Flood routing in long channels: Alleviation of inconsistency and discharge dip in Muskingum-based models

The Muskingum parameters are often expressed as a function of discharge and channel properties (Cunge, 1969; Chow et al., 1988) by referring to the coefficients of the advection-diffusion (A-D) equation or the equivalent parabolicized Saint Venant equation. As extensively investigated in this paper by Taylor series expansion, the Muskingum model and its variants are found tangibly inconsistent to the A-D equation. In addition, these Muskingum-based models experience the effect of negative outflows, i.e., the well-known dip phenomenon. To avoid these problems.. which are both present in the conventional routing procedure, this paper introduces an extra term to the traditional Muskingum storage function, that is then linked to Gill's concept of initial storage. Through this technique, not only the dip phenomenon but also the model inconsistency can be alleviated, and a fairly satisfactory outflow prediction can thereby be achieved. A proper time to employ the extra term is sought by a convolution integral which is a result of the Laplace transformation applied to the Muskingum model. It also represents the analytical expression of outflow discharge. The convolution integral enables us to trace the origin of discharge dip and quantify the shape variation of outflow hydrographs. With the convolution integral, compact models for computing the maximum flow dip, dip, and its occurrence time, t(c), are also offered in this study. As a concluding example, routings by the traditional Muskingum model, Gill's procedure, and the newly developed algorithms having the extra term are performed in a long channel reach of 90 kilometers to test the robustness of each model.