Numerical scheme for solving a porous Saint-Venant type model for water flow on vegetated hillslopes

Hillslope hydrology is a very important part of research based on watershed hydrology. In this study, we focus on water flow over a soil surface with vegetation in a hydrographic basin. We introduce a partial-differential-equation model based on the general principles of fluid mechanics where the unknowns are the depth and velocity of water. The effect of vegetation on the dynamics of water is explained in terms of porosity (a quantity that is related to the density of vegetation) that is a function defined over the hydrological basin. Using a Finite Volume scheme for discretization in space, we introduce an ordinary-differential-equation system that constitutes the base of the discrete model that we are working with. We discuss and investigate several properties of this model that have a physical relevance. Finally, we perform different quantitative validation tests by comparing numerical results with exact solutions or with laboratory-measured data. We also consider some qualitative validation tests by numerically simulating the flow on a theoretical vegetated soil and on a real hydrographic basin.