A new criterion for the evaluation of the velocity field for rainfall-runoff modelling using a shallow-water model

The aim of this work is to evaluate the accuracy of physically based models using the 2D Shallow Water Equations (SWE) for the simulation of rain-induced runoff. A key expectation of these models is to be able to reproduce properly both hydrograph at the outlet of the watershed and the associated velocity field within the spatial domain for a given rain event. However, this ability can be very challenging in mountainous environments, which include heterogeneous and high values of bottom slope and transient rain inputs. In this paper, we focused on both the error due to the numerical resolution of the SWE for low water depths runoff on steep slopes and that due to the physical representation of friction on the bottom for various water depth to roughness height ratios. Four test cases have been used, at different scales, to evaluate the model in comparison with experimental data. The first test case (from Kirstetter et al. (2015)) was a rain induced flow in a straight channel. It allows to evaluate the error made for the chosen numerical resolution of the SWE and its impact on the velocity distribution in the domain (metric scale). A criterion defining an acceptable numerical error has been proposed. A second test case (from Cea et al. (2014)), has been chosen for being also metric scaled and in the right range of the numerical error criterion. It allowed to evaluate the ability of a new friction law (adapted from Lawrence (1997)) to represent the measured velocity field in several inundation regimes at the metric scale. The third test case was a 40 m 2 real plot studied by Tatard et al. (2008). On this test case, measured velocities have been compared with the simulated ones. Thanks to an infiltration source term added to the mass conservation equation, hydrographs at the outlet of the plot have been well reproduced, as well as the main part (including the highest values) of the measured velocities. In particular, the velocities are well represented when the criterion given in the first test case is satisfied. Finally, the model is applied to a real watershed with an area of 1 km(2) on which hydrographs at the outlet have been successfully reproduced for two extreme rainy events (Le Bouteiller et al., 2015). It appeared that both physical and numerical error were low in the hydrographic network and that the local velocities are correctly reproduced there, unlike in the hillslopes where the velocities are not well reproduced even if the mass fluxes are correct.