Flow-Duration Curves

Manual of Hydrology: Part 2. Low-Flow Techniques

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1542-A
Flow-Duration Curves

By JAMES K. SEARCY

Manual of Hydrology: Part 2. Low-Flow Techniques

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1542-A

Methods and practices of the Geological 

Survey

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1959
UNITED STATES DEPARTMENT OF THE INTERIOR 

WALTER J. HICKEL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

First printing 1959 

Second printing 1963 

Third printing 1969
CONTENTS

Page

Abstract_______________________________________________________ 1

Introduction._____________________________________________________ 1

Preparation of data._____---____-_----___________--________________ 3

Period used___________________________________________________ 3

Time units__________________________________________________ 4

Class intervals--__---_--____----_-_________-_-________________ 6

Compiling flow-duration data. __________________________________ 7

Use of form 9-217c________________________________________ 7

Use of form 9-217d________________________________________ 9

Presentation of data.______________________________________________ 9

Tabular presentation_________________________________________ 9

Graphical presentation_________________________________________ 10

Type of paper___________________________________________ 11

Discharge units.__________________________________________ 11

Long-term flow-duration curves from short-term records._______________ 12

Selection of the index station_____-_______-_______________-______ 14

Establishing the relation___-_---__-___--_____-________--___-__ 14

Adjusting the short-term record_________________________________ 16

Estimation of the flow-duration curve._______________________________ 17

Hydrologic significance of the flow-duration curve _____________________ 21

Shape of the curve___________________________________________ 22

Mean.__-_-_____---_---_____-_-________--_____________-______ 22

Median ______________________________________________________ 22

Mode_--___.-___-________________________________._ 22

Uses of the flow-duration curve__--------_-__-__---_-_-_______-_-_-__ 23

Studying the effect of geology on low flows. ______________________ 24

Water-power studies___________________________________________ 26

Stream-pollution studies._____------______-___-__-_-_____-_-____ 29

Quality-of-water studies._-__-----_________-_-_-_________-_-____ 29

Variability indexes_____-____-___--_-__-____-_------_-_-__-__--___ 30

Lane's variability index. ____-_---_-__-__-___-_-___-.______-_-__ 30

Reliability of an individual station record___________________________ 32

References cited._--__-_______--_--_____--_--___-_-_-_-____--_----- 33

in
IV CONTENTS

ILLUSTRATIONS

FIGURE 1. Duration curve of daily flow, Bowie Creek near Hattiesburg,

Miss., 1939-48.. _..___.._..._._____.__.____._._._..._. 2

2. Duration curves of daily, monthly, and annual flows, Bowie

Creek near Hattiesburg, Miss., 1939-48-__._-__---_______ 5

3. Curve for changing class intervals on flow-duration dHa from

Bowie Creek near Hattiesburg, Miss., 1939-48. ___________ 6

4. Duration of daily discharge, Bowie Creek near Hattiesburg,

Miss., for year ending Sept. 30, 1943, form 9-217c__-___-_. 8

5. Summary of duration of daily discharge, Bowie Cr?ek near

Hattiesburg, Miss., 1939-48, form 9-217d________________ 10

6. Comparison of flow-duration curves for three nearby stations 

in northeastern Georgia, water year 1952. A, In terms of discharge ; B, In terms of ratio to mean flow._______________ 13

7. Correlation between Kankakee River and Iroquoh River,

based on discharge of equal percent duration____________ 16

8. Duration curves of daily flow, Iroquois River near Chebanse,

111..-._______________________________________________ 18

9. Correlation between Kankakee River and Iroquoi^ River,

based on 10 discharge measurements, 1946-50___-____.__ 20

10. Relation of duration and frequency curve.__________________ 23

11. Geologic map of area in southern Mississippi having approximately uniform climate and altitude.__________________ 25

12. Flow-duration curves for selected Mississippi streams, 1939-48. 27

13. Flow-duration curve applied to hydropower study ___________ 28

TABLES

Page 

TABLE 1. Class limits for discharges on flow-duration table._._.___...__ 7

2. Discharge of equal percent duration on two rivers in Illinois. __ 15

3. Days of concurrent discharge, Kankakee River and Iroquois

River stations, 111., 1946-50_._____________-_-__.________ 19
MANUAL OF HYDROLOGY: PART 2, LOW-FLOW TECHNIQUES

FLOW-DURATION CURVES

By JAMES K. SEARCT

ABSTRACT

The flow-duration curve is a cumulative frequency curve that show the 

percent of time specified discharges were equaled or exceeded during a given 

period. It combines in one curve the flow characteristics of a stream throughout 

the range of discharge, without regard to the sequence of occurence. If the 

period upon which the curve is based represents the long-term flow of a stream, 

the curve may be used to predict the distribution of future flows for v'aterpower, water-supply, and pollution studies.

This report shows that differences in geology affect the low-flow ends of flowduration curves of streams in adjacent basins. Thus, duration curves aro useful in appraising the geologic characteristics of drainage basins.

A. method for adjusting flow-duration curves of short periods to represent 

long-term conditions is presented. The adjustment is made by correlating the 

records of a short-term station with those of a long-term station.

INTRODUCTION

Flow-duration curves have been in general use since about 1915; 

their theory has been discussed by Foster and others. (See Iht of 

references.) This chapter describes the methods used by the Geological Survey to construct flow-duration curves from streamflow data 

and is a revision of instructions originally prepared by W. D. Mitehell 

and W. B. Langbein, for use in the Survey only, and later modified 

by C. H. Hardison.

The flow-duration curve (fig. 1) is a cumulative frequency curve 

that shows the percent of time during which specified discharges were 

equaled or exceeded in a given period. For example, in the period 

1939-48, the daily mean flow of Bowie Creek (fig. 1) was at leas4; 144 

cubic feet per second during 90 percent of the time.

The flow-duration curve is the integral of the frequency diagram. 

Perhaps a simpler concept of the flow-duration curve is that it is 

another means of representing streamflow data combining in one 

curve the flow characteristics of a stream throughout the ranges of
MANUAL OF HYDROLOGY: PART 2, LOW-FLOW TECFTC1QUES

10,000

9000

8000

7000

6000

5000

1000

900

800

700

600

500

400

\

\

Y

Mirrmum observed

0.050.10.2 0.5 12 5 10 20304050607080 90 95 9899 99.5 99.8 99.9 99.99

PERCENT OF TIME INDICATED DISCHARGE WAS EQUALED OR EXCEEDED

FIOURK 1. Duration curve of daily flow, Bowie Creek near Hattiesburg, Miss., 1939-48.

discharge. Although the flow-duration curve does not show the 

chronological sequence of flows, it is useful for many studies.

To prepare a flow-duration curve, the daily, weekly, or monthly 

flows during a given period are arranged according to magnitude, 

and the percent of time during which the flow equaled or exceeded 

the specified values is computed. The curve, drawn to average the 

plotted points of specified discharges versus the percent of time during which they were equaled or exceeded, thus reprsents an average 

for the period considered rather than the distribution of flow within 

a single year.

In a strict sense, the flow-duration curve applies only to the period 

for which data were used to develop the curve or to the period to 

which the curve is adjusted. If streamflow during tH period on 

which the flow-duration curve is based represents the long-term flow 

of the stream, the curve may be considered a probability curve and 

used to estimate the percent of time that a specified discharge will be 

equaled or exceeded in the future.

The flow-duration curve provides a convenient means for studying 

the flow characteristics of streams and for comparing or°, basin with 

another. Various uses of the flow-duration curve are discussed later.
FLOW-DURATION CURVES 3

PREPARATION OF DATA

The two principal methods used to construct flow-duration curves 

are (1) the calendar-year method (Barrows, 1943, p. 137-143 and 

Saville and Watson, 1933, p. 408-411) and (2) the total-period 

method.

In the calendar-year method, the discharges for one year are ranked 

according to magnitude (order number 1, 2, 3 * * *). This process 

is repeated for each year of record. The discharges for each order 

number are averaged. A block diagram is plotted with the abscissa 

in time units and the ordinate in discharge units. If a day is the 

time unit, the first item plotted is the average of the annual maximum 

days for the period of record. A percent-of-time scale can be constructed for the abscissa, if desired. The calendar-year method gives 

lower values for the high discharges and higher values for th°» low 

discharges than the more accurate total-period method.

In the total-period method, all discharges are placed in classes according to their magnitude. The totals are cumulated, beginning 

with the highest class, and the percentage of the totaled time is computed for each class. The data are then plotted with the discharge 

as the ordinate and the time in percent of total period as the abrcissa.

The Geological Survey uses the total-period method and the discussion which follows is restricted to this method.

PERIOD USED

All complete years of record can be used to prepare a flow-duration 

curve; records for partial years should be excluded. The years for 

which records are complete need not be consecutive, but the records 

used should be for years in which physical conditions in the basin, 

such as artificial storage, diversions, or other manmade influences, 

were essentially the same. The double-mass curve, which is discussed 

in another chapter of this manual, is useful for checking the consistency of records to be used for constructing flow-duration curves.

The data for the flow-duration curve are usually prepared on a 

water-year basis, the same basis as that on which the Geological Survey publishes streamflow records. The use of the water year (which 

ends September 30) in analyses of the flow-duration curve usually 

divides a low-flow period. This division is of no consequence for long 

records, but a flow-duration curve based on the water year of lowest annual flow might not represent a combination of flows as low as that 

which actually occurred in a 12-month period. When the flow-duration curve is used to study the variations in streamflow from year to 

year, yearly curves are prepared for climatic years beginning April 1.

In some western streams, snowmelt during a few months provides 

practically all the flow available for use during the year. For such
4 MANUAL OF HYDROLOGY: PART 2, LOW-FLOW TECFOTQUES

streams, a "flow year" of a few months' length might be a more practical basis for flow-duration curves than a 12-month year.

Flow-duration curves for selected portions of a long record are 

used in adjusting the flow-duration curves for short records to the 

period of the long record. Flow-duration curves for th-*, Geological 

Survey standard 25-year period, water years 1921-45, are useful for 

comparing streamflow in different parts of the United States; the 

period eventually will be changed to the 30 years 1931-60, to conform 

with the practice agreed upon by the World Meteorological 

Organization.

When streamflow has been regulated by storage or diversions, or 

when the regimen varied during the period of record, a cnrve for the 

total period has little meaning. The period used for th<»- flow-duration curve depends on the purpose of the curve. If the curve is to be 

used in a hydrologic analysis of the characteristics of natural flow, 

only the unregulated period of record should be used. If the curve 

is to be used as an indication of the flow that may be expected in the 

future with a continuation of present conditions, only tt e record obtained during a period when facilities for regulation and pattern of 

regulation have been constant should be used. A duration curve 

based on the record of a short period of regulated flow and adjusted 

to a longer period by correlation with an unregulated stream (explained in a later section) shows the flow to be expected under the 

long-period hydrologic conditions, but in the pattern of regulation 

that existed during the short period. With the adjusted curve as a 

base, one can make allowances for expected additional regulation or 

for changes in the pattern of regulation.

TIME UNITS

The choice of a time unit, such as the day, the week, or the month, 

is largely a matter of weighing the accuracy of the flow-duration 

curve against the work involved in its preparation. Cne primary 

use of such duration curves is to show the characteristics of flow. 

The details of the variations in flows are obscured if the time unit is 

long. For most streams, the monthly discharges are ur satisfactory 

for showing the variation in flow, and duration curve^1 of annual 

mean discharges would have but little use because ther range in 

variation is comparatively small and because only a fev7 values are 

available for defining the curve.

A study in North Carolina (Foster, 1934, p. 1236) showed differences as great as 35 percent between a duration curv^ based on 

monthly mean discharges and one based on daily mean discharges. 

Weekly mean discharges have been used (Saville and Watson, 1933,
FLOW-DURATION CURVES 5

p. 407) for certain relatively stable streams. Weekly means, however, are not generally available in published records, and tl ^ advantage, if any, of using the weekly mean is offset by the additional 

computations required.

The effect of varying the time unit (fig. 2) is not the same for 

all streams. Where the flow from day to day is almost uniform, as 

in the St. Lawrence River, the daily and weekly duration curves 

would be nearly identical, and the monthly duration curve would 

not differ greatly from the daily curve. On the other hand, if the 

stream is "flashy," with sudden floods lasting only a few hours or 

days, the daily and weekly curves will differ appreciably, and the 

monthly curve will differ considerably from the daily curve. Daily 

discharges have been used almost exclusively in recent studies

After a smooth curve is drawn, the fact is often overlooked that 

the mean discharge for the selected time unit was used as th°. discharge that was grouped into the class intervals. Thus, if the nonth 

were chosen as the time unit, the correct statement would be that "the 

monthly mean discharge was at least 150 cubic feet per second for 

90 percent of the months". The daily mean flows during part of the 

90 percent of the time probably were less than 150 cubic feet per

PERCENT OF TIME INDICATED DISCHARGE WAS EQUALED OR EXCEEDED

FIGURE 2. Duration curves of dally, monthly, and annual flows, Bowle Creek near Hattiesburg, Miss., 1939-18.

353-858 O - i
6 MANUAL OF HYDROLOGY: PART 2, LOW-FLOW TECHNIQUES

second. This distinction is of minor importance when the day is 

chosen as the time unit, unless the stream has a large diurnal 

fluctuation.

CLASS INTERVALS

The class intervals should provide from 20 to 30 we1 !-distributed 

points on the curve. The extreme points should be so selected as 

barely to include the extremes of daily discharge for the period of 

record. Table 1 shows recommended class intervals in cubic feet per 

second for ranges in discharge from 1 to 5 log cycles. Form 9-2l7c 

(fig. 4) gives an example of how table 1 is used for a stream whose 

range in discharge is almost within two log cycles. Class intervals 

recommended for 2 log cycles were chosen in preference to those 

recommended for 3 log cycles because the 2-cycle intervals give more 

points.

When flow-duration data are computed using different sets of class 

intervals, the data can be combined by plotting a curve of the cumulated days against the discharge for the entire period that was computed by one set of class intervals. The cumulated days for the other 

set of class intervals is picked from the curve. Semilogarithmic 

paper with a finely divided arithmetic scale is recommended for this 

purpose. (See fig. 3.) Discharges are plotted on the logarithmic

300 400 500 600 800 1000 2000 3000 4000 6000 8000

DISCHARGE, IN CUBIC FEET PER SECOND

FIGURE 3. Curve for changing class intervals on flow-duration data f~om Bowie Creek 

near Hattiesburg, Miss., 1939-48.
FLOW-DURATION CURVES

scale, and cumulated days are plotted on the arithmetic scale. The 

points are connected by straight lines. A smooth curve based on the 

points would be as likely to harm the results as to improve them. 

Data plotted in figure 3 are from figure 5.

TABLE 1. Class limits for discharges on flow-duration table

Range in daily discharge

1 log cycle

10

11

12

13

14

15

16

18

20

22

24

26

28

30

33

36

40

45

50

55

60

65

70

75

80

90

100

2 log cycles

10

12

14

17

20

25

30

35

40

45

50

60

70

80

100

120

etc.

3 log cycles

10

15

20

25

30

40

50

60

80

100

150

etc.

4 log cycles

10

15

20

30

40

50

70

100

150

etc.

5 log cycles

10

15

20

30

50

70

100

150

etc.

NOTE. Table shows sequence of numbers for five ranges in 

discharge. Locate decimal point and starting discharge to 

suit conditions. In general, use cycle closest to observed 

ranee. Where considerable data have been tabulated at a 

station by using other rules for class intervals, additional data 

would normally be tabulated by using the class intervals 

previously used.

COMPILING FLOW-DURATION DATA

A digital computer or a similar machine can be used to compile 

the flow-duration data. The machine furnishes the same kinds of 

data as are shown in figure 5, including the data in columns headed 

"total," "total days," and "percent of time." If a machine is not 

available, the method shown in figures 4 and 5 is recommended for 

compiling the flow-duration data.

USE OF FORM 9-217C

Form 9-2l7c, shown in figure 4, is designed for arranging the 

daily flows during 1 water year, in classes according to magnitude. 

The lower limit of each class interval (from table 1) is tabulated in 

the discharge column.

For the daily 'discharge of each day of gaging-station record, a 

tally mark is made in the appropriate monthly column on th°. line
8 MANUAL OF HYDROLOGY: PART 2, LOW-FLOW TECTHSITQUES

representing the highest flow that the flow of the particular day 

equals or exceeds. The tally marks for each month are counted to 

ensure that one tally mark was made for each day of the month. The 

tally marks in each class are counted and the number is entered in

UNITED STATES DEPARTMENT OF THE INTERIOR

GEOLOGICAL. SURVEY 

WAT*R RESOURCES OIVIWON

DwaUoi ubie of -

for tho rear endtag Sept. SO, . Drainage aroa, .......3-Q:t............... i<ure 

DISCHARGE

Cb

100

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170

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It-jOOO

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5,000

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SHOWN IN FIRST TWO COLUMNS AND LESS THAN THAT SHOWN ON NEXT LINE

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4. Duration of dally discharge, Bowie Creek near Hattlesburg. Miss., for year 

ending Sept. 30, 1943, form 9-217c.
FLOW-DURATION CURVES 9

the column for the year. The total of the yearly column should equal 

the number of days in the year.

An alternate method of compiling form 9-2l7c is to go through a 

month, counting the days in each class and showing the count by a 

numeral instead of several tally marks. A check mark is placed by 

each day when it is counted; this reduces the number of items to be 

scanned for successive class intervals.

USE OF FORM »-217D

The yearly totals, by classes, are transferred from the yearly forms 

(fig. 4) to the summary form (fig. 5), by using a column for each 

year. Column headings, "total," "total days," and "percent of time," 

are added at the end of the yearly totals, and the period to which 

the totals apply is indicated. Class totals are cumulated from the 

bottom upward, and the percent of the total number of days is computed for each class summation and entered in the percent-of-time 

column. The columns "total days" and "percent of time" represent 

the time that the discharge shown in column 2 was equaled or exceeded. Column 1 is reserved for listing the equivalent of column 2 

in other units, such as cubic feet per second per square mile, million 

gallons per day, or ratio to mean discharge.

When additional years of record become available, the total days 

in each class interval for those years can be added to the existing 

summary table to obtain a total for the entire period.

PRESENTATION OF DATA

The results of a flow-duration study may be presented in tabular 

or in graphical form. The tabular presentation has the advartage 

of being more compact than the graphical presentation, but it is somewhat less clear.

In either form of presentation, the title should show the time interval (such as "daily flow"), the name of the gaging station, and the 

period represented by the data.

TABULAR PRESENTATION

Tabular arrangements of flow-duration data may show eithe~ the 

discharge for a given percent of duration or the percent of tim? for 

a given discharge. An arrangement showing the discharge corresponding to given percents of duration is better suited to hydrologic 

comparisons than is an arrangement showing the percent of time 

for a given discharge.

Some compilations of flow-duration data show the yearly class 

totals as shown in figure 5; others show skeleton tables with only 

enough points given to define a duration curve.
10 MANUAL OF HYDROLOGY: PART 2, LOW-FLOW TEGE"NTQTJES 

GRAPHICAL PRESENTATION

The curve in figure 1 shows the duration of daily flow of Bowie 

Creek near Hattiesburg for the water years 1939-48. The abscissa 

should be labeled as shown in this figure, rather than just "percent

UNITED STATES DEPARTMENT OF THE INTERIOR

GEOLOGICAL SURVEY 

WATER RESOURCES DIVISION

Dontfon table amoury of .

for tho years. Drainage .gqg»ro mfles

DISCHARGE

 

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120

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170

200

250

300

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5,000

6,000

7,000

8^000

10,000

12,000

lit-, 000

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NUMBER Or DAYS WHEN DISCHARGE WAS EQUAL TO OR GREATER THAN THAT 

SHOWN IN riRST TWO COLUMNS AND LESS THAN THAT SHOWN ON NEXT LINE

1939

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FIOUBK 5. Summary of duration of daily discharge, Bowie Creek near Hattiesburg, Miss.,

1939-48, form 9-217d.
FLOW-DURATION CURVES 11

of time." When the curve represents basic data for the period of 

record, as in this figure, plotted points are shown and a smooth curve 

is drawn by eye to fit the data. When the duration data have been 

adjusted to represent a longer period, a smooth curve alone i? sufficient. If the curve is merely one step in a hydrologic analysis, 

straight lines between successive points are preferable because they 

prevent introduction of differences due to personal judgment.

TYPE OF PAPER

Flow-duration curves are plotted on rectangular-coordinate paper, 

logarithmic paper, arithmetic-probability paper, or logarithmicprobability paper. For general use the logarithmic-probability 

paper is recommended.

The advantage of using a logarithmic scale for the discharge 

(ordinate) is apparent when it is noted that normally 3 or 4 log cycles 

are required for the range in discharge. An arithmetic scale that 

would accommodate the range in discharge of most streams would 

be undesirably small for all except the highest discharges.

An arithmetic scale for the percent of time is considered easier to 

use, but such a scale provides poor definition at the extremities, 

where the slope of the flow-duration curve changes rapidly. Because of the difficulty of determining the area under a curve plotted 

on a logarithmic-probability paper, the rectangular coordinates are 

often used in hydroelectric-power studies and similar studies.

A logarithmic scale for the percent of time expands one end of the 

curve but undesirably condenses the other end.

The probability scale expands both ends of the flow-duration curve. 

Data that are normally distributed plot as a straight line on probability paper. As the logarithms of discharge are more normally 

distributed than the discharge itself, the logarithmic-probability 

paper tends to straighten out the flow-duration curve.

DISCHARGE UNITS

For tabulating flow-duration data, to express discharge in cubic 

feet per second is most convenient because the daily discharges are 

published in those units. But the discharge data can be converted 

to any other desired unit for plotting the flow-duration curve. For 

example, if the duration curve is to be used for a study of flow at 

the gaging-station site, the ordinate can be in cubic feet per second, 

in million gallons per day, or in some function of discharge such as 

kilowatts corresponding to the net head and plant efficiency available at the site. If the duration curve is to be used for comparing 

flow characteristics among streams, to express discharge in cubic
12 MANUAL OF HYDROLOGY: PART 2, LOW-FLOW TECHNIQUES

feet per second per square mile or in ratio to average flov7 is suitable 

for the ordinate scale.

To express discharge either in cubic feet per second per square 

mile or in ratio to mean flow eliminates the effect of size of drainage area; but to express discharge in ratio to mean flov7 eliminates 

also the effect of differences in mean annual runoff per square mile. 

For many streams, comparisons based on cubic feet per second per 

square mile are almost identical with comparisons based on ratio to 

mean flow. However, among stream basins in which precipitation 

differs greatly, flow-duration curves based on ratio to mean flow are 

much closer together than those based on cubic feet per second per 

square mile. For example, in figure 6, flow-duration curves for 

three nearby stations in northeastern Georgia are shown for the 

water year 1952. Figure QA shows the flow-duration curves in cubic 

feet per second per square mile, and figure QB shows the same flowduration curves in ratio to mean flow. No attempt was made to 

determine the average precipitation in each drainage ba^in, but the 

precipitation total for the water year 1952 is given for one 

precipitation station in each drainage basin.

Discharge per square mile and ratio to mean annual flow are useful 

conversions for hydrologic comparisons, but care should H taken not 

to imply that the stream with the highest yield per square mile is 

the best source of supply regardless of the size of its drainage area, 

or that the flow varies uniformly over the drainage basin.

LONG-TERM FLOW-DURATION CURVES FROM SHORTTERM RECORDS

Very seldom do all the gaging-station records in a given area cover 

concurrent periods. If records are to be compared with each other, 

they must represent, or be adjusted to, concurrent periods, in order 

that differences between the records will be due to differences in climatic or drainage-basin characteristics and not to the fact that the 

records cover different periods of time. Furthermore, flow-duration 

curves based on short records are unreliable for predicting the future 

pattern of flow, but they can be made reliable by adjusting them to 

represent longer periods.

Several methods (Mitchell, 1950, p. 12-18) are available for adjusting short-term records to long-term records. In the method now 

used by the Geological Survey (sometimes called the index-station 

method), a relation is established between two stations for the short 

period of concurrent record by plotting a graph of the discharges for 

given duration points at one station against the corresponding discharges at the other station. The graph for the short period is also
FLOW-DURATION CURVES 13

A. (Duration curves in cubic feet per 

second per square mile

^s

q 

\

o\ -\

\

B

 * 

>

fe\t

V

D

\

\

1

*.

p

urol

fr

I

ion (

*

urvi

X \N

X

* in

\

ratk

^

> tc

Oi

^ .

sL

-^

> n

EXPLANATION

-T^-Q-Chattooga River near Cloyton, Ga; 

drainage area, 203 sq. miles 

Precipitation ot Clayton, Go.; 70.86 in.

-A <y Panther Creek neor Toccoa, Go.; 

drainage area, 31.4 sq. miles 

Precipitation at Toccoa, Go., 61. 79 in.

 Q South Beaverdom Creek at Dewy Rose, 

Go.; drainage area, 36.6 sq. miles 

Precipitation at Cartton Bridge, Ga; 

43.61 inches

^

two

4

>

n ft

1

v?^

ow

feu^ 

^bS0>ss

3~" '

 ^

- o

^

^ *N «>* -^

~ 

' D.I 0.2 0.5 1 2 5 10 2Q 30 40 50 60 70 80 90 95 98 99 99.5 99.9 99.99

PERCENT OF TIME INDICATED DISCHARGE WAS EQUALED OR EXCEEDED

FIGURB 6. Comparison of flow-duration curves for three nearby stations in north-eastern

Georgia, water year 1952.

353-858 O - 69 - 3
14 MANUAL OF HYDROLOGY: PART 2, LOW-FLOW TECHNIQUES

assumed to represent the relation between the stationr for a long 

period. If the assumption is true, the flow available 50 percent of 

the time at the long-term station can be used to enter the curve of 

relation (based on the short period of record), in ordQ-r to obtain 

the adjusted (to long term) flow available 50 percent of the time at 

the short-term station. Adjusted flows for other percents of time 

at the short-term station can be obtained in the same manner.

SELECTION OF THE INDEX STATION

In order for the index-station method to be valid, both the index 

station and the short-term stations must be influenced by similar climatic occurrences. Although the drainage basins of these stations 

need not have concurrent rains, each basin should have tbs same likelihood of receiving rain. Thus, a station in the rain shadow of a 

mountain could hardly be used to adjust the records of a stream on 

the opposite side of the mountain.

The index station used to adjust the record for a short-term station 

should be carefully selected. A gaging station whose record has previously been used in a regional low-flow analysis usually serves as 

a good index station. A station on the same stream rs the shortterm station is usually a better index station than one or a stream in 

an adjacent basin. The index station and the short-term station must 

have a sufficient period of concurrent records to establish a usable 

relation. Distance from other stations is a factor in selecting the 

index station. Eecords from gaging stations nearby, other factors 

being equal, provide better relations than records from remote stations, but usable relations have been established between stations as 

far apart as 50 miles, when long periods of concurrent record were 

available on which to establish the relation.

ESTABLISHING THE RELATION

A flow-duration curve is prepared for the short-term record (fig. 

8), and a flow-duration curve for the corresponding period is prepared for the long-term record. The points are connected by straight 

lines rather than by a smooth curve, in order not to introduce personal interpretation of the data.

The discharges for about 15 percent-duration points, ranging from 

0.5 to 99.5 percent (table 2), at the short-term station are plotted 

on logarithmic paper against the discharges for the sr.me percentduration points at the long-term station. A percent-duration point 

is the point on the flow-duration curve that corresponds to a specified 

percent of time.
FLOW-DURATION CURVES 15

A straight line or lines (connected by a transition curve at sharp 

breaks) are drawn through the plotted points, with the following 

suggestions serving as guides :

1. Straight lines or a smooth curve should be drawn in preference 

to a wavering line connecting all points.

2. On logarithmic paper, the upper end of the curve of Nation 

is often a 45-degree line and is usually on the drainago-area 

ratio (equal-yield) line or parallel to it. If the two streams 

differ in their high-flow characteristics, the upper end of the 

curve of relation will depart from a 45-degree line.

3. If the geologic characteristics of the basins differ, the lower 

points will define a relation other than a 45-degree

TABLE 2. Discharge of equal percent duration on two rivers in Illinois 

Discharge, in cubic feet per second

Percent duration

99.5.--.- -- 

QO

98 

95. -- __ . - 

90 

80 

70 

60 

60.. 

W.. ....................................

W... .......... .........................

Vi... ...................................

\Q........... ...........................

6...... .................................

2_ 

1..... ..................................

Q.5.. ...................................

Kankakee 

River at 

Motnence 

1946-50

542

CEQ

566

618

700

882

1,080

1,280

1,580

1,930

2.500

3,440

4 (Ufl

5,200

6.380

7,180

8,000

Iroquois River 

near Chebanse 

1946-50

46

49

53

67

102

188

334

525

750

1,150

1,990

3,220

5.300

7,160

9,700

12,300

14,600

Kankakee 

River at 

Momence 

1924-50

432

453

508

578

658

822

970

1,130

1,370

1,680

2,100

2,770

3,940

4,800

5,780

6,600

7,210

Iroquo'" River 

near C'lebanse, 

adjusted to 

192 < -50

23

26.5

37.5

54

80

150

240

380

580

880

1,320

2,130

4,000

5,780

7,800

10,300

12,100

4. Little weight should be given to the points at the extreme upper 

or lower end when the line is drawn. Comparison of curves 

of relation based on successive 5-year periods shows that an 

apparent sharp break in the relation near the extremes is 

likely to be balanced by a break in the opposite direction 

during the next 5-year period.

To demonstrate the method, records from two streain-gagingr stations in northeastern Illinois are used. The Iroquois River near 

Chebanse (record for water years 1924-50) is used as the short-term 

station with the assumption that its record covers only the water 

years 1946-50. The record for the index station Kankakee Ri^er at 

Momence (record for water years 1916-50) is assumed to cover only 

the period 1924-50, so that the extension of the short-term station 

record may be compared with the actual record.
16 MANUAL OF HYDROLOGY: PART 2, LOW-FLOW TECHNIQUES

The flow-duration curve for 1946-50 at Kankakee River at Momence 

is not shown, but table 2 gives the discharge for equal percent duration picked from the curve.

The curve of relation in figure 7 is plotted from the data in the 

second and third columns of table 2.

ADJUSTING THE SHORT-TERM RECORD

The discharges for various percent-duration points at the long-term 

station (col. 4, table 2) are used as the argument in the curve of relaS RIVER NEA CHEBANSE, YEARS I946~I950 ILL., WATER 

DISCHARGE,IN CUBIC FEETPER SECOND

TOO o

* 

0

o

20 

in

/

/

/

E

/

7

iqi

j'

<s

f

10

/

/

'

/

y

'

e

1

9

"d-^/ 

/ }

/

r

/

O

D percent.

P~-95 percent

-«3S P<

/f

7

^

/

^

/

'

/

!/

j^20 percent r~

50 percent

100 200 500 1000 2000 5000 10,000

DISCHARGE; IN CUBIC FEET PER SECOND 

KANKAKEE RIVER AT MOMENCE, ILL., WATER YEARS 1946-1950

FIGUBB 7. Correlation between Kankakee River and Iroquola River, basec1 on discharge of

equal percent duration.
FLOW-DURATION CURVES 17

tion (fig. 7) to obtain the adjusted discharge at the short-term station 

for the particular percent-duration point. For example, the discharge 

of the Kankakee River (long period, 1924-50) at the 50-percent-duration point is 1,370 cf s. Entering 1,370 cf s, on figure 7, we find that the 

discharge (adjusted to the long period, 1924-50) of Iroquois River 

at the 50-percent duration point is 580 cfs. The discharge for other 

percent-duration points is obtained in the same way, and the adjusted 

values are plotted to define the adjusted curve (fig. 8).

Figure 8 shows the actual flow-duration curve for the short period 

(1946-50) and the flow-duration curve adjusted to the long period 

(1924-50). For comparison, the curve for the long period (1934-50) 

based on actual record is shown also. The flow-duration curve adjusted to the long period compares favorably with the actual longperiod record.

ESTIMATION OF THE FLOW-DURATION CURVE

Frequently there is need for flow-duration data on streams for 

which there are no gaging-station records. When the low-flow end 

of the duration curve is of prime interest, estimates based on runoff 

per square mile of nearby gaged areas are seldom reliable unless it is 

known that the ground-water geology of the two areas is the. same.

The effect of basin geology on streamflow can be evaluated by several discharge measurements made during periods of base flow. The 

discharge measurements at the base-flow observation point should 

preferably be made over a period of several years, but, regardless of 

when they are made, estimates of low flow based on base-flow measurements are much more reliable than those based on the runoff per 

square mile of nearby gaged areas.

The use of base-flow measurements for estimating a flow-duration 

curve is explained by an example for which the Kankakee River and 

Iroquois River stations are used. In this example we assume that 

the Iroquois River near Chebanse is ungaged, but that 10 base-flow 

measurements have been made. The Kankakee River at Momence 

is used as the index station.

Concurrent daily discharges for two days each year during the 

period 1946-50 are used as discharge measurements. (See table 3.) 

The selected days are days of base-flow periods when little change in 

discharge occurred at either station. The same base-flow criteria 

would be followed when discharge measurements are made on ungaged areas, except that the rate of change in discharge could be 

observed only at the gaging station. Local inquiry will usually 

reveal the time of the last rain on the ungaged area.
18 MANUAL OF HYDROLOGY: PART 2, LOW-FLOW TECHNIQUES

50,000 

40,000 

30,000

20,000

8000 

7000 

6000 

5000 

4000

3000

o

S 2000

S 

at

0.

tuj . . -^ 

u. 1000

y 900

p 800 

° 700 

~ 600 

g 500

§ 40° 

5 

300

200

100 

90 

80 

70 

60 

50 

40

30 

20

c 

«

x

s.

N

X

^

X.

\

N

N

V\ V

\

1

^

>\**

\ \

N

\

[$

\N\

^

A

^

,

\

,\

^

A

v

EXPLANATION

-o o Actuol flow, 1946 -50 

 « -r 1946-50 curve extended to,l924-50 

 Actuol flow, 1924- 50

 * *- Ettimofed curve, 1924- 5O based on 

10 dischorge measurements made duri 

1946-50

\

i V,

a;

V

\

^

^

+ V

V.

C\

y

^

\

\ \

\

\

; N

V

\1

=>-.

k. 

\>

\

~^ ^

\

X

~~,

V

19

\

1 | 2 S S 8 8 88 8888888 88 8 8 S g $ g

I-H ci co ^ in u5 f*^ oo cf> CTI (}{ 5> 5* 5> 5> c

PERCENT OF TIME INDICATED DISCHARGE WAS EQUALED OR EXCEEDED

FIGURE 8. Duration curves of daily flow, Iroquois River near Che^anse, 111.
FLOW-DURATION CURVES 19

TABLE 3. Days of concurrent discharge, Kankakee River and Iroquois River

Stations, III., 1946-^50

Date

Aug. 31,1946

Sept. 20, 1946

July 16,1947

Aug. 24, 1947

Aug. 26,1948

Sept. 28, 1948

Sept. 1,1949

Sept. 25, 1949

Jan. 1, 1950

Sept. 1,1950

Discharge, in cubic feet

per second

Kankakee

River at

Momence, 111.

600

525

1,280

780

682

562

712

630

4,460

1,040

Iroquois

River at

Chebanse, 111.

56

48

306

93

107

61

107

63

4,880

168

The base-flow measurements (days of concurrent discharge) are 

plotted on figure 9 to establish a relation between the ungag°d site 

(Iroquois River near Chebanse) and the index station (Kankakee 

River at Momence). It is obvious that the measurements group near 

the lower end of the curve of relation and that lines of variour slopes 

could be drawn to average the group of measurements. Thus, the 

measurements serve to fix the location of a line of relation but not 

its slope.

The characteristics of relations between gaging-station records furnish a basis for drawing the curve of relation when only the position 

of the curve at its lower end is known. An equal-yield line (drainage-area ratio), drawn as a 45-degree line on logarithmic paper 

through the plot of the drainage areas, serves as a convenient guide 

for this purpose. Although the curve of relation diverges from the 

equal-yield line at low flow, for many streams it tends to become parallel to the equal-yield line at higher flows, because, at high flow, storm 

runoff predominates and the base flow, affected by geology, is a negligible part of the total flow. However, when one basin contains 

enough storage, either on the surface or in the ground, to distribute 

the effect of storm precipitation over several time units, and tr-^ other 

basin contains comparatively little storage, the upper end of the line 

of relation deviates from the equal-yield line toward the station with 

less storage. (See fig. 7.)

For many streams in the eastern part of the country, a line through 

the base-flow measurements intersects the equal-yield line at a discharge about l!/£> times that of the mean discharge. The actual point 

of intersection is fairly stable for a given area and should preferably 

be located by correlating several gaging-station records. The point 

of intersection of the equal-yield line and low-flow line is ured as a 

pivot point to draw a line through the average of the low-flor^ meas-
20 MANUAL OF HYDROLOGY: PART 2, LOW-FLOW TECHNIQUES

urements. In figure 9, a discharge iy2 times the average discharge 

for the 1924-50 period at Momence was used as the pivot point. The 

relation based on low-flow measurements can be used to estimate 

a flow-duration curve in the same way that figure 7 was us?.d to extend 

a flow-duration curve.

10,000

5000

2000

1000

500

200

50

20

10

Eqiwl yield

I.SQr

/

L

100 200 500 1000 2000 5C^X) 10,000

DISCHARGE, IN CUBIC FEET PER SECOND 

KANKAKEE RIVER AT MOMENCE, ILL., DRAINAGE AREA 2,340 SQUARE MILES

FIGURE 9. Correlation between Kankakee River and Iroquols River, based on 10 dlcharge

measurements, 1946-50.
FLOW-DURATION CURVES 21

In the example, index-station discharges for the 17 percent-duration 

points in column 4 of table 2 are used as the argument in figure 9 to 

obtain discharges for the corresponding percent-duration points at 

the ungaged site. The data thus obtained are plotted on figure 8 so 

that the estimated flow-duration curve might be compared with the 

duration curve based on actual record. Although in the example the 

duration curve is estimated throughout its range, reports pubUshed 

by the Geological Survey usually show only the lower half of the 

curve, because the relation at higher flows may not follow the equalyield line. It will be seen in figure 7 that the upper line of relation 

between the stations used in the example actually crosses the equalyield line. The result of assuming an equal-yield relation is se°.n on 

figure 8 by the departure of the estimated curve below 10 percent of 

the time.

The line of relation should not be extended to flows much lower 

than those measured at the ungaged site unless one has a thorough 

knowledge of the geology of the two drainage basins. Some lines of 

relation have a second break at an extremely low flow, particularly 

when one stream goes dry and the other is perennial.

HYDROLOGIC SIGNIFICANCE OF THE FLOW-DUPATION CURVE

The water measured at a gaging station is the surface outflow of 

the drainage basin above a specified point on the stream. Thus, the 

streamflow record integrates the effects of climate, topography, and 

geology, and gives a distribution of runoff both in time and in iragnitude. When the flows are arranged according to frequency of occurrence and a flow-duration curve is plotted, the resulting curve shows 

the integrated effect of the various factors that affect runoff.

It is important to keep in mind that the flow-duration curve is an 

average curve for the period upon which it is based. To say that a 

flow-duration curve based on a 15-year record represents the distribution of the yearly flow is incorrect. The flow lower than that which 

was equaled or exceeded 96.7 percent of the time might have occurred 

during one 6-month period of a 15-year drought. Such a flow would 

not be expected, on the average, 3.3 percent of the time each year, but 

would be expected, on the average, 50 percent of the time during one 

year of each 15-year period. The flow-duration curve for the period 

is often supplemented by flow-duration curves for the year of lowest 

runoff and the year of maximum runoff.
22 MANUAL OF HYDROLOGY: PART 2, LOW-FLOW TECHNIQUES

SHAPE OF CURVE

As the shape of the flow-duration curve is determined by the hydrologic and geologic characteristics of the drainage area, the curve may 

be used to study the characteristics of a drainage basin or to compare 

the characteristics of one basin with those of another. A curve with 

a steep slope throughout denotes a highly variable stream whose 

flow is largely from direct runoff, whereas a curve with a flat slope 

reveals the presence of surface- or ground-water storage, which tends 

to equalize the flow. The slope of the lower end of the duration 

curve shows the characteristics of the perennial storage in the drainage basin; a flat slope at the lower end indicates a large amount of 

storage; and a steep slope indicates a negligible amourt. Streams 

whose high flows come largely from snowmelt tend to have a flat 

slope at the upper end. The same is true for stream? with large 

flood-plain storage or those that drain swamp areas.

The statistics of the duration curve are discussed in papers by 

Foster (1924, 1934), Slade (1936), Beard (1943), and others. The 

work of Elderton (1953) also contains valuable information on the 

statistical and mathematical basis of the duration curve. Such a 

discussion is beyond the scope of this chapter.

MEAN

The area under the flow-duration curve is a measure of the discharge available 100 percent of the time. Dividing the area by 100 

(base of the curve 100 percent of the time) gives the average ordinate which, multiplied by the scale factor (see sectior on waterpower studies), is the mean discharge. Similarly, the area under 

a portion of the curve, divided by the percent of time of that portion, 

represents the mean flow during the particular percent of time. This 

property of the curve has important applications in seme studies, 

but finding the scale factor of a flow-duration curve plotted on logarithmic-probability paper is somewhat complicated.

MEDIAN

The median flow is the curve value at 50 percent of the time.

MODE

The point of inflection of a flow-duration curve plotted on rectangular-coordinate paper occurs at the modal flow. This inflection point 

can be detected on figure 10, where the frequency curve of a hypothetical stream has been plotted with its flow-duration curve. However, the inflection point of a flow-duration curve is usually not
FLOW-DURATION CURVES 23

definite enough for determining accurately the modal flow from the 

flow-duration curve and, when logarithmic and probability scales 

are used for plotting, the mode does not fall at the apparent inflection point.

The modal value has been suggested (Meyer, 1928, p. 119) as an 

appropriate "normal" flow, as it is the flow that occurs most often.

USES OF THE FLOW-DURATION CURVE

As early as 1908, Mead (1908, p. 184^-189) presented flow-duration 

curves in cubic feet per second per square mile for six Michigan 

rivers to show their similarity, and, at the same time, to point out the 

error that might result from estimating the flow of an ungaged stream. 

Mead states, "This form of diagram represents the best basis for the 

comparative study of streamflow for power purposes where storage 

is not considered, and where the continuous power of the passing 

stream is to be investigated."

FLOW INCREASING * 

FIGURE 10. Relation of duration and frequency curve.
24 MANUAL OF HYDROLOGY: PART 2, LOW-FLOW TECHNIQUES

It was not until about 1915, however, that flow-duration curves came 

into general use in the United States, and about 1920 the Geological 

Survey adopted the flow-duration curve as a basis for defining rates 

of flow to be used in computing waterpower statistics.

In 1930 the International Advisory Committee on Eating of Kivers 

adopted certain percent-duration points for quoting power statistics. 

W. G. Hoyt (1934, p. 1240-1243) stated that "It is doubtful whether 

agreement could have been reached on the basis of any defined rates 

of flow other than those obtained through the use of the duration 

curve."

One of the earliest uses of the flow-duration curve was for waterpower studies. The subject is discussed by many writers, among 

whom are Barrows (1943, p. 137-192), Hickox and Wessenauer 

(1933), and Foster (1934).

Beard (1943) and Pettis (1934, p. 1237-1240) have applied the upper end of the flow-duration curve to flood studies.

More recently the flow-duration curve has been used for preliminary investigations of water supply, location of industrial plants, 

pollution studies, and many other purposes. Simple examples of 

a few uses of the flow-duration curve are given in the following 

sections.

STUDYING THE EFFECT OF GEOLOGY ON LOW FLOWS

The flow-duration curve is a valuable medium for studying and 

comparing drainage basin characteristics, particularly the effect of 

basin geology on low flows. Except in basins with a highly permeable surface, the distribution of high flows is governed largely by 

the climate, the physiography, and the plant cover of the b".sin. The 

distribution of low flows is controlled chiefly by the geology of the 

basin. Thus, the lower end of the flow-duration curve is a valuable 

means for studying the effect of geology on the ground-water runoff 

to the stream. Where the stream drains a single formation, the 

position of the low-flow end of the curve is an index of the contribution to streamflow by the formation. The effect of geology on low 

flow in Ohio has been discussed by Cross (1949), Cross and Bernhagen (1949, p. 5), and Schneider (1957).

Six streamflow records for southern Mississippi are us°d to illustrate the effect of geology on low flow. The area selected has a 

fairly uniform climate and little difference in elevation. The map 

showing outcrops of the principal formations (fig. 11) is adapted 

from plate 2, Water-Supply Paper 576 (Stephenson rnd others, 

1928). Details, such as variations within the prinicpal formation 

outcrop and outcrops within the immediate vicinity of the streams, 

have been omitted.
FLOW-DURATION CURVES 25

Descriptions of the formations (from pi. 2, WSP 576) are as 

follows:

Citronelle formation_Sand, gravel, and clay.

Catahoula sandstone_Irregularly bedded sand, sandstone, and clay.

Vicksburg group___Limestone, marl, clay, and sand.

Jackson group_____Basal portion. Moodys Branch formation shells embedded in quartz sand and glauconite; upper portion Yazoo clay clay, more or less calcareous, with 

some sand and marl.

90'

32°

FIGURE 11. Geologic map of area In southern Mississippi having approximately uniform

climate and altitude.
26 MANUAL OP HYDROLOGY: PART 2, LOW-PLOW TECHNIQUES

Figure 12 shows flow-duration curves for six southerr Mississippi 

streams plotted from data in "Surf ace Waters of Mississip r>i" (Anderson, 1950). Curves 1, 2, and 3 represent streams draining the Citronelle formation outcrop; curves 4, 5, and 6 represent streams that 

contain outcrops from the Catahoula sandstone, the Vickrburg group, 

and the Jackson group.

It is apparent from figures 11 and 12 that streams draining the 

same geologic formations have more nearly the same lor^-flow duration in cubic feet per second per square mile than nearby streams 

draining different geologic formations. For example, ttn flow-duration curve for Bowie Creek (curve 2 on figures 11 and 12) more 

nearly resembles the curve for Bogue Chitto (curve 1) than that of 

the nearby Leaf Eiver near Collins (curve 4).

Where the yield per square mile to streamflow from adjacent 

geologic formations differs greatly, the index stations, to be used 

with discharge measurements at base-flow observation points, should 

measure single geologic formations whenever possible. In some 

drainage basins containing several formations, it may be possible to 

estimate the average yield per square mile of the whole basin by 

adding the proportional contribution of each geologic frrnation.

This discussion has been oversimplified to illustrate the principles 

involved. Even if the geologic formation is the same throughout a 

basin, other factors, such as variations in permeability of the formation, the character of the underlying formation, and the depth of incision of the stream, affect the low-flow characteristics at a particular 

point on a stream. For example, although the streams represented 

by curves 1, 2, and 3 (fig. 12) all drain from the Citronelh formation, 

curve 3 differs greatly from curves 1 and 2.

This discussion of the effect of geology on streamflow provides a 

warning against estimating low flow from ungaged arnas without 

carefully studying the area and making base-flow measurements at 

several different points. In such a study, knowledge of the geology 

of a basin can seldom be used to make quantitative estimates of lowflow potential of streams, but it will help to explain the differences.

WATERPOWER STUDIES

The flow-duration curve is used for preliminary studies of plant 

capacity, economic feasibility of projects, and pondago (Barrows, 

1943; p. 159-191 and Foster, 1934; p. 1230-1234). Its use in a 

multiple-plant development is discussed by Hickox and Wessenauer 

(1933).

As a simple example, assume that a "run of the river" hydroelectric plant is at a site with a net head of 100 feet and that the
FLOW-DURATION CURVES 27

flow-duration curve is the one shown in figure 13. The tertative 

turbine capacity is 6,000 cfs. The preliminary study requires an 

estimate of the average yearly output in million kilowatt-hours at 

the wheel shaft for primary and for secondary power, assuming 80- 

percent turbine efficiency, a load factor of 100 percent, and no loss or 

waste of power.

The area below the duration curve and the turbine capacit 17 line 

represents the flow available for producing power. Flow in excess of 

6,000 cfs cannot be used by the turbine. The scale factor (value of 

each small square) can be determined by computing the number of 

squares (500) that represents 1,000 cfs, 100 percent of the tiim. In 

figure 12 the scale value of each square is 1,000 (cfs) divided l~y 500 

(squares), or 2 cfs for each square.

 \

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EXPLANATION 

Bogue Chitto near Tylertown 

Bowie Creek near Hattiesbur; 

Homochitto River at Eddicet 

Leaf River near Coll ins

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6 Strong River at D'lo

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PERCENT OF TIME INDICATED DISCHARGE WAS EQUALED DR EXCEEDED

FIGURE 12. Flow-duration curves far selected Mississippi streams, 1039-48.
28 MANUAL OF HYDROLOGY: PART 2, LOW-FLOW TECITNIQUES

The average yearly output at the wheel shaft for 1 cfs, with 80- 

percent efficiency and with a 100- ft head, is computed as follows:

62.4 X 100 X. 80X6535 ,

=59,338 ( kilowatt-hourr )

The upper limit of primary power (power available continuously) is 

the horizontal line through the minimum flow (800 cf? in this example). Thus, the primary power is 800 (cfs) X 59,338=47.5 million

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PERCENT OF TIME INDICATED DISCHARGE WAS EQUALED C 

FIGURE 13. Flow-duration curve applied to hydropower stud]

100 

v * EXCEEDED
FLOW-DURATION CURVES 29

kilowatt-hours yearly. The discharge available to produce secordary 

power is represented by the area bound by the upper limit of primary 

power, the turbine-capacity line, and the flow-duration curve. This 

area is determined by planimetering, by counting squares, or by 

scaling. In the example, the area is 1,380 squares and the flow is 

2,760 cfs (2X1,380). The secondary power is thus

59,338X2,760=163.8 million kilowatt-hours yearly 

STREAM-POLLUTION STUDIES

To illustrate the use of the flow-duration curve in a preliminary 

analysis of the degree of treatment required, the following data are 

assumed :

1. No contamination above point under investigation.

2. Allowable BOD (biochemical oxygen demand) for stream 

below the disposal plant is 4 ppm.

3. The allowable BOD (4 ppm) may be exceeded not more than 

1 percent of the time, on the average.

4. Flow equals or exceeds 10 cfs 99 percent of the time.

5. Sewage flow is 1,000,000 gallons per day (1.55 cfs).

6. BOD of untreated sewage is 200 ppm. 

Compute the degree of treatment required : 

The allowable BOD below disposal plant outlet =4 ppmX(10

cfs +1.55 cfs).

The BOD of the sewage =200 ppm X 1.55 cfs. 

The degree (D) of BOD not removed by treatment 

(DX 200X1.55) must not exceed the allowable (4X11.55) or

4X 11 55F=0.15 or 15 percent. Thus, 85 percent cf the

BOD must be removed by the sewage disposal plant. 

QUALITY-OF- WATEB STUDIES

The quality of surface streams is often shown by duration curves 

of sediment, turbidity, hardness, or some other quality-of -water characteristic. These duration curves are obtained in a manner similar to 

that described for flow-duration curves.

If the quality-of- water data are insufficient for direct computation 

of some descriptive statistics, such as frequency distribution, annual 

loads, annual average concentrations, and standard deviations, flowduration curves can be used sometimes to make approximations. The 

suitability of this technique is dependent on the correlation of the 

quality characteristics against stream discharge. The error of the 

approximation includes both the error of the flow-duration curv«s and 

the error of estimate for the correlation.
30 MANUAL OF HYDROLOGY: PART 2, LOW-FLOW TECF3SJTQUES

For each sample, the quality-of-water characteristic to be shown 

is plotted against the stream discharge at the time of collection. A 

rating curve of quality-of-water characteristic versus discharge is 

drawn to average the plotted points. The duration curve of streamflow is converted to a curve showing frequency of a specified quality 

characteristic by looking up the discharge at several percent-duration 

points and obtaining the corresponding value of the qua! ity-of-water 

characteristic from the rating curve. These values are plotted against 

the appropriate percent of time to obtain points for drawing the 

quality-of-water frequency curve. The quality-frequency curve 

developed thus is divided into convenient segments or groups, and 

the desired statistical description is computed in the customary 

manner.

VARIABILITY INDEXES

An important characteristic of the flow of a stream is its variability. Variability of streamflow is the result of variability in precipitation as modified by basin characteristics. Storage, either on 

the surface or in the ground, serves to reduce the variability of flow. 

As an illustration of the range in variability of streams, the May 

1955 flow of a station in Michigan was among the lowest 25 percent 

of record although the flow was 93 percent of the median, whereas 

at a station in California the flow was not among the lowest 25 percent of record although it was only 10 percent of the median.

The slope of the flow-duration curve is a quantitative measure of 

the variability. Several indexes of this slope have been used. In 

1920 the Geological Survey adopted the flow available 50 percent of 

the time (q50 ) and the flow available 90 percent of the time (q90 ) as 

standards of flow for waterpower statistics. The q90 is a measure of 

the prime power and the q50 is an index of the power potential with 

storage. Together, the two indicate the variability of f ow.

LANE'S VARIABILITY INDEX

Lane and Lei (1950) introduced an index of variability, which 

was defined as the standard deviation of the logarithms of the stream 

discharge. On log-probability paper this index represents the fall 

(in terms of log cycles) of the duration curve in one standard deviation. The index can be estimated by scaling the value from a plot 

on logarithmic paper, or it can be computed as follows:
FLOW-DURATION CURVES 31

1. From the duration curve, pick values of discharge at 10-percent intervals 

from 5 percent to 95 percent of the time.

2. Look up the logarithms of these discharges and compute the standard deviation of the logarithms (index of variability) : 

a. Obtain mean of the logarithms.

b. Compute differences of each of the 10 logarithms from the meen. 

c. Square the differences, 

d. Obtain the sum of the squares, 

e. Divide the sum of the squares by 9. 

f. Extract the square root of the results of step e.

Lane and Lei (1950) discussed the use of their variability index 

in studying streamflow characteristics and proposed that an estimated 

variability index be used with an estimated mean annual discharge 

to produce a synthetic flow-duration curve. They found that large 

drainage areas tended to have lower values of variability than small 

ones.

Mitchell (1957, p. 161-166) found that, for Illinois, Lane's variability index differed considerably from one region to another, yet 

when the values of the index were plotted on a map, a consistent 

pattern was observed. The map of Illinois showing regional values 

of the variability index was presented for use in deriving synthetic 

duration curves for unmeasured areas.

A synthetic duration curve based on an estimated variability index 

and an estimated median flow may be accurate for the portion of the 

flow-duration curve that plots as a straight line on log-probrbility 

paper. Likewise, a duration curve, drawn as the average of the 

duration curves of nearby gaged streams draining areas thot are 

apparently similar, may be reliable considerably below the median 

point but may depart radically at the extreme low end. TH end 

points of the flow-duration curve cannot be accurately determined 

from the slope of the straight portion of the flow-duration curve. 

This statement can be verified for the low-flow end by studying the 

flow-duration curves of figure 12 and the location of the gaging stations on figure 11. Although the end portions of the flow-duration 

curve occupy a sizable space on probality paper, the percentage of 

time during which the inaccurately determined flows occur is small, 

and for some studies a synthetic curve may be- suitable. When the 

extreme low end of the flow-duration curve is important, bas<vflow 

measurements should be made at the unmeasured site and correlated 

with the concurrent discharge at a stream-gaging station. (S°-e the 

section on estimation of the flow-duration curve.)
32 MANUAL OF HYDROLOGY: PART 2, LOW-FLOW TECHNIQUES 

RELIABILITY OF AN INDIVIDUAL STATION RECORD

A gaging-station record, however long, represents only a small sample of the long-term flow characteristics of the stream. Tl ^ reliability 

of the record for predicting the behavior of the stream depends upon 

the accuracy and consistency of the record and upon how well the 

period of record represents the long-term flow of the stream.

The need for accurate records of basic data is obvious, but consistency of the record often does not receive enough emphasis. A 30- 

year record collected during 10 years of natural flow conditions, 10 

years of irrigational or storage developments, and 10 years under 

changed basin conditions cannot serve as the basis for predicting future flows. Other major changes in the basin also affect the record. 

Changes in gage location or measuring section, even without appreciable change in drainage area, can affect the consistency of the record. 

This is particularly true of areas underlain by limestone, where low 

flows vary widely within short reaches of the same stream. Before analysis, records should be examined for consistency and the drainage 

basin history should be studied for information on developments 

that might affect the consistency of the record.

Even though a gaging-station record is accurate and1 consistent, 

it may not be representative of the long-term flow of the stream. This 

is particularly true of short-term records, which may include a predominance of dry or of wet years. Duration curves for such records 

should be adjusted to a long-term period by correlation vlth records 

at a long-term gaging station.

If no long-term gaging station record for an area is available, a 

regional analysis of all the short-term records for the area will improve the reliability of each of the flow-duration curves. A regional 

analysis of flow-duration data consists of (a) establishing a relation 

(as in fig. 7) between each individual station and a "pivot" station, 

(b) transferring the flow-duration curve of each individual station 

to the pivot station by using the curve of relation, (c) drawing an 

average or composite curve for the pivot station, and (d) transferring the composite curve of the pivot station back through the curve 

of relation to the individual station.

Kegional analysis modifies the individual gaging-station record 

so that a chance occurrence of an event, such as a local she ^er during 

an extreme drought in a particular drainage basin, will not unduly 

affect the analysis. Likewise, regional analysis takes into account 

the failure of such an event to occur within a particular drainage 

basin when the event did occur within other basins in the region.
FLOW-DURATION CURVES 33 

REFERENCES CITED

Anderson, I. E., 1950, Surface water of Mississippi: Mississippi Geol. Survey

Bull. 68, 338 p. 

Barrows, H. K., 1943, Water power engineering: 3d ed., New York, McGrawHill Book Co., 791 p. 

Beard, L. R., 1943, Statistical analysis in hydrology: Am. Soc. Civil Enfineers

Trans., v. 108, p. 1110-1160. 

Cross, W. P., 1949, The relation of geology to dry-weather stream flow in Ohio:

Am. Geophys. Union Trans., p. 563-566. 

Cross, W. P., and Bernhagen, R. J., 1949, Ohio stream-flow characteristics 

Pt. I, Flow duration: Ohio Dept. Nat. Resources, Div. Water Brll. 10. 

Elderton, W. P., 1953, Frequency curves and correlation: 4th ed., Washington,

D.C., Barren Press, 272 p.

Foster, H. A., 1924, Theoretical frequency curves and their application to engineering problems: Am. Soc. Civil Engineers Trans., v. 87, p. 142-303.

 1934, Duration curves: Am. Soc. Civil Engineers Trans., v. 99, D. 1213-

1267. 

Hickox, G. H. and Wessenauer, G. O., 1933, Application of duration curves to

hydro-electric studies; Am. Soc. Civil Engineers Trans., v. 98, p. 1276-1308. 

Hoyt, W. G., 1934, Discussion of duration curves by H. A. Foster: Am. Soc.

Civil Engineers Trans., v. 99, p. 1240-1243.

Lane, E. W. and Lei, Kai, 1950, Stream flow variability: Am. Soc. Civil Engineers Trans., v. 115, p. 1084-1134. 

Mead, D. W., 1908, Water power engineering: 1st ed., New York, McGraw-Hill

Book Co., 787 p. 

Meyer, A. F., 1928, The elements of hydrology: 2d ed., New York, John Wiley

and Sons. 

Mitchell, W. D., 1950, Water-supply characteristics of Illinois streams: Illinois

Dept. Public Works and Bldgs., Div. Waterways, 311 p.

 1957, Flow-duration of Illinois streams: Illinois Dept. Public Wor^s and

Bldgs., Div. Waterways, 189 p. 

Pettis, C. R., 1934, Discussion of duration curves by H. A. Foster: Am. Soc.

Civil Engineers Trains., v. 99, p. 1237-1240. 

Saville, Thorndike, and Watson, J. D., 1933, An investigation of flow-drtition

characteristics of North Carolina streams: Am. Geophys. Union Trans.,

p. 406-425. 

Schneider, W. J., 1957, Relation of geology to streamflow in the upper Little

Miami basin: Ohio Jour, of Science, v. 57, No. 1, p. 11-14.

Slade, J. J., Jr., 1936, An asymmetric probability function: Am. Soc. Civf Engineers Trans., v. 101, p. 35-104. 

Stephenson, L. W., Logan, W. N., and Waring, G. A., 1928, The ground-water

resources of Mississippi: U.S. Geol. Survey Water-Supply Paper P76.

U.S. GOVERNMENT PRINTING OFFICE : 1969 O - 353-858