Research papersOn mass and momentum conservation in the variable-parameter Muskingum method

- Mass and momentum conservation for finite-length rectangular and trapezoidal channel reaches.
- Formulation of the relative momentum error for a channel reach.
- Inter-comparison of the Saint-Venant (SV) solutions with the Mike11, SOBEK and HEC-RAS models.
- Mass and momentum errors for the SV solutions and the VPM are comparable.
- Application of the VPM to the river Moselle with discharge and water levels simulations.

In this paper we investigate mass and momentum conservation in one-dimensional routing models. To this end we formulate the conservation equations for a finite-dimensional reach and compute individual terms using three standard Saint-Venant (SV) solvers: SOBEK, HEC-RAS and MIKE11. We also employ two different variable-parameter Muskingum (VPM) formulations: the classical Muskingum-Cunge (MC) and the revised, mass-conservative Muskingum-Cunge-Todini (MCT) approach, whereby geometrical cross sections are treated analytically in both cases. We initially compare the three SV solvers for a straight mild-sloping prismatic channel with geometric cross sections and a synthetic hydrograph as boundary conditions against the analytical MC and MCT solutions. The comparison is substantiated by the fact that in this flow regime the conditions for the parabolic equation model solved by MC and MCT are met. Through this intercomparison we show that all approaches have comparable mass and momentum conservation properties, except the MC. Then we extend the MCT to use natural cross sections for a real irregular river channel forced by an observed triple-peak event and compare the results with SOBEK. The model intercomparison demonstrates that the VPM in the form of MCT can be a computationally efficient, fully mass and momentum conservative approach and therefore constitutes a valid alternative to Saint-Venant based flood wave routing for a wide variety of rivers and channels in the world when downstream boundary conditions or hydraulic structures are non-influential.