This project focuses on the formal derivation of a unified model for groundwater and surface water flow, combining the shallow-water model and the Dupuit--Forchheimer model. The primary goal of the model is to describe the dynamics of the water table, which represents the level below which the medium is fully saturated, while disregarding unsaturated regions. The unified shallow-water/Dupuit--Forchheimer model provides a single set of equations across the domain. It transitions to the shallow-water model in areas where the porous medium is absent and to the Dupuit-Forchheimer model when the water table remains below the surface. The derivation is carried out in two steps: first, a vertical integration separates the ground and surface media, followed by the application of a low-permeability limit in the ground layer. The model ensures conservation of both mass and energy. A numerical scheme that preserves these properties at the discrete level, along with several numerical experiments, demonstrates the robustness and efficiency of the approach.
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