A mass conservative and water storage consistent variable parameter Muskingum-Cunge approach

The variable parameter Muskingum-Cunge (MC) flood routing approach, together with several variants proposed in the literature, does not fully preserve the mass balance, particularly when dealing with very mild slopes (<10<sup>−3</sup>). This paper revisits the derivation of the MC and demonstrates (i) that the loss of mass balance in MC is caused by the use of time variant parameters which violate the implicit assumption embedded in the original derivation of the Muskingum scheme, which implies constant parameters and at the same time (ii) that the parameters estimated by means of the Cunge approach violate the two basic equations of the Muskingum formulation. The paper also derives the modifications needed to allow the MC to fully preserve the mass balance and, at the same time, to comply with the original Muskingum formulation in terms of water storage. The properties of the proposed algorithm have been assessed by varying the cross section, the slope, the roughness, the space and the time integration steps. The results of all the tests also show that the new algorithm is always mass conservative. Finally, it is also shown that the proposed approach closely approaches the full de Saint Venant equation solution, both in terms of water levels and discharge, when the parabolic approximation holds