of storage for Cochrane-Gainesville 

Date

(1)

1940 

June 28._-_.-_-_-_ _ _ ... _

29 

30- --- ... - 

July 1 

2 

3 

4 

5 -. - -..

6... .. .-. _ _ _

8-.----.... - -..

9... ----- ... .-.

10. .-.--. -..-_-__ .

11--.---...-. ----------

12

13-. ....... --..

14 

15 

16_- . _ .---. ..-._.__--

17__. _ . 

18

19 ..., -....

20 ---- . .........

21-. . . .--. .._.

22 

23 -- 

24.............. ......

25.

26._. _._..... ......

27.....................

28 

Partial inflow (cfs) discharge

Noxubee

(2)

294 

590 

1,480 

1,420 

1,030

2,050 

6,400 

6,900 

6,980 

7,180

7,900 

11, 100 

22, 400 

20,400 

19, 500

18, 500 

14, 700 

11, 500 

10, 800 

9,840

8,620 

6,030 

6,100 

6,160 

6,540

6,420 

5,030 

3,760 

936 

846 

410

Sipsey

(3)

243 

264 

318 

296 

432

1,100 

1,590 

1,570 

1,550 

1,760

2,020 

2,030 

2,410 

2,810 

3,630

4,780 

5,760 

6,460 

7,020 

6,690

5.280 

5, 040 

4,230 

3,500 

3,550

3, 040 

2,230 

I, "850 

1,810 

1,670 

1.500

Noxubee plus 

Sipsey

(4)

537 

854 

1,798 

1,716 

1,462

3,150 

7,990 

8,490 

8,530 

8,940

9,920 

13, 130 

24, 810 

23, 210 

23, 130

23, 280 

20, 460 

17, 960 

17, 820 

16, 530

13, 900 

11, 070 

10, 330 

9,660 

10, 090

9,460 

7,260 

5,610 

2,746 

2,516 

1,910

Column 4 

times 1.16

(5)

623 

991 

2,086 

1,990 

1,696

3,654 

9,268 

9,850 

9,895 

10, 370

11, 500 

15, 200 

28, 800 

26, 900 

26, 800

27, 000 

23, 700 

20, 800 

20, 700 

19, 200

16, 100 

12, 800 

12,000 

11, 200 

11, 700

11,000 

8,420 

6, 510 

3,185 

2,920 

2,220

Cochrane

(6)

1,360

2,770 

3,880 

4,710 

4,680

14, 500 

29, 000 

31, 400 

36, 100 

40, 400

42, 000 

41,700 

40, 400 

36, 800 

36, 100

36, 700 

38, 700 

40, 500 

40, 300 

39, 300

36, 700 

33, 700 

30, 900 

27,200 

23, 300

20, 000 

12, 700 

7,670 

3,790 

3,610 

3,360

Total in

discha

(cfs

(7)

1

3

4

4

5

5

5

6

6

6

6

6

6

6

5

5

4

4

3

3

3

2

1
90 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES 

501 

20 40 60 80 100 

STORAGE, IN THOUSANDS OF CUBIC 

FEET PER SECOND PER DAY

FIGURE 35. Relation of mean gage height to storage in Cochrane-Oainesville reach.

The slope of the stage-storage curve, As: Ah, is shown plotted against 

the mean stage in figure 36.

The above operation should be plotted for several floods, and 

an average stage-storage relation adopted. Major differences for one 

or more floods indicate that backwater conditions may exist or that 

there are errors in ratings or stage data.

The treatment of the above reach is simplified somewhat, because 

the storage volume is treated as a function of the straight average of 

the stage at each end of the reach. In reaches where a large percentage of the storage volume is near one end of the reach, the stages 

generally have to be weighted. Records from gages on the tributaries 

should be given some weight in determining the mean stage in the 

reach. These weights may be assigned from a knowledge of the 

physical characteristics of the channel or may be determined by trial 

and error.

To test the stage-storage relation, the outflow hydrograph is 

computed from the inflow hydrograph and from the record of stage at 

both ends of the reach (table 2). Column 2 lists the daily mean gage 

height in the reach, which is obtained by averaging the observed mean 

daily stages at both ends of the reach. The change in storage per foot 

of gage height is listed in column 3. This change is equivalent to the
STORAGE AND FLOOD ROUTING 91

50

40

20

10

0246 8 10 12

CHANGE IN STORAGE, IN THOUSANDS OF CUBIC FEET PER

SECOND PER DAY PER FOOT OF GAGE HEIGHT

FIGURE 36. Relation of change in storage per foot of gage height to mean gage height for CochraneGainesville reach.

surface area (see equation 3) and is the abscissa of figure 36 associated 

with the ordinate which is the mean stage from column 2. Listed in 

column 4 is the change in stage in the reach during the time interval 

selected, which is 1 day in this example. The product of the change 

in storage per foot of stage (column 3) and the stage per day (column 4) 

is shown in column 5. The outflow which is listed in column 7 is equal 

to the inflow minus the change in storage. A comparison of the computed and the actual outflow is shown in figure 37.

12 16 20 24

JULY 1940

FIGURE 37. Comparison of actual and computed hydrographs for Tombigbee River at Gainesville, Ala.
92 MANUAL OF HYDROLOGY! FLOOD-FLOW TECHNIQUES 

TABLE 2. Example of routing for Cochrane-Gainesville reach

Date

(1)

1940 

July 2..-.-.....

3 

4____.__

5.- ..

fi .

7 --

8 .

9._. ______

10.-. ....

11-. . .

12.________

13 

14 ______ _

15.- .

16 _______

17 

18.__________

19__-__._____

20 

21... ________

22 . 

23 _ _____

24. __________

25 _______

26 ________

27 -

Mean 

daily gage 

height (ft) 

(feet)

(2)

8.3 

14.1 

25.1

28.7

31.4 

34.1 

35.7 

37.1 

38.9

40.0 

40.4 

40.6 

40.9 

41.2

41.4 

41.4 

41.0 

40.4 

39.6

38.3 

36.6 

34.0 

29.8 

23.3 

16.3 

10.3

AS 

Aft 

(1,000 cfsdays)

(3)

0.2 

.7 

1.8 

2.1

2.8 

4.0 

4.9 

6.0 

7.7

8.6 

9.1 

9.2 

9.5 

9.9

10.1 

10.1 

9.6 

9.1 

8.2

7.1 

5.6 

3.9 

2.4 

1.6 

.9 

.4

Aft 

A. 

(feet per 

day)

(4)

+0.6 

+10.9 

+7.2 

+3.0

+2.7 

+2.3 

+1.2 

+1.8 

+1.5

+.80 

+.30 

+.20 

+.40 

+.30

+.10 

-.30 

-.40 

-.80 

-1.00

-1.4 

-2.2 

-3.1 

-5.3 

-7.4 

-6.4 

-4.2

AS

AT 

(cfs-days)

(5)

+120 

+7, 630 

+13, 000 

+6, 300

+7, 560 

+9, 200 

+5, 880 

+10, 800 

+11,600

+6,880 

+2, 730 

+1, 840 

+3, 800 

+2, 970

+1,010 

-3, 030 

-3,840 

-7,280 

-8,200

-9,940 

-12, 300 

-12, 100 

-12,700 

-11,800 

-5,760 

-1,680

Total 

inflow

(cfs)

(6)

6,380 

18,200 

38.300 

41,200

46,000 

50,800 

53,500 

56,900 

69,200

63, 700 

62,900 

63, 700 

62,400 

61,300

61,000 

58,500 

52,800 

46,500 

42,900

38,400 

35,000 

31,000 

21,100 

14,200 

6,980 

6,530

Outflow (cfs)

Computed

(7)

6,260 

10,600 

25, 300 

34,900

38, 400 

41,600 

47, 600 

46, 100 

57, 600

56,800 

60, 200 

61,900 

58, 600 

58, 300

60,000 

61,500 

56, fiOO 

53,800 

51,100

48,300 

47, 300 

43,100 

33,800 

26,000 

12, 700 

8,210

Actual

(8)

6,160 

12,200 

27,100 

34, 400

37, 900 

42,000 

45, 200 

48, 100 

55,700

61,700 

61,000 

60,100 

59,300 

58,600

58,200 

58,100 

56,900 

55,100 

52,900

50,200 

46,900 

41,800 

34,300 

23,800 

14, 700 

8,280

DISCHARGE-STORAGE METHOD OF FLOOD ROUTING

Most methods of flood routing now in use define the storage volume 

in terms of the inflow and the outflow rather than the stage. The 

simplest, but least accurate, routing procedures define the reach 

storage in terms of the outflow only. If the storage in the reach is 

plotted against the outflow, an irregular loop is developed rather than 

a single-valued graph. The loop reflects the influence of wedge 

storage. A much more satisfactory definition of storage was introduced by McCarthy (written communication, 1938) who used both 

the inflow and the outflow rates, to express storage as a function of 

the weighted mean flow through the reach, as follows:

Storage = K[z7+(l x)O] (4)

7 = inflow rate at a given time 

O = outflow rate at a given time

K = slope of storage- weigh ted discharge relation and has the dimension of 

time

x = & dimensionless constant which weights the inflow and outflow.

This method, which is known as the Muskingum method, assumes 

that the water-surface profile is uniform and unbroken between the
STORAGE AND FLOOD ROUTING 93

upstream and downstream points on the reach, that the stage and discharge are uniquely defined at these two places, and that K and x are 

sensibly constant throughout the range in stage experienced by the 

flood wave.

The factor x (equation 4) is chosen so that the indicated storage 

volume is the same whether the stage is rising or falling. For spillway discharges from a reservoir, x may be shown to be zero, because 

the reservoir stage and hence the storage are uniquely defined by the 

outflow; hence, the rate of inflow has a negligible influence on the 

storage in the reservoir at any time. For uniformly progressive flow, 

x equals 0.50, and both the inflow and the outflow are equal in weight. 

In this wave no change in shape occurs and the peak discharge remains 

unaffected. Thus, the value of x will range from 0 to 0.50 with a 

value of 0.25 as average for river reaches. No way is known for 

determining the value of x from the hydraulic characteristics of a 

channel system in the absence of discharge records.

The factor K has the dimension of time and is the slope of the 

storage-weighted discharge relation, which in most flood problems 

approaches a straight line. Analysis of many flood waves indicates 

that the time required for the center of mass of the flood wave to 

pass from the upstream end of the reach to the downstream end is 

equal to the factor K. The time between peaks only approximates 

the factor K. Ordinarily, the value of K can be determined with 

much greater ease and certainty than that of x. Equations (2) and 

(4) may be combined into:

or

n = . -. T . , (K-Kx-Q.fat)

2~ (K-Kx+Q.5M) 2+ (#-&+0.5A£) 1 -.

(6) 

or simplified as

where x and K have the same meaning as in equation 4 and:

At = time unit of computation in fractions of a day 

/i, 72 = total instantaneous inflow in cubic feet per second to a reach

at the beginning of successive time units (At) .

0i, 02= instantaneous outflow in cubic feet per second at the beginning 

of successive time units (At)

NOTE: Instantaneous values at the beginning of successive time units are used to 

derive the values of x and K . Average values of discharge for the successive 

time intervals may be used in applying equation (7) to a set of data.
94 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

The numerator of equation (5) is the storage increment, which is 

equal to the inflow minus the outflow in cubic feet per second days, 

whereas the denominator is the corresponding weighted-flow increment 

in cubic feet per second. Equation (6) gives 02 in terms of 3 "routing 

coefficients" and 3 known discharges: II , I2 , and Oi. The routing 

coefficients may be computed from known values of x and K. Equation (7) reduces the routing procedure by the Muskingum method to 

tabular multiplication and addition.

SELECTION OF TIME INTERVAL

The time increment, A£, between successive values of inflow or 

outflow should be greater than 2Kx to avoid negative values of <?0 . 

For some reaches, a value of A£ greater than 2Kx will be too long to 

define the hydrograph adequately. In such reaches, trial routings 

should be made, using different values of x through n subreaches of 

travel time, Ks=At, such that nKs is equal to the value of K for the 

entire reach. The value of x which satisfactorily reproduces the 

outflow hydrograph would then be used in subsequent routings 

through the subreaches.

COMPUTATION OF STORAGE VOLUME

The storage volume is computed from equation (1) and from 

a record of the inflow and outflow hydrograph. Starting at a time 

when the inflow and outflow are about equal, that is, where both are 

equal to base flow, successive values of inflow and outflow are cumulated. If the inflow is from several sources such as main stem, 

tributary, or local, then the combination of these is used as the inflow. 

Local inflow from an ungaged area may be estimated from rainfall 

, records or may be estimated on the basis of the flow from a drainage 

area of similar size in the region. A check should be made to determine whether the measured inflow and outflow volumes and the 

estimated unmeasured volume are in balance. Minor adjustments 

may be made to the estimated unmeasured inflow, but if major adjustments seem necessary, possible errors in the main stream discharge records should be investigated. This may be done by choosing additional reaches above and below the main reach, and computing the storage volumes for these reaches for the same floods. 

It should then be apparent whether either of the main stream records 

is in error, and if so, which one. The difference between the cumulated 

inflow and the cumulated outflow represents the volume of storage 

in the reach at that time. An example of the computation procedure 

is given in columns 2 to 6 of table 3.
TABLE 3. Computation of storage for Andalusia-Brooklyn 

Date

(1)

1944

17

18 -- --- - 

19

20.. _.__ - -.. - 

21 --.--- ------- --------

22 ...... __ __ .........

^&....... ........... ..................

24 - - . -----

25 ___ . - -

26 --....-..-.--.-.------ -------

27

28

29

30

31 - - - 

April 1

2

3

4

5 - - -

6

8 - . ,-

9

Partial inflows (cfs)

Andalusia

(2)

2,510 

3,140 

4,170 

7,000 

9,600

10,100 

12,300 

21,000 

29,800 

29,800

29,500 

24,000 

14,600 

18,400 

19,300

18,500 

22,000 

22,000 

14,200 

10,100

7,950 

6,340 

5, 410 

4,210 

4,170

Thad

(3)

552 

766 

947 

1,630 

2,520

2,280 

2,560 

7,250 

8.390 

8,390

5, 440 

3,190 

2,440 

3,790 

5,860

10,000 

5,950 

3,300 

2,120 

1, 570

1,220 

998 

830 

718 

671

McKenzie

(4)

698 

1,030 

1,480 

2,210 

3,400

4, 550 

5,080 

8,420 

16,400 

16,900

10,300 

5,680 

4,120 

6,630 

15, 500

18,800 

11,500 

6,210 

2,690 

1,930

1,480 

1,170 

925 

800 

742

Local

(5)

500 

2,710 

4,570 

5,890 

6,070

4,020 

6,630 

9,330 

5, 370 

2,650

2,650 

1,590 

500 

5,860 

4,520

1,840 

1,840 

2,320 

1,500 

1,120

786 

786 

666 

500 

500

Total in

(cfs)

(6)

4

7

1

16

2

2

26

46

60

57

47

34

2

34

45

4

4

3

2

1

1

9

7

6
96 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES 

EVALUATION OF THE CONSTANT WEIGHTING THE INFLOW AND OUTFLOW

The value of x for a reach is determined by a trial and error procedure. A value of x is assumed and the weighted outflow-inflow discharge is computed. The weighted discharge is then plotted against 

the corresponding volume of storage in the reach. If the correct 

value of x has been chosen, the storage loop should close and the plot 

usually approximates a straight line through the medium and high 

discharge range. An example of the procedure is given in table 4 

and in figure 39 and will be discussed subsequently.

The value of x ranges from 0 for reservoir-type storage to 0.5 for 

uniformly progressive flow. For natural stream channels, a common 

value of x is 0.25. In determining the value of x, great accuracy is 

not required because the method is relatively insensitive to the value 

of this coefficient.

EVALUATION OF THE SLOPE OF THE STORAGE-WEIGHTED DISCHARGE

RELATION

The coefficient K has the dimension of time and is the slope of the 

weighted-discharge storage plot described in the preceding section. 

The value of K thus determined is the average K for the reach, but if 

tributaries enter the reach, probably not all the inflow will rise and 

fall simultaneously. In such streams the storage will be influenced 

by the variation in inflow and a more accurate evaluation of its 

effect can be made by separately routing each inflow from its point 

of entry into the reach to the lower end of the reach. To accomplish 

this, it is necessary to determine individual coefficients for each 

component of the flow. This may -be approximated by the following 

equations:

=

and

Kc=MKm (9)

where K average K for the reach in units of days 

Km K per mile

Kc = individual K for separate routing of inflows 

Qc = total inflow at one point during a given flood 

M = distance in miles from point of inflow to lower end of 

reach.

When determining the value of K for several reaches of the main 

stem of a river, a plot of K and the length of the reach is useful in 

determining the value of K for subreaches.
STORAGE AND FLOOD ROUTING 97

The time interval represented by K is equivalent to the time required for an elemental discharge wave to traverse the routing reach. 

In the absence of complete discharge records, the value of K for 

natural stream channels may be approximated by dividing the 

length of the reach by the mean velocity in the channel and a coefficient, C, that varies with the shape of the channel:

Channel Coefficient, C

Wide rectangular._________________________________________________ 1. 67

Wide parabolic.___________________________________________________ 1. 44

Triangular. _ -.----______..__.._..-_____._.__________._.--_.-_-.--. 1.33

The value of K in natural channels is also about equal to the time 

interval between the peak flow at successive points along the channel.

EXAMPLE OP COMPUTATION PROCEDURE

The computation of routing coefficients for a reach on the Conecuh 

River in Alabama is presented as an example. This reach is composed of the Conecuh River from Andalusia to Brooklyn, Pigeon 

Creek from Thad to its mouth, and Sepulga River from McKenzie, 

to its mouth (see fig. 38). The inflow at Andalusia, Thad, and 

McKenzie, and the outflow at Brooklyn are gaged. All other interGaging station

FIGURE 38. Andalusia-Brooklyn reach on Conecuh River.
98 MANUAL OF HYDROLOGY! FLOOD-FLOW TECHNIQUES

vening inflow is ungaged and is classed as local inflow. The drainage 

areas for the various sources are:

Drainage area 

(square miles)

__.. 1,300

Point

Andalusia- -___-.____________-_-_________-__---_-__-------_

McKenzie-_______________________________________________________ 296

Thad____________________________________________________________ 470

LocaL.__..________._.__._....______._._....______-___--________- 334

Brooklyn_________________________________________________________ 2, 400

DETERMINATION OF STORAGE

Records for the flood of March-April 1944 were used in computing 

storage. Local inflow was estimated on the basis of rainfall and runoff per square mile of other streams in the region. Since the streams 

rise fairly slowly, the time unit used was the day. In table 3, the 

daily inflows at points in the reach are listed in columns 2, 3, 4 and 5. 

The total daily inflow is listed in column 6 and the daily outflow at 

Brooklyn is listed in column 8. The cumulative inflow and the 

cumulative outflow are listed in columns 7 and 9, respectively, and 

the difference in cumulative inflow and cumulative outflow, which is 

equal to the storage in the reach up to midnight of each indicated 

day, is listed in column 10.

DETERMINATION OF CONSTANTS

In table 4, the total instantaneous inflow at 12 p.m. is listed in 

column 2 and the outflow discharge and total storage in the reach 

at 12 p.m. are listed in columns 3 and 4. The weighted discharge for 

various assumed values of x was computed by using equation (10) 

and is listed in columns 5 to 7.

weighted discharge=^/-f(l x)0 (10)

The weighted discharge-storage plots are shown on figure 39. 

The storage loop is closed and the relation best approximates a

406080 020406080

020406080 0204060 

STORAGE. IN THOUSANDS OF CUBIC FEET PER SECOND PER DAY

FIGURE 39. Relation of storage to weighted discharge for different values of x.
STORAGE AND FLOOD ROUTING 99

straight line when £=0.2. The average slope of the weighted discharge-storage relation is equal to K. On the graph for z=0.2 the 

slope averages about 2.0 days, hence, for the reach as a whole, £=0.2 

and K=2.0 days.

TABLE 4. Computation of weighting factor x for Andalusia-Brooklyn reach,

Conecuh River

Date

(1)

1944 

March 16 ______ . .........

17.....................

18. - ._- 

19. ..-. . .

20.....................

21..- ............. --..

22. 

23. 

24...-.- . ........ .

25 

26 

27 . 

28 

29. 

30 . 

31. -

April I... ... ... ___ ... ... ...

2 

3 

4

5 

6 

7

8 

9 

Total 

inflow 

12 p.m.

(cfs)

(2)

5,870 

9,310 

12,900 

20,500 

21,000

23,400 

32,500 

55,400 

62,700 

52,600

43,200 

25,200 

22,800 

41,200 

50,400

45, 300 

38, 800 

27,000 

16, 200 

12,400

10, 200 

8,080 

6,010 

5,050

Total 

outflow 

12 p.m.

(cfs)

(3)

4,180 

6,970 

7,560 

14, 200 

18,300

18,500 

21,300 

29, 300 

39, 700 

48,700

53,300 

48, 700 

37,100 

35,800 

35,800

35,800 

42,700 

44, 100 

35, 400 

25,200

16,400 

11, 500 

9,380 

7,860

Storage 

volume 

12 p.m. 

(1,000 cfsdays)

(4)

0.1 

2.1 

6.0 

13.0 

17.6

20.3 

27.5 

48.1 

74.0 

87.1

83.8 

66.4 

46.4 

45.3 

54.4

68.4 

71.2 

60.4 

40.8 

25.5

16.3 

11.9 

9.5 

7.2

Weighted discharge

r=0.1

(5)

4,350 

7,200 

8,090 

14,800 

18, 600

19,000 

22,400 

31,900 

42,000 

49,100

52,300 

46, 400 

35,700 

36,300 

37, 300

36,800 

42,300 

42,400 

33,500 

23,900

15,800 

11,200 

9,040 

7,580

a: =0.2 

(6)

4,520 

7,440 

8,630 

15, 500 

18,800

19, 500 

23,500 

34,500 

44,300 

49,500

51,300 

44,000 

34,200 

36,900 

38,700

37,700 

41,900 

40,800 

31, 600 

22,600

15,200 

10,800 

8,710 

7,300

Z=0.3

(7)

4,690 

7,670 

9,160 

16,100 

19,100

20,000 

24,700 

37,100 

46,600 

50,000

50, 300 

41, 700 

32,890 

37,400 

40,200

38,600 

41,500 

39,000 

29, 690 

21, 400

14, 590 

10,500 

8,370 

7,020

The inflow, which enters the reach at four places, necessitates the 

determination of a separate K for each component inflow as is shown

in the following table:

K for each component inflow 

[Equations 8 and 9 used]

Point

Andalusia

Thad_____. ..___.___.____

McKenzie

Local

Total__--__-_______

Miles to 

Brooklyn 

M

20

25

25

12

Total discharge 

during flood 

Q 

(cfs-days)

350, 000

83, 380

148, 640

74, 720

656, 740

QXM

7, 000, 000

2, 084, 500

3, 716, 000

896, 640

13, 697, 140

KC

1. 9

2. 4

2. 4

1. 1

X

0. 2

. 2

. 2

. 2

days per mile.

Kf=MK
100 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

From the value of x and K for each component of the inflow, the 

flood-routing coefficients for each inflow may be computed from 

equation (6).

To test the routing coefficients, the outflow hydrograph is computed 

from the record of inflow discharge. These computations, along 

with the routing coefficients used, are shown in table 5. At the

.Ul 

Li) liCS 

IT 

< 

I 

O 

CO

Q

18 22 26 

MARCH 1944

3 7 

APRIL 1944

11

FIGURE 40. Comparison of computed and actual hydrograph, Conecuh River near Brooklyn, Ala.

beginning of the rise the inflow and outflow are assumed to be equal. 

Then, since /i, /2 and #1 are known, 02 may be computed from equation (7).

The total of the computed outflows is compared with the actual 

outflow on figure 40.

The procedure used in testing the coefficients may be used also in 

routing other floods through the reach in either the upstream or 

downstream direction.
TABLE 5. Example of routing for Andalusia-Brooklyn re

[02=C0 I2+Cl h+C2 01 ] 

[02 =0.20 (4570)+0.52 (2710)+0.28 (942)=2,590 cfs Local inflow to B

Date

1944

n. ...................

is....................

19.. -----------------

20.--.-.....--........

21----.. ...... --------

22. ............. -.-...

23--. _________________

24._._ ________________

25-__. -._____-_._._...

26__._ ________________

27__._ ________________

28. _-___.-_-____..____

29___. ________________

30___. .___.__-_-_-____

31_._________________.

April 1__. _________________

2

3___. ________________

4. . ........... .-.-.--

5..-.------ _________

6___. ________________

7---. ................

8-... ----------------

9_____. ..............

Local inflow to Brooklyn

i =0.20 if =1.1 

C,= .20 Ci = .52 

C2 = .28

Inflow

500 

2,710 

4,570 

5,890 

6,070

4,020 

6,630 

9,330 

5,370 

2,650

2,650 

1,590 

500 

5,860 

4,520

1,840 

1,840 

2,320 

1,500 

1,120

786 

786 

666 

500 

500

Outflow

500 

942 

2,590 

4,280 

5.480

5.490 

4,9.50 

6,700 

7,800 

5,510

3,450 

2,660 

1,670 

1.900 

4,480

3,970 

2.440 

2, 100 

2,100 

1.590

1.180 

896 

793 

668 

547

Thad to Brooklyn

I =0.2 K =2.4 

C0 = .01 C2 = .58 

Ci= .41

Inflow

552 

766 

947 

1,630 

2,520

2,280 

2,560 

7,250 

8,390 

8,390

5,440 

3,190 

2,440 

3,790 

5,860

10, 000 

5,950 

3,300 

2,120 

1,570

1,220 

998 

830 

718 

671

Outflow

552 

554 

645 

778 

1,150

1.720 

1.960 

2,260 

4,370 

6,060

7,010 

6,330 

5,000 

3,940 

3,900

4,760 

6,920 

6,490 

5,140 

3,870

2,900 

2,190 

1,690 

1,330 

1,070

McKenzie to Brook

i =0.2 K =2.4 

C0 = .01 C2 = .58 

Ci= .41

Inflow

698 

1,030 

1,480 

2,210 

3,400

4,550 

5,080 

8,420 

16,400 

16, 900

10,300 

5,680 

4,120 

6,630 

15, 500

18, 800 

11, 500 

6,210 

2,690 

1,930

1,480 

1,170 

925 

800 

742

Outfl

1

1

1

1

1

1

1
102 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES 

RESERVOIR-STORAGE METHOD

If the storage in a pond or reservoir can be related solely to the 

outflow discharge, as schematically shown in figure 31, then the general equation (2) can be rearranged to permit rapid calculation of the 

outflow hydrograph.

At

(11)

The only data needed to use this method are the inflow hydrograph 

and the storage-outflow relation. The latter may be determined by 

combining the stage-outflow relation and the stage-storage relation.

2 S1 

To use the method conveniently, values of -rr+0 are computed and

LA v

plotted against corresponding values of 0. Selected values are 

shown in table 6 to illustrate the computation.

2 S TABLE 6. Selected values of --\-O for &t=15 minutes

Stage (feet)

10.29_-____.---_--__.____._._. 530

10.3______________________________ 535

10.4_.._.__.________....___...___. 680

10.6._----_------_--_________.____ 1, 050

10.8.. --------_-------._______.._. 1, 550

11.0.-----_--------_.___..__._____ 2, 240

11.2_______,__________________ . 3,210

11.4. ________________ 4,580

11.6.--_---.--------.____._._...__ 6, 430

11.8-_.__._.-----__----_........._ 8, 800

12.0.-----_-_--------__-___-_._.__ 11, 900

12.2______________________________ 16, 800

12.4 ____________________________ 23,400

12.6__----------------__-____-____ 31,600

12.8______________________________ 41, 800

13.0____________________ _ 53,600

13.2______________________________ 67, 900

13.4______________________________ 86, 200

13.6. -----_----__________.________ 111, 000

13.8- ____________________________ 143, 000

14.0 ___________________________ 190, 000

14.2____________________ _ 256,000

14.4 ____________________________ 352, 000

14.6______ _____ 493, 000

14.8______________________________ 658,000

15.0 ____________________________ 952,000

Storage 

(cubic feet)

Outflow discharge (cfs)

0

. 01

. 32

2. 55

7. 50

14. 4

21. 6

28. 9

36. 3

44. 3

53. 0

62. 0

71. 5

81. 5

92. 0

103

114

126

138

150

163

177

191

205

219

233

H+0 (cfs)

1. 18

1. 20

1. 83

4. 88

10. 9

19. 4

28. 7

39. 1

50. 6

63. 9

79. 4

99. 3

123

152

185

222

265

318

385

468

585

746

973

1,300

1. 680

2. 350
STORAGE AND FLOOD ROUTING 103

A sample of the routing computations is given in table 7. The 

computations begin at a known value of outflow and inflow. The 

steps then are as follows:

1. Entries in columns 1 and 2 are known. The first entry of outflow in column 

5 is given.

2. Column 3 contains successive sums of values of inflow in column 2 for the 

particular time period under consideration and for the previous period.

2S 3. Table 6 is entered with the known value of O, and the initial value of -rr+0

is selected for column 4 of table 7.

4. From the value in column 4 subtract twice the value in column 5 and enter 

the result in column 6.

5. Add the value in column 6 to the value in column 3 of the succeeding time 

period and enter the result in column 4 for the new time period under 

consideration.

6. The new outflow discharge in column 5 is again obtained from the relation

2S of + 0 with O in table 6.

7. The steps 4, 5, and 6 are then repeated until the entire outflow hydrograph 

is generated.

8. The stage corresponding to this outflow is then selected from table 6 and 

entered in column 7 of table 7.

TABLE 7. Routing through a reservoir

Time

(1)

0:00 

:15 _ ___________

:30 

:45 ... . . ............. _

1:00- ----------------

1:15 -- __- _ --_ 

1:30 ..

1:45... ....... ... .... .._....

2:00 _-_-___ .._-

2:15 

2:30 

2:45 --__----_-__- .-.-

3:00 --- --------_-_. _

3:15. -----_-.. ........... _-_

3:30 

3:45 ...

4:00 

4:15 - _._

4:30 

4:45..---------.--, _

5:00------------------.--..

5:15 

5:30 

5:45-------------.......--..

6:00 

Inflow 

cfs

(2)

0

1.44

4.34

8 90

18.7

38.8

106

216

9Q1

320

325

309

260

911

1QO

165

129

116

106

% A

88.0

80.2

/1+/2

(3)

0

1.44

5.78

13.24

27.6

57.5

144.8

ooo

507

611

645

634

495

446

399

353

310

O74

246

ooo

202.4

184 4

%+°

(4)

1 18

2.62

6.72

n

OQ

22.9

46.0

10Q Q

303

CfM

1,128

1,366

1,661

1,683

1,598

1,324

1,158

QQO

CAO A

461.0

Outflow 

cfs

(5)

0

84

A f\A

Q Oft

17.2

33.5

71.6

161

185

208

214

220

220

219

217

212

206

199

191

181

2?-o

At

(6)

1.18

-1.36

-4.72

-11.5

-21.0

1Q 4

57

242

483

732

1,116

1,223

1,284

1,164

1,050

760

440.4

292.8

163.0

Gage 

height 

(feet)

(7)

10.29

10.47

10.67

10.82

11.08

11.52

12.40

13.35

13. 97

14.31

14.50

14.64

14.73

14.80

14.81

14.81

14.80

14.77

14.70

14.61

14.51

14.40

14.26

14.04

15:00 

15:15....--.....-.--... .

15:30 - _ _

15:45 .

16:00-------.-----. ...

Total-.------...........

0.75

.47

99

0

0

4,504

1.82

.69

.22

0

2.61

2 1 7

1.78

1.40

.83

.30

.11

4,504

1.09

1.18

1.18

10.47

10.43

10.39

10.33

10.29
104 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

The relation + 0 versus 0 may be used in either a graphic or a

tAt

tabular form. If a number of peaks are to be routed through the 

reservoir, the tabular form is more convenient because machine 

computation can be more readily utilized.

In table 7 the computations are shown only for the first 6 hours 

and the final hour. Figure 41 is a plot of the entire inflow and

FIGURE 41. Inflow and outflow hydrographs for reservoir.

outflow hydrographs. The sums of the ordinates of the two hydrographs, at the bottom of table 7, are equal.

The method can be used also to determine the inflow hydrograph 

if the outflow hydrograph is known.

SELECTED REFERENCES

Kinnison, H. B., Conover, L. F., and Bigwood, B. L., 1938, Stages and flood

discharges of the Connecticut River at Hartford, Conn.: U.S. Geol. Survey

Water-Supply Paper 836-A, p. 1-18. 

Langbein, W. B., 1940, Channel storage and unit hydrograph studies: Am.

Geophys. Union Trans., pt. 2, p. 620-627. 

Myer, O. H., 1941, Flood routing on Sacramento River: Am. Geophys. Union

Trans., pt. 1, p. 118-124. 

Rutter, E. J., Graves, Q. B., and Snyder, F. F., 1939, Flood routing: Am. Soc.

Civil Engineers Trans., v. 104, p. 275-294. 

Johnstone, Don, and Cross, W. P., 1949, Elements of applied hydrology: New

York, Ronald Press Co., 276 p. 

U. S. Department of the Army, Corps of Engineers 1953, Routing of flood

through river channels: Eng. Manual, pt. 114, chap. 8. 

U. S. Geological Survey, 1952, Kansas-Missouri floods of July 1951: U.S. Geol.

Survey Water-Supply Paper 1139, 239 p.

U.S. GOVERNMENT PRINTING OFFICE : I960 O S32668