Geoscience Reference
In-Depth Information
Practical implementation: linear or nonlinear?
In the standard application of the Muskingum method the parameters
K
and
X
are usually
considered as constants, and treated as characteristics of the channel reach in question.
However, the physically based expressions (7.53) and (7.54) show how in reality
K
and
X
depend on the reference rate of flow
Q
0
, once
x
has been decided upon. As long as
the actual flow rates
Q
are only small deviations from the reference flow
Q
0
, a linearized
algorithm can be expected to perform well. Flood waves normally involve large deviations,
and since the Muskingum method is essentially linear, the question arises what value should
be assigned to
Q
0
in Equations (7.53) and (7.54) to ensure optimal results. A few studies
have focused on this issue.
Actually, the form of (7.53) and (7.54) opens up the possibility of applying the Musk-
ingum method in a nonlinear way. Cunge (1969) already suggested that
Q
0
be adjusted to
allow extrapolation of the Muskingum parameters beyond the range of previously observed
events. Miller and Cunge (1975, p. 226) subsequently treated the parameters as functions
of time
K
dA
c0
) the values cor-
responding to the flow rate
Q
at that time and adjusting them at every time step of the
computation; they applied this technique to a channel with a compound cross section. In the
same vein, Koussis (1978) proposed an adjustment of
K
at each computational time step,
by means of the uniform rating curve to estimate (
dQ
0
/
dA
c0
) in (7.53); but while
X
can
also be easily adjusted as a function of
Q
, he found from his analysis of wave propagation
on the Rhine, that the results tend to be relatively insensitive to the exact value of
X
and
that therefore a constant value should be adequate. On the other hand, Ponce and Yevjevich
(1978) concluded that the overall difference between a linear and nonlinear application of
Equations (7.53) and (7.54) in (7.37) is usually quite small. In addition, they obtained better
results using an average value of
Q
to represent
Q
0
, but opined that the use of the peak value
of the hydrograph as
Q
0
might be easier to implement in practice.
=
K
(
Q
(
t
)) and
X
=
X
(
Q
(
t
)), by assigning to
Q
0
and (
dQ
0
/
Example 7.4. Application of the MCD method
Consider again the inflow and outflow hydrographs of Example 7.2 and illustrated in
Figure 7.11. Assume that these flow rates took place in a river channel with an effective
width of
B
c
=
=
0.035. The length of the reach, that is the distance between the inflow and outflow section
was taken as
170 m, an effective slope
S
0
=
0.0004 and an effective roughness
n
=
x
23 km. In Example 7.2 the Muskingum parameters were found to be
=
=
K
0.30. It is easy to check that Equations (7.53) and (7.54) produce
the same values of these parameters with an assumed reference value of the discharge
rate of roughly
Q
0
=
1.99 h and
X
2000 m
3
s
−
1
; by means of the Gauckler-Manning equation (5.41)
this can be shown to correspond with a water depth
h
0
=
6.14 m and a reference velocity
92ms
−
1
. The hydrograph calculated with these values of
K
and
X
has already
been compared in Figure 7.10 with the observed outflow hydrograph. The sensitivity of
the MCD method to the value of the assumed reference discharge rate
Q
0
can be tested by
carrying out the calculations with two different values, say
Q
0
=
V
0
=
1
.
1500 and 2500 m
3
s
−
1
.
1500 m
3
s
−
1
, in the same river channel (5.41) produces a velocity
In the case of
Q
0
=
1.71 m s
−
1
V
0
5.17 m. With these values one obtains
with (7.53) and (7.54) the parameter values
K
=
and a water depth
h
0
=
=
2.24 h and
X
=
0.34. In the case of
