EPA-670/2-75-046
May 1975
RAINFALL-RUNOFF RELATIONS ON URBAN AND RURAL AREAS
By
Ernest F. Brater
James D. Sherrill
University of Michigan
Ann Arbor, Michigan 48104
Grant No. R-800941 (11040 DRS)
Program Element No. 1BB034
Project Officer
David J. Cesareo
Storm and Combined Sewer Section (Edison, N.J.)
Advanced Waste Treatment Research Laboratory
National Environmental Research Center
Cincinnati, Ohio 45268
NATIONAL ENVIRONMENTAL RESEARCH CENTER
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
CINCINNATI, OHIO 45268
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REVIEW NOTICE
The National Environmental Research Center—
Cincinnati has reviewed this report and approved
its publication. Approval does not signify that
the contents necessarily reflect the views and
policies of the U.S. Environmental Protection
Agency, nor does mention of trade names or com-
mercial products constitute endorsement or recom-
mendation for use.
ii
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FOREWORD
Man and his environment must be protected from the adverse effects
of pesticides, radiation, noise and other forms of pollution, and
the unwise management of solid waste. Efforts to protect the
environment require a focus that recognizes the interplay between
the components of our physical environment—air, water, and land.
The National Environmental Research Centers provide this multi-
disciplinary focus through programs engaged in
o studies on the effects of environmental contaminants on
man and the biosphere, and
o a search for ways to prevent contamination and to recycle
valuable resources.
As part of these activities, the study described here investigated
the factors which control the relationship between storm rainfall,
snow melt, and the resulting storm runoff, including the effects
of urbanization on the runoff process.
A. ¥. Breidenbach, Ph.D.
Director
National Environmental
Research Center, Cincinnati
iii
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ABSTRACT
A procedure was developed for estimating the frequency of storm runoff
of various magnitudes from rainfall and/or snowmelt on small drainage
basins in various stages of urbanization. The study was based pri-
marily on the analysis of storm runoff events on real basins varying
in size from 0.02 to 734 sq mi. The method is based on applying unit
hydrographs to precipitations of various frequencies .after deducting
infiltration and retention. A concurrent study with an analytical
drainage basin model provided additional understanding of the effects
of some parameters. The unit hydrograph-infiltration capacity con-
cept was selected as the most accurate practical method for predict-
ing storm runoff. It was found that the form of the unit hydrograph
could be related to drainage basin size and degree of urbanization as
measured by population density. Other characteristics of the drainage
basin are much less important. The form of the unit hydrograph stays
relatively constant for various durations and magnitudes of input as
long as the duration of input is smaller than a critical time which
can also be related to the size and population density of the basin.
As the population increased from rural to highly urbanized peak dis-
charges for the same runoff became as much as ten times greater. In-
filtration capacity was found to vary seasonally. The prediction of
flood frequency by this procedure is fully operable for Southeastern
Michigan. For application to other areas some hydrograph analysis
must be made.
This report was submitted in fulfillment of Project Number R-80ti941
(formerly 11040 DRS), by The University of ffichigan, under the
(partial) sponsorship of the Environmental Protection Agency. Work
was completed as of July 1973.
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CONTENTS
Review Notice
Foreword
Abstract • ' .
List of Figures
List of Tables
Acknowledgments
Sections
I Conclusions
II Recommendations
III Introduction
IV The State-of-the-Art
V Collecting Data
VI Hydrograph Analysis
VII Infiltration Capacity
VIII Unit Hydrographs
IX Mathematical Model
X Frequency Studies
XI Time-Intensity Rainfall Patterns
XII Predicting Flood Magnitudes and Frequencies
XIII References
Page
ii
iii
iv
vi
viii
ix
1
4
6
7
10
19
32
38
69
71
76
85
94
v
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LIST OF FIGURES
No. . . . . Page
1 Atypical hyetographs and hydrographs. 11
2 Complex hyetograph and hydrograph. 20
3 Relation between slope and discharge on ground
water depletion curve. .22
h Relation between slope and discharge on ground
water depletion curve. '• 23
5 Surface runoff vs. precipitation. 26
6 Surface runoff vs. precipitation. 27
7 Surface runoff vs. precipitation. 28
8 Seasonal variation of infiltration capacity. 33
9 Five unit hydrographs from the same basin. : 39
10 Reproduction of a complex hydrograph. : ^1
11 Definition sketch. h2
12 Unit hydrograph peaks vs.,area (original data). k8
13 Unit hydrograph peaks vs. area with uniform slopes. k-9
ih Unit hydrograph peaks vs. population density. 50
15 Significant period of rise vs. area. 51
l6 Significant period of rise vs. area with uniform density. 52
vi
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LIST OF FIGURES' (Concluded)
No. ,
17 Significant period of rise -vs. population density.
18 Period of rise vs. area.
19 Period of rise vs. area with uniform slope.
20 , Period of rise vs. population density.
21 Unit hydrograph peak vs. area-design curves.
22 Significant period of rise vs. area-design curves.
2J Period of rise vs. area-design curves.
2k Width at 75 percent of peak vs. area-design curves.
25 Width at 50 percent of peak vs. area-design curves.
26 Width at 25 percent of peak vs. area-design curves.
27 Width of base vs. area-design curves.
28 The effect of snowmelt on frequency.
29 The effect of length of record on frequency.
JO Final seasonal frequency curves.
31 Precipitatidn-rduration expressed as percent
of 2k-hr rainfall.
32 Intensity-duration expressed as percent of 2k-br rainfall.
33 Typical seasonal hyetographs expressed as
percent of 2^-hr rainfall,,
31). Area-depth curves.
35 Synthesized and observed unit hydrographs for Red Run.
36 Predicted and observed flood frequencies for Red Run.
Page
53
55
56
62
63
6k
65
66
67
68
73
7^
75
77
78
82
8k
87
92
Vll
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TABLES
No.
1
2
5
6
7
8
9
10
11
Flood Runoff Data
Computations of Surface Runoff and Unit Hydrograph
Ordinates for Rain of Spril 1, 1959, on Plum Brook
Computation of Infiltration Capacity
Unit Hydrograph Parameters
Legend for Figures 12-20
Equations and Statistical Parameters for Lines
on Figures 12, 13, 15, 16, 18, and 19
Maximum Rainfall for Various Durations
• Hyetograph Ordinates in Percent of 2^-Hour Rainfall
per Hour
Twenty-Four Hour Rains of Various Frequencies
Computation of Volume of Surface Runoff for a
50-Year Winter Rain
Computation of Composite Hydrograph for 50-Year
Winter Rain
Page
12
36
37
57
79
83
86
89
91
viii
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ACKNOWLEDGMENTS
The financial support of the Environmental Protection Agency is grate-
fully acknowledged. The assistance of Messrs. Robert M. Buckley and
David J.. Cesareo> Project Officers, has been most appreciated.
The project was initiated in ~L96k by means of a small grant from the
University of Michigan Graduate School. In 1965 a grant was obtained
from the National Institutes of Health. This grant was continued to
this time under the sponsorship of FWQA and EPA. During the period
1966-1971 additional support was provided by the Michigan Department of
State Highways and the U.S. Bureau of Public Roads.
Many graduate students worked on the project. •Their services made the
project possible. Particular acknowledgment is made to Dr. Suresh K.
Sangal who not only worked on the project but contributed greatly to
the research through his Ph.D. dissertation.
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SECTION I
CONCLUSIONS
The unit hydrograph-infiltration capacity method has been developed by
this research program to the point where it provides very satisfactory
estimates of peak discharge and if needed entire flood hydrographs, of
various frequencies for drainage basins of all sizes and all degrees
•of urbanization. The procedure consists of forming runoff hydrographs•
of various frequencies by applying the unit hydrograph for that basin
size and population density to the precipitation excess. The,precipi-
tation excess is precipitation plus snowmelt of a selected frequency
minus retention and infiltration on the permeable portions of basins
and minus retention on the impermeable portions. The methods de-
scribed in this report provide the most accurate method of determining
required capacities of storm sewers, culverts, bridges and other flood
carrying structures known to the writers.
The specific conclusions are summarized below:
1. The unit hydrograph peaks as well as their time characteristics
such as their periods of rise and widths at various fractions of the
peak discharge can be correlated with watershed areas and population
density to provide statistically significant relations which enable
hydrologists to estimate the runoff characteristics of ungaged areas.
These relationships were derived from the analysis of hundreds of
flood hydrographs from 53 drainage basins from five states. The areas
of these basins vary from 0.02 to ?4j sq mi and the population den-
sities cover a range from less than 100 to more than 14,000 persons/
sq mi.
2. The effect of urbanization is primarily in the production of flood
hydrographs of much shorter duration and higher peaks. For example as
the population density changes from 100 to 13,000 persons/sq mi the
peak rate of surface runoff for a given total surface runoff becomes
about 10 times greater while the time parameters decrease to about one
tenth of the values for rural areas. For the same population increases
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the increase in total runoff due to increased impermeable area is only
about 10 percent in Southeastern Michigan.
3. Tflien the 16 basins located in Southeastern Michigan* were studied
separately the correlations showed lower unit hydrograph peaks for low
population densities than the corresponding relation for all 53:ba-
sins. It is suspected that the differences are related to the nature
of the drainage networks. The use of local results improved the accu-
racy in Southeastern Michigan for low population basins over the use
of the general curves.
4. Eleven basins located in Texast exhibited a tendency toward higher
peaks for given areas and populations then the trends of the other 53
basins which included many Texas basins at other locations. Further
research will be required to determine the cause of this anamolous
behavior. However, based on the behavior of the other basins it ap-
pears likely- that the drainage systems in these 11 basins have been
developed in advance of the population growth.
5. Based on the analysis of about 200 flood hydrographs produced by
rains equal to or greater than >1 in., the average summer and .winter
infiltration capacities in Southeastern Michigan are O.kd and 0.10 in./
hr, respectively.
6. The effect of the impermeable area was taken care of by developing
the concept of hydrologically significant impermeable area which was
found to vary linearly with population density from about 1 percent for
1000 persons/sq mi to about 10 percent for 7500 persons/sq mi.
7. The maximum value of retention on the permeable portion of the
basins is about 0.2 in. The average in summer is approximately! 0.15
in. and in winter 0.10 in. An estimated retention of 0.05 in. on the
impermeable areas gave satisfactory results.
8. An analysis of the frequency of one day' occurrences of winter
rain plus snowmelt gave values about 2 percent higher than the values
for the same frequency determined in the conventional manner from pre-
cipitation records, where snowfall is included along with rain.
9« Typical large rainstorms in Southeastern Michigan have a nearly
symetrical time distribution with the maximum hour near the center of
*These 16 basins were part of the 53 basins which formed the basis
for the general relations.
tThese 11 basins were not among the 53 used for developing the gen-
eral relations.
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the rain storm. In winter the maximum hour contains about 24 percent
of the rain and in summer the maximum hour usually provides about 55
percent of the total storm rainfall. Information was obtained per-
mitting the formation of typical time-intensity precipitation patterns
from 2h-hr rains of any frequency. •
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SECTION II
EECOMMEMDATIOHS
The evidence provided by this research shows that storm runoff from
most drainage basins behaves in a consistant manner when basin size
and population density are taken into account. However, there were a
number of basins which did not follow this pattern. It is believed
that an investigation of these anomolies would be very productive in
improving our knowledge of the runoff process. The results to this
time indicate that most differences could be explained if more were
known about the density and efficiency of the drainage systems. It
is suspected that some drainage systems may be developed in advance
of population increases whereas in other locations the drainage sys-
tem may lag behind population increases. Related to this same problem
would be land use. For example, highly industrialized areas may not
show population densities which are consistant with their storm sewer
system. Perhaps in the future s©me measures of drainage efficiency
could be used in place of population density as a measure of urbani-
zation.
Although there is no substitute for field investigations and the use
of real rainfall and runoff records to find these answers, it is be-
lieved that an extension of the mathematical model to more complex
drainage networks would be of nearly equal value in establishing a
better knowledge of this process.
Research should be continued to determine more accurately the relation
between point rainfall and average rainfall on areas of various sizes
for the same frequencies. The form of precipitation time-intensity
pattern should also be studied at other locations as should the effect
of including snowmelt with precipitation in frequency studies. ;
In general it is strongly recommended that hydrograph analysis be
carried on systematically to determine the various parameters such as
infiltration capacities, retentions, hydrologically significant im-
permeable areas, and unit hydrograph characteristics for basins in
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various locations. Population densities and'drainage patterns should'
also be determined. These data will be needed to provide accurate de-
sign procedures even if in the future other methods are found to be
more satisfactory than the one suggested here.
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SECTION III
INTRODUCTION
The objective of this project was to gain a better understanding of
the factors which control the relationship between rainfall or snow-
melt and the resulting storm.runoff and to determine the effect of
urbanization on this runoff process. The benefits would include the
prevention of flood damage by means of improved design of storm sewers
and. waterways and would provide the data needed for the improved de-
sign and operation of facilities for control of pollution due to storm
water and/or combined sewage. A basic approach was adopted which
would provide a better understanding of the surface runoff process in
rural and urban areas while concurrently examining and testing known
methods of predicting flood peaks. The development of any practical
procedure for predicting peak discharges obviously needed to be based
on known surface runoff events. Therefore, the largest portion of
the effort was gathering and analyzing storm runoff events from' small
drainage basins at various stages of urbanization. Data were gathered
and analyzed from 69 basins located in five states and varying in size
from 0.02 to kj$ sq mi. In addition.to the testing and development of
practical flood prediction procedures a simple mathematical model was
developed in which runoff could be computed'for various rainfall inputs.
The model served to check and extend information derived from the study
of actual rainfall and runoff data.
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SECTION IV
THE STATE-OF-THE-ART
An intensive study of procedures and literature dealing with runoff
processes was carried out at the beginning of the research. In Addi-
tion, to studying the literature, the Principal Investigator visxted
centers and researchers working on this phase of hydrology in the U.S.
and Europe. The review of the subject included a reevaluation of
methods which had been used in the past as well as those in current
usage. In a general way, these procedures can be separated into the
following categories; statistical methods, procedures utilizing em-
pirical equations or curves, storage-routing procedures and unit hydro-
graph procedures. The later two categories refer to the way in which
the surface runoff hydrograph is formed; the total volume of runoff
being usually determined by the use of the infiltration capacity con-
cept.
Statistical methods utilize flood records on a particular basin.
Sometimes they are applied regionally by assuming similar rainfall,
snowmelt and drainage basin characteristics. These methods are usu-
ally limited in accuracy because of lack of sufficient records to make
a significant statistical analysis as compared with methods using rain-
fall as input in which frequency can be determined from the much longer
precipitation records. They lack the flexibility to determine the ef-
fect of urbanization or other watershed changes. The various empirical
methods also require a fairly long period of records to determine the
necessary constants in particular locations and again they do not lend
themselves to modification for changing conditions.
Most of the attention was given to methods which combined the deter-
mination of flood volumes by deducting .infiltration from precipitation
and forming flood hydrographs by storage routing or unit hydrographs.
These procedures provide models which attempt to represent the -actual
runpff process.
Peak flood discharges are controlled by two independent processes.
-------
One process determines the time rate of input to the system and the
other establishes the response of the system which is the flood hydro-
graph. The system input is rainfall plus snowmelt minus infiltration
and other minor retentions. This process is dependent on the frequency
and nature of the rain and snowmelt as well as the variations of the
infiltration capacity of the soil. Since the concept of infiltration
capacity was first introduced1 a number of investigators have studied
the variations of infiltration capacity with time during a rain2~5 as
well as its seasonal variation.6-8 Its variation from place to place
has also been noted.9,10 ^ factors which control infiitrati6n capa_
city are quite well understood but quantitative values for any region
must be obtained from the analysis of rainfall and surface runoff
events on drainage basins within the region. For example, previous
work on this project has provided more than 200 values of infiltra-
tion capacity for Southeastern Michigan from which average seasonal
values have been determined.5
The second process which deals with the response of.the system and
controls the shape of the flood hydrograph is a function of the input
and of the physical characteristics of the drainage basin. Knowledge
gained from this aspect of the runoff process is not limited to re-
gional conditions but can be expected to be applicable to all loca-
tions. The important input parameters are intensity, duration and
spacial distribution of rain and snowmelt. The physical characteris-
tics are those of the flow system. The flow system consists of a
series of surfaces which receive the input and contribute to the
stream system. Throughout the system, flow is spatially variable and
unsteady. For simplified basins the flow can first be computed over
the land surfaces1!-^ and then through the channels.15,16 combining
the two phases provides a useful mathematical model.17,18 For prac_
tical flood predictions a simpler approach is essential. Various
methods of routing input through storage have been proposed 19-21 The
storage-discharge relation is usually derived from the recession side
or a hydrograph. In some cases the Input is transferred to the outlet
by a convolution process after estimating the time of travel from vari-
ous segments of the drainage area. All of these procedures have been
explored in this research program. The unit hydrograph procedure has
shown the best results when considering simplicity as well as sensi-
tivity to urbanization and other physical parameters.
The unit hydrograph idea was first ProPosed22,25 f large watershed
but it is also a useful tool for small watersheds.^ Many investi-
gators have studied the effect of drainage basin characteristics on
unit hydrographs.25-32 previous research has shown'that drainage
basin size is one of the most important factors in influencing the
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shape of the unit hydrograph.? This project has provided much more
evidence that the area is one of the most important factors0;*-(>n>
and has produced quantitative evidence that population density is a
very important parameter.
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SECTION V
COLLECTING DATA
It was clearly recognized from the beginning of this research that any
effort that would provide a better understanding of surface runoff pro-
cesses and eventually lead to a dependable method of predicting runoff
from rainfall would require the analysis of many rainfall and/or snow-
melt and runoff events from many different.sizes and types of drainage
basins. &
A great deal of effort was devoted to searching for available runoff
and rainfall records. It became obvious early in the work that the
effect of urbanization would be one of the most important factors in
the relations dealing with hydrograph shape. Many runoff records from
urbanized areas are unpublished and must be obtained by copying from
the original hydrographs where they are stored. Many were not pub-
lished in sufficient detail and therefore'it was necessary to work from
the original gage charts or tapes in the U.S.G.S. offices to obtain the
records. £1 the case of Red Run, a highly urbanized basin in South-
eastern Michigan, it was also necessary to combine the runoff from the
gaging station with the discharge diverted into the sewer system. An-
other part of the analysis consisted of the determination of the
weighted hourly precipitation for each event and the computation of
snowmelt when it occurred. Each event was then plotted as shown in
Figure 1. It is estimated that the collection of data along with the
hydrograph analysis mentioned above required about 70 percent of the
total effort.
In Table 1 is presented a list of the watersheds that were studied
The location, the U.S.G.S. number, area, population density and the
number of rain gages are also shown in the table. ' The number of drain-
age basins studied was 69, varying in size from 0. 02 to 73^ sq mi with
54 basins having areas of less than 20 sq mi. Population densities
vary from less than 100 to more than 36,000 persons/sq mi. The num-
ber of hydrographs analyzed for each basin is also shown. The total
number of hydrographs analyzed was 1620.
10
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Q-
0.50
0.25
0
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PLL//W BROOK
Figure 1.
2345
APRIL, 1959
Typical hyetographs and hydrographs.
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SECTION VI
HYDROGRAPH ANALYSIS
Hydrograph analysis on perennial streams for.the purpose of determining
the surface runoff, infiltration capacity, the duration of precipitation
excess, and the form of the surface runoff hydrograph requires the sep-
aration of surface runoff from ground water discharge or base flow.- The
method conventionally used was based to some extent on judgment. Fur-
thermore, it has been commbn practice to neglect initial retention (the
portion of the rain which is intercepted by vegetation or the ground
surface and never becomes infiltration or surface runoff) and the ef-
fect of runoff from impervious areas on the computations 'of infiltra-
tion capacity. Therefore, one of the initial goals of this project was
to develop objective methods of carrying out this operation including the
effect of retention and impermeable area.
SEPARATION OF GROUND WATER DISCHARGE FROM SURFACE RUNOFF
The first step in hydrograph analysis is to select lines such as b]_b2 in
Figures 1 and 2, to separate ground water discharge from surface runoff.
Various logical^subjective selections of this line could be made. Usually
the range of reasonable locations where this line could be established is
not great enough to create substantial differences in the computed values
of either the, infiltration capacity or the unit hydrograph ordinates.
However, in the interest of better uniformity the following method was
derived. There is little difficulty in selecting the point bL where sur-
face runoff begins, but locating the point b2 where surface runoff ends
is more difficult. Consequently, it was decided to make use of the ground
water depletion curve to determine the location of such points. The
ground water depletion curve is the hydrograph of river discharge during
a time of no precipitation and it depicts a rate of decrease in discharge
that is much smaller than that of the recession side of a flood hydrograph
such as sb2 in Figures 1 and 2. If point b2 correctly locates the' end of
surface runoff, then a ground water depletion curve for the basin should
closely fit the hydrograph to the right of b2 but should depart from the
hydrograph to the left of b2. This is illustrated by the ground water
19
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depletion curves gdc in Figures 1 and 2.
The practice of deriving ground water depletion curves by graphically
fitting together portions of the curve permits the exercise of consider-
able judgment. Consequently, a more objective procedure was sought.
The method selected is based on the idea that insofar as there exists a
consistent relationship between ground water discharge and storage on the
drainage basin, there* must also exist a consistent relationship between
discharge (Q,)> which is the ordinate of a point on the depletion curve,
and its slope (AQ/AT), and that the equation for this relationship can
be solved to obtain the equation for the ground water depletion curve.
Values of Q and AQ/AT were read from only those portions of the hydro-
graphs preceded by a period of at least 3 days during which there was
neither rain nor snowmelt. Examples, of plotted values of AQ/AT versus Q
are shown in Figures J. and k. In all river basins for which this deri-
vation has been made, the relationship between AQ/AT and Q appeared to
be linear. For example,.the numerical value of the linear correlation
coefficient (r) for the group of points in Figure k is 0.93, which is
much greater than the value of 0.7 required to indicate that there is
only one chance in 1000 that linearity is a fortuitous occurrence. The
equations relating AQ/AT to Q were derived by the method of least squares.
The equation for Plum Brook (Figure k) is
dQ/dT = 0.102 + 0.12UQ
Solution of this differential equation yields
- 1 PkT1
Q = (Q + 0.82)e ^ - 0.82
(1)
(2)
in which Qo is the value of Q selected for T = 0. This is the equation
of the ground water depletion curve. A segment of this curve is plotted
as line gdc in Figure 1. • .• ,
A point of interest regarding the relation between AQ/AT and Q is that
none of the straight lines such as those in Figures 3 and k pass through
the 0,0 coordinate. There seems to be no physical reason why they should.
On the other hand, the data are probably not sufficiently accurate to give
any significance to the location of the lower end points of these lines.
Having eliminated most of• the subjectivity from selecting point b2, it
remains to decide on the form of the line connecting b-j_ and b2. Logical
deduction would suggest that the ground water discharge would continue to
recede for a time beyond b-j_ and then rise in an s curve to point b2. How-
ever, until this process is better defined quantitatively, it was decided
to use a straight line connecting points b]_ and b2. The straight line has
21 .
-------
1.0
0.8
0.6
o
Of
0.4
0.2
0
0
1 1
Rouge at
Farmington
I/
4 6
Q INCFS
8
Figure 3. Relation between slope and .discharge
on ground water depletion curve.
10
22
-------
2.4
2.0
>-
-------
the advantage of providing a consistent method of making this separation
and eliminating any imaginative approach, even though' it is based on
logic.
Once the line bjbg is established, the surface runoff may be computed by
finding the area between this line and the hydrographs. The example shown
in Figure 1 is complicated by the occurrence of a second rain before the
surface runoff from the preceding one has ended. In such cases, a reces- .
sion line such as xy is drawn having the same form as the recession sbg.
After the initial retention and the runoff from the impermeable area are
taken into consideration, the average value of infiltration capacity (pav)
is obtained by trial, assuming that the precipitation excess (Pe) is equal
to the surface runoff.
RETENTION
Before overland flow begins, or during its early stages, a small portion
of the initial rainfall is stored and permanently abstracted from surface
runoff by interception and surface or depression storage. The intercep-
tion^evaporates, and the depression storage either evaporates or infil-
trates after the end of rainfall. The interception is abstracted from
the beginning of rainfall, whereas depression storage accumulates only
after the rain intensity exceeds the infiltration capacity. However, for
the purpose of flood prediction it is convenient to combine the two ab-
stractions. In this discussion, the total of these .two abstractions will
be referred to as retention (R). There is evidence1)- that interception
continues to accumulate until the rainfall has reached 2 or more inches,
and it seems logical to assume that depression storage may also increase
toward some top limit as the rain continues at high intensities. There-
fore, small rains are not likely to fulfill the maximum possible abstrac-
tions. The values of retention that have been estimatedQin the past
presumably refer to the upper limit. Tholin and Keif er have suggested
values of retention of 1/16 in. for pavements and lA in. for grass land.
For small paved areas, values reported by Vies sman35 range from O.Ol)- to
0.10 in. Values of this order of magnitude could be considered as in-
consequential when dealing with large floods. However, for smaller stream
rises, variations in the selected magnitudes of this factor produce sig-
nificant differences in the computed values of infiltration capacity.
For example, for the hydrographs shown in Figure 1, values of fav com-
puted first by neglecting this abstraction and then by assuming .a value
of 0.2 in. have the ratio of about 3 to 2, respectively, for the .first
stream rise and k to 1 for the second stream rise.
Although quantitative values of depression storage cannot be determined
directly from the analysis of individual hydrographs, it is possible to
24
-------
gain a reasonably good idea, of the maximum values of retention by ar-
ranging the data for total precipitation and surface runoff in the manner
shown in Figures 5, 6, and 7. In these figures Line A is drawn near the
left side to represent the condition of 100 percent surface runoff which
could only occur if there were no infiltration (f = 0) and no retention
(R = 0). If the vertical and horizontal coordinates were drawn to the •
same scale this would be a ^-degree line. Any points falling near Line
A represent stream rises which must have occurred when the infiltration
capacity on the permeable portion of the basin was very low.*
Another line, Line B, is then drawn through the points nearest to Line A
and above and to the left of the main body of points. The magnitude of
the retention can be estimated by assuming that points falling on Line B
have zero infiltration and that all of the precipitation becomes either
surface runoff or retention. Therefore the retention is the vertical
distance between Lines A and B. It will be seen that for Plum -Brook and
Big Beaver Creek (Figures 5 and,6) nearly all of the points fall below
and to the right of the lines representing an R of 0.2 in., thus indi-
cating that the probable maximum value of R is slightly more than 0.2 in.
For Red Run (Figure 7) more points fall above the 0.2-in. line than in the
cases of Plum Brook or Big Beaver Creek. However, it- may be noted that
most of these points are designated in Figure 7 by black circles which
represent winter rains which occurred within 15 hr after antecedent pre-
cipitation. Therefore, the portion of the retention caused by intercep-
tion on vegetation was very low and there is also a good possibility that
there was residual retention from the previous rain. The four summer
storms falling between Line A and" B in Figure 7, designated by triangles,
were very small rainstorms (P < 0.2 in.) for which the retention capacity
may not have been filled. It may be inferred, therefore, that the value
of R on Red Run for summer rains greater than 0.2 in.' and not preceded by
a rain within 15 hr is approximately 0.2 in., but that it may be smaller
for winter rains.
Although the maximum value of R appears to be about 0.2 in., it is clear
that the average value during large rains is less. The determination of
typical average values to be used in flood prediction was carried out
in the analysis of the runoff hydrographs by assuming several values of
R and computing the time of beginning of precipitation excess, recognizing
*The significance of the points pn these graphs which lie toward the
lower right, near the line labelled "SRO from A± only," is discussed in a
later section of this report. The points in the central portion of the
graph represent stream rises during which there occurred.both surface
runoff from and infiltration into the permeable portion of the basin.
25
-------
CO
UJ
O
O
ID
UJ
O
s
oc
ID
CO
Plum Brook
oNov.thru Apr. No Antec. Precip.
„ Precip. within
15 hrs.
May thru Oct.
No Antec. Precip.
SRO from A; only, (A-,/A = 0.1)
0
.5 1.0 1.5 2.0
STORM PRECIPITATION, INCHES
Figure 5. Surface runoff vs. precipitation.
26
-------
CO
LLJ
1C
O
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CO
LU
n:
o
uu
U_l n
o .3
CO
Nov. thru April
No Antec. Precip. —
Nov. thru April
Precip. within 15hrs
May thru October
Precip. within 15hrs
May thru October
No Antec. Precip.
.5 1.0 1.5 2.0
STORM PRECIPITATION, INCHES
Figure 7- Surface runoff vs. precipitation.
28
-------
that the initial precipitation excess would become retention. For each
storm the value was selected which produced the best time relation be-
tween the beginning of precipitation excess and the beginning of surface
runoff. For all rain storms over 1 in. in magnitude the values of R for
the permeable portion of the basins varied from zero to 0.2 in. and the
average value for summer was 0.15 in. and for winter 0.09 in. The latter
value was rounded to 0.10 in. for practical application to flood predic-
tion. • For the impermeable portion of the basins a value of 0.05 in. gave
satisfactory results.
A value of retention of 0.15 in. determined as described above represents the
weighted average of the retention on the impermeable portion of the basin
(R^) and the retention on the permeable portion of the basin (Rp)- If a
value is assumed for Ri? then the value of Rp can be computed from- the
following equation: *
RA =
R A
P P
(3)
in which A is the total area of the basin, and Ai and Ap are the areas of
the impermeable and permeable portions, respectively. The solution will
be illustrated for Red Run by assuming Rj_ = 0.05 in. and making use of
the fact (demonstrated later) that ^ is 10 percent of the total area.
Then
0. 15A = 0.05A.
+ R A
p p
and
0.15 ?= 0.05A./A + R A /A
i p p
Since A.J/A = 0.1 and A /A = 0-9,
le is presented to illustrate t
is found to be 0. 16 in. This exam-
ple is presented to illustrate that even for a basin in which 10 percent
of the area is impermeable, there is only a small difference between R
and Rp, probably less than the uncertainty in the estimated value of R.
For less urbanized areas such as Plum Brook and Big Beaver Creek, this
difference would, of course, be even smaller.
IMPERMEABLE AREAS
.36
Shortly after the time when Hortori' recognized the now obvious fact that
surface runoff is produced when precipitation intensity ex-ceeds the infil-
tration capacity of the soil, studies made by the senior author on storm
runoff from small watersheds seemed to indicate a flaw in this concept. 2^"
Typical flood hydrographs were being observed on watersheds having such
permeable soils that surface runoff was assumed to be impossible. It was
29
-------
only after surveys of the stream channel area were made that it became
clear that the precipitation falling on the streams themselves and the
immediately adjacent banks produced unit hydrographs exactly similar to
those from adjacent, less permeable areas. This discovery indicated that
a thorough study of the characteristics of infiltration capacity must
take into consideration the nearly 100 percent runoff from water surfaces
and other impervious areas. Similar findings have since been reported
for other drainage basins.-^'
Hie extent of the effectively impervious area is of particular interest
in this research, because it appears obvious that this is one of the im-
portant factors whereby urbanization influences storm runoff. It is rec-
ognized that the effectively impervious portion of the basin may be larger
during wet seasons than during dry periods. This might be caused by in-
creased area of water surfaces on the basin but also by portions of the
permeable portions of the basin that might become nearly impervious after
prolonged rainfall or snowmelt. It was decided to approach the problem
by attempting to determine what portion of ,the drainage basin acted as an
impermeable area under all conditions. This area, expressed as the per-
centage of the total area, has been called the 'hydrologically significant
impermeable area, ' (HSIA.). (HSIA = 100 (A^A), where Ai is the imperme-
able area and A is the total area of the drainage basin). It was reasoned
that this area could be computed from hydrographs produced during periods
of very high infiltration capacity when the entire hydrograph of surface
runoff can be attributed to the rainfall minus retention (P - R.^) on the
HSIA..
Although such hydrographs can be found from a search of the records, they
can also be isolated readily by plotting the surface runoff against pre-
cipitation for all stream rises of record, as shown in Figures 5, 6, and
7. The hydrographs, which are of the type mentioned above, provide the
points on the lower right side of Figures 5, 6, and 7, near the line
labeled 'SRO from A$_ only. ' The slope of this line (Ai/A) multiplied by
100 gives the value of the HSIA.. The two drainage basins for which, such
data are plotted in Figures 5 and 6 are relatively unurbanized when com-
pared with highly populated areas. They contain farm land, some scat-
tered housing along main roads, and several industrial parks. The popu-
lation density of Plum Brook is about 700/sq. mi and that of Big Beaver
Creek approximately 800/sq. mi. (Population density data were compiled by
the Detroit Metropolitan Area Regional Planning Commission.) It may be
seen that the lines drawn for an HSIA of 1.0 percent of the basin area
and for an assumed value. of retention on the impermeable area (R^) of
0.05 in. form envelopes that include nearly all of : the lowest points.
(if RJ were assumed to be zero, the line for the same hydrologically
significant area would have the same slope but would pass through the
30
-------
origin. ) A similar plotting for Red Run, shown in Figure J, reveals
an HSIA area of about 10 percent of the drainage basin. The popula-
tion density of the Red Run basin is approximately 7500. Thus, it is
indicated that, for these particular basins, the HSIA increases in
about!the same way as the population density. As additional rainfall
and runoff data are analyzed for other drainage basins, the relation of
the HSIA to other measures of the degree of urbanization, as well as to
natural physical characteristics of the drainage basin, will be examined.
The small amount.of HSIA in the Plum Brook and Big Beaver watersheds makes
little difference in the computed values of fav when there is a substan-
tial contribution of surface runoff from the pervious portions of a drain-
age basin. However, even such small amounts of HSIA as 1.0 percent can,
if ignored, lead to the computation of meaningless values of fav for those
summer or autumn floods produced'entirely by runoff from impervious areas.
The importance of taking into consideration the runoff from even a small
percentage of impervious areas is illustrated by the stream rise for the
River Rouge at Birmingham shown in Figure 2. It may be .seen that the sur-
face runoff started on October 6 at OJOO, whereas the rainfall excess did
not begin until 0600. This is illogical, unless the runoff from the HSIA
is included as a separate item. If the HSIA. were 1.8 percent, which is typi-
cal for that population density, then the rainfall occurring before 0600
would have produced a runoff from the impermeable area of the amount shown
by the hatched area in Figure 2. The beginning of surface runoff from
precipitation excess on the permeable area would then be at 0700, or an
hour later than the beginning of rainfall excess, and the entire hydro-
graph becomes a reasonable output from the precipitation pattern.
Values of HSIA were determined for 12 basins in Southeastern Michigan.
TOien plotted against population density the trend of the points was
closely represented by the following equation
HSIA = l-38Pd
where Pfl is the population density in thousands of persons per square
mile and HSIA is in percent of total area. •
31
-------
SECTION VII
niFILTRATION CAPACITY
The prediction of the peak flow resulting from a,specific input from
rainfall and/or snowmelt requires knowledge of the portion of the
total input which will be abstracted as infiltration and surface re-
tention. Of these abstractions, infiltration is by far the most impor-
tant. Unlike the shapes of the unit hydrographs which depend primarily
on the areas of the drainage basins and on other physical character-
istics of the basins which are not related to a particular geographical
region, the* infiltration capacity(f) depends to a large extent on the
soil in the particular location where flood predictions are to be made.
Therefore one of the principle parameters sought from hydrograph ^analy-
sis is the infiltration capacity and its seasonal variation.
The determination of infiltration capacity for a runoff event may be
described with reference to Figure 1. The amount of surface runoff
which is the hatched area for each stream rise is computed first, then
the average infiltration capacity during a storm is computed by trial
by finding the value which makes the precipitation excess, Pe on the
rain intensity diagram, eq.ua! to the surface runoff, after taking into
consideration retention and runoff from impermeable portions of the
basin. Sketching this value of infiltration capacity on the hyeto-
graph as shown in Figure 1, then establishes the duration of precipi-
tation excess which becomes an important parameter in the formation
of the surface runoff hydrograph. Infiltration capacity decreases
during rain storms as illustrated by the two rains in Figure 1 and it
also varies seasonally as shown in Figure 8.
Because of the variation of f with time during a rain the actual value
during a particular rain depends on whether there has been recent ante-
cedent precipitation. The scatter of points in Figure 8 is largely due
to this factor. The only method of determining the average infiltra-
tion capacity of a particular drainage basin is by means of hydrograph
analysis. This type of analysis yields the average value of infiltra-
tion capacity ( fay) for each stream rise. The variation of f with
32
-------
>
CO
-P
•H
t)
o
o
•H
•8
!4.
s
•H
o
•H
-P
03
•H
o
CQ
03
0
CO
CO
LCi
LPk
'AllOVdVONOIlViiniJNI
33
-------
time can be obtained from a closely spaced series of stream rises as
shown by an example in the next section on "Unit Hydrographs." ,
Each of the more than 200 points in Figure 8 was determined by hydro-
graph analysis on the 16 basins in Southeastern Michigan. These
basins have quite similar soils and therefore the infiltration capacity
did not vary greatly from one basin to. another. The average value for
summer which included June, July, August, and September was 0.^ in./hr
and the average for winter, which included the other eight months, was
0.10 in./hr. The values should not be used in other regions. Enough
hydrographs must be analyzed in any region to establish the order of
magnitude of infiltration capacity. The computed value of infiltra-
tion capacity depends to some extent on whether retention and runoff
from impermeable areas are included in the computations. The latter
factor becomes particularly important for highly urbanized areas where
the impermeable area becomes large. In the analysis, the total sur-
face runoff (SRO) is taken to be equal to the surface runoff from im-
permeable portion of a basin (SROj_) -plus that from the permeable portion
(SROp). These values are defined by the following equations
= p -
'(5)
SRO = P - F - R
P P
where P is the average precipitation, R. is the retention on the im-
permeable area, F is the total infiltration and Rp is the retention on
the permeable area. A detailed description of the determination of the
impermeable area and the retention has been presented in the section on
Hydrograph Analysis.
The computations for determining the infiltration capacity for the
first stream rise in Figure 1 are presented here to illustrate the
procedure.
Area of Drainage Basin, A = 22.9 sq mi
Impermeable area, Aj_ = .02A '.
Retention on impermeable area, R^ = 0.05 in.
Retention on permeable area, Rp = 0.15 in.
Weighted average precipitation, P = 1.35 in.
The first step is the computation of the surface runoff. This requires
that a line separating surface runoff from gfoundwater discharge be drawn
-------
such as b^bg in Figure 1. The procedure used to do this was explained
earlier in this report in the section on hydrograph analysis. In the
example shown in Figure 1, it was also necessary to draw the line xy
to separate the hydrograph produced by the first rain from one resul-
ting from the second rain. This is done by sketching a line having
the same form as sb2 which represents the same release of storage as
would have occurred at the end of the first hydrograph. Ordinates from
the first hydrograph were tabulated for 2-hr intervals as shown in col-
umns 1 and 2, Table 2. The surface runoff in inches on the basin is
then computed from the summation of column 2 by multiplying by the num-
ber of second in 2 hr dividing by the area of the basin and multiplying
by 12 as shown as follows.
SRO =
x 2 x 3^00
22.9 x 5280 x 5280
x 12 = .^68 in.
The surface runoff from the impermeable area (SROj_) is computed using
Eg.. (5) and then converting to inches on the entire basin by multi-
plying by the ratio of the impermeable area to the total area as follows
SRO = —
(7)
= ' .02 (1.35 - .05) = -026 in-, on A
The surface runoff from the permeable area (SROp) is then the difference
between the total surface runoff and SROi and is (.1)68 - .026) = .¥)2.
expressed in inches on the total basin. This value is converted to
inches on the permeable portion of the basin as follows:
SRO = j-x(M2) =
P
= .1451 in. on A
The total precipitation excess on the permeable area (Pep) is then ob-
tained by adding the retention (Rp) as follows:
• = SRO + R
ep P P
= .14-51 + .15 = .601 in. on A
P
(8)
-------
Table 2. COMPUTATIONS OF SURFACE RUNOFF AND UNIT HYDROGRAFH
ORDIMTES FOR RAIN OF APRIL 1, 1959, ON PLUM BROOK
(Drainage Area = 22.9 sq mi)
Number of
2-hour
Intervals
1
2
3
4
5
6
7
8
9
10
11
12
15
14
15
16
17
18
19
20
21
22
23
2k
25
26
27
28
Average rate of
surface runoff,
cfs
28
116
230
294
308
320
(324)
320
308
280
240
190
148
120
96
80
66
56
48
4o
36
32
26
22
18
14
10
7
3
3456
aTypical computation of unit hydrograph
28 cfs
Unit hydrograph.,
cfs/sq. mi. /in.
2.6
10.8
21.5
27.4
28.8 .
29.9
(30.3) (peak)
29.9
28.8
26.1
22.4
17.7
13.8 :
13.8
9-0
7.5
6.2
5-2
4.5
3.7
3.4
3.0
2.4
2.1
1.7 '
1.3
0.9
0.7
0.3
ordinate :
in
22.9 SOL mi x 0.468 in.
36
-------
The infiltration capacity of the permeable area is now computed by
trial. The final trial computations are shown in Table 3 in which
hourly precipitation is given in column 2, the assumed infiltration
capacity in column 3 and the precipitation excess in column 4. The
total of column 4 multiplied by 60/60 to convert from inches per hour
to inches is 0.600 which agrees with the value of Pep computed above.
Therefore the infiltration capacity of the permeable portion of the
basin is 0.114 in./hr. This procedure applies to simple hydrographs
which can readily be assigned to a particular, rain. For more complex
storms the method involves an application of the unit hydrograph and
will be described later.
Table J. COMPUTATION OF INFILTRATION CAPACITY
Hours
17 •
18
19
20
21
22
23
24
25
26
2?
28
Precipitation
intensity,
±n./nr.
.00
.07
.26
.16
.18
.32
.25
.03
.ok
.01
.03
.00
Infiltration
capacity,
in./hr.
.114
.111*.
.114
,.114
.114
.114
.114
.114
.114
.114
.UL4
.114
Precipitation
excess,
in . /hr .
-
' -
0.146
0.046
0.066
o. 206
0. 136
-
-
-
-
-
0.600
37
-------
SECTION VIII
UNIT HYDROGRAPHS
By far the major portion of the research effort was deyoted to studying
unit hydrograph characteristics. This procedure in its simplest form is
based on the assumption that the important characteristics of a surface
runoff hydrograph for any basin are essentially constant if the duration
of 'precipitation excess is less than some critical value, that the ordi-
nates of this hydrograph vary linearly with the magnitude of rainfall
excess and that various complex rainfall or snowmelt inputs can be trans-
formed into a complete hydrograph by a linear additive convolution
process.
The unit hydrograph is obtained from a surface runoff hydrograph pro-
duced by a precipitation excess having a duration less than some critical
duration to be defined later by taking the surface runoff ordinates for
successive arbitrarily selected uniform time intervals and converting
them to cfs per square mile per inch of rainfall-excess. This is done by
dividing the ordinates in cfs by the area of the drainage basin in square
miles and by the rainfall excess in inches as illustrated in Table 2.
The ordinates of a unit hydrograph could also be expressed in percentage
of total surface runoff. In this form the graph is often referred to as
a distribution graph. This latter form is not used in this report. The
example shown in Table 2 is for the first of the two hydrographs of Fig-
ure 1. In Figure 9 are shown five unit hydrographs from this same basin.
These five show typical variations for a basin. It should be noted that
they are relatively consistent even though the total surface runoff for
the largest of the five was about four times the smallest as shown by the
values of surface runoff tabulated in Figure 9. The average unit hydro-
graph is also shown in Figure 9. ::
The most important characteristic of a unit hydrograph is its peak be-
cause this is the value used to predict peak flood flows. However, in
order to construct a complete flood hydrograph the entire set of unit hy-
drograph ordinates and abscissas must be known as shown in tabular form
in columns 1 and 3 in Table 2 or graphically in Figure 9. A very useful
-------
PLUM BROOK
(Area 22.9 Sq-Ml.)
40
30
X
Z
•X
S
in
20
10
SURFACE
DATE RUWOFF (IN-)
—O— April 29,1956 1.008
—A— MAY 6.1956 -283
—O— JULY 11.1957 -378
—0- April 1.1959 .468
—O— JUNE 16,1960 -423
Ave. Unit Hydrograph
30 40 SO
TIME (Hrs.)
60
10 20
Figure 9. Five unit hydrographs from the same,basin.
70
39
-------
application of the unit hydrograph is in determining the progressive vari-
atipn of infiltration capacity during a closely spaced series of storms.
This process also provides a test of the accuracy of the unit hydrograph
procedure by applying the unit hydrograph for a drainage basin to a com-
plex rain storm in which the contributions from various portions of the
rainfall excess must be added taking into account the sequential time of
occurrence of the periods of rainfall excess and comparing the computed
hydrograph with the actual hydrograph. A typical example is shown in
Figure 10. When a unit hydrograph is applied to successive portions of.
a long complex series of rains such as those in Figure 10, the infiltra-
tion capacity is computed for each separate stream rise and adjusted by
trial until the summation of the overlapping hydrographs fits the actual
hydrograph. The curve of infiltration capacity (fav) shown in Figure 10
is based on five values of fav determined for the five bursts of rainfall.
The volumes of precipitation excess (Pe) are shown by the hatched areas
of the hyetograph above the infiltration capacity curve. The derived hy-
drograph shown by the dashed line was computed by applying the average '
unit hydrograph for that basin five times to the five values of precipita-
tion excess (Pe) and adding overlapping ordinates. This latter process is
sometimes called convolution.
For the purpose of relating unit hydrograph shape to basin characteristics
the average unit hydrographs for each basin was defined in terms of the
peak (q-pA.), the period of rise (Tr), the significant period of rise (tr),
the width of the graph at the base (W0) and at 25, 50, and 75 percent of
the peak discharge (W25, ¥50, and ¥75). These parameters are defined
graphically in Figure 11. The average values of. these parameters for 53
of the basins* are presented in Table k. The numbers in column 1 of Table
k provide a convenient method of identifying each watershed by reference
*Sixteen of the 69 basins listed in Table 1 are not included in Table 4.
Basins l£ and 59 had not experienced enough large rains to define the unit
hydrograph. Basins 5, 16, and k2 were omitted because the flood hydro-
graphs were double peaked thus making it difficult to define their shape.
The double peaks appear to be caused by highly urbanized areas in the
lower reaches of an otherwise less urbanized basin. Eleven basins in the
Austin (kk-k6), Dallas (Vf-5l), and San Antonio (60-62) regions were
omitted because they displayed runoff characteristics which indicated high
urbanization even though their values of Pd are very low. It is suspected
that the storm drainage systems of these eleven basins may have been de-
veloped in advance of anticipated urban growth. As with the double peaked
basins, a detailed study of the drainage system is expected to reveal the
causes of the anomolous behavior. This additional research was beyond the
scope of this project.
40
-------
29
April 1956
Figure 10. Reproduction of a complex hydrograph for the flood
of April 27-50, 1956 on Plum Brook (area = 22.9 sq mi).
41
-------
CO
LU
u
I
<
o:
u_
o
CO
UJ
Centroid of Precipitation
Excess
Precipitation Excess
0.75qDA
0 0.50qDA
or MPA
Lag
Duration of Precipitation
Excess
Period of Rise
Significant Period
of Rise
O.IOqeA
'PA
L
-W0
TIME
Figure 11. Definition sketch.
42
-------
cu
~ a1 ft ca
CO -i-l
•• _ - '^-•S h
CQ a1 c! co c!
ca ca O CH <8- CH
as .H o o .o
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a
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o;
t3 _ rr\ (TV p o p t—
s 3i^:^^'7d 3
i _
ft IfN N^\ C*- VD K^\ C*-
~ oj p H H ir\ ir\ tov
"Eb
•H OJOOOOrHHO O
OOOVO
t~-HIAQ>
tr\OCO-*
b~ |0v OJ
o
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8O O O O ON VD O
OJVD OOOOOJONO
IPvVD^XO ONrH C^-ITNCQ O
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(Ov 10, '"^ -^
VD
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o c9 o
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1T\
HHrHHOJOJOJlfN
CO
o
ir\
43
-------
PM
K
1
O
I
EH
•p
^
^
PM
vo o o vo t- KN
o o o o o o
*. |^_ *-v |Si__ ^-*
-* KN VO H H C-
O O O H H O
^IS^rr0 tITNCVIVO H-* HVO
iH r- 1 OJ rH
i-l -=r rH
OJCOiHOONCO
rHrH
rHH
O 0 O H H H
t— C— KN t—
VO CO CO H
OJ O H KN
KN H CO H H CO
KN OJ -3- -* ON in
O O O O O* O
Si
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O t- CV1 CVJ
ir\ vo -=)- -*
OJ H [Ov ^r^
o
KN
O O O O O O
•H
CO
fl
0)
•O
aj
H
I
ft
-------
1
1
5J
O IT\
O • C- IS- CO N"\ C—
ir\ ir\ H o oj H
tc\ O ,O
tc\ H o r^\ vo vo
ON ON fTN. H KN OJ
ir\ b- ir\ ir\
CO 00 OJ O\ O\ OJ
vo ir\ ipv H tr\ -=*•
rH i — 1
VD
O O 00 H O O
IOv -4" O -* CO H
hO\ K"\ H H H
N^\
o ITS ir\ ^r^ o ipi
OJ O OJ CO ON C-
tf\ vo ro\ o OJ OJ
o o o x~\ t<~\ ir\
O lf\ O CO CO OJ
-* -* t^v O H OJ
t- vo.. .
CO VD t- VO t--*
lf\ ITN VO OJ N^\ O>
H -* H H
CT\ VO IA O -* O
K% t- CO rH ON H
O\ -H- CO -*
OJ ON VO O N^ ON
O -* IT\ iH OJ H
OJ H
S3RRK®
H O
OJ fe1^
H H
0^
SJ^
O O
ir\ H
«-\ J-
o
-P ON. O
ca H
pj
d)
Q
a
0
•H
-P O O
S H OJ
Pi
O
P^
0
•s
•H If N O
<3) H H
a H oj
0)
43
^
O O
VD O
OJ OJ
SN.S
»OJ
0
CO CO
H
-H-VO
C~-
lfNOHH-4-HHOJHOJH o
^wss&sswz®
oo o cvj i^*\ co cvi OJ K^\ cvj ^c\ OJ i — 1
\c\ o^ ^^ ^? ^5 o c^ ^\ ^\ oo
"H"IA^^""""CUH
•0
fl ITN o ir\ ir\
•HOlS-OOOirNOirNOlf\OJOJ
Si^^ ^o}^. SI o7^
OONVO -f}"^Q CO ITNKA
IT\ IT\ OJ OJ -=T t— OO H 1P\ IP\ r— 1 N~N.
f^O-^3^^O.H
r^oJOJOJOJOJOJOJK^^f^lr^vo
45
-------
to Table 1. All of the parameters are used to define, the shape of a unit
hydrograph. However, tr is also the critical time characteristic which is
related to the maximum duration of rainfall excess which will produce a
single unit hydrograph. •
It was previously discovered that for a group of watersheds from within
the same large watershed system there was a relatively consistant rela-
tionship between the two most important unit hydrograph characteristics,
their peaks and periods of rise, and the areas of the drainage basins.9
The research reported here provided an opportunity to determine if such
relationships, exist for a wide variety of watersheds from different re-
gions. The relations between unit hydrograph peak and area and between
the various time parameters and area were found to be quite consistant
for all watersheds if the degree of urbanization was accounted for. At
this stage in the research the population density (Pd) seems to be a very
significant factor in expressing the degree of urbanization. Satisfactory
practical relationships were developed for varying population densities.
This factor appears to be much more important than such factors as water-
shed shape, channel slope, or roughness.
The investigation of the effect of population density was carried out by
arranging the basins in the three groups shown in Table ij-. The first
group "High Population Density" consists of 15 basins having population
densities greater than 5,590 persons/sq mi. The average value of Pa for
this group of 13,300 persons/sq mi. The second group in Table if- "Low
Population Density" consists of 23 basins having population densities
less than 1,200 persons/sq mi, the average for this group being 539
persons/sq. mi. Finally Table It- shows 15 watersheds under the heading
"Intermediate Population Density" for which values of Pd were in the range
1,400-4,700 and the average Pd was 2,689 persons/sq mi. ,
The unit hydrograph peaks are shown plotted against Pd in Figure 12. The
symbols used in Figures 12-20 are explained in Table 5. Best fit equa-
tions based on the use of a least squares analysis were derived for the
three groups of points representing three degrees of urbanization in Fig-
ure 12. These are Eqs. (9), (15), and (13) in Table 6 and the straight
lines represented by these equations are shown in Figure 12. Table 6 also
shows the number of points used to derive each equation, the average P,
for each graph and the linear correlation coefficient (r). All values of
r are well above the value needed to indicate a one percent chance that
the linear relationship is accidental. Also shown in Figure 12 is a
dashed line which represents Eq. (ll). In the derivation of Eq.;(ll) two
points representing basins 25 and 28, both from the Huntington Bayou area'
of Houston, were omitted. These points fell well below the other high Pd
points as can be seen in Figure 12. It might therefore be inferred that
46
-------
Symbol
A
D
O
A
0
0
Table 5. LEGEND FOR FIGURES 12-20
Pd*? 5590, Pd Av. = 13300
All basins
Pd g 1200, Pd Av. = 539
All basins
1400 ? Pd § 4700, Pd Av. = 2763 or 2689
All basins
Pd g 1200, Pd Av. = 610
.Michigan basins
Pd =£ 1200, Pd Av. = 488
All basins except Michigan
1400 S Pd S 2700, Pd Av. = 2038
Michigan basins
1400 =S Pd ^ 4700, Pd Av. = 2926
All basins except Michigan
Michigan basins
All basins except Michigan
Values obtained from Figures 13,
16 and 19 at A = 10 sq. mi
for all drainage basins.
Values obtained from Figures 13,
16 and 19 at A = 10 sq mi
for Michigan basins only.
Figures
12, 13, 15, 16,
18, 19
12, 15, 18 .
12, 15, 16, 13,
19
13, 16, 19
13, 16, 19
13
13
14, 17, 20
14, 17, 20
14, 17/20 ~
14, 17, 20
*P,, = Population density in persons/sq. mi.
47
-------
CS
•8
•d
bo
•H
03
Q)
03
0)
P)
s-
I
I
OJ
.§
Vd
'( b) >|V3d
1INP
48
-------
oo
v/d
•uiniAfbsrSJD -'( b)>iv3d
ilNfl
49
-------
O
CD
O_
t
W
a
d
o
•H
•8
H
I
ft
0)
ID -0
Q_ +D
£ s
D_ ta
0)
•H
•U|/'!W*S/'SJO
°wd
' V
>|V3d
UNfl
50
-------
05
0>
0>
w
o
•H
JH
(D
A
O
•H
•H
ra
H
0)
'(
jo aomd INVOIJINOIS
51
-------
'( J
dO
o
1NVOHIN9IS
52
-------
'(
3SId JO
1NVOWN9IS
53
-------
sjnoH'(Jl) 3SIH JO
54
-------
'(Jl) 3SIH dO
55
-------
05
-p
ra
a
§
-H
—• to
00 >
Z a)
LoJ co
f~\ -rl
S~i
8
0)
°WJ
'( 1) 3Siy dOQOI^d
56
-------
Table 6. EQUATIONS AND STATISTICAL PARAMETERS FOR LIKES
ON FIGURES 12, 13, 15, 16, 18, AND 19
N = Number of drainage basins
Pd = Population density in persons
per square mile
r = Linear correlation coefficient
(1)
Eq. No.
9
10
11
12
13
14
15
16
17
18
19
20
21
22.
23
24
25
26
27
28
29
30
V
V
V
V
V
V
V
V
t
r
t
r
t
r
t
r
t
r
t
r
t
r
t
r
T
r
T
r
T
r
T
r
T
r
T
r
(2)
Equation
- 448
= 834A >445
= 835A"'400
= 1032A~*
= 1034A~*
= 184A-458
- 400
= 162A * '
= 375A-558
- 4OO
= 269A 'w
= . .274A
= .274A'457
= .325A
. „
= .325A'^'
486
= 1.292A
= 1.447A'457
= .572A'655
= .849A'457
= .383A'425
= .384A-457
.446
= .449A
= . Vi8A
= 1.688A'504
- 1.968A'457
(3)
N
15
15
13
13
23
23
15
15
13
13
15
15
23
- 23
14
14
13
13
. 15
15
23
23
(4)
Av. Pd
13,300
13,300
14,250
14,250
539
539
2,689
2,689
14,250
14,250
13,300
13,300
539
539
• 2,763
2,763
14,250
14,250
13,300
13,300
53.9
539
(5)
r
-0.81
-0.92
-0.77
-0.95
0.87
0.81
0.75
0.90
0.91
0.85
0.78
57
-------
Table 6 (concluded). EQUATIONS AND STATISTICAL PARAMETERS FOR
LINES ON FIGURES, 12, 13, 15, 16, 18, AND 19
(1)
(2)
(3)
(4)
(5)
31
32
33
34
35
36
37
38
39
40
Tr = .835A'655
T = 1.196A'457
r
Michigan Basins
= 65. OA~'25
q^ = 112A"400
q^ = 159A"'595
-.
Pfi
tp - 3.78A'26
t = 2.18A'457
I* . *
Tr - 4.18A'54
T ,= 3.06A'457.
14
14
Only
9
9
4
4
9
9
.9
9
2,763
2,763
610
610
2,038
2,038
610
610
610
610
: 0.88
0.95
0.99
0.90
0.88
58
-------
some unusual conditions exist on these two basins which can only be de-
termined from more detailed investigation. If these two points are •jgmit-
ted the average population density of this group of points becomes ll|.,25Q
persons/sq mi.
Hie lowest group of points in Figure 12 is for drainage basins having pop-
ulation densities less than 1,200 persoris/sq mi, and with an average den^" -
sity of 539 persons/sq mi. These points are plotted with a distinguishing
symbol which may be identified in Table 5- '
It is significant to note that the slopes of these lines as determined by
a least squares analysis, are -Q.kkS and - .kl.6 for the upper two lines
which represent highly populated basins and -O.lf-58 for the lowest line
which was derived for low values of P^. -A similar set of unit hydrographs
peaks derived from the mathematical model (discussed in the next section)
produced a slope of -O.k when plotted in the same fashion.
The center group of points representing an average population density of
2,689 when fitted by least squares (Eq. (15)) produced a somewhat steeper
line having a slope of -0.558. One explanation for this difference in
slope is that the development of a -drainage system is probably not always
a gradual process related directly to gradual changes in population den-
sity but it may lag behind or jump ahead of population increases thus
creating anomolous situations. Because of the fact that there is so much
evidence that the slopes of these lines should, be about -0.il-, the least
squares method was applied again to find the best fit with a slope of -O.U.
Five curves for which this was done, are shown in Figure 1J. Three of
these represent the upper three groups of points from Figure 12, the
fourth and fifth lines represent the low population basins (average popu-"
lation density is 610) and intermediate population basins (average popu-
lation density is 2,038) in Southeastern Michigan. The line representing
Eq. (llj-) for all of the low population basins coincides with the line rep-
resenting Eq. (36) for the intermediate population basins in Southeastern
Michigan. The equation numbers are shown in Figure 13 to enable the
re'ader to locate the corresponding equations in Table 6. The average pop-
ulation densities are also shown on the graphs. The lines for Southeastern
Michigan were included' in order to provide for the most accurate applica-
tion to that region. These lines fell somewhat below lines representing
all of the low population basins. The difference between'the more general.
lines and the one for Southeastern Michigan will be discussed in the next
portion of the report.
Although interpolation between the lines showing unit hydrograph peaks for
various population densities might be carried out on Figure 13, a new pa-
rameter q which eliminates area as a variable was found more useful
59
-------
for this purpose-. This factor is defined by the following equation
.40
V
in which qp^ is the unit hydrograph peak for any area A in cfs/sq' mi/in.
and QpAo is the corresponding unit hydrograph peak for a selected: base
area AQ. The value <3pA0 ^s obtained for any basin from its value of q_^
by means of Eq. (kl} or by following along a line such as those in Figure
13 to the base area A . In this case AQ was chosen as 10 sq mi but any
other size of basin could be selected for this purpose. The resulting
points may then be plotted against population density as shown in Figure
14. It will be seen that the trend is quite clear and that the basins
from Southeastern Michigan (white diamonds) follow a slightly different
trend than those from other areas (black triangles). Also shown in Figure
J.h are values of P^ vs. qpA0 read from the six straight lin'es in Figure
13, two of which are coincident. The symbol 0 is used for the four points
derived from all 53 basins and the symbol 0 for the points from the two
lines representing the 16 Southeastern Michigan basins.
The solid curve shown in Figure lit- is drawn to best represent Southeastern
Michigan. A more general curve such as the dashed line represents all
basins. The curve in Figure li)- was used to derive design curves for South-
eastern Michigan.
The relation of the unit hydrograph time parameters to area and population
density were analyzed and correlated in the same manner as the peak dis-
charges. The values of tr, the significant period of rise (see Figure
11), are plotted against area in Figure 15. The top line represents
the "Low Population Density" basins and the lower two lines represent
"High Population Density" basins. These lower two lines differ only in
the omission of two data points from the dashed line as previously de-
scribed. Equation numbers are shown to aid in identifying the correspon-
ding equations and statistical parameters in Table 6.
The slopes of these lines are 0.^22, O.kk^, and 0'. hQ6, respectively,
•which are again similar to the slope derived from unit hydrographs de-
veloped by the model which was 0.^37- Also, as in Figure 15 the inter-
mediate population basins showed a different trend. Figure 16 shows
the best fit lines converted to a common slope of 0.^37 and Figure 17
gives the relation of
with population density. The correlation
coefficients and equations for the regressions are shown in Table 6.
Similar correlations for Tr are presented in Figures 18, 19, and 20.
The equations and statistical parameters are shown in Table 6. The
60
-------
corresponding derivations for WQ, Wgc, ¥,-Q, and ¥7^ are not presented
in this report but the final results are -given as described in the
following paragraph.
In order to provide a more convenient way of deriving the form of the
unit hydrograph for any area and population density curves such as those
shown in Figures lU, 17, and 20 were used to provide values of qpA, tr,
Tr, ¥75* ¥50* W25> and W0 for an area of 10 scl mi for various population
densities. Lines with the proper slope were then constructed as shown
in Figures 21-27 for round numbered values of population density. These
curves make interpolation much easier than in the original graphs. Re- .
suits from streams in Southeastern Michigan were given the most weight
in deriving these curves. Therefore the curves apply more accurately
there than elsewhere. The use of these curves will be demonstrated in
the section titled Predicting Flood Magnitudes and Frequencies.
61
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68
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SECTION IX
MATHEMATICAL MODEL
It became apparent early in the work that an analytical approach to the
surface runoff problem would be an exceedingly valuable goal if it could
be developed well enough to simulate natural watershed responses from
precipitation inputs. Consequently, a mathematical model was constructed
which fulfills these requirements to a certain point..(the next step is
to incorporate more complex drainage networks into the model). The model
simulates the runoff process in a conceptual watershed consisting of a
plane rectangular basin with a stream flowing in the middle. The model
performs a two phase transformation on the rainfall excess input. For
the first or land phase, the kinematic wave theory is used to transform
the precipitation excess into an overland flow hydrograph. For the
second or channel phase, the complete equations of motion are used to
transform the overland flow hydrograph, which is now taken as the lat-
eral inflow, to the main channel, into the output from the watershed.
These equations are formulated in a direct implicit method using a
fixed rectangular grid, and the resulting system of simultaneous non-
linear finite difference equations is solved using the generalized
Newton-Raphson procedure on an IBM 360-67 digital computer. The model
functions well even in the difficult conditions of very low initial
flows (less than 1 percent of peak flow). The results are independent
of any reasonable downstream boundary conditions imposed.
The model was investigated first to determine the effect of the dura-
tion of rainfall excess ('tQ) on the peak discharge (Qp) and on such time
parameters as Tr, t , W5Q, ¥?5 (Figure 11) which define the form of the
unit hydrograph. This was done by running the model for various values
of t with the total input (D) held constant. These runs indicated
that the form of the unit hydrograph varies little when t0 < tp/2 and
for t < t the variation is small for practical purposes. The.meaning of
t is shown-in Figure 11. These results agree with those obtained from
natural watersheds where tr was used instead of t . The values of tr and .
t-n were found to be nearly the same in natural watersheds and tr is more
useful and easier to obtain.
69
-------
The second set of runs was made holding to to constant at a value less than
tp and varying the input D. It was found that the relation between peak
discharge.and input was expressed by the following equation
CD
G
If G is unity the relation between peak and volume is linear. For
trapezoidal channels the value of G was found to be about l.h whereas
for parabolic channels with the sides concave downward G was about 1.2.
It is believed that this latter shape tends to simulate natural rivers
and flood plains much more clearly than does a trapezoidal channel.
If the roughness is allowed to increase with increasing depth natural
conditions are simulated even more closely and the value of G in the
model was reduced to 1.05- These findings helped explain why many natu-
ral basins have a linear relation between peak and total input. The
effects of roughness, slope, and channel length on the value of ;G were
found to be much less important than changing the shape of the channel.
In addition to providing a better understanding of the storm runoff
process as described above,the model provides qualitative information
on the effect of certain parameters which are likely to have only a
limited range in gaged natural watersheds. For example, under urban
conditions some watersheds are changed artificially so that they are
much more slender than typical natural basins. The model enables one
to determine how such parameters as unit hydrograph peak vary as, the
shape of the model watershed is changed from nearly square to very
slender. Such results can be used to indicate what trends can be ex-
pected for the same parameters in real watersheds. Specific refer-
ences to comparisons of specific phases of the runoff process in the
model and in natural watersheds are given at appropriate places in
this report.
70
-------
SECTION X
FREQUENCY STUDIES
The determination.of the winter rainfall frequency is complicated in
Michigan by the fact that recorded precipitation may be snowfall rather
than rainfall and should therefore be excluded. Furthermore, rapidly
melting snow whether or not accompanied by rainfall produces precipi-
tation excess and such events should be included in the frequency anal-
ysis. Fortunately, including snowfall tends to compensate for leaving
out snowmelt in the frequency analysis of precipitation, however, the
extent to which they do so needed .to.be determined. The frequency in-
vestigation was made by analyzing 535 station-years of records from
16 stations in Southeastern Michigan. The recorded daily precipita-
tion and average temperature were determined from records of the
U.S. Environmental Data Service. For each winter season at each
weather station a continuous record was developed of the water equiva-
lent of snow on the ground by adding snowfalls and deducting daily
snowmelt. The recorded precipitation was considered to be snowfall
when the average of the daily maximum and minimum temperature was 32°
or lower. Snowmelt from rain was computed by means of Eq. "
P (T -32)
M =
and snowmelt resulting from atmospheric heat was computed by the degree-
day method using the following equation
M = 0-05 (T -3*0
9*
In these equations M is the water derived from melt in inches per day,
T is the average of the daily maximum and minimum temperature and P
is the 2U-hr rainfall. Examination of these two equations will show
that melt due to rainfall tends to be very small compared with that
71
-------
caused by atmospheric heat. Although Eq. (W) only indirectly includes
the significant amount of heat transferred to the snow by condensation
it has given satisfactory results in accounting for snowmelt in Michigan
based on the analysis of several hundred hydrographs from many drainage
basins. . . •
The recorded daily rainfalls used in this analysis have been corrected
to a 2U-hr basis by adding half of the largest rain which occurred on
an adjacent day. This procedure is based on the fact that the maximum
2k hr of rain could have included an amount recorded for an adjacent
day varying from zero to the full recorded value. Shown in Figure 28
are winter frequencies for rainfall plus snowmelt, for rainfall plus
snowfall (conventional recorded precipitation) and, for comparitive
purposes, the values for rainfall alone are also shown. In the lower
frequencies, up to about 30 years, the curve of rainfall plus snowmelt
is about 2 percent above the curve for rainfall only. The fact that
the correct curve (rainfall plus snowmelt) falls only 2 percent above
the conventional precipitation curve is accounted for by the fact that
including snowfall while excluding snowmelt nearly equals the effect
of including snowmelt while excluding snowfall. This is fortunate be-
cause a great deal of time and expense is involved in computing snow-
melt for a large number of years. It is, of course, not known, whether
the error would be this small in other latitudes or at other geographi-
cal locations at the same latitude. A frequency study was also niade
for 1950 station-years of conventional rainfall to determine if 535
station-years was long enough to provide a good estimate of frequency.
The 535 station-years used in the snowmelt studies are included in the
longer period of records and all stations are in a meteorologically
homogeneous area in Southeastern Michigan and Northwestern Ohio. The
longer period of record included enough larger rains to raise the! curve
an average of 2 percent for recurrence intervals up to 30 years. : The
two curves are shown in Figure 29. ' ;
It can then be assumed that the best estimate of the frequency of 'winter
rain plus snowmelt can be obtained by drawing a curve about 2 'percent
higher than the conventional curve for the 1950 station-years of record.
This corrected curve is shown in Figure 30 along with the curve giving '
frequencies of summer rains as determined from the 1950 station-years
of record. The winter curve of Figure 30 differs from the winter curve
in Figure 29 in that after adding 2 percent for snowmelt the month of
October has been included with the winter months and eliminated from'
the summer months. This was done because infiltration capacities (dis-
cussed previous^) and time-intensity rainfall patterns (discussed in
next section) in October were found to be more like the winter months
than the summer months.
72
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SECTION XI
TIME-INTENSITY RAINFALL PATTERNS
The analysis of rainfall and runoff for natural drainage basins smaller
then about 100 sq mi as well as for all sizes of urbanized basins requires
the use of rain intensities for time intervals smaller than 2k hr. The
unit hydrograph-infiltration capacity method of flood frequency prediction
requires the use of time intervals as small as one hour on natural basins
of about 0.2 sq mi, and for urbanized drainage basins of about 10 sq mi.
When drainage basins are smaller than about 0.2 sq mi for rural areas or
about 10 sq mi for densely populated areas the rainfall intensities must
be broken down to smaller than hourly time intervals. Although fre-
quency studies can be made for durations less than 2k hr there is much
less information available for these shorter durations. Therefore,
greater accuracy can usually be achieved by determining frequencies for
24-hr rains and then finding relationships between shorter duration
rains and 24-hr rains. This was done in two ways.38 One method was to
find the values of maximum continuous precipitation of various durations.
The other method was to determine the chronological time distributions of
rain storms about the maximum hour of rain. •
The maximum continuous accumulated rainfall was determined for durations
from 1 to 24 hr for 80 summer rains and for 44 winter rains. The' rains
were selected from a rain gage network located in Southeastern Michigan
maintained by the Detroit Metropolitan Area Planning Cpmmission and the
U.S. Environmental Data Service. Only rains having a 24-hr rainfall equal
to or greater than 1.5 in. were used, and the same rain storm was not used
twice even though it may have covered more than one of the rain gages.'
The period covered was the thirteen years from I960 through 1972.' The re-
sults expressed in percent of 24 hr rains along with the standard devia-
tions for each duration are shown in Table 7. The rather large standard
deviations indicate the considerable variations among individual values
for each duration. The values of P versus t are plotted in Figure 31.
Also shown in Figure 31 are values of P derived from U.S. Weather: Bureau39
frequency studies. In Figure 32 the same data are shown as'P/t versus t.
In this form the ordinates are percentage of 24-rain divided by the
76
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duration in hours and the curves have the form of the commonly used
intensity-duration curves. The equations for these curves have the fol-
lowing form
P/t
(t•+ B)
n
(WO
in which A, B and n are constants. If this equation is expressed in
logarithmic form (Eq. 45) it can be reduced to a straight
log P/t = log A - n log (t + B)
(45)
line by the proper selection of B. The optimum value of B was determined
by applying the method of least squares to derive linear equations for a
series of values of B and selecting the equation for which the linear cor-
relation coefficient was a maximum and -the standard deviation from the
regression was a minimum.
The resulting equations are, for summer,
P/t
and for winter
112.0
(t + 1.0)
1.026
(46)
P/t =
68..0-
For each of these equations the linear correlation coefficient was very
near unity thus indicating a highly linear relationship. The correspond-
ing standard deviation from the regression indicated deviations ifrom the
derived equations of less than 2 percent. Among the individual values the
most interesting item of information determined from this analysis is that
for summer rains the maximum hour of rain included on the average 55 per-
cent of the 24-hr total while for winter the rainfall during the maxi-
mum hour was about 24 percent of the 24-hr rainfall. Separate analyses
by months indicated that this change takes place quite abruptly at the
ends of the two seasons. Additional analyses made by dividing the rains
into two groups according to size indicated that the intensity patterns
were independent of magnitude of rainfall.
80'
-------
Typical hyetograph ordinates (P^/t) can be derived from these curves or
from Equations (k6) and (kj} by assuming that the rain for any duration
is the maximum for that duration and also contains all of the maximum
shorter durations. A convenient way to do this is to express time (t) in
terms of uniform time intervals (At). Then the rain expressed as percent
of 2^-hr rain for a duration NAt is NAt (P/t)jj-. If one subtracts the
rain for a duration which is smaller by the amount At ((H-t)At (P/t) ),
the difference is the rain during the Nth time intervals, ((P^/At )jj A t ).
In equation form this relationship becomes
(Ph/At)NAt = NAt (P/t)N - (N - l)At
Values of (P/t) are computed from Equations (ij-6) and (^7), (P/t)N being
the value of (P/t) at time t = NAt and (P/t)u_i being the value at time
t = (N-l)At. Then, dividing by At the following simple' relationship is
obtained
(PhAt)H
= N(P/t)N- (N- 1) (P/t)
Computed values of P/t and Ph/At are .shown in Table 8. These values
are plotted as typical hyetographs in Figure 53- In order to change
the ordinates to rain-intensity in in./hr it is only necessary to
multiply each one by the 2k/hr rainfall of the desired frequency.
The order in which the various intensities are arranged in Figure 33 is
important if the period of rainfall excess is longer than can be, converted
to runoff with a single application of the unit hydrograph. Therefore the
nature of typical time-intensity patterns was studied. This was done by
arranging the 80 summer rains and kk winter rains previously described in
tabular form so that the maximum hours coincided. It was found that on
the average the rainfall patterns tended to be symmetrical. The actual
volumes of precipitation were about equal before and after the maximum .
hour when averaged for all the rains. The-second largest hour occurred
somewhat more often after the peak hour than before for the summer rains
and an equal number of times before and after the maximum hour for the
winter rains. Based on this study, the hyetographs shown in Figure 33 may
be considered as typical for Southeastern Michigan. The use of these
curves will be demonstrated in the section on Predicting Flood Magnitudes
and Frequencies.
All'rainfall frequency data discussed so far in this section are from in-
dividual rain gages and are usually referred to.as .point rainfall. Before
81
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82
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Table 8. HYETOGRAEH ORD MATES IN PERCENT OF 24-HOUR RAINFALL PER HOUR
Duration
In hours
(t)
1
2
3
4
5
6
7
8
9
• 10
11
12
13
1H.'
15
Summer
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84
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SECTION XII
PREDICTING FLOOD MAGNITUDES AND FREQUENCIES
The procedure presented in this report is based on the idea that rainfall
and/or snowmelt can be determined for any.desired duration and frequency
and that peak surface runoff rate or, if needed, the entire surface run-
off hydrograph can be constructed by use of the unit hydrograph taking
into account infiltration, the contribution from impermeable areas and
retention. For perennial streams this predicted hydrograph must be
superimposed on a typical base flow. This procedure has been worked out
and is presented here for Southeastern Michigan. The information
dealing with precipitation frequency, infiltration capacities, base flows
and HSIA are derived from the large group of basins for which hydro-
graphs were analyzed in Southeastern Michigan. The design curves for
estimating unit hydrograph form were derived from basins located in many
locations including Southeastern Michigan. However, as discussed in Sec-
tion VIII more weight was given to results from Southeastern Michigan
for low values of population density. The curves based on the average
data for all basins would therefore give slightly different results for
rural areas. In presenting the steps used in predicting flood peaks for
a given watershed a numerical example will be presented. The example
selected is Red Run, a 36.5 sq mi basin located in Southeastern Michigan
on which the population density is 7500 people/sq mi. It is assumed that
a knowledge of a range of flood magnitudes is desired. Therefore a num-
ber of solutions are made in order to provide a set of points through
which a curve relating flood magnitude to frequency may be plotted.
1. A number of values of 2^--hr summer point rainfall and winter point
rainfall plus snowmelt of various frequencies are determined from Figure
30 and tabulated in Table 9-
2. Also shown in Table 9 are the corresponding values of average pre-
cipitation on 36.5 sq mi. The conversion factors for changing point
rainfall to average values were determined from Figure 3*4- for an area of
36.5 sq mi and for durations which were arbitrarily selected to cover the
time required for 95 percent of the rain to fall. Reference to Table 8
85
-------
Table 9. TWENTY-HOUR HOUR RAINS OF VARIOUS FREQUENCIES
WINTER
Frequency
inyr
1
2
5
10
25
50
100
At a
point
1.52
1.86
2.29
2.62
3-13
3-39
3-71
On 36.5
sq mi
1.1*5
1-77
2.18
2.50
2.98
3.23
3-53-
SUMMER
At a
point
1.6p
2.01
2.55
2.97
3.62
IK 02
^.57
On 36.5
sq mi
1.5U
; 1.93
: 2,11-5
2.85
3.^8
I 3.86
4.38
and Eqs. (5) and (7) shows that the duration would be 10 hr for summer
rains and 18 hr for winter rains. Then the conversion factors from Fig-
ure 3^ are .960 for summer and .953 for winter.
3. A typical unit hydrograph is derived for the basin. The unit hydro-
graph characteristics determined from Figures 21-27 for an area of 36.5
sq, mi and a population density of 7500 people/sq mi are as follows:
2
q,pA = 138 cfs/mi/in.
tr =
T =
W0 =
2.6 hr
3-7 hr
2.8 hr
If.l hr
6.9 hr
22.0 hr
The unit hydrograph shown in Figure 35 is constructed by first placing the
peak at the time of rise (Tr). Then the significant period of ri;se (tr)' is
used to sketch in the rising side of the hydrograph. Next, the hydrograph
widths at 75 percent, 50 percent, 25 percent of peak plus the base width
are used to complete the recession side of the hydrograph. Minor modifi-
cations may be necessary to form a unit hydrograph having an area of 1 in.
Also shown in Figure 35 is the average unit hydrograph derived from run-
off records on Red Run. The close approximation of the actual unit hydro-
graph by the one synthesized from generalized curves should not always be
expected. It happens that the Red Run unit hydrograph characteristics
86
-------
140 -
120 -
100 -
cr
co
CO
& 80
LU
O
% 60
40
20
0
0
Unit Hydrograph Obtained
From Rainfall and
Runoff Records
Synthesized Unit
Hydrograph
1
10 20
TIME, Hours
Figure 35. Synthesized and observed unit hydrographs for Red Run.
87
-------
fall near the best fit lines for the highly urbanized basins.
k. A typical hyetograph of hourly precipitation is derived for summer
and winter rain using the arrangement shown in Figure 33 and numerical
values from Table 8. An example computation is shown in Table 10 for a
50-yr winter rain. The numerical values of the ratio of 1-hr rain to
2^-hr rain are arranged in a typical time sequence in the top line of
Table 10. Line 2 shows the corresponding values of hourly precipitation
obtained by applying the ratios in row 1 to the 50-yr 2l<-hr winter rain.
The magnitude of the rain was. given as 3.23 in. in Table 9.
5« The precipitation excess (Pe) is then computed for each hour by de-
ducting the average hourly winter infiltration which was found to be
0.10 in. for Southeastern Michigan. (See Figure 8.) These values are
given in line 3 of Table 10.
6. In line h, the winter retention (0.10 in.) is deducted from the first
2 hr of Pe. The values in line h are in inches on the permeable area.
In line 5 these are converted to inches on the entire basin by multiply-
ing by (l - HSIA/100). The value of HSIA is obtained from Eq. (50). This
HSIA = 1.38 Pd
(50)
equation was derived for basins in Southeastern Michigan. HSIA is ex-
pressed in percent of total area and Pd in thousands of persons/sq mi.
7. The surface runoff from impermeable area is the total precipitation
given in line 2 converted to inches on the total area by multiplying by
HSIA/100. These values are shown in line 6 of Table 10. The retention
on the impermeable area, 0.05 in., was accounted for by the first several
hours of precipitation which are not included in Table 10. '
8. Line 7 is the sum of the hourly contributions to surface runoff from
the permeable area (line 5) and from the impermeable area (line 16). These
are combined into success 2-hr inputs to the drainage system in :line 8.
This was done because it was shown earlier that the effective period of
rise (tr) is 2.6 hr and therefore the unit hydrograph can be applied to
inputs having durations as long as 2.6 hr. The use of 2-hr increments
is therefore convenient and reduces the computation time over that re-
quired if 1-hr intervals were used. '
9- The next step is to convert the 2-hr increments of surface runoff
88
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input to hydrographs by means of the unit hydrograph. This is illustrated
in Table 11 for the 50-yr winter rain. The first two columns in Table 11
show the coordinates of the unit hydrographs as read from Figure 35. At
the top of columns 3> ^-> 5> and 6 are shown the successive 2-hr incre-
ments to surface runoff derived in Table 10. The values of surface run-
off rates shown in these four columns are obtained by multiplying by
corresponding unit hydrographs ordinates and by the area, 36.5 sq mi.
Column 7 is the summation of Column 3> ^-, 5, and 6 and represents the
predicted surface runoff hydrograph with a peak of 6631 cu ft/sec.
10. The actual peak discharge is obtained by adding a typical ground
water discharge. This value is obtained from Eq. (51) in which Q,™ is
ground water discharge in cfs and A is the area of the drainage basin in
GW
= .6A
(51)
square miles. This equation was derived from a study of typical ground
water discharge in Southeastern Michigan. The value for Red Run which
has an area of 36.5 sq mi is 22 cfs thus making the estimated total peak
discharge 6653 cfs.
11. The operations described in steps (ll) and (12) are then repeated
for all of the rains shown in Table 9. The results are plotted in Figure
56.
12. A curve for total frequency is then obtained by combining the summer
and winter curves and is shown in Figure 36. This is done by adding the
probabilities which are the reciprocals of the frequencies for floods of
selected magnitudes as shown by the following equation in which T^, Tg and
T.
T
_!_
Ts
(52)
TT are the winter, summer, and total frequencies, respectively. The
total frequency is then computed by rearranging Eq. (52) to obtain Eq.
(53)« Such values of Trp then form the final frequency curve.
(53)
90
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12,000 —
Figure 36".
2 4 6 8 10 20 50
FREQUENCY IN YEARS
Predicted and observed flood frequencies for Red!Run.
100
92
-------
The frequency- curve for Red'Run computed from Eq. (53) is shown in Figure
36. Also plotted on this figure are the observed frequencies from Ik yr
of records for Red Run. The higher values of frequency in this series are
not reliable due to the shortness of the period of records but the lower
values should be reasonably correct and serve as a check with lower por-
tion of the total frequency curve derived from precipitation and snow-
melt. Such remarkable agreement cannot always be expected. It stems in
part from the fact that the unit hydrograph characteristics of Red Run
fall near to the average lines for high population densities in Figures
1^ 17, and 20. However, the close agreement serves to verify the concept
that rainfall frequency can be used to predict flood frequency if a sound
hydrological approach,is used in computing runoff from rain and snowmelt.
93
-------
SECTION XIII
REFERENCES
1.
2.
3-
Horton, R. E., "Surface Runoff Phenomena," Publ. 101, Edwards Bros.,
Ann Arbor, Michigan, 1935-
Horton, R. E., "An Approach Toward a Physical Interpretation on In.
filtration Capacity," Soil Sci. Soc. Proc., 19^0. ' I
Swartzendruber, D. and M. R. Huberty, "Use of Infiltration Equation
Parameters to Evaluate Infiltration Differences in the Field," Trans.
Am. Geophys. Union, v. 39, February 1958, p. 84.
Betson, R. P., "What is Runoff?," J. Geophys. Res., v. 69, April 15,
p.
5.
6.
8.
9.
10.
Thames, J. L. and S. J. Ursic, "Runoff as a Function of Moisture-
Storage Capacity, J. Geophys. Res., v. 65, February I960, p. 651.
Horner, W. W. and C. Leonard Lloyd, "Infiltration-Capacity Values
as Determined from a Study of an Eighteen-Month Record at Edwards -
ville, 111., Trans. Am. Geophys. Union, Part II, 19^0, pp. 522-5^1.
Wisler, C. 0. and E. F. Brater, "Report on Floods on the Rouge
River," Report to Wayne County Board of Road Commissioners, .August
1957. I
Brater, E. F. and J. D. Sherrill, "Prediction of Magnitudes and
Frequencies of Floods in Michigan," Report to Michigan Department of
State Highways and U.S. Bureau of Public Roads, 1971.
Wisler, C. 0. and E. F. Brater, "Hydrology," John Wiley & Sons, Inc.,
New York, 2nd Ed., 1959.
Ursic, S. J. and J. L. Thames, "Effect of Cover Types and Soils on
Runoff in Northern Mississippi," J. Geophys. Res., v. 65, February
I960, p. 663.
94
-------
11. Kueligan, G. H. , "Spatially Variable Discharge Over a Sloping Plane,"
Trans. Am. Geophys. Union, Part VI, 19^-.
12. Izzard, C. F., "The Surface Profile of Overland Flow," Trans. Am.
Geophys. Union, Part VI, 19^-.
13. Woo, D. C. and E. F. Brater, "Spatially Varied Flow from Controlled
Rainfall," J. Hydraulics Div. , ASCE, v. 88, November 1962, p. 51.
lU. Woolhiser, D. A. and J. A. Liggett, "Unsteady One-Dimensional Flow
Over a Plane — The Rising Hydrograph, "• Water Resources Research,
v. J., no. 3, 1967.
Lin, P. N. , "Numerical Analysis of Unsteady Flow in Open Channels,"
Trans. Am. Geophys. Union, v. 33, April 1952, p. 226.
Lin, P. N. and Ching Seng Fang, "Streamflow Routing with Application
to North Carolina Rivers," rReport No. -17, University of North
Carolina, Chapel Hill (January 1969).
Sangal, S. , "The Surface Runoff Provess During Intense Storms,"
Doctoral Dissertation, Department of Civil Engineering, The Univer-
sity of Michigan, 1970.
Tholin, A. L. and C. J. Keller, "Hydrology of Urban Runoff," Trans.
ASCE, v. 125, I960, p. .1308.
Holtan, H. N. and D. E. Overton, "Storage-Flow Hysteresis in Hydro-
graph Synthesis," J. Hydrology, v. II, No. h, April 1965, PP- 309-
323.
Crawford, Norman H. and Ray K. Kinsley, "Digital Simulation in
Hydrology: Stanford Watershed Model IV," Technical Report, No. 39,
Department of Civil Engineering, Stanford University, 1966.
. Laurenson, E. M. , "A Catchment Storage Model for Runoff Routing, "
J. Hydrology, v. II, 196^, p.
15.
l6.
17.
18.
19.
20.
21.
22.
23.
2k.
Sherman, • L. K. , "Stream Flow from Rainfill by the Unit Hydrograph
Method," Eng. News-Record, v.' 108, 1932, p. 501.
Bernard, 'Merrill M. , "An Approach to Determinate Stream Flow,"
Trans. ASCE, v. ' 100, 1935, P-
Brater, E. F., "The Unit Hydrograph Principle Applied to Small
Water-Sheds," Trans. ASCE, v. 105, 19^0, p. 1151+.
95
-------
25.
26.
27-
28.
29.
50.
31.
32.
Snyder, Franklin F. "Synthetic Unit-Graphs," Trans. Am. Geophys.
Union, Part I, 1938, p.
Taylor, A. B. and H. E. Schwartz, "Unit Hydrograph Lag and; Peak
Flow Related to Basin Characteristics," Trans. Am. • Geophys. Union,
v. 33, no. 2, 1952, p. 235. i
O'Kelley, J. J. "The Employment of Unit Hydrographs to Determine
the Plans of Irish Arterial Drainage Channels," Proc. Inst. Civil
Engrs., v. k, Part III, 1955, p. 365. ' ;
O'Kelley, J. J., "A Unit Hydrograph Study with Particular Reference
to British Catchments," Proc. Inst. Civil Engrs., v. 1?, 1960,
p.
33.
35.
36.
37.
Gray, Don M. , "Synthetic Unit Hydrographs for Small Watersheds,"
J. Hydraulics Div., ASCE, July 196l, pp. 35-5^. j
Eagleson, Peter S., "Unit Hydrograph Chacteristics for Sewered
Areas, " J. Hydraulics Div. , ASCE, Marcy 1962, pp. 1-25.
Wu, I. Pai, "Design Hydrographs for Small Watersheds in Indiana, "
J. Hydraulics Div., ASCE, November 1963, p. 35.
Espey, William H., Jr., Carl W. Morgan, and Frank D. Masch, "A Study
of Some Effects of Urbanization on Storm Runoff from a Small Water-
shed, " Center for Research in Water Resources, Department of Civil
Engineering, The University of Texas (1965).
Brater, E. F., "Steps Toward a Better Understanding of Urban, Runoff
Processes," Water Resources Research, v. k, no. 2, April 1968, pp.
335-3^7.
Brater, E. F. and S. Sangal, "Effects of Urbanization on Peak Flows,"
University of Texas, Water Research Symposium on the Effects of
Watershed Changes on Streamflow, University of Texas Press, Austin
1969.
Viessman, W. , Jr., "The Hydrology of Small Impervious Areas!," Water
Resources Research, v. 2, no. 3, 1966, p.
Horton, R. E., "The Role of Infiltration in the Hydrological Cycle,"
Trans. Am. Geophys. Union, v. 1^, 1933, p. kk6. ;
Betson, R. P., "What is Runoff?," J. Geophys. Res. v. 69, April 15
, p. 15ln.
96
-------
38. Brater, E. F., S. Sangal, and J. D, Sherrill, "Seasonal Effects on
Flood Synthesis/' Water Resources Research, A. G.U., v. 10, no. 3,
June 1971)-.
39- U.S. Weather Bureau, "Rainfall Frequency Atlas of the United States
for Durations from 30 Minutes to 2k Hours and Return Periods from 1
to 100 Years,," Technical Paper Wo. bo, Washington, B.C., May 1961.
97
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
I. REPORT NO.
EPA-670/2-75-046
2.
3. RECIPIENT'S ACCESSION-NO.
4. TITUE AND SUBTITLE
RAINFALL-RUNOFF RELATIONS ON URBAN AND RURAL AREAS
5. REPORT DATE
May 1975; Issuing Date
6. PERFORMING ORGANIZATION CODE
r. AUTHORfS)
Ernest F. Brater and James D. Sherrill
8. PERFORMING ORGANIZATION REPORT NO.
9, PERFORMING ORGANIZATION NAME AND ADDRESS
Department of Civil Engineering
University of Michigan
Ann Arbor, Michigan 48104
10. PROGRAM ELEMENT NO.
1BB034; ROAP 2;1ATB; TASK 008
11.-J6®WF-RAGWGRANT NO.
11040 DRS
12. SPONSORING AGENCY NAME AND ADDRESS
National Environmental Research Center
Office of Research and Development
U.S. Environmental Protection Agency
Cincinnati, Ohio 45268
13. TYPE OF REPORT AND PERIOD COVERED
Final |
14. SPONSORING AGENCY CODE
10. SUPPLEMENTARY NOTES
16. ABSTRACT
A procedure was developed for estimating the .frequency of storm runoff of various
magnitudes from rainfall and/or snowmelt on small drainage basins in various stages
of urbanization. The study was based primarily on the analysis of storm runoff
events on real basins varying in size from 0.02 to 734 sq mi. The method is based
on applying unit hydrographs to precipitations of various frequencies after deducting
infiltration and retention. A concurrent study with an analytical drainage basin
model provided additional understanding of the effects of some parameters. The unit
hydrograph-infiltration capacity concept was selected as the most accurate practical
method for predicting storm runoff. It was found that the form of the;unit hydro-
graph could be related to drainage basin size and degree of urbanization as measured
by population density. Other characteristics of the drainage basin are much less
important. The form of the unit hydrograph stays relatively constant for various
durations and magnitudes of input as long as the duration of input is smaller than a
critical time which can also be related to the size and population density of the
basin. As the population increased from rural to highly urbanized, peak discharges
for the same runoff became as much as ten times greater. Infiltration capacity was
found to vary seasonally. The prediction of flood frequency by this procedure is
fully operable for Southeastern Michigan. For application to other areas some
hydrograph analysis must be made.
7.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS
c. cos AT I Field/Group
*Watersheds
*Mathematical models
*Runoff
*Urban areas
*Surface water runoff
Rural areas
Unit hydrograph
Southeastern Michigan
Hydrograph analysis
Infiltration/retention
Infiltration capacity
13B
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RELEASE TO PUBLIC
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108
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98
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