The Muskingum method is a commonly used hydrologic routing method in situations requiring a variable storage-discharge relationship (Chow et al., 1988). The Muskingum method models the storage volume of flooding in a river channel using a combination of wedge and prism storage (see schematic below). The key parameters in Muskingum routing are K (travel time) and X (weighting coefficient). The value of X depends on the shape of the wedge storage to be modeled, and the value of X ranges from 0 for reservoir-type storage to 0.5 for a full wedge. In natural streams, X is between 0 and 0.3 with a mean value near 0.2 (Chow et al., 1988). K is the time required for an incremental flood wave to traverse its reach, and it may be estimated as the observed time of travel of peak flow through the reach (Chow et al., 1988). If observed inflow and outflow hydrographs are available for a river reach the values of K and X can be determined to provide the best fit (or narrowest loop) relative to the observed flows (Veissman and Lewis, 2003).
The Muskingum method assumes that water surface in the reach is a uniform unbroken surface profile between upstream and downstream ends of the section. It also assumes that K and X are constant through the range of flows (Veissmann and Lewis, 2003). The Muskingum parameters (K and X) are best derived from stream flow measurements and are not easily related to channel characteristics.
To overcome these limitations, Cunge (1969) proposed a method based on the Muskingum method which provides an approximate solution of the kinematic wave equations. This method, Muskingum-Cunge method, is one of the most recommended techniques for general use. It is classified as a hydrologic method, yet it gives results comparable with hydraulic methods (Veissman and Lewis, 2003). In the Muskingum-Cunge method, the K parameter is the travel time for a wave to travel the routing reach length and is dependent on the celerity and the reach length. The celerity is velocity with which a variation in flow rate travels along a channel (i.e. the wave speed). The value for X is also calculated from channel and discharge characteristics. As a result, the values for K and X both change with respect to time and space in the Muskingum-Cunge method.
The reach container in this model, "Rch1", implements a modified Muskingum channel routing with constant parameters. The implementation presented here follows that of Neitsch et al. 2005. As mentioned, the key parameters in Muskingum routing are K (travel time) and X (weighting coefficient). The travel time K is estimated as a function of the celerity, the speed with which a new flow rate propagates downstream which is similar to the Muskingum-Cunge method except that K does not vary with distance along the reach. K is calculated as part of "Rch1" container calculations based on estimates of the reach's celerity during bankfull and 10 percent of bankfull conditions.
It is necessary to enter a bankfull discharge, Manning's N parameter, slope, length, width, number of segments to represent the reach, an averaging weighting coefficient (X), and weighting coefficient for celerity. The weighting coefficient for celerity is the weight that should be given to the celerity calculated for the bankfull discharge. The weighting coefficient should be between 0 and 1. A weight of 1 uses the bankfull discharge celerity; a weight of zero uses only the celerity calculated for discharge at 10 percent of bankfull. In any case, the weighted estimate of celerity is used for all routing, regardless of changes in inflow. This differs from variable parameter Muskingum Cunge routing, where the celerity is calculated with each change in flow. The modified Muskingum method given here was selected because it is simple and it conserves mass completely. A common problem in variable parameter schemes is mass conservation.
Chow, V.T., Maidment, D.R., and Mays, L.W. (1988). Applied Hydrology. Tata McGraw-Hill Education (reprint edition). 572 pages.
Cunge, J.A. (1969). On the subject of a flood propogation method (Muskingum method), Journal of Hydraulics Research, International Association of Hydraulics Research, v. 7, no. 2, pp. 205-230.
Neitsch, S.L., Arnold, J.G., Kiniry, J.R., and Williams, J.R. (2005). Soil and Water Assessment Tool (SWAT): Theoretical Documentation, Version 2005.
Veissman, W. and Lewis, G.L. (2003). Introduction to Hydrology. 5th Ed. Prentice Hall: Upper Saddle River, NJ. 612 p.