
It is reasonable to assume that the segments of
duration DT of the rainfall hyetograph are straight lines.
The accuracy of this assumption is greater when the
increments of time DT become smaller.
Two conditions have to be satisfied at the point of
the intersection A:
(a) rainfall intensity at A = infiltration rate at A
(b) accumulated mass precipitated until A = accumu
lated mass infiltrated until A.
From the condition (a):
or
so
Kt
i(I) +
[id + i)
t =  i in
id) +
fo  fc
[id + i)  id)]  f
fo  fc
if and only if the argument of the logarithm is between
0.0 and 1.0.
From the condition (b):
f£,^
60 T
60K
(le"Kt) =
x
60
(3)
(4)
where mi(I) = mass precipitated until a time (IDT)
and i(I) = ordinates of the rainfall intensity curve.
Each time increment DT can be divided in as many n
increments as desired. Therefore x = DT/n where n = 1,2,..
The computer will solve equation (3) for 1=1 and
2DT „ = (n1) DT. fQr M and
x = 0 x =
x u, x
n
X=
x
DT
x = 0 , x = — ,
n
fn—
x = 
x
DT
n
; and so on . The time t
which will be found for each combination of I and x will
be substituted (together with the corresponding values of
I and x) into equation (4) .
238

When equation (4) is satisfied, then the computation
will stop and the values of t, I, and x will be stored.
The computation of the actual infiltration curve will
follow, and the stored value of t will be used to start
the calculation of the surface retention.
239

5. SURFACE RETENTION
Surface retention includes interception, depression
storage, and evaporation during a storm.
Interception by vegetation may range from 0.01 to 0.5
inches in forest areas, but is not usually significant in
urban drainage.
Evaporation is of little significance for the short
rainfall duration encountered in urban storm drainage
design.
Therefore, for simplicity of the model, interception
and evaporation will be assumed negligible for an urban
area.
Some investigators have assumed that all depressions
must be filled before overland flow begins. The real
situation is that almost immediately after the beginning
of the rainfall excess, the smallest depressions become
filled and overland flow begins. Most of this water in
turn fills larger depressions, but some of it follows an
unobstructed path to the collecting gutters.
Linsley, Kohler, and Paulhus [14] recommended an
exponential relationship
V
1  e
N(P  F)
(5)
where V = volume of water in depression storage at any
time throughout a storm, in inches
S, = total depression storage capacity of the basin,
in inches
P = accumulated volume of water precipitated in
inches
F = accumulated volume of water infiltrated in
inches
N = storage constant
240

Assuming that the initial increment of rainfall
excess is completely retained by depressions:
dV
d(p  F)
= 1 when (P  F) is near zero.
Then by differentiation can be found that
N _ 1
^\j _ ^i"
(6)
If the depression storage supply is s (in/hr), then:
•t
s dt = V
0
where t is time in hours from the beginning of the rainfall
excess .
Then
or
s =
s =
d
dt
__d
dt
P  F
Sd
(P  F) e
P  F
but
therefore
dt
(P  F) = (i  f)
(in/hr)
P  F
s = (i  f) e
(7)
Of course on impervious areas where infiltration is
zero and also accumulated volume of water infiltrated does
not exist, equations (5) and (7) become
241

V
3d
1  e
s = i e
Horton [15] stated that the depression storage
"commonly ranges from 0.125 to 0.75 inches for flat areas
and from 0 .5 to 1.5 inches for cultivated fields and for
natural grass lands or forests". On moderate or gentle
slopes he estimated that pervious surface depressions
"can commonly hold the equivalent of 0.25 to 0.5 inch depth
of water and even more on natural meadow and forest land".
Hicks [16] has used depression storage losses of 0.20
inch for sand, 0.15 inch for loam, and 0.10 inch for clay,
considering high rates of rainfall and runoff.
Recent gagings show depression storage to be about
0.3 inch in forest litter, 0.2 inch in good pasture, and
0.05 to 0.10 inch in smooth cultivated land [17].
If there are no data available, the overall de
pression storage capacity can be assumed equal to 0.25
inch on pervious urban areas which usually present
individual depressions with depths from 0 to 0.5 inches.
The overall depression storage on impervious urban areas
can be assumed equal to 0.0625 inch with the depth of de
pressions ranging from 0 to 0.125 inches. These assump
tions, when there is lack of data about the basin under
consideration, seem reasonable from observations made
during rainfalls [4], [9] , [18] .
A typical computer output of the developed computer
program (Appendix III) simulating infiltration and
depression storage supply is presented graphically in
Figure 138.
Subtracting the calculated values of infiltration
and depression storage supply from the rainfall intensity
at any time, the rainfall excess which produces the over
land flow is obtained.
242

O
CM
O
O
*
o
243

6. OVERLAND FLOW
En route to a channel or gutter/ the water is desig
nated as overland flow.
Because of the complex nature of flow, models are
generally simplified to two dimensions. This was the
approach used by Keulegan (1944) who first worked on the
spatially varied unsteady flow problem.
Horton [19] and Izzard [3] studied unsteady flow
across sloping plane surfaces. However, Horton's equation
for overland flow has limited experimental support, and
Izzard^s dimensionless hydrograph is limited to situations
where iL < 500, where i is the rainfall intensity in in/hr
and L is the length of overland flow in ft.
In the following the build up of the flow over a
sloping plane is treated starting with the partial differ
ential equations of momentum and continuity governing the
twodimensional overland flow on a plane surface with
vertical inflow (rainfall excess).
Subtracting retention and infiltration from the rain
fall intensity, the overland flow supply is obtained. As
this supply increases, a sheet of water builds up over the
surface and the water starts to flow toward the collecting
gutter. This movement of water is overland flow and the
volume of water on the surface is the surface detention
Continuity Equation
Considering flow of unit width on a surface with
small bottom slope (sin B = 0) and uniform velocity dis
tribution, the continuity equation can be written:
J II I Id)
mi
^
Figure 139. OVERLAND FLOW
244

~Sx
.
9A
£Ft
= i Ax  f Ax  s Ax
or
la
9x
9t
= ifs
because Area A = y • 1, where y is the depth of flow in ft,
g is the discharge in cfs/ft of width, and (i  f  s) is
the inflow or supply rate in cfs/sq.ft.
Momentum Equation
The momentum equation can be written in the following
form, neglecting the raindrop momentum due to the impact of
the raindrops on the surface
9V
Tt
9V
8y _
9x
= (ifs)
PY
+ g sinO (9)
where V is the velocity of flow
r is the shear stress at the bottom
o
p is the water density
b is the width of flow perpendicular to the direc
tion of flow. For overland flow b» y, so £ = 0.
In comparison with gravity, the rainfallinfiltration
process has a negligible effect upon the flow dynamics [20]
Therefore the term (ifs) can be omitted from the
momentum equation.
An order of magnitude analysis for overland flow can
lead to the conclusion that free surface slope and inertia
terms in equation (9) are negligible in comparison with
those of bottom slope and friction.
245

Equation (9) then reduces to the wellknown relation
for steady uniform flow in a wide channel:
TQ = Y y sine (10)
where y is the .specific weight of water.
Laminar Flow
For laminar flow Newton's law of viscosity applies:
dv
dy'
relating the dynamic viscosity y and the shear stress T
at a distance y' from the bottom. Equation (10) can be
written for any laminar layer:
(11)
= Y
~ y1 ) sin9
or
T =^~ (y  y') sine
where v is the kinematic viscosity in (ft /sec)
Then from equations (11) and (12):
dv = g
sin9
_
V
_ gsinB
(y  y1 )
(yy. 
v %*J 2 '
Therefore the average velocity V will be:
V =
so q =
1 P
7
Jo
Vdy =
(12)
3v
(13)
246

Turbulent Flow
For turbulent flow, Manning's equation can be used,
considering the hydraulic radius R equal to the depth y.
q = 1..49
n
(14)
with q (cfs/ft width) and y (ft).
Equations (13) and (14) are of the general form
q = K ym (15)
(a) for laminar flow!
m = 3 and K =
[— i
[sec.
_
ft
2. 2
with g in (ft/sec ) and V in (ft /sec)
(b) for turbulent flow:
m = and K = 1.49
Vsine
n
sec
Equation (15) is a good approximation for discharge
of surface flow for practical purposes, supported by
Keulegan [21], Chow [22], and Eagleson [20].
Criteria of Flow Regime
Horton [23] believed that the Reynolds criterion is
not satisfactory for sheet flow over relatively rough
surfaces. He found that the point of equal velocities
represents the minimum amount of energy capable of main
taining turbulent flow.
Equating the velocities for laminar and turbulent
flow, the following relationship is obtained:
gsin9
v 3
1.49 2/3
n y
or
2/3
32.17 n y
VsinG
2/3
247

Substituting the above in Manning's equation.we get:
v
V =
4.83 n2 y2/3
(16)
The flow cannot be turbulent if the velocity is less
than that given by equation (16) .
Therefore because q = Vy, turbulent flow occurs if:
q <
v
1/3
(17)
4.83 n
otherwise the flow is laminar .
Izzard (1946) believed the overland flow to be
turbulent for:
i L > 500
where i = supply rate (in/hr)
L = length of overland flow (ft)
But the experimental data of Izzard and Augustine and
of Ekern (1950) support the hypothesis that rainfall
causes the flow to be turbulent even when i L «: 500.
Field investigations have shown that overland flow in
thin sheets may be entirely turbulent, or partly turbulent
and partly laminar. For smooth surfaces, it appears that
laminar flow exists with the possibility of changing from
laminar to turbulent, and vice versa, within a short
distance,. For subdivided flow through grass, most of the
experimental data indicate that the flow is laminar to
some extent and may be represented by a condition of 75%
turbulent flow, in which case the profile of overland^
flow can be assumed parabolic [24] with an equation like
(15), having an m = 2.
Also experimental measurements of surface detention
show that high intensity rainfalls often yield Reynolds
numbers that indicate turbulent flows.
However, because the division between the laminar and
turbulent ranges is difficult to establish and because of
the nature of urban areas, the turbulent range equations
have been finally selected in the development of the
248

Cincinnati Urban Runoff Model.
Overland Flow Equations
Overland flow can be defined by the system:
9t
ax
(8)
= Ky11
(15)
These equations constitute a Kinematic wave problem
which has been solved by Eagleson [20] for s = 0, i and f
constants in time and space, using the method of
characteristics.
Here, an attempt has been made to solve equations (8)
and (15) considering rainfall intensity, infiltration, and
depression storage supply as variables of time.
The only rigorous general methods for simulating un
steady overland flow are finite difference techniques for
the numerical solution of the governing partial
differential equations. However, the accuracy to be
gained by using so laborious methods for overland flow is
still subject to question because of the limited accuracy
of the basic field data. •
Therefore an analytical method based finally on an
empirical relationship between outflow depth and detention
storage [8] has been chosen in connection with a routing
equation.
From the continuity equation (8), at equilibrium:
ay _
Tt ~
0
and q = rx
(18)
where r = (i 
2
is the overland flow supply rate in
(cfs/ftz) and q is the discharge at equilibrium in cfs/ft of
width at any section x on the flow plane. Therefore the
outflow at equilibrium is q = rL.
249

Also as Wolf [25] stated, the change in discharge as
a function of x on a uniformly sloping plane must be zero,
before local equilibrium is reached. Therefore the depth
at any point on the plane prior to local equilibrium
(Figure 140) is
y = rt
(19)
Figure 140. OVERLAND FLOW ON A UNIFORMLY
SLOPING PLANE
Between time t = 0 and t = t (where t = time to
s s
equilibrium in sec), the total inflow will be equal to the
total runoff (outflow) plus the surface detention (storage)
in units ft /ft of width.
The total inflow during this time will be:
I = r L t
(20)
The surface detention will be:
ydx
but equation (15) gives y = )/ and also the discharge
at equilibrium from equation (18) is q = rx
SO
/'
Jo
dx
250

(21)
Accepting that the total runoff is some fraction (a)
of the total inflow:
R = arLt
but also R
_r
~4>
qdt
From equation (19), from t
at x = L should be:
v = (—) v
V *
Then equation (15) becomes:
_, m T, / t»m m
q = Ky = K (^—) y
e
Therefore equating (22) and (23):
m
arLt = K
(22)
(23)
0 to t = te/ the depth
e tmdt
m
from where arL =
m + 1
but
qe » rL =
m
so
m + 1
Then the total runoff will be R =
(24)
(25)
The necessary time to reach equilibrium now can be
found from the equation: I = D + R substituting the ex
pressions of equations (20) , (21) , (25). Therefore:
251

rLt =
rje
K
from where
K
1/m
rLt
(26)
5 I/IT
For turbulent flow has been found m=j and K1.486 n
where S is the slope in ft/ft equal to the sine for small
slopes.
Then from equation (24): a = 0.375
From equation (21): Dg =
From equation (26): t =
0.493 r
°'6
1 ' 6
0.01315 L°'6 n°'6
Where: D = detention storage at equilibrium in ft /ft of
e width
t = time to reach equilibrium in minutes
e
2
r = overland flow supply rate in cfs/ft
S = slope of the plane in ft/ft
L = length of overland flow in ft
n = Manning's roughness coefficient
Detention storage in inches depth per unit area and
overland flow supply rates in inches per hour _ are more easy
to visualize. Therefore changing in these units:
De =
0.009792
n°  6 L° ' 6
0.94 L
S
0.6
0.3
(27)
0.4 C0.3
JT O
252

where now DQ (in/unit area), tQ (min), r (in/hr), L (ft),
S (ft/ft). ,
The rate of discharge from overland flow based on
Manning's equation is:
a = 1'486 v5/3 S1/2
^ y to
where q in cubic feet per second per foot of width and y
the depth in feet at the lower edge of the flow plane.
Rewriting Manning's equation with q in inches per
hour per unit area, y in inches, and L in ft:
q .
l/2
(28)
The most satisfactory empirical relationship found
between outflow depth and detention storage for repro
ducing experimental hydrographs [8] is:
y =
D 1.0
+ •0.6 (5)
e
(29)
where y is the outflow depth in inches
DS is the detention storage required at equilibrium
for the current rate of inflow in inches per unit
area, calculated from equation (27)
D is the current detention storage in inches per
unit area
During recession flow, when D is less than D, the
ti
ratio D/Dg in equation (29) is assumed to be one .
Then equation (28) becomes
q
= 1020.7 sl/2 D5/3 r
with units q (in/hr/unit area)
0.6
5/3
(30)
L (ft) and D, D (in/unit area)
253

To determine the overland flow runoff hydrograph, a
storage routing procedure is used.
The storage equation can be written:
where r is the inflow (overland flow supply, known from
the infiltration model), q is the outflow, dD is the change
in detention storage, and dt is the time increment in
minutes.
This equation can be written in an incremental form
form [26]
:1 + r2 _ ql + q2 _
60
D2  D]
At
or
D
60
AT
(31)
where the subscripts 1 and 2 indicate values at the begin
ning and the end of the routing increment At.
Combining equation (31) with equation (30) we get:
D.
—
At
nL
D
,5/3
D.
+ 60
At
(32)
where D can be found for each corresponding r from
e
equation (27).
Now all the known values are at the lefthand side of
equation (32). Starting at the first time increment with
q_ =0, D, =0 the value of D« is found from equation (32)
and then q2 is found from equation (30). Then the calcu
lated values of q2, D2 become q^ and DI for the subsequent
period and the above process is repeated giving the hydro
graph of the overland flow in inches per hour which can be
directly compared to the rainfall rates.
254

However, in order to calculate the discharge entering
into the gutters due to the overland flow supply, it is
easier to have available the overland flow in cubic feet
per second per foot of width. This can be done by
multiplying the values of q found from the routing pro
cedure by a factor, i. e.
cfs
in/hr
ft width
43200
unit area
Where L is the length of the overland flow.
This method has the advantage in using the Manning ' s
roughness coefficient for which there are, a lot of experi
mental data available for different ground covers (see
Appendix I) .
This method agreed satisfactorily with the well known
Izzard's method [3], [27]. An example output is presented
in Figure 141. Overland flow rates by using Izzard's
method are less during the beginning of the rainfall excess
and higher near the peak than these calculated by using the
Urban Runoff Model. But finally the volume of water which
ran off is approximately the same (areas under the overland
flow curves) . The differences can occur due to the fact
that Manning's n =0.35 may not well correspond to Izzard's
C = 0.06.
Several computer runs have shoxvn that the Cincinnati
Urban Runoff Model simulating the overland flow, finally
comes out with a volume of water ^ which ran off, approxi
mately equal to the total volume of water which was
available for overland flow. Differences of less than 5%
proved that the model follows adequately the continuity
principle.
A brief sensitivity analysis has been done for the
overland flow model which proved ever more its accuracy.
An example is presented in Figure 142. For bigger
Manning's roughness coefficients, the peaks of overland
flow hydrographs tend to be higher and to occur at earlier
times. For bigger lengths of overland flow the peaks tend
to be lower and to occur at a later time, resulting in a
flatter overland flow hydrograph.
The described overland flow model offers simplicity
and calculating speed compared with other more exact
methods, while attempting to maintain a reasonable
approximation to physical behavior.
255


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257

7. GUTTER FLOW
Gutters will gather overland flow in a continuous
fashion along their lengths, to feed the catchment's inlets
and catch basins.
Experiments by W. I. Hicks, introducing equal
increments of flow at regular intervals along the length of
a gutter, showed the difference between the watersurface
slope and the construction grade. The upstream water slope
was much flatter. ;
The type of flow in roadside gutters is spatially
varied unsteady flow with increasing discharge.
In this type of flow, the energy loss is of relatively
high magnitude and uncertainty, and is due to the turbulent
mixing of the added water and the water flowing in the
channel [26]. The momentum equation will be found more
correct than the energy equation in solving this problem.
Considering the x component of the Velocity of the
lateral inflow negligible, the general differential
equation for spatially varied unsteady flow, based on the
momentum principle, is:
_iY_ + Y
3x g 3x
4. _Y _iQ + I 8V
gA 9x g at
(33)
where S = slope of the gutter bottom
Sf = slope of total head line
Q,V,y are the discharge, velocity and depth at a
certain cross section of the gutter
g = acceleration of gravity
The total head line slope Sf will be computed from
Manning's equation, i. e.:
where
Sf =
C
2.22 A2 R4/3
(34)
n = Manning's roughness coefficient
A = area of gutter cross section
258

Also
R = hydraulic radius of gutter cross section
/ Qx_ _Q 3Q Q2 3A
V 3V = J2 3
g 3x gA 3x
..
gA
__ .
2 3x
therefore ^ _3V = _Q^ _3Q _ Q^T
d ,} %r v ^ *v <
y O'i xrA ^ °X — •» J
3 3x
(35)
where
T = width of the water surface.
V JQ = _Q 3C2
3x _,2 3x
Also:
By using equations (35) and (36), equation (33)
becomes:
(36)
S  S   n 
° f " 3X gA3
+ 2
Q 9Q
(37)
The continuity equation for unsteady flow with lateral
inflow gT , is:
'
(38)
The lateral inflow qT , available from the overland
flow, will be constant along the length of the gutter for
a time interval. So qT is a function of t but not of x.
Because only the discharge entering the sewer catch
basins at the downstream end of the gutters is required,
equation (37) can be discarded and the Q at the end of the
gutters can be obtained by using only equation (38).
For an urban area the lengths of the gutters are
relatively small. Also the overland flow supply in cfs/ft
is small. Therefore, in a time increment, the change of
the depth in the gutter will be very small in comparison
with the other terms of equation (38) (see Appendix I).
Omitting the term
T equation (38) becomes:
259

3Q
=
and integrating: Q = q • L + Q
(39)
where QQ is the discharge entering the gutter from an up
stream gutter, otherwise zero (Figure 143)
q =q
^
Figure ]k3. GUTTER FLOW
Once the hydrpgraph of the discharge entering into an
inlet (inflow hydrograph) is obtained from equation (39)
it will be routed through the sewer until the next inlet
downstream, where a new inflow hydrograph will be added to
the routed one and the summation will be routed downstream,
and so on.
260

8. ROUTING THROUGH LATERAL AND MAIN SEWERS
In Chicago, Tholin and Keifer [4] used a timeoffset
method for conduit routing because of its simplicity com
pared with other routing methods and because more refined
procedures (storage routing) had not led to significantly
different results.
Storage routing can be applied if satisfactory dis
chargestorage relationships are available. This makes
necessary the computation of instantaneous backwater
curves. Since only the rate of change in storage is
necessary to solve the storage equation, it is considered
expedient to assume a uniform flow condition for each dis
charge rate [22] and compute the conduit volume occupied
by the flow.
Other methods used in Europe, like the semigraphical
method of Hauff and the Italian storage method, are
relatively easy to use, but are based on broader
assumptions [28].
Generally there are many points of entry (inlets)
into a lateral sewer. As points of entry for a main sewer
are considered the junctions with the lateral sewers.
Considering a lateral sewer of uniform flow and
crosssection, for each inlet a hydrograph of the inflow
is available from the gutter flow. It seems more
reasonable to use a hydrograph simplified as constant
discharge rates for each time increment, rather than a
smooth curve dischargetime relationship (Figure 144).
Q
DT
Figure 144. TRANSFORMED HYDROGRAPH
261

Assuming a uniform flow condition for each discharge
rate, Manning's equation can be used to find the depth and
the velocity of flow into the sewer for each discharge.
So the velocity with which each water volume element is
moving downstream will be known.
The average velocity with which the total inflow
volume (represented by the area under the hydrograph curve)
moves downstream is considered to be the weighted average
of all the partial velocities with respect to the
corresponding volumes, i. e. multiply each volume element
by its velocity, sum the products and divide the sum by
the total volume of the water which moves downstream.
Once the average velocity of flow is found and knowing
the length of the conduit between inlets, the time needed
for the inflow water volume to travel this distance can be
found.
Each inflow hydrograph is shifted in time without
changing in shape. This has been found satisfactory and
convenient, producing only a slightly higher peak rate of
flow occurring at a somewhat later time than other more
exact methods.
Therefore, for a lateral sewer of uniform slope and
crosssection, the computer shifts the first inflow hydro
graph the time required to reach the second inlet. ;Then,
it adds the new inlet hydrograph and the shifted one and
shifts the summation hydrograph until the next inlet, and
so on until it reaches a critical point.
Changes in slope or changes in crosssection are con
sidered as critical points. After a critical point,_the
procedure is repeated from the beginning, with the_first
inflow hydrograph the one found at the critical point.
This continues until the junction of the lateral sewer
under consideration with a main sewer.
The routing through the lateral sewers is done first,
in order to find the hydrographs entering into the main
sewer at the junctions. These hydrographs are used as
inflow hydrographs for the main sewer and the same method
is followed to find the hydrographs at selected points of
the main sewer.
Overflow of the sewer system is not allowed. The
262

computer compares continuously the maximum capacity of the
conduits and the routed discharges. If the capacity of a
conduit is not sufficient, the computer indicates that in
the output. This justifies the purpose of the present
"Model" to design a sewer system or improve an existing
one. In case of:.overflow, the crosssection of the in
sufficient conduit or its slope or both have to be changed
and the computer program rerun.
; • It is assumed that the Manning's roughness coefficient
remains constant as the depth changes (actually the rough
ness coefficient decreases as the depth of flow increases).
For practical purposes also, it can. be assumed that
the maximum discharge of a circular conduit does occur at
the full depth, because the depth for maximum discharge is
sp.close to the top that there is always a possibility of
slight backwater to increase this depth closer or equal to
the full depth [22.3 .
Time steps, for the computer solution, may be chosen
to coincide with the spacing of the.ordinates of the
inflow hydrographs for convenience. However, 1 minute
intervals will facilitate the addition of the hydrographs.
Let us suppose now that the discharge entering into
the sewer during a time increment At is Q, taken from the
inflow hydrograph. In order to find the velocity with
which the volume (Q . At) is traveling downstream, Manning's
equation is used: , . . .
R2/3 sl/2
(40)
where
Q
discharge in' cf s
n = roughness coefficient of the conduit (see
Appendix I)
A = crosssectional area of flow in ft2
R = hydraulic radius in ft
S  slope of the invert of the conduit in ft/ft
The present method is used for sewers of circular and
rectangular cross section. Other shapes can be transformed
•into these basic two cross sections.
263

(A) Circular shape:
A =
(x  sinx) (41)
sinx
" P 4 x
where x in radians and
0 ^. x =^ 2rr
The maximum capacity of the pipe is (full flow):
0.46417 n8/3 1/2
Q = ' ' L) D
~max n
For partially full flow conditions equation (40) becomes:
 sinx)
5/3
x
13.566 Q n
~ rs8/3 . ci/:
(42)
The riaht hand part of equation (42) is known, so
equation (42) can be solved by trial to find the angle x.
Knowing x, the area of flow A is obtained from equation
(41) and the velocity of flow V is obtained from the
equation:
V
 (ft/sec)
(B) Rectangular shape
H
I
5 (A)
L PI — 4
(43)
A = Dy
P = D + 2y
R = Dy/(D + 2y)
264

The maximum discharge passing through the conduit
(full flow, y = H) is:
Q.
0.9361 (D H)
5/3
a/2
max
n
(D + H)
2/3
For partially full flow, equation (40) becomes
5/3
Q n
(D + 2y)
2/3
1.486 S1/2 D5/3
(44)
The right hand part of equation (44) is known, so can
be solved by trial to find the depth of flow y (in ft).
Then the velocity of flow will be
V =
(ft/sec)
(45)
If an inflow hydrograph consists of N time increments
At with the corresponding discharges Q, , Q« , .. ., Q,, ,
then the average velocity with which the total inflow
volume travels downstream until the next inlet is:
Vav=(VlQl At+V2Q
At+Q2 At+...+QN At)
or
= (V1Q1
where V, , V~ , ... are the velocities with which the
volumes Q, At, Q2 At, ... travel downstream. 'These
velocities are given from equations (43) or (45) ..
265

9. FIRST CASE STUDY ON SIMULATION OF URBAN STORM WATER
RUNOFF BY USING THE CINCINNATI URBAN RUNOFF.MODEL
A catchment of 10 acres in,Chicago, Illinois, has been
selected to verify the Cincinnati Urban Runoff Model be
cause detailed data were available and because it is
typical of the Chicago street pattern and may be typical of
some other cities.
The catchment has been divided into six basic types of
subcatchments:
Type
Length Manning's
Description (ft)* Slope* n
I Dense blue grass turf,
pervious
II Dense blue grass turf,
pervious
III Dense blue grass turf,
pervious
IV Street pavement,
impervious
V Alley pavement,
impervious
VI Common composition
roofing, impervious
16
40
70
17
8
12
0.0100 0.350
0.0100 0.350
0.0100 0.350
0.0200 0.013
0.0375 0.013
0.6670 0.012
*Parallel to the direction of overland flow
The 10 acres catchment constitutes 5.37 acres (53.7%)
of pervious areas and 4.63 acres (46.3%) of impervious
areas.
Building roofs are not discharging through the down
spouts on the lawn areas, but are connected directly to the
underground sewers. To reduce the amount of computations,
six roofs were considered directly connected with each
catch basin.
The garage roofs, because they discharge onto backyard
lawns, were considered a part of the pervious areas.
266

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The lateral sewer was of circular cross section with
a diameter of 3 ft, a slope of 0.0045 ft/ft, and a
Manning's roughness coefficient of 0.013.
The subdivision of the catchment into subcatchments,
the sewerage, and the directions of overland flow and
gutter flow are shown in Figure 145.
Figure 146 represents the catchment as a system of
geometric elements, the types of overland flow, gutters,
catch basins, and sewer system. The way of numbering the
gutters, the points of entry into the sewer system, and the
numbering of the pipes is shown on Figure 146.
The computer program for the Cincinnati Urban Runoff
Model is inserted in Appendix III. Complete information
for the preparation of the data cards is in Appendix IV,
and the data cards for the Chicago 10acre catchment are
presented in Appendix V.
A 181minute design storm was applied to the drainage
basin and the computations were made at 5minute
intervals. A detailed computer output for the present
case study is available at the Division of Water Resources,
Department of Civil Engineering, University of Cincinnati.
For the purpose of comparison, the runoff hydrograph
at the outlet of the Chicago 10acre catchment was calcu
lated by three methods: the Cincinnati Urban Runoff Model,
the EPA Storm Water Management Model, and the Chicago
Method. The resulted hydrographs are plotted in Figure
147 together with the 181minute design storm hyetograph
which was applied to the drainage basin.
Comments on the Results of the First Case Study
1. The Chicago Method, the EPA Storm Water Manage
ment Model, and the Cincinnati Urban Runoff Model, calcu
lated that the peak would occur at about the same time,
i.e., 75 minutes from the start of the rainfall as shown
in Figure 147.
2. The Chicago Method and the EPA Storm Water Manage
ment Model calculated a peak of 18.2 and 22.6 cfs respec
tively. The Cincinnati Urban Runoff Model calculated a
peak of 23.0 cfs, which agreed satisfactorily with the EPA
Storm Water Management Model. However, the latter divided
the Chicago 10acre catchment into 80 subcatchments, 40
gutters, and 4 pipes, while the Cincinnati Urban Runoff
269

CHICAGO, 10ACRE TRACT
DESIGN STORM
80 100
TIME (MINUTES)
ISO 190
22
20
18
16
? 14
o
S 12
S 10
O
8
6
it
2
0
CINCINNATI
URBAN RUNOFF MODEL
EPA STORM WATER
MANAGEMENT MODEL
CHICAGO METHOD
20
60
80 100
TIME (MINUTES)
IfaO 190
Figure
RAINFALL HYETOGRAPH AND CALCULATED RUNOFF HYDROGRAPHS
CHICAGO, 10ACRE TRACT
DESIGN STORM
270

Model used repeatedly 6 basic types of subeatchments, 40
gutters, and 8 pipes, therefore reducing the computer time
to about 5.5 minutes using the IBM 360/65 computer of the
University of Cincinnati.
Comparing the peaks obtained by the three above
methods with the Rational Formula Q = ciA , the following
is obtained:
c = coefficient of runoff equal to 0.40 for
relatively flat residential areas, 30%
impervious [29]
A = catchment area, 10 acres
i = average rainfall intensity in inches per hour
for a duration equal to the time of concen
tration of the catchment
Q = peak flow rate, in cfs
During the rainfall used in the previously presented
study case/ 2.50 inches of water precipitated on the
catchment. The duration of the rainfall was 181 minutes.
This rainfall has a return period of 10 years for the
Chicago area [30]. Then the average rainfall intensity,
for a time of concentration equal to 15 minutes and a re
turn period of 10 years, is i = 5 inches per hour [31].
Therefore, Q=0.4x5xlO=20 cfs, which agrees
satisfactorily with the peaks in Figure 147.
3. The three methods gave a comparable rising limb.
However, the Chicago Method and the Cincinnati Urban Runoff
Model calculated a lower recession than that obtained from
the EPA Storm Water Management Model. This can only be
accounted for by a larger infiltration loss. Actually the
Chicago Method and the Cincinnati Urban Runoff Model com
pute the mass curve for infiltration loss. Thus, the
infiltration rate tends to be satisfied at any given time
even though there is not sufficient rainfall at that
moment, under the condition that the accumulated mass
precipitated is bigger than the accumulated mass infil
trated.
4. The EPA Storm Water Management Model maintained
the mass continuity within 0.1% of the rainfall. No such
information was available on the Chicago Method for com
parison. The Cincinnati Urban Runoff Mathematical Model
maintained the mass continuity within an acceptable 2% of
the rainfall as has been shown by a series of case studies
performed at the University of Cincinnati. Manpower and
computer time did not justify an effort for more accuracy.
271

10. SECOND CASE STUDY ON SIMULATION OF URBAN STORM WATER
RUNOFF BY USING THE CINCINNATI URBAN RUNOFF MODEL
The Oakdale Avenue drainage basin in Chicago, Illinois,
has been selected to verify the Cincinnati Urban Runoff
Model because rainfall and runoff have been periodically
measured and recorded since 1959 by the Chicago Department
of Public Works, Bureau of Engineering [32].
The Oakdale Avenue catchment is an urban drainage area
12.9 acres in size (approximately 2 1/2 blocks long by one
block wide) and consists entirely of residential dwellings.
The catchment has been divided into six basic types of
subcatchments:
Type Description
I Grassed, pervious
II Grassed, pervious
\
III Grassed, pervious
Grassed, pervious
IV
V
VI
Street pavement,
impervious
Common composition
roofing, impervious
Length
(ft)*
35
35
50
23
15
11
Slope*
0.010
0.010
0.010
0.010
0.020
Manning ' s
n
0.350
0.300
0.350
0.350
0.013
0.667
0.012
*Parallel to the direction of overland flow
The 12.9 acres catchment constitutes 7.05 acres
(54.7%) of pervious areas and 5.85 acres (45.3%) of im
pervious areas.
Building roofs and garage roofs are draining directly
into the underground sewers.
The existing 30inch diameter combined sewer was in
stalled in 1958 and replaced a smaller combined sewer of
inadequate storm water capacity. The slopes of the 30inch
lateral sewer vary from 0.0045 upstream to 0.0030 down
stream, and the Manning's roughness coefficient is
considered to be 0.013.
The plan of the 12.9acre Oakdale Avenue drainage
272

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273

basin is presented in Figure 148. In order to use the
Cincinnati Urban Runoff Model, the Oakdale Avenue drainage
basin has been divided into 6 basic types of subcatchments,
52 gutters and 9 sewer pipes.
Two storms were applied to the Oakdale Avenue drainage
basin, the second storm of July 2, 1960 having a duration
of 66 minutes and the storm of July 7, 1964 having a
duration of 86 minutes. The computations were made at
1minute intervals. A detailed computer output for this
case study is available at the Division of Water Resources,
Department of Civil Engineering, University of Cincinnati.
For the storm of July 2, 1960 the runoff hydrograph
at the outlet of the Oakdale Avenue drainage basin was
calculated by three methods: the Cincinnati Urban Runoff
Model, the EPA Storm Water Management Model and the RRL
Method. The resulting hydrographs are plotted in Figure
149 together with the recorded hydrograph and the 66
minute storm hyetograph.
For the storm of July 7, 1964, the runoff hydrograph
at the outlet of the Oakdale Avenue drainage basin was
calculated by three methods: the Cincinnati Urban Runoff
Model, the RRL Method and the Chicago Method. The
resulted hydrographs are plotted in Figure 150 together
with the recorded hydrograph and the 86minute storm hye
tograph .
Comments on the Results of the Second Case Study
(a) Second Storm of July 2, 1960:
1. The two peaks calculated by the three methods
occurred at about the same time, i.e. around 38 and 62
minutes from the beginning of the rainfall. The recorded
first peak, however occurred sometime earlier, 32 minutes
from the beginning of the rainfall. The second peak was
not recorded.
2. The Cincinnati Urban Runoff Model, the RRL Method
and the EPA Storm Water Management Model calculated a first
peak of 18.13 cfs, 14.30 cfs and 15.60 cfs respectively
and a second peak of 10.58 cfs, 6.90 cfs and 12.50 cfs
respectively. The first peak calculated by the Cincinnati
Urban Runoff Model (18.13 cfs) was closer to the recorded
peak^ (17.40 cfs) because it considered a reduced infil
tration capacity due to the wet antecedent condition of
the soil caused by the previous rain which occured earlier
274

u
I J 1
OAKDALE AVENUE. BASIN
SECOND STORM OF JULY 2. I960
10 20 30 kO 50 60
TIME (MINUTES)
70
80
90
100
19
18
17
16
15
\k
13
12
*. 11
«/i
u.
ii 10
LU
I 9
5
w>
o
7
6
RECORDED
CINCINNATI URBAN
RUNOFF MODEL
EPA STORM WATER
IAGEMENT MODEL
10
20
30
50 60
TIME (MINUTES)
70
80
90
100
Figure H9. RAINFALL HYETOGRAPH. CALCULATED AND OBSERVED RUNOFF HYDROGRAPHS
OAKOALE AVENUE BASIN, CHICAGO ,
SECOND STORM OF JULY 2, I960
275

i 3
§ 0
OAKDALE AVENUE BASIN
STORM OF JULY 7. 196U
40 50
TIKE (MINUTES)
80
90
90
TIME (MINUTES)
Figure '50. RAINFALL HYETOGRAPH, CLACULATED AND OBSERVED RUNOFF HYDROGRAPHS.
OAKDALE AVENUE BASIN, CHICAGO
STORM OF JULY 7, 196*4
276

the same day. Therefore, some overland flow was generated
from the pervious areas and contributed to the first peak.
In comparison, the RRL Method without taking into con
sideration flow from pervious surfaces calculated lower
peaks.
3. The three methods gave a comparable first rising
limb. However, the RRL Method calculated a lower re
cession than that obtained by the other two methods,
producing a smaller total runoff than that recorded.
(b) Storm of July 7, 1964:
•1. The two peaks predicted by the Cincinnati Urban
Runoff Model and the Chicago Method occurred at'about the
same time, i.e. around 17 and 36 minutes after the begin
ning of the storm. The two peaks predicted by the RRL
Method both delayed for about 3 minutes. The recorded
peaks occurred 20 and 37 minutes after the beginning of
the storm. Therefore the three methods agreed quite satis
factorily in the timing of the peaks.
2. The Cincinnati Urban Runoff Model, the RRL Method
and the Chicago Method calculated a first peak of 3.95 cfs,
6.00 cfs and 6.00 cfs respectively, and a second peak of
8.83 cfs, 11.40 cfs and 12.13 cfs respectively. Again the
peaks predicted by the Cincinnati Urban Runoff Model
(3.95 cfs and 8.83 cfs) were closer to the recorded peaks
(4.2 cfs and 9.6 cfs). By using the Cincinnati Urban
Runoff Model, the storm under consideration generated a
very small amount of overland flow from pervious areas 30
minutes after the beginning of the rainfall and for 10
minutes only. Since the RRL Method does not take into con
sideration flow from pervious areas and since the overland
flow from pervious areas predicted by the Cincinnati Urban
Runoff Model was very small, a better agreement between
these two methods was expected.
3. The three methods calculated comparable rising
limbs for both peaks. However the two recessions calcu
lated by the Cincinnati Urban Runoff Model were consider
ably lower than the recessions predicted by the other two
methods.
277

11. CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY
The Cincinnati Urban Runoff Model consists of five
submodels:
1. The Infiltration Model uses Horton's equation to
find the shape of the proper infiltration capacity curve.
If data from the urban basin under study are not available,
the use of Jens' Infiltration Capacity Curves is recom
mended. A method has been developed for the computer to
shift the infiltration capacity curve in time, so Horton's
equation can be used when i < f .
2. The Depression Storage Model uses an equation
proposed by Linsley, Kohler, and Paulhus, and a new
equation has been developed from it relating rates of de
pression storage versus time.
3. The Overland Flow Model has been developed using
the continuity equation, the momentum equation, and an
equation by Crawford relating the depth of flow at the
downstream end of the overland flow plane with the surface
detention. Several computer runs showed a satisfactory
degree of sensitivity of this model to changes of ground
slope, ground cover, and length of overland flow.
4. The Gutter Flow Model used the continuity
equation for steady flow since the change in depth with
respect to time was found negligible. The validity; of
this simplification is shown in Appendix I.
5. The Routing Model routes the flow through the
lateral and main sewer by shifting an inflow hydrograph
from one inlet to the next, adds the shifted hydrograph
with the new one, and proceeds downstream. This solution
does not reduce the peak flows, but it has been proved to
compare sufficiently close to the solution obtained from
more rigorous methods (i.e., backwater curves computation).
A computer program has been developed and combines
the five mentioned submodels in the sequence with which
they appear above.
It is believed that although a great deal of time
has been spent in the development of the presented Mathe
matical Model, each step can and should be improved when
more field data are available for verification.
Since depression storage reduces significantly the
278

overland flow, especially from pervious areas, more re
search is needed to determine the proper depression storage
to be used in the Model.
Other factors reducing the overland flow which can be
considered in the future are: evaporation, interception,
and wetting of surfaces.
More exact storage routing procedures can be de
veloped to route the flow through the gutters and the
;lateral and main sewers. However, each improvement of the
Mathematical Model has to be justified by manpower and
computer time needed.
Further research is recommended concerning the
capacity of the stormwater gutter inlets, because during
hea\sy rainstorms there exists a carryover flow (i.e., the
portion of the gutter flow that does not go into an inlet).
If the water table of an urban basin is high, then
groundwater infiltration or seepage through pipe joints,
broken pipes, cracks, or openings in manholes and similar
faults, should be taken into account. Research can be
oriented to find a relation between pipe material, age of
the sewer system, and groundwater infiltration or seepage.
The Cincinnati Urban Runoff Model as presented is
considered to be preliminary and will be improved as
additional testing is accomplished with actual field data.
279

APPENDIX I
DIMENSIONAL ANALYSIS OF GUTTER FLOW
Gutters are usually triangular channels with the one
side approximately vertical (this depends on the shape of
the curb). As can be seen in Figure 151, a typical
battered curb has a height of 5.5 inches and if the water
is flowing with this depth then it is covering the corres
ponding half of the roadway.
<*>
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5" One course .concJ
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Curb grade?^ fr
t
X
0
BATTERED CURB
Figure 151.
CONCRETE ROADWAY WITH
BATTERED CURB DETAIL
The Rational formula gives:
. A
q = c i E
where q is the discharge in cfs/ft width
A is the contributing area in acres
b is the width of overland flow (ft)
c is runoff factor < 1
i is average rainfall intensity in inches per
hour for a storm duration equal to the time of
concentration of the area which provides the
overland flow. Usually i is not less than 1
nor more than 3 inches per hour for urban
areas.
280

Through the assumption of orders of magnitude/ it is
possible to show that the term [T^] in equation (38) on
page 259 can be neglected.
It seems reasonable to assume: [the symbols " = ORD ( )"
mean "is of the order ( )"]
T = ORD (10°) ft
Ay = ORDUO"1) ft
At =' ORD(10) sec
c = ORD (10"1)
i = ORD (10°) in/hr
A = ORD (10°) acres
b = ORD(10 ) ft
then the ratio
t AZ
At = 1. Ay/At
q c i
= ORD(10 2)
Therefore the term [Tj can be neglected from equation
(38) as being small in comparison with the rest of the
terms of this equation.
As an illustration, for a certain rain, two values of
overland flow have been considered in a 5 minute inverval,
namely q.^ = 0.00069 cfs/ft and q2 = 0.00115 cfs/ft at the
beginning and the end of the interval. The length of the
gutter was 100 ft with a slope 0.03 ft/ft and a Manning's
coefficient 0.017. The slope of the street pavement was
1:27.
The corresponding depths of flow have been found by
Manning's equation to be y1 = 0.0774 ft and y_ = 0.0852 ft,
The average surface width of the flow was T = 2.195 ft.
Then TIT T = 0.000057
which is a 6% of the average
inflow q.
281

APPENDIX II
MANNING'S ROUGHNESS COEFFICIENT
1. For Overland Flow
Manning's n for overland flow can be estimated from
the table below [8], or from other more detailed sources •
[22].
Watershed
Cover
Smooth asphalt
Asphalt or concrete paving
Packed clay
Light turf
Dense turf
Dense shrubbery & forest
litter
Manning's n For
Overland Flow
0.012
0.014
0.030
0.200
0.350
0.400
When there is a mixture of ground cover in a sub
catchment of area A, then a harmonic mean Manning's rough
ness coefficient n can be obtained from the expression:
A
n
B_
n.
(A  B)
n2
where B is a part of the subcatchment area having a
Manning's roughness coefficient n, and the rest of the
subcatchment area (A  B) has a roughness coefficient equal
to n2 [8] .
The same procedure can be used if an area has more
than two different kinds of ground cover.
2. For Pipe Flow
For a combined sewer system, the same coating of
slimes form on the inside of all materials used for the con
struction of the sewers. This means that all materials used
in the construction of combined sewers will have the same
hydraulic roughness coefficient [31]. In Manning's formula
that coefficient is n = 0.013. Proof of this was demon
strated by extensive tests at the Ohio State University
[33].
282

APPENDIX III
COMPUTER PROGRAM FOR THE CINCINNATI URBAN RUNOFF MODEL
The volume of computations required by the Cincinnati
Urban Runoff Model would have been too formidable without
the use of an electronic computer. A digital computer IBM
1130 with an available storage of 16K was used to run parts
of the present program. Because usually more storage is
required, the whole program was run on the digital computer
IBM 360/65, of the University of Cincinnati, which has a
maximum storage capacity of 512K. The computer program is
written in FORTRAN IV language.
The differences in programming for the IBM 1130 and
the IBM 360/65 are only:
1
2
3
4
IBM 1130
job cards
READ (2 , . . .)
WRITE (3, ...)
Only arithmetic IF
statements
IBM 360/65
job cards
READ (5, ...)
WRITE (6, ...)
Both logical and arith
metic IF statements
The program presented here consists of three parts,
the main program and two subroutines, namely:
I/ MAIN program:
2/ GUTFL subroutine:
3/ PIROU subroutine:
for infiltration, depression storage,
and overland flow
for the gutter flow
for the routing through the lateral
and main sewers
283

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21
"CINCINNATI URBAN RUNOFF MODEL"
COMPUTER PROGRAM
*********************** ************ *******************
INFILTRATION AND DEPRESSION STORAGE IK PERVIOUS AREAS
*«#*»****##*##*******#*************«*************#***
DIVIDE THE TOTAL RAINFALL TIME IN N EQUAL SMALL INCREMENTS
READ N+l VALUES OF AVAILABLE RAINFALL HYDROGRAPH ( I M/HR )
EACH TIME INCREMENT DT IS IN MINUTES
AT TIME ZERO 1=0. AT THE END OF THE TIME PERIOD I=N
RAINFALL INTENSITY AT TIME ZERO IS R01FOR 1=0) .^
RAINFALL INTENSITY AT A TIME T=DT*I IS RItl)
AT A TIME T=DT*I . INFILTRATION CAPACITY IS F(I) AND ACTUAL
INFILTRATION IS AFU) IN (IN/HR)
THE CONSTANT RATE AT WHICH F(I) APPROACHES ASYMPTOTICALLY
AS TIME CONTINUES»IS FC
THE RATE OF INFILTRATION CAPACITY WHEN TIME=0(I.E. 1=0) IS FO
CK IS A POSSITIVE CONSTANT
DIFFERENCE BETWEEN RAINFALL AND ACTUAL INFILTRATION S(I)
ACCUMULATED MASS PRECIPITATED UNTIL A TIME DT*I IS RMII) INCHES
ACCUM^MASS ACTUALLY INFILTRATED UNTIL A TIME DT*I IS AFM(I) INCHES
WHERE RMtl) OR AFM1I) INDICATE THE MASS FROM TIME DT*(I1) UNTIL
TIME DT*I »PLUS THE PREVIOUS MASS
DIFFERENCE BETWEEN MASSES UNTIL A TIME DT*I IS P(I) IN INCHES
VOLUME OF WATER IN DEPRESSION STORAGE IS VS ( I ) INCHES
TOTAL DEPRESSION STORAGE CAPACITY SD INCHES
DEPRESSION STORAGE SUPPLY IS Dfl) INCH/HOUR
SS(I) IS THE RATE OF ( RA INI NF I LTRDEPRESS ION STORAGE)
PPU) IS THE ACCUMULATED MASS OF ( RA I N INF I LTRDEPRESS ION STORAGE)
COMMON RK70) ,AF(70) iS<70> »RMt70) tAFM(70> ,P ( 70 ) , D < 70 ) , VS I 70 ) ,SS
1(70) tPP(70) ISSIM170) tPPIM(70) iVSIMt70) .DIM (70) .DATE(S) »DC(70) »OE(7
20) »OVQ(250i 30) »OVQW(70) »NEW( 30) «Q!NL( 70.20) .QINLOI20) t IW(250 1 »V(25
30) «QLEAV<250) »N,DT,NN«N INLtQINP ( 250 »20 )
WR1TE16.1)
FORMAT ( ' 1 ' )
READ! 5*21) (DATE (I) »I=1«8>»DT.N
FORMAT(8A4»F5t2«I5)
READ (5»4) RO
FORMAT (F5. 2)
READ(5»5) (RI(I)tI=l»N)
FORMAT (12F5t2)
READ (5»6) FC.FO.CK.SD.SDIM
FORMATOF1045.2F8.4)
READ(5t20)AN
FORMAT (F5. 2)
WRITEC6*501)
FORMAT ( 5X » ' *******************************************************
l****«*i,/,5X. "SIMULATION OF INFILTR.AND DEPRESSION STOKAGE IN PERV
2 IOUS AREA" »/»5X.'*** **********************************************
20
501
WRITE (6.221 (DATE! 11,1=1.5)
22 FORMAT (5X. "RAIN OF ' , 6X « 5 A4 , // )
WRITE(6.23)(DATEII)»I=6t8)
23 FORMAT(5X» "TIME OF RAIN START • i3X ,3A4 •// I
284

WRITE (6t605)
605 FORMAT!1IX.•FC'»8X»•FO1»9X»'CK')
WRITE(6»606)FC»FO»CK
606 FORMAT(10X,F4.2»6X,F4.2«6X«F6.4,//)
WRITE(6«607) SD
607 FORMAT<5X«'TOTAL DEPRESSION STORAGE CAPACITY OF THE'»/»5X*'SUBCATC
1HMENT IS (IN INCHES) ='t15X.F8.4t//)
IF (ROFO) 10»7»7
7 DO 8 I=liN
8 AF(I)=FCM (FOFC)*(2.718**(CK*DT*I)>>
SO=RO~FO
DO 9 I«=1»N
9 S(1)=RI(I)AF+RI < IU ) /2.0)*OT/60.0
DO 13 1=1iN •
13 AFMfI)=((FC/60.0)*DT*I) + ((< FOFC ) / (60.0*CK.))*(1tO(2.718**<CK*DT*
II))))
DO 14 I=1»N
14 PU )=RMf I )AFM< I )
OD=SO
SSO=0.000
NCON=0.0
DO 800 I=1*N 
IFIRKI)AF(I)) 900.900»799
900 0(11=0.000
NCON=1«0
VS(
SS(
PP(
GO
)=SD
)=S(I)
)=P(I1SD
TO 800
799
798
IF(NCON) 798*798*900
800
15
16
17
18
805
806
DU =S( I)*(2.718**(P( I )/SD) )
VS( )=SD*(1.012.718**tPtI1/SD)))
SSI )=S(I)D TT«RI( I)*AF(I)»D(I 1*SS!I )
FORMAT!5X«F8.2.6XjF6.3»8X.F6.3»8X«F6.3*8X*F6.3)
WRITE(6,805)
FORMAT('1')
WRITE(6»S06)
FORMAT(5X*'TIME FROM MASS MASS DEPRESSION
1DIFFERENCE'»/.5X,'KAIM START PRECIPITATED
INFILTRATED
STORA
285

2GE (MRMIDS)'»/*5X.'(MINUTES) (IN) (IN)
3 (IN) (IN) '»/)
00 807 1*1»N
TT«DT*I
807 WRITEt6»808) TT.RM(I)»AFM(I)»VS(I)»PP(1/
808 FORMAT(5X«F8,2.8XiF6.3»9X,F6.3»7X.F6.4»7X»F6.3i
GO TO 118
C IF INFILTRATION CAPACITY GREATER THAN RAINFALL INTENSITY
C AT THE BEGINNING OF THE RAINFALL (WHERE TIKE=0 AND 1=0) THEN
C INFILTRATION CAPACITY CURVE STARTS AFTER A TIME ( I*DTtX*L) FROM
C THE BEGINNING OF THE RAINFALL
C WE HAVE DIVIDED DT IN (NN+1) EQUAL INCREMENTS X(L FROM 1 TO NN)
c THE SHIFTED INFILTRATION CAPACITY CURVE WILL INTERSECT THE
C RAINFALL INTENSITY CURVE AFTER A TIME TJ=J*DT+Y AFTER THE
C BEGINNING OF THE INFILTRATION CAPACITY CURVE.OR AFTER A TIME
C (I*DT+X#L) FROM THE BEGINNING OF THE RAINFALL
C THE I AFTER WHICH THE SECTION OF THE CURVES TAKES PLACE IS(ISEC)
19 XsDT/AN
9999
299
00 299 L=1«NN
ALLL=< (RI ( I )+*(X*L/60.0)
B2*(FC*TJ/60.0)K ( FOFC )*( 1. 0 (2.718** tCK.*TJ ) ) ) / ( 60.0*CK.»
9998
300
301
100
LSEC=L
B12T=ABS(B1B2)
IF(B12T0.01) 301»301i299
CONTINUE
DO 300 I=2.N
DO 300 L=1»NN
RMl I )=RM( I1) + {(RI(I)+RHI1> > /2 .0 )*DT/60.0
ALLL=((RI(I)+(X*L*(RI(IH)RI < I )) /DT 1FC) /( FOFC) )
IFIALLL.GT.O.ANDtALLL.LT.l. ) GO TO 9998
GO TO 300
TJ*tl.O/CK>*ALOGt (R I ( I >KX*L*< RI t 1 + 1 1RI t I) 1 /DT )FC )/( FOFC ))
B1=RM+RI (I ))/(2.0*DT) ) )*tX*L/60. 0)
B2(FC*TJ/60.0)M (FOFC ) * 1 1. 01 2.718**(CK.*T J ) ) )/ ( 60. 0*CK.) )
ISEC=I
USEC=L
B12T>=ABS(B1B2)
IF(B12T0.01) 301»301»300
CONTINUE
BEFORE THE SECTION
AFO=RO
S0=0»0
000.000
550=0.000
DO 100 I=1»ISEC
AF(1)=RI(I)
S(I)=0«0
D(I)=0.000
ssm=o.ooo
286

AT THE SECTION
IISEC=ISECH
RISEC=(LSEC*X*(RI( II.SEORH I SEC) J/DTl+RK I SEC)
AFSEC=RISEC
SSEC=0.0
DSEC=0.000
SSSEC=0.000
AFTER THE SECTION
IISEC=ISECH
DO 101 I=IISEC»N
AFU) = FC+()))
101 sm=RI(I )AF(I)
BEFORE THE SECTION
DO 102 I=2»ISEC
102 RM(I)=RM( Il) + «Rim+RI(ll))/2.0)*DT/60.0
DO 103 I*1«ISEC
AFM(I)=RM(I)
VSU>=0.0000
PPU)=0.000
103 P(I)=AFM+RI < 11 ) )/2 .0 )*DT/60«0
DO 105 I=IISEC«N
105 AFM(1) = {FC/60.0)*{(I1SEC)*OTX*LSEC+TJ) K (FOFC)/(60«0*CK))*(
1(2»718«*(C<*((IISEC)*DTX*LSEC+TJ))11
DO 205 1=1 ISEC»N
205 P(I)=RM(I)AFM(I)
NCON=0.0
DO 810 I=IISEC»N
IFIRIfI)AF(I)) 901»901»795
901 D(I!=0.000
NCON=1«0
VS(
sst
pp(
GO
)=SD
)=StI)
)=P(I1SD
0 810
795
794
VS(
SS<
IF1NCON) 794t794i901
=St I)*(2.718**(P(
)=SD*<1.0<2.718**<P(I 1/SD)))
)=S(I)DfI)
PPII)=P(1)VStI)
810 CONTINUE
T«=DT*I SEC+X*LSECTJ
WRITE(6»106) T
106 FORMAT(5X»'INFILTRATION CAPACITY CURVE
1INNING OF THE RAINFALL (IN MINUTES)
WTT = DT*ISECiX*LSEC
WRITE(6»108)WTT
STARTS AFTER THE1t/»5Xf'BEG
••»5X»F8.2.//)
287

r
c
c
c
c
c
c
c
c
c
c
c
108 FORMATI5X."INFILTRATION CAPACITY CURVE INTERSECTS THE'«/.5X,'RAIN
1FALL INTENSITY CURVE AFTER THE BEGINNING*»/»5X."OF THE RAINFALL
2 (IN MINUTES) ='»12X»F8«2«//)
WRITE (6.110)
110 FORMAT!5X»'TIME FROM RAINFALL ACTUAL DEPRESSION
10VERLAND «»/.5X.*RAIN START INTENSITY INFILTRATION STOR.SUPP
2LY FLOWSUPPLY'./»5X,'(MINUTES) (IN/HRJ UN/HR) (
31N/HR) (IN/HR) '«/>
W=0«00
WRITE(6»111)W.RO«AFO»OD»SSO
111 FORMATt5XtF8.2»6X»F6.3»8X»F6t3.8X.F6.3»8X»F6.3)
DO 112 I*1»ISEC
WW"=DT*I .
112 WRITE(6»113)WW»RI(I)»AF(I)»D(I)»SS(I)
FORMAT(5X»FS.2 »6X.F6.3 » 8X *F6•3i8X•F6 4 3»8 X »F6•3)
WRITE(6»114)WTT»RISEC»AF5EC»DSEC.SSSEC
FORMAT(5X»FS.2.6X»F6.3»8XtF6.3tSX.F6.3«8X.F6.3)
DO 115 I=IISEC»N
WWW»DT*I
WRITE(6»116) WWW.RItI)»AF(I)»DtI)»SS(I)
FORMAT!5X.F8.2»6X»F6.3»8XtF6.3i>8X»F6.3»8X»F6.3)
WRITE(6»120)
FORMAT!'!')
WRITE!6»121)
FORMAT!5X,'TIME FROM MASS MASS DEPRESSION
1D!FFERENCE',/»5X»'RAIN START PRECIPITATED INFILTRATED STORA
2GE * tMRMIDS)'»/.5X,'(MINUTES) (IN) (IN)
3 (IN) (IN) './)
DO 122 I=1»ISEC
WW=DT*I
KRITE(6*123) WW»RM(I).AFM!I)»VS(I)»PP!I)
FORMAT!5X»F8.2«8X«F6.3»9X.F6.3.7X.F6.4»7X»F6.3)
WRITE(6.124)WTT»RMSEC«AFMSE.VSSEC»PPSEC
FORMAT!5X»F8.2»8X.F6.3»9X»F6.3»7X»F6«4.7X.F6t3)
DO 125 I=IISEC.N
WWW«DT*I
WRITE(6. 126) WWW.RM!I)»AFMtI)»VS(I)»PP(I)
FORMAT(5X»F8.2.eXtF6.3»9X,F6.3.7X,F6.4,7X.F6t3)
WRITE (6.119)
FORMAT ('1')
********#**********************************************
INFILTRATION AND DEPRESSION STORAGE IN IMPERVIOUS AREAS
**#********************************#*******«***********
INFILTRATION CAPACITY IS ZERO
VOLUME OF WATER IN DEPRESSION STORAGE IS VSIM(I) INCHES
TOTAL DEPRESSION STORAGE CAPACITY IS SDIM (INCH)
DEPRESSION STORAGE SUPPLY IS DIM!I) INCH/HOUR
SSIM(I) IS THE RATE OF IRAINFALLDEPR.STORAGE)
PPIM(I) IS THE ACCUMULATED MASS OF (HAINDEPR.STORAGE)
WRITE(6»500)
500 FORMAT < 5Xt'*********#***********«*******•********»*****************
1*#******« ,/,5X.'SIMULATION OF INFILTR.AND DEPRESSION STORAGE IN IM
2PERVIOUS AREA1 t/«5Xt '#**********#********•**#***#*#*******•»********
t//)
113
114
115
116
120
121
122
123
125
126
118
119
288

CAPACITY IS ZERO'.//)
WRITE<6»25>
25 FORMAT(5X«'INFILTRATION
26 FORMAT!5X.'TOTAL DEPRESSION STORAGE CAPACITY OF THE••/»5X«'SUSCATC
1HMENT IS (IN INCHES) =*»15X»F8.4 . //)
DO 28 1=1»N
DIM
SSIM(I)=RI(I1DIM1I I
PPIMt I )=RMC I JVSIMU)
CONTINUE . '
WRITE(6.29)
FORMAT(5X»'TIME FROM RAINFALL DEPRESSION OVERL.FLOW't/,5X
INTENSITY STOR.SUPPLY SUPPLY1•/i5X»'(MINUTES)
(IN/HR)

READ(5i649)IA»INDW«ALW«ARW»SW
649 FORMAT(2I2t3F7.4)
WRITE(6.652>
652 FORMAT ( /»5X» ' **•*****#*****#*********#***********#**#****#****»»*##
1**' I
IFUNDW1) 653»654»653
653 WRITE(6«655)IA
655 FORMAT(5X»'OVERLAND FLOW SIMULATION OF IMPERVIOUS AREA NUMBER «»I2
1)
GO TO 657
654 WRITE(6i656)IA
656 FORMAT(5Xi'OVERLAND FLOW SIMULATION OF PERVIOUS AREA NUMBER '»I2)
657 WRITEI6.658)
658 FORMAT 15X »'**»*******»*»******•******»*****#*#**********************
!«*//)
WRITE(6»659)ALW
659 FORMAT(5X»'LENGTH OF OVERLAND FLOW IN FEET «««F7.2>
WRITE(6»690) SW
690 FORMATOX.'SLOPE AT THE DIRECTION OF FLOW (FT/FT) ='»F7.4)
WRITE<6»691) ARW
691 FORMAT(5Xt'MANNINGS ROUGHNESS COEFFICIENT
AAA=510.35*(SW**0.5)/

IF851»852,852
851 CONTINUE
852 DC(I)=G
OVQW( I )=(2.0*AAA)*(DC( I >**1.6666>*( ! 1.0+ < AA* ( DC t I)**3.0) J )** 1.6666
1)
OVQ( I »IA) =ALW*OVQW( I ) /43200.0
829 TIME=DT*I
WRITE (6 ,85 3) TIME,OVOW( I ) ,OVQ( I , I A ) »DE ( I I • DC < I ) »SS t I >
853 FORM AT { 7X.F6.2 » 11X»F7 .4 » 14X »F8 .6 » 10X ,F8. 5 » 10X»F8.5»10X,F6.3)
NWN=N+8
NSTAR=N+1
DO 661 I=NSTAR»NWN
ssm=ss
IF(STBSTA) 664,665,665
CONTINUE
DO 679 10=1,20000,1
WD=ID
G«=WD/100000.0
STB«( ( (BA*G)t(ABA*(G**4.0) ) ) **1 .6666 lt 1 ABB*G )
IF(STBSTA) 679,665,665
CONTINUE
291

1650 DO 646 IDnl«2000»l
) **1»6666 )+ ( ABB*G >
G=WD/1000.0
STB» ( ( (BA*G)+{ ABA*(G**4.0)
IF(STBSTA) 646»665i665
CONTINUE
DC(K)=G
1F(DE(K)DC«) ) 666.667,667
IF(INOWl) 5060.1651.5060
IFtSW0.5) 5027.5026*5026
DO 668 10=1.20000.1
WD=ID
G*WD/10000.0
STB ( AAA* t G**l . 6666 ) *AB ) + ( ABB*G )
IF/43200.0
292

SUBROUTINE GUTFL
C
c »#*********#***»******
C GUTTER FLOW PROGRAMM
C *#*##»***•****#********
C
C NINL IS THE NUMBER OF INLETS
C N IS THE NUMBER OF TIME INCREMENTS
C NW IS THE NUMBER OF SUBCATCHMENTS
C THE PROGRAM WILL TREAT INLETS ACCEPTING A NUMBER OF
C NBRAN SYSTEMS OF GUTTERSiAND DIRECT CONNECTIONS
C Nl IS THE NUMBER OF THE FIRST UPSTREAM GUTTER OF
C EACH GUTTER SYSTEM
C N2 IS THE NUMBER OF THE LAST DOWNSTREAM GUTTER OF
C EACH GUTTER SYSTEM
C ITYPL IS THE NUMBER OF SUBCATCHMENT FEEDING A GUTTER
C FROM THE LEFT SIDE
C ITYPR IS THE NUMBER OF THE SUBCATCHMENT FEEDING A GUTTER
C FROM THE RIGHT SIDE
C GLEN IS THE LENGTH OF THE GUTTER
C IROOF IS THE NUMBER OF HOUSE 'ROOFS DIRECTLY CONNECTED WITH AN INLET
C IWRF IS THE NUMBER OF THE TYPICAL HOUSE ROOF SUBCATCHMENT
C ROOFL IS THE WIDTH OF A ROOF IN FEET
COMMON RIt70).AF(70>,S<70),RM(70) »AFM<70) tP ( 70 ) .0(70 ) »VS t 70 ) »SS
1 (70) »PP{70) »SSIM(70) tPP!M(70) »VSIM(70) »DlM<70) i DATE (8 ) .DC ( 70 ) ,DE ( 7
20) »OVQ(250»30) iOVQW(70> .NEW (30) »QINL( 70.20) «OINLO(20) »IW(250).V(25
30) »OLEAV(250) tN.DT.NN.N INL ,G INP < 250»20 )
READt5«?000) NINL
3000 FORMAT I 15)
DO 3004 I IN = ltNINL
DO 3004 1=1 «N
3004 QINL( I » I IN 1=0.0
DO 4001 I !N=1»NINL
READ<5»3009) NBRAN
3009 FORMAT f 15)
DO 4010 11=1, NBRAN
READ(5,3010) N1»N2
3010 FORMAT (215)
DO 4002 J=N1»N2
READ(5»3011) ITYPL.ITYPR.GLEN
3011' FORMAT(2I5.F10.2)
DO 4002 I=1.N
QGUT=(OVO( I«ITYPL)+OVQ< I .ITYPR) )*GLEN
4002 QINL( I »IIN)=OINL( I. I IN1+QGUT
4010 CONTINUE
READI5.3013) IROOF, IWRF .ROOFL
3013 FORMAT (215. F5. 2)
GTIMO=0.00
QINL01 II,M)=0. 00000
DO 4005 1=1 .N
HERE QGUT SHOUS FLOW ENTERING THE INLET
OGUT*IROOF*2.0*ROOFL*OVQ( I , IWRF)
QlNLt I . IIN)=QINL( I »I INI+QGUT
4005 CONTINUE
4001 CONTINUE
DO 5890 NINI=1»6
WRITE(6.5500)
5500 FORMAT t ' 1 ' )
N15=5*NINI4
N16*5*NINI
FROM DIRECT CONNECTIONS
293

WRITE(6,5555)(I,I=N15,N16)
5555 FORMAT(5X,1TIME FROM DISCHARGE
1 DISCHARGE',/.5X,'RAIN START
2ERING ENTERING'»/,5X.' IN
3INLET ',12,' INLET '.12,' INLET
4 IN CFS IN CFS IN CFS
IFIN16NINL) 5800,5800.5810
N16=NINL
WRITEI6.3014) GTI MO,IQINLOt1 IN),IIN = N15»N16>
FORMAT(6X»F7.2i4X«F8.5»4(3X»F8.5))
DO 5850 1=1.N
GTIMSI*DT
WRITE(6.5820) GTIM.tQINL
5810
5800
3014
5820
5850
THIS IN ORDER TO FACILITATE THE ADDITION OF THE HYDROGRAPHS
DURING THE ROUTING THROUGH THE LATERAL AND MAIN SEWERS
KK=DT/DDD
NN=KK*N
DO 6901 IIN=1.NINL
DO 6901 1=1.N
JJ=N+2I
JJJ=N+1I
QINL(JJ,HN)=QINLUJJ»IIN>
6901 CONTINUE
DO 6900 I IN = 1.NINL
6900 QINLU »I IN)=OtOOOOO
DO 6912 I IN=1,NINL
11*0
00 6911 1=1,N
DO 6911 K=1,KK
11=11+1
QINPdltllNl
1NL(I,IIN)1)
6911 CONTINUE
6912 CONTINUE
DO 6922 NINI=1»6
WRITE(6,6916)
6916 FORMAT('!')
N15*5*NINI4
N16=5*NINI
WRITE(6,6917) { I , I=N15.N16)
6917 FORMAT(5X,«TIME FROM DISCHARGE DISCHARGE DISCHARGE DISCHARGE
1 DISCHARGE',/,5X,'RAIN START ENTERING ENTERING ENTERING ENT
ENTERING' ,/,5X, «
IN
12
INLET '.12.'
IN CFS IN
6918.6918.6919
INLET
CFS
6919
6918
6921
6920
6922
6923
2ERING
3INLET '
4 IN CFS
IF(N16NINL)
N16=NINL
DO 6920 1=1. NN
TRTIM=DDD*( 11)
WRITE(6.6921) TRTIM. (QlNPt I »J> »J=N15.N16)
FORMAT (6X«F7»2«4XiF8»5»4J3X«F8»5) )
CONT 1MUE
IFIN16NINL) 6922.6923.6923
CONTINUE
RETUR'N
END
INLET '.12.' INLET '.12.'
',I2./.6X.'MINUTES IN CFS
IN CFS'.//)
294

SUBROUTINE PIROU
C **#*»»******#*****'***#****#**#**********
C ROUTING THROUGH LATERAL AND MAIN SEWERS
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
909
917
907
905
912
913
925
906
USE OF MANNING'S EQUATION TO FIND THE AVERAGE TIME NEEDED
FOR A DISCHARGE HYDROGRAPH TO TRAVEL DOWNSTREAM UNTIL A
POINT ( INLET tMANHOLE ) WHERE A SECOND DISCHARGE HYDROGRAPH
ENTERS THE SEWER SYSTEM. ADD THE SHIFFTED AND THE NEW
HYDROGRAPH AND CONTINUE UNTIL THE POINT FOR WHICH WE SEEK
THE OUTFLOW HYDROGRAPH
«AREA' IS THE AREA OF FLOW IN SQUARE FEET
'SLOP' IS THE SLOPE OF THE INVERT IN FT/FT
«ALEN' IS THE LENGTH BETWEEN TWO CRITICAL POINTS ( INLETS »
MANHOLES* CHANGE IN SLOPE OR DIAMETER OR SHAPE)
(WHEN THERE IS A CHANGE IN SLOPE OR DIAMETER OR SHAPE WE
CONSIDER THERE AN HYPOTHETICAL MANHOLE WITH ZERO INFLOW)
«PIPN' IS THE MANNING'S ROUGHNESS COEFFICIENT OF THE PIPE
'Y1 IS THE DEPTH OF FLOW IN THE CONDUIT IN FEET
•HR' IS THE HYDRAULIC RADIUS IN FEET
«DIA' IS THE DIAMETER OF THE SEWER PIPE IN FEET OR ITS WIDTH
IF IT IS RECTANGULAR
«H« IS THE HEIGHT OF RECTANGULAR SEWER IN FEET
14=1 INDICATES CIRCULAR PIPE*I4=2 RECTANGULAR CONDUIT
NINL=NUMBER OF INLETS. MANHOLES AND CHANGES IN SLOPE AND DIAMETER
NPIPE=NUMBER OF PIPES
COMMON RI (70) »AF(70) »S(70) »RM<70) «AFM(70) tP(70)«D(70) iVS(70) »SS
1(70) «'PP(70) .SSly,(70) .PPIM(70) »VSIM(70) .DIM I 70) .DATE(8) *DC(70) »DE(7
20) iOVQ( 250.30! iOVQW(70 ) »NEW( 30) .01 NL( 70.20) »QiNLO(20) .IW(250)»V(25
30) *QLEAV(250) »N »DT »NN »N INL»Q I NP (250*20 1
READ<5*909) NPIPE
FORMAT (15)
NIN=NINL1
DO 1050 1IN=1*NIN
J=IIN
WRITE(6»917)
FORMAT ( '!« )
READ (5*907) 14 .SLOP.P IPN «DI A.H* ALEN
FORMAT! I 5 »F10.6 *F 10.4.2F5.3 »F10«3 >
KK=DT/1.0
NN«=KK*N
IFU41) 950*905*950
QMAX=(0.46417*

SAK
X*SA/1000.0
BB"=nxSIN(X> )**1.6666)/(X»*0.6666>
IF ) 955*951*951
955 IW(I)«=1
GO TO 960
951 IWU)«0
B«(QINP(IiIIN>*PIPN)/U.486*CSLOP**0«5)*JDIA**l»6666) )
M«=H*1COO«0
DO 959 K=1.M
SA=K
Y=SA/1000.0
BB«=(Y**1« 66661/1 ( DIAH 2.0*Y) )«*0.6666)
IF(BBB) 953»953«959
959 CONTINUE
953 V(I )=OINP( I«I IN!/(DIA*Y)
960 CONTINUE
970 SUK«:0.0
DO 1000 I=1«NN
SUM=SUMftV( I )*QINP( I • UN) )
SE=SE+QINP( I»IIN).
1000 CONTINUE
AVERV=SUM/SE
AVT=ALEN/(AVERV*60«0)
1AVT=AVT
DIF=AVTIAVT
IF(DIK0«5) 991.992*992
AVERT=IAVT
GO TO 993
AVERT=IAVT+1
IAVET=AVERT
DO 994 I=1.IAVET
OLEAVt I 1=0.000
IIAVT=IAVET+1
NNEW=NNKAVET
991
992
993
994
DO 995 I=IIAVTtNNEW
KLIIAVET
OLEAVt H=OINP(KL« I IN)
DO 996 I=NEEWiNNEW
OINPtI »IIN)=0.000
995
996 VU)=0»00
296

997
WRITE16.997) J
FORM AT t 28X» '•»*************#*« »/»28X» 'PIPE
NUMBER '»I2i/.28X»'**»
INVERT IN FT/FT IS
PIPE IN FEET IS
COEFFICIENT IS
IF(I41) 968,969,968
969 WRITE16.967) DIA
967 FORMAT15X»'CIRCULAR CONDUIT OF DIAMETER
WRITE<6»964) ALEN
964 FORMAT(5X,'THE LENGTH OF THE
l',F7.2)
WRITE(6»963) PIPN
963 FORMAT(5X,'THE MANNINGS ROUGHNESS
l'.F7.5)
WRITE(6»998)OMAX
998 FORMAT(5X,'MAXIMUM CAPACITY OF THE CONDUIT IS (CU.FEET/SEC)
1 «»F6.2)
WRITEI6.999) AVERV
999 FORMAT(5X.'AVERAGE VELOCITY OF ENTERING DISCHARGES (FT/SEC)
1 '.F5.2)
WRITE(6,1001) AVERT
1001 FORMAT<5Xt=0.000
1070 Vtl)=0«00
DO 1074 I=1,NNEW
1074 OLEAVfI)=QLEAV(I)+QINP(I»NINLI
1069 DO 1003 I=1.NNEW
TIMP=(11)*1.0
IF(lWtl)l) 1072.1080,1072
1072 WRITE(6.1004) TIMP.OINP

Comments on the Developed Computer Program
i
1. The Program does not take into account the dry
weather flow. If information and measurements are avail
able from the urban area under consideration, the dry
weather flow simply can be taken as an additional input for
the routing through the sewers.
2. The SUBROUTINE subprogram "PIROU" has been written
based on the assumption that only one lateral sewer drains
the catchment under consideration. This is usually true
for catchments of about 10 acres area.
If a broader subdivision of the whole basin will
be preferred, then the catchments probably will be drained
from a network of lateral sewers. In cases like this the
SUBROUTINE "PIROU" has to be replaced by the program pro
posed on the next page, which deals with a pipe network.
This program will use SUBROUTINE "PIROU" in the place of the
statements 6, 9, 12, and 15. In Figure 152 a complex pipe
network is presented with constant inflows at the inlets.
Figure 152 shows the way of numbering the inlets and the
pipes, necessary for the computer input.
298

Computer Program Dealing with a Pipe Network
DIMENSION QINL(100)«IPIPS(100)»I PIPE(100)»PIPQ(100)
C
C
C
50
51
150
11
100
20
21
7
10
8
12
102
13
NUMB.OF DOWNSTREAM LAST POINT OF
N NUMBER OF INLETS OR CHANGES IN.
NN NUMBER OF PIPES
READ(5»50)N»NN»MAIN
FORMAT(315)
READ(5»51) (QINLU) »I = 1»N)
FORMATt12F5.2)
READ(5»3) dPIPS(J)»IPIPE(J)»J»ltNN)
FORMAT(2I5> ......
WRITE(6»150)
FORMAT(10X.'DISCHARGE AT PIPE NUMBER&
TOTQ=0.0
TEMPQ=0«0
00 1 I=1»MAIN
DO 2 J*1»NN
IFdPIPE(J)I) 2»5»2
K=IPIPS(J)
PIPQ(J)=OINL(K)
TEMPQ=PIPQ(J)
TOTQ=TOTQ+TEMPQ
WRITE<6»100)TOTQ»J
FORMAT(10X»F9.2»10XiI3)
GO TO 21
TOTQ=TOTQ+TEMPQ
TEMPQ=0.0
IPIPS(J)=1
GO TO 2
IDUM=J
IFdPIPS(J)l) 6>6»7
IFdPIPSl IDUM)'1J 9»9»10
JK=IDUM
DO 8 JJ=1»NN
IDUM=JJ
IF(IPIPEtJJ}IPIPS(JK))8»7»8
CONTINUE
GO TO 12
K=IPIPS(JK)
PIPQ(JK)=QINL(KJ
TEMPQ=TEMPQ+PIPQ(JK>
IPIPS(JK)=1
IFMPIPEI JKJI )10»11»10
K=IPIPS(JK)
PIPQ(JK)=QINL(K)
TEMPQ=TEMPQ+PIPQ ( JK)
WRITE(6»102)TEMPO*JK
FORMAT(10X,F9.2»10X»13)
IF(IPIPE

IP(IPIPSUJ)IPIPEUK) )
15 K=IPIPS(JJ)
PIPQfJJ)=QINL(K)
TEMPQ=TEMPQ+PIPO(JJ)
WRITE(6»103)TEMPQ»JJ
103 FORMATUOX»F9.2»10X,I3)
JKSJJ
IFUPIPE(JK)I) 13*20,13
14 CONTINUE
2 CONTINUE
1 CONTINUE
CALL EXIT
END
II
I
113
13
13
i
114
115
(<)
C2)
19
112 
\k
!
i.
no
\
(S)
V
C?)
— 116
117
5 C5)
16
17
I
i
t
K
tu
ui
M
(7) 6
Z
s:
Figure 152. COMPLEX PIPE NETWORK WITH
CONSTANT INFLOWS AT THE INLETS
300

APPENDIX IV
DATA CARDS PREPARATION
A. Rainfall Data Cards
I. card 1 (8A4, F5.2, 15)
Variable Example
Names Values
col. 120 Date of storm
(day, month, year)
col. 2132 Hour of start of storm
(hours, min.,AM or PM) DATE
col. 3337 Time interval between
intensity values
DATE 24 SEPT 1970
5:35 AM
DT 5.00
col. 3842 Number of time steps
to be calculated N
II. card 2 (F5.2)
col. 15 First value of rainfall
intensity RO
III. Rainfall intensity cards, each (12F5.2)
Intensity,in in/hr • RI
B. Infiltration Data Cards
I. card 1 (3F10.5, 2F8.4)
col. 110 Const, infiltr. rate
(in/hr) _ FC :
col. 1120 Initial infiltr. rate
(in/hr)' FO
col. 2130 Decay rate of infil.
(1/min) CK
col. 3138 Retention storage of
pervious areas (in) SD
col. 3946 Retention storage of
impervious areas (in) SDIM
36
0.20
0.53
3.00
0.0697
0.250
0.0625
301

II. card 2 (F5.2)
Variable
Names
Example
Values
col. 15 A number to divide DT
in smaller intervals
(see statement 19. If
DT = 5.0 min, a logical
value of AN is 20.0 or
30.0. Then the new
intervals will be of
0.17 min. each)
C. Overland Flow Data Cards
I. card 1 (15)
col. 15 Number of subcatchments
to be simulated
II. One card for each subcatchment
(212, 3F7.4)
col. 12 Number of the subcatch
ment
col. 34 Indication number (1 for
pervious, 2 for imper
vious)
col. 511 Length of area at the
direction of flow (feet)
col.1218 Manning's roughness coef.
col.1925 Slope of the area at the
direction of flow (ft/ft)
D. Gutters Data Cards
I. card 1 (15)
col. 15 Number of entry points
into the sewer system
(Data cards of type II, III, IV, V to
be repeated for each entry point)
AN
NW
IA
INDW
ALW
ARW
SW
NINL
30.0
16.0
0.35
0.01
302

II. card 2 (15)
Variable
Names
col. 15
Number of gutter
branches leading into
the entry point
(Data cards of type III, IV to be
repeated for each gutter branch)
III. card 3 (215)
col. 15 Number of the first up
stream gutter segment
of the gutter branch
col. 610 Number of the last down
stream gutter segment
of the gutter branch
(Data cards of type IV to be repeated
for each gutter segment)
IV. card 4 (215, F10.2)
col. 15 Number of the subcatch
ment providing flow to
the gutter segment from
the left
col. 6^10 Number of the subcatch
ment providing flow to
the gutter segment from
the right
col. 1120 Length of the gutter
segment in feet
V. card 5 (215, F5.2)
Example
Values
NBRAN
Nl
N2
ITYPL
ITYPR
GLEN
5
297.0
col. 15 Number of roofs directly
connected with entry
point IROOF
col. 610 Number of subcatchment
corresponding to the 
roof type IWRF
col. 1115 Length of the roofs (ft) ROOFL
6
30.0
303

E. Pipes Data Cards
I. card 1 (15)
col. 15 Number of pipes
II. one card for each pipe
(15, F10.6, F10.4, 2F5.3, F10.3)
col. 15 Indication number, 1
for circular pipe, 2
for rectangular
col. 615 Slope, of the pipe
invert
col. 1625 Manning's roughness
coefficient
col. 2630 Diameter of circular
pipe or width of rec
tangular pipe (ft)
^col. 3135 Height of rectangular
pipe (0.0 for circular)
(ft)
col. 3645 Length of pipe (ft)
Variable
Names
NPIPE
DIA
H
ALEN
Example
Values
14
SLOP
PIPN
1
0.0045
0.013
3.0
0.0
120.0
304

APPENDIX V
DATA CARDS FOR THE FIRST CASE STUDY
24
0.20
0.20
5.30
0.35
30.0
6
1 1
2 1
3 1
If 2
$ 2
6 2
9
2
1
3
1
3
5
If
0
if
5
2
6
If
7
ii
8
2
12
If
9
2
10
It
11
if
12
2
12
if
13
2
1ft
It
15
if
16
2
12
ft
17
SEPTFMBER 1,970 5 35 AM 5.0 36,
0.20
5.25
0.25
0.53
16.
I»0.
70.
17.
8.
12.
2
5
\i
if
3
1
6
5
If
6
2
7
2
8
it
6
9
ft
10
2
11
2
12
ft
6
13
It
1ft
2
15
2
16
ti
6
18
O.?0 0.20 0.20 0.20 0.42 0.42 0.50 0.62 0.87 1.63
2.80 1.75 1.25 0.88 0.80 0.67 0.50 0.47 0.47 0.47
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
3.00 0.0697 0.2500 0.0625
0 0.35 0.01
0 0.35 0.01
0 0.35 0.01
0 0.013 0.02
0 0.013 0.0375
0 0.012 0.667
297.0
IftO.O
297.0
IftO.O
30.0
105.0
105.0
105.0
105.0
30.0
105.0
105.0
105.0
105.0
30.0
105.0
105.0
105.0
105.0
30.0
305

5
19
3
1
21
3
1
23
5
0
It
25
2
26
E}
27
1}
28
2
12
29
2
30
It
31
It
32
2
12
L
1
33
2
3J
35
36
2
12
2
37
5
k
39
3
1
0
8
1
1
1
1
1
1
1
1
3 297.0
20
5 297.0
I* 140.0
22
5  297.0
k 140.0
21*
3 297.0
6 30.0
25
l» 10 5. 0
26
2 105.0
27
2 105.0
28
I* 105.0
6 30.0
29
U 105.0
30
2 105.0
31
2 105.0
32
I* 105.0
6 30.0
33
l» 105.0
2 105.0
35
2 105.0
36
it 105.0
6 30.0
38
3 297.0
1 11*0.0
1(0
5 297.0
4 11*0. 0
6 30.0
0.001*5 0.013 3.0 0.0
0.001*5 0.013 3.0 0.0
0.001*5 0.015 3.0 0.0
O.OOU5 0.013 3.0 0.0
O.OOU5 0.013 3.0 0.0
0.001*5 0.013 3.0 0.0
0.001*5 0.013 3.0 0.0
0.001*5 0.013 3.0 0.0

120.0
210.0
210.0
120.0
120.0
210.0
210.0
120.0
306

SECTION VII
MATHEMATICAL MODEL FOR SIMULATION OF
RUNOFF QUALITY FROM AN URBAN WATERSHED
1. INTRODUCTION
This study is devoted to the development of a mathe
matical formula for the concentration of pollutants at any
location within a combined sewer system of an urban water
shed. A stochastic process is used to describe the removal
of surface pollutants by the runoff water and inertial
effects of solids during motion are taken into account. The
pollutants are routed from the inlets through the lateral
and main sewers.
2. PREVIOUS WORK
A lot of information concerning urban runoff pollution
is available, but only a few studies have included data on
both runoff quantity and quality in sufficient detail so
that the quality of direct surface runoff can be evaluated
as a function of runoff intensities. .
A recent study under the sponsorship of the Environ
mental Protection Agency, Water Quality Office by a con
sortium of contractors—Metcalf & Eddy, Inc., the University
of Florida, and Water Resources Engineers, Inc. [9] 
provides probably.the most comprehensive runoff quality
model currently available. The model presented here, as
developed at the University of Cincinnati, has similarities
with the above referenced model; however, a major difference
is that an integral solution has been developed instead of
a stepwise solution.
3. DEVELOPMENT OF THE MATHEMATICAL MODEL
FOR URBAN RUNOFF QUALITY
Wastewater, surface dust and dirt, catch basin trap
pings, and air pollutant residues are the major sources of
storm runoff pollution.
The pollutants in this study are classified as soluble
and nonsoluble. The following are soluble: BOD, COD,
307

total hydrolyzable phosphates, various forms of nitrogen,
and various forms of coliforms; while suspended solids are
recognized as nonsoluble.
3.1 Mathematical Formula for Surface Pollutant Removal
Surface pollutants consist of street litter and
dustfalls that appear on the surface at the start of an
oncoming storm.
The following assumptions are made as the basis for
the development of the mathematical formulation:
a. Decaying effects of pollutants due to chemical
changes or biochemical degradation during the runningoff
period are neglected.
b. The amounts of pollutants percolating into the
soil by infiltration are neglected.
c. The rate of removal of pollutants by runoff water
is assumed to be proportional to the amount of pollutant
remaining, and to the runoff intensity.
d. Because the distribution of air pollutants in the
atmosphere is nonuniform, spatially and unsteady, varying
from time to time, it is difficult to formulate the amount
which runs off in rainfall; therefore, this source has not
been evaluated separately but is considered to be included
in the overall runoff constituents.
The amount of a pollutant remaining on a runoff
surface at a particular time, the rate of runoff at that
time, and the general characteristics of the watershed can
be related in a simplified form as follows:
dP
dt
KqP
(D
Rearranging,
dP
Integrating,
= Kqdt
rp
I dP
J "*"'
•'•n
Kqdt =
308

In ^ =  K
o
qdt =  K V.
where
Vt = J qdt
P KV
— = 6
P =
(2)
In equations (1) and (2) , '
PQ = amount of pollutant on the surface drainage
area at the start of an oncoming storm
P = amount of pollutant remaining on the surface
at time t
g = runoff intensity at time t
Vt = accumulated runoff water volume up to time t
e = the base of natural logarithms
K = a constant characterizing the drainage area.
Thus, the remaining amount of pollutant at any instant
on the surface is assumed to be decreasing exponentially as
a function of the accumulated runoff volume.
Probabilistically, P/PQ = e~KVt can be considered as
the probability that a polluting particle on the surface
will still remain at its original position when .storm runoff
water volume has accumulated to V. at time t.
Consequently,
(3)
where 0 is the probability that a polluting particle will
be removed from its original position after time t. Differ
entiating equation (3) with respect to V:
309

* f(V) =
(4)
where f (V) is the probability density function for 4>. The
integral of f(V) over all possible values of V is unity:
1
f (V) dV =
Ke~KV dV
too
= eKVl = 1
Jn
It is assumed that the function f(V) is nowhere negative,
and is singlevalued.
Therefore the probability that a particle is removed
from its original position during the time from ^ to t2/.
with corresponding accumulated volume of runoff from ^
V«, is
to
Pr (V, £ V ^ V,) =
•*• •«. ^*
KV KV
Ke KVdV=e KV
=KVi_eKV2
V
Assuming that once a soluble pollutant is picked up by
runoff water it will flow into an inlet ,the amount of
pollutants which are flushed into a sewer inlet is :
Figure 153 schematically represents the removal of soluble
surface pollutants by runoff water.
For the transport of s'olids by runoff water, the drag
force exerted by flowing water on a particle varies as the
square of the flowing velocity, i.e.
Hence, the portion of the solids that are removed from
their original positions and flushed into a sewer inlet is
assumed to be proportional to the square of the runoff
intensity.
r = X q'
(6)
310

LU
>
h

In equation (6), r is the fraction of the removed solids
that is carried off into a sewer inlet at the instant when
the runoff intensity is q; X is a proportionality factor.
Thus, the amount of solids that are flushed into a
sewer inlet during the same time interval from t, to t0
JL £
with mean runoff intensity
 ql + q2
q = ^ , 'is
P0 X q
(e
KVl
eKV2)
(7)
Thus, with the previous assumptions and this formula, it is
obvious that the closer a particle is to the inlet, the
smaller the runoff rate necessary to wash it into the inlet,
and the larger the particle, the greater the runoff rate
required.,
Metcalf & Eddy, Inc., et al, [9] found that a rainfall
rate of 0.14 in/hr would begin to move 20 micron material,
and a rate of 0.65 in/hr would begin to move 40 micron
material. Also a runoff rate of 0.01 in/hr would transport
20 micron material and a rate of 0.03 in/hr would transport
40 micron material.
If the sewer inlet is a catch basin, then the retention
of solids in the trap should be taken into consideration.
In its relation to sedimentation, the sizeweight
composition of a suspension can be expressed by the
frequency distribution of the settling velocities of constit
uent particles [35].
A hypothetical cumulative distribution of settling
velocities of a suspension carried into a sewer inlet is
presented in Figure 154.
Thus, with the size distribution of the flushedin
suspension and the flowing rate of runoff, u , the portion
of the flushed—in suspension that is retained in the catch
basin can be estimated at any instant during the running
off period.
The proportion 19 of particles settling with
velocities u
is retained in its entirety. The remainder
312

100
CtL
o
r 6o
O
o
o
U
o
 20
10
0.02 0.04 0.06 0.08
SETTLING VELOCITY V (CM/SEC)
0. 10
Figure 154. HYPOTHETICAL FREQUENCY DISTRIBUTION AND CUMULATIVE
DISTRIBUTION OF SETTLING VELOCITIES OF A SUSPENSION
313

de
reaches the bottom, and the overall retention portion be
comes
 eo)
U
u
de
(8)
Since the catch basin is generally short in the
direction of flow, hence 1  6 is the portion of the
suspension that is retained, and 9 is that portion which
is carried by runoff water into a sewer. Consequently,
during the time interval from t.. to t2 the amount of solids
that are flushed into a sewer is
Po X q2 (e
while the amount that is retained in a catch basin is
(9)
(10)
3.2 Significance of K Value
In all these derivations K is a very important para
meter presumably characterizing the drainage area.
Suppose that a continuous random variable x has a
probability density function f (x) .
The expectation E (x) of the random variable x is de
fined as [36] :
E
(x) = f x f(x) dx
•J —00
which may be interpreted as the center of gravity of the
probability density function f (x) .
In this study, the probability density function is
f (V) = Ke~KVt
314

E(V) =
V f (V) dV =
V K e~KV dV
d(VKe KV) = KVde~KV + e KV d(KV) = K2 Ve~KV dV + Ke~KV dV
KVe~KV dV = i Ke~KV dV  d(KVe~KV)
E(V) =
i\
KVe • dV
I

= 0.1
K = 4.6
The factors that will affect the K value presumably
are;
(a) size and shape of a drainage area,
(b) slope of gutters,
(c) types of storm water inlets,
(d) time of concentration of a drainage area,
(e) kind of pollutant to be transported.
3.3 Removal of Pollutant from Catchbasins
Catchbasins historically have been constructed on in
lets to combined sewers and storm drains for the purpose of
removing heavy grit and detritus which might otherwise
settle out in the piping system and providing a water seal
to confine sewer gas.
The trapped pocket of liquid and solids in which
organic materials undergo decomposition between storms con
tributes a substantial amount of pollutants to urban runoff
water.
The American Public Works Association [37] determined
the way the soluble pollutants in a catchbasin at the start
of a storm are flushed into the sewer. They experimented
by adding 15 to 45 pounds of sodium chloride dissolved in
water to a catchbasin containing 353 gallons. Water from a
hydrant was discharged through a hose and water meter to
the gutter near the catchbasin. Samples were taken from
the effluent when various quantities of water up to 1685
gallons had been added to and passed through the catchbasin,
Figure 155 presents the cumulative percent of salt dis
charged as a function of gallons of liquid added.
A formula regressioned by Metcalf and Eddy, Inc. [9]
to fit the'curve in Figure 155 came out to be
R =
1.0  e
100
(12)
316

100
80
o
o:
o
 60
O
o
Q_
1
LU
O
o:'
LU
Q.
20
e
Theoretical values with
perfect mixing
Observed values at
flow rate of
1 cu. ft./min. O
4 cu. ft./min. 4
7 cu. ft./min. 9
353
706
GALLONS OF LIQUID ADDED
1059
Figure 155.RELATIONSHIP OF FLOW INTO CATCHBASIN AND
REDUCTION OF CONCENTRATION ON SALT (NaCl)
317

In equation (12),
R = percent of salt removed,
V = cumulative inflow volume to the catchbasin,
and
G = trapped volume of liquid in a catchbasin.
A scrutiny of equation (12) reveals the similarity of
this empirical equation and the equation derived previously
for soluble pollutant removal in the form of = 1 e~KVt ,
equation (3). Hence, it might be inferred that K value is
inversely proportional to the size of a drainage, area.
If Y is the amount of soluble pollutants in a catch
basin at the start of a storm, then
R = Y
V,
1.0  e
1.6G
will be the amount of soluble pollutant in the catchbasin
to be flushed out into the sewer when the runoff volume
During the time interval when the
, the per
centage of removal during that time interval is
has accumulated to V .
volume of runoff has accumulated from V, to V
•!•
V,
R =
1.6G
(13)
For solids removal from a catchbasin, the portion of
the solids flushed out into the sewer by the incoming
runoff could be estimated from Figure 154. Assuming that
an amount of solids Z is retained in a catchbasin at the
instant the flushing velocity is UQ , the amount of solids
to be flushed out into the sewer is Z
Z (1
9Q)
is to be retained.
& while the amount
o
3.4 Routing of Soluble Pollutants through Lateral and Main
Sewers
For simplicity in routing soluble pollutants through
sewer systems, it is assumed that the decaying effects due
to chemical and biochemical changes are negligible.
318

The soluble pollutant is assumed to travel at current
velocities along the distance from where it enters the
inlet to any point downstream in the sewer system.
Routing takes account of the volume of runoff 'water
and amount of pollutant removed into an inlet from the
farthest inlet area upstream.
At the next inlet point, the contributions from the
next inlet area are added to the previously routed
pollutant from upstream. This process is continued at
successive points.
The total amount of soluble pollutant at the outlet
end of the lateral sewer at the mean time, from t^ to t^
can be expressed mathematically as
n
n
W =
01
KV,
KV,
 e
n
5>i
1.6G
 e
1*6G
(14)
The concentration of the routed soluble pollutant at
the outlet end of the lateral sewer is
c =
W
r
/
Jt
Qdtf
(15)
In equations (14) and (15)
n
x.
"""
estimated amount of pollutant due to
rainfall contamination in the air, and
n is the number of inlet contributions
routed.
n
z
oi
KV
tl~tr
KV
 e
t2~tr
319

total amount of surface soluble pollutant flushed into in
lets by runoff water, and P . denotes the amount of soluble
pollutant on the surface drainage area served by the ith
inlet, and t is the travel time from the ith inlet to the
outlet end.
n
Y
v
tltr
1.6G
V
 e
t0t
2 r
1.6G
total amount of soluble pollutant flushed out from catch
basins.
r
AI
c q dt = total amount of pollutants in dry
weather flow from tn to t9, and c
L «£ S
is the concentration of the pollu
tant in dry weather flow, and q
S
is the dry weather flow rate.
q dt = dry weather flow volume monitored
at the outlet end from t, to t~.
Q dt = runoff volume at the outlet end
during the routing period from
to t .
Routing of soluble pollutants through the main sewer
follows the same method as that used for the lateral sewers
by taking the outlet pollutants of the laterals as the
individual contributions to the main sewer.
Diagrammatical representation of the routing of sol
uble pollutants through a sewer system is shown in Figure
156.
3.5 Routing of Settleable Solids in Sewer Systems
Heavy solids in sanitary sewage water and in runoff
water are swept down the sewer invert like the bed load of
streams. Light materials float on the water surface.
320

Latera
Sewer
ith
inlet
Mai n
Sewer
\
Outlet end
of lateral
sewer
V. Cumulat i ve
I runoff volume
Time
Lateral Sewer ,
Outlet Hydrograph]
Figure 1 56. D1AGRAMMAT1C REPRESENTATION OF
SOLUBLE POLLUTANT ROUTING
321

When the velocity in the conduit falls, the heavy solids
are left behind as bottom deposits; when velocities rise
again, gritty substances are picked up once more and moved
along in heavy concentrations.
This section of the study is devoted to the estimation
of the amount of sediments in sewers to be scoured from the
sewer bottom or to be deposited on the sewer bottom with
respect to the varying velocities involved in the water
transport in sewers.
The channel velocity [35] initiating scour of parti
cles deposited in sewers is defined as
V =
1.486
n
R'
1/6
B (ss  1)
1/2
(16)
In equation (16)
n = Manning's n roughness coefficient
R = hydraulic radius of the conduit
d = particle diameter
B = shield magnitude of sediment
characteristics
s = specific gravity of particles
s
Manning's formula for flow velocity is
,2/3 1/2
V =
n
R
(17)
After equating equations (16) and (17) and solving for d:
d =
B (ss  1)
(18)
In equation (18), S is the slope of the invert of the
sewer; B varies from 0.04 for unigranular sand to 0.06 or
more for nonuniform sticky materials; and s is taken as
s
2.70. Equation (18) is used to find the critical diameter
of sediments to be deposited or scoured up.
Case I. Sediment Uptake: that portion of sewer sedi
ments that has particle sizes greater than or equal to the
critical diameter found from equation (18) will remain at
the bottom of the sewer. Those particles with diameters
smaller than the critical diameter will be scoured up and
transported downstream.
322

Case II. Sediment Deposition: that portion of the
suspension in the incoming flows with particle diameters
greater than or equal to the critical diameter found from
equation (18) will settle to the bottom of the sewer. Those
particles with diameters smaller than the critical diameter
will be transported farther.
The sewer sediment characteristic is approximated by a
plot prepared by Metcalf and Eddy, Inc. [9] and shown in
Figure 157.
Routing of solids in sewer systems can be accomplished
by applying the continuity equation of mass transport,
starting from the farthermost upstream point of the sewer
and moving downstream.
Routing procedures for settleable solids take the
following steps: ,
(a) Calculate the average velocity in each conduit.
(b) Calculate the hydraulic radius in the conduit of
concern.
 (c) Determine the critical diameter of particles by
equation (18).
(d) Determine the fraction of particles with diameters
greater than that calculated in step (c).
(e) Determine sediment scour and deposition by con
sidering the mass balance in the sewer conduit
considered.
Figure 158 shows the contributing settleable suspen
sion flow sources in a sewer conduit considered for
routing of solids.
M, = amount of sediments at the sewer bottom at
time t, ,
x, = fraction of M, with particle diameters greater
than or equal to the critical diameter calcu
lated in equation (18),
M9 = amount sediments at the sewer bottom at time
fc2 '
323

1.00
0.80
lo.60
E 0.40
o
<
u0.20
0.00
5.0
PARTICLE DIAMETER (mm)
10.0
Figure 157.SIEVE ANALYSIS PLOT FOR SEWER SEDIMENT
(AFTER METCALF & EDDY, INC.)
Settleable Suspension Flow
Rate F_ From Runof
Settleable Sus m
pension Flow Rate
F From Upstream
Settleable Suspension Flow Rate
F_ From Incoming Sewage
Figure 158.MASS FLOW IN A SEWER CONDUIT
324

At = t2  t1
Fl = settleable suspension flow rate from sewer
conduit immediately upstream,
F2 = settleable suspension flow rate of incoming
sewage,
F3 = settleable suspension flow rate of incoming
runoff,
x2 = fraction of settleable suspension flows with
particle diameters greater than the critical
diameter calculated in equation (18),
VQ = current volume of wastewater in the sewer
conduit under consideration,
QQ = incoming flow rate at the sewer conduit con
sidered.
Then the total amount of sediment at the sewer conduit
under consideration at time t0 is
M2 = Ml
X
At
(19.).
The concentration of the suspension which will flow
into the next sewer conduit is
Concentration =
(1.0  x,) M
   
V
+
(1.0  x,) (F, + F0 + F_)
2 1.2 3
Q
o
4. APPLICATION OF THE MODEL
At the present stage, equation (15) is tested by con
sidering the drainage area as a whole, instead of being made
up of a number of individual inlet drainage areas with
individual inlet hydrographs which are routed through the
system and combined with other inlet areas as the routing
proceeds downstream. Contribution of the pollutants from
the air was neglected, and catchbasin,contributions were
included in surface loadings. Dry weather flow was assumed
,to have a constant rate and a constant level of quality
during the whole period of storm runoff. Furthermore, in
325

these tests the K value is chosen as 4.6, based upon the
assumption that after a runoff of 1/2 inch, 90% of the
surface pollutants are removed from the whole drainage
area.
The Laguna Street combined sewer draining a steep
drainage area of 370 acres in San Francisco [9] was se
lected as a demonstration site for the verification of the
model. One storm occurred there on March 10, 1967 yielding
a total runoff of 0.22 inch, and another storm occurred 5
days later with a total runoff of 0.33 inch. Both storms
were analyzed assuming a dry weather flow of 1.5 cfs with
a 250 mg/1 BOD concentration. The surface loading of BOD
at the start of the storm was estimated to be 3.85
pounds/acre. Tables 65 and 66 tabulate the computations,
and Figures 159 and 160 show the computed results and the
measured values. The results reveal a good agreement of
the trends of the variation of BOD concentrations during
the whole period of runoff.
The "Bloody Run" combined sewer watershed of approxi
mately 2380 acres in Cincinnati, Ohio, was also selected
to test the model. A storm occurred there on April 1, 1970
with a total runoff of 0.10 inch. The dry weather flow
was at 1.5 cfs with BOD at a level of 300 mg/1. The
surface BOD loading was estimated to be 5.65 pounds/acre.
The results of computations and trends of variations of
BOD concentrations are shown in Table 67 and Figure 161.
326

San Francisco Lagunn Street
'March 10, 1967 Storm
TABLE 65
Time
20:35
21:05
21:20
21:35
21:50
22:05
22:35
22:50
23:05
23:20
23:35
23:50
24:05
24:25
q
in/hr
Av
in
V
in
^V,^.^
Po'D
^V*83.8
Di luted
• Sewage
BOD
Result
BOO
mg/1
0.0
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
,02
,02
03
045
055
02
025
03
13
21
13
08
04
0
0
0
0
0
0
0
0
.01
.005
.0075
.01125
.014
.01
.00625
.0075
0.0449&
0.
,01
0.02172
0.
015
0.03165
0.
0.
0.
0.
0.
0.
0225
03375
04775
05775
064
0715
0.0325
0.
1040
0.0525
0.
1560
0.0325
0.
1890
0.02
0.
2090
0.013
0.
2190
0.
0.
o.
o.
0.
0.
0.
0.
0.
0.
04547
05340
03609
02173
02526
09994
13297
0676
03685
01719
76.1
74.0
72.0
68.5
64.8
61.5
59.0
57.3
52.3
43.0
37.4
31.4
22.5
41.6
41.6
29.4
20.4
17.0
41.6
29.4
7.5
4.7
7.5
4.7
11.9
22.7
117.7
115.6
101.4
88.9
81.8
103.1
88.4
64.8
57.0
50.0
42.1
43.3
45.2
327

^o.oo.
I
C0.10
u_
u.
o
§ 0.20
250
200
? 150
£
T 100
o
m
50
Reported BOD
Computed BOD
r
TIME
Figure 159. SAN FRANCISCO LAGUNA STREET COMBINED SEWER
March 10, 196? Storm
328

San Francisco Laguna Street
March 15, 1967 Storm
TABLE 66
Time
20:15
20:30
20:45
21:00
21:15
21:30
21:45
22:00
22:15
22:30
22:45
23:00
23:15
23:30
23:45
24:00
q
in/hr
0
0
0
0
0
0
'
0
0
0
0
0
0
0
0
0
.02
.03
.025
.015
.01
.005
.0
.06
.205
.21
.08
.12
.19
.18
.15
Av
in
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.005
.0075
.00625
.00375
.0025
.00125
.0
.015
.05125
.0525
.02
.03
.0475
.045
.0375
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
V
In
.0
.005
.0125
.01875
.0225
.025
.02625
.02625
.04125
.0925
.1450
.1650
.1950
.2425
.2875
.3250
D
eKVe' t
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
02274
03314
02676
01569
01031
00511
0
05909
17372
14020
04511
0&034
08004
06128
04222
Po.D
1^*83.8
77.4
75.0
73.0
71.2
70.0
69.5
0,0
67.2
57.5
45.5
38.4
34.2
28.7
23.1
19.2
Di luted
 Sewage
BOD"
41.6
29.4
34.5
52.6
71.4
ni.o
250.0
1.5.6
4.8
4.7
11.9
8.05
5.15
5.43
6.5
Result
EGD
mg/1
119.0
104.4
107.5
123.8
141.4
180.5
250.0
82.8
62.3
50.2
50.3
42.3
33.9
28.5
25.7
329

0.00
Q
O
CO
0.10
0.20
200
150
1 100
50
o
o
o
CM
Reported BOD
Computed BOD
o
CM
O
o
CM
o
CM
O
O
CM
CM
TIME
o
CM
CM
O
O
O
CO
CM
Figure 160. SAN FRANCISCO LACUNA STREET COMBINED SEWER
Storm of March 15, 196?
330

Cincinnati Bloody Sewer
' April 1, 1970 Storm
TABLE 6?
Time
5:05
X • *' «^
5:12.5
5:20
5:27.5
5:30
5:37.5
5:50
5:57.5
6:05
6:12.5
6:20
6:27.5
q £
in
0.
0.
0.
0,
0.
0.
0.
0.
0.
0.
0.
06
107
123
118
102
087
074
056
03*1
026
022
0
0
0
0
0
'0
0
0
0
0
0
^v
in
.0075
.013375
.015375
.01*475
.01275
.010875
.009375
.007
.00*125
.00325
.00275
0
0
0
0
0
0
0
0
0
0
0
0
V
in
.0
.0075
.020875
.036250
.05100
.06375
.07*»625
.084
.091
.09525
.09850
.10125
KV
e
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
D
UKV2
03391
057&5
06203
05552
0*f505
03639
02994
02153
0127**
00957
00799
P .D
o
£V*57.0
113.0
108.0
101.0
94.0
88.5
83.7
80.0
77.0
75.0
7*i.O
72.6
Di luted
Sewage
BOD
62.6
35.0
30.5
31.8
36.8
42.2
50.8
67.2
110.0
I*t5.0
170.0
RGSLI] t
BOD
ir.g/ 1
175.6
1*O.O
131.5
125.8
125.3
125.9
130.8
144.2
185.0
219.0
242.6
331

u.
u.
o
0.00
0.02
0.0k
'0.06
0.08
0.10
0.12
300
200
o
§ 100
Reported BOD
Computed BOD
LA
O
r
o
LA
o
oo
o
CS
oo
TIME
Figure 161CINCINNATI BLOODY RUN SEWER
Storm of Apr!1 1 , 1970
332

5. CONCLUSIONS AND SUGGESTIONS
FOR FURTHER STUDY
Conclusions
1. The quality of urban runoff can be modeled from
.urban runoff hydrographs and pollutant contributions from
the associated environments.
2. The amount of surface pollutants in an inlet
drainage area removed during the time that the volume of
runoff water increases from V. ' "
to V» is expressed by
KV.,
 e
KV2
and correspondingly the amount of soluble pollutants in a
catchbasin is indicated by
v, v. \
1.6G
\
 e
1.6G
3. The concentration of soluble pollutants at any moni
toring station in the drainage area can be found from
equation (15) after finding W from equation (14). Equation
(15) is repeated below.
c =
X!
W
Suggestions for Further Study
The present study is confined to theoretical consider
ations of pollutant, removal and transport in the drainage
system. Verification or modification is anticipated as
field data become available.
A determination of the surface pollutant removal con
stant K as a function of the physical characteristics of
various drainage areas would be worthwhile research.
Experimental determination on the proportionality
factor A in equation (7) for the removal of solids into .an
inlet by runoff water would make the formula for solid
removal more reliable.
333


SECTION VIII ;
ACKNOWLEDGMENTS
A list is given below of various people in the Division
of Water Resources at the University of Cincinnati who have
contributed significantly to this research project.
Dr. Herbert C. Preul •
Associate Professor of
Civil Engineering
Dr. Louis M. Laushey,
Head, Civil Engineering
Department
Constantine Papadakis
Research Assistant
A. S. Rashidi
Research Assistant
T. C. Wu
Research Assistant
J. S. Chien
Research Assistant
D. D. Dasnurkar
Research Assistant
Paul F. Carlson
Research Assistant
David L. Maase
Research Assistant
R. T. Kuo
Research Assistant
John B. Malouf
Research Assistant
Principal Investigator
and Project Director
Acting Project Director,
Summer period, 1970
Project Engineer and Co
ordinator (also contrib
uted M.S. thesis on "Cin
cinnati Urban Runoff Model")
Project Engineer
Spring, 1970
Laboratory Analyst
and Field Assistant
Laboratory Analyst (also
contributed M.S. thesis on
"Mathematical Model for
Urban Runoff Quality")
Laboratory Analyst,
Civil Engineer
Field & Office Engineer
Field & Equipment Engineer
Laboratory Analyst,
Civil Engineer
Laboratory Analyst,
Civil Engineer
Appreciation is extended to the following officials of
,the Water Quality Office of the Environmental Protection
Agency for their assistance and cooperation during the
335

course of the Project:
Mr. Darwin R. Wright, Project Manager, Storm and Com
bined Sewer Pollution Control
Branch
Mr. Robert L. Feder, Director of Regional Research and
Development Office
Mr. William A. Rosenkranz, Chief, Storm and Combined
Sewer Pollution Control Branch
336

SECTION IX
REFERENCES
1. Gregory, R. L. and C. E. Arnold
"Runoff  Rational Runoff Formulas"
Trans. Amer. Soc. Civil Eng., Vol. 96, 1932.
2. Hicks, W. I.
"A Method of Computing Urban Runoff"
Trans. ASCE, Vol. 109, 1944.
3. Izzard, C. F.
"Hydraulics of Runoff from Developed Surfaces"
Proceedings, 26th Annual Meeting, Highway Research
Board, Vol. 26, 1946.
4. Tholin, A. E. and C. J. Keifer
"The Hydrology of Urban Runoff"
Trans. ASCE, Vol. 125, 1960.
5. Kaltenbach, A. B.
"Storm Sewer Design by the Inlet Method"
Public Works, Vol. 94, 1963.
6. Eagleson, P. S.
"Unit Hydrograph Characteristics for Sewered Areas"
Journal of the Hydraulic Division, ASCE, Vol. 88,
No. HY2, March, 1962.
7. Terstriep, M. L. and J. B. Stall
"Urban Runoff by Road Research Laboratory Method"
Jour, of Hydraulic Division, ASCE, Vol. 95,
No. HY6, November, 1969.
8. Crawford, N. H. and R. K. Linsley
"Digital Simulation in Hydrology: Stanford Watershed
Model IV", July 1966, Tech. Rep. 39, Stanford Univer
sity.
9. Metcalf and Eddy, Inc.; University of Florida',1 Water
Resources Engineers, Inc.
"Storm Water Management Model"
Draft Report; Environmental Protection Agency, Water
Quality Office; September, 1970.
10. Horton, Robert E.
"An Approach Toward a Physical Interpretation of In
filtration Capacity"
Soil Science Society Proceedings, Vol. 5, 1940.
337

11. ASCE  Manual of Engineering Practice 28
"Hydrology Handbook".
1949
12. Jens, Stifel W.
"Drainage of Airport Surfaces  Some Basic Design
Considerations", ASCE Transactions, Vol. 113, 1948.
13. U. S. Soil Conservation Service
"National Engineering Handbook" Section 4,
Hydrology, Supplement A
14. Linsley, R. K., M. A. Kohler, and J. L. H. Paulhus
"Applied Hydrology"
15. Horton, R. E.
"Surface Runoff Phenomena. Part I, Analysis of the
Hydrograph", Horton Hydrol. Lab. Pub. 101,1935.
16. Hicks, W. I.
"A Method of Computing Urban Runoff"
Tfans. ASCE, Vol. 109, 1944.
17. ASCE Manual of Practice No. 9
"Design and Construction of Sanitary and Storm Sewers"
1969.
18. Keifer, C. J. and H. Hsien Chu
"Synthetic Storm Pattern for Drainage Design"
ASCE Proceedings, HY4, August 1957.
19. Horton, R. E.
"The Interpretation and Application of Runoff Plot
Experiments, with Reference to Soil Erosion Problems"
Proc. Soil Sci. Soc. Amer., Vol. 3, 1938.
20. Eagleson, P. S.
"Dynamic Hydrology"
McGrawHill, 1970.
21. Keulegan, G. H.
"Spatially Variable Discharge over a Sloping Plane"
Trans. Amer. Geophysical Union, Pt. VI, 1944.
22. Chow, Ven Te
"Open Channel Hydraulics"
McGrawHill 1959.
23. Horton, R. E., H. R. Leach, and R. Van Wiet
"Laminar Sheet Flow" '
Trans. Amer. Geophysical Union, Vol. 15, 1934.
338

24. Horner, W. W. and S. W. Jens
"Surface Runoff Determination from Rainfall without
Using Coefficients"
Trans. ASCE, Vol. 107, 1942.
25. Wolf, P. O.
"The Influence of Flood Peak Discharges of Some Meteo
rological, Topographical, and Hydraulic Factors"
In srnational Assoc. of Scientific Hydrology, General
Assembly of Toronto, Tome III, 1958, pp 2634.
26. Henderson, F. M.
"Open Channel Flow"
1969.
27
28
29
30
31
Izzard, C. F.
"The Surface Profiles of Overland Flow"
Trans. Amer. Geophysical Union, Part VI, 1944.
Papadakis, C.
"Sewer Systems Design" p,
Athens, Greece, 1969.
45 and 96
Horner, W. W. and F. L. Flynt
"Relation between Rainfall and Runoff from Small Urban
Areas", Trans. ASCE, Vol. 101, p. 140, 1936.
U. S. Weather Bureau Technical Paper No. 40
Chart 26, p. 59, May, 1961.
American Concrete Pipe Association
"Concrete Pipe Field Manual"
1966
32. Tucker, L. S.
"Oakdale Gaging Installation, ChicagoInstrumentation
and Data"
ASCE Urban Water Resources Research Program, Technical
Memorandum No. 2, August 15, 1968.
33. Cosens, K. W.
"Sewer Pipe Roughness Coefficients"
Sewage and Industrial Wastes Journal, January, 1954.
34. Yarnell, D. L.
"Rainfall IntensityFrequency Data"
Miscellaneous Publication No. 204, U.S.D.A., Washing
ton, 1935.
339

35. Pair, G. M., J. C. Geyer, and D. A. Okun
"Water and Wastewater Engineering" Vol. I
John Wiley and Sons, New York 1966
36. Gutman, Irwin and S. S. Wilks
"Introductory Engineering Statistics"
John Wiley & Sons, Inc., 1965
37. Federal Water Pollution Control Administration
"Water Pollution Aspects of Urban Runoff"
by the American Public Works Association, 1969
340

Accession Nt.mihor
Subject /*"io/r/&. Group
SELECTED WATER RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
Organization
University of Cincinnati; Division of Water Resources;
Department of Civil Engineering; Cincinnati, Ohio 45221
Title
URBAN RUNOFF CHARACTERISTICS
10
22
Author(s)
Preul , Herbert C.
and
Papadakis, Constantine
•IJL Project Designation
EPA Rebearcli Grant No. 11024 DQU
2] Note ^
Citation
23
Descriptors (Starred First)
*Storm Water, *Runoff, "'Overflows, ''Watershed Characteristics, *Hydrographs,
*Pollutographs, "'Management Model, Theoretical Models, Computer Program,
Sequential Sampling, Pollution Analyses.
25
Identifiers (Starred First)
Storm V/ater Management, Model Testing
•yj Abstract
This i s an interim report on investigations for the development of a comprehensive
storm water management model
Detailed information on the watershed characteristics and data on runoff quantity
and quality have been compiled from a one year study of a combined sewer watershed
of approximately 2380 acres in Cincinnati, Ohio. Collection of these data is
planned to continue over a several year period. The information collected will be
used to test and develop practical storm water management models.
Abstractor
Herbert C. Preul
Institution
Un 5 ve r s 5 ty of Cin.c i_nn_a_tj ;_Dj_yJ sion of Water Resources
WR:I02 (REV JULY 1969)
WHSIC
SEND TO: 'YVATEIR RESOURCES SCICNTIFIC INFORMATION
U.S. DECP ARTMENLT OF THE INTERIOR
WASHINGTON. D. C. 20140
U. S. GOVERNMENT PRINTING OFFICE: 1979 — 657060/5335


Continued from inside front cover.
11022
11023
11020
11023
— 08/67
09/67
12/67
05/68
11031 08/68
11030 DNS 01/69
11020 DIH 06/69
11020 DES 06/69
11020 06/69
11020 EXV 07/69
11020 DIG 08/69
11023 DPI 08/69
11020 DGZ 10/69
11020 EKO 10/69
11020 10/69
11024 FKN 11/69
11020 DWF 12/69
11000 01/70
11020 FKI 01/70
11024 DDK 02/70
11023 FDD 03/70
11024 DMS 05/70
11023 EVO 06/70
11024 06/70
11034 FKL 07/70
11022 DMU 07/70
11024 EJC 07/70
11020 08/70
11022 DMU 08/70
11023 08/70
11023 FIX 08/70
11024 EXF 08/70
Phase I  Feasibility of a Periodic Flushing System for
Combined Sewer Cleaning
Demonstrate Feasibility of the Use of Ultrasonic Filtration
in Treating the Overflows from Combined and/or Storm Sewers
Problems of Combined Sewer Facilities and Overflows, 1967
(WP2011)
Feasibility of a StabilizationRetention Basin in Lake Erie
at Cleveland, Ohio
The Beneficial Use of Storm Water
Water Pollution Aspects of Urban Runoff, (WP2015)
Improved Sealants for Infiltration Control, (WP2018)
Selected Urban Storm Water Runoff Abstracts, (WP2021)
Sewer Infiltration Reduction by Zone Pumping, (DAST9)
Strainer/Filter Treatment of Combined Sewer Overflows,
(WP2016)
Polymers for Sewer Flow Control, (WP2022)
RapidFlow Filter for Sewer Overflows
Design of a Combined Sewer Fluidic Regulator, (DAST13)
Combined Sewer Separation Using Pressure Sewers, (ORD4)
Crazed Resin Filtration of Combined Sewer Overflows, (DAST4)
Stream Pollution and Abatement from Combined Sewer Overflows •
Bucyrus, Ohio, (DAST32)
Control of Pollution by Underwater Storage
Storm and Combined Sewer Demonstration Projects 
January 1970
Dissolved Air Flotation Treatment of Combined Sewer
Overflows, (WP2017)
Proposed Combined Sewer Control by Electrode Potential
Rotary Vibratory Fine Screening of Combined Sewer Overflows,
(DAST5)
Engineering Investigation of Sewer Overflow Problem 
Roanoke, Virginia
Microstraining and Disinfection of Combined Sewer Overflows
Combined Sewer Overflow Abatement Technology
Storm Water Pollution from Urban Land Activity
Combined Sewer Regulator Overflow Facilities
Selected Urban Storm Water Abstracts, July 1968 
June 1970
Combined Sewer Overflow Seminar Papers
Combined Sewer Regulation and Management  A Manual of
Practice
Retention Basin Control of Combined Sewer Overflows
Conceptual Engineering Report  Kingman Lake Project
Combined Sewer Overflow Abatement Alternatives 
Washington, D.C.

•it.
