 07 Nov 2022
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 DarkLight
Muskingum Method
 Updated on 07 Nov 2022
 1 Minute to read
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 DarkLight
It can be shown that the outflow and storage are related in the form of an infinite series as follows:

where a, x and n are constants unknown at the outset. 
If this series is truncated in the second term, Equation (1) can be expressed as:


From this equation, dS/dt can be obtained and replaced in the continuity equation to obtain the following storage expression :


Replacing (1/a)(1/n) by k and taking n as unity we obtain the storage equation for the Muskingum method, that is:

where: k is a storage constant and expresses the ratio of storage to discharge and has the dimension of time. It can be approximated by considering the travel time through the reach the constant x considers the relative importance of inflow and outflow in determining storage. For many cases x lies between 0.0 and 0.3 
If Equation (4) is rewritten as:


and substituted in the discretised storage equation (see Storage Equation equation (2)) we obtain an explicit equation for the outflow at the end of the time step where all terms on the righthand side are known:

where:

Combining the equations for C_{0}, C_{1} and C_{2} gives :


Also, all the coefficients must be positive in order to obtain valid results.
It is obviously important to choose appropriate values for k and x. If data from measured flood hydrographs are available, equation (5) can be used to plot storage as a function of the weighted inflow and outflow by using different values of x as shown in Storage Equation (Figure 1).
The value of x that gives the curve that is closest to a straight line is assumed to be the most appropriate value to use. The value of the parameter k comes from the slope of the straight line.