Hydrological Theory
Simulation of Large Catchments


From the MIDUSS Version 2
Reference Manual  Chapter 7
(c) Copyright Alan A. Smith Inc. 
Rainfallrunoff
simulation requires certain assumptions with respect to the level of
discretization to be employed and the
parameter values to be used for the subareas. When modelling very
large watersheds a compromise is necessary between using subareas
that are too small or too large. Small subcatchments impose cost
penalties in data preparation and computation effort. Large areas
present problems in assigning values to parameters – such as overland
flow length  that are a reasonable representation of the physical
system.
The ratio of channel
travel time to subarea response time varies widely for areas that are
close to or distant from the outflow point. This variation results in
a diffusion of the flow peaks from individual areas and accounts for a
significant part of the basin lag. In large catchments this basin lag
can equal or exceed the overland flow travel time and attempts to
represent this by distorting overland flow parameters are subjective,
unrealistic and storm specific. The process can be better represented
by convoluting the overland flow response function with the derivative
of the timearea diagram for the total watershed. The resulting
modified response function can then be convoluted with the effective
rainfall to yield a good approximation to the runoff hydrograph for
the total area.
Numerical experiments
suggest that the unwieldy process of double convolution can be
approximated by using a single convolution of overland flow and
rainfall and then routing the resulting hydrograph through a
hypothetical linear channel and linear reservoir. The latter have lag
times that are related to the maximum conduit travel time through the
drainage network. MIDUSS uses some preliminary guidelines to estimate
the lags and partially automate the process.
This section describes
the process used and compares a typical example with a fully
discretized simulation. It must be
emphasized that the suggested method is preliminary and would benefit
from further testing of either real or idealized cases to verify or
improve the guidelines.
Example of a
Large Catchment
The process is
described with reference to the catchment area in Figure 727. The
runoff obtained from the discretized
version will be compared to the approximate 'lumpedparameter’
version. In Figure 727 the area in hectares of the subareas is
shown in italics and in parenthesis in the lowerright corner of each
rectangular area.
Figure 727 –
Discretized version of a large catchment
(Areas shown in hectares as (12) in lower right corners)
The system is
subjected to a 5year storm represented by a 360 minute Chicago
hyetograph with a total depth of 50.45 mm (the MIDUSS default
values). All the subareas are assumed to have the same overland flow
characteristics with the exception of area, i.e.
Overland flow length = 45 m
Overland slope = 2.0 %
Percent impervious = 30 %
Pervious roughness n = 0.25
Impervious roughness n = 0.013
With this simplifying
assumption, the response functions of the subareas will have the same
time parameters and vary only with respect to the area. Thus, for a
triangular response function:
The drainage network
is composed of pipes with a gradient of approximately 0.4%. The
output file from the discretized
simulation is called ‘Large1.out’ and can be found in the ..\Samples\
folder of the Miduss98 directory.
The distribution of
areas relative to the outflow point can be represented by a timearea
diagram as illustrated in Figure 7‑ 28. From Figure 727 you will
note the rather circuitous (and unrealistic) drainage path of areas 1,
2 and 3. This gives rise to the late contribution of the furthest
upstream 32 ha which in turn makes the TimeArea diagram depart from
the reasonably linear shape which is apparent over the first 15
minutes. This feature makes the approximation more challenging.
Figure 728 – TimeArea
diagram for the catchment of Figure 727
The time to
equilibrium Te
is 23.21 minutes from node #1 to the outflow point at node #22. This
is the time at which the entire catchment is contributing and is a
function of the drainage network. The time of concentration
tc
and therefore the timebase
tb
of the response function, is a characteristic of the overland flow.
Both quantities are also dependent on the magnitude of the storm.
The relative magnitude
of Te
and
tc
is an important parameter in determining the limit for lumped
representation of a catchment. If
Te /
tc << 1.0 then it is
likely that overland flow dominates the runoff process and thus the
overland flow response function is a reasonable approximation for the
entire catchment. In large catchments,
Te /
tc is larger (although
still probably less than 1.0) and channel/pipe routing will play an
important role in determining the shape and peak of the runoff
hydrograph.
Combining
Overland Flow and Drainage Network Routing
The combined effect of
overland flow routing and drainage network routing can be obtained by
convoluting one response function with the other. The right side of
Figure 729 shows a triangular response function being convoluted with
the derivative of the timearea diagram to produce a modified response
function. This is then convoluted with the hyetograph of effective
rainfall to produce the modified runoff hydrograph in the lower right
corner of the figure.
The unwieldy process
of double convolution can be approximated by the process shown on the
left side of the Figure 729. The normal overland flow response
function is convoluted with the effective rainfall to produce a
‘lumped’ runoff hydrograph. This assumes that all the subareas
contribute to runoff simultaneously. This is then routed through a
linear channel and linear reservoir. If appropriate values can be set
for
Kch
and
Kres
the resulting hydrograph should be a close approximation of the
modified runoff hydrograph.
Figure 729 –
Representation of the Lag and Route method
The total lag
Ktot is defined as the
sum of the two components
Kch
and
Kres,
i.e.
The distribution of
the total lag between the two constituent parts is defined by a
fraction r
(0.0
< r < 1.0) as follows.
and
Estimating the
Lag Values
Results from a limited
number of numerical experiments suggest that some correlation exists
between:
·
The total lag
Ktot and the ratio of
time to equilibrium to time of concentration (Te/tc),
and
·
The fraction
r =
Kres/Ktot and the basin
time to equilibrium
Te
As preliminary
guidelines the following relationships are used in MIDUSS.
Values for the
fraction r =
Kres/Ktot
are based on a curve which – for the data
analyzed – approaches an asymptotic value of about 0.55 as shown in
Figure 730.
Figure 730 – Empirical
curve defining r = Kres/Ktot = f(Te)
The data of Figure
730 is contained in a small data file called ‘LagRout1.dat’ which
resides in the Miduss98 folder. This file can be edited or updated as
and when further numerical experimental results are available. The
trends suggested are interesting but inconclusive. More experiments
are required to test the sensitivity of the identified parameters to
factors such as:
·
Shape and duration of storm
·
Size and shape of the
catchment
·
Choice of overland routing
model
·
The rainfall loss model
employed.
Until such time as
further test results are available you should use this feature with
caution. When possible, it is useful to carry out a comparison
between the Lag and Route approximation and a typical
discretized simulation to provide a
measure of confidence in the method. The next section describes the
results obtained for the catchment of Figure 727.
Comparison of
Discretized and Approximate Results
Refer to the output
file …\MIDUSS\Samples\Large1.out for details of the test described
here.
The fully
discretized simulation produced a peak
runoff of 19.136 c.m/s.
The lumped catchment
runoff is found to have a peak of 26.488 c.m/s
The Lag and Route
command is then used with MIDUSS default values for all quantities
with the exception of the catchment area aspect ratio which is set at
2000m/1400 m or 1.43 and the average pipe slope of 0.4%. The form is
shown in Figure 731 below. The longest drainage path estimated by
MIDUSS is 3397 m whereas scaling the reach from node #1 the length is
3900 m. The underestimate is close to 13% and is due to the
circuitous route from node #1 to node #5. The error will result in a
slightly higher peak flow for the reduced peak flow that is shown as
20.217 c.m/s. A graphical comparison of
the results is shown in Figure 732. Apart from the overestimated
peak flow the approximation is reasonable.
If the Stream length
is entered as 3900 m as a result of scaling the drawing (Figure 7_27)
the result is improved. The peak of the approximate runoff is reduced
to 19.745 c.m/s with no measurable change
in the general agreement between the discretized
and approximate runoff hydrographs. The effect of the change in
stream length is summarized in the Table below.
Stream length
(m) 
Kch
(min) 
Kres
(min) 
Ktot
(min) 
r 
Qpeak
(c.m/s) 
3397 
1.984 
6.033 
8.017 
0.753 
20.217 
3900 
2.725 
6.508 
9.233 
0.705 
19.745 
Discretized
simulation 
19.136 
By checking the output
file you will also see that continuity is respected and the total
runoff volume is given as 6.2605 ham in all cases.
You can experiment
with this example by running the file ‘Large1.out’ in automatic mode.
After generating the database Miduss.Mdb, navigate to the
Hydrograph/Start new tributary command following completion of the
discretized simulation. Change the
command from ‘40’ to ‘ 40’ and then use the [RUN] button in the
Automatic Control Panel to run up to that point. You can then step
through the ‘lumped’ runoff calculation and the Lag and Route
approximation using the [EDIT] command button or in Manual mode
Figure 731 – Using the
Lag and Route command
Figure 732 – Comparison
of the Lag and Route approximation with the
discretized runoff.
