NOMENCLATURE
(Notation used in Appendix I not included)
A = Waterway area in square feet; or
drainage area in acres or in square
miles.
a = Waterway area in square feet.
al, a2, a3, . . = Subdivided drainage areas in acres.
b = A constant.
C = A coefficient; a runoff coefficient de-
pending on characteristics of drain-
age areas.
D = drainage area in acres or in square
miles; time interval in hours.
D.D. = Drainage density, or total length
of visible channels per unit drainage
area, in feet per acre.
F = A frequency factor.
FF = A frequency factor.
H = Fall of the drainage basin from the
farthest point on the watershed to
the outlet of runoff, in feet.
I = Rainfall intensity in inches per
hour.
K = A coefficient; a lag-time factor
equal to L/-/S .
K, = A lag-time factor equal to
A0.3/S/ D.D.
k = A physiographic factor.
L = Length of stream or of drainage
basin in miles or in feet.
Lo = Total length of visible channels in
a drainage basin in feet.
LF = A land use and slope factor.
M = Drainage area in square miles.
m = An exponent.
N = Runoff number or hydrologic soil-
cover complex number.
n = An exponent.
P = Peak discharge in cubic feet per
second (c.f.s.); peak discharge of a
unit hydrograph.
p = Numerical percentage rating on the
Myers scale.
Q = Discharge in c.f.s.; 10-year dis-
charge in c.f.s.; direct runoff in
inches.
Qb = Base flow in c.f.s.
Qd = Computed design peak discharge
in c.f.s.
Q.ax = Maximum peak discharge in c.f.s.
q = Discharge in c.f.s. per acre in the
Btirkli-Ziegler formula.
q, = Peak discharge in c.f.s.
R = Rainfall or rainfall excess in inches;
a rainfall factor.
IR, = Rainfall excess or direct runoff in
inches.
ReĆ½ = Rainfall excess in inches at Urbana,
Illinois.
R,, = Rainfall in inches at Urbana, Illi-
nois.
RF = A rainfall factor.
S = Average ground or channel slope in
feet per 1,000 feet, in feet per foot,
or in per cent.
S, = Average land slope of watershed in
per cent.
T = A slope factor.
Tb = Base time of a triangular hydro-
graph in hours.
T, = Time of concentration in hours.
T, = Time from beginning of direct run-
off to peak flow in hours.
t = Duration of rainfall or rainfall
excess in minutes or in hours.
t, = Time of concentration in hours.
t,' = Time of concentration in hours for
sub-areas.
th = Time in hours since rainfall excess
began.
to = Lag or the time interval in hours
from center of mass of rainfall
excess to center of mass of runoff.
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
tp = Lag time in hours; the time interval
from center of mass of rainfall to
the resulting runoff peak; the time
of rise of the peak flow in an in-
stantaneous unit hydrograph.
V = Total volume of runoff in acre-feet.
W = A rainfall factor; weight.
X = A runoff factor equal to R,,/t; dif-
ference in elevation in feet of a
stream-bed at the culvert site and
at 0.7L upstream, in which L is the
length of the stream.
x = Part of length of drainage basin in
miles for sub-areas.
Y = A climatic factor equal to 1.008R/
R.; difference in elevation in feet of
a stream-bed at 0.7L upstream and
at the headwater, in which L is the
length of the stream.
Z = A peak-reduction factor equal to
Pt/I.008A.
I. INTRODUCTION
A. SIGNIFICANCE AND PURPOSE OF THE STUDY
It is estimated that over 15%/ of the total cost
of the development of modern highways across tile
country is spent on the construction and mainte-
nance of minor drainage structures such as culverts
and small bridges. In view of this high percentage
of expenditure there is great need for improvement
in the design method for economical determinations
of the water-carrying capacity of these structures.
Current methods employed by most highway
engineers for the determination of waterway areas
involve the use of empirical approaches such as the
Talbot formula. Such approaches do not generally
encourage consideration of the many significant
factors involved in a given problem, but cause these
factors to be treated in a lump, usually by means of
a coefficient. In the use of an empirical approach,
moreover, there is danger that the limitations are
often overlooked or ignored. Some judgment and
experience are therefore necessary for the applica-
tion of the empirical method, particularly in the
selection of the proper coefficients. Also, an inex-
perienced designer using the empirical approach has
a tendency to overdesign the structure.
The purpose of the present study is to develop a
simple but scientific procedure for the use of engi-
neers in establishing economical and adequate
waterway areas of small drainage structures. The
procedure would not rely so heavily on the judg-
ment of the designer as does an empirical approach.
It could therefore be used easily by relatively inex-
perienced designers. The method developed can be
applied to any area for which hydrologic data are
available. The procedure presented is based on data
for Illinois and is therefore especially applicable
to this state.
The method thus developed should be useful not
only to highway engineers, but also to railroad and
agricultural engineers who design drainage struc-
tures, to practicing hydraulic engineers, and to hy-
drologists dealing with small rural drainage basins.
B. DEVELOPMENT OF THE STUDY
This study was started in the fall of 1952. En-
gineers of the Illinois Division of Highways were
not satisfied with existing methods (based on em-
pirical formulas) of designing the waterway open-
ings of culverts and other drainage structures. They
recommended a thorough analytical investigation
of the problem by the Department of Civil Engi-
neering as one phase of the Illinois Cooperative
Highway Research Program at the University of
Illinois. By 1957 the Bureau of Public Roads had
become interested in the study and was participat-
ing in the project.
In the beginning, a compilation was made of
existing formulas and available literature related
to the subject. It was followed by a nation-wide
survey of drainage structure design practices
adopted by different state highway agencies. A
critical review of existing methods for the determi-
nation of waterway areas was then made. In the
meantime work was also continued in two direc-
tions: one was the collection of hydrologic data
and their analysis and the other was the exploration
of available analytical methods for hydrologic anal-
ysis. The outcome of this investigation, which con-
sidered all significant hydrologic factors involved
in the problem, was the development of a procedure
for the determination of design peak discharge of
small drainage basins for the design of waterway
openings. This procedure was proposed as a prac-
tical solution to the problem; it was later recon-
sidered for further improvement and then modified
and simplified. For the use of practicing engineers,
a design chart was prepared for the proposed
method for the conditions in the State of Illinois.
As a result of the intensive study, seven pre-
liminary reports were produced and submitted to
the members of the Project Advisory Committee
and to the sponsors of the project. After reviewing
these reports, the Committee recommended that the
proposed method should be made available to stu-
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
dents of hydraulic engineering and to hydraulic
engineers. At its meeting of December 8, 1960, the
Committee recommended the preparation and pub-
lication of this final report summarizing the pre-
vious preliminary reports and presenting a compre-
hensive picture of the work done on the project.
C. SCOPE OF THE STUDY
This study deals with the determination of peak
discharge from small rural drainage basins in Illi-
nois, because such a determination is required for
an economical design of waterway areas of highway
culverts and small bridges.
This study consists of the following major
phases:
(1) A compilation of existing formulas for wa-
terway area determination.
(2) An extensive review of available literature
on the subject and compilation of an annotated
bibliography.
(3) A historical review of engineering studies
and methods of waterway area determination.
(4) A survey by questionnaire of design prac-
tice employed by different state highway agencies
in the United States.
(5) Collection and analysis of available hydro-
logic data for small rural drainage basins.
(6) Development of a scientific, simple, and
practical method for the determination of waterway
areas and its simplification for practical design
purposes.
II. A HISTORICAL REVIEW
A. GENERAL
A survey of the literature reveals that the engi-
neering studies on the problem of waterway area
determination started as early as a century ago
when the surveyor of London, John Roe, prepared
a drainage table for sewer sizes and slopes in 1852.
Records show that the studies in the United States
began about three-quarters of a century ago, when
the problem was first recognized by sewerage engi-
neers. About 25 years later the railroad engineers
began to be interested in it and continued to be so
during the subsequent 30 years. Then there was a
short period of recess before the highway engineers
and the water and soil conservation workers started
to investigate the problem. These investigations
have continued over the last 40 years.
A list of historical events in the chronological
development of engineering studies on waterway
area determination is presented later in Section
I I-. The detailed review and discussion in the fol-
lowing articles are more or less in close chronolog-
ical order with the listed events.
B. THE MYERS FORMULA
Major E. T. C. Myers, Chief Engineer of the
Richmond, Fredericksburg, and Potomac Railway
shortly after the Civil War, is believed to be the
first American railroad engineer to propose the use
of a formula as a guide for determining waterway
areas. His formula was first presented by Clee-
mann"'* in a paper before the Engineers' Club of
Philadelphia and was published in the Club's Pro-
ceedings in 1879.
The Myers formula appears in the following
form:
A = C/D (1)
in which A = area of waterway in square feet
D = drainage area in acres
C = coefficient recommended to be 1.0 as
a minimum for flat country, 1.6 for
Suillci r(lil,L iiiiI, ir i a tn pai.eitlu 's Ire fr l o 'i"Apljedix 1 - Hl fer-
Ctu it, d."
hilly compact ground, 4.0 as a max-
imum for mountainous and rocky
country, and higher values in excep-
tional cases.
Cleemann suggested that the coefficient C be
derived from careful and judicious gagings at char-
acteristic points within the region under treatment
and be applied liberally. Also, the formula should
be applied only to small structures, probably be-
cause the formula results in openings which are too
small for large drainage areas. The formula was
found satisfactory for regions adjacent to the line
of the Richmond, Fredericksburg, and Potomac
Railroad in the State of Virginia. Hence, it was
used widely by railroad engineers in the New Eng-
land states and generally in the eastern part of the
United States.
The Myers formula received many comments
after its publication. Among the comments, Well-
ington's editorial"2' is typical:
It is natural for fallible man to wish to reduce every-
thing to rule, even if it be only a rule of thumb. The
responsibility of the individual is much diminished if he
has something of that kind to lean on, and in so doubtful
a matter as the proper size of culverts, this is especially
natural. It is well, however, to be certain that we are
not simply making a rule where there is no rule, and so
laying the foundation of future trouble, and we must
confess to doubts as to whether this is not the case with
the various formulas for proportioning the waterway for
culverts . . . when in addition the probable variations in
maximum rainfall and possible future changes in the con-
dition of the surfaces are considered, we cannot but re-
gard the proportioning of culverts by a formula as
entirely futile. Even in the much simpler, because more
regular and determinable, problem of proportioning the
size of city sewers, many engineers claim that safety can
only be assured by comparison with experience with as
many similarly situated sewers as possible and then tak-
ing care not to overload the sewer after it is built; and
with much reason. For culverts, if we were called upon
to suggest a formula, we could do no better than this:
Estimate the necessary area as carefully as possible by
existing evidences of maximum flow, which let equal A.
Then will -/8A equal the proper area for the culvert.
In more popular language: "Guess at the proper size
and double it." We apprehend that this formula will
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
give far more satisfactory and trustworthy results than
that which our correspondent quotes (Myers' formula),
or any other which purports to be of general application
to a problem subject to such extremely diverse condi-
tions.
It is apparent that the Myers formula was not
rational enough to be considered as a rigid rule,
and furthermore there was the danger of its being
abused rather than used intelligently and properly.
However, this formula has had two significant con-
sequences: to stimulate the development of other
formulas, and to inspire the later use of a device
called the "Myers scale" by C. S. Jarvis in connec-
tion with the study of floods at the U. S. Bureau of
Public Roads and U. S. Geological Survey.
In 1926, Jarvisc'3 modified the Myers formula
and broadened its use by the introduction of the
Myers scale. The Modified Myers formula is written
Q = 100 p\/M
q 'I -
SA
in which A = drainage area in acres
S = average slope of ground in feet per
1,000 feet
I = average rate of rainfall in inches per
hour during the heaviest storm
C = coefficient depending on nature or rel-
ative imperviousness of ground sur-
face, equal to 0.31 for an average
condition, 0.20 for rural sections, 0.25
for farm country, 0.30 for village with
lawns and macadam streets, 0.65 for
ordinary city streets, and 0.75 for
paved streets and built-up business
blocks.
Professor Talbot derived his formula as fol-
(2) lows:
in which Q = discharge in c.f.s.
M = drainage area in square miles
p = numerical percentage rating on the
Myers scale.
The advantage of the Myers scale is that it
furnishes a standard by which the flood flow char-
acteristics in different streams can be roughly com-
pared. In order to assist in visualizing the flood
potentialities of the various regions within the
United States, maximum flood flows have been ex-
pressed in per cent on Myers scale as experienced
at widely scattered stream gaging stations in this
country."4 The use of the Myers scale is ingenious,
but it was soon found to be too simple to be an
index representing the complicated nature of flood
flow.
C. PROFESSOR TALBOT'S RENOWNED FORMULA
A year after Wellington's comment on the
Myers formula, Professor A. N. Talbot'" of the
University of Illinois published (1887) his well-
known formula for determining the waterway area
of culverts, which has since been very widely
adopted in the United States. In deriving his for-
mula, Professor Talbot made use of the Biirkli-
Ziegler formula.(') The latter is a storm water
runoff formula published by the Swiss engineer
Biirkli-Ziegler in 1880 and then introduced into
American practice by Hering(') in 1881.
The Biirkli-Ziegler formula for discharge q in
c.f.s. per acre is written as
Since by this formula (Eq. 3) the quantity of dis-
charge per acre varies inversely as the fourth power of
the area drained, the volume of discharge from the whole
area will vary as A , or A'; and, assuming the same
velocity through the culvert as in the stream above, the
opening will vary likewise. This assumption will be true
when the grade of the culverts is the same as that of the
stream above and when the smaller coefficient of friction
in the culvert over that of the channel itself is counter-
acted by the resistance to entering the culvert. We may
then write
a = C /A or
Area of water-way in sq. ft. =
C 7/(Drainage area in acres)3
for which the coefficient C must be determined.
By comparison with the formula of Biirkli-Ziegler and
with the flood flow of streams up to several of 77 square
miles area, I conclude that for rolling agricultural coun-
try subject to floods at time of melting of snow, and with
the length of valley three or four times the width, 1/3 is
the proper value of C. If the stream is longer in propor-
tion to the area, decrease C. In districts not affected by
accumulated snow, and where the length of the valley is
several times the width, 1/, or /6 or even less may be
used. C should be increased for steep side slopes, espe-
cially if the upper part of the valley has a much greater
fall than the channel at the culvert.
In any case, the judgment must be the main depend-
ence, the formula being a guide to it. On a road already
constructed the C may be determined for the character
of surface along that line by comparing the formula with
the high-water mark of a known drainage area. Experi-
ence and observation on similar water-courses is the most
valuable guide. A knowledge of the action of streams of
similar situations in floods and of the effects of peculiar
II. A HISTORICAL REVIEW
formations and slopes is of far more value than any
extended formula.
In a subsequent discussion of his paper, Pro-
fessor Talbot proposed that "for steep and rocky
ground, C varies from %2 to 1."
Concerning the difficulty of developing a ra-
tional formula to determine waterway areas, Pro-
fessor Talbot listed the following considerations:
(1) The variation of the rate of rainfall in different
localities.
(2) Paucity of data, since records are generally given
as so much per day and rarely per hour, while the dura-
tion of the severe storm is not recorded.
(3) The melting of snow with a heavy rain.
(4) The permeability of the surface of the ground,
depending upon the kind of soil, condition of vegetation
and cultivation, etc.
(5) The degree of saturation of the ground and the
amount of evaporation.
(6) The character and inclination of the surface to the
point where the water accumulates in the water-course
proper.
(7) The inclination or slope of the water-course to the
point considered.
(8) The shape of the area drained and the position
of the feeders.
He emphasized that "any formula will be ap-
proximate, that the estimation of the values of the
different conditions entering into the subject will
be almost wholly a matter of judgment, so that the
formula must be considered more as a guide to the
judgment than as a working rule."
For estimating the discharge of a stream flow
in large drainage areas, Professor Talbot recom-
mended the Ch6zy formula for waterway area de-
termination.
An investigation of the Talbot formula reveals
the following points of interest: The formula was
derived with special reference to areas under 77 sq.
mi. in size, although it has been applied to an area
as large as 400 sq. mi. Generally, the results are
much too high for large areas. Since this formula
was based on the runoff data of a large number of
observations in the Midwest, it does not take into
account the variation in intensity of rainfall, ve-
locity of flow, and frequency factor when applied to
other localities. Studies on results obtained by this
formula indicate that the maximum rainfall for
these observations was probably about 4 in. per hr.,
and the velocity in the observed cases was variable
but less than 10 ft. per sec.
Because of its simplicity the Tablot formula has
been widely used either in its original form or with
modified coefficients to meet local conditions. The
formula has been presented by charts'8- " o) and
tables,"'" and slide rules for solving it have been
developed."2'
From the modern hydrologic and hydraulic
viewpoint, the Talbot formula gives only a very
crude answer to the problem. The formula assumes
that tie waterway area is directly proportional to
the discharge which varies with the %-power of the
drainage area. This is not accurate for a reliable
design of numerous drainage structures being built
nowadays. The relationship between the waterway
area and the drainage area is far more complex than
the i-power law; it depends on many physical
characteristics of the drainage basin, as well as on
the various hydrologic and hydraulic factors in-
volved in a given problem.
D. EARLY CONTRIBUTIONS BY SEWERAGE
ENGINEERS
Sewerage engineers are interested in waterway
area determination primarily for the purpose of
designing storm sewers. The Biirkli-Ziegler formula
mentioned in the previous article is one of the
earliest contributions by sewerage engineers.
One of the well-known contributions by sewer-
age engineers is the rational formula, which was
developed primarily for estimating rates of runoff
from urban areas. The origin of this formula is
somewhat obscure. In American literature, the
formula was first mentioned in 1889 by Emil
Kuichling."3' The runoff coefficient in the formula
was derived by him from measurements of rainfall
and of the flow in the sewers of Rochester, New
York, during the period from 1877 to 1888. Ac-
cording to Dooge,"14) the principles of the method
were explicit in the work of Mulvaney(5) in 1851.
In England it is often referred to as the Lloyd-
Davis method and hence by implication ascribed to
his paper of 1906.0'6
The rational formula is
Q = CIA
in which Q = discharge in c.f.s.
C = runoff coefficient depending on char-
acteristics of the drainage basin
I = rainfall intensity in inches per hr.
A = drainage area in acres
Many formulas have been proposed for esti-
mating the rainfall intensity in the rational formula
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
(see Appendix I). The general form may be writ-
ten as
KF,
I (F+b' (6)
(t + b)"?
in which t = duration of rainfall in minutes
F = frequency factor indicating the fre-
quency of occurrence of the rain-
fall
K, b and m, n = coefficient, constant, and exponents,
respectively, depending on condi-
tions which affect the rainfall in-
tensity
When using the rational formula, one assumes
that the maximum rate of flow, due to a certain
rainfall intensity over the drainage area, is pro-
duced by that rainfall which is maintained for a
time equal to the period of concentration of flow
at the point under consideration. This is the time
required for the surface runoff from the remotest
part of the drainage basin to reach the point being
considered. In other words, the critical duration of
rainfall t in the rainfall intensity formula (Eq. 6)
should be equal to the time of concentration.
Values of C commonly recommended for design
purposes are as follows:
Type of Drainage Area
Business:
Downtown areas
Neighborhood areas
Residential:
Single-family areas
Multi units, detached
Multi units, attached
Suburban
Apartment dwelling areas
Industrial:
Light areas
Heavy areas
Parks, cemeteries
Playgrounds
Railroad yard areas
Unimproved areas
Streets:
Asphaltic
Concrete
Brick
Drives and walks
Roofs
Runoff Coefficient, C
0.70-0.95
0.50-0.70
0.30-0.50
0.40-0.60
0.60-0.75
0.25-0.40
0.50-0.70
0.50-0.80
0.60-0.90
0.10-0.25
0.20-0.35
0.20-0.40
0.10-0.30
0.70-0.95
0.80-0.95
0.70-0.85
0.75-0.85
0.75-0.95
Type of Drainage Area
Lawns:
Sandy soil, flat, 2%
Sandy soil, average, 2-7%
Sandy soil, steep, 7%
Heavy soil, flat, 2%
Heavy soil, average, 2-7%
Heavy soil, steep, 7%
Runoff Coefficient, C
0.05-0.10
0.10-0.15
0.15-0.20
0.13-0.17
0.18-0.22
0.25-0.35
The above values were reported by a joint
committee of the American Society of Civil Engi-
neers and the Water Pollution Control Federation
in "Design and Construction of Sanitary and Storm
Sewers," ASCE Manuals of Engineering Practice
No. 37 and WPCF Manual of Practice No. 9, 1960.
They are applicable for storms of 5-year to 10-year
frequencies. Less frequent higher-intensity storms
will require the use of higher coefficients because
infiltration and other losses have a proportionally
smaller effect on runoff.
From the data of the American Railway Engi-
neering Association, Dr. J. A. L. Waddell""' has
compiled values of C for different watershed sizes
as follows:
Drainage Value
Area of
sq. mi. C
1,00)0 .... 0.95
2,000).... 0.82
3,000. ... 0.74
4,000.... 0.66
5,000 ... 0.61
6,000. .. 0.56
7,000.... 0.52
Drainage Value
Area of
sq. mi. C
8,000.... 0.49
9,000 ... 0.46
10,000.... 0.43
11,000.... 0.40
12,000.... 0.38
13,000.... 0.36
14,000.... 0.34
Drainage Value
Area of
sq. mi. C
15,000.... 0.32
16,000.... 0.30
17,000.... 0.29
18,000 .... 0.28
19,000.... 0.26
20,000.... 0.26
21,000 .... 0.25
These values are for average drainage condi-
tions in the United States and do not apply to
drainage basins which have exceptional runoffs.
The latter cases must receive individual considera-
tion.
The rational formula is based on a number of
assumptions. According to Krimgold,'") the as-
sumptions are:
(1) The rate of runoff resulting from any rainfall in-
tensity is a maximum when this rainfall intensity lasts
as long or longer than the time of concentration.
(2) The maximum runoff resulting from a rainfall
intensity, with a duration equal to or greater than the
time of concentration, is a simple fraction of such rain-
fall intensity; that is, it assumes a straight line relation
between Q and I, and Q = 0 when I = 0.
(3) The frequency of peak discharges is the same as
that of the rainfall intensity for the given time of con-
centration.
(4) The relationship between peak discharges and
size of drainage area is the same as the relationship
between duration and intensity of rainfall.
II. A HISTORICAL REVIEW
(5) The coefficient of rinoff is the same for stormis
o1f variolls I'frequ ciivi .
(it The coefficient of riunoff is the same for ill slornms
onl givenl watershed.i
It is believed that these assumptions might
conceivably hohl for paved areas with gutters and
sewers of fixed dlimensions and hydraulic character-
istics. The formula has thus been rather popular
for the design iof d(rainage systems in urban areas
iand airports. The exactness and satisfaction of
these assumptions in application to other drainage
basins, however, have been questioned. In fact,
lmany hyli ologists"' "') have called attention to
the inadeqluacy of thii miethod. Bernard(2'" hiad
attempted to modify tle rational formula, but his
solution is hardly practicable for design purposes.
Anot her study by Gregory and Arnold'21 resulted
in a general rational foirmula, taking into account
such factors as basin shape and slope, tlihe pattern
of the stream system, and tile elements of channel
flow. However, tihe coimplexity of the miethod
hinders its wide application.
)ther well-known formulas developed by sewer-
age engineers includes the Hawksley formula of
1857, the Adams forinmula of 1880, the McMath
formula of 1887, the Hering formula of 1889, the
Pariimely fiorlmula of 1898, and the (Gregory formula
of 1907. These formullas anld Imanly others are listed
in Appendix 1.
E. DRAINAGE TABLES OF RAILROAD ENGINEERS
Various drainage tables have been developed for
tlie determination of wvaterway areas. These tables,
generally prepared froml the actual stream flow
data, give the size of waterway for a given drainage
area. The most popular and frequently quoted
drainage table is the )Dun iraterway table or Dun
ldrainagc table (Table 1) prepared by James Dun,
former (Clief Engineer of the Santa Fe railroad
system. The table was first published in 1906,(22
but Professor NV. I). Pence'"' of the University of
Wisconsin pointed out that the forerunner of this
table was a drainage table of a somewhat smaller
range issued in 1897 in blueprint fornm.
According to D)un,'-''
The accomlpalnying table has ),(ben in use oil the Sanita
Fe Systeml for the past 15 years for proportioning water-
ways. Il general, we have found this table to be suffi-
cient, and particularly up to drainage areas of 5 square
miles. In 1I8)3, how\ever, we noticed soume floods in Cenl-
tral Kan:iis which exceeded the tiables from 200 to 300
per cent. Also ill t(h yeiar 1905 we haid a series of foods
in the vicinity of Fort Madison, Iowa, that far exceeded
our tables. In one case, where the drainage area is about
150 square miles, the area of waterway was about 12,X)0
square feet, and the current was so swift as to scour out
the stream to a( depth of 40 feet. I believe, however,
that these floods are rare exceptions and that it would
not pay a railway company or anyone else to undertake
to provide for them.
The table referred to is based upon observations taken
by nme and others unlder my jurisdiction on floods in NMis-
souri, Kansas, Indian Territory and Texas. The section
of waterway at the contracted part of different streams
was accurately measured from tilme to ime as floods
occurred andt the table was made up from these data.
Wherever possible, cross-sections were taken in the larger
t reanis at points where rock bluffs canie in on both sides
and where the streain has a rock bottom, thus eliminat-
ing tlie question of sco(urT. This, however, was not pra:'-
ticabile in every case.
The Dun table was prepared from observations
taken along the line of the Santa Fe railroad systeim
in Missouri, Arkansas, Kansas, and Indian Terri-
tory. This region in general is colmposed of steep
rocky slopes, and l)ercolation is a small percentage
of the total rainfall. For other sections of the
Santa Fe line, I)un gave in his table coefficients to
be applied to the basic values listed. The use of
the table can alpparently be extended to other
regions of complarable conditions.
Dun was of the opinion that his data could not
be expressed by a formula for practical use. How-
ever, Purdon('4) has derived an approximate water-
way area formula based on D)un's data:
a = A (240 - 12/ A )
in which a = waterway area in square feet
A = drainage area in square miles
This formula applies to drainage areas of more
than 16 sq. mi. For small areas the waterway areas
should be increased to allow for drift, etc.
The Dun data have also been expressed'"23 by a
curve of logarithmic plotting. Two curious breaks
can be found on the curve at drainage areas of
about 1 sq. mi. and 4 sq. mi., respectively. They
are probably due to the abrupt change in nature of
the rainfall intensity or in basin characteristics, or
to some other unknown reasons.
Other notable drainage tables developed in early
days and used in railroad engineering practice are:
(1) The "Table of Minimum Cross-Section
Areas for Waterways of Culverts and Small
Bridges" of the Pittsburg and Lake Erie Railroad.
In this table the runoff was calculated by the
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
Table 1
The Dun Drainage Table
Atchison, Topeka & Santa Fe Railway System (1906)
Areas Areas of Waterway
Drained
in Square Missouri Cast Pipe. Box and Arch Percentage of Column 2
Miles and Banks over 15 Culverts.
Kansas Ft. Use 80 Per 1st Fig. Diam. Illinois Indian Texas New
Cent 2d. Fig. Bench Territory Mexico
1 2 3
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
4.2
4.4
4,6
4.8
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10
11
12
13
14
15
16
17
18
19
20
22
Areas Areas of Waterway
Drained
in Square Missouri Percentage of Column 2
Miles and
Kansas Illinois Indian Texas New
Territory Mexico
2.0
4.0
6.0
7.5
9.0
10.5
12.0
13.5
15
16
25
32
38
44
51
56
62
66
70
74
78
81
85
88
91
94
97
100
110
120
130
140
150
160
170
180
190
200
220
240
260
280
300
321
340
357
373
388
403
417
430
443
455
483
509
533
556
579
601
622
641
660
679
710
740
775
805
835
865
890
920
945
970
1,015
1-24 in.
1-24 "
1-30
1-36 "
1-42
1-42
1-48
2-36
2-3 "
2-36
2-48 "
3-42 "
3-48 "
4 5 6 7 8
2 xl B
2 x2
2 x3
2%x3
3 x3 " East of South of Use Use
3 !x3 " Streator Purcell Column Column
3 x4 " use 60 use Texas 2 2
2%x3 " per cent Column
2%x3
3 x3
3 x4
6 x4 A
6 x5
6 x5% "
8 x4% "
8 x5
8 x6
8 x6
8 x6% "
10 x4% "
10 x5
10 x5% "
10 x6
10 x6% "
10 x6% "
10 x6% "
12 x5
12 x5 " 98%
12 x6 105 98%
12 x7 105 98%
12 x8 105 98%
14 x6% " West of North of 105 98%
14 x7 Streator Purcell 105 98%
16 x6% " use 80 use Col- 105 98%
16 x7 " per cent umn 2 105 98%
16 x7 " 105. 98
16 x8 105 98%
18 x7 105 98%
18 x8 " 105 98%
18 x9% " 105 98%
20 x8 "105 98M
20 x9 105 98%
20 x9% 105 98%
22 x8% 105 98%
22 x9 105 98%
24 x8% " 105 98%
24 x9 105 98%
28 x7 "105 97
28 x7% " 105 97
28 x8 "105 97
28 x8% " 105 97
28 x9 105 97
28 x9% " West of North of 105 97
28 xl0 " Streator Purcell 105 97
32 x7% " use 80 use Col- 105 97
32 x8 " per cent umn 2 105 97
32 x9 105 97
32 x10 " East of South of 105 97
32 x1l " Streator Purcell 105 97
32 xll " use 60 useTexas 105 97
32 x12 " per cent Column 105 93%
32 x12% " 105 93%
32 x13 " 105 93%
105 93%
105 93%
Bridges de- 105 93%
signed to 105 93%
provide area 105 93%
according to 105 94
circumstances 105 94
105 94
105 94
105 94
105 94
I
The above classification by states is for convenience only, and merely denotes the general characteristics of topography and rainfall.
Column 2 in this table is prepared from observations of streams in Southwest Missouri, Eastern Kansas, Western Arkansas and the southeastern
portions of the Indian Territory. In all this region steep, rocky slopes prevail and the soil absorbs but a small percentage of the rainfalls. It indicates
larger waterways than are required in Western Kansas and level portions of Missouri, Colorado, New Mexico and Western Texas.
(
1 2 5 6 7 8
24 1,060 110 94
26 1,100 110 92
28 1,140 110 92
30 1,180 110 92
32 1,220 East of South of 110 92
34 1,255 Streator Purcell 110 92
36 1,290 use 60 use Texas 110 91
38 1,320 per cent Column 110 91
40 1,350 110 91
45 1,435 110 91
50 1,510 110 89%
55 1,580 115 89%
60 1,650 115 89%
65 1,720 115 88
70 1,780 115 88
75 1,840 115 88
80 1,900 115 86%
85 1,960 115 86%
90 2,015 115 86%
95 2,065 115 86%
100 2,120 120 85
110 2,220 120 85
120 2,315 120 85
130 2,405 125 83%
140 2,500 125 83%
150 2,580 130 82
160 2,665 130 82
170 2,745 130 80%
180 2,820 130 80%
190 2,900 130 79
200 2,970 130 79
220 3,115 West of North of 130 77%
240 3,245 Streator Purcell 130 77%
260 3,370 use 80 use Col- 130 76
280 3,495 per cent umn 2 130 76
300 3,615 130 74%
325 3,770 130 74%
350 3,900 130 73
375 4,035 130 73
400 4,165 130 71%
450 4,385 130 70
500 4,610 130 68%
550 4,825 130 67
600 5,030 130 65%
650 5,230 130 64
700 5,420 130 62%
750 5,610 130 61
800 5,800 130 59%
850 5,890 130 58
900 6,080 130 56%
950 6,230 130 ....
1,000 6,380 130
1,100 6,705 West of North of 130
1,200 6,960 Streator Purcell 130
1,300 7,230 use 80 use Col- 130
1,400 7,480 per cent umn 2 130
1,500 7,725 130
1,600 7,960 East of South of 130
1,700 8,195 Streator Purcell 130
1,800 8,390 use 60 use Texas 130
1,900 8,625 per cent Column 130
2,000 8,820 130
2,200 9,240 130
2,400 9,605 130
2,600 9,970 130
2,800 10,320 130
3,000 10,640 130
3,500 11,445 130
4,000 12,160 130
4,500 12,825 130
5,000 13,500 130 .
5,500 14,080 130
6,000 14,520 130
6,500 15,140 130 ...
II. A HISTORICAL REVIEW
Biirkli-Ziegler and McMath formulas (Appendix I)
with maximum rainfall equal to 3 in. per hr. and a
runoff coefficient of 0.3. The assumed maximunm
velocity for culvert running full is 6 ft. per see.,
that is, the maxinmum discharge in c.f.s. is equal to
6 times the opening in sq. ft.
(2) The "Drainage Table of El Paso and South
Western Railway" compiled under the direction of
James Dun:
Table 2
Drainage Table of El Paso and South Western Railway
Drainage Area Waterways Drainage Area Waterways
sq. mi. sq. ft. si. 11i. s1q. ft.
0.12 18.8 4.6 353
0.22 31.4 5.8 428
0.40 51.0 7.2 508
0.70 74.0 9.5 628
1.15 102.0 12.4 75
1.55 134.0 15.8 878
2.00 170.0 19.9 978
2.50 209.0 24.6 1,088
3.50 280.0 30.2 1,199
In this table, the waterway area is that which is
supposed to be adequate on steep rocky slopes
where very little rainfall is absorbed. The area
should be multiplied by a coefficient greater than
1 for exceedingly mountainous country and less
than 1 for comparatively level country. The coeffi-
cients may vary from 0.50 for flat country with
porous soil to 1.50 for rocky mountain gorges.
(3) The table of "Data for Concrete Arches and
Waterway Areas, 1908" for Missouri, Kansas and
Texas Railway. This table was constructed on the
basis of the Talbot formula, using C = 1.1 for
steep, 0.85 for medium, and 0.60 for flat lands.
(4) The "Table of Areas Drained by Culverts
and Bridges" by Mississippi River and Bonne Terre
Railway, prepared under the direction of James
Dun for use in the Boston Mountain country of
Northwest Arkansas. This table was constructed
more or less on the basis of the original Dun drain-
age table. Where the land is not so rugged and
water collects more slowly, the size of opening
should be reduced somewhat, as judgment directs.
The maximum velocity was assumed to be 7 ft.
per sec.
(5) The table for "Dimensions of Pipe and
Culvert Openings" by Mobile and Ohio Railroad.
This table was derived from the Talbot formula in
which the coefficient C varies from 0.2 for level
land, 0.4 for rolling land, and 0.6 for hilly land to
0.8 for mountainous regions. It was recommended
that no drain less than 24-in. diameter be used and
that all drains over 4 feet in diameter be of con-
crete.
F. DEVELOPMENT OF NUMEROUS
OTHER FORMULAS
Since tile problem of waterway area determina-
tion was of interest to the sewerage and railroad
engineers, a great number of methods involving the
use of formulas, tables, and charts were developed
and proposed for design purposes. This develop-
ment almost reached its climax during the time the
American Railway Engineering and Maintenance
of Way Association held its Twelfth Annual ('on-
vention at the Congress Hotel, Chicago, Illinois, on
March 21 to 23, 1909. During the convention, the
Sub-Committee on Formulas for Waterways pre-
sented a report 26) which contains a description of
formulas for waterways. In Appendix A of this
report, "Waterway for Culverts" by W. D. Pence,
a brief historical account, compilation and compari-
son of formulas, permissible velocity, and other
features are given. Appendix B lists data on the
maximum flood flow of stream in various sections
of the United States. Appendix C contains an index
to literature on the subject of waterways for cul-
verts and allied topics. This report was published
in 1911 together with an earlier report presented
at the Tenth Annual Convention of the Association.
This report123' contains important discussions and
reviews of current practice with reference to the
methods of dimensioning waterways that had ap-
peared in publications2', 22) of two representative
technical societies. Together they contain a very
comprehensive survey of the methods and prac-
tices current at the time, as well as many authori-
tative comments on the subject.
The problem of waterway area determination
has also been investigated in other countries. Be-
sides the well-known Biirkli-Ziegler formula de-
veloped in Switzerland, there were the Chamier
formula of 1898 in London, the Possenti formula of
1881 in Vienna, the Lillie formula of 1924 in Lon-
don, the Lauterberg formula of 1887 in Germany,
the Craig formula of 1868 in England, the Wood
formula of 1917 in New Zealand, and the Kresnik
formula of 1886 in Vienna. Even recently (1951)
the Russian Scientific Academy formula was pub-
lished in U.S.S.R. and the Ribeiro formula was
published in Brazil. These formulas and many
others are listed in Appendix I.
An early study of waterways for culverts, which
included a compilation of formulas, was carried out
as a thesis investigation by A. F. Gilman and G. W.
Chamberlain in 1909-1910, under the direction of
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
Professor W. D. Pence, at the University of Wis-
consin. A digest of the thesis material was incor-
porated in a report'23' in 1909.
At a later date (1934), T. M. Munson(27' listed
36 formulas and 4 sets of curves used at the time
of his study in determining runoff and sizes of
highway and railroad drainage structures.
Many of the published formulas, such as the
Biirkli-Ziegler formula and the Talbot formula,
have been very popular for many years. Based on
local experience, various coefficients were developed
to satisfy the local conditions. However, most engi-
neers have never been satisfied with the wide range
of aecuracy provided by these formulas.
G. EFFORTS OF AGRICULTURAL ENGINEERS
Since the formation of the Soil Erosion Service,
the forerunner of the Soil Conservation Service, in
1933, and the establishment of the CCC (Civilian
Conservation Corps) camps in the following year,
the soil and water conservation engineers began to
work actively in the construction of many drainage
structures on small drainage basins. This stimu-
lated the need for a reliable and practical method
to determine design discharges for these structures.
One of the early efforts made to satisfy this need
consisted of the well-known Ramser curves.(21) It
is believed that these curves paved the way for the
development of the refined procedure of waterway
area determination later used by the Soil Con-
servation Service and the Bureau of Public Roads.
Ramser presented three sets of curves, giving
runoff in c.f.s. for drainage basins of different char-
acteristics and of sizes up to 30 acres for 10-year
rainfall frequency and from 30 to 1,000 acres for
10- and 50-year rainfall frequencies. The discharge
shown by the curves is for the region covering
Pennsylvania, Ohio, Indiana, Illinois, Wisconsin,
Minnesota, South Dakota, Nebraska and parts of
their neighboring states. For other regions the
value may be multiplied by a factor of rainfall
intensity ratio.
The Ramser curves were computed by the ra-
tional method with values of C and of times of con-
centration based largely on the results of measure-
ments made in 1918 on six small drainage basins,
1.25 to 112 acres, near Jackson, Tennessee.(29) Rain-
fall intensities for various durations and the 10-
and 50-year frequencies were taken from Meyer's
book on hydrology.(30) The curves were extended
beyond Meyer's ranges after the Yarnell rainfall
intensity frequency data were published in 1935.(31)
However, actual experience showed that Ramser
curves could not be applied over a wide range of
conditions as are encountered in the United States.
The curves were later gradually abandoned because
more information was needed on values of C, and
also because the estimation of the time of concen-
tration is uncertain and hence would affect the
accuracy of the results.
In seeking a reliable coefficient of runoff to be
used in the rational method, the Soil Conservation
Service established runoff and experimental drain-
age basin studies'32) in widely separated locations
as early as 1938. The studies covered a broad range
in topography, soils, vegetal cover, and tillage
practices. The results obtained indicated that the
coefficient of runoff and the time of concentration
could be employed very reliably in the rational
method. However, the assumptions underlying the
method were found to be inadequate and inappli-
cable to small rural drainage basins. Character-
istics and conditions of such drainage areas, as well
as of the channels, are greatly affected, not only by
the amounts and intensities of rainfall, but also by
other climatic factors and by land use, tillage, and
cropping practices. In the meantime, Krimgold('O
made a study on the relation of peak rates of runoff
to rainfall intensities for drainage basins of various
sizes and locations, but failed to show the signifi-
cance of such a relation. He also pointed out that
the frequency of runoff cannot be the same as the
frequency of rainfall intensity which is assumed
in the rational method (Section II-D). He sug-
gested a frequency study of the recorded peak
discharges and then derived the frequency curves
for the Claypan Prairies333' and other areas.
The Soil Conservation Service later developed a
procedure known as the Cook method'"4 after
Howard L. Cook, in which the working curves are
based on the results of runoff studies undertaken by
the Bureau of Agricultural Engineering in its cen-
tral district and on representative formulas of
flood flow and runoff coefficients then in use in the
North Central States. The method has been modi-
fied by M. M. Culp and others.(35' In this pro-
cedure, the probable maximum peak discharge from
a given drainage basin is computed as the product
of three factors, namely, the peak flow P in c.f.s.,
the rainfall factor R, and the frequency factor
F; or
Q = P XRXF (8)
II. A HISTORICAL REVIEW
Table 3
Runoff Producing Characteristics of Drainage Basins with Corresponding Weights W
(The weights are shown in brackets)
Runoff Producing Characteristics
(100)
Extremell
(40)
Steep, rugged, terrain, with
average slopes generally
above 30 per cent
(20)
No effective soil cover;
either rock or thin soil
nmantle of negligible infiltra-
tion capacity.
(20)
No effective plant cover;
bare except for very sparse
cover.
(20)
Negligible: surface depres-
sions are few and shallow;
drainage-ways steep and
smllal; no ponds or marshes,
(75)
Hitgh
(30)
Hilly, with average slopes
of 10 to 30 per cent
(15)
Slow to take up water; clay
or other soil of low infiltra-
tion capacity, such as heavy
gumllbo.
(15)
Poor to fair; clean-cultivated
crops or poor natural cover;
less than 10 per cent of
drainage area under good
cover.
(15)
Low; well-defined system of
smnall drainage-ways; no
polnds or marshes.
(50)
Normal
(20)
Rolling with average slopes
of 5 to 10 per cent
(10)
Normal, deep loam with
infiltration about equal to
that of typical prairie soil.
(10)
Fair to good; about 50 per
cent of drainage area in
good grassland, woodland,
or equivalent cover; not
more than 50 per cent of
area in clean-cultivated crops.
(10)
Normal; considerable surface-
depression storage; drainage
system similar to that of
typical prairie lands; lakes,
ponds and marshes less than
2 per cent of drainage area.
(25)
Low
(10)
Relatively flat land, with
average slopes of 0 to 5
per cent
(5)
High; deep sand or other
soil that takes up water
readily and rapidly.
(5)
Good to excellent, about 90
per cent of drainage area
in good grassland, wood-
land, or equivalent cover.
(5)
High; surface-depression
storage high; drainage
system not sharply defined;
large flood-plain storage
or a large number of lakes,
ponds or marshes.
The peak flow P is estimnated from a chart in
Figure 1. It depends on runoff producing character-
istics which are measured by tie sumllation of
weights as shown in Table 3. For a given drainage
area, the runoff producing characteristics are evalu-
ated by the sum of the weights, IW, counted for
different conditions. The peak discharge P is then
determined with this value of )XV from the chart.
Tile rainfall factor R varies with location as indi-
cated in the map of Figure 1. Tile frequency factor
F is 1.00 for 50-year frequency, 0.83 for 25-year
frequency, and 0.71 for 10-year frequency. For
example, the area of a drainage basin in Pike
County, Illinois, is 440 acres. The runoff producing
characteristics of the drainage basin are evaluated
as follows:
Basin Characteristics
Relief: slightly rolling with
average slopes of 5 to 16%
Soil Infiltration: normal
Vegetal Cover: fair
Surface Storage: normal
however, other methods of runoff determination
should be used to aid the judgment in arriving at
proper peak rates.
In recent years the U. S. Agricultural Research
Service las developed a method of hydrograph syn-
thesis for estimating flow characteristics from the
physiographic features of small drainage areas.t")
10000
7500
5000
2500
/000
750
4500
Wleights, W
- 250
20
10 X
10
10
XW = 50
From the chart, with 1WV = 50 and drainage
area of 440 acres, P is found to be 750 e.f.s. From
the rainfall factor map, R = 0.9. For a frequency
of 25 years, F = 0.83. Therefore the peak discharge
Q = 750 X 0.90 X 0.83 = 560 c.f.s. In applying
this method to drainage areas larger than 600 acres,
Droinage area, ocres
Figure 1. Chart and map for peak fow determination
by the Cook method
)esignation
of Basin
Characteristics
Relief
Soil Infiltration
Vegetal Cover
Surface Storage
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
The method involves (1) estimation of a char-
acteristic lag time from readily determined basin
parameters, (2) use of the basin lag time to predict
the hydrograph peak rate for an assumed total
volume of runoff, and (3) synthesizing the entire
hydrograph using the lag time, the estimated peak
rate, and a standard dimensionless hydrograph.
In developing the method of hydrograph syn-
thesis, multiple correlations of lag time with vari-
ous combinations of basin and channel slopes and
lengths, drainage density, shape, and size were
made. Of some 50 such multiple correlations a
formula for lag time was derived. The lag time t,
is defined as the time interval measured from the
center of mass of a block of intense rainfall to the
resulting peak of the hydrograph. The formula for
t,, in hours proposed by the Agricultural Research
Service is
t, = 1.75Ko0e'6 (9)
where Ko is a lag-time factor or
A0.3
K,= D.D.(10)
in which A = drainage area in acres
So = average land slope of the drainage
basin in percentage
D.D. = drainage density, this is total length
Lo of visible channels divided by
drainage area A, expressed in feet per
acre.
The lag time was found to be a major determi-
nant of the hydrograph shape, and hence a correla-
tion of the lag time with the ratio of the peak rate
of runoff to the total runoff volume was obtained.
Thus,
q, = 90.8 (11)
tp
in which q, = peak discharge in c.f.s.
V = total runoff volume in acre-feet
t, = lag time in hours
A generalized dimensionless hydrograph was
also developed. The coordinates of the hydrograph
are expressed respectively in the ratio of discharge
to peak discharge and in the ratio of time to lag
time. By means of this hydrograph and the esti-
mated t, and q, by the above-mentioned formulas,
the synthesized hydrograph can be constructed.
The development of the above method is based
on the analysis of rainfall and runoff records for
14 experimental drainage basins in Arizona, New
Mexico, and Colorado. The results were found to
be satisfactory for these regions. For other regions,
new correlation formulas may be necessary. The
method generalizes the rainfall pattern, soil condi-
tion, land use, and other factors. It ignores the
frequency factor which should be considered in the
computation of the total runoff volume.
The U. S. Soil Conservation Service has pro-
posed another method of hydrograph synthesis for
developing design hydrographs.(': In brief, this
method involves the following steps:
(1) Take a maximum probable 6-hour point
rainfall amount for the appropriate geographical
location of the structure.
(2) Modify the 6-hour point rainfall amount to
account for size of drainage area above the struc-
ture in accordance with a given synthetic rainfall
depth-area relationship.
(3) Develop a rainfall hyetograph for the modi-
fied 6-hour point rainfall in accordance with a
given synthetic hyetograph distribution pattern.
(4) Determine the hydrologic soil-cover com-
plex number of the drainage basin above the struc-
ture. The number shows the relative value of the
hydrologic soil-cover complexes as direct runoff
producers. The higher the number, the greater the
amount of direct runoff to be expected from a
storm. The numbers for various land uses, crop
treatments, hydrologic condition, and hydrologic
soil groups were prepared using data from gaged
drainage basins with known soils and cover.
The determination of the soil-cover complex
number is done with reference to both soil cover
and soil type.
The soil cover, as described from the hydro-
logic point of view, is given as either good, fair, or
poor, depending on the infiltration capacity. A soil
cover of high, medium, or low infiltration capacity
is described as being of good, fair, or poor condi-
tion respectively.
The soil types are classified on the basis of
intake of water at the end of long-duration storms
occurring after prior wetting and opportunity for
swelling, and without the protective effects of vege-
tation. The major hydrologic soil groups are:
Type A (lowest runoff potential) includes deep
sands with very little silt and clay; also deep,
rapidly permeable loess.
Type B includes mostly sandy soils less deep
than type A, and loess less deep or less aggregated
II. A HISTORICAL REVIEW
than type A, but the group as a whole has above-
average infiltration after thorough wetting.
Type (C complrises shallow soils and soils con-
taining considerable clay and colloid, though less
than those of type I). This type has below-average
infiltration after pre-saturation.
Type I) (highest runoff potential) includes
mostly clays of high swelling per cent, but the type
also includes some shallow soils with nearly imper-
meable sublhorizons near the surface.
A classification of about 2,000 major soils of
continental United States into the above four types
was made available by the Service.('3)
(5) Determine the direct runoff Q in inches by
the following formula:
( -- + 2
Q = +.. -0 (12)
R + N - 8
in which R = rainfall in inches from step (3)
N = hydrologic soil-cover complex number
from step (4)
Apparently, Equation 12 becomes invalid when
R < (200/N-2). For a value of R equal to 200/
N-2, Q will be zero. For R values greater than this
quantity, the Q versus R relationship is good. For
R values less than this quantity, Q has a positive
value, even when R equals zero. Obviously, the
equation must be modified so that when R < (200/
A-2), Q is taken as zero. In the practical applica-
tion, the relationship would probably never be ap-
plied in this lower range. Therefore, the equation
is not valid for these lower values.
The above equation was derived by plotting
storm rainfall versus direct runoff for observed
floods and correlating the results with the field
hydrologic soil-cover complex numbers for an aver-
age antecedent moisture condition.
(6) Correct the direct runoff values obtained in
the previous step for high or low antecedent condi-
tions if the design criterion is not for an average
antecedent condition.
(7) Obtain the direct runoff from the previous
step for uniform time intervals in the synthetic
hyetograph.
(8) Compute the time to peak (T,), base time
(Tb), and peak discharge (q,) of a triangular hy-
drograph for the direct runoff in each time interval
of the hyetograph by the following equations:
7, + 0.67T,
2
(13)
in which T,i = time from beginning of direct runoff
to peak in hours
D = time interval of effective rainfall in
hours
T, = time of concentration in hours
Tb = 2.67T,,
in which T,,
and
(14)
base time of the triangular hydro-
graph
ql =
484AQ
(15)
in which ,, = peak rate of flow in c.f.s.
A = drainage area in square miles
Q = direct runoff in inches
(9) Add all the triangular hydrographs and thus
obtain a composite hydrograph. The latter is the
design outflow hydrograph for the drainage basin.
The Soil Conservation Service method is too
complicated. It appears that certain details in this
method can be simplified or modified in order to
arrive at a simple method that can be used for
practical design of waterway areas.
Another activity of agricultural engineers has
been the provision of basic hydrologic information
for use in the design of small drainage structures.
The U. S. Agricultural Research Service has re-
cently published valuable hydrologic data for small
agricultural drainage basins:
"Monthly Precipitation and Runoff for Small
Agricultural Watersheds in the United States,"
since July, 1957.
"Annual Maximum Flows from Small Agricul-
tural Watersheds in the United States," since
June, 1958.
"Selected Runoff Events for Small Agricultural
Watersheds in the United States," since Janu-
ary, 1960.
These data should be of great use to the future de-
velopment of methods for waterway area determi-
nation.
H. DEVELOPMENTS BY HIGHWAY ENGINEERS
Since the beginning of the period of rapid ex-
pansion of highway constructions at about 1920,
highway engineers have been much interested in the
unavoidable problem of waterway area determina-
tion for their drainage structures. At the early
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
-EXAMPLt-
A culvert site in the Imperial Valley is 3 miles downstream
and 900 feet lower than the critical point on the watershed.
The tributary basin has an area of 2.0 square miles and
average ratio of run-off to precipitation is estimated at 60 percent.
The funicular thru H.900 ft, L- 3 mi., GOQ, A Z tsq.mi ,
and k-60 intersects Q 1700 sec.-ft. the required design discharge.
41 minute 5torm of 2t. inches per hour.]
~ LIMIT DESIGN DISCHARGE-
Based on a flood of 50-100 year frequency and
intensity - frequency data -
ST
u 3-
U
z
5-
500-
........ 000_-
5000 -
10000-
ACREs
n
a
EQUATIONS
L -0. 1(6 T o' s e,
P-l.00AL QA oo
p Range of K Volues
S I Thsfactrssnouldh mdifiMd
- tinrtion and anlitid d00
avincihsacrof terain I
90-95 Imperious surfaces
- 80-90 Steep barrens
60-80 Rolling barrens
50-70 Flat barrens
W 40-65 Rolling meadow
35-60 Deciduous timberland
25-50 Conifer himberlind
-o115-40 Orchird oo
I5-4u uplbno Tarms ,
10-30 Vallaj farmland,'
too k / o' 00-
1 ,0 -0 1-
S- zooo so-
B
-5
-i0
SQ. Mi.
JOINT DEPARTMEuETAL EP90
iP peP~IOeaa,&cE o WPT t f is
00. ON, TH
Ć½Ć½0- -i0O CULVERT A"PRACTICE
1941
CHART "A"
(Foq SMALL BAsINs ONLY) -1
NOMOooAPH Fon CAtCULATION OF DESION DISCHARGE
'ROM
H ~ The fall of the basin from watershed to culvert siteifct)
L - The length of channel from watershed to culvert site mi.
G- A Geographical classification.
A - The area of the basin tributary to the culvert.
k - The estimated percentage of run-off to precipitation.
V.ow. -24-40
Figure 2. Chart for computation of design discharge by the California method
Source: California Division of Highways
stage, the highway practice in designing drainage
structures was based almost entirely on the experi-
ence of railroad engineers and some sewerage engi-
neers. Even at the present time, many highway
engineers are still using the Talbot formula and the
Dun table, which were developed primarily for
railroad engineers, or the rational formula, which
was proposed for sewerage engineers.
The interest in waterway area determination by
highway engineers is reflected through the many
papers published by highway agencies and pre-
sented in highway engineering conferences. Most
of them recommend the use of formulas, charts, and
modified rational methods. One of the first compre-
hensive studies was made by the Oregon State
Highway Conmmission in 1934.01) This bulletin
contains a general description of the economic de-
sign of waterway areas and the results of research
by the Oregon Highway Department during the 15
years preceding the time of publication. It treats
both hydrologic and hydraulic aspects of the
problem.
Rowe and Thomas of California State Depart-
ment of Highways presented a new formula for
design discharge in the form of a nomograph in
1942.("S This study was later expanded and revised
under the supervision of Rowe and published in a
bulletin entitled "California Culvert Practice."*'3"
The nomograph is shown in Figure 2. In comput-
ing a design discharge from the nomograph, five
factors are considered: channel slope and length,
rainfall intensity-frequency, drainage area, and
basin texture. This California method is based on
the rational formula and hence is subject to the
same criticisms as that formula.
Other early paplers on the subject of waterway
areas for highway engineers in the hydrology field
include those by Houkt4"" in 1922, by Springer'4"'
presented to Road School at Purdue University in
1931, by Greve'42' in 1943, by Mavis(43) in 1946,
- OO
- tOO
10-
s :
H0U01
-j
-5000
i id t ll th fl r s l a
I
L
.... F3Fjr_
,,
II. A HISTORICAL REVIEW
2.5 50 75 10 50 /00 500
Drainage area,acres /000
Figure 3. Peak rates of runoff for drainage basins under 1,000 acres
(from Bureau of Public Roads Manual, August, 1951)
drainage basins established by the Soil Conserva-
tion Service supplied most of the runoff data('.' for
small agricultural drainage basins of less than 1,000
acres with different types of land use in the humid
region of the United States, ranging from Maryland
to Nebraska and south to Texas. Based on the
statistical analysis of tlese peak rates of runoff and
the Yarnell rainfall intensity data, the Bureau de-
veloped a method, the BPR method, which consists
of tile use of two charts as shown in Figure 3 and
Figure 4.1'11, :,"
The procedure involved in the BPR method is
similar to the Cook method of the Soil Conservation
Service described previously. In this method, the
design peak discharge from a given drainage basin
is computed as the product of four factors; namely,
the rainfall factor RF, the land use and slope factor
LF, the frequency factor FF, and the peak rate of
runoff Q for mixed cover in humid regions with a
frequency of 25 years and rainfall factor of
unity, or
Q,,,i,,, = sIl'FX L x FF X Q
(16)
Figure 4. Rainfall factors- use with Figure 3 in estimating
peak rates of runoff (from Bureau of Public Roads Manual,
July, 19511
by Merrel.(44 and Exulml(4' presented to the Ohio
Highway Engineering Conference in 1951, by Iz-
zard(4"''4 in 1951 and 1952, and by Bossy(41' in
1952.
The research engineers of the Bureau of Public
Roads have recognized the fact that the determina-
tion of waterway areas should be considered in two
steps. The first is to estimate the peak discharge
of a given frequency and the second is to determine
the physical dimensions relating to the culvert site
and thus to find the size of culvert required. In
developing the research work, the Bureau had the
cooperation of the U. S. Soil Conservation Service
and the U. S. Geological Survey. The experimental
The rainfall factor is obtained from Figure 4. The
land use and slope factor and the frequency factor
are to be selected from the table in Figure 3. The
discharge Q is to be selected from the curve in
Figure 3 corresponding to the given drainage area
in acres. This curve is applicable to localities where
the 25-year frequency of one-hour rainfall has an
intensity of approximately 2.75 in. per hr. For
other localities the peak rate of runoff should ie
increased or decreased in proportion to the one-hour
rainfall intensity fo: this frequency. This correc-
tion is to be applied by means of the rainfall factor.
It, is understood that the land use factors given
in Figure 3 are most reliable for steep land since
all of the observed data were for steeply sloping
land. The factors for flat and very flat land are
estimated from limited data on the effect of slope
on rates of runoff from test plots under simulated
rainfall with allowance for increased channel stor-
age. It is therefore noted that land use and slope
factors for flat and very flat land slopes are subject
to revision when more observed data become
available.
Like the Cook method the BPR method appears
to be practical and simple. However, the land use
and slope factor is still subject to a certain amount
of personal judgment. Furthermore, since the
method is based on the data of different geographic
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
locations, its application to a specific region will
give only a general estimate of the design discharge.
In order to consider the climatic and topo-
graphic features of a special region, the multiple
correlation method has also been used to determine
discharge from drainage areas over 100 acres. Pot-
ter<"'l 2) of the Bureau of Public Roads applied the
method to the analysis of runoff data from 51
drainage basins in the Allegheny-Cumberland Pla-
teau ranging from 100 to 350,000 acres in size. A
correlation of the runoff data to topography of the
drainage basins as well as to rainfall data from 89
stations in the region resulted in a correlation
formula:
Q = 0.038 A''70 T-0.554 TV (17)
in which Q = 10-year discharge in c.f.s.
A = drainage area in acres
W = a given rainfall factor
T = slope factor given by
0.7L
- 7T=.
0.3L
V/YO0.3T
in which L = length of stream in miles
X = difference in elevation in feet of the
streambed at the culvert site and at
0.7L upstream
Y = difference in elevation in feet of the
streambed at 0.7L and at the head-
water
The problem of waterway area determination is
continually of keen interest to highway engineers.
This is indicated by various research projects on
this subject sponsored by many state highway
agencies either with or without the cooperation of
the Federal highway agency. For example, a study
made by Kentucky Department of Highways is re-
ported by E. M. West and W. H. Sammons.('53
Many states are installing stream gages at the cul-
vert sites for collecting data for future analysis and
use in the determination of culvert areas.
I. CLASSIFICATION OF EXISTING METHODS
From a comprehensive study of available litera-
ture in this investigation, the existing methods of
hydrologic determination of waterway areas may
be classified into the following categories:
(1) Method of Judgment. By this method, the
determination of waterway areas is dependent on
practical experience and individual judgment. The
judgment developed by the engineer is invariably
guided by personal observation and general infor-
mation collected on the ground such as flood height,
size of channel, openings in the vicinity carrying
the same stream, etc. This may be a satisfactory
method if the judgment is good. However, the dis-
advantage lies in the fact that no judgment can be
perfect because the conditions vary so greatly from
problem to problem. Also, the method is not valid
for beginners or for those who have little practical
experience.
(2) Method of Classification and Diagnosis. By
this method, drainage areas are classified and pre-
scribed for different sizes and kinds of openings, the
limits for each opening allowing variations to be
made according to the local conditions, topography,
slopes, soil, rainfall, etc. This method has some
advantages if a table is prepared with reference to
a specific territory so that due allowance can be
made for the variable rainfall conditions and the
prevailing regional characteristics of the territory.
A glance at the table serves to indicate the general
class of opening required. The final determination,
however, will still be dependent on individual judg-
ment and a personal examination of the area.
(3) Method of Empirical Rules. An empirical
rule of thumb is usually developed to replace judg-
ment. Such methods were used frequently in the
early days, but have now become almost obsolete
because of their crudeness and the development of
better methods.
(4) Method of Formulas. By this method, a
formula is developed to determine the waterway
area. In Appendix I, a compilation of a large nun-
ber of formulas is presented. The formulas range
from simple to complex ones; many like the Talbot
formula(") are still very popular in engineering
practice. The use of a formula may generally cast
a certain amount of scientific glamour on the
method. The greatest merit of formulas is their
function of serving as a guide to determine quickly
the general range of the probable minimum, maxi-
mum, and average values. The method can also be
considered as practicable and serviceable for rough
calculations. However, the disadvantage of the
method is the uncertainty involved in the selection
of the proper coefficient in most formulas in order
to meet closely the conditions of the problem under
consideration.
(5) Method of Tables and Curves. Instead of
formulas, tables and curves are sometimes prepared
to serve the same purpose. The Dun table(22) is a
II. A HISTORICAL REVIEW
prominent example. The simplicity of this method
is its chief advantage. However, the table or curve
is too simple and does not usually include the con-
sideration of the many variables involved in the
problem.
(6) Method of Direct Observations. This method
involves making careful field surveys of drainage
area and stream characteristics and then making a
precise hydrologic analysis and hydraulic study.
Finally, it is used to arrive at the required size and
shape of the waterway which will carry off the
water quickly and without causing either scouring
or deposition in the channel.
(7) The Rational Method. This is a method
which is based on the rational formula, such as the
California method(3") or the Gregory and Arnold
method.(21)
(8) Method of Correlation Analyses. This
method involves the correlation of important hy-
drologic factors by statistical analysis. The result
may be represented by a formula or nomograph for
practical applications. The Cook method,'4) the
BPR method,'4""- "" and the Potter's multiple corre-
lation method("' "2) are examples.
(9) Method of Hydrograph Syntheses. By this
method, the hydrograph theory is used to derive a
synthetic hydrograph for design purposes. The U. S.
Agricultural Research Service('"' and Soil Conser-
vation Service"'' have developed such methods.
J. CHRONOLOGICAL DEVELOPMENT
From a historical review of the literature and
the study of existing methods, it appears that the
development of engineering studies of hydrologic
determination of waterway areas could be described
by a series of significant events. A list of such
events is:
1852 Preparation of a table expressing the relation
between the diameter and slope of a circular
outlet sewer and the size of its drainage area
by John Roe, Surveyor of the Holborn and
Finsbury Sewers, London, after numerous ob-
servations of the storm discharges.
1857 Presentation of the Hawksley formula in a
"Report of Commission on Metropolitan
Drainage, London." The original formula
seems to have been established at some time
between 1853 and 1856.
1879 Presentation of the Myers formula by T. M.
Cleemann in a paper before the Engineers'
Club of Philadelphia. The formula was then
published in "Railroad Engineers' Practice,
Discussion on Formulas" in the Proceedings
of the Club."' This is the first known water-
way formula by an American author, Major
E. T. D. Myers, Chief Engineer of the Rich-
mond, Fredericksburg, and Potomac Railway
in Virginia.
1880 Publication of the Biirkli-Ziegler formula in
a report'"' by the Swiss hydraulic engineer,
A. Biirkli-Ziegler, Switzerland.
1881 Introduction of the Biirkli-Ziegler formula to
the American technical literature by Rudolph
Hering in a report on "Sewerage Works in
Europe," to the National Board of Health. '7
1886 Discussion of the Myers formula and the use
of waterway formulas in general by Welling-
ton in an "Editorial" in the Railroad Ga-
zette.'(
1887 Publication of the Talbot formula by Pro-
fessor A. N. Talbot of the University of
Illinois. ()
1889 Publication of the rational method for esti-
mating rates of runoff from urban areas by
Emil Kuichling.'3)
1897 Publication of a report of Committee on
Waterway for Culverts by W. G. Berg.'2")
1897 Issuance of a drainage table for the Atchison,
Topeka, and Santa Fe Railway in blueprint
form. This table covers a range of drainage
areas from 0.1 to 1,000 sq. mi., and was later
expanded and appeared as the well-known
Dun Waterway Table in 1906.
1898 Publication of a discharge formula by George
Chainier.(t4)
1906 Publication of the Dun Waterway Table by
James Dun, Chief Engineer of the Santa Fe
System.(22)
1909 Publication of Reports of Committee No. I
on Roadway in the Proceedings of American
Railway Engineering and Maintenance of
Way Association,(23) including "The Best
Method for Determining the Size of Water-
ways," pp. 967-978, and a "Digest of Current
Practice" collected from the members of the
Association, pp. 978-1022.
1910 Completion of a thesis investigation on wa-
terway for culverts by A. F. Gilman and
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
G. W. Chamberlain in 1909-1910,'") under
the direction of Professor W. D. Pence, Uni-
versity of Wisconsin.
1911 Publication of a Report of Sub-Committee
on Formulas for Waterways of Committee
No. 1- On Roadway, in the Proceedings of
American Railway Engineering and Mainte-
nance of Way Association,(261 in which vari-
ous formulas for waterways were discussed.
1922 Publication of a paper by Ivan E. Houk,(4"'
emphasizing that each opening requires indi-
vidual design and formulas are makeshifts.
1926 Introduction of the Myers scale and the mod-
ified Myers formula by C. S. Jarvis."(' '""
1931 Publication of a paper by G. P. Springer on
waterways for culverts and bridges.'"'
1933 Recommendation of Ramser's Curves for the
design of soil and water conservation
works.(28)
1934 Publication of a comprehensive discussion on
design of waterway areas for bridges and cul-
verts by C. B. McCullough,(" describing
economic design of waterway areas and re-
sults of research by the Oregon State High-
way Department in the preceding 15 years,
including both hydrologic and hydraulic
studies.
Listing of 36 formulas and 4 sets of curves
used at that time for determining runoff and
sizes of highway and railroad drainage struc-
tures by T. A. Munson.(27'
1940 Publication of the Cook method by the U. S.
Department of Agriculture.(141
1943 Publication of a paper by F. William (ireve
on bridge and culvert flow areas.(42)
1944 Publication of the first edition of California
Culvert Practice by the State of California,
Department of Public Works, Division of
Highways, describing the California method
of culvert opening design.("")
1946 Publication of a paper by 1). B. Krimgold on
the hydrology of culverts, introducing the
U. S. Soil Conservation Service's practice to
highway drainage problems.(''
1949 Publication of the BPR method in Highway
Practice in the United States of America by
Public Roads Administration.(4)
1951 Publication of a paper by Carl F. Izzard, "'(
outlining some ideas about how the latest
knowledge of hydrology and hydraulics could
be applied to the design (if highway drainage
structures.
Publication, of a paper by J. F. Exum on
waterway areas for culverts and small
bridges.("')
Revision of the discharge-area curve in the
BPR method in Figure 1 of Hydraulic Infor-
mation, Circular No. 1,("") by the U. S. Bu-
reau of Public Roads. The revision was based
on a statistical analysis of actual records of
runoff on small agricultural drainage basins.
1952 Presentation of a paper by Tate )alrym-
ple"1)' on hydrology in design of bridge
waterways, describing the U. S. Geological
Survey's approach to the design of highway
bridge waterways. The paper was first pre-
sented to Raleigh Engineers' Club, Raleigh,
North Carolina, on February 11, 1952.
Presentation of a paper by Carl F. Izzard(47
to the American Society of Civil Engineers
at New Orleans Convention on March 5-7,
1952. The paper describes the latest practice
of estimating peak discharges for the design
of highway bridges and culverts.
Presentation of a paper by Herbert G.
Bossy (4' on simple methods for hydraulic
design of culverts at the 1952 annual meeting
of the Southeastern Association of Highway
Officials.
1955 Publication of a report by E. M. West and
W. H. Sammons"(,) of the Kentucky High-
way Department on the study of runoff from
small drainage areas and the openings in at-
tendant drainage structures.
1957 Publication of Monthly Precipitation and
Runoff for Small Agricultural IWtersheds in
the United States by the IT. S. )Department of
Agriculture, Agricultural Research Service. 0(
1958 Publication of Annual Maximum Flows from
Small Agricultural Watersheds in the United
States by the U. S. Department of Agricul-
ture, Agricultural Research Service."8)
Publication of a paper by Franklin F. Sny-
der(1"' describing the method developed by
U. S. Army Corps of Engineers for the deter-
mination of peak discharges from small
drainage basins of a given frequency.
1959 Publication of a paper by R. B. Hickok,
II. A HISTORICAL REVIEW
R. V. Keppel, and 13. H. Rafferty,1'"' describ-
ing hydrograph synthesis for small drainage
basins.
1960 Publication of Selected Runoff Eients for
Small Agri'culturaIl 1Vatrshieds in the United(
States 1)y the U. S. Department of Agricul-
ture, Agricultural Research Service.(''"1
Publication of a paper by Neal E. Min-
shall'('" on a synthetic metlod of predicting
runoff froni small experimental drainage
basins.
Publication of a pamphllet by W. 1). Pot-
ter,'") describing a correlation analysis of
peak discharges from small drainage basins.
III. A SURVEY OF DESIGN PRACTICE
A. THE QUESTIONNAIRE
In order to assess the current methods for the
determination of waterway areas as an important
reference to the present investigation, a nation-wide
survey was conducted by sending a questionnaire
to all State Highway Departments in the United
States in June 1953. Replies were received from 43
of the 48 states.
As supplements to this survey, additional infor-
mation was obtained from publications or other
indirect sources, and reference was also made to the
report of an earlier survey conducted in 1943 by
Tilton and Rowe'"" of the California Division of
Highways.
In the questionnaire used in the 1953 survey,
the following major items concerning the design of
waterway areas were requested:
(1) The recommended design frequency of drain-
age structures.
(2) The methods of calculation of the design dis-
charge and the determination of the size of culvert
and bridge openings.
(3) The consideration of the computation of cul-
vert slope and sections.
(4) The consideration of the computation of
head losses through culverts.
(5) The consideration of the hydraulic effects of
bridge piers and approach conditions, such as the
backwater effects.
The first two items are related to the hydrologic
design of drainage structures and the other items to
the hydraulic design. Since the present investiga-
tion is concerned primarily with the hydrologic de-
sign of the determination of waterway areas, the
main objective of the survey was essentially cov-
ered by the first two items.
It is obvious that the results obtained from the
survey cannot be considered as conclusive because
the information received was in general incomplete
and inaccurate and it represents only the practice
in 1953. Nevertheless, the findings of the survey
have furnished general knowledge about the design
practice at that time and such knowledge was found
valuable in the present investigation.
B. THE DESIGN FREQUENCY
From the replies to the questionnaire regarding
the policy of design frequency, the following major
findings were obtained:
(1) The design frequency and its use varied in
different states. There was no definite rule for de-
termining the design frequency. Generally speak-
ing, the design frequency depended mainly on the
size, type, importance, and location of the struc-
ture. However, in most cases, the importance of the
structure depended on economic and social factors.
(2) In several states, the design of important
drainage structures was based on historical floods
or independent investigations of the structures, but
not on design frequency. The historical floods were
usually obtainable from the records of the U. S.
Geological Survey. If the flood record was not
available at the site of the structure under consid-
eration, independent, investigation became neces-
sary for a proper determination of the design flood.
(3) For culverts, small bridges, and the drainage
structures in secondary highway systems, the de-
sign frequency varied widely from 5 to 100 years.
The design frequency most commonly used was 25
years.
(4) For bridges, large culverts, and the drainage
structures in primary highway systems, the design
frequency varied from 5 to 200 years. The design
frequency most commonly used was 50 years.
(5) Most replies do not specify clearly whether
the frequency referred to is the rainfall frequency
or the runoff frequency. It is known in hydrology
that the two frequencies are not identical. How-
ever, it may be generally assumed that many cases
imply the rainfall frequency rather than the runoff
frequency, because even at the present time runoff
data in most small drainage basins are not enough
to assure any reliable determination of the runoff
frequency.
III. A SURVEY OF DESIGN PRACTICE
Table 4
Design Frequency for Culverts, Small Bridges, and Drainage Structures
No. State Frequency of Design Flood
5 10 15 20 25 30 35 40 50
Alabama
Arizona
Arkansas
California
Colorado
Connecticut
Delaware
Florida
( ;eorgia
Idaho
Illinois
Indiana
Iowa
Kansas
Kentucky
Louisiana
Maine
Maryland
Mlassac rhulset
Michigan
M innesota
Mississippi
Missouri
Montana
Nebraska
Nevada
New Hlampshire
New Jersey
New Mexico
New York
North Carolina
North 1)akota
Ohio
Oklahoma
Oregon
Pennsylvania
Rhode Island
South Carolina
South I)akota
Tennessee
Texas
Utah
Vermont
Virginia
Washington
West Virginia
Wisconsin
Wyonling
in Secondary Highway
Historical
100
Systems
Indepndendet 'Type of
Investigation Structures
S
-. s
Culver ts.
Small bridges,
DIrainage structlres in secondary highw\ ay systemn.
Minor drainage structures in general.
No head on crown of the rulvert.
Balanced design w hich is defined as that combination of conduit section, shape, textiure, and gradient with entrance and outlet appurtenances
which will just pass a 100-yr. flood without interruption of traffic and without serious damage to structure, embankment or abutting property.
A summary of the reported design frequencies
adopted in different state highway agencies is given
in Tables 4 and 5.
C. HYDROLOGIC DESIGN PRACTICE
From the replies to the questionnaire regarding
the hydrologic design of waterway areas, the major
findings were as follows:
(1) Of the 43 states that supplied information,
25 states, or 58%, used the Talbot formula directly
or with modification. Ten states, or 23%, used the
rational formula; 8 states, or 19%, used the Biirkli-
Ziegler formula; 3 states, or 7%, used their own
formulas; and 9 states, or 21%, used miscellaneous
formulas.
(2) Of the 43 states. 5 states, or 12%, used tlhe
Dun table; 8 states, or 19%., used the Bureau of
Public Roads charts; and 5 states, or 12%, used
their own charts or tables.
(3) Of the 43 states, 11 states, or 26%c, used
U. S. Geological Survey's flood data or information
for the design.
(4) Most states which did not have a formula,
table, or chart prepared for their own design pur-
pose used several existing methods in order to ob-
tain the most appropriate design. For the design of
large or important structures, technical aid from
other agencies, such as the U. S. Geological Survey,
were frequently sought, and special independent in-
vestigations were usually made.
- - - - - -x
. . . . X
. . . . X
. . . . X
X X
- - - - - x
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
Table 5
Design Frequency for Bridges, Large Culverts, and Drainage Structures in Primary Highway Systems
Irequency of I)Dsign Flood
Ilistoriial Inde pendiet 'Typle of
I'loow I nvestiiation St ructu11res
i 10 15 20 25 30 35 40
50 100 200
Alabama
Arizona
A rkansas
California
( Colorado
('Connel(ticot
Dvlawaiv
I )e'iti ft 1
Illinois
Indiana
iKent nky
Maine
Mi il soti i
Mi,is isppi
Misiouri
Mi ontallna
NcIraska
Nevada
New i iampshire
New .hJrs-y
New Mexico
Now Y'o,'k
I( tllllr Ylalll
North Caroilina
North Dakota
Oklahma
VI isylvaiii
IltO Islal
South Carolina
Weasliintoin
West Virginia
Wiisconsin
Wvoniine
Iege'll C = Iairg' 'ug'lverts.
I = Bridlgs.
P = Irainage striucturs in priinar' sy stiem.
( = Draiageiii strctliiris ill ge'iieral.
* = Plis frehboard for drift.
(5) Many states were engaged in independent
studies for developing better methods of design. At
least ten states had developed either charts, formu-
las, or manuals for design purposes. Three states,
Illinois, Kentucky, and Missouri, were active in
continuous investigation for an adequate solution
of the problelm.
A sunlnlary of the reported hydrologic design
practice is given in Table 6.
D. HYDRAULIC DESIGN PRACTICE
From the replies to the questionnaire regarding
the hydraulic design practice, the major findings
were as follows:
(1) Based on the information available, 25
states indicated that tlhe considered the slope in
the culvert design. Most states which considered
the slope tried to place the invert of the culvert
along the natural slope of the stream or ground. In
those states where the land is flat, the slope of the
floor of the culvert was generally set to achieve a
safe velocity of flow through the culvert.
(2) Based on the information available, 16
states indicated that they considered the head loss
in the design and 12 states ignored it. In certain
cases the lhead loss was considered but minimized
or avoided in the design.
(3) Based on the information available, 13 states
indicated that they considered the backwater and1
apptroach conditions, 7 states ignored tliem, and 4
states avoided theim.
(4) The design references used by lmost states
No. State
------------~ ~~~-
- -
III. A SURVEY OF DESIGN PRACTICE
Table 6
Summary of Hydrologic Design Practice
No. State
1 Alabama
2 Arizona
3 Arkansas
4 California
5 (olorado
6 Connecticut
7 I)elaware
8 Florida
9i (heorgia
10 Idaho
11 Illinois
12 Indiana
13 Iowa
14 Kansas
15 Kentucky
1M Louisiana
17 Maine
18 Maryland
19 M\assachusetts
20 Michigan
21 Minnesota
22 Mississippi
23 Missouri
24 Montana
25 Nebraska
26 Nevada
27 New Hampshire
28 New Jersey
29 New Mexico
30 New York
31 North Carolina
32 North Dakota
33 Ohio
34 Oklahoma
35 Oregon
36 Pennsylvania
37 Rhode Island
38 South Carolina
39 South Dakota
40 Tennessee
41 Texas
42 Utah
43 Vermont
44 Virginia
45 Washington
46 West Virginia
47 Wisconsin
48 Wyoming
Total
% of 43 States
Formula
Talhot Rational Biirkli- State's Miscellaneous
Ziegler Own
x
x
Meyer
lMelatih. .uller
MeMatli
Wentworth
California
x Meyer
x
10 8 3 9
23 19 7 21
Table or Chart U.S.(.S. Independent No
Data Investigation Information
)un B.P.R. State's
Own
5
12
8 5
19 12
11 2
26 5
(a) "Basic Principles of Highway Drain-
age," Hydraulic Information Circular No. 1, Au-
gust, 1951, and "Special Problems in )rainage,"
Channels, by Sherman
.1. Posey. New York:
1941.
M. Woodward and Chesley
John Wiley and Sons, Inc.,
dic Information Circular No. 2, September (c) Information or aid given by the U. S.
,both prepared by the U. S. Bureau of Pub- Geological Survey.
tds. A summary of the reported hydraulic design
(b) Hydraulics of Steady Flow in Open practice is given in Table 7.
Hy!dra
1, 1951
lie Roa
~---- '
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
Table 7
Summary of Hydraulic Design Practice
No. State
1 Alabama
2 Arizona
3 Arkansas
4 California
5 Colorado
6 Connecticut
7 Delaware
8 Florida
9 Georgia
10 Idaho
11 Illinois
12 Indiana
13 Iowa
14 Kansas
15 Kentucky
16 Louisiana
17 Maine
18 Maryland
19 Massachusetts
20 Michigan
21 Minnesota
22 Mississippi
23 Missouri
24 Montana
25 Nebraska
26 Nevada
27 New Hampshire
28 New Jersey
29 New Mexico
30 New York
31 North Carolina
32 North Dakota
33 Ohio
34 Oklahoma
35 Oregon
36 Pennsylvania
37 Rhode Island
38 South Carolina
39 South Dakota
40 Tennessee
41 Texas
42 Utah
43 Vermont
44 Virginia
45 Washington
46 West Virginia
47 Wisconsin
48 Wyoming
Total
Head loss
Considered
0.5 to 1%
Natural
Computed
Natural or ignored
Natural
Natural
Natural
Considered
Considered
Natural
Ignored
Considered
Considered
Considered
Considered
Considered
Considered
Natural slope
Natural slope
Natural slope
Considered
Usually ignored
Ignored
Variable
Considered
Ignored
Ignored
(onsidered
Ignored
Considered
Usually ignored
Ignored
Considered
Considered
Considered
Considered
Avoided
Ignored
Considered
Ignored
Considered
No strict design Ignored
Considered
Considered Considered
Considered Considered
Overcome friction Considered
Natural slope Ignored
No less than 0.5% Ignored
Minimized
25 Considered 14 Considered
12 Ignored
2 Avoided or
Minimized
Backwater and
Approach Condition
Remarks
No information
No information
No information
Considered Use Nagler formula for backwater
computation
Usually ignored Use Manning formula for discharge
Most streams are small
Use B.P.R. Manual
Variable Use Kutter or Manning formula
Considered Aid from U.S.(.S.
Considered Use Manning formulai
Ignored
Considered Studies under way
Considered
No information
No information
Use B.Pl.R. Manual
Considered
Us(e Kutter formula
Avoided Use Manning formula
Usually ignored
Ignored Use Portland Cement Association formula
No information
Use Manning formula and B.P.R. charts
Considered Use standard publications
Considered
Considered Use Manning formula, and Univ. of
Iowa. Univ. of Minnesota, and B.P.R.
formulas
Avoided
Usually ignored
Considered
Ignored
Considered
Avoided
(onsidered
Ignored
Considered
Avoided
No information
Use B.P.R. Charts
New studies made
No information
Aided by U.S.(.S.
No information
Use own mianual
No information
Use B.P.R. Manual
Use of B.P.R. Charts and nomographs
Use own chart
* Use Manning and other formulas
13 Considered
7 Ignored
4 Avoided
~~
--
IV. HYDROLOGIC PRINCIPLES, DATA, AND ANALYSES
A. HYDROLOGY OF A DRAINAGE BASIN
For the design of small drainage structures, the
peak discharges under consideration are those of
runoff from small drainage basins.
liydrologic investigations have shown that there
is a significant difference between the small and the
large drainage basin."4'" ". For a small basin the
rates and amounts of runoff are dominantly influ-
enced by the physical condition of soil and cover
over which man has some control. Thus, more at-
tention in hydrologic study is given to the basin
itself. For a large basin the channel storage effect:
becomes very pronounced and more attention is
given to the hydrology of the stream. In the hydro-
logic study of large basins, direct measurements of
runoff at individual stream locations are generally
used, and often extralolated and extended. For
small basins, on the other hand, it has been neces-
sary to use a statistical "sampling" procedure be-
cause the task of gaging every small drainage basin
where runoff data might be required would be im-
practicable. In this procedure reliance is placed
largely on sampling of upstream areas with exten-
sion of the findings to other areas by means of
rainfall-runoff relations, and also on evaluation of
the climatic and physiographic factors influencing
runoff.
Strictly speaking, it is difficult to distinguish a
small drainage basin from a large drainage basin
using only the size of the watershed as a criterion.
Frequently two basins of the same size may behave
entirely differently from the hydrologic viewpoint.
One drainage basin may show prominent channel
storage effects, like most basins of large size, while
the other of the same size may manifest strong
influence of the land use, like most basins of small
size. In other words, a distinct characteristic of tlhe
small basin is the fact that the effect of overland
flow rather than the effect of channel flow is a
dominating factor affecting the peak runoff. Con-
sequently, a small basin is very sensitive both to
high-intensity rainfalls of short duration and to
land use. On large basins, the effect of channel
storage is so pronoun.ced thlat such sensitivities are
greatly suppressed. Therefore, a small drainage
basin may be defined ais one that is so small that its
sensitivity to hiyh-intensity rainfalls of short dura-
tions and to land use is not supprcssed by the chan-
nel characteristics. By this definition, the size of a
small basin may he from a few acres to 1,000 acres,
or even up to 50 sq. mi. The upper limit of the area
depends on the condition at which the above-
mentioned sensitivity becomes practically lost due
to the overwhelning channel-storage effect. For the
present study, however, a limit of 6,000 acres was
adopted as a criterion of small drainage basins for
practical purposes.
From the hydrologic point of view, the runoff
from a drainage basin can be considered as a prod-
net in the hydrologic cycle, which is influenced by
two major groups of factors: climatic factors and
physiographic factors. Climatic factors include
mainly the effects of rain, snow, and evapotranspi-
ration, all of which exhibit seasonal changes in ac-
cordance with the climatic environment. Physio-
graphic factors may be further classified into two
kinds: basin characteristics and channel charac-
teristics. Basin characteristics include such factors
as size, shape, and slope of drainage area, pernmea-
bility and capacity of ground water reservoirs,
lresence of lakes and swaimps, land use, etc. Chan-
nel characteristics are related mostly to hydraulic
properties of the channel which govern the move-
ment. and configuration of flood waves and develop
the storage capacity. It should be noted that the
above classification of factors is by no means exact
because many factors are interdependent to a cer-
tain extent. For clarity, the following is a list of
the major factors:
Climatic factors
(1) Rainfall
(a) Intensity
Ib) Duration
(ce Time distribution
(d) Areal distribution
(e) Frequency
(f) Geographic location
Snow
Evapot-ranspirat ion
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
Physiographic factors
(1) Basin character
(a)
ristics
Geometric factors
1. Drainage area
2. Shape
3. Slope
4. Stream density
(1)i Physical factors
1. Land use or cover
2. Surface infiltration condition
3. Soil type
4. (Geological condition, such as tlhe
permeability and capacity of
ground water reservoir
5. Topographical condition, such as
the presence of lakes and swamps
(2) Channel characteristics
(a) (Carrying capacity, considering size
and shape of cross section, slope,
and roughness
(b) Storage capacity
In midwest areas, high peak discharges from
small drainage basins are usually caused by rain-
falls of short duration largely due to thunderstorms
in summer months. A part of the precipitation is
lost through the process of interception, evapotrans-
piration, and infiltration. The remaining portion
which eventually becomes runoff is known as tlhe
rainfall excess and it is usually expressed in inches.
The proportion of the rainfall excess to the total
precipitation depends on climatic factors such as
the rainfall as well as on physiographic factors such
as antecedent moisture condition of tile ground,
type of surface soil and subsoil, and vegetation.
These factors vary largely with seasons.
As mentioned previously, land use plays an
important role in runoff phenomenon on a small
drainage basin because the flow on the watershed
is mostly of the overland type. Theoretically, tlhe
variables governing overland flow are the same as
those governing ordinary hydraulic flow of the
same type. For turbulent overland flow, which is
often the case in nature, these variables are the
depth of water, the slope of the ground, and the
surface roughness coefficient. The depth of overland
flow at a given time and at a given point in a
drainage basin is further governed by the length of
overland flow, duration of excess rainfall, rainfall
intensity during the time of rainfall excess, volume
of depression storage, infiltration capacity, and ini-
tial detention. All of these variables control tile
behavior of surface runoff and hence also the mag-
nitude of the maximum runoff. Many analytical
and experimental investigations have been under-
taken to develop laws and formulas for the deter-
mnination of overland flow. However, these laws and
formulas are found less than satisfactory when ap-
plied to natural drainage basins because of the
complexity, uncertainty, and variability of basin
characteristics. It seems at present that the best
approach to evaluation of the variables controlling
the overland flow, as well as other hydrologic phe-
nomnena of a drainage basin, is the use of stattistcal
and hydrologic analysis of all factual data that
have been collected from the watersheds.
It is generally considered that the runoff from
a given drainage basin is composed of two parts:
a base flow and a direct runoff which is produced
by the rainfall excess. The runoff, influenced by all
the physiographic and climatic factors described
above, is not steadyl but varies with time, reaching
a maximum or peak discharge and then falling off.
The curve of the discharge versus time is defined
as the hydrograph for the drainage basin under
investigation at its outlet, where the runoff is
measured. The hydrograph can be regarded as an
integral expression of the physiographic and cli-
matic characteristics that govern the relations be-
tween rainfall and runoff of the particular basin.
It shows the time distribution of runoff, defining the
complexities of the basin characteristics by a single
empirical curve. By separating the base flow, the
remaining hydrograph is the direct runoff hydro-
graph. The latter is to be discussed below.
For the purpose of hydrologic analysis, Sher-
man'""' has introduced the idea of a unit hydro-
graph or unitgraph. By definition, the unit
hydrograph of a drainage basin is a hydrograph of
direct runoff resulting from 1 inch of rainfall excess
generated uniformly over the basin area at a uni-
form rate during a specified period of time. In
applying the method of unit hydrograph according
to this definition, it should be noticed that the
following major assumptions are implied.
(1) Tle rainfall excess is uniformly distributed
within its duration or specified period of time.
(2) The rainfall excess is uniformly distributed
throughout the whole area of the drainage basin.
(3) The ordinates of the hydrographs of a com-
mon time base are directly proportional to the total
amount of runoff represented by each hydrograph.
IV. HYDROLOGIC PRINCIPLES, DATA, AND ANALYSES
Under the natural condition of rainfall and
drainage basins, these assumptions can hardly be
satisfied perfectly. However, the results obtained
by using unit hydrograph analysis have been found
acceptable for practical purposes. Although the
unit hylrograph was originally developed for large
drainage basins, many investigators(67, '6) have sub-
sequently shown that it is applicable also to small
drainage basins as well. It is known that there are
some exceptional cases which do not support the
use of unit hydrographs for small drainage basins,
but in the majority of the cases the results obtained
are acceptable within practical limits of error. The
method of unit hydrographs is therefore considered
practical also for small drainage basins.
The hydrologic quantities can be treated as sta-
tistical variables, and in their study reference
should be made to the frequency of their occur-
rence. The frequency is defined here as the average
time interval for such a quantity to be equalled or
exceeded. In a hydrologic study of runoff from
small drainage basins, the determination of the
runoff frequency is important. For numerous small
drainage structures accommodating floods from
small drainage basins along highways and on farm
lands, the frequency determination offers a logical
basis for establishing a design policy and makes
feasible a rational economic analysis for the design
purpose. It should be noted that the nature of these
structures is different from that of large drainage
structures designed for large floods, such as the
spillway of a dam, flood walls, etc. The failure of
small structures does not usually involve loss of
human life, nor does it cause the catastrophe often
connected with the failure of large structures. The
design of small structures can therefore be justifi-
ably made on the basis of a calculated risk deter-
mined by a frequency analysis.
B. SOURCES OF HYDROLOGIC DATA
The data used in the present study include the
rainfall, runoff, and other hydrologic data that are
available for the State of Illinois and its neighbor-
hood. The sources of these data are as follows:
1. Rainfall
In the State of Illinois, rainfall records of long
periods are published by tlie U. S. Weather Bureau
for first-order stations at Chicago, Cairo, Peoria,
Moline, and Springfield. The excessive precipita-
tion data for short durations at these places for the
period 1881-1896 were published in the annual re-
ports of the Chief of the Weather Bureau 1896-
1897. Data for the years 1897 through 1934 were
published in the annual reports of the Chief of the
Weather Bureau. For the years 1935 through 1949
the data were published in the issues of the United
States Meteorological Yearbook. For 1950 and suc-
ceeding years excessive precipitation data are pre-
sented in the annual issues of the Climatological
Data, National Summary.
The data for the intensity-duration-frequency
of the excessive precipitation were prepared by
Yarnell.'"" A revision of the data was published
recently by the U. S. Weather Bureau.(', 71) De-
tailed analyses of the Chicago and Urbana
data'(71'' 7 are also available. However, the results
obtained from these later analyses indicate that
they are in general much lower in value than the
Yarnell data. As valuable references to this inves-
tigation other rainfall frequency data 73, 74, 7, 76, 77)
for Illinois are also available.
A comprehensive compilation of maximum pre-
cipitation point data was published by the Weather
Bureau.(T. ') The probable maximum precipita-
tions based on a hydrometeorological analysis were
published by the Hydrometeorological Section of
the Weather Bureau.(s8, "8) Unofficial data and data
from Weather Bureau Cooperative Stations are also
available for comparative studies.
2. Runoff
The runoff data from large drainage basins in
Illinois have been studied and analyzed by the U. S.
Ceological Survey. The unit hydrographs'82) and
flood frequencies('" for basins larger than 10 sq. mi.
have been published.
The runoff data from small drainage basins in
Illinois are limited. The U. S. Agricultural Re-
search Service (abbreviated as ARS, formerly com-
bined with the Soil Conservation Service or SCS)
has maintained experimental plots and small single-
crop drainage basins at Urbana, Dixon Springs,
and Joliet. For securing information on the drain-
age properties of the soil and cover, precipitation,
runoff, and soil loss, data from plots at Urbana and
Dixon Springs were obtained from the ARS project
at the Agricultural Engineering Department, Pur-
due University, Lafayette, Ind. No records for
,Joliet were included, however, because these plots
are not equipped to measure rates of runoff, and
moreover, very little runoff has occurred since the
initiation of the plot study.
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
For small natural drainage basins, the ARS has
the projects at Alhambra, Edwardsville, Elmwood,
and Monticello. However, only the data at Monti-
cello are continuous and suitable for analyses.
Such data were obtained from the Department of
Agricultural Engineering, University of Illinois.
At the final stage of the present investigation,
observed data for runoff from small agricultural
drainage basins in the United States',7',s, 60o be-
caine available, and such data were used to great
advantage.
3. Hydrologic Soil Types
The University of Illinois Agricultural Experi-
lnent Station has made a comprehensive study on
the types of soil in Illinois. From this informa-
tion'8"' the hydrologic properties of different soil
types in Illinois were identified. The State Office
of the U. S. Soil Conservation Service in Chanm-
paign, Illinois also supplied the data on hydrologic
soil groups for Illinois.
C. RAINFALL ANALYSIS
The frequency data of rainfall intensity at Ur-
bana, Illinois are used as a basis for this study.
The data were developed mainly by the analysis in
an earlier study at the Departmlent of Civil Engi-
neering of the University of Illinois. (72 Data for
rainfall of long durations were adjusted by the
maximumn daily precipitation reported by the Illi-
nois State Water Survey.'"' The rainfall amount
frequency data at Urbana for frequencies up to 100
years is shown in Figure 5.
The maximum recorded rainfalls of various du-
rations in Illinois and its vicinity were also studied.
They are listed in Table 8. The maximum recorded
rainfalls at four rainfall stations in Illinois and ten
close-by stations in the neighboring states were ob-
tained from the U. S. Weather Bureau Technical
Report No. 2.0(7 The data at Urbana were ob-
tained from the Illinois State Water Survey.(7"'
For durations of 1, 2, 3, 6, 12, and 24 hours, the
maximum recorded rainfalls in Illinois, Indiana,
and Ohio were obtained from the U. S. Weather
Bureau Technical Report No. 15.(') From these
data, the maximum recorded rainfalls for all listed
durations in Illinois and in Illinois and its vicinity
were selected.
Table 8 also lists the probable maximum pre-
cipitations (PMP) at Urbana which were interpo-
lated from the all-season envelope for areas of 10
Table 8
Maximum Recorded Point Rainfalls in inches in Illinois and Vicinity
5
Cairo, Ill. 0.63
Month/Day 7/7
Year 1915
Chicago, Ill. 0.64
Month/Day 7/15
Year 1906
Davenport, Iowa 0.64
Month/Day 7/17
Year 1939
Dubuque, Iowa 0.80
Month/Day 7/9
Year 1919
Evansville, Ind. 0.51
Month!Day 9/15
Year 1934
Hannibal, Mo. 0.56
Month/Day 7/7
Year 1915
Keokuk, Iowa 0.69
Month/Day 5/22
Year 1899
Madison, Wis. 0.60
Month/Day 9/1
Year 1937
Milwaukee, Wis. 0 79
Month/Day 8/29
Year 1939
Peoria, 111. 0 73
Month/Day 8/17
Year 1925
Royal Center,
Ind. 0.65
Month/Day 8/9
Year 1930
St. Louis, Mo. 0.60
Month/Day 7/9
Year 1942
Springfield, II. 0.66
Month/Day 7/23
Year 1917
Terre Haute,
Ind. 1.15
Month/Day 7 7
Year 1915
Urbana, III. 0 62
Month/Day 7/8
Year 1942
Summarized 0.73
max. in Illinois
Max. in IIl.-
Max. in Indiana&
Max. in Ohio-
Max. in 11. and 1.15
vicinity
Probable max.
(PMP) at
Urbana, Ill.
Minutes
15
1.33
7/30
1913
1.31
9/13
1936
1.43
7/17
1939
1.54
7/9
1919
1.19
8/10
1908
1.39
8/18
1906
1.33
8/1
1932
1.41
8/8
1906
1.34
8/6
1942
1.26
7/2
1931
1.11
7/9
1925
1.39
8/8
1923
1 41
7/23
1917
1 38
7/7
1915
1.39
6/22
1931
1 41
24
5.69
10/3
1910
6.19
8/2
1885
5.18
7/13
1889
5.48
9/8
1927
6.94
10/5
1910
5.83
9/3
1926
5.88
6/28
1933
5.31
9/7
1941
5.76
6/22
1917
5.52
5/18
1927
3.23
5/18
1927
8.78
8/15
1946
5.94
6/4
1917
5.60
9/14
1931
4.61
5/25
1921
6.19
3.25
3.20
3.65
1.32 1.54 2.56 3.67
8.35 11.47
6.80 6.94
6.07 6.35
8.35 11.47
24.0 28.2 30.5
* Reported in U.S.W.B. Technical Paper No. 15.(9M
sq. mi.(1" These were derived by a theoretical hy-
dromcteorological method which involves an anal-
ysis of air-mass properties (effective precipitable
water, depth of inflow layer, temperature variation,
winds, etc.), synoptic conditions prevailing during
the recorded storms in the region, topographical
features, season of occurrence, and location of the
respective areas involved. However, the rainfall
amounts thus derived are extremely high, and
therefore are somewhat unrealistic and impractical
for design purposes. They are listed only for the
purpose of comparison.
IV. HYDROLOGIC PRINCIPLES, DATA, AND ANALYSES
./ .25 .5 .75 1 5 10 50
Duration in hours
Figure 5. Rainfalls of different frequencies and maximum rainfalls
There are some unofficial data in Illinois which
are higher than the official maximum recorded rain-
falls given in Table 8. These data, supplied by
Mr. F. A. Huff of the Illinois State Water Survey
in a letter dated June 24, 1957, are as follows:
(1) Field observation values:
(a) 16.54 in. at East St. Louis and 13.75 in.
at Belleville for 9.5 hr. duration on June 14-15,
1957 (recorded by a station of the East Side Levee
and Sanitary District).
(b) 12.5 in. at Ocoya for 6 hr. duration on
,July 8, 1951 (unofficial).
(c) 12.5 in. at Belvidere for 6 hr. duration
on July 18, 1952 (unofficial).
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
(d) 9.75 in. at Aurora and 11.45 in. at Wa-
terman for 24 hr. duration on October 9-10, 1954
(recorded by the State Water Survey gage).
(e) 11.00 in. at Bellflower for 12 hr. dura-
tion and 14.0 in. total in 36 hr. on May 26-27, 1956
(unofficial).
(2) Maximum daily amounts at Weather Bu-
reau Cooperative Stations:
(a) 9.15 in. at Galva, August 20, 1924.
(b) 10.48 in. at Aurora, October 9-10, 1954.
(c) 8.41 in. at Rockford, July 18-19, 1952.
(d) 8.24 in. at Mascoutah, August 16, 1946.
(e) 7.50 in. at Grafton, July 9, 1942
The fact that some high values of unofficial ob-
servation have not been registered by the long-
range Weather Bureau first-order stations indi-
cates that the probability of occurrence of these
values at a particular location is small. For the
purpose of designing minor drainage structures, the
official data should constitute a practical and rea-
sonable upper limit.
The above data are all plotted in Figure 5.
Comparing these data with the frequency curves
described previously as shown in the figure, the fre-
quency of the Urbana maxima is approximately in
the order of 50 years, and the Illinois maxima imay
be in the order of 100 years for durations less than
30 minutes and of about 1,000 years or more for
higher durations.
In a study made by the Illinois State Water
Survey,(6) it was found that the State of Illinois
may be divided into four geographical sections of
generally similar precipitation climate. Annual and
seasonal maximum precipitation data used in the
study were from 39 stations for the 40-year period
1916-55. By several methods of frequency analysis
isohyetal maps were developed for a 50-year re-
currence of one-day precipitation. This isohyetal
pattern and the isohyetals of mean annual precipi-
tation indicated that it is climatologically satisfac-
tory to divide the entire state into four sections:
northwest, north central, south central, and south-
east. In each section the average value of 50-year
one-day precipitation was computed as shown in
Figure 6. The sectional values in extreme north-
western and southern Illinois are high. These areas
coincide with the Rock River Hills and the Shawnee
Hills, the most prominent hill regions in the state,
which may have some augmenting effect on storm
precipitation.
In the present investigation tile 50-year one-day
Figure 6. Fifty-year one-day precipitation isohyetals, averages,
and conversion factors for four climatological sections of Illinois
precipitation value of 5.9 at Urbana is taken as an
index. The ratios of the sectional values to the
value at Urbana are computed as shown in Figure
6. These ratios are conversion factors for the four
geographical sections. Thus, the average precipita-
tion value in a section is equal to the product of
the Urbana value and the conversion factor for the
section. As mentioned before, the Urbana maxima
fit the 50-year frequency agreeably, and all fre-
quency curves in Figure 5 appear to have a similar
trend of variation. Therefore, the conversion fac-
tor may be considered applicable to all Urbana
frequency curves.
Should more reliable rainfall data become avail-
able, greater accuracy may be gained by expressing
the climatic factor by isocontours. With the lim-
ited information available at present, however, the
IV. HYDROLOGIC PRINCIPLES, DATA, AND ANALYSES
average sectional values are believed to provide
more realistic values than isocontour values. The
stuldy made by the Illinois State Water Survey
indicated that the sensitivity of isohyetal patterns
to sampling variation of data is remarkably large.
With insufficient data the orientation and distribu-
tion of the isohyets thus determined would be there-
fore very uncertain.
It may be noted in Figure 6 that Urbana is
located in the north central region. The conversion
factor for this region is 0.92 while the theoretical
conversion factor should be 1.00. Because of the
uncertain sampling variation described above, such
difference may be ignored for practical purposes.
However, the regional conversion factors are aver-
ages. For any location away from the middle zone
of the region, an interpolated value of the conver-
sion factor may be obtained if so desired.
The rainfall data at Urbana were observed at
the site of the rain gage and are therefore referred
to as "point-data." The maximum rainfall occurs
at a point or over a small area at the center or
centers of a storm. Outside the center the rainfall
amount decreases so that the average amount over
a larger area becomes less with increase of this
area. The pattern of rainfall distribution in a
storm is very irregular and indefinite. For small
drainage basins under consideration (less than
6,000 acres), the rainfall reduction is small. For
practical purposes, it may be assumed that the
average rainfall over the basin area is equal to the
point value. Therefore, to be on the safe side, no
correction for areal distribution is made.
D. DEVELOPMENT OF UNIT HYDROGRAPHS
As mentioned previously the derivation of unit
hydrographs was originally based directly on ac-
tually measured hydrographs fulfilling as nearly as
possible the assumptions of the method. There are
however other methods for the determination of
unit hydrographs which have been developed in
later years. Three methods for the development of
a unit hydrograph will be discussed. They are
(1) direct derivation from an observed hydrograph
or hydrographs, (2) synthesizing observed hydro-
graphs from a number of drainage basins, and
(3) building-up of hydrographs on the basis of
theory.
1. The Method of Direct Derivation
A unit hydrograph can be developed by direct
(a)
(b)
(c)
q,
3
b
Figure 7. Derivation of unit hydrograph by the S-curve method
derivation from an observed hydrograph or hydro-
graphs. This method is described in most hydrology
textbooks. In selecting a hydrograph, care should
be taken that assumptions involved in the unit
hydrograpli theory should be satisfied as closely as
possible. A hydrograph resulting from an isolated
storm of nearly uniform distribution in space and
time is the most desirable. When a hydrograph ex-
hibits several closely related peaks as a result of
the occurrence of multiple storms, single-peaked
hydrographs should be first segregated in the unit
hydrograph analysis. A method of hydrograph
segregation is described by Mitchell.(82)
From the hydrograph of a single storm, a unit
hydrograpl can be derived. The duration of this
unit hydrograph is defined as being equal to the
duration of the rainfall excess of the storm. From
the same data, unit hydrographs of other durations
may be derived by the so-called S-curve method
first suggested by Morgan and Hullinghors.(87' The
theoretical S-curve is a hydrograph that is pro-
duced by a continuous rainfall excess at a constant
rate for an indefinite period. The curve assumes a
Continuous rainfall excess
at a rate of x in. per hr
S-Curve
t hr Continuous rainfall excess at
-: a rate of x in. per hr
- ~-v//////////////////////////////
SPosition of initial S-Curve
'I
Offset S-Curve
xl
I -Position of initial S-Curve
4 ---- Offset S-Curve
I I. I ly = Difference in S-Curve
l I ordinates
S-- Un/i hydrograph for t hr
duration
Ay__
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
/0.000
7500
5000
2500
750
2500
Q-o
250
'/Vn
.10 .25 .50 .75 1.0
t. Duration in hr
Figure 8. Relationship between the unit
and its duration for the Boneyard dr
Champaign-Urbana, Illino
deformed S-shape and its ordinate
proach the rate of rainfall excess
S-curve can be constructed graphic
up a series of identical unit hydro
intervals equal to the duration of t
from which they are derived. The
numerical example of the S-cu
described in a report by Chow.(72l
After the S-curve is constructed
graph of a given duration may b
lows (Figure 7): Assume that th
duced by a continuous rainfall exc
rate of x in. per hr. Then advar
position of the S-curve for a per
desired duration in t hr. and call tl
offset S-curve. The difference betw
of the original S-curve and the ol
vided by xt, should result in the de
graph.
When a number of unit hydrog
durations are obtained, a curve of
peak discharge against its duratio
in a logarithmic scale. Experienc
use of actual data in plotting such
than derivation by theory, which
dition of linearity. Therefore, wl
observed hydrographs for variou
available, it is advisable to deri
hydrographs from these hydrograp
their peak discharges against th
k
2.5 5.0 75 10 ungaged drainage basins by synthesizing a num-
ber of representative unit hydrographs in a given
hydrograph peak region was first proposed by Snyder.'8-) However,
ainage basin,
ais his study was made primarily for the Appalachian
highlands. In Illinois, a synthetic study of 58 unit
hydrographs for large drainage basins has been
s ultimately ap- made by Mitchell,'2) resulting in three synthetic
as a limit. The S-curves (Table 9) for basin areas of 80, 500, and
ally by summing 1,200 sq. mi. respectively. The ordinates of these
graphs spaced at curves are expressed in per cents. For an equilib-
he rainfall excess rium runoff the rate is equal to the rate of rainfall
procedure and a excess. Considering a rate of rainfall excess equal
re method ar to 1 in. per t hours and the drainage area as A
acres, the equilibrium runoff is equal to 1.008A/t
, the dunit hydro- c.f.s. The abscissas of these curves are expressed in
Sderived as fol- lags. The lag t, is equal to the time interval in hours
e S-curve is pro- from the center of mass of rainfall excess to the
ess at a constant center of mass of runoff.
nce or offset the In a frequency study of floods in Illinois by
iod equal to the Mitchell,(8" an expression for to was found as
his S-curve as an
een the ordinates
ffset S-curve, di-
sired unit hydro-
graphs of various
unit hydrograpl
n can be plotted
e has shown the
a curve is better
assumes the con-
ien a number of
is durations are
ve separate unit
hs and then plot
ie corresponding
to = k DoĀ° (19)
in which k is a physiographic factor varying from
0.60 to 1.82 with an average value of 1.05 and D is
the watershed area in square miles.
Applying the S-curve method described previ-
ously, the unit hydrograph peak discharges for
various durations for drainage basins of the three
sizes mentioned above can be computed from the
synthetic S-curves. The ratio of the unit hydro-
graph peak discharge P to 1.008A/t, or Pt/1.008A,
is defined as the peak-reduction factor Z, or
Pt
Z Pt (20)
= .008A (20)
To illustrate the effect of the duration on the
durations. For example, the unit hydrograph peak
discharges for a number of observed hydrographs
for the Boneyard Creek drainage area at Cham-
paign-Urbana, Illinois, are plotted against the
duration as shown in Figure 8. The resulting curve,
despite the scattering of the plotted data, usually
represents a better relationship than that obtained
by the S-curve method from a single hydrograph.
When a curve showing the relationship between
the unit-hydrographl peak and the duration is avail-
able, the value of a unit hydrograph peak discharge
P can be interpolated from this curve for any given
duration within the range of the plot.
2. The Method of Synthesizing
A method of developing unit hvdrographs for
IV. HYDROLOGIC PRINCIPLES, DATA, AND ANALYSES
Table 9
Ordinates of Synthetic S-Curves in Per Cent of Equilibrium Runoff for Drainage Basins
of 80, 500, and 1,200 sq. mi. in Illinois
Time in Basin Area, sq. mi.
Terms of 80 500 1,200
Lag t.
0.05 0.12 0.23 0.600
0.10 0 72 0.83 1.62
0.15 2.02 1.80 2.88
0.20 4.17 3.17 4 49
0.25 7.27 1. 95 6.51
0.30 11.10 7.16 9.14
0.35 15. 8 0.80 12.14
0.40 20.41 12.95 15.49
01.45 25. (4 16.43 19.09
0.50 30 89 20.26 22 83
0 55 35.83 24.4 f 26 71
0.00 40.47 28.97 30.04
0. i5 44.82 33.65 34.01
0.70 48.89 38.45 38,57
0.75 52. 9 13.12 42 48
0.80 56.23 47.64 46.33
0 85 59.52 51.91 50.10
0190 62.57 55.89 53.76
0.95 65.)39 59. 58 57.29
1.00 1i7.99 62.97 60.60(
Time in
TPermls of
Lag to
1.05
1.10
1.15
1.20
1.25
1.30
1 .35
1. 40
1.45
1 50
1.55
I . 60
1.605
1.70
1.75
1.80
1 85
1.90
1.95
2.00(X
Basin Area, sq. ii.
80 500 1,200
70.38
72.57
74.57
76.39
78.04
79.52
80.88
82.11
83.24
84.29
85.27
80619
87.00
87.88
88.65
89.38
90 07
90.72
91 33
91.90
66.10
019.02
71.76
74.33
76.73
78.96
81.02
82.91
84.(63
86.18
87.58
88.84
89.97
9011 99
91.91
92.73
93.47
94.13
94.73
95 27
63.85
66.8.4
69.061
72.18
74.57
76,78
78.81
80.68
82.41
84.00
85.47
86.83
88.08
89.22
90.27
91.2:3
92.10
92.89
93 61
95.26(
Time in
Ter'ms of
Lag to
2.10
2.15
2.20
2.25
2.30
2.35
2.40
2.45
2.501
2.55
2.600
2.05
2.70
2.75
2.80
2.85
2. 90
2.95
3. 00
Basin Area, sq. Imi.
80 500 1,200
95.75
96.19
96.591
96.95
97.27
97.55
97.81
98.04
98.25
98.44
98.01
98.76
98 89(
99 01
99.12
99 22
99.31
99(.39
99.46
99.52
9 . 84
95.36
9(5.84
96.261
96.065
96.9()
(97.30
97.58
97.83
98.05
98 25
98.43
98 59
98 74
98.88
99.00
99. 11
99.21
99 30
9!. 38
Time in Basin Area, sq. mi.
Terms of 80 500 1,200
Lag to
3.05 98.51 99.58 99.45
3.10 98.68 9(. .03 99.51
3.15 98.84 99.(18 99.57
3.20 98.9(9 99.72 99.602
3.25 99.13 99.76 99.66
3 30 99.26 99 79 99 70
3 35 99.38 99.82 99.74
3 40 99.49 99.85 99.78
3.45 99.58 99.87 99.81
3.501 99.60 99.89 99.8-4
3.55 99.73 99.91 99.86
3.60 99 79 99.9 2 99.88
3. 65 99.84 91)9.93 99.90
3.70 99.88 99.94 99.92
3.75 (99.91 99.95 99.94
3.80 99 94 99.90 99.96
3.85 99 91 99.97 99.97
3.90 99.98 99 98 99.98
3, 95 99.99 99.(99 99.99
4.00 100.00 100.00 100.00
lIeak discharge, the peak-reduction factor can be
plotted as the dimensionless ratio of t/ti. The re-
sults obtained for the synthetic unit hydrographs
of the three areas are presented in Figure 9.
A study of the curves in Figure 9 indicates that
they appear to converge to a limiting position as
the drainage area decreases. For small drainage
basins, therefore, it can be assumed that the corre-
sponding curve would lie somewhere above the
curve for 80 sq. ni.
3. The Method of Building-up
This method derives a unit hydrograph by
breaking up the drainage area into a number of
segments and calculating the contributing flow of
each segment. The method will be described with
reference to a hypothetical circular drainage basin.
The fictitious circular basin (Figure 10) is
.5
N
.075
.050
/n01
.01 .025 .05 .075 ./ .5 5
,/o
Figure 9. Dimensionless plot of unit hydrograph peak
against duration
assumed to have a length L, which is measured
along the stream, and to le exlosed to ta uniform
and continuous rainfall excess of intensity of 1 in.
per t hours. The slope from the upstream end at A
to the point of concentration 0 is S.
As the rain starts to fall over the whole basin,
the water reaching 0 will be in the beginning only
thatl falling in its inlnediate vicinity, say in area
Circular
Rectangular
Triangular - equilateral - Triangular
Figure 10. Hypothetical drainage basins
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
Table 10
Computation of the Dimensionless S-Curve for a
Hypothetical Circular Drainage Basin
z/L t,'t. a a/11.95 I
(1) (2) (3) (4) (5)
0.1 0.17 0.24 0.021 0.021
0.2 0.29 0.77 0.065 0.086
0.3 0.40 0.96 0.080 0.166
0.4 0.50 1.26 0.105 0.271
0.5 0.59 1.48 0.124 0.395
0.6 0.68 1.54 0.129 0.524
0.7 0.76 1.74 0.145 0.669
0.8 0.85 1.68 0.141 0.810
0.9 0.92 1.38 0.115 0.925
1.0 1.00 0.90 0.075 1.000
11.95 1.000
a,. It will, however, gradually increase as the water
from upstream points on the basin arrives. Eventu-
ally, when water from A reaches 0 in some time t,
after the commencement of the storm, tie cdis-
charge at 0 will be maximum.
For small drainage basins, Ramser(291 has de-
termined the time of concentration by noting the
time required for the water in the channel at the
gaging station to rise from the low to the maximum
stage as recorded by the water-stage recorders.
Based on these and other data, R. R. Rowe (from
a private correspondence dated March 22, 1957)
obtained an equation for the time of concentration
t, in hours as follows:
t = (11.9=L'/H)Ā·.=") (21)
in which L is the length of basin area in miles,
measured along the watercourse from the gaging
station and in a direct line from the upper end of
the watercourse to the farthest point on the drain-
age basin, and H is the fall in feet of the basin from
the farthest point on the basin to the outlet of
runoff. The equation was developed by empirical
compromise of data from many sources. The lower
values of t, obtained by Equation 21 closely follow
Ramser's data of 1927 as reduced by Kirpich.1'9'
Above 30-min. concentrations, use was made of a
summary of many unit hydrograph computations
for bridge and culvert design by several engineers
from the Bridge Department of the California Divi-
sion of Highways. Equation 21 can be modified and
expressed as
0.77
to = 0.00013 (L/VS ) (22)
in which L is in feet and S = H/L or approxi-
mately the average slope of the drainage basin in
feet per foot.
Now divide the circular basin into ten parts
with areas a1, a2, a,, etc., in acres. The division is
made by dividing the stream length into ten equal
o
o
0
0
I
(0
I
0
S'c,, or 'h4,
Figure 11. S-curve and unit hydrographs for a hypothetical
circular drainage basin
parts, and arcs passing through the divided points
are drawn with their common center at the outlet O.
In case of a real drainage basin, division should be
made in accordance with the topographic condition.
Let tG' be the time of concentration for any sub-
area a and x be the length of the subdivided basin,
then Equation 22 gives
= 0- = 0 0.77(x/
t,' = 0.00013 (x/v S )
Dividing Equation 23 by Equation 22, and assum-
ing the slope of the sub-areas equal to the slope
of the area,
(24)
t _ (. )0.77
toL
The computation for an S-curve of this basin is
given in Table 10. Column 1 of this table gives
values of x/L. Column 2 gives values of t,'/t, as
computed by Equation 24. Column 3 gives the
measured subdivided areas. Column 4 gives the
proportioned sub-areas so that the sum of the areas
is equal to unity. A value in this column is equal
to the value in column 3 divided by the sum of the
values in column 3, or 11.95. Column 5 gives the
cumulative values of column 4. Since the uniform
rainfall excess is assumed to be of an intensity of
1 in. per t hr., the runoffs in c.f.s. from the sub-areas
are 1.008a,/t, 1.008a,/t, 1.008as/t, etc. These are
the rates that runoff is being generated on the up-
land parts of the sub-areas. Their summation is
predicated on those rates combining in phase as they
progress downstream to an equilibrium value of
1.008A/t. To attain this equilibrium the duration
of rainfall excess must continue at the constant rate
for a period equal to or greater than t, for A.
IV. HYDROLOGIC PRINCIPLES, DATA, AND ANALYSES
Table 11
Computation of the Built-Up Unit Hydrograph for a
Hypothetical Circular Drainage Basin
Duration t = 0.11
I/I S-curv e Offset Unit Moment
S-icurve Hydro-
graph
Ordinates
0) (2) (3) (4) (5)
0.1 ;0.008 0.008 0.000)8
0.2 0.025 0.008 0.017 0.0034
0.3 0.088 0. 025 0,063 0.0189
0. 0 1. fii 0088 0.078 0.0312
0,5 0. 270 0.166 0.104 0.0520
0.6 0.405 0 270 0.135 0.0810
0.7 0.561 0 405 0.156 0.1092
0.8 0.725 0.561 0.164 0.1313
0.9 0.895 0 725 0.170 0.1530
1.0 1.(000 0.805 0.105 0.1050
1.000 0000 0.0000
1.000 0.6858t1
0.5x 0. lt = 0.0500f,
Lag = t,, = 0.63580,
Therefore, tl1e values il column 5 are the cumula-
tive runoffs expressed in fractions of 1.008A/t. By
plotting these cumulative runoffs against the values
in column 2 and smoothing the plot, a dimensionless
S-curve can be obtained as shown in Figure 11. The
times of concentration t,' marked along the abscissa
of the curve are actually equal to the times since
the beginning of the rainfall excess, when the flows
arrive at the gaging station. It is possible therefore
to replace them by a continuous time unit t,, which
is independent of tile mode of subdivision of tile
area.
After the S-curve is constructed, a unit hydro-
graph of duration equal to t may be constructed
by the method described previously. For simplicity
of computation, the S-curve ordinates are read from
the S-curve in Figure 11 for every tenth of tf,/t,.
and the values are listed in columns 1 and 2 in
Table 11.
Considering a duration of the unit hydrogralph
equal to t = 0.1 t,, the computation of the unit
hydrograph is shown in columns 1 to 5 of Table 11.
Column 1 gives values of th,/t, in tenths. Column 2
gives values of S-curve ordinates in fractions of
1.008A/t from Figure 11. Column 3 is the offset
values of S-curve ordinates for a time interval
equal to the unit hydrograph duration t. Column 4
gives the difference between values in column 2 and
column 3. This column gives the ordinates for tile
unit hydrograph. According to tile definition of a
unit hydrograph, the sum of these ordinates should
be equal to a correslonding rainfall excess of 1
inch. Since the intensity of rainfall excess is 1/t and
the duration is t, the amount of rainfall excess is
equal to 1. The ordinates of the unit hydrograph
are therefore expressed in fractions of 1.008A/t.
-< (0
S.50
i.25
/0
.050
.050
0
t
Triangulor, equ
_ I-- _ __
SCircular
(V Circulor
Triangular.
equilateral
-T--
4
-Q] Rectangular
m Ip oes
T
rf- For droinge basins of 80sq miles in Illinois
.050.075.10 .25 .5
'o
.75 1.0
2.5 5.0
Figure 12. Dimensionless plot of unit hydrograph peak
against duration for hypothetical drainage basins
This unit is the same as that used for the S-curvc.
The peak discharge of the unit hydrograph is found
to be 0.170 of 1.008A/t. The unit hydrograph is
plotted as shown in Figure 11.
The lag to of the unit hydrograph is equal to the
time interval between the center of mass of rainfall
excess to the center of mass of runoff. The center
of mass of rainfall excess is equal to 0.5t = 0.5 X
0.1I,. = 0.05t,. from the beginning of the rainfall
excess. The center of mass of runoff may be com-
puted as shown in column 5 of Table 11. The values
in this column are the moments of unit hydrograph
ordinates about t;, = 0, or equal to the products of
values in column 1 and the corresponding values in
column 4. The sum of the values is equal to 0.6858t..
The moment arm is equal to 0.6858t,/1.000 =
0.6858t,.. Thus, the lag should be equal to t, =
0.6858t, - 0.05t,. = 0.6358t,.. Hence, t/t,, = 0.lt/
0.6358t,, = 0.157. This ratio corresponds to the
abscissa of the plots in Figure 9.
From the above computation for a unit hydro-
graph duration equal to 0.1t,, a point with coordi-
nates of Z = 1.008A/Pt = 0.170 and t/t, = 0.157
can be plotted in Figure 12. Points with coordinates
for other durations can also be computed and plot-
ted, resulting in a dimensionless curve of unit
hydrograph peak versus duration. The unit hydro-
graphs for the hypothetical circular basin for other
durations are shown in Figure 11.
Similarly, such plots for hypothetical rectan-
gular and triangular basins (Figure 10) have also
been computed as shown in Figure 12. A compar-
ison of these curves indicates the effect of the shape
I
I i Ā·
------
^
---~
1
____ __T
I
'ZIC-j
I
I I
I j [j
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
Table 12
Computation of the Unit Hydrograph Ordinates for a Storm
of 6-Hour Duration Having a Uniform Rainfall Distribution
ih Ih/l S-curve S-curve Unit
in hr. Ordinates Offset Iydro-
graph
Ordinates
(1) (2) (3) (4) (5)
0.5 0.05 0 12 0.12
1. 00.10 0.72 0.72
1.5 0.15 2.02 2.02
20 0.20 4.17 4.17
25 0.25 7.27 7.27
3 0 0.30 11.10 11.10
3.5 0.35 15.48 15.48
4.0 0.40 20.41 20.41
4.5 0.45 25.64 25.64
5.0 0.50 30.89 30.89
5.5 o 0.55 35.83 35.83
6.0 0 60 40.47 40.47
6.5 065 44.82 0.12 44.70
7.0 0.70 48.89 0.72 48.17
7.5 0.75 52.69 2.02 50.67
8.0 0.80 56.23 4.17 52.06
8.5 0.85 59.52 7.27 52.26
9.0 0.90 62.57 11.10 51.47
9.5 0.95 65.39 15.48 49.91
10.0 1.00 67.99 20.41 47.58
etc. etc. etc. etc. etc.
Note: Values in columns 3, 4, and 5 are in units of
per cent of 1.008A/P.
of a basin. Apparently, the rectangular shape with
its width narrower than the length has a high re-
tention effect and hence shows low unit hydro-
graph peaks. On the other hand, the triangular
shapes produce higher peaks and the circular shape
gives average peak values.
The unit hydrograph peak vs. duration curve
developed in Figure 9 for drainage basins in Illinois
of 80 sq. ini. in size is also plotted in Figure 12
for comparison. This curve fits in a position be-
tween the circular and rectangular shapes.
It should be noted that the built-up method is
based on a number of hypotheses which are unre-
alistic in actual drainage basins. Therefore, the re-
sults of computation by this method will have more
qualitative than quantitative significance. Because
of the hypotheses the theoretical results may de-
viate considerably from the actual values.
E. VARIATION OF RAINFALL INTENSITY
IN A STORM
In the derivation of a unit hydrograph, a con-
stant value of rainfall intensity is assumed within
the period of the rainfall excess. The actual distri-
bution of rainfall intensity, however, is rarely uni-
form in a storm. Based on a study of several
typical rainfall distributions, the U. S. Soil Con-
servation Service has recommended an average pat-
tern of rainfall distribution for use in general
cases.(31) This pattern as given in the top part of
Table 13 is based on the maximum amount of rain-
fall experienced for durations of up to 6 hours.
This SCS pattern of rainfall distribution and the
synthetic S-curves in Illinois (Table 9) by the U. S.
Geological Survey will be used in the following
analysis for the study of the effect of the time
variation of rainfall intensity upon runoff peak diQ-
charges. Since the pattern of infiltration loss is
unknown, it will be assumed that the SCS pattern
of rainfall distribution is approximately equal to
that of the rainfall excess. It should be noted there-
fore that the analysis will be made only on a hypo-
thetical basis, assuming t/t, = 0.60 and t = 6 hr.,
then to = 10 hr. Using an average value of k =
1.05, Equation 19 gives D = 43 sq. mi. The analy-
sis is thus applicable to the watershed of a size
equal to 43 sq. mi., and it is given below.
Based on the USGS data, the hydrograph ordi-
nates for a storm of 6-hr. duration, having uniform
rainfall distribution, are computed as shown in
Table 12. In the table, column 1 gives the time t,
in hours, column 2 gives t,./to, column 3 gives
S-curve ordinates from the USGS data (Table 9),
column 4 gives the offset S-curve ordinates, and
column 5 gives the unit hydrograph ordinates which
are the differences between the values in columns
3 and 4. The unit hydrograph ordinates are ex-
pressed in per cent of 1.008A/t. It can be seen that
for t = 6 hr. the peak discharge is 52.3%i of
1.008A/t.
Table 13 shows the computation of the hydro-
graph ordinates for a storm of 6-hr. duration hav-
ing a pattern of rainfall distribution recommended
by the SCS. In the table, column 1 gives the time
th in hours, column 2 gives th/to, column 3 gives the
ordinates of unit hydrograph for a duration of t =
0.5 hr. as derived from the USGS data in a similar
way to that used in Table 12, columns 4 to 15 give
runoff contributed by different rainfall depths in
the storm, and column 16 gives the resulting hydro-
graph. The peak discharge is found to be 4.6129%
of 1.008A/t, in which t = 0.5 hr. This is equivalent
to 4.6129 X 12 = 55.4% of 1.008A/t if t = 6 hr.
From the above computation, the peak discharge
for uniform distribution of rainfall intensity was
found to be 52.3% of 1.008A/t with t = 6 hr., and
for the non-uniform distribution recommended by
the SCS it is 55.4% of 1.008A/t with t = 6 hr.
Therefore, the effect of the non-uniform distribu-
tion according to the pattern recommended by the
SCS is to increase the peak discharge about 6.0%.
It should be reminded that this analysis is based on
an average condition of t/to = 0.6 and D = 43
IV. HYDROLOGIC PRINCIPLES, DATA, AND ANALYSES
Table 13
Computation of the Hydrograph Ordinates in Per Cent of Equilibrium Runoff for a Storm of 6-Hour Duration
Having a Rainfall Distribution Pattern Developed by the Soil Conservation Service
tI I/t,, Ordinate- of
ill hr. 0.5 hr. Unit
Ilydroniehl,
11) (2) (3)
0 i .0 0 12
1 .0 .10 0.410
1 .5 . 1 1 30
2.0 .20 2. 15
2.5 .25 3. 10
3 0 .30 3. 83
3 5 ..3i 1.38
4 0 40 1 !13
4.5 .15 5>.23
5.0 .50 5i25
S.5 .55 1 91
. (1 .G10 14 .l
6.5 6. 5 I 135
7 0 .70 4 107
7.5 .75 3. 80
8 0 .80 3 5.4
8.5 .85 3.29
9.0 90 3.05
9 5 .95 2.82
10 0 1 00 2.60
10.5 1.05 2.39
11.0 1.10 2.19
11.5 1.15 2.00
12.0 1.20 1.82
12.5 125 1 65
etc. etc. etc
Rainfall Distribution, Expressed as Ratio to Total Rainfall Driiing a 6-Ilour Period, for Ilalf-hour Tinme
Incremlents
.035 045 .055 .095 .370 .105 075 .055 .050 .040 .040 .035
(1) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
.0042
.011455
.0753
.10861
.1340
.1534
.1726
.1833
,1838
.1730
.1623
.1522
.1124
.1330
.1240
.1152
.1068
.0988
.0910
.0838
.0766
.0700
.0636
.0578
etc.
.00I)1i;
.0330(
.0715
.0726;
.1706
.2107
.2110
.27101
.2880
.28!90
2720
.2550
.2395
.2240
.2090
.1948
.1810
.1680
.1551
. 1430
.1315
.1204
.1100
etc.
.044411
.2220
.4810
.796(10
1 118
1 117
1 620
1 824
1 937
1 942
1 824
1 717
1 .610
1 .505
1 -105
1 310
1.218
1 130
1 044
0.961
0 885
etc.
.009()
.04150
.0975
. 11)13
.2325
.2875
. 3290
S3700
.3927
.3940
.3700
.3480
.3262
.3060
.2850
.2656
.2465
.2290
.2115
etc.
0)(tili
.0330
.0715
.1183
.1705
.2108
.2-110
.2716
.2880
.2890
.2720
.2550
.2395
.2240
.2090
.1950
.1810
.1680
etc.
.1-148
.0240
.05201
.)0860
.1240
.1533
.1752
.1973
.2095
.2100
.1978
.1856
.1740
1630
1521
.1418
etc.
.0048
.0240
.0520
.0860
.1240
.1533
.1752
.1973
.2095
.2100
.1978
.1856
.1740
.1630
.1521
etc.
.0042
.0210
.0455
.0753
.1086
.1340
.1534
.1726
.1833
.1838
.1730
.1623
.1522
.1424
etc.
sq. . i. By assuming other values of t/t , different
results vwould be obtained. However, it is believedl
that the effect of an average non-uniform rainfall
distribution in most cases. will not increase the peak
discharge by more than 10%.
Iit order to support the above analysis, a theo-
retical study by Rihards""'' may be (quoted. He
assumned two extreme cases ( of linear rainfall distri-
bution: one with a heavy intensity at tie begin-
ning and zero at the end, and one with zero inten-
sity at tlhe beginning and a maxilmuml at tlhe end.
The results of the study are shown in Table 14. It
will be noted that with the increase of the duration
t (or the time of concentration for maximum runoff
as implied in Richards' study), the effect of the
time variation of the rainfall intensity upon runoff
peaks is nearly constant. For small drainage basins,
the duration is generally less than 6 hr. The cor-
responding effects due to the two extreme cases are
about +13% and -20% respectively. These cases,
however, occur infrequently.
For a more comprehensive understanding on tlhe
effect of the rainfall intensity distribution, it would
be necessary to analyze all available storm data in
the Midwest of various durations and intensities,
as well as the infiltration data. The work would
not only be laborious but also unjustified, because
tle data vary in a wide range, and the magnitude
of the effect is small. In the present study, it is
therefore assumed that the average effect of non-
unifornl distribution of rainfall intensity increases
the peak discharge due to uniform distribution by
about 6.0%. This is, of course, more or less arbi-
trary but it is considered to be a reasonable value.
In order to include this effect of non-uniform dis-
tribution of rainfall intensity in the determination
of a peak discharge, the rainfall intensity used in
the computation assuming a uniform distribution
may be increased by the same per cent.
Table 14
Effect of Rainfall Distribution Pattern on Peak Discharge
(Qm.a) According to Richards'"'
t lrs.
1
3
5
7
10
25
50
100
Rainfall Distribution Pattern
Delayed
%0m. .
88.5
81.9
79.7
78.5
77.6
76.8
76.0
75.6
Uniform
%Qma..
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
Advanced
%Qm..
106.5
112.0
113.2
113.9
115.2
116.1
116.7
117.5
(16)
.0042
.0264
.0791
.1782
.3781
.7042
1 2644
1.85196
2.5593
3.1165
3.6328
4. 1()6
4 4406
4.6129
4.6075
4.5336
4.3946
4.2153
4.00612
'3.7766
3.5296
3.2818
3 0418
2.8123
2.5961
etc.
V. DEVELOPMENT OF THE NEW METHOD
A. THE PROPOSED FORMULA
An intensive study of the many existing methods
and a survey of current practice reveals that a de-
sirable method for the determination of peak dis-
charge for waterway opening design must satisfy
the following requirements:
(1) It should consider the major climatic condi-
tions in the area of the given drainage basin.
(2) It should consider the major physiographic
conditions in the area of the given drainage basin.
(3) It should consider either a definite design
frequency or a limiting design condition.
(4) It should be based on sound and simple
hydrologic principles so that the practicing engineer
can use it with confidence and understanding.
(5) It should be simple and practical so that a
beginner can use it readily with ease.
(6) It should depend less on personal judgment
and more on logical lrocedure so that the result
will be relatively consistent among determinations
by different individuals.
In order to satisfy the above requirements, a
method has been derived which utilizes the concept
of unit hydrographs and is based on unit hydro-
graph synthesis. It also makes use of principles and
concepts discussed in Section IV and in Sections
II-G and II-H. The development of the method is
as follows:
The direct peak runoff from a drainage basin
may be computed as a product of the rainfall excess
and the peak discharge of a unit hydrograph, or
Q = ReP (25)
in which Re = rainfall excess in inches for a given
duration of t hrs.
P = unit hydrograph peak in c.f.s. per
inch of direct runoff for the duration
t hr. of rainfall excess
From Equation 20, using the concept of peak-
reduction factor Z:
1.008A Z(26)
P = - t (26)
Then, substituting Equation 26 in Equation 25,
1.O08R.A Z
(27)
In this expression the factor 1.008Re/t may be re-
placed by the product of two factors: X and Y.
The factor X is a runoff factor, expressed by
S= -Re.- (28)
in which Re, = rainfall excess at Urbana, Illinois,
increased by 6.0% to allow for the
effect of variable rainfall distribu-
tion in the duration t
The factor Y is a climatic factor. Assuming Re/Re,
= R/Ru, this factor represents
1.0081
(29)
in which R, = rainfall in inches at Urbana, Illinois
R = rainfall in inches at other location
R/Ru = conversion factor for converting the
rainfall at Urbana to that at other
places in Illinois
Consequently, Equation 25 may be written as
Q = AXYZ
(30)
If the base flow at the time of the peak discharge
is Qb, then the design peak discharge is
Qd =Q + Qb
(31)
The factors involved in the proposed formula
(Equation 30) will be discussed in the following
articles.
B. FACTORS AFFECTING RUNOFF
The factors affecting runoff considered in the
proposed method can be divided into two groups.
One group affects directly the amount of rainfall
excess or direct runoff and it consists mainly of
land use, surface condition, and soil type, and the
amount and duration of rainfall. The other group
affects the distribution of direct runoff and it in-
% J
)
V. DEVELOPMENT OF THE NEW METHOD
Table 15
ction of Runoff Number N
Sirface Condition
Sele
L,and [l's or Cover
Fallow
Row crops
Smalll grains
I,vcginws (closed-drillidt
ior 1brolrdust) or
rotation (imeadow
Pasture or rang".
IMefadow (leralnlnlt)
IoodNls (fillirl woodI lots)
Farmntlads
Hoads
F'orest
Itnper- iotS sit faet'
St raillit row
Straiilit row
Contoured
(ontoulred and terraced
Straight row
Contoured
Contoured and terraced
Straight row
Contoured and terraced
Poor
Normal
(ood
(Contoillod. poor
C(ontoIured, norlimal
(Contoured, good
Nornial
Sparse or low tran-
spiration
Normal
Dense or high tran-
spirationt
Normal
Dirt
Hard surface
Very sparse or low
transpiration
Sparse or low tran-
spiration
Norlial
Dense or high tran-
spiration
Very denst or high
transpiration
eludes tile size and shape of the drainage basin, tihe
land slope, and a time measure of detention effect
such as the lag time. This distribution of direct
runoff is expressed in terms of the unit hydrograplh.
There may be a certain interdependence exist-
ing between the two groups of factors described
above. This interdependence is however unknown
and for practical purposes it may be assumed that
it does not affect the relationship between the direct
runoff and rainfall excess. This assumption forms
the basis upon which Equation 25 is established.
After an extensive search for information perti-
nent to the study of the runoff condition in small
drainage basins, it was found that the data used in
the method of hydrograph synthesis by the U. S.
Soil Conservation Service (Section II-G) can be
used for the evaluation of the rainfall excess or
direct runoff. These data, however, should be mod-
ified in order to meet the present purpose.
For the present investigation, an average hydro-
logic condition of drainage basins was assumed.
This condition represents the average conditions of
antecedent moisture content and groundwater influ-
ence during the optimum time of the year.
The hydrologic soil-cover complex numbers used
by SCS to describe different land uses, surface con-
ditions, and soil types were modified with the aid
Soil Type
B C
86 91
80 87
77 83
73 79
76 84
74 82
71 79
75 83
72 81
70 78
79 86(
69 79
(il 74
67 81
59 75
35 70
58 71
66 77
60 73
t5 70
74 82
82 87
84 90
75 86
68 78
60 70
52 62
14 54
100 100
Table 16
Hydrologic Soil Groups in Illinois
(Iroup A-- Soil Series
Perks
Plainfield
\Morocco
Clhute
Kilbourne
(Group B --Soil Series
Iandes
Sharon
Bold
XM auitee
Biggs
Billctt
W(orttlen
Tailula
I I agener
Littletonl
Disco
Iorllzo
Smlluner
C(roup (' Soil Series
Petrolia
Arnbraw
Beallcoup
Elliott
Miillbrook
Shilolt
Ilerrick
Group D)-- Soil Series
Jacob
Iarwin
Denny
Osceolla
Okaw"
WV'ynoos
Rinard
Sable
Atterbery
)Drtsdten
St. ltria es
Clinton
loona
Srlni;>
Druntriiiiier
Lisbon
Beaver
Clary
Il:co
V\tar.saw
Aslhknii
Virden
Varna
Bluford
Stoy
Bogota
Ava
Luikin
Iantouil
Bryce
DeSoto
Whitson
Flora
Cartni
Brenton
Caplron
Miami
Birkbeck
Ringwood
Tanma
Saybrook
Sidell
Proctor
MAiseatinte
Flanagan
Ellison
Fayett'
Virgil
Fay
Westville
Starks
( rantsburg
And res
Schapville
(lllinois)a
Clarence
)enrock
Illey
Racoon
(isnt
SThi is s wlere Illinois plac's thlse soils. Iowa lha grouped Sawinill
ani Sclhalpville differently.
of other data and renamned the runoff number, N.
Table 15 lists the runoff numbers thus obtained.
Tihe soil types are classified by SCS in accordance
with the runoff characteristics of the material into
four hydrologic soil groups A, B, C(, and D as de-
fined in Section II-G.
For the convenience of identifying the soil types
in Illinois, a map (Figure 13) was prepared in
accordance with a study made at the University of
Illinois.s14) On this mlapl, different hydrologic soil
types are clearly indicated. In certain parts of the
State where the soil has a variable textured, strati-
fied alluvium, the hydrologic soil type is described
as "variable" and its runoff number may be taken
as the average of the runoff numbers of the types
A, B, C, and D.
For a more specific identification of the soil
type, a list of "Hydrologic Soil Groups for Illinois"
has been prepared by the State Office of the U. S.
Soil Conservation Service. This list is given in Ta-
ble 16, showing various names of soils found in
Illinois grouped under the four soil types.
For a more or less homogeneous runoff condi-
tion, the runoff number can be obtained directly
from Table 15 with the aid of the soil type map
(Figure 13) or the soil type list (Table 16). For a
composite runoff condition, a weighted runoff num-
her should be determined. For example, when a
Port Byron
Joy
Biggrukillh
Mt. Carmll
Fall
Alvin
Nletea
Decolr a
Hopper
Tinula
Seaton
Allisoni
Sawi iill
(Illinois)a
E'dington
Oronc(e
O'Fallon
Richvieu
Be eriick
Iosiller
Hankin
Brooklyn
Milrovy
Niota
Mones
Weir
Htenry
Co)ll
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
0 /0 30
5 20 40
mr/es
Figure 13. Soil types in Illinois
[
E
I
E
I
V. DEVELOPMENT OF THE NEW METHOD
drainage basin contains 37.4% of impervious area
and the remaining area of meadow, the weighted
runoff number is computed as follows:
Cover Percentage Runoff No. Product
Impervious 37.4 X 100 = 37.4
Surface
'Meadow
62.6 X 58 = 36.3
Sum = 73.7
The weighted runoff number is 73.7.
After the runoff number is determined, the value
of R,, or rainfall excess for a given rainfall depth
can be computed or found directly from the chart
in Figure 14 developed by the U. S. Soil Conserva-
tion Service.('7) The chart shows the relationship
between the direct runoff (Re) and the rainfall (R)
for different runoff numbers (N). The curves in
this chart were computed by Equation 12 which is
rewritten using the present notation as:
1( 200 2
, = (32)
+800 ()
R+ -8
V
For example, with N = 73.7 and R = 3 in., the
direct runoff from either the chart or Equation 32
is Re = 0.89 in.
C. DETERMINATION OF RUNOFF FACTOR X
When the runoff number is determined, the rain-
fall excess or direct runoff for a given rainfall can
l)e either computed by Equation 32 or determined
from the curves in Figure 14. Then, the runoff fac-
tor X for duration t of the rainfall can be computed
1by Equation 28.
For rainfalls of 5-year frequency at Urbana,
Illinois, the computation for the runoff factor X is
given in Table 17. In this table, column 1 gives the
assigned rainfall duration t in hours. Column 2
gives the corresponding rainfall in inches obtained
from the rainfall frequency curves in Figure 5, with
an increase of 6.0% to account for the effect of non-
uniform distribution of rainfall in an average storm.
Columns 3 to 11 give the runoff factor X for runoff
numbers N = 100 to 60. The value of X is com-
puted by Equation 28. For example, when t = 0.1
hr., the rainfall for 5-year frequency from Figure 5
R, rainfall in inches
Figure 14. Relationship between rainfall and rainfall excess for various runoff numbers
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
is 0.52 inches. With an increase of 6.0%, the rain-
fall is 0.55 inches. For N = 95, the rainfall excess
from Equation 32 or Figure 14 is 0.30. By Equa-
tion 28, X = 3.00.
Similarly, computations of runoff factor X for
frequencies of 10, 25, 50, and 100 years are given
in Tables 18, 19, 20, and 21, respectively. The colm-
puted values are represented graphically in Figures
Table 17
Computation of Runoff Factor X for 5-Year Frequency
D)ura- Rain- Runioflf Factor A for equal to
tuin fall ,100 9l 90 85 80I 75 70 65
in hrs. in in.
(1) (2) (3) (4) (5)5 (6) (7) (8) (9) (10)
0.10 0.55 5.50 3.00 1.50 0.40 0.00 0.00 0.00 0.00
0.20 0.85 4.25 2.50 1.30 0.70 0.25 0.00 0.00 0.00
0.30 1.07 3.67 2.20 1.20 0,77 0.37 0.07 0.00 0.00
0.40 1.19 2.98 1.88 1.15 0.73 0.40 0.13 0.03 0.00
0.50 1.31 2.62 1.72 1.08 0.70 0.42 0.18 0.08 0.00
0.75 1.53 2.04 1.39 0.93 0 63 0.41 0.23 0.13 0.04
1.00 1.68 1.68 1.18 0.82 0.57 0.38 0.23 0.14 0.07
1.25 1.78 1.43 1 02 0.72 0.50 0.35 0.21 0.14 0.07
1.50 1.87 1.25 0.90 0.65 0.45 0.32 0.20 0.13 0.07
2.00 2.02 1.01 0.76 0.55 0.40 0.29 0.20 0.13 0.08
2.50 2.11 0.84 0.64 0.46 0.35 0.26 0.17 0.11 0.08
3.00 2.20 0.73 0 57 0.42 0.32 0.23 0.16 0.11 0.07
4,00 2.35 0.59 0.46 0.35 0.26 0.20 0.14 0.10 0.07
5.00 2.46 0.49 0.40 0.30 0.23 0.17 0.12 0 09 0.06
6.00 2.56 0.4 04 0.34 0.26 0.20 0.16 0.11 0.08 0.0(
7 00 2.67 0.38 0.31 0.24 0.18 (014 0.10 0.08 0.07
8.00 2.72 0.34 0.27 0.22 0.17 0.13 0.09 0.07 0.05
15, 16, and 17. The value of runoff factor X for
given frequency, runoff number, and duration can
be readily interpolated from the curves in these
graphs.
D. DETERMINATION OF CLIMATIC FACTOR Y
The climatic factor Y is represented by Equa-
tion 29. This is equal to the conversion factor
Table 18
Computation of Runoff Factor X for 10-Year Frequency
I)ur1a-
tion
in brs.
(1)
0.10
0.20
. 30
0 40
0.50
0.75
1 00
1 25
1 .50
2 00
2. 50
3 00
4 00
6 00
6. 0(1
7.00
8 00
li1noff1 IFator X for .N iqral to
90 85 80 75 70 65
/ .5 / 5 /0
t. Duration in hr f. Duration in hr
Figure 15. Runoff factor X for 5- and 10-year frequencies
Figure 16. Runoff factor X for 25- and 50-year frequencies
V. DEVELOPMENT OF THE NEW METHOD
shown in Figure 6 multiplied by 1.008. Figure 18 is
a map which shows boundaries of sections and the
average value of the climatic factor for each section.
E. DETERMINATION OF PEAK REDUCTION
FACTOR Z
The peak-reduction factor Z represented by
Equation 20 is equal to the ratio between the peak
discharge of a unit hydrograph due to the rainfall
of a given duration t and the equilibrium runoff or
tle runoff of the same rainfall intensity continuing
indefinitely. Tn Figure 9, this ratio is shown to be
a function of the ratio between duration t and lag
t,. For the purpose of developing the proposed
method, however, the value of Z is to be represented
by a function of the ratio between duration t and
Table 19
Computation of Runoff Factor X for 25-Year Frequency
Table 20
Computation of Runoff Factor X for 50-Year Frequency
Dura- Rain-
tion fall 100
in hrs. in in.
(1) (2) (3)
0.10 0.83 8.30
0.20 1.31 6.55
0.30 1.64 5.47
0.40 1.87 4.68
0.50 2.05 4.10
0.75 2.34 3.12
1.00 2.54 2.54
1.25 2.66 2.13
1.50 2.77 1.85
2.00 2.90 1.45
2.50 3.02 1.21
3.00 3.09 1.03
4.00 3.24 0.81
5300 3.36 0.67
6.00 3.46 0.58
7.00 3.53 0.51
S. (X) 3. 62 0.15
Itnoff Iactor X for N erqal to
90 85 80 75 70 65
(8) (9) (10) (11)
0.00 0.00 0.00 0.00
0.40 0.15 0.00 0.00
0.67 0.40 0.17 0 00
0.75 0.50 0.25 0.18
0.80 0.54 0.34 0.20
0.73 0.51 0.35 0.20
0.66 0.47 0.33 0.20
0.58 0.42 0.30 0.18
0.53 0.39 0.28 0.18
0.44 0.33 0.24 0.15
0.39 0.29 0.21 0.14
0.34 0.25 0.20 0.13
0.28 0.21 0.16 0 11
0.24 0.18 0.14 0.10
0.21 0.16 0.12 0.09
0.18 0.15 0.11 0.08
0.17 0.13 0.10 0.08
Runoff Factor X for N equal to
90 85 80 75 70 65
.5 /
I, Duroaion in hr
Figure 17. Runoff factor X for 700-year frequency
Rain-
Sfall
in in.
(2)
0.73
1.17
1.47
1.66
1.81
2.08
2.24
2.36
2.45
2.61
2.72
2.77
2.93
3.04
3.14
3.22
3.31
Dukra-
tion
in s
(1)
0.10
0.20
0.30
0,40
0.50
0.75
1.00
1.25
1.50
2.00
2.50
3.00
4.00
5.00
6.00
7.00
8 00
/C
5
X
5 /0
Figure 18. Climatic factor Y for climatological sections in Illinois
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
Table 21
Computation of Runoff Factor X for 100-Year Frequency
Rain-
fall
in ill.
(2)
1 .-12
2.03
2.24
2.77
2 .89
3 01
3.21)
3.28
3 41
3 58
3. 8
3.7915
R. 95
lllnolt Iaetor X for V equal to
910 85 80 75 70
lag time t,,. The relationship between lag and lag
time will be discussed later in this article.
1. Time Measure of Detention Effect
As stated in Section V-B, one of the physio-
graphic characteristics defining the hydrograph of
a drainage basin is a time measure of detention
effect. This time element may be the time of con-
centration (t,.) in the rational method (Section
II-D), the lag time (t,,) used in the method of hy-
drograph synthesis by the U. S. Agricultural Service
as defined by Equation 9, or the lag (I,) used by
the U. S. Geological Survey as defined in Equation
19 for conditions in Illinois.
In search for a suitable time measure of deten-
tion effect in tle proposed method, the lag time t,,
was adopted. This lag time has been defined before.
For an instantaneous unit hydrograph this lag time
is equal to the time of rise from the beginning of
runoff to the runoff peak. The instantaneous unit
hydrograph is a hypothetical unit hydrograph
whose duration of rainfall excess approaches zero
as a limit, while maintaining a fixed amount of
rainfall excess equal to 1 inch.
It may be pointed out that the lag time so de-
fined may not exactly correspond to the classical
concept of "time of concentration." For natural
drainage basins of large size and of complex drain-
age pattern, runoff water originating from the most
remote portion may and usually does arrive at the
outlet too late to contribute to the peak flow. Ac-
cordingly, the lag time will generally be less tlan
tihe time of concentration for a given basin. For
small drainage basins with simple drainage pat-
terns, lag time may be very close to the time of
concentration. Moreover, critical peak flows from
small basins are usually caused by rainfall due to
thunderstorms of short duration. The durations are
relatively so short that the mass of rainfall excess
is practically concentrated near the beginning of
the rise of hydrograph. Thus, the resulting unit
hydrograph may approach an instantaneous unit
hydrograph.
For small agricultural drainage basins, as men-
tioned previously in Section IV-D3, Ramser(2" has
determined the time of concentration by noting the
time required for the water in the channel at the
gaging station to rise from the low to the maximum
stage as recorded by the water-stage recorders.
Thus, Ramlser's time of concentration may be very
close to the lag time, or t, = t,,. Thus, Equation 22
becomes
t, = 0.00013 K-"77
(33)
in which K = L/V'S is a lag-time factor, where
L in feet measured up to tile basin boundary and
,S in feet per foot are defined in Equation 22.
lhe lag time used in the method of hydrograph
synthesis by the U. S. Agricultural Research Serv-
ice, as described previously in Section II-G by
Equation 9, is measured from the center of mass
rainfall block to the peak discharge. For small
drainage basins, however, this may be close to the
rise of the instantaneous unit hydrograph, that is
to the lag time defined in this study. This conclu-
sion is supported by the results of a study of small
drainage basins in arid-land.(3') In that study, it
was found that the lag time depends mainly on the
hydrograph shape and its relation to basin physiog-
raphy and it is practically independent of the rain-
fall duration. In other words, the lag times of
hydrographs of different durations for a given basin
are practically all the same because theoretically
they belong to the same instantaneous unit hydro-
graph of the basin.
2. Verification of Lag-Time Equations by Data
In order to verify the lag-time relationships
represented by Equations 9 and 33, the data from
a number of small drainage basins in the midwest-
crn area are used. These data are shown in Table
22 covering drainage basins in Illinois, Indiana,
Ohio, Iowa, Wisconsin, Missouri, and Nebraska.
Except for basins at West Salem, Madden Creek,
and Hurricane Creek, Illinois (obtained from U. S.
Geological Survey), the data were obtained from
publications by the U. S. Agricultural Research
Service.: n"'' The data used in the analysis include
20 drainage basins and 53 storms or 60 runoff peaks.
For each storm and basin, a unit hydrograph
V. DEVELOPMENT OF THE NEW METHOD
and the corresponding instantaneous unit hydro-
graph were derived; and thus, the duration, lag
time, and unit hydrograph peak were determined.
Most of the storms selected for analyses have a
well-defined single peak of runoff. In cases of mul-
tiple peak runoff, the peaks of the hydrograph were
first separated before making a unit hydrograph
analysis.
For the derivation of an instantaneous unit
hydrograph, the methods developed by O'Kelly,' '
Nash,(12' '', ) or Dooge('" may be used. As it can be
shown that the integration of an area covered by
an instantaneous unit hydrograph is proportional
to the ordinate of an S-curve which is a hydrograph
due to a uniform rainfall intensity of continuous
duration. In other words, the instantaneous unit
hydrograph represents the slope of the S-curve. The
time at the point of inflection on the S-curve corre-
splonds to the time of the peak flow of the instan-
taneous unit hydrograph. Therefore, an approximate
procedure to determine the lag time is to construct.
the S-curve of a given direct-runoff hydrograph and
then to locate its point of inflection.
Table 22 shows the computed values of lag-time
factors Ko by Equation 10 and K, defined in Equa-
tion 33, and the average lag time for each drainage
basin. The average lag time is taken from the com-
putation in Table 23.
It should be noted that the slope So used for
computing the lag-time factor Ko is the average
land slope. The original land slopes given in the
publications of the U. S. Agricultural Research
Service(1" 6"') are shown in percentage of the drain-
age area lying in each slope class. For example, the
slopes of Watershed W-I at Edwardsville, Illinois,
are given as "63% in 0-1.5% class; 21% in 1.5-
4%o; 9% in 4-7%; and 7% in 7-12%." The aver-
age land slope is therefore 2.21%.
The slope S used for computing the lag-time
factor K may be determined by many methods.""'
Generally, the length of the channel is measured
beyond the upper end of the clearly discernible
stream channel to the drainage divide and then
the difference in elevation between this point on the
ridge line and the outflow point of interest in the
channel is divided by the length. In the present
determination, the profile of the stream was plotted
using the elevation as the ordinate and the distance
up the principal channel as the abscissa. Then, a
straight line through the gaging point was fitted to
the stream profile so that the area between the
straight line and the stream profile lying below the
line was equal to that lying above it. The slope of
this "straight line of best fit" was taken as the
channel slope. As a comparative study, broken lines
of best fit were also tried, and the weighted average
of the slopes was taken as the channel slope, but
the results showed no significant differences. The
single straight line of best fit was therefore used
owing to its simplicity.
The computed lag-time factors are plotted
against the lag time as shown in Figures 19 and 20.
In Figure 19, Equation 9 is plotted as a straight
line. Similarly, in Figure 20, Equation 33 is plotted
as a straight line. It appears that the data do not
satisfactorily fit either Equation 9 or Equation 33.
Table 22
Computation of Lag-Time Factors
No. State Drainage Basin Size A Visible
acres Channel
Length
ft.
1 2 3 4 5
1 Illinois W-l, Edwardsville 27.22 1,650
2 W-4, Edwardsville 28!9.8 19,800
3 W-1-A, Monticello 82.0 3,530
4 W-l-B, Montirello 45.5 2,053
5 West Salem 969( (i,600
(6 Madden Creek, 992 6,440
Edwards County
7 lHurricane Creek 92.2 3,500
Trib., Witt
8 Ohio W-97, Colshoton 4,580 52,800
9 W-183 Coshocton 74.2 3,4(X)
10 W-196, ('oslhoton 303 19,000
11 W-l, lamilton 187 5,390
12 Missouri W-3, Betlany 4.85 1,660
4.48 1,660
13 Wisconsin W-l, Fenniniore 330 8,200
14 W-2, Fenninore 22.8 560
15 W-4, Fennimtore 171 4,000
16 W-l, Colby 345 4,490
17 Indiana W-5, Lafayette 2.87 280
18 W-6, Lafayette 2.79 240
19 Iowa Ralston Creek, 1,920 64,240
Iowa City
20 Nebraska W-3, llastings 481 30,500
I)rainage
Density
ft/ac
6
61.0
68.32
43.05
.45.13
6.5 1)
6 .49
38.0
11.52
45.8
29.7
28.8
340
370
24.8
24.6
23.4
13.0
97.6
86.0
33.4
63.4
Land Slope K. =
S/,% AO."
.S'., /D.D.
7 8
2.21 0.156
5,37 0.128
1.56 0.366
1.15 0.407
0.60 5.07
1.87 1.664
2.00 0.315
17.21 0.215
15.86i 0.034
16.20 0.063
4.59 0.195
7.72 0.011
(av.)
5.97 0.191
5.92 0.087
4.96 0.191
2.47 0.648
1.93 0.072
2.26 0.065
10.25 0.163
5.30 0.151
Channel
Slope Length
S%/ L, ft.
9 10
1 51 1,461
1.29 5,765
0. 66 2,120
0.54 2,750
0 51 9,926
0.58 8,848
0.60 1,900
0.53 26,100
0.65 3,180
3.70 4,460
1.15 5,000)
6.25 675
2.03 5,780
4.75 1,000
2.17 3,270
0.76 6,575
1.41 l566
1.48 585
0.60 21,600
0..55 9,000
K = Av. Lag Time
L t, in hr.
11 12
1.19 104 0.315
5.08x104 0.443
2.62x 10 0.576
3.75x1 0 0.436
13.82x10' 1.872
10.69x 10 1.809
2.45x 10 0.625
35.8x10( 1.900
1.22x 10 0.240
2.32x104 0.310
4.66x104 0.286
0.27x 10 0.117
4.05x 104 0.419
0.458x104 0.116
2.15x104 0.267
7.55x 10' 0.417
0.418104 0.139
0.48x 10 0.165
27.9x1(0 1.422
12.1x101 0.136
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
Table 23
Computation of Lag Time and Peak-Reduction Factor Z
N.o. Stitt IV l Dainage t(Ba-.i. Size A
avrcs
I)Dat- if Storiu,
1 2 4 5
1 Illinois W-I, Ilwardlvill 27.22 May 27, 1938
June 21, 1942
NMarl'h 31, 19152
March 31, 11152
.IJly 2, 1!.052
W I. ldiwvardlsvill, 2811.8 Mlay 27, 1938
himn 21. 19142
March 31. 11152
lMarch 31, 19152
.)uly 2, 19152
3 W-1-A, ,Monlircllo 82,0 Orl. 21, 11119
Jlily 9, 19l51
. 9ly, 11)51
I WV-I-B, 1Montir4llo 1.5.5i ()Oct. 21, lltlI
.5 WVsi Sali )ll 0 .July 18, 1958
.Ily 1i, 11158
(i Mael.n (rek, 1992 J.lyl 18, 1!158
Ilwarils County iuly 11), 1!1.58
7 lil rli.an'i ('r.k 112.2 Aplril 5. 1958
Trililitary, Witt. May 1),10, 111519
AUig. (1 )195)
A iig. (, 111511
8 Ohio W-117. (Coslhoto ,.580 hJune .I, 1114
.ilnn 28, 19157
9I W-183 Cosholton 74.2 J.li o 11, 1111
Augi. 1(1, 11147
Se)t. 1, 1950
Stuit. 1, 1950
.hlun 12, 19157
10 W-1i11 (Coshlocton, 303 Jlun 16, 194i6
Alui. 71, 19117
SI)pt. 1, 11150
Sepl. I, 191150
.Illni 12. 19157
11 W-l, I lanilton 187 .hily 4, 11)391
.IJly 7, 11943
12 Mlissouri W-3, liethaliy 1.85 May 21. 11933
4.18 Oct. 19. 11134
.hiune 17, 113.5
13 Wisconsin W-1, Feniiiiiinore 330 Aug. 12, N1-43
.iuly 1, 1!944
.llline 28, 194.5
11 W-2, I'eniirnore 22.8 .lline 28, 1114.5
.,lne 24, 11)911
15 W-1, Fenniinor, 171 July 11, 1914
Juhine 28, 1!9415
Jiune 24, 1919)
10( W-1, Colby 3:15 July 28, 191)9
17 Indiana W-5, l'afayetto. 2.87 .July 5, 1943:
Juan, (1), 19I16
iiJune 24, 195110
18 W-(l, I.afayett( 2.79 J.ily 5, 19i13
.hlnie 11), 9IIi
.1l111 24, 19150
191 Iown Italslon Creek, 1,926 .lillne 1, 19'3:
Iowa C('y luily 1, 19501
July 18, 19.56.
20 Nebraska W-3, I Iastings .181 .Jiine 20, 11139
.liine 7, 191537
hJuly 15, 1957
Dluration I.ag T'imn t U nit
t t '7 Ilvydrogaphi
hours hours l'eak I' c.f.s.
6 7 8
0..150 l 0.1167 0. i)9 121 2
0.2167 0 517 0 .51 3.57-1
0..550 0. 38( 1. 590 3!9 80
0.383 0.229) 1I 675 .56 11
0.317 0.277 1. 145 62 .05
av. 0.315
0 1533 0.413 1 .292 508. I
0 417 0. 453 0.920 4110 7
0.I5.50 0.467 1.1711 333. 8
0).450 (10 5(i7 0.71-4 1419
0.3(i7 0.313 I 173 I1. 9i
iv. 0.I443
1 167 0.5!15l 0.280 12 . 5
0 217 0.7011 0.30( 13!. 7
0.2(1) 0.I 23 1. 73 (17 I
a\. 0. 5711
0.150 0 .136 0.314 51. 5
av. 0.43(6
0.333 1.1120 0. 171 15). I
0.1)17 1 .25 . 0(0.i .503 8
ak. 1.872
0.333 1 .817 0. 1791) .10
I) 1(17 1.75'0 0.015 1)96 0
iav. 1.80H(
0.233 0.583 0. I00) 113 8
0. 117 O.i33 0. 213 158 2
(II 1)) 0 733 0.(182 1 I 8
02.50 W. 55)0 0. 155l 1 II 8
ai. 0. 625
0. 1167 2.250 0.017 3,(X)5
S.(183 1.51) 1.087 2,0301
av. I .!X)
0.1533 0.2(17 2.0()( 122 7
0.2(0) 0.250 0.1) 8) 185.
0.733 0.233 3 .113 87 0
0 1 5I) 0.217 2.308 129 2
0 350 0. 233 I .50() 151 .2
av. 0 2410
0.1117 0.323 2.011) 3!13
0.217 10.341 0.i3.5 1173
0. 11117 0.310 2.150 360
0 3117 0.310 1 180 5:32
0.317 0. 2117 1.187 11917
av. 0.310
0.21) 0.317 0.1132 177
0. 117 0.255 1'.(154 37()
av. 0.28(1
0. 067 0.142 0.471 53:.8
0.18:3 0.083 2.21X) 19 5
0.083 0.125 0(.167 43.
av. 0 117
0.150 0.375 0.41X) 678
0.1117 0..545 0.3061 4191
0. 133 0.338 0.314 748
av. 0.41!)
0. 1() 0.082 1.22L 120. 8
0.250 0.150 16.117 72 1
av. 0.11(1
0.083 0.375 0.222 318
0.13:3 0.183 0.727 483
0. 3(1 0.242 1.242 :1711
av. 0. 267
0.233 0.417 0.5538 342
av. 0.417
0.067 0.1(Xi 0. 667 30.11
0.033 0).200 0.167 12.(
0.133 0.117 1.1-13 1610
av. 0.13i9
0.0(17 0.1(17 0.41 I 11
0.117 0.208 0..5(1 13.
0. 133 0.120 1.111 11.
av. 0. 11. 5
0.250 1.3(17 0.183 1,710)
1.1117 1 .l100 I. 155 178
0.333 1 .5(11 0.222 1,778
av. 1.122
0.250 0.625 (0.-I(0 612
0.717 0.6011 1.102 3911
0,217 (1.(633 ( I.312 7711
av. o).(tli
Z Points
for
lhtt ing
10 11
0 .6(12 l(a)
0.318 1(1))
0 7!7 1(c)
0. 78.5 I(d)
0.717 19')
.!) 27 2(a)
0.702 2(11)
0. (129 2(v)
0.(1 6913 2(d)
0 1111 2(1 )
( 187 3(a)
0 3117 3(1h)
( 1112 3(')
0. 1(18 4
0. 155 .i5(:
0.10811 5(h)
0. 152 (ia)
0.083 Ii(i))
0.28(1 7(a)
11.28- 7(1,)
0. 103 7(v)
0.31 I 7(d)
0. 108 8(a)
0.738 8(b))
0.875 !l(a)
0. 119i !l(h)
0.85)1 i9(c)
O. til !9(d)
0.722 9 )(
0.8.58 ll)(a)
0,7851 10(c)
0.6311 10(Id)
0.723 10((.)
0.4150 11 (a)
1.328 11(I))
0.738 12(u)
0.71)0 12(1)
0.7!17 12(c)
0.305 13(ia)
0.2461 13())
0.291) 13(()
0..52.5 1l(a)
0.785 -1(b))
0.1153 15(a)
0.372 151b)
0.1151 1.5(c)
0.221) 16
0.701 17(a)
0. I14 17(1h)
0.7315 17(c)
0.383 18(a)
0.557 18(1))
0.525 18(81)
0.220 19(a)
0.811 11())
0.30.5 1Il)<)
0.31.5 20(a)
0.1585 20(01)
0.3 14 211(.)
(2
(1,.
I Iii ~ c 1...ih
M illtiph' h )...
M tiltiple h-nkh
MulltiphPeaklh~
NI tilt i)l eak
Mlipl Peak.
Do. (i( '.,
V. DEVELOPMENT OF THE NEW METHOD
.025 .05.075./ .5
Ko = A S3
(A in acres; S in per cent; D.D. in feet per a
Figure 19. Verification of tp-Ko relationship
/10
5
./
5 10 .075
.05C
rcre )
.025
.0/0
In Figure 20, however, the plotted points do not
scatter as broadly as they do in Figure 19, and they
indicate a well-defined trend. Therefore, a line of
best fit was drawn and its equation was found to be
t,= 0.00054 (L/VS)o."4
(L in feet ;Sin feet per foot)
Figure 20. Verification of tn-K relationship
(34)
When S is expressed in per cent, the above equation
becomles
t , = 0 .0 0 2 3 6 ( L / V~ S ) o .6 4 ( 3 5 )
where L is in feet.
In deriving the line of best fit the method of
least squares was used to compute two regression
lines, one taking abscissas as independent variables
and the other taking ordinates as independent vari-
ables. The line of best fit is the mean of the two
regression lines; that is, its slope is equal to the
mean slope of the regression lines and it passes
through the centroid of the plotted data. For
practical applications, Equation 35 is shown by a
chart in Figure 21.
3. Conversion of Z Factor
In Figure 9 the factor Z is shown as a function
of t/t, where t is the duration of rainfall excess in
hours and to is the lag. The lag can be converted
to lag time by establishing a relationship between
the lag to and lag time tp. This relationship can be
established theoretically by using the concept of
instantaneous unit hydrograph.
Figure 22b shows the sketch of a unit hydro-
graph for the duration t of rainfall excess, with the
lag to and the lag time t, marked on it. This unit
hydrograph was developed from an S-curve and the
same S-curve in an offset position for a delay of
rainfall excess equal to t as shown in Figure 22a.
The principle and procedure for the development
of the unit hydrograph have been described in Sec-
tion IV-D. Figure 22c shows the instantaneous
unit hydrograph derived from the same S-curve.
The ordinates of the instantaneous unit hydrograph
are proportional to the slope of the S-curve as men-
tioned previously. It can be seen from Figure 22c
that the lag time t, for instantaneous unit hydro-
graphs is equal to the time of rise to the peak.
In the present discussion, the S-curve derived by
the U. S. Geological Survey from the data for
drainage basins of about 80 sq. mi. in Illinois was
used.'(82 This is plotted in Figure 23 with the
abscissa expressed in terms of the lag to. The in-
stantaneous unit hydrograph was derived from the
slope of the S-curve. It can be seen from Figure 23
that the relationship obtained between to and t, is
as follows:
t, = 0.47 to
(36)
From the above relationship, the curve for factor
Z in Figure 9 can be readily modified to produce a
new curve of factor Z versus the ratio t/t,. Both
the original and new curves are shown in Figure 24.
4. Verification of Factor Z by Data
In order to check the corrected curve for factor
Z, the data for the verification of the lag time were
.10
.01
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
used. The computation for factor Z by Equa-
tion 20 is shown in Table 23.
The computed values of Z are plotted against
t/t, as shown in Figure 25. The converted curve
based on the unit hydrograph for drainage basins
of an average size of 80 sq. mi. in Illinois is plotted
in dashed line. Obviously, the plotted data do not
fit the converted curve but are shown above the
latter. This is reasonable, because the drainage
basins under consideration are much smaller than
80 sq. mi. Figure 9 shows that the curve for smaller
basins has a higher unit hydrograph peak or a
higher value for factor Z.
To fit the plotted data, a curve in full line was
constructed in Figure 25. It should be noted that
1pz.00236(L1 .64
1000,
000
1.100, 1000 100, 11011 000
e 000, e -ooo
sko '0;'
ZĆ½--;i"A Od ;;-o
pe .oo -'Ć½".001---o L00,
loo
k oo,
Z A I-loo"
.0000 ko 0
000 1 10000 60
loll '.00 11000 Ć½0
o
"ooo"o
..' 1.10 '000 10000,
1--oo .0, 0-0
.001 1
'00Ć½ 10Ć½ -.oolrooĆ½Ć½
1-1 L10
.10
.01010,0 000,
'ooo- o'- 000-
the fitted curve is the converted curve displaced
horizontally except its upper portion where Z ap-
proaches unity. Theoretically, Figure 22 indicates
that t should not be greater than 2t,. Otherwise,
the peak discharge would occur before the end of
rainfall excess. At t = 2t, and greater, the unit
hydrograph should reach and maintain a maximum
value. In other words, Z - 1 at t/t,, Ā 2. The up-
per portion of the curve was therefore drawn to
satisfy this condition.
It is believed that the general relationship be-
tween Z and t/t,, should follow the trend of the
converted curve, whereas the horizontal shifting of
the data is due to the size of basin areas under
consideration. In addition to the basin size effect,
L. length along channel, ft
Figure 21. Determination of lag time by Equation 35
V. DEVELOPMENT OF THE NEW METHOD
(a)
(b)
(c)
Time
Figure 22. Schematic relation between the duration t, lag t.,
and lag time tp,
the scattering of the plotted data may be mainly
due to the errors involved in the accuracy of tlhe
data as well as in the processing of the data, such
as the uncertain flow separation in the case of mul-
tiple-peak hydrographs. In four cases: 3c, 8a, 12a,
and 16 of Tables 22 and 23, there are strong evi-
/20
80
S40
0
CIL
Q
.8
.4
I
.010
Time in terms of to
Figure 23. The S-curve and instantaneous unit hydrograph
for drainage basins of an average size of 80 sq. mi.
in Illinois based on U.S.G.S. data
.05.0
JL
)75.I .25 .50 .75 I
Figure 25. Relationship between Z and t/t,
5 tO
/o or/ tip
Figure 24. Modification of peak-reduction factor Z
dences of inconsistency due to either the inaccuracy
of the data or the way it was processed. Hence,
these cases were excluded from the plotting. In all
other cases, the errors involved are believed to be
compensative. Therefore, despite the scattering,
fitting with an average curve is justifiable.
F. DESIGN CRITERIA
From the nation-wide survey of design practice
on waterway area determination in different state
highway agencies, described in Section III, it was
found that there have been no definite criteria con-
cerning the climatic and physiographic conditions
of drainage basins forming the basis for the design
of waterway openings of drainage structures. In
the proposed method it is suggested that adequate
design criteria be considered and established as a
first step towards the design of the structures. For
this purpose, a definition of the design discharge to
be used by the proposed method is given as follows:
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
The design discharge is the maximum dis-
charge that would occur under an average physi-
ographic condition of the drainage basin due to
rainfall of a given frequency and various dura-
tions and due to base flow.
In Illinois the design of culverts for highway
drainage structures is handled in the highway dis-
tricts. The design of bridges is handled in tlhe
Bridge Office at the Bureau of Design. The dis-
tricts, with the exception of District 10, have been
using the Talbot formula almost exclusively in tlhe
determination of the required waterway openings
for the design of culverts. In District 10, with the
District office at Chicago, there are two designing
agencies; the regular District design office and the
Expressway design office. Both offices reported
making use of both the Talbot formula and the
rational formula in culvert design. The regular
District design office reported that when the ra-
tional formula is applied, 5-year and 10-year fre-
quencies are used. The use of a 10-year frequency
is limited to depressed sections where flooding is
possible. In view of the lack of a universal criterion
for the design frequency, values of runoff factor X
for frequencies of 5, 10, 25, 50, and 100 years were
prepared for use in the proposed method. The
choice of a certain frequency in a design remains to
be a matter of policy. From the rainfall analysis
described in Section IV-C, the maximum recorded
rainfall at Urbana, Illinois, was found to have a
frequency of the order of 50 years. It may appear
reasonable to set the 50-year frequency as a prac-
tical upper limit for the design of culverts.
The definition of the design discharge signifies
it as the maximum discharge due to rainfall of
various durations and due to base flow. Accord-
ingly, different durations must be assigned in the
computation by the proposed method until a maxi-
mum value of discharge is obtained. This value
is then taken as the design discharge.
It is important to point out that the duration
referred to in the determination of runoff factor X
is the time interval within which the maximum
depth of rainfall of a given frequency would occur.
Therefore, this duration is not necessarily equal to
the duration of the entire storm or to the duration
of rainfall excess. Unless the true duration of rain-
fall excess is available and used in the analysis of
rainfall frequencies, we can only assume that the
maximum depth of rainfall occurs uniformly within
the designated duration, and thus for this block of
uniform rainfall the duration is equal to the dura-
tion of rainfall excess. Since thunderstorms of high
intensities and short durations usually are the cause
of the peak flows from small drainage basins, this
assumption is reasonably justified. If the true
duration of rainfall excess were considered, the
rainfall amount for a given frequency would be
reduced. Therefore, the assumption is on the con-
servative side.
G. DESIGN PROCEDURE BY THE PROPOSED
METHOD
To facilitate the application of the proposed
method, all information necessary for the computa-
tion is shown in a design chart, Figure 26. For a
drainage basin of given size, runoff condition, and
location in Illinois, the procedure of computing the
design discharge is as follows:
(1) From the soil type map in the design chart
(or Figure 13), determine the soil type.
(2) From the runoff number table (or Table
15), determine the runoff number for the soil type
and the given cover and surface condition. If the
drainage basin has composite soil types and cover
and surface conditions, a weighted runoff number
should be computed.
(3) Assign a certain rainfall duration t.
(4) From the curves for runoff factor X (or
Figures 15 to 17), determine the value of X for the
assigned duration, the given frequency and the
runoff number.
(5) From the chart for climatic factor (or
Figure 18), determine the value of Y.
(6) From the chart for lag time (or Figure 21),
determine the value of t,, for the given length and
slope of the stream.
(7) Compute the ratio t/tp.
(8) From the curve for factor Z (or Figure 25)
determine the value of Z for the computed t/t,.
(9) Compute the discharge by Equation 30 or
Q = AXYZ.
(10) Repeat the steps for other assigned dura-
tions.
(11) Plot the computed discharges against as-
signed durations. The largest computed discharge
is the design discharge Q,.,x.
(12) If the stream is perennial, the base flow
Qb should be estimated and added to the discharge
determined in step 10. The design discharge is then
Qd = Qu.ax + Qb.
V. DEVELOPMENT OF THE NEW METHOD
The above procedure is illustrated by the fol-
lowing example:
Example. Determine the design discharge for a
highway culvert from the following data:
(a) Location: FA Route 14, Section IX-1,
Station 802 + 25
Williamson County, Illinois
(b) Land use: (Cover- 3% pasture or
range;
33% row crop; and
64% small grain.
Surface condition--Normal,
straight row
(c) Basin Area: 30.0 acres
(d) Channel length: 1,550 ft.
(e) Channel slope: 2.6%
(f) Design frequency: 50 years
(g) Base flow: None
Solution. From the given location the following
are obtained from the design chart:
Soil type: D
Climatic factor: Y = 1.10
The runoff number for the given land use is
found from the runoff number table in the design
chart. The weighted runoff number is computed as
follows:
Land use
Pasture
Row crop
Small grain
Runoff
Percentage number Product
3.0% X 84 = 2.5
33.0% X 90 = 29.7
64.0% X 88 = 56.4
Sum = 88.6
Say, N = 89
For L = 1,550 ft. and S = 2.6%, tile lag-time
curves give t,, 0.19 hr.
Now assign t = 0.10 hr. Since the design fre-
quency is 50 years and the runoff number is 89, the
curve for runoff factor X gives X = 2.30.
Since t = 0.10 hr. and t, = 0.19 hr., t/t, = 0.53.
The curve for peak-reduction factor Z gives Z =
0.41. Thus, the discharge is 30.0 X 2.30 X 1.10 X
0.41 = 31.1 c.f.s.
Similarly, the discharges for other assigned
values of t can be computed. The whole computa-
tion is shown in Table 24. A plot of the discharge
against the assigned t is then constructed as shown
in the accompanying figure. The maximum dis-
charge Qmax = 72.6 c.f.s. In practical applications,
the value of Qm.a can be easily estimated from the
column of Q values for the assigned durations, and
hence a plot of Q vs. t is not necessary. Also, after
Given Data
Location: FA Route 14, Sec. IX-1
Sta. 802 + 25
Williamson County, Ill.
Land use: 3% Pasture
33% Row Crop
64% Small Grain
Normal, Straight Row
Basin Area, A = 30.0 Acres
Channel Length, L = 1,550 ft.
Channel Slope, S = 2.6%
Design Frequency: 50 yrs.
Base Flow, Qb = 0 c.f.s.
Chart Values
Soil Type: D
Runoff Number, N = 89
Climatic Factor, Y = 1.10
Lag Time, t, = 0.19 hr.
Q,... = AX'YZ; Qd = OQ.. + Qb
Computation
t hr. t/lp X
z
0.1 0.53 2.30 0.41
0.2 1.05 2.50 0.66
0.3 1.58 2.38 0.87
0.4 2.11 2.20 1.00
0.5 2.64 2.05 1.00
0.6 3.16 1.88 1.00
0.7 3.69 1.75 1.00
0.8 4.21 1.62 1.00
0.9 4.74 1.51 1.00
1.0 5.27 1.41 1.0X)
Q, = 72.6 + 0 = 72.6 c.f.s.
Q
31.1
54.4
68.3
72.6
67.7
62.0
57.8
53.5
49.8
4(.5
80
70---
60
so ----- - s--
50
40
.8 1.0
some practice, fewer assigned values of t would be
necessary for the determination of Qmax.
Since the base flow is zero, the design discharge
is 72.6 + 0 = 72.6 c.f.s.
H. MERITS OF THE PROPOSED METHOD
There are several major merits that can be
stated for the proposed method. They are as
follows:
(1) The method has an analytical basis since it
is developed from sound hydrologic principles.
Therefore, the method is rational and the user, if he
wishes, can follow through the procedure of the
development and thus develop an insight to the
underlying hydrologic principles.
(2) The method is based on the available data
which are adequate to the local conditions under
consideration.
Table 24
Computation of a Design Discharge
I _ _ I _ _ I_ _ _ _ _ I __ _
0 .2 .4 .6
I. Duration in hr
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
Cover
Surface condition
N,
A
Follow Straight row 77
Row crops Straight row 70
Contoured 67
Contoured a terraced 64
Small groins Straight row 64
Contoured 62
Contoured a terraced 60
Legumes or rota- Straight row 62
tion meadow Contoured 60
Contoured 8 terraced 57
Posture or range Poor 68
Normal 49
Good 39
Contoured, poor 47
Contoured, normal 25
Contoured. good 6
Meadow, permanent Normal 30
Woods, form Sparse 45
wood lot Normal 36
Dense 25
Farmsteads Normal 59
Roads Dirt 72
Hard surface 74
Forest Very sparse 56
Sparse 46
Normal 36
Dense 26
Very dense 15
Impervious surface /00
Step Procedure
/ Read soil type from (a)
2 With cover, surface condition, and soil type read N, runoff number, from (b)
3 With N. frequency,and assigned I, read factor X from (c) or (d)
4 Read Y. climatic factor, from (e)
5 With drainage basin data read tp from (f)
6 Compute the ratio l/tp
7 With t/tp read factor Z from (g)
8 Compute discharge by Eq 0 =AXYZ
(O=discharge in cfs) (A=drainoge area in acres)
runoff number
Soil type
B C D
86 91 94
80 87 90
77 83 87
73 79 82
76 84 88
74 82 85
7/ 79 82
75 83 87
72 8/ 84
70 78 82
79 86 89
69 79 84
6/ 74 80
67 8/ 88
59 75 83
35 70 79
58 7/ 78
66 77 83
60 73 79
55 70 77
74 82 86
82 87 89
84 90 92
75 86 9/
68 78 84
60 70 76
52 62 69
44 54 6/
/00 /00 /00
V. DEVELOPMENT OF THE NEW METHOD 63
I 5 i 5 /0
/ 5
I. Duration in hr ,. Duration in hr
5
.5
Z
.05
.0/
05 .1
L, Length along channel, ft 'p
Figure 26. Chart for the determination of design discharge for small rural drainage basins in Illinois
5
I
.5
<
.05
.01
5
/
.5
X
.l
.0/
.01
.
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
(3) The criterion for the determination of a:
design discharge is clearly defined.
(4) Although the result obtained by tlie method
ilay still require sonme professional supervision or
review for final adoption in a design, the method
will provide a unique solution or produce close
:uswers by different individuals.
(5) The procedure of the lmethod is simplified(
by a design chart so that practicing engineers can;
use it with great case.
(6) The method can be readily improved lby
further verification with the accumulation of rain-
fall and runoff data and the field explrience. Since
the methlod is based on analytical principles, hlie
improvement will not change the basic schemie, but
will involve only the modification of the curves amd
charts which depend on the quantitative part of
lie data.
Like most metrhods in scientific and engineering
works, the proposed method lhas disadvantages as
well as advantages. The mnajor disadvantage is
thel fact t:hat tlle design discharge tlhus determinled
is based on a given frequency of railfall instead of
runoff. This shortcoming is entirely due to the lack
of suitable data of runoffrequency fr or small drain-
age basins. Tihe analysis ol a large quantity of
suiitable runoff data may overcome this problem
in tie Ifuture.
VI. FUTURE STUDIES
The present investigation has led to the develop-
ment of a practical procedure for the determination
of peak discharges for the design of drainage struc-
tures on small rural drainage areas. Although tile
procedure illustrated in this report is prepared for
design conditions in Illinois, the concept of the
method is universally applicable to other states
provided adequate data in these states are available
for similar analysis and development.
Generally speaking, hydrologic data for small
drainage basins are meager as compared to the
data for large basins. In the present study, how-
ever, the data were available for the development
of the proposed method, but they were not of suffi-
cient quantity to justify a refinement of the method
by correlation analyses of the data. When a sub-
stantial amount of data becomes available in the
future, correlation analysis should be applied to
the data in order to refine the functional relation-
ships between various factors considered in the
method. As data become abundant, a program of
computation by digital computers should be de-
veloped for the selection and analysis of the data
to be used in the formulation of the design charts.
Simplifying the tedious processing and derivation
of instantaneous unit hydrographs will be of great
value. In connection with this future study, a
comprehensive investigation on various theories
that have been proposed for the concept of instan-
taneous unit hydrographs is necessary. From such
an investigation it will be possible to select the best
theory suitable to the present purpose. For a physi-
cal understanding and interpretation of some
assumptions underlying the unit hydrograph prin-
ciple, a hydraulic model of drainage basins designed
for different geometric elements and physiographic
features should prove most useful. Such a model
should be capable of testing, for instance, the prin-
ciple of superposition that basically constitutes the
conventional method of unit hydrographs.
When suitable rainfall data become available,
a further study on rainfall variables should be in
order for an improvement of the proposed method.
The rainfall variables include such items as the
frequency range of variation of the effect of ante-
cedent moisture condition upon runoff, the relation-
ship between the durations of gross rainfall and
rainfall excess (Section V-F), and the areal and
time distributions of rainfall. Similarly, sufficient
quantitative information on physiographic condi-
tions of drainage basins will enable an investigator
to perform a frequency analysis of the runoff num-
ber used in the proposed method. At present, the
runoff number based on an average condition of
runoff is assumed.
Because of the shortage of runoff data pertain-
ing to small drainage basins, the design frequency
used in the proposed method is based on the data
of rainfall intensities (Section V-F). When more
runoff data for small drainage basins are accumu-
lated in the future, there will be a need to perform
a frequency analysis of the runoff data and to
adopt a design criterion on the basis of runoff fre-
quencies. The family of curves for the runoff
factor will have to be modified accordingly, while
the general concept of the proposed method may
remain the same.
VII. APPENDICES
APPENDIX I. A COMPILATION OF FORMULAS FOR WATERWAY AREA DETERMINATION
A. GENERAL
In engineering literature there are numerous
empirical and semi-empirical formulas which have
been developed to determine the waterway areas
of culverts and bridges for various hydrologic and
geographic conditions. A number of these formulas
give an answer to the required waterway area di-
rectly; while others provide the maximum or design
values of rainfall and runoff by which the desired
waterway area may be determined.
In this appendix 102 such formulas are pre-
sented. Informative remarks and references for
each formula are given, furnishing further infor-
mation concerning the original development of the
formula and its background. However, this conm-
pilation of formulas is by no means complete, as
the listed formulas are only those which have been
popular primarily in English-speaking countries.
Formulas are attractive to practicing engineers
because they are simple in form and easy to use.
However, the user may often ignore the conditions
for which the formula was developed, thus pro-
ducing erroneous answers. It is apparent that
formulas are so many in number and vary so much
in results that it is impossible to reconcile the
widely divergent answers which would be obtained
by applying a number of them to any particular
problem. Each formula has its particular purpose,
but none is suitable for general application. For
example, the popular Talbot formula which is based
on data obtained from Middle Western United
States is often applied to other parts of the country
having entirely different hydrologic conditions.
Such indiscriminate uses of the formulas should be
avoided.
Despite the fact that most of these formulas
were not intended for general application, attempts
are usually made to broaden the application by
means of radical changes in coefficients. It is there-
fore necessary in each case to select a proper coeffi-
cient for the problem under consideration. Such a
step requires practical experience and sound judg-
ment. A good selection depends on a thorough
knowledge of the conditions for which the formula
was derived and also on a correct understanding of
the limitations to which the formula should be sub-
jected. Unfortunately, the limited applicability of
such formulas is not always well understood, and
the personal factors involved in selecting proper
coefficients vary greatly. Consequently, the attrac-
tiveness of the simplicity of such formulas often
leads to much abuse and results in errors in design.
Furthermore, modern hydrology discusses nu-
merous factors which govern the rainfall on any
drainage basin and the runoff from it. Some of the
empirical formulas achieve their simplicity by neg-
lecting a number of factors which may be vital to
the establishment of a correct answer. Other
formulas are complicated and contain certain
factors which may be insignificant and difficult to
evaluate. Therefore, a study of these formulas and
the factors involved in them is desirable in evalu-
ating appropriate factors which must be considered
in the development of a rational procedure for
waterway area determination.
B. CLASSIFICATION OF FORMULAS
The 102 formulas collected in this appendix are
classified into five groups as follows:
Group 1. Waterway Area Formulas. In these
formulas the waterway area is expressed directly
in terms of the drainage area. A coefficient is
generally used to take care of the variation in con-
ditions which affect the waterway area. The gen-
eral form of the formulas is
a = C F(A)
in which a is the waterway area, C is the coefficient,
and F(A) is a function of the drainage area A. The
most common form of F(A) is
F(A) = A"
VII. APPENDICES
in which n is an exponent ranging from 0.5 to 1.0.
The value of C varies from 0.2 to 4.0.
A collection of 12 formulas is given in this
group.
Group 2. Simple Flood Formulas. In these
formulas the flood discharge is expressed directly in
terms of the drainage area. The waterway area is
then obtained by dividing the computed discharge
by an assigned desirable or safe value of the ve-
locity of flow V through the waterway opening,
giving
Q
a = -V
The general form of the formulas is
Q = C F(A)
in which Q is the discharge, C is a coefficient de-
pending on conditions which affect the discharge,
and F(A) is a function of the drainage area. The
most common form of F(A) is again F(A) = A".
The value of C and n varies widely.
A collection of 30 formulas is given in this
group.
Group 3. Rainfall Intensity Formulas. These
formulas are used to compute the rainfall intensity.
The discharge is computed by the rational formula
Q = CIA
in which Q = discharge in c.f.s.
C = percentage of runoff depending on
characteristics of the drainage basin
I = rainfall intensity in inches per hour
A = drainage area in acres
The general form for rainfall intensity formulas
K 'Fn
(1 + b)'
in which
K, b, and n, n, = respectively coefficients and ex-
ponents depending on conditions
which affect the rainfall inten-
sity.
F = frequency factor indicating the
frequency of occurrence of the
rainfall
t = duration of the storm in minutes
which is equal to the time of
concentration
The forms of rainfall intensity formulas are
generally of the following three types:
Type I: I = K
Type II: I = K
Ut
Type III: I = -- -
Vt + b
A collection of 24 formulas is given in this
group.
Group 4. Frequency Formulas. These for-
mulas express the discharge in terms of basin char-
acteristic parameters and the frequency of occur-
rence. The formulas are generally developed by
means of a frequency analysis of the flood data.
The general form of the formulas is
Q = a + b F(T)
in which a and b are constants, and F(T) is a func-
tion of the recurrence interval T in years. The
recurrence interval is defined as the average inter-
val of time within which the magnitude of the flood
will be equalled or exceeded once on the average.
The waterway area for a given design frequency
is obtained by dividing the computed discharge of
the corresponding frequency by an assigned de-
sirable or safe velocity of flow through the drainage
structure.
A collection of 5 formulas is given in this group.
Group 5. Elaborate Discharge Formulas.
These formulas express the discharge in terms of a
number of factors indicating climatic variations
and characteristics of drainage basin. These for-
mulas are generally developed by the rational for-
mula or by the method of multiple correlation. The
general form of these formulas is
A = F (D, W, L, S, F . . )
in which D, W, L, S, F . . . are factors under con-
sideration.
A collection of 31 formulas is given in this
group.
C. GROUPING OF FORMULAS
The following formulas are compiled under
various groups; within each group the formulas are
listed alphabetically.
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
Group 1. Waterway Area Formulas
1-1. The C. B. and Q. formula
1-2. The Fanning formula
1-3. The Hawksley formula
1-4. The Modified McMath formula
1-5. The Myers formula
1-6. The Peck formula
1-7. The Purdon-Dun formula
1-8. The Ramser formula
1-9. The Talbot formula
1-10. The Tidewater (Virginia)
Railway formula
1-11. The Wentworth formula
1-12. The Yule formula
Group 2. Simple Flood Formulas
2-1. The Beale formula
2-2. The Bahadur formula
2-3. The C. B. and Q. Railroad
formula
2-4. The Cooley formula
2-5. The Dickens formula
2-6. The El Paso and S. W. Railway
formula
2-7. The Elliot formulas
2-8. The Fanning formula
2-9. The Frizell formula
2-10. The Ganguillet formula
2-11. The Gray formula
2-12. The Horn formula
2-13. The Inglis formula
2-14. The Italian formulas
2-15. The Kresnik formula
2-16. The Kuichling formulas
2-17. The Lauterberg formula
2-18. The McCrory and Others
formula
2-19. The Metcalf and Eddy formula
2-20. The Meyer formula
2-21. The Morgan Engineering Co.
formulas
2-22. The Modified Myers formula
2-23. The Murphy and Others
formula
2-24. The Nagler formula
2-25. The New Kuichling formula
2-26. The O'Connell formula
2-27. The Ryves formula
2-28. The Schnackenberg formula
2-29. The U.S.G.S. formulas
2-30. The Williams formula
Group 3. Rainfall Intensity Formulas
3-1. The Allen-Babbitt formula
3-2. The Bleich formulas
3-3. The Brackenbury formula
3-4. The de Bruyn-Kops formula
3-5. The Bureau of Survey formula
3-6. The Clarks formula
3-7. The Dorr formula
3-8. The Gregory formula
3-9. The Hendrick formula
3-10. The Hill formula
3-11. The Horner formula
3-12. The Institution of Civil
Engineers' formula
3-13. The Kuichling formula
3-14. The L. .. Le Conte formula
3-15. The Metcalf and Eddy formula
3-16. The Meyer formula
3-17. The Nipher formula
3-18. The Sherman formula
3-19. The Schafmayer formula
3-20. The Steel formula
3-21. The Talbot formula
3-22. The Webster formula
3-23. The Whiney formula
3-24. The Williams formula
Group 4. Frequency Formulas
4-1. The Crcager formula
4-2. The Fuller formula
4-3. The Horton formula-A
4-4. The Horton formula-B
4-5. The Lane formula
Group 5. Elaborate Discharge Formulas
5-1. The Adams formula
5-2. The Desson formula
5-3. The Boston Society of Civil
Engineers' formula
5-4. The Burge formula (See: The
Dredge formula)
5-5. The Biirkli-Ziegler formula
5-6. The Chamier formula
5-7. The Craig formula
5-8. The Cramer formula-A
5-9. The Cramer formula-B
5-10. The Dredge or Burge formula
5-11. The Gregory formulas
5-12. The Gregory and Arnold
formula
5-13. The Gregory and Hering
formula
VII. APPENDICES
5-14. The Grunsky formula-A
5-15. The Grunsky fornmula-B
5-16. The Hawksley formula
5-17. The Hering formula
5-18. The Iszkowski formula
5-19. The Kinnison and Colby
formulas
5-20. The Lauterberg formula
5-21. The Lillie formula
5-22. The McMath formula
5-23. The Parmley formula
5-24. The Pettis formula
5-25. The Possenti formula
5-26. The Protodiakonov formnula
5-27. The Rhind formula
5-28. The Ribeiro formula
5-29. The U.S.S.R. NTK-NKPS
formula
5-30. The U.S.S.R. Scientific
Academy formula
5-31. The Switzer and Miller formula
5-32. The Walker formula
D. NOTATION
The following is a list of notation used in the
formulas which may differ from those used in
the original presentation of the formulas. When the
units are different from those given in the list, they
will be specified under each individual case.
a = Waterway area in square feet.
b, bl, b,, b = (Coefficient or parameter.
C, C1, C2, c.3 = (Coefficient or parameter, depending
on basin characteristics, hydrologic
condition, or other factors.
A, At, A2 = Drainage area in acres.
Ak = Drainage area in square kilometers.
A'k= Forested part of the drainage area
in square kilometers.
1), D1 = Drainage area in square miles.
d = Diameter of a circular sewer in
inches.
F = Frequency factor indicating the oc-
currence of a storm or flood.
G = Geographic factor.
II = Average water content, of snow in
millimeters of depth.
I = Rainfall intensity in inches per hour.
1,; = Design rainfall intensity in centi-
meters per minute.
i = Permeability of soils in centimeters
per minute.
K = Coefficient or parameter in rainfall
intensity formula.
k = Climatic factor.
L = A length on the drainage basin in
miles, feet, or kilometers measured
along the main stream or nearly so;
it may be the greatest length of the
drainage basin, the straight line dis-
tance from point of discharge to
center of gravity of basin, the length
of the outlet channel from the edge
of the drainage area to the outlet,
the length of path of raindrops from
farthest point of drainage area to
point where discharge is considered,
the length of the stream from its
source to the point of observation,
the length of sectors of drainage
area, or the average distance which
water travels to the outlet.
m = A dividing factor.
N = Length of sewer in feet per foot of
fall.
II, 1i, n72, 13 = Exponents.
P = Proportion of impervious surface.
Q = Discharge in c.f.s.
Q,,, = Maximum discharge in cubic meters
per second.
Q..ax = Maximum discharge in c.f.s.
Q,, = Average discharge in c.f.s.
Qr = Recorded discharge in c.f.s.
q, qi = Discharge in c.f.s. per square mile.
q -.n.x = Maximum discharge in c.f.s. per
square mile.
R = Raim'all depth in inches or meters;
it may be the mean, normal or maxi-
mum value in an hour, a day, a year
or in record; or direct runoff in inches.
R, = Maximum one-day rainfall in inches.
R, = Recorded one-day rainfall in inches.
r = Parameter for forestation.
S, 8S, S2 = Slope or fall of drainage area or
stream expressed in ratio or in feet
per thousand feet or in feet per mile.
s, s1, S2 = Median altitude of the drainage area
above a datum or the outlet, in feet.
T = Recurrence interval in years.
T, = Exceedance interval in years.
t = Duration of the rainfall in minutes,
or the time of concentration in min-
utes when the rational formula is
used, or time for absolute maximum.
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
,.. = Shortest time for infiltration during
a 24-hr. intensive rainfall.
V = The average velocity of flow in feet
per second or in meters per second
when specifically stated.
W = Average width of the drainage basin
in miles.
0 = Angle of sectors of drainage area in
degrees.
E. GROUP 1: WATERWAY AREA FORMULAS
1-1. The C. B. and Q. formula
0.46875 A
a
3 + 0.079 VA
or 300 D
3 + 2-\/])
Remark: This formula was derived from the Chi-
cago, Burlington and Quincy Railroad formula
12-3) for flood discharge, assuming a velocity of
flow equal to 10 ft. per see.
Reference: Thaddeus Merriman and T. H. Wiggin,
Aimerican Civil Engineers' Handbook (5th ed.;
New York: John Wiley and Sons, 1930), p. 2010.
1-2. The Fanning formula
(1) Assuming a velocity of 4 ft. per see.,
a = 0.23 Al/"
or a = 50 D'1"
(2) Assuming a velocity of 10 ft. per sec.,
a = 0.09 A-1"
or a = 20 DJ'/
Remark: The formulas were derived from the
Fanning formula (2-8) for flood discharge.
References: Thaddeus Merriman and T. H. Wiggin,
American Civil Engineers' Handbook (5th ed.;
New York: John Wiley and Sons, 1930), p. 2009.
George H. Bremner, "Areas of Waterways for
Railroad Culverts and Bridges," Journal, Western
Society of Engineers, Vol. 11, No. 2 (April, 1906),
p. 139.
1-3. The Hawksley formula
ird2
a=
4
in which log d = 3 log A + log N + 6.8
in which log d = - 10 - -.Ā, ---_-- -
10
where d = diameter of circular sewer in inches
N = length of sewer in feet per foot of fall
Remarks: (1) This formula was first published in
a Report of Comnmission of Metropolitan Drain-
age, 1857, London.
(2) The formula is believed to have been estab-
lished between 1853 and 1856. It was developed to
express analytically the relationship between the
diameter and slope of a circular outlet sewer and
the size of its drainage area. This relationship was
found from numerous observations of storm dis-
charges in sewers and then represented by a tabular
form in 1852 by John Roe, Surveyor of the Holborn
and Finsbury sewers, London.
(3) A maximum rainfall intensity of 1 in. per
hr. was probably assunmel.
Reference: Emil Kuichling, "Storm Water in Town
Sewerage," Transactions, Association of Civil Engi-
neers, Cornell University, Vol. 1 (1893), p. 46.
1-4. The Modified McMath formula
a = 0.5908A'"
Remark: This was derived from the McMath
formula (5-22) for conditions at St. Louis, Mis-
souri, with C = 0.75, I = 2.75, S = 15, and a ve-
locity of flow of 6 ft. per sec.
Reference: Thaddeus Merriman and T. H. Wiggin,
American Civil Engineers' Handbook (5th ed.;
New York: John Wiley and Sons, 1930), p. 2010.
1-5. The Myers formula
a = C A"-
in which C = 1 as a minimum for flat country
= 1.6 for hilly compact ground
= 4.0 as a maximum for mountainous
and rocky country
Remarks: (1) The formula was applied satisfac-
torily to the water courses mostly on the line of the
Richmond, Fredericksburg and Potomac Railroad
(State of Virginia).
(2) It was generally found that the formula
gives values too large for drainage areas less than
about 1 sq. mi. in flat country or 0.5 sq. mi. in
mountainous regions.
(3) The formula was first published in a paper
read by Cleemann before the Engineers' Club of
Philadelphia in 1879.
Reference: T. M. Cleemann, "Railroad Engineers'
Practice. Discussion of Formulas," Proceedings,
Engineers Club of Philadelphia, Vol. I (April 5,
1879), p. 146.
VII. APPENDICES
1-6. The Peck formula
A
a = C
in which C = 4 for very mountainous country,
where slopes of hills and mountains
are steep and abrupt
= 6 for ordinary flat rolling country,
such as in most agricultural regions
Remarks: (1) This formula was introduced by
R. M\. Peck for design of structures on the Missouri
Pacific Railway and St. Louis, Iron Mountain and
Southern Railway.
(2) This formula appears to have no great merit.
Reference: W. G. Berg, "How to Ietermine Size
and Capacity of Openings for Waterways," Conm-
mittee Reports for 1896-97, presented at the 7th
Annual Convention of the Association in Denver,
Colorado, October, 1897, and pub)lished in Proceed-
ings, Association of Railway Superintendents,
Bridges and Buildings, Vol. 7 (1897), pp. 86-100.
1-7. The Purdon-Dun formula
a 640 240 -12 )
Remarks: (1) This formula is based on James
Dun's waterway area data and the data collected
from the land adjacent to the Santa Fe Railway.
(2) Purdon found that the formula fitted the
I)un data for drainage areas of more than 16 sq. mi.
For small areas an arbitrary addition was made to
allow for drift, etc.
Reference: C. 1). Purdon, "Discussion on Flood
Flow Characteristics," Transactions, American So-
ciety of Civil Engineers, Vol. 89 (1926), p. 1090.
1-8. The Ramser formula
a = C 130 7-00
A + 600F/
in which C = 1.4 for cultivated hilly land
= 1.0 for cultivated rolling land
= 0.8 for pasture in hilly land
= 0.6 for pasture in rolling land
= 0.4 for woods in hilly land
= 0.3 for woods in rolling land
Remarks: (1) This formula was recommended for
areas not larger than 800 acres.
(2) For fan- or square-shaped drainage basins
the results should be multiplied by 1.25.
(3) The formula was designed primarily for
drop-inlet culverts. The values computed by the
formula are used for vertical drops through culverts
up to 5 feet. The values should be multiplied by
0.71 for drops through culverts up to 10 feet and
by 0.58 for drops through culverts up to 15 feet.
Reference: Q. C. Ayres, Recommendations for the
Control and Reclamation of Gullies, Bulletin 121
(Ames, Iowa: Iowa State College Engineering Ex-
periment Station, 1935).
1-9. The Talbot fornula
a = C A"'.
in which C = 1.00 for mountainous region
= 0.66 for very hilly country
= 0.50 for hilly country
= 0.33 for rolling country
= 0.25 for gently rolling country
= 0.20 for flat land
Remark: The values listed above are commonly
employed. For originally recommended values and
discussion of the formula see Section II-C.
Reference: A. N. Talbot, "The Determination of
Water-Way for Bridges and Culverts," Selected
Papers of the Civil Engineers' Club, Technograph
No. 2, University of Illinois (1887-88), pp. 14-22.
1-10. The Tidewater (Virginia) Railway formula
a = 0.62 A"''
Remark: The computed area may be increased
30% for streams having a flat fall, and 20% for
double openings.
References: Tables and Data for Estimates and
Comparisons (blue print), Tidewater Railway, Ro-
anoke, Virginia. January, 1905.
"Report of Sub-(Committee of Roadway Com-
mittee No. 1, Bulletin 131," Proceedings, American
Railway Engineering and Maintenance of Way As-
sociation, Vol. 12, Pt. 3 (March, 1911), p. 499.
1-11. The Wentworth formula
a = A%;
Remarks: (1) This formula was derived for design
on the Norfolk and Western Railway. It was found
to fit the conditions along that line satisfactorily.
It has also been used in the southeastern states.
(2) For very flat ground with less rainfall, the
results obtained by this formula are generally too
large. Wentworth observed that 60%A of the com-
puted area may be taken as the lower limit.
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
Reference: Letter to the Editor, Railroad Gazette,
1903, p. 57.
1-12. The Yule formula (Also known as the Indiana
State Highway Iepartment formula)
a = C A':
in which C =0.3 to 0.7 for flat country
= 0.7 to 1.3 for rolling country
= 1.3 to 2.2 for hilly country
Remarks: (1) This formula was given only for
conditions of topography and rainfall similar to
those in Indiana. It was not proposed for arid
regions, for areas having large average annual rain-
fall, or for mountainous country.
(2) The formula was developed from a study by
the Indiana State Highway Department since 1927.
Reference: Robert B. Yule, "Bridge Waterway
Area Formula Developed for Indiana," Civil Engi-
neering, Vol. 20, No. 10 (October, 1950), pp. 26
and 73.
F. GROUP 2: SIMPLE FLOOD FORMULAS
2-1. The Beale formula
Q = C A"7'
in which C = 1,600 for unforested area
= 1,400 to 1,000 for forested area in the
central provinces of India
Remark: This is an adaptation of the Dickens for-
mula (2-3) to suit the conditions of the Western
Ghats in the Bombay Presidency from the observed
discharges on the Nira Canal.
Reference: G. R. Hearn, "The Effect of Shape of
Catchment on Flood Discharge," Proceedings, The
Institution of Civil Engineers, Vol. 217 (1923-24),
p. 289.
2-2. The Bahadur formula
Q = C D[0.02- (1/14)log D]
in which C = 1,600 to 2,000
Remark: This formula was proposed by Nawab
Jung Bahadur for drainage basins in Hyderabad-
Deccan area in India.
Reference: V. B. Priyani, The Fundamental Prin-
ciples of Irrigation Engineering (Anand, India:
Charotar Book Stall, 1957), p. 48.
2-3. The C. B. and Q. Railroad formula
59.2 A
37.9 + /A
S 3,000 D
3 + 22/V_
3+2VD
Remark: This formula was used for culvert design
by the Chicago, Burlington and Quincy Railway
Company.
Reference: G. H. Bremner, "Areas of Waterways
for Railroad Culverts and Bridges," Journal, West-
ern Society of Engineers, Vol. 11, No. 2 (April,
1906), p. 139.
2-4. The Cooley formula
Q = 2.43 A = 180 D%
Q = 2.70 A, = 200 D%
Reference: G. H. Bremner, "Areas of Waterways
for Railroad Culverts and Bridges," Journal, West-
ern Society of Engineers, Vol. 11, No. 2 (April,
1906), pp. 139 and 170.
2-5. Tile Dickens formula
Q = CAo.7
Q = C,Do".,
in which C = 1.56 or (, = 200 for Madras Presi-
dency, India
= 3.91 or C, = 500 for Central Prov-
inces, India
C = 6.45 or C, = 825 for Bengal and
Bihar, India
C = 9.37 or C, = 1,200 for Upper Kaveri,
India
C = 17.2 or C, = 2:200 for Gadamatti,
India
C = 6.6 or C, = 850 for average condi-
tions
Remark: Other values of C and C, suggested by
S. K. Gurtu in Proceedings, The Institution of Civil
Engineers, Vol. 217 (1923-24), p. 386, are as follows:
C = 11.0-15.6 and C, = 1,400-2,000 for bare
drainage basins, covered with precipitous
hills (Class I)
C = 7.8-9.4 and C, = 1,000-1,200 for basins
with hills on the skirts, with undulating
country below up to the outfall (Class II)
C = 6.3-7.8 and C, = 800-1,000 for undulating
country, with hard indurated clay soil
(Class III)
C = 1.6-4.7 and C, = 200-600 for flat, sandy,
absorbent, or cultivated plains (Class IV)
References: C. H. Dickens, "Flood Discharge of
Rivers," Professional Papers on Indian Engineer-
VII. APPENDICES
ing, Thomason College Press, Roorkee, India, Vol.
II (1865), pp. 133-136.
G. R. Hearn, "The Effect of Shape of Catch-
ment on Flood-Discharge," Proceedings, The Insti-
tution of Civil Engineers, Vol. 217 (1924), p. 288.
S. K. Gurtu, "Correspondence on Flood-
Discharge," Proceedings, The Institution of Civil
Engineers, Vol. 217 (1924), p. 386.
2-6. The El Paso and S. WV. Railway formula
Q = 60 A",-
Remark: This is practically the same formula de-
veloped by Joseph P. Frizell, which lie obtained
from records of flow over the Holyoke Dam, Mas-
sachusetts, for a period of 50 years, and published
in his book on hydraulics.
Reference: "Report of Sub-Committee of Roadway
Committee No. 1, Bulletin 131," Proceedings,
American Railway Engineering and Maintenance
of Way Associrtion, Vol. 12, Pt. 3 (March, 1911),
p. 499.
2-7. The Elliot formulas
(1) For swamps and wet lands in Northeastern
Arkansas
24
ND
+ 6) 1)
=( 0. 0.00937) A
A: ~ +
soils cast of Crowleys Ridge, and 200% for the
slopes of Crowleys Ridge.
(2) The second formula specifies that the soils
are absorptive and easily drained.
(3) The third formula was given to areas of 200
sq. mi. and less.
(4) These formulas were used for rough approx-
imations. The results should be checked for local
conditions.
References: (C. (. Elliot, Engineering for Land
Drainages (New York: John Wiley and Sons, Inc.,
1919), pp. 198-199.
A. E. Morgan, "A Preliminary Report on the St.
Francis Valley Drainage Project in Northeastern
Arkansas," U. S. Departmlent of Agriculture Cir-
cular 86, Office of Experimental Stations, 1919, p. 20
(for the first formula).
2-8. The Fanning formula
Q = 0.92 A%
or Q = 200 D%
Remark: The formula was derived from a rela-
tively small number of observations on American
rivers.
Reference: J. T. Fanning, Practical Treatise on
IIater Supply Engincering (New York: D. Van
Nostrand Co., Inc., 1878), p. 66.
2-9. The Frizell formula
(2) For, swamps and other wet lands of the
Upper Mississippi Valley
Q = + 3.63) D
or
Q = -92- + 0.00568) A
(3) For satisfactory drainage areas in North
(entral Illinois
Q=( --673
S19.2 + VDI
Q ( 26.6
Q (= 1.+ / I
486 + -\1A
- 11.3) D
- 0.0177) A
Remarks: (1) The first formula was used to com-
pute the discharge from the low flat alluvial lands
in the preliminary drainage investigation in North-
eastern Arkansas. The results may be increased
50C% for the more rolling and less sandy land in the
east part of Mssiissippi County, 100% for the clay
Q = 61.3 A"''
Q = 1,550 Do-s
Remark: This formula is converted from the orig-
inal form q - 17.35 V 8006/D) for maximum flood
rate in c.f.s. per sq. mi. on the Connecticut River.
The general form is q = q, V I),/D where q, is
the observed maximum flood rate in c.f.s. per sq. mi.
and I, is the corresponding drainage area in sq. mi.
Reference: J. P. Frizell, Water Power (3rd ed.;
New York: .John Wiley and Sons, Inc., 1905), p. 41.
2-10. The iGanguillet-Kutter formula
56.2 A
78.7 + v/A
Q = 1,421 D
3.11 + v -/
Remark: This formula was developed for Swiss
streams in 1869.
Reference: C. S. Jarvis, "Hydrology," Section 2 of
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
Handbook of Applied Hydraulics, edited by C. V.
Davis (1st ed.; New York: McGraw-Hill Book
Co., Inc., 1942), p. 109.
2-11. The Gray formula
Q = 0.049 A'".
( = 3,77.U 1) '7'
Remark: The original form is Q = 5.89 D%, where
Q is the discharge in c.f.s. per acre and D is the
drainage area in sq. mi.
Reference: "Report of Sub-Conmnittee of Roadway
Committee No. 1. Bulletin 131," Proceedings, Amer-
ican Railway Engineering and Maintenance of Way
Association, Vol. 12, Pt. 3 (March, 1911), p. 499.
2-12. The Horn formula
Q = A(10.45 - In A)
or Q = 640 D(4 - In D)
Remark: This formula was proposed for maximum
discharge in c.f.s. from drainage areas less than 20
sq. mi.
Reference: G. R. Hearn, "The Effect of Shape of
Catchment on Flood-1)ischarge," Proceedings, the
Institution of Civil Engineers, Vol. 217 (1924), pp.
279-280.
2-13. The Inglis formunla
7,000 1)
Remark: The formula was developed by Sir C. C.
Inglis for fan-shaped drainage basins in Bombay
State, India.
Reference: V. B. Priyani, The Fundamental Prin-
ciples of Irrigation Engineering (Anand, India:
Charotar Book Stall, 1957), p. 48.
2-14. The Italian formulas
71.8 A
(1) 7.87 + A
or
1,819 D
Q =
0.311 +V D
103.0 A
(2) Q =
7.87 + /A -
S 2,600 D
0.311 + D\/
Remark: The first formula was developed for
northern Italy and the second formula for small
brooks in the same region.
Reference: Emil Kuichling, Annual Report on the
Barge Canal, New York, 1901.
2-15. The Kresnik formula
71 A
7.84 +-/A
in which C = a maximum of 3.07 for the Pliessnitz
River near Bertschodorf, Germany
= up to 6.0 for rivers outside of Europe
Remark: Tables and charts have been prepared for
the values of C for a number of rivers. They may
be found in Hydraulic Structures by A. Schoklitsch
I English translation: (New York: The American
Society of Mechanical Engineers, 1937), Vol. 1, pp.
58-601.
Reference: "Allgemeine Berechnung der Wass.er-
profile und Gef:illsverhiiltnisse fiur Fliisse und
Kanile." Technischc Vortriige und Abhandlungen,
No. 8, Vienna, 1886.
2-16. The Kuichling formulas
=( 44,000
SA + 108,800
42A
Q( 127,000 1 l
A + 23,700 86.57
For frequent
floods
For rare
floods
Remarks: (1) The formulas were developed for
floods on streams similar to Mohawk River in New
York State. Kuichling noted th thtthe formulas are
applicable to hilly or mountainous regions, such as
are found in New England, Middle and North At-
lantic States, and are probably also applicable to a
rolling country having a clayey surface soil.
(2) The original forms of the above formulas are
44,000
-D + Ā±170 +20
and
127,000
q = + 7.4
D + 370
in which q = discharge in c.f.-. per sq. mi. and D =
drainage area in sq. mi.
(3) These formulas apply to drainage areas
larger than 100 sq. mi. For drainage areas less than
100 sq. mi., the corresponding formulas are
25,000
q = 125 + 15
+ 125
35,000
q = D + 10
D + 32
as published in American Sewerage Practice by
Leonard Metcalf and Harrison P. Eddy, (Vol. I;
VII. APPENDICES
New York: McGraw-Hill Book Co., Inc., 1914),
p. 250.
Reference: Einil Kuiclling, Annual Report on the
Barge Canal, New York, 1901, p. 848.
2-17. The Lauterberg formula
Q = A ( .000039t + 0.0008275)
\ 6 + 0.0000039 A
2-20. The Meyer formula (also known as the Min-
nesota Flood Flow formula)
A = 100 C F D"-"
in which C = a coefficient depending on
slope, character of soil, and
phy. Recommended values
as follows:
different
topogra-
of C are
Q = D ( 615 + 0.53)
=l ( 6 +-_ 0.0025 -D +-
Remark: The formula was developed from floods
due to continuous heavy rain of 3 to 4 days of
duration at an average rate of 2 in. per day.
Reference: Emil Kuichling, Annual Report on the
Barge Canal, New York, 1901.
2-18. The McCrory and Others formulas
(1) Q = 0.159.A'5/
(2) Q = 90( +
(A 64
Ā» 3( ,4t00
(3) Q -= A +32,000
128
Remarks: (1) These formulas wee re preared for
swamllps and wet lands at particular locations.
(2) The first, formula was given for Cypress
Creek Drainage District, Arkansas.
(3) The second formula was given tentatively
for certain Louisiana districts with no considerable
storage in bayous and ditches.
(4) The third formula was proposed tentatively
for certain districts in the Everglades of Florida.
Reference: Drainage Investigation, U. S. I)eplart-
ment of Agriculture Bulletin 198, Publications of
the Office of Experimental Station, Professional
Papers of U. S. Department of Agriculture, 1915.
2-19. The Metcalf and Eddy formula
Q = 3.95 A''7
. = 440 1. 7*:
Remark: This formula was developed to suit drain-
age areas of 6,400 to 160,000 acres near Louisville,
Kentucky, in connection with studies for the flood-
water discharge of Beargrass Creek, Louisville,
Kentucky.
Reference: L. Metcalf and H. P. Eddy, American
Sewerage Practice (Vol. I; New York: McGraw-
Hill Book Co., Inc., 1941), p. 251.
(haracter of
Drainage Basin
1. Very flat agricultural
or timber land with
some marshes and
swamps.
2. Relatively flat agri-
cultural or timber
land with some marshec
and ponds.
3. Gently rolling agri-
cultural or timber
land full of lakes, pont(
and marshes connected
by poorly defined
water courses.
-. Relatively flat agri-
cultural or timber
land of fairly uniform'
slope, without lakes
and ponds.
5. Slightly undlulating
agricultural or timber
land without lakes or
ponds; or distinctly
rolling to hilly agri-
cultural or timber
land, with lakes and
ponds.
6. Gently rolling agri-
cultural or timber
land without lakes or
ponds.
7. Iistinctly rolling to
hilly agricultural or
timber land without
lakes and ponds; or
hilly agricultural or
timber lands with steep
slopes and lakes, ponds
and marshes in valleys.
Sandy Loam Clayey
Soil
0.35
Soil
0.40 0.50
0.45 0.50 0.60
0.50 0.60 0.75
0.60 0.70 0.85
0.70 0.80 1.00
0.85 1.00 1.25
1.10 1.50 2.00
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
Character of S
Drainage Basin
8. Hilly agricultural or
timber land with steep
slopes barely admit-
ting of cultivation;
without lakes, ponds
or marshes.
9. Very hilly timber or
brush-covered land,
slopes too steep for
cultivation; ravines
and gullies with oc-
casional small ponds
or marshes.
10. Very hilly timber or
brush-covered land
with much rock out-
cropping; ravines anti
gullies, occasional
small ponds and
marshes.
11. Very hilly to rugged
country with much
rock outcropping; scat-
tered timber; occasional
small ponds and
marshes.
andy
Soil
2.25
Loam Clayey
Soil
3.00 4.00
3.50 4.50 6.00
5.00 6.00 8.00
9.00 10.00
12.00
12. Rugged to precipitous ... 15.00
rocky country with
practically no soil cover;
small timber and
brush; ravines and
gullies; no lakes, ponds,
or marshes to retard
runoff.
F = frequency factor recommended as follows:
For a flood to be expected Factor F
Once in 10 years 0.85
Once in 25 years 1.00
Once in 100 years 1.40
Remarks: (1) This formula was developed by A. F.
Meyer with the aid of C. M. Halseth in the Depart-
ment of Drainage and Waters, State of Minnesota,
particularly for Minnesota conditions.
(2) The formula was found to apply fairly well
to floods which have occurred at many stations on
the Mississippi and Ohio rivers.
References: E. V. Willard, Drainage Areas of Min-
nesota Streams and Method of Estimating Probable
Flood Flows, Conmmissioner, Department of Drain-
age and Waters, State of Minnesota, October, 1922.
Adolph F. Meyer, Elements of Hydrology (New
York: John Wilcy and Sons, Inc., 19281, pp. 369-
371.
2-21. The Morgan Engineering Co. fornuila'
(1) Q =
/ 380
V = I)
(2) =
( 2.80
,2, -\/D
+ A
S88.82 )
+ 7.2) D
Remarks: (1) The first formula was used for the
Cache River Drainage District.
(2) The second formula was used for Mississippi
County, Arkansas.
(3) These formulas were used by the Morgan
Engineering Company of Memphis, Tennessee, in
their design of most drainage structures.
(4) These formulas were given for swamps and
wet lands.
Reference: D)aniel W. Mead, Hydrology (New
York: Mc( rawl-Hill Book Co., Inc., 1950), p. 666.
2-22. The Modified Myers formula
Q = 10,000 C A"'
in which ( = coefficient representing variable per-
centage expressing the ratio of max-
imum flood to an assumed extreme
maximum flood.
= 5% for the Colorado River below
the Gila Junction at Yuma
= 4% for the Nile
= 50% for the Amazon
= 21% for the Mississippi at Cairo,
Illinois
= 32% for the Ohio at Paducah, Ken-
tucky
= 10% for the Gila both at Yuma and
Florence, Arizona
= 27%1 for the Salt River, before res-
ervoir controls at the mouth
and at Roosevelt, Arizona
=117% for Salado Creek, Texas
VII. APPENDICES
= 50% for Miami, Dayton, Ohio
= 40% for Otay, Lower Otay Dam,i
California
Remark: The value of 100C is equivalent to tlhe
so-called "Myers scale" (Section II-B). Values of
the Myers scale for major floods in the United
States may be found in Applied Hydrology by
R. K. Linsley, Jr., M. A. Kohler, and J. L. H.
Paullus (New York: McGraw-Hill Book Co., Inc.,
1949), pp. 227-242.
Reference: C. S. Jarvis, "Flood Flow Characteris-
tics," Transactions, American Society of Civil En-
gineers, Vol. 89 (1926), p. 994.
2-23. The Murphy and Others formula
Q ( 46,790 + 1 A
A + 205,000 42.7
S= ( 40 + 15 1)
D \ + 320 --
Remark: This formula was developed for streams
of the northeastern United States from which Mur-
phy had collected the data.
Reference: E. C. Murphy and others, "Destructive
Floods in the United States in 1904," U. S. Geolog-
ical Survey Water Supply and Irrigation Paper No.
147 (1905), p. 189.
2-24. The Nagler formula
Q = 2.84 A%
Q = 210 1)
Remark: This formula was developed for the 50-
year flood to be expected in Iowa streams.
Reference: F. A. Nagler, "A Survey of Iowa
Floods," Proceedings, Iowa Engineering Society,
1928, pp. 48-68.
2-25. The New Kuichling formula
0.065 A (396,800 + A)
15,360 + A
or
41.6 D (620 + ))
24 + 1)
in which Q = maximum discharge
Remark: Kuichling indicated that this formula ap-
plies to river basins in the Southern Atlantic States,
and it is based on the greatest observed discharges
of the Potomac River at Point of Rocks, Md., New
River at Radford, Va., the Catawba River at Rock
Hill, N. C., the Little Tennessee River at Judson,
N. C., Can Creek at Bakersville, N. C., and numer-
ous other streams which exhibit somewhat smaller
rates of discharge than the preceding ones. It may
be regarded as applicable to mountainous and hilly
drainage basins having areas of not more than
10,000 sq. mi. in the part of the country indicated.
Reference: Emil Kuichling, "Discussion on Flood
Flows," Transactions, American Society of Civil
Engineers, Vol. 77 (1914), p. 649.
2-26. The O'Connell formula
Q = / 458 (4.58 + A) - 45.8
Remark: This formula was said to have been based
on studies of rivers in Europe, India, and America,
and to be best adapted to small drainage basins.
Reference: P. O. L. O'Connell, "Relation of the
Freshwater Floods of Rivers to Areas and Physical
Features of Their Basins; and on a Method of
Classifying Rivers and Streams, with Reference to
the Magnitude of Their Floods, Proposed as a
Means of Facilitating the Investigation of the Laws
of Drainage," Proceedings, Mlinutes of the Institu-
tion of Civil Engineers, Vol. 27 (1868), pp. 206-210.
2-27. The Ryves formula
Q = C A:
in which C - local coefficient depending on the rain-
fall, soil and slope of the district
= 9.1 for upper India
Remarks: (1) In the original form the drainage
area is in sq. mi. and the values of C in India are
as follows:
C = 450 within 15 miles of the coast
C = 560 within 15 to 100 miles inland
C = 675 for a limited area near the hills
(2) This formula is used extensively in India.
Reference: K. R. Sharma, Irrigation Engineering
(Punjab, India: Ramna Krishna and Sons, 1944),
p. 575.
2-28. The Schnackenberg formula
Q = 20,000 A0O.
Remark: This formula was developed for extremely
high runoffs for the worst New Zealand conditions.
Reference: E. C. Schnackenberg, Extreme Flood
Discharges, New Zealand Institution of Engineers,
1949.
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
2-29. The U.S.G.S. formulas
Region
North Atlantic Slope
South Atlantic and
Eastern Gulf of Mexico
drainage
3. Ohio River Basin
4. St. Lawrence River Basin
5. Hudson Bay and Upper
Mississippi drainage
6. Missouri River Basin
7. Lower Mississippi River
Basin
8. Western Gulf of Mexico
drainage
9. Colorado River Basin
10. The Great Basin
11. Pacific Slope Basin in
California
12. Pacific Slope Basins in
Washington and Upper
Columbia River Basin
13. Snake River Basin
14. Pacific Slope Basins in
Oregon and Lower
Columbia River Basin
Formula
= 190 A0.5
= 250 A0.5
= 230 A.0
= 1,020 A0.as
= 230 A".4"
Q = 130 A".5
Q = 250 A".5
Q = 34.5 A".77
(below
2,550,000
acres)
Q = 104,000 A"0'3
(above
2,550,000
acres)
Q = 99 AO.5
Q = 26 A0."
(below 256,000
acres)
Q = 15,000
(above 256,000
acres)
Q = 200A0.5
Q = 180 AIO
Q = 0.51 A0.83
Q = 229 AO-5
Remark: These formulas were developed from sep-
arate enveloping curves of peak streamflow for
each of the 14 regions used by the U. S. Geological
Survey for publication of streamflow data.
Reference: R. K. Linsley, Jr., M. A. Kohler, and
J. L. H. Paulhus, Applied Hydrology (New York:
McGraw-Hill Book Co., Inc., 1949), p. 580.
2-30. The Williams formula
C_
n C a
in which thle coefficients C and n are as follows:
Locality Coeffi- Drainage area
(ients Less than 10-20,000
10 sq. mi. sq. mi.
North-East United States C 1,4( 2,400
n 0.75 0.54
Mississippi Valley C 2,500 4,800
n 0.75 0.47
Rocky Mountains C 1,900 3,600
a 0.75 0.45
Pacific Coast, U.S.A. C 1,625 2,700
A 0.75 0.53
Uplands in British Isles C S00 1,200
n 0.75 0.54
Western India C 2,700 4,600
a 0.75 0.52
North-East India ( 1,400 1,700
n 0.75 0.65
Remark: The coefficients for the United States are
based on flood records listed in the paper "Flood
Flow Characteristics" by C. S. Jarvis, Trans-
actions, American Society of Civil Engineers, Vol.
89 (1926), pp. 985-1032. For the British Isles, they
are basled on flood records reported by a Committee
on Floods of the Institution of Civil Engineers. For
Western India, they are based on records of floods
in the Bombay Presidency. For North-East India,
they are based on papers presented before the Insti-
tution of Civil Engineers by Sir Gordon Hearn,
G. L. Lillie, and E. L. Glass and the papers by
W. A. Buyers before the Institution of Engineers,
India.
Reference: G. B. Williams, Storage Reservoirs
(London: Chapman and Hall, 1937), p. 71.
G. GROUP 3: RAINFALL INTENSITY FORMULAS
3-1. The Allen-Babbitt formula
200
I=
t+ 20
Remark: This formula was derived by H. E. Bab-
bitt from Kenneth Allen's 25-year frequency curve
based on a 51-year record observed at Central
Park, New York City. See "The Prediction of
Probable Rainfall Intensities" by Kenneth Allen,
Engineering News Record, Vol. 86, No. 14 (April 7,
1921), p. 588.
Reference: Handbook of Culvert and Drainage
Practice (Middletown, Ohio: Armeo Drainage
Products Association, 1945), p. 180.
3-2. The Bleich formulas
C1
The reciprocal formula: I = -
T + b
The exponential formula: I = 0_
1 12
VII. APPENDICES
The modified exponential formula:
I = .- _ _ . (3 -1
(t + b3)"'
in which the parameters are as follows:
Storm
1-year
2-year
5-year
10-year
25-year
50-year
C,
76.4
96.3
128.0
168.0
261.0
291.0
C,
10.11
12.38
18.10
18.10
24.80
28.00
n2
0.529
0.530
0.576
0.530
0.526
0.530
C,
38.85
41.62
60.53
63.75
1,468
4,201
n:,
0.842
0.813
0.835
0.795
1.342
1.513
Remark: These formulas are said to be applicable
to New York City.
Reference: S. D. Bleicl, "Rainfall Studies for New
York," Transactions, American Society of Civil
Engineers, Vol. 100 (1935), pp. 618-619 and 621.
3-3. The Brackenbury formula
23.95
I .= + 0.154
t + 2.15
Remark: This formula was derived for Spokane,
Washington.
Reference: R. A. Brackenbury, "Construction of a
Large Sewer in Spokane," Engineering Record, Vol.
66, No. 6 (August, 1912), p. 156.
3-4. The de Bruyn-Kops formula
= K
I _ _ K. . .
t+b
in which K = 191 and b = 19 for maximum storms
K = 163 and b = 27 for storms occurring
once in 2 years
K = 141 and b = 27 for storms occurring
once a year
Remark: This formula was developed for Savan-
nah, Georgia, in 1908, and based on U. S. Weather
Bureau Records, 1899-1906.
Reference: J. de Bruyn-Kops, "Notes on Rainfall
at Savannah, Georgia," Transactions, American So-
ciety of Civil Engineers, Vol. 60 (1908), pp. 248-
257.
3-5. The Bureau of Survey formula
K
I-
in which K = 27 for a maximum storm
K = 18 for rainfalls of high intensity
K = 9 for ordinary storms
Remarks: (1) This formula was developed for
Philadelphia, Pennsylvania as given in the Annual
Report of the Bureau of Survey, 1911.
(2) It is based on a rainfall record of 25 years.
Reference: Leonard Metcalf and Harrison P. Eddy,
American Sewerage Practice (Vol. I; New York:
McGraw-Hill Book Co., Inc., 1914), p. 22.
3-6. The Clarke formula
in which K = 54 for storms to be expected each year
K = 162 for storms to be exceeded once in
8 years
K = 324 for storms to be exceeded once in
15 years
Remark: The first two values of K were good
representations of the New York rainfall records.
Reference: E. W. Clarke, "Storm Flows from City
Areas, and Their Calculation," Engineering News,
Vol. 48, No. 19 (1892), p. 388.
3-7. The Dorr formula
t + 30
Remarks: (1) This formula was developed for
basis of storm sewer design at Boston, Massachu-
setts, in 1892.
(2) Coefficients for the rational formula to be
used with this formula vary from 0.15 to 0.90,
which were selected to suit the conditions the dis-
tricts may attain in 25 or 30 years.
Reference: C. W. Sherman, "Maximum Rates of
Rainfall at Boston," Transactions, American So-
ciety of Civil Engineers, Vol. 54 (1905), p. 179.
3-8. The Gregory formula
K
in which K = 12 and n = 0.5 for ordinary severe
storms
K = 6 and n = 0.5 for winter storms
K = 32 and n = 0.8 for the maximum
storm
Reference: C. E. Gregory, "Rainfall and Runoff
in Storm Water Sewers," Transactions, American
Society of Civil Engineers, Vol. 58 (1907), p. 475.
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
3-9. The Hendrick formula
K
t+b
in which K = 300 and b = 25 for heavy storms
K = 105 and b = 10 for basis of design
Remark: This formula was proposed for Baltimore,
Maryland.
Reference: Calvin W. Hendrick, "Design andt
Construction of the Baltimore Sewerage System,"
Engineering and Contracting, Vol. 36, No. 6 (Au-
gust, 1911), p. 160.
3-10. The Hill formula
120
t+15
Remark: This formula was developed for Chicago,
Illinois in about 1907 and based on the U. S.
Weather Bureau observations on important storms,
1889-1910, inclusive.
Reference: Leonard Metcalf and H. P. Eddy,
American Sewerage Practice (Vol. I; New York:
McGraw-Hill Book Co., Inc., 1914), p. 224.
3-11. The Horner formula
56
I=
(t + 5)0.85
Remark: This formula was developed for St. Louis,
Missouri, and based on the Weather Bureau record
of excessive rains from 1873-1909.
Reference: W. W. Horner, "Modern Procedure in
District Sewer Design," Engineering News, Vol. 64,
No. 13 (September, 1910), p. 329.
3-12. The Institution of Civil Engineers' formula
8
IJ--8-
t+I
in which I = extreme rainfall intensity in inches per
hour likely to cause "catastrophe"
t = duration of storms in hours
Remark: This was developed for the British Isles,
and adopted by the Committee of the Institution of
Civil Engineers on "Floods in Relation to Reservoir
Practice." For rainfalls likely to cause normal
maximum floods, one-half the figure arrived by this
formula was considered sufficient.
Reference: G. B. Williams, Storage Reservoirs
(London: Chapman and Hall, 1937), p. 20.
3-13. The Kuichling formula
I= K
t +
in which K = 120 and b = 20
K = 106 and b = 13
Remarks: (1) This formula was developed for the
basis of design at Boston, Massachusetts, in 1905.
(2) This formula was found suitable for heavy
rainfalls near New York City.
References: Emil Kuichling, "Discussion on Maxi-
mum Rates of Rainfall," Transactions, American
Society of Civil Engineers, Vol. 54 (1905), p. 195.
C. E. Gregory, "Rainfall and Runoff in Storm
Water Sewers," Transactions, American Society of
Civil Engineers, Vol. 58 (1907), p. 475.
3-14. The L. .. ,Le Conte formula
Remark: This was found suitable for San Fran-
cisco, California.
Reference: L. J. Le Conte, "Discussion on Maxi-
mum Rates of Rainfall," Transactions, American
Society of Civil Engineers, Vol. 54 (1905), p. 198.
3-15. The Metcalf and Eddy formula
K
I-
0-.5 + b
in which K = 15.5 and b = 0 for Boston, Massa-
chusetts.
K =14 and b = 0 for Louisville,
Kentucky.
K =19 and b = 0 for New Orleans,
Louisiana.
K = 84 and b = 4 for Denver, Colorado.
Remark: This formula was proposed in 1911 as a
basis of design for maximum rainfalls.
Reference: L. Metcalf and H. P. Eddy, American
Sewerage Practice (Vol. I; New York: McGraw-
Hill Book Co., Inc., 1914), pp. 222, 224, 227-228.
3-16. The Meyer formula
K
t+b
t + b
in which the coefficients K and b are listed as fol-
lows:
t0.5
VII. APPENDICES
Storm Frequency, Years
2 5 10 25 50
180 220 276 355 450
24.5 27 32 40 50
131 171 214 252 289
21 23.5 26 28 30
96 122 150 181 216
16 18 19.5 21 23
84 108 132 160 186
16 17.5 19 20 21
75 90 105 126 152
13 13 13 14 16
Remark: The rainfall stations used by Meyer as
basis for his formula cover the cities which are
grouped in five regions as follows:
Group 1 ..... Galveston, New Orleans,
Jacksonville
Group 2 ..... New York, Philadelphia, Wash-
ington, 1). C., Norfolk, Raleigh,
Savannah, Atlanta, Little Rock,
Fort Worth, Abilene, Bcnton-
ville, St. Louis, Kansas City, Lin-
coln, Des Moines.
Group 3 ..... Boston, Albany, Pittsburgh, Elk-
ins, Asheville, Knoxville, Meii-
phis, Cairo, Indianapolis, Cincin-
nati, Cleveland, Detroit, Grand
Haven, Chicago, Madison, St.
Paul, Moorhead, Yanton, Dodge.
Group 4 ..... Duluth, Escanaba, Buffalo,
Rochester.
Group 5 ..... De)cnver, Bismark.
Reference: Adolph F. Meyer, Elements of Hydrol-
ogy (2nd ed.; New York: John Wiley and Sons,
Inc., 1928), pp. 191-200.
3-17. The Nipher formula
360
I=-
t
Remark: This formula was based on a study of the
rainfall intensity record of St. Louis, Missouri for
a period of 47 years.
Reference: Leonard Metcalf and H. P. Eddy,
American Sewerage Practice (Vol. I; New York:
McGraw-Hill Book Co., Inc., 1914), p. 220.
3-18. The Sherman formula
Regional (Coffi-
Group cients
1 K
b
2 K
b
3 K
b
4 K
5 K6
b
Storm frequency,
years
2
5
10
(15) *
20
(25)*
K
102
138
166
(182)
193
(203)
* Inlirpoi lted frlm iii llcaftiayr's data.
Remarks: (1) This formula was developed for the
Chicago area in 1938.
(2) This formula was intended for applications
to storms not exceeding 120-minute duration.
Reference: A. J. Schafmayer and B. E. Grant,
"Rainfall Intensities and Frequencies," Transac-
tions, American Society of Civil Engineers, Vol. 103
(1938), p. 355.
3-20. The Steel formula
K
t+b
in which K and b are coefficients depending on
regions as shown in the following umap (Figure 27)
and table:
Fr'
in which K = 38.64 and n = 0.687 for maxiinuin
stonms
luency, Coeffi-
Years cients 1 2
2 K 206 140
b 30 21
5 K 247 190
b 29 25
10 K 30() 230
b 36 29
25 K 327 260
6 33 32
50 K 315 350
b 28 38
100 K 367 375
b 33 36
Regions
4
70
13
97
16
111
16
170
27
187
24
220
28
K = 25.14 and n = 0.687 for basis of
design of maximum
storms
K = 18 and n = 0.5 for extraordinary
conditions
Remark: This formula was applied to Chestnut
Hill, Boston, Massachusetts.
Reference: C. W. Sherman, "Maximum Rates of
Rainfall at Boston," Transactions, American So-
ciety of Civil Engineers, Vol. 54 (1905), pp. 178-
179.
3-19. The Schafmayer formula
K
I -
t+b
in which thle coefficients are as follows:
e(1
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
age Treatment (New York: John Wiley and Sons,
Inc., 1940), p. 40.
3-23. The Whiney formula
0.(i
(t + 4)2/'
Remark: This formula is )based on excessive pre-
cipitation data at nine cities in Northeastern United
States.
Reference: S. Whiney, "I)iscussion on Maximum
Rates of Rainfall," Transactions, American Society
of Civil Engineers, Vol. 54 (1905), p. 203.
3-24. The Williams formula
Remarks: (1) These formulas were pre)pared fromi
Yarnell's data under the direction of Professor
E. W. Steel.
(2) The formulas give the average maximum
precipitation rates for durations up to 2 hours to lie
expected for the particular area and frequency.
Reference: E. W. Steel, Water Supply and Sewer-
age (New York: McGraw-Hill Book Co., Inc.,
1947), pp. 350-352.
3-21. The Talbot formula
1=
in which K = 105 and b =
K = 180 and b =
K = 360 and b =
+b
15 for ordinary
maximum
30 for maximum occur-
ring once in about fif-
teen years
30 for maximum
exceeded two or three
times per century
Remark: These formulas were developed for areas
east of the Rocky Mountains.
Reference: A. N. Talbot, "Rates of Maximum
Rainfall," Technograph No. 5, University of Illinois
(1891-92), p. 103.
3-22. The Webster formula
12
t0.6
Remark: This formula was said to be applicable
to ordinary conditions for Philadelphia, Pennsyl-
vania.
Reference: Harold E. Babbitt, Sewerage and Sew-
K
t+b
in which t = duration of the rainstorm in hours
b = 0.75
K = 3 for maximumi probable in short
periods
K = 6 for maximum probable in 100 years
K = 9 for maximum probable in long
periods of time
Remark: These formulas were developed for East-
ern India. However, the results are probably on tle
low side and would certainly he so for some parts
of India, and still more so for Ceylon.
Reference: G. B. Williams, Storage Reservoirs
(London: Chapman and Hall, 1937), p. 20.
H. GROUP 4: FREQUENCY FORMULAS
4-1. The Creager formula
In 0. 1 T
3
In 0. IT
3 _I
in wlich C = coefficient depending upon the char-
acteristics of the drainage area, equal
to 6,000 for areas most favorable to
large floods
Remark: The coefficient C has to be determined by
the judgment of the engineer.
Reference: W. P. Creager and J. D. Justin, Hydro-
electric Handbook (New York: John Wiley and
Sons, Inc., 1927), p. 55.
4-2. The Fuller formula
Q .ax = (1 + 21)D .)
Q= CDO.[ 2 - e-30I.0 " (
Q 9.. - .3 --. 1 -
VII. APPENDICES
in which Q = Qa (1 + 0.8 log T)
Qav = C D8
Remarks: (1) In this formula, Qa~ is the average
of the annual 24-hour flood discharge, Q is the
probable greatest average discharge for 24 consecu-
tive hours during a period of T years, or the maxi-
mum 1-day flood, and Qm,,, is the maximum flood
discharge; all quantities are in c.f.s.
(2) The expression for Qmax is approximate as it
was based on only 26 available records of the
maximum flood and the 24-hour flood. The maxi-
mum flood does not necessarily occur on the same
day as the 24-hour flood.
(3) This formula was developed primarily for
eastern streams of the United States.
Reference: W. E. Fuller, "Flood Flows," Transac-
tions, American Society of Civil Engineers, Vol. 77
(1914), p. 567.
4-3. The Horton formula - A
Qav= Qmax (1 - e-bTr.)
in which Qmax = maximum possible flood
Qav = average magnitude of flood
Te = average exceedance interval of an
event of magnitude Q
b and n = factors varying with locality
Remarks: (1) The value of Q,,m is to be assumed,
and Q,, b, and n are to be determined from ob-
served frequency data.
(2) From data of flood records for 70 years for
the Connecticut River at Hartford, Connecticut,
(1843-1917), Qax = 1.82Q,,, b = 0.255, and n=
0.54.
Reference: R. E. Horton, "Discussion of Flood
Flow Characteristics," Transactions, American So-
ciety of Civil Engineers, Vol. 89 (1926), p. 1086.
4-4. The Horton formula - B
Q = 4,021.5To .2
in which Q = flood discharge equalled or exceeded
in an average interval of T years
Remarks: (1) This formula was developed for
streams in Eastern Pennsylvania.
(2) Special cases were also developed as fol-
lows:
Q = 30T'"2D for the Neshaming (D = 1,393).
Q = 30 TO.27D for the Perkiomen (D = 152.0).
Q = 40 TO"I2D for the Tohickon (D = 102.2).
Reference: R. E. Horton, "Discussion of Flood
Flows," Transactions, American Society of Civil
Engineers, Vol. 77 (1914), p. 665.
4-5. The Lane formula
Q = Clog(T + b)
in which T = period in years in which a given Q
will be equalled or exceeded
C and b = constants
Remark: This formula was developed for New
England streams.
Reference: C. S. Jarvis, "Hydrology," Section 2 of
Handbook of Applied Hydraulics, edited by C. V.
Davis (New York: McGraw-Hill Book Co., Inc.,
1942), p. 111.
I. GROUP 5: ELABORATE DISCHARGE FORMULAS
5-1. The Adams formula
12 5
Q=CAI A21
in which C = 1.035
I - 1.0 or maximum intensity of rainfall
in inches per hour
S = slope in feet per thousand feet
Remark: This formula was developed from the
fundamental expression for a circular conduit flow-
ing full, and the assumption that one-half of the
precipitation, I inches per hour, will reach the sewer
at the time of maximum discharge.
References: J. W. Adams, Sewers and Drains for
Populous Districts (New York: D. Van Nostrand,
1880).
Emil Kuichling, "Storm Water in Town Sewer-
age," Transactions, Association of Civil Engineers,
Cornell University, Vol. I (1893-94), p. 41.
5-2. The Besson formula
Q = CA" -- I T G A"
For any drainage area,
Qmax = -r R C
R, C2
in which Qmax = maximum discharge
Qr = recorded discharge
R,, = maximum one-day rainfall
R, = recorded one-day rainfall
C = coefficient equal to the product of
the precipitation
R in inches, the topographic factor
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
T, and a factor G for ground sur-
face conditions. C, is for maxi-
mum condition and C, for re-
corded condition.
n = exponent which has been given
values varying from 0.5 to 0.83.
Reference: F. S. Besson, "Maximum Flood Flow
Prediction," The Military Engineer, Vol. 25, No.
143 (September and October, 1933), p. 424.
5-3. The Boston Society of Civil Engineers'
formula
Q = CR \/ A
or Q = C.,R \/ D
SVA _ 1,290 VD
in which C, = - and C, = ere
t is the time in hours of the flood
period
C, = 2.4 to 4 and C. = 60 to 100 for flat
streams with relatively large channel
pondage
, = 4 to 24 and (, = 100 to 500 for ordi-
nary conditions
(' = 20 to 40 and C, = 500 to 1,000 for
lmountainous regions
R = total flood runoff, inches on drainage
area
= 3 in. for occasional floods in New
England (25 to 75 yr. frequency)
= 6 in. for rare floods in New England
(50 to 200 yr. frequency)
= not over 8 in. for maximum floods in
New England
Remarks: (1) This formula gives the total runoff
and is based on floods in New England.
(2) This formula is based upon a concept that
peak flows tend to vary directly with the total
volume of flood runoff.
Reference: "Report of the Committee on Floods,
March 1920," Journal, Boston Society of Civil En-
gineers, Vol. 130 (September, 1930), p. 297.
5-4. The Burge formula (see The Dredge formula)
5-5. The Biirkli-Ziegler formula
Q = C(A - A
in which (C is a coefficient depending upon the char-
acter of the ground surface as follows:
Description of Area
Values of C
For densely built up area where streets,
walks and yards are paved and the re-
maining area is practically all roof area as
in downtown districts .................. 0.75
For areas adjacent to downtown district
where streets and alleys are paved and
yards sm all .......................... 0.70
For densely built-up residence districts
where streets are plaved and houses are
close together ........................ . 0.65
For ordinary residential areas ........... 0.55-0.65
For areas having small yards and a me-
dium density of population ............. 0.45 0.55
For sparsely built-up areas or those hav-
ing large yards.........................0.35 0.45
For suburbs with gardens and lawns and
macadamized streets ................... 0.30
For parks, golf courses, etc. covered with
sod and having no pavements........... 0.20
I = average rate of rainfall in inches per hour dur-
ing the heaviest storm, varying from 1.0 to 3.0
and conmmonly using 2.75 in the Middle West
Remarks: (1) This formula was first published in
"The Greatest I)ischarge of Municipal Sewers,"
(Grisste Abflussmnengen bei Stidtischen Abzugs-
kanilc) by the Swiss hydraulic engineer, A. Biirkli-
Ziegler, in Zurich in 1880. In 1881, it was intro-
duced to American technical literature in a report
on "Sewerage Works in Europe" by Rudolph Her-
ing to the National Board of Health.
(2) It is based on the expression of Hawksley's
formula.
(3) In deriving the formula, observations ex-
tended up to slopes of 10 feet per 100, but were
limited to small areas of less than 50 acres. The
formula was found especially applicable to sewer
design.
Reference: Rudolph Hering, "Sewerage Systems,"
Transactions, American Society of Civil Engineers,
Vol. 10 (1881), p. 362.
5-6. The Chamiier formula
Q = 5 C I A"-7
0r Q = 640 CI D"'."'
in which ( = coefficient of surface discharge, giv-
ing the proportion of rainfall that
VII. APPENDICES
may be expected to flow off the sur-
face
I = anticipated greatest rainfall inten-
sity in inches per hour for a duration
equal to the time of concentration
Remark: This formula was tested by Chamier on
streams in New South Wales along the Cootamun-
dra-Gundagai Railway having drainage areas of
from 20 acres to 400 sq. mi.
Reference: George Chainier, "Capacities Required
for Culverts and Flood-Openings," Proceedings,
Institution of Civil Engineers, Vol. 134 (1898), pp.
313-323 (p. 319).
5-7. The Craig formula
Q = 440 C W In - --
in which L = mean length of the drainage area in
miles
IV = mean width of the drainage area in
miles
C = (C,VR
where (C, = coefficient of discharge
V = velocity towards the culvert in feet
per second
R = depth of rainfall in inches
Remarks: (1) Generally, C = 0.68 to 1.95, varying
with rainfall and topography.
(2) The formula is based on Indian records.
Reference: J. Craig, "Maximum Flood-Discharge
from Catchment Areas, with Special Reference to
India," Minutes of Proceedings, Institution of Civil
Engineers, Vol. 80 (1884-1885), p. 201.
5-8. The Cramer formula-A
C CI R A S1'3
S57,600 + (27,000,000 C1 R A)1/'4
in which C = 186 for rough, natural basins of
rivers.
C = 697 for smooth, comparatively level.
and impervious areas such as well-
built cities.
R = mean annual rainfall in inches.
C, = a coefficient depending on the total
area A in acres, the flat area AI in
acres, which is likely to be inun-
dated in freshets, and the mean
annual rainfall R, such relation be-
ing expressed by:
Ā 1 *709 Ai
C = 1 - sin -tan-' 709 A- for the
AR9
simple case when A1 is distributed
in an approximately uniform man-
ner throughout the whole basin, and
C1 = 1 - sin -tan- 1,418 A if A
AR
is concentrated only at the lower
end of the basin.
S = the mean slope and declivity of the
whole basin in feet per thousand; or
(s8 + s2)/2L, where si is the average
altitude in feet, s2 the altitude of the
point of discharge, and L the average
distance in feet traveled by the
water from the boundaries of the
watershed to the point of measure-
ment.
Remarks: (1) The original form of the formula is
C3 R3 m A (S"2)'1
9 + (0.0658 m R3a A)1/'
in which C3, R3, m, A and S2 correspond to C, R,
C1, A, and S respectively, and A is expressed in
square miles.
(2) The original form was reduced by Kuich-
ling, for conditions on Mohawk River, to
80.6
1 + 0.1347 (A)L/
Reference: "Report of Sub-Committee of Roadway
Committee No. 1, Bulletin 131," Proceedings,
American Railway Engineering and Maintenance of
Way Association, Vol. 12, Pt. 3 (March, 1911), p.
498.
5-9. The Cramer formula-B
Q = 640 RC A%
Reference: Emil Kuichling, Annual Report on the
Barge Canal, New York, 1901.
5-10. The Dredge or Burge formula
2.03 A
L- 2/i
or
_ 1,3001)
L2/3
Remarks: (1) This formula, based on Indian flood
records, was used for the Madras Railway, India.
(2) If the area is taken as a rectangle having
dimensions of L by (1Ā½)L in miles, then the for-
mula reduces to Q = 1,030 D%.
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
Reference: J. T. Fanning, A Practical Treatise on
1]'ater-Sitpply Enginmcrinc (New York: i). Van
Nostrand Co., Inc., 1878), p. 66.
5-11. The Gregory formulas
Q = CI S' ; A 0.8-
in which CI = 2.8 for impervious surface; and
Q = 105 C (84 N/yAS + 25)
in which C = 0.10 to 0.54
Remarks: (1) This formula was developed for use
in New York, 1907.
(2) The formula was based on the rational
formula.
Reference: C. E. Gregory, "Rainfall and Runoff in
Storm Water Sewers," Transactions, American So-
cicty of Civil Engineers, Vol. 58 (1907), p. 474.
5-12. The Gregory and Arnold formula
S(3,600 t) 4"" ( r (
Q =- (1,000)-) -" L ) , ) (2) ~ "
in which C = coefficient representing the ratio of
the rate of runoff to the rate of rain-
fall
C, =constant, depending on shape of
drainage area and its manner of con-
centration. C, = v/V, in which v =
average velocity of water in feet per
second in traversing the length L
and V = velocity of Q at outlet of
drainage area, based on average val-
ues of C, and S for the area
C2 = constant, depending on shape and
condition of main channel of flow
I = average rainfall intensity, in inches
perl hour, for a period of t hours
L = length, in feet, which water must
traverse in running from the most
remote portion of the drainage area
to its outlet.
S = fall, in feet per 1,000 feet of main
channel of flow
n = a positive fractional exponent in
rainfall intensity formula
t = time of concentration in hours
n, = 1/(4 - n)
Remark: This formula was derived by the rational
formula with several important basin factors being
considered.
Reference: R. L. Gregory and C. E. Arnold, "Run-
off- Rational Runoff Formulas," Transactions,
American Society of Civil Engineers, Vol. 96
(1932), pp. 1038-1099.
5-13. The Gregory and Hcring formula
Q = C I A0rx 3 S.27
Remarks: (1) This formula was deduced by
Charles E. Gregory in 1907 from diagrams of run-
off to be expected in New York City prepared in
1889 by Rudolph Hering.
(2) The value of CI = 1.02 for suburban areas
to 1.64 for metropolitan areas.
Rleferences: C. E. Gregory, "Rainfall and Runoff
in Storm Water Sewers," Transactions, American
Society of Civil Engineers, Vol. 58 (1907), p. 458.
Leonard Metcalf and H. P. Eddy, American
Scwcrage Practice (Vol. I; New York: McGraw-
Hill Book Co., Inc., 1914), p. 235.
5-14. The Grunsky formula- A
For maximum urban storm-water flow:
5CIA
Q = V t-
For maximum stream flow from large areas:
3,200 CI A
0V=
Q ~ -- -
For general applications:
C, IA
Q = t"
Q-
in which C = coefficient as function of time = 60/
(60 + C, / t)
I = maximum rainfall in 1 hour, based
on the California record
t = critical time in minutes for continu-
ance of rainfall
C, = 0.5 for impervious areas
C, = 5.0 for mountainous areas
CI = 20.0 for rolling country
C, = 50.0 for flat country
Ci = 250.0 for sandy regions
C., = 3,500 and n = 0.5 for impervious
areas
C2 = 3,300 and n = 0.6 for mountainous
areas
C2 = 3,000 and n = 0.7 for rolling country
C, = 2,100 and n = 0.75 for flat country
C, = 600 and n = 0.8 for sandy regions
Remark: This formula was based on California
records.
VII. APPENDICES
Reference: C. E. Grunsky, "Rainfall and Runoff
Studics," Transactions, American Society of Civil
Engineers, Vol. 85 (1922), p. 67.
5-15. The Grunsky formula - B
C Ctl A
Qmax - -
in which QM,,x = maximum rate of discharge
t = time of concentration in hours
C = 0.586 and n = % for less than
0.33 hr.
= 0.782 and n = 1 for t greater
than 0.33 hr. and less than 64 hrs.
= 1.562 and n= 2/ for t greater
than 64 hrs.
C 1/(1 + C, ( t), where C, is a
factor dependent on the surface
conditions of the drainage basin
C, = 0.013 for impervious areas
= 0.25 for mountains
= 0.40 for rolling country
- 1.3 for flat country (ordinary
soil)
= 6.5 for sandy regions
Remark: The values of C, were suggested for
ordinary conditions in a temperate climate. They
should be increased in localities where the ground
may be frozen or water-logged, or where the maxi-
mum runoff occurs when heavy rain falls on snow.
Reference: R. L. Gregory and C. E. Arnold, "Run-
off -Rational Runoff Formulas," Transactions,
American Society of Civil Engineers, Vol. 96
(1932), p. 1146.
5-16. The Hawksley formula
Q = CAI AI-
in which C = 0.7
I = 1.0 or maximum intensity of rainfall
in inches per hour
Remark: This formula was derived from the
Hawksley original formula (1-3).
Reference: Emil Kuichling, "Storm Water in Town
Sewerage," Transactions, Association of Civil Engi-
neers, Cornell University, Vol. I (1892-93), p. 47.
5-17. The Hering formula
RVA RVA
640 L or 640 t
in which R = total runoff in inches during a storm
V = mean velocity of the stream
L = length of the river
t = time of concentration
Remark: In the original form of the formula, the
drainage area is expressed in square miles.
Reference: T. M. Cleemann, "Railroad Engineers'
Practice, Discussion on Formulas," Proceedings,
Engineers' Club of Philadelphia, Vol. I (1879), p.
146.
5-18. The Iszkowski formula
Q,, = (0.022 Ct + b C.,) R D
in which Q,,, = probable maximum flood discharge
in cubic meters per second
R = mean annual depth of rainfall in
meters
D = drainage area in square kilometers
C', = 0.20 for very flat, sandy, or swampy
areas to 0.65 for high mountainous
areas
C, = 0.035 for very permeable land cov-
ered with vegetation to 0.70 for
impervious rocky or frozen land,
without active vegetation, and cov-
ered with snow which will increase
the runoff by melting
b = 7.88 for D = 10 square kilometers
to 0.65 for D = 100,000 square
kilometers as set forth in a table
from which Kuichling has deduced
the approximate relation:
S 0.59 (11,150 + D)
818 + D
Remarks: (1) This formula was converted to Eng-
lish units by Kuichling as follows:
For a hilly territory with slightly permeable
soil and sparse vegetation:
S0.568 R ( ,129.5 D)
315.8 + D
For mountainous territory with rocky or frozen
soil:
S 0.848 R (4,147.4 + D)
Q. 315.8 + D
where Qnax = maximum discharge in cubic feet per
second
R = mean annual depth of rainfall in
inches
D = drainage area in square miles
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
(2) This formula was proposed by R. Iszkowski,
Chief Engineer of the Austrian Ministry of Public
Works in 1884. It was called an "induction formula
for estimating the normal and flood discharges,
based on the characteristics of the watershed."
Reference: Emil Kuichling, "Discussion of Flood
Flows," Transactions, American Society of Civil
Engineers, Vol. 77 (1914), p. 648.
5-19. The Kinnison and Colby formulas
D0l.95
Q = (0.000036 s.2.+ 124) -.
P)O.86
Q = (0.0344 st'-+ 200) Lo
Do.95
Q = (0.0595 .s'5+ 342) 0.7
no.9o
Q = (0.128 8s.'+ 1,800) L-O
LO.7
for minor
floods
for major
floods
for rare floods
for maximum
floods
in which Q = the peak discharge in c.f.s.
s = the median altitude of the drainage
basin in feet above the outlet
P = the percentage that lake, pond, and
reservoir surface is to the total
drainage area
L = average distance in miles which wa-
ter from runoff uniformly distributed
over the basin must, travel to the
outlet
Remark: These formulas were developed by the
U. S. Geological Survey for the Commonwealth of
Massachusetts.
Reference: H. B. Kinnison and B. R. Colby, "Flood
Formulas Based on Drainage Basin Character-
istics," Transactions, American Society of Civil En-
gineers, Vol. 110 (1945), p. 871.
5-20. The Lauterberg formula
Q = 0.0001285 C R A
in which Q = average discharge of a river in c.f.s.
R = total annual rainfall in inches
C = 0.20 for marshy soil
= 0.25 for level plains
= 0.30 for rolling land
= 0.35 for low hills
= 0.45 for hilly country like the Ar-
dennes, the Odenwald and the Eifel
= 0.55 for the Black Forest and the
Vosges
= 0.70 for high rocky mountains
Remarks: (1) This formula is based upon data
collected in Switzerland.
(2) In the original form, the drainage area is in
square kilometers and a coefficient of 0.03171 is
used.
Reference: Abstract in the Minutes of Proceedings,
Institution of Civil Engineers, Vol. 149 (1887), p.
392, from Lauterberg's Schweizerische Striimabflus-
singen (Bern: Huber and Co., 1876).
5-21. The Lillie formula
Q = VRC X(O L)
in which Q = discharge in c.f.s. at the moment of
peak flood
V = standard mean velocity in feet per
second
R = 2 + annual rainfall/15
C = 1.1 + log L
L = length of sectors of drainage area in
miles
0 = angle in degrees, at the discharge
point, of the sections into which the
catchment is divided. The sections
are in fan shape having a common
center meeting at the discharge point.
Remark: This formula was developed with refer-
ence to rivers in India.
Reference: G. E. Lillie, "Discharge from Catch-
ment-Areas in India, as Affecting the Waterways
of Bridges," Proceedings, Institution of Civil Engi-
neers, Vol. 217 (1924), p. 309.
5-22. The McMath formula
Q =CAI -
in which C = 0.20 for rural sections
= 0.30 for macadamized streets
= 0.75 for paved streets
= 0.75 for St. Louis, Missouri
I = 1.9 to 2.75 for maximum intensity of
rainfall in inches per hour; the latter
value was used for St. Louis.
S = slope of the ground surface in feet
per thousand; a value of 15 being
recommended for St. Louis
Remark: This formula was proposed for St. Louis,
Missouri.
Reference: R. E. McMath, "Determination of the
Size of Sewers," Transactions, American Society of
Civil Engineers, Vol. 16 (1887) p. 183.
VII. APPENDICES
5-23. The Parnley formula
Q = C I S'/4 A/,'
in which C = 0 to 1
I = rainfall intensity in in. per hr. for a
period of 8 to 10 minutes for the
Walworth Run River, Cleveland.
Use I =4 in. per hr. for the most
violent storms and also for the fur-
ther damage caused by the prevail-
ing direction of the storms.
Remark: This formula was used for the Walworth
Sewer of Cleveland, Ohio.
Reference: W. C. Parmley, "The Walworth Sewer,
Cleveland, Ohio," Transactions, American Society
of Civil Engineers, Vol. 55 (1905), p. 345.
5-24. The Pettis formula
Q =C (R W)1.2
in which Q = approximately the 100-year flood
peak in c.f.s.
C = 310 for the humid regions east of the
Mississippi and along the Pacific
Coast
= 100 for level prairie region of Illinois
= 200 for the semi-arid Rocky Moun-
tain region such as Colorado
= 40 in desert region
R = 100-year 1-day rainfall in inches
V = average width of the basin in miles
as determined by dividing the area
of the basin by the length of the
stream, neglecting minor sinuosities
Remarks: (1) This formula, also known as the
"width formula," is designed for use on unregulated
basins having areas greater than 100 but less than
40,000 sq. mi.
(2) It was found that the formula was applied
successfully to basins of the areas stated, ranging
from twice as long as they are wide to others which
are about nine times as long as they are wide.
(3) This formula was first given as
Q =328 (R W)'-25
in which R is the maximum rainfall in inches to be
expected on the basin within a period of six days,
on the average of once in 100 years. In 1934, the
formula was revised to use a general coefficient C
in place of 328 as shown above.
References: C. R. Pettis, Major, A New Theory of
River Flow, Corps of Engineers, U. S. Army, Balti-
more, Maryland, 1927.
C. R. Pettis, "Flood Probability Formula Modi-
fied to Simplify Application," Engineering News-
Record, Vol. 112 (June, 1934), pp. 804-805.
C. R. Pettis, "Relation of Rainfall to Flood
Run-off," Military Engineers, Vol. 28, No. 158
(1936), pp. 94-98.
5-25. The Possenti formula
SAR
Q=C R (A2+ A
in which C = coefficient with an average of 1.72
A, = flat area in acres
A2 = hilly area in acres
R = depth of 24-hr. rainfall in inches
L = length of the stream from its source
to the point of observation in miles
Remark: This formula was found satisfactory for
mountain streams of moderate size in the Appen-
nines.
Reference: W. E. Fuller, "Flood Flows," Transac-
tions, American Society of Civil Engineers, Vol. 77
(1914), pp. 564-617.
5-26. The Protodiakonov formula
Q,, = 16.67 (Idk - i) Ak
in which L, = design rainfall intensity in centimeters
per minute
k = climatic factor equal to the ratio of
the maximum rainfall intensity at the
given watershed to that at the center
of European Soviet Union
i = permeability of soil in centimeters per
minute to be determined experimen-
tally
The design rainfall intensity is computed by
I, = -
1 + 0.06t
in which t = duration of rainfall in minutes which
is equal to the time of concentration
or
t = 16.67(+L+ L
in which L, = length of the channel in kilometers
L, = half width of the drainage area in
kilometers
V, = velocity of flow in the channel in
meters per second
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
V, = velocity of overland flow in meters
per second
Remarks: (1) This formula was developed by
M. M. Protodiakonov and recommended for the
determination of runoff cue to intense storms on
defrozen soils in summer as stated in "The Specifi-
cations for Normal Runoff" of the U.S.S.R. Public
Transportation Committee in 1931 and Transpor-
tation Design Committee in 1938 and 1939.
(2) The value of Id is for a frequency of 500
years. For the design of culverts and small bridges,
the recurrence interval is in the order of 50 to 100
years. The corresponding design discharge should
be equal to the computed discharge divided by a
factor m, or
Qso-loo =
the values of in are as follows:
All types of small bridges 1.25
Culverts, inverted siphons, siphons, em-
bankments, drainage ditches with pos-
sible ponding or overflowing 1.50
Temporary structures 1.75
Drainage ditches without ponding 2.00
Reference: V. V. Lebedev, Hydrology and Hy-
drometry Manual (Soviet Hydrometeorological
Publication, 1955).
5-27. The Rhind formula
CSR D"
L
in which C = coefficient depending on R/L
S = average fall in feet per mile of bed
in a length of 3 miles above the point
of discharge
R = greatest annual rainfall
L = greatest length of drainage basin
n = a variable index
Remark: The formula appears to be founded on
observation of rivers with drainage areas exceed-
ing 41 sq. mi.
Reference: G. R. Hearn, "The Effect of Shape of
Catchment on Flood-Discharge," Proceedings, In-
stitution of Civil Engineers, Vol. 217 (1924), p. 289.
5-28. The Ribeiro formula
17FCC Akv\R
"" - \/A,1T
SAk + 10
L2
in which C1 = 1.28 - 0.07 L2
Ak
Ak = drainage area considered, in square
kilometers (total or partial water-
shed)
L = length of path of raindrop in kilo-
meters, from farthest point of the
watershed to point where flood is to
be computed
Qn = maximum flood discharge in cubic
meters per second
F = frequency factor, equal to 1.00 for a
maximum storm
R = mean (weighted) average annual
rainfall in inches over the watershed
C = 1.00 for pastures of cultivated
ground with vegetation
= 1.25 for barren or rocky slopes
= 0.75 for highly wooded ground
Reference: George Ribeiro, "Formula Presented
for Estimating Flood Discharges," Civil Engineer-
ing, Vol. 21, No. 11 (November, 1951), p. 661.
5-29. The U.S.S.R. NTK-NKPS formula
Qm = G C Ak
in which G = geographical factor, being equal to
the maximum design rainfall inten-
sity in centimeters per minute at the
site of the drainage basin, divided
by 16.67
C = coefficient depending on permeability
of soils, runoff conditions, and chan-
nel length and slope
Remarks: (1) This formula applies to drainage
areas less than 4G sq. km. when G > 15 and less
than 60 sq. km. when G < 15.
(2) For unfrozen porous soils (less than 1 meter
thick of sand, gravels, or limestone with crevices)
the computed discharge should be reduced by 50%.
When there is influence of forest on the drainage
basin, the reduction should not be greater than 20%
for drainage areas greater than 20 sq. km. When
the soils are impervious (compact clay or crystal-
line rocks with no crevices) or frozen, the computed
discharge may be increased 20%.
(3) Experience has shown that the discharge
computed by the formula is usually too high. Ac-
cording to the All-Soviet Standards (GOST), the
discharge thus computed should be multiplied by
0.7.
(4) This formula was proposed by the Technical
Research Committee of the U.S.S.R. Public Com-
munication Committee (NTK-NKPS) in 1928. It
VII. APPENDICES
has been used by the Design Division of tlhe
U.S.S.R. Department of Interior, Bureau of High-
way Administration, for the design of culverts and
small bridges.
Reference: V. V. Lebedev, Hydrology and Hy-
drometry Manual (Soviet Hydrometeorological
Publication, 1955).
5-30. The U.S.S.R. Scientific Academy formula
C H5/4 r5/4 C13/4 IJ/171 W /4
- 3 t/4 L3/4
3 tq,/4 L3/4
in which Q,, = Maximum discharge in cubic meters
per second
C = coefficient for maximum discharge
(for full maximum discharge or
Q1oo, C = 1)
H = average water-content of snow in
millimeters before melting and be-
coming runoff as indicated by a long
period observation (to be obtained
from a snow isohyetal map)
t, = shortest time for infiltration during
a 24-hour intensive rainfall (to be
obtained from an isohyetal map)
r = parameter for forestation, computed
by tile formula--- where
1 + AsI'/Ak
Ak' in square kilometers is the part
of the area Ak that is forested and
Ak is the sub-drainage area under
consideration
C, = roughness coefficient, equal to 6.5 for
areas without forest and 5.0 for for-
ested areas
S = average slope of the main channel
counting from the upstream edge of
the drainage area to the culvert
W- = average width of the drainage area
or AI/L
L = length of the outlet channel from the
edge of the drainage area to the
outlet
A, = total drainage area in square kilo-
meters
Reference: M. F. Sribnyi, "Method of Determin-
ing Maximum Flood Discharge from Its Relation
to the Area of the Watershed," Russian Scientific
Academy News (Isv. AN SSSR otd. tekh nauk)
No. 1 (January, 1952), p. 151.
5-31. The Switzer and Miller formula
Q=RC W"
in which Q = 24-hr. flood in c.f.s.
R = rainfall in inches
V = mean width of drainage basin in
miles, obtained by dividing the area
of drainage basin in square miles by
the length of tile main stream in
miles
C = 80
n= 1.5
Remarks: (1) The formula is based on a study of
47 rivers in the United States.
(2) When Q is expressed for peak flows in c.f.s.,
then C = 135 and n = 1.4.
Reference: F. G. Switzer and H. G. Miller, Floods
(Engineering Experiment Station Bulletin No. 13),
Cornell University, December, 1929, p. 11.
5-23. The Walker formula
CRD
L5/6
in which C = 4 to 30, being a maximum for drain-
age basins having impervious sur-
faces, little storage, steep slopes, lit-
tle vegetation, direct alignment of
waterways, etc., and minimum for
previous surfaces, much storage, flat
area, much vegetation, and water-
ways with irregular and meandering
alignment. Most values of C range
between 8 to 20 for average condi-
tions. A general average of C is
about 12.
R = mean, or normal, annual rainfall in
inches over the entire basin
L = straight line distance in miles from
point of discharge to center of grav-
ity of the basin
Reference: Thomas Walker, "Flood Discharge
Formulas," American Railway Engineering Associ-
ation Bulletin, Vol. 24, No. 248 (August, 1922),
pp. 23-26.
APPENDIX II. REFERENCES CITED
(References marked * are also annotated in Appen-
dix III)
1*. T. M. Cleemann, "Proper Amount of Water-
Way for Culverts," Proceedings, Engineers'
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
Club of Philadelphia, Vol. 1 (April 5, 1879),
pp. 146-149.
2*. A. M. Wellington, "Culvert Proportions," Edi-
torial, Railroad Gazette, Vol. 18 (September,
1886), pp. 629-630.
3. C. S. Jarvis, "Flood Flow Characteristics,"
Transactions, A4merican Society of Civil Engi-
neers, Vol. 89 (1926), pp. 985-1032.
4. C. S. Jarvis, "Average and Maxinum River
Iischarges," Appendix I in Low Dams, Na-
tional Resources Committee, 1938, pp. 393-426.
5*. A. N. Talbot, "The Determination of Water-
Way for Bridges and Culverts," Selected Papers
of the Civil Engineers' Club, Technograph No.
2, University of Illinois, 1887-8, pp. 14-22.
6. A. Biirkli-Ziegler. The Greatest Discharge of
Municipal Sewers (Grosste Abflussmenge bei
Stiidtischen Abzugkaniilen). Zurich: Orell,
Fiissli & Co., 1880.
7. Rudolph Hering, "Sewerage Works in Europe,"
Report to the National Board of Health, 1881,
and "Sewerage Systems," Transactions, Amer-
ican Society of Civil Engineers, Vol. 10 (1881 1,
pp. 361-384.
8. W. C. Curd, "Discussion of the Best Method
for Determiining the Size of Waterways," Ap-
pendix B of Report of Committee No. 1- on
Roadway, Proceedings, Amnerican Railrway En-
gineering and Maintenance of lWay Association,
Vol. 10, Pt. II (1909), pp. 991-993. The chart
entitled "Logarithmic Diagram for Area of Wa-
terway Required by Talbot's Formula" is be-
tween pp. 992 and 993.
9*. Bill Turner, "Graph for Talbot's Formula,"
Engineering News Record, Vol. 126, No. 25
(June 19, 1941), p. 971.
10. "Diagram for Solution of Talbot Formula for
Culvert Sizes," Handbook of Drainage and
Construction Products, Armco Drainage and
Metal Products Inc., 1958, p. 227.
11. C. B. McCullough, "Design of Waterway Areas
for Bridges and Culverts," Oregon State High-
way Commission Bulletin No. 4, July, 1934.
12*. C. B. Coe, "Talbot's Formula by Slide Rule,"
Civil Engineering, Vol. 9, No. 8 (August, 1939),
pp. 496-497.
13. Emil Kuichling, "The Relation Between the
Rainfall and the Discharge of Sewers in Popu-
lous Districts," Transactions, American Society
of Civil Engineers, Vol. 20 (1889), pp. 1-56.
14*. J. C. I. Dooge, "The Rational Method for
Estimating Flood Peaks," Engineering, (Lon-
don), Vol. 184 (September 6 and 20, 1957), pp.
311-313, 374-377.
15. T. J. Mulvaney, "On the Use of Self-Registering
Rain and Flood Gauges in Making Observa-
tions of the Relations of Rainfall and of Flood
Discharges in a Given Catchment," Transac-
tions, Institution of Civil Engineers of Ireland,
Vol. 4, Pt. 2 (1850-51), p. 18.
16. 1). E. Lloyd-Davies, "The Elimination of Storm
Water from Sewerage Systems," Minutes of
Proceedings, Institution of Civil Engineers,
Vol. 164 (1906), p. 41.
17*. D. B. Krimgold, "On the Hydrology of Cul-
verts," Proceedings, 26th Annual Meeting,
Highway Research Board, Vol. 26 (1946), pp.
214-226.
18. W. W. Horner and S. W. Jens, "Surface Runoff
Determination from Rainfall Without Using
Coefficients," Transactions, American Society of
('iiil Engineers, Vol. 107 (1942), pp. 1039-1075.
19. W. 1). Potter, "Surface Runoff from Agricul-
tural Watersheds," Surface Drainage, Highway
Research Board Report No. 11-B, 1950, pp.
21-35.
20. Merrill Bernard, "Modified Rational Method
of Estimating Flood Flows," Appendix A in
Low Dams, National Resources Committee,
1938, pp. 209-233.
21. R. L. Gregory and C. E. Arnold, "Runoff-
Rational Runoff Formulas," Transactions,
American Society of Civil Engineers, Vol. 96
(1932), pp. 1038-1099.
22. G. H. Bremner, James Dun, J. W. Alvord, A. N.
Talbot, and others. "Symposium on Methods of
Determining the Size of Waterway for Bridges
and Culverts," Journal, Western Society of
Civil Engineers, Vol. 11 (April, 1906), pp. 137-
190.
23*. W. D. Pence, "The Best Method of Determin-
ing the Size of Waterways" (Appendix B of
Report of Committee No. 1 on Roadway),
Proceedings, American Railway Engineering
and Maintenance of Way Association, Bulletin
108, Vol. 10, Pt. II (1909), pp. 967-1022.
24. C. 1). Purdon, "Discussion on Flood Flow Char-
acteristics," Transactions, American Society of
Civil Engineers, Vol. 89 (1926), p. 1090.
VII. APPENDICES
25*. W. G. Berg, "How to Determine Size and Ca-
pacity of Openings for Waterways," Report of
the 7th Convention, American Railway Bridge
and Building Association, 1897, pp. 86-100.
26. "Report of Sub-Committee of Roadway Com-
mittee No. 1, Bulletin 131," Proceedings, Amer-
ican Railway Engineering and Maintenance of
Way Association, Vol. 12, Pt. 3 (March, 1911),
pp. 481-528.
27*. T. M. Munson, "Formulas and Methods in
General Use by Engineers to Determine Runoff
from Watershed Areas," Texas Engineer, Vol.
4, No. 4 (April, 1934), p. 7.
28*. C. E. Ramser, "Brief Instruction on Methods
of Gully Control," U. S. Department of Agri-
cultural Engineering (mimeographed), August,
1933, pp. 16, 21-23, 26-27.
29*. C. E. Ramser, "Runoff from Small Agricul-
tural Areas," Journal of Agricultural Research,
Vol. 34, No. 9 (May 1, 1927), pp. 797-823.
30. A. F. Meyer. Elements of Hydrology (1st ed.).
New York: John Wiley and Sons, Inc., 1917.
Table 15.
31. D. L. Yarnell, "Rainfall Intensity Frequency
Data," U. S. Department of Agriculture Mis-
cellaneous Publication No. 204, August, 1935.
32*. D. B. Krimgold, "Runoff from Small Drainage
Basins," Agricultural Engineering, Vol. 19, No.
10 (October, 1938), pp. 439-446.
33*. D. B. Krimgold and N. E. Minshall, "Hydro-
logic Design of Farm Ponds and Rates of Run-
off for Design of Conservation Structures in the
Claypan Prairies," U. S. Department of Agri-
culture, Soil Conservation Service Research Re-
port SCS-TP-56, May, 1945.
34. C. L. Hamilton and H. G. Jepson, "Stock-Water
Development: Wells, Springs, and Ponds,"
Farmers' Bulletin No. 1859, U. S. Department
of Agriculture, 1940, p. 39.
35. "Engineering Handbook," U. S. Soil Conserva-
tion Service, Milwaukee Office, 1942.
36*. R. B. Hickok, R. V. Keppel, and B. R. Raf-
ferty, "Hydrograph Synthesis for Small Arid-
land Watersheds," Agricultural Engineering,
Vol. 40, No. 10 (October, 1959), pp. 608-611,
615.
37. "Hydrology," Supplement A, Section 4 of Engi-
neering Handbook, U. S. Department of Agri-
culture, Soil Conservation Service, 1957.
38*. R. R. Rowe and R. L. Thomas, "Comparative
Hydrology Pertinent to California Culvert
Practice," California Highways and Public
Works, Vol. 20 (September, 1942), pp. 6-11.
39. California Culvert Practice. Division of High-
ways, Department of Public Works, State of
California, 1st ed., 1944, 2nd ed., 1953.
40*. I. E. Houk, "Hydraulic Design of Bridge Wa-
terways," Engineering News Record, Vol. 88,
No. 26 (June 29, 1922), pp. 1071-1075.
41*. G. P. Springer, "Adequate Waterways for Cul-
verts and Bridges," Proceedings, 17th Annual
Road School at Purdue University, January
19-23, 1931, Purdue University Engineering
Bulletin, Vol. 15, No. 2 (March, 1936).
42*. F. W. Greve, "Hydrological Factors in the
Design of Culverts and Small Bridges," Roads
and Bridges, Vol. 81, No. 4 (April, 1943), pp.
32, 66, and 68.
43*. F. T. Mavis, "Reducing Unknowns in Small
Culvert Design," Engineering News Record,
Vol. 137, No. 2 (July 11, 1946), pp. 51-52.
44*. J. C. Merrell. "Hydrology for Highway Struc-
tures," Proceedings of the Ohio Highway En-
gineering Conference, April 3-5, 1951 (Engi-
neering Experiment Station Bulletin No. 145).
Columbus, Ohio: The Ohio State University,
1951. Pp. 64-77.
45*. J. F. Exum. "Waterway Area for Culverts and
Small Bridges," Proceedings of the Ohio High-
way Engineering Conference, April 3-5, 1951
(Engineering Experiment Station Bulletin No.
145). Columbus, Ohio: The Ohio State Uni-
versity, 1951. Pp. 61-63.
46*. C. F. Izzard, "Importance of Drainage Design
in Highway Construction," Proceedings of the
3rd California Conference on Street and High-
way Problems, Institute of Transportation and
Traffic Engineering, University of California,
1951.
47*. C. F. Izzard, "Peak Discharge for Highway
Drainage Design," Transactions, American So-
ciety of Civil Engineers, Vol. 119 (1954), pp.
1005-1015.
48*. H. G. Bossy, "Simple Methods for Hydraulic
Design of Culverts," Proceedings, Southeastern
Association of Highway Officials, October, 1952,
pp. 34-46.
49. "Highway Practice in the United States of
BULLETIN 462. HYDROLOGIC DETERMINATION OF WATERWAY AREAS
America," Public Roads Administration, 1949,
pp. 136-137.
50*. "Basic Principles of Highway Drainage," Hy-
draulic Information Circular No. 1, U. S. Bu-
reau of Public Roads, 1951.
51*. W. D. Potter, "Rainfall and Topographic Fac-
tors that Affect Runoff," Transactions, Ameri-
can Geophysical Union, Vol. 34 (1953), pp.
68-73.
52*. W. D. Potter, "Use of Indices in Estimating
Peak Rates of Runoff," Public Roads, Vol. 28,
No. 1 (April, 1954), pp. 1-8.
53. E. M. West and W. H. Sammons, "A Study of
Runoff from Small Drainage Areas and the
Openings in Attendant Drainage Structures,"
University of Kentucky Highway Research
Laboratory Report No. 2, July, 1955.
54*. George Chamier, "Capacities Required for
Culverts and Flood Openings," Proceedings, In-
stitution of Civil Engineers, Vol. 134 (1898),
pp. 313-323.
55*. C. S. Jarvis, "General Formula for Water-
ways," Engineering World, Vol. 29 (August,
1926), pp. 79-83.
56*. Tate Dalrymple, "Hydrology in Design of
Bridge Waterways," North Carolina Engineer,
July, 1952.
57. "Monthly Precipitation and Runoff for Small
Agricultural Watersheds in the United States,"
U. S. Department of Agriculture, Agricultural
Research Service, since July, 1957.
58. "Annual Maximum Flows from Small Agricul-
tural Watersheds in the United States," U. S.
Department of Agriculture, Agricultural Re-
search Service, June, 1958.
59*. F. F. Snyder, "Synthetic Flood Frequency,"
Proceedings of the American Society of Civil
Engineers, Journal of the Hydraulics Division,
Vol. 84, No. HY5, Pt. 1 (October, 1958), pp.
1-22.
60. "Selected Runoff Events for Small Agricultural
Watersheds in the United States," U. S. Depart-
ment of Agriculture, Agricultural Research
Service, January, 1960.
61*. N. E. Minshall, "Predicting Storm Runoff on
Small Experimental Watersheds," Proceedings
of the American Society of Civil Engineers,
Journal of Hydraulics Division, Vol. 86, No.
HY8, Pt. 1 (August, 1960), pp. 17-38.
62*. W. D. Potter, "Peak Rates of Runoff from
Small Watersheds," Hydraulic Design Series
No. 2, U. S. Department of Commerce, Bureau
of Public Roads, U. S. Government Printing
Office, April, 1961.
63*. G. A. Tilton and R. R. Rowe, "Culvert Prac-
tice in California," Proceedings, Twenty-third
Annual Meeting, Highway Research Board,
Vol. 23 (1943), p. 205.
64. V. T. Chow, (Chairman) and others, "Report
of the Committee on Runoff, 1955-56," Trans-
actions, American Geophysical Union, Vol. 38,
No. 3 (June, 1957), pp. 379-384.
65. V. T. Chow, "Hydrologic Studies of Floods in
the United States," in Floods, Vol. III of Sym-
posia Darcy, International Association of Sci-
entific Hydrology, Publication No. .2, 1956,
pp. 134-170.
66. L. K. Sherman, "Stream Flow From Rainfall
by the Unit-Graph Method," Engineering News-
Record, Vol. 108 (April 7, 1932), pp. 501-505.
67*. E. F. Brater, "The Unit Hydrograph Principle
Applied to Small Watersheds," Transactions,
American Society of Civil Engineers, Vol. 105
(1940), pp. 1154-1178.
68. D. L. Yarnell, "Rainfall Intensity-Frequency
Data," U. S. Department of Agriculture Mis-
cellaneous Publication 204, 1936.
69. "Rainfall Intensity-Duration-Frequency Curves
for Selected Stations in the United States,
Alaska, Hawaiian Islands, and Puerto Rico,"
U. S. Weather Bureau Technical Paper No. 25,
December, 1955.
70. "Rainfall Intensity-Frequency Regime, Part 4
--The Ohio Valley," U. S. Weather Bureau
Technical Report No. 29, 1957.
71. V. T. Chow. Frequency Analysis of Hydrologic
Data with Special Application to Rainfall In-
tensities (Engineering Experiment Station Bul-
letin No. 414). Urbana, Ill.: University of
Illinois College of Engineering, 1953.
72. V. T. Chow, "Hydrologic Studies of Urban Wa-
tersheds, Rainfall and Runoff of Boneyard
Creek, Champaign-Urbana, Illinois," Univer-
sity of Illinois Civil Engineering Studies, Hy-
draulic Engineering Series No. 2, November,
1952.
73. F. A. Huff, "Rainfall Intensity-Frequency Data
for Champaign-Urbana, Illinois," Illinois State
Water Survey Division Circular No. 28, 1949.
VII. APPENDICES
74. F. A. Huff and J. C. Neil, "Frequency Rela-
tions for Storm Rainfall in Illinois," Illinois
State Water Survey Division Bulletin 46, 1959.
75. "Rainfall Frequencies," Illinois State Water
Survey Division Technical Letter No. 1, 1959.
76. "Rainfall Frequencies for 5 to 60 Minutes,"
Illinois State Water Survey Division Technical
Letter No. 4, 1960.
77. R. H. Shaw, G. L. Barger, and R. F. Dale,
"Precipitation Probabilities in North Central
States," North Central Region Publication No.
115, Agricultural Experimental Stations of Illi-
nois, Indiana, Iowa, Kansas, Michigan, Minne-
sota, Missouri, Nebraska, North Dakota, Ohio,
South Dakota, Wisconsin and the U. S. Depart-
ment of Agriculture cooperating, University of
Missouri Agricultural Experiment Station, June,
1960.
78. "Maximum Recorded United States Point
Rainfall for 5 Minutes to 24 Hours at 207 First
Order Stations," U. S. Weather Bureau Techni-
cal Paper No. 2, April, 1947.
79. "Maximum Station Precipitation for 1, 2, 3, 6,
12, and 24 Hours," U. S. Weather Bureau Tech-
nical Paper No. 15, Part XX: Indiana, (1956);
Part XXI: Illinois, (1958); and Part XXII:
Ohio, (1958).
80. "Maximum 24-Hour Precipitation in the United
States," U. S. Weather Bureau Technical Paper
No. 16, January, 1952.
81. J. T. Riedel, J. F. Appleby, and R. W. Schloe-
mer, "Seasonal Variation of the Probable
Maximum Precipitation East of the 105th
Meridian for Areas from 10 to 1,000 Square
Miles and Durations of 6, 12, 24, and 48
Hours," U. S. Weather Bureau and Corps of
Engineers Hydrometeorological Report No. 33,
April, 1956.
82. W. D. Mitchell, "Unit Hydrographs in Illinois,"
U. S. Geological Survey and Illinois Division
of Waterways, 1948.
83. W. D. Mitchell, "Floods in Illinois: Magni-
tude and Frequency," U. S. Geological Survey
and Illinois Division of Waterways, 1954.
84. H. L. Wascher, J. B. Fehrenbacher, R. T. Odell,
and P. T. Veale, "Illinois Soil Type Descrip-
tions," AG-1443, Department of Agronomy,
Agricultural Experiment Station, University of
Illinois, 1950.
85. S. A. Changnon, Jr., "Second Progress Report,
Illinois Cooperative Project in Climatology, 1
July 1955 through 30 June 1956," Illinois Water
Survey Circular No. 57, 1956.
86. F. A. Huff and J. C. Neill, "Frequency Rela-
tions for Storm Rainfall in Illinois," Illinois
State Water Survey Bulletin 46, 1959.
87. R. Morgan and D. W. Hullinghors, "Unit Hy-
drographs for Gaged and Ungaged Watersheds"
(Unpublished manuscript), U. S. Engineers
Office, Binghamton, New York, July, 1939.
88. F. F. Snyder, "Synthetic Unit-Graphs," Trans-
actions, American Geophysical Union, Vol. 19
(1938), pp. 447-454.
89*. Z. P. Kirpich, "Time of Concentration of
Small Agricultural Watersheds," Civil Engi-
neering, Vol. 10, No. 6 (June, 1940), p. 362.
90. B. D. Richards. Flood Estimating and Control.
London: Chapman and Hall, 1955.
91. J.. . O'Kelly, "The Employment of Unit Hy-
drographs to Determine the Flows of Irish
Arterial Drainage Channels," Proceedings, In-
stitution of Civil Engineers, Vol. 4, Pt. III
(August, 1955), pp. 365-412.
92. J. E. Nash, "Frequency of Discharges from Un-
gaged Catchments," Transactions, American
Geophysical Union, Vol. 37, No. 6 (December,
1956), pp. 719-725.
93. J. E. Nash, "The Form of the Instantaneous
Unit Hydrograph," Comptes-Rendus et Rap-
ports, Assemblge de Toronto 1957, International
Association of Scientific Hydrology, Vol. 3,
Publication No. 45 (1958), pp. 114-121.
94. J. C. I. Dooge, "A General Theory of the Unit
Hydrograph," Journal of Geophysical Research,
Vol. 64, No. 2 (February, 1959), pp. 241-256.
95. B. L. Golding and I). E. Low, "Physical Char-
acteristics of Drainage Basins," Proceedings,
American Society of Civil Engineers, Journal
of Hydraulics Division, Paper 2409, Vol. 86,
No. HY3 (March, 1960), pp. 1-11.