Geoscience Reference
In-Depth Information
purpose Equation (7.11) can be approximated as
1
2
(
Q
i1
+
1
2
(
Q
e1
+
Q
i2
)
t
−
Q
e2
)
t
=
S
2
−
S
1
(7.35)
in which the subscripts 1 and 2 refer to the beginning and the end of the time period
t
.
Upon substitution of (7.15) this becomes
1
2
(
Q
i1
+
1
2
(
Q
e1
+
Q
i2
)
t
−
Q
e2
)
t
=
K
[
XQ
i2
+
(1
−
X
)
Q
e2
]
−
[
XQ
i1
+
(1
−
X
)
Q
e1
]
}
(7.36)
{
After collecting the terms, (7.36) can be written as a coefficient equation
Q
e2
=
c
0
Q
i2
+
c
1
Q
i1
+
c
2
Q
e1
(7.37)
in which
−
2
X
+
t
/
K
2
X
+
t
/
K
c
0
=
K
,
c
1
=
and
−
+
/
−
+
/
2(1
X
)
t
2(1
X
)
t
K
2(1
−
X
)
−
t
/
K
c
2
=
(7.38)
2(1
−
X
)
+
t
/
K
with the obvious requirement that (
c
0
+
c
1
+
c
2
)
=
1.
Constraints on the parameters
In practical applications, the parameters in the Muskingum method must satisfy a num-
ber of constraints, if it is to perform well. When the method was originally developed not
much attention was paid to this issue, as the basic underlying assumptions were not fully
understood; therefore, the method sometimes produced unreasonable results (such as neg-
ative flow rates). The values of the weighting parameter
X
, of the time step
t
, and of the
length of the channel reach
x
, affect the outcome of the calculations, so that some attention
should be given to their choice.
(i)
As already discussed earlier, Equation (7.25) indicates that
X
should not exceed 0.5;
indeed values in excess of 0.5 would indicate that the flood peak magnitude increases
as it moves downstream; this never occurs in situations where the lumped kinematic
approach is applicable. Moreover, a negative value of
X
would indicate in (7.15) that
a larger inflow rate into the reach results in a smaller storage. Thus, one can constrain
X
as follows
0
≤
X
≤
0
.
5
(7.39)
(ii)
The Muskingum method involves several time scales, namely the finite time step of
the numerical solution
t
, the time of travel through the channel or lag
K
, and the
characteristic life time of the incoming flood wave, say its time to peak
t
p
. To allow
sufficient resolution of the temporal behavior of the flood wave, it stands to reason
that
t
should be small compared with the life time of the incoming flood. Therefore
it is usually assumed (see Jones, 1981; Ponce and Theurer, 1982) that
t
≤
at
p
(7.40)
in which
a
is a number of the order of 4 to (preferably) 5.
