Hydrological Theory
Calculating Runoff
SWMM  RUNOFF Algorithm 

From the MIDUSS Version 2
Reference Manual  Chapter 7
(c) Copyright Alan A. Smith Inc. 
Figure 722 
Representation of the SWMM/RUNOFF algorithm.
The U.S. EPA SWMM
model is made up of a number of large program modules. One of these 
the RUNOFF block  is used to generate the runoff hydrograph from a
subcatchment. In MIDUSS, the ‘SWMM Method’ option uses a similar
algorithm with the limitation that only the Horton or Green and
Ampt infiltration equations are supported.
The method employs the
surface water budget approach and may be visualized as shown in Figure
722. The incident rainfall intensity is the input to the control
volume on the surface of the plane; the output is a combination of the
runoff Q
and the infiltration
f.
Considering a unit breadth of the catchment the continuity and dynamic
equations which have to be solved are as shown in equations [744] and
[745].
where
L
= overland flow length
B
= catchment breadth
CM
= 1.0 for metric units
1.49 for Imperial or
US customary units
n
= Manning roughness coefficient
y_{d}
= surface depression storage depth
Rewriting [744] with
q = Q/B
and then substituting for
Q
by means of [745] yields the single equation:
or
If the depth on the
plane at the start and finish of the time step
Dt
is represented by
y_{1}
and y_{2}
respectively an equation for
y_{2}
can be developed using the following approximations.
for y>0
Equations [747] ‑
[749] are solved using a Newton Raphson
method to yield a solution for
y2
which is then used to obtain a value for
Q.
It can be shown
(Smith, 1986a, see references) that the algorithm developed above is
equivalent to convoluting the storm rainfall with a
Dirac
dfunction
and then routing the resulting 'instantaneous' runoff through a
nonlinear reservoir with storage characteristics given by:
where
In equation [750] 'A'
is the catchment drainage area and other terms are as defined
previously (see equations [741] and [745]).
Three points of some
significance arise with respect to the ‘SWMM Method’ option.
(1) Considering the method to be
equivalent to routing the instantaneous runoff through a nonlinear
reservoir, it follows that the peak of the outflow must lie on the
recession limb of the inflow. Consequently the time to peak for
pervious and impervious fractions will not differ significantly and
the total runoff will not exhibit the double peaked hydrographs which
are sometimes encountered with the ‘Rectangular’ or ‘Triangular SCS’
options.
(2) The form of equations [744]
and [745] implicitly assumes that the depth of flow over the plane is
quasiuniform. This overestimates the volume on the plane and will
usually result in overattenuation of the peak runoff.
(3) Since infiltration is assumed
to continue over the entire surface after cessation of rainfall as
long as the average depth is finite, the recession limb of the runoff
hydrograph will generally be much steeper than for the ‘Rectangular’
or ‘Triangular SCS’ options. In practice, after cessation of
rainfall, the surface water tends to concentrate in pools and rivulets
so that the area over which the infiltration continues is likely to be
much less than the total area A. A more realistic
representation of the infiltration after the storm is likely to be
intermediate between the two extreme cases represented by the ‘SWMM
Method’ method on one hand and the ‘Triangular SCS’ or ‘Rectangular’
method which employs the concept of effective rainfall. This feature
is sufficiently important that a detailed example is presented in the
following section in order to illustrate the fundamental difference
between the methods.
An Example of the SWMM Runoff Algorithm
The object of this example is to compare
the overland flow that is generated by the ‘SWMM Method’ option with
that which would be obtained using an effective rainfall approach.
For simplicity we shall assume a catchment of 5.0 ha with no
impervious area and no depression surface storage. The storm used is
a 3rd quartile Huff storm with a total rainfall depth of 30 mm
occurring in 60 minutes. Infiltration will be modelled by the Horton
method with the following parameter values:
·
n
= 0.25
·
f0
= 40 mm/hour
·
fc
= 20 mm/hour
·
K
= 0.25 hours
·
y_{d}
= 0
To simulate the SWMM
algorithm using an effective rainfall approach we shall make use of
equation [7.50] to define a nonlinear reservoir through which the
instantaneous runoff is routed. This hydrograph can be created by
convoluting the effective rainfall with an impulse (also known as a
Dirac
dfunction)
which can be simulated by specifying a very short overland flow
length.
The steps are
summarized as follows. You may find it instructive to run this
example on your own computer as you read through the steps.
·
In the Time Parameters use 2
minute timesteps and a storm duration of
60 minutes.
·
Define the Huff storm; use 30
mm rainfall; 60 minutes duration; 3rd quartile.
·
The impervious
characteristics are not important but we must use the Horton method.
Set n = 0.015 and the other parameters to zero.
·
The first catchment 101 is
used to represent the Dirac
dfunction
so use the following parameters.
Area = 5.0 ha
Length = 0.1 m
Slope = 2.0 %
Percent impervious = 0
For
the infiltration parameters use the Horton method with:
fo
= 40 mm/hour
fc
= 20 mm/hour
K
= 0.25 hour
yd
= 0
The peak effective
rainfall intensity is found to be 68.279 mm/h. Use the ‘Rectangular’
option since this most closely approximates an impulse. The peak
runoff is 0.948 c.m/s. A few seconds with
a calculator will confirm that for an area of 5 hectares this is
equivalent to 68.279 mm/h.
Figure 723 – Statistics of the Diracd
response hydrograph
·
We want to route this runoff through an imaginary pond
with stage discharge characteristics as given by [7‑ 50]. Use the
Hydrology/Add Runoff command to define the inflow to the pond to
be used in step (7).
·
The final step to simulate the SWMM hydrograph is by
routing the instantaneous runoff through a nonlinear reservoir. For
the data used in this example, the value of
C in equation [750] works
out to be 1115.39. We now define a pond with discharges ranging from
0.0 to 0.25 in increments of 0.025  i.e. 11 stages. Figure 725
shows the result of a pond design. The value of each storage volume
is given by 1115.39 x Q^{0.6}.
The peak outflow is found to 0.246 c.m/s
In Figure 725, the hydrograph has been extended to 130 minutes.
Figure 724 ‑ Design of
a hypothetical pond.
·
The next step is to generate
the ‘SWMM method’ hydrograph, so define another catchment with the
same parameters as in step (4) but with a flow length of 50 m. The
infiltration options are the same as before. The peak runoff is found
to be 0.248 c.m/s.
The similarity in peak
flows is promising, but the true test is to compare the plotted
hydrographs. Figure 724 shows the ‘SWMM Method’ and simulated SWMM
hydrographs
The rising limbs of
the two hydrographs are in good agreement apart from a slight lag of
about 2 minutes which is the shortest 'impulse' that MIDUSS can create
when
Dt
is 2 minutes. However, immediately following the cessation of the
effective rainfall the ‘SWMM Method’ recession limb drops more
steeply. This is due to the fact that the surface water budget method
assumes that infiltration continues as long as there is excess water
on the pervious surface whereas the effective rainfall approach 
which produced the longer curve in Figure 725  assumes that
infiltration stops at the end of the effective rainfall. The two
recession limbs start to diverge at
t
= 50 min. which marks the end of the effective rainfall hyetograph.
Figure 725 ‑ Comparing
the SWM HYD and simulated SWMM hydrographs.
Figure 726 ‑ Statistics
of the ‘SWMM Method’ hydrograph.
This difference serves
also to explain the anomaly that appears in the hydrograph statistics
screen when using ‘SWMM Method’ option. Figure 726 shows the
summary statistics obtained at the end of step (7) and you will note
that the runoff volume (296.2 c.m) is much
less than that for the effective rainfall volume of 636.38
c.m. (from Figure 723).
This may not always be
the case and you should repeat this experiment with a finite value for
depression surface storage  say 2 mm or 100 c.m
over the 5 hectares of area. You will find that the effective
rainfall volume is reduced by exactly 100 c.m.
The infiltration and runoff volume are also reduced by amounts which
add up to 100.0 c.m. less the volume still
trapped in surface depressions when the calculation was ended. If
continued long enough, this too would have infiltrated thus balancing
the books properly. Hence the name 'surface water budget'.
