Derivation of weakly hydrodynamic models in the Dupuit–Forchheimer regime

The current study is dedicated to the formal derivation of a hierarchic of asymptotic models that approximate the groundwater waves problem within the Dupuit–Forchheimer regime, over a regular, non-planar substratum. The derivation methodology employed bears resemblance to the techniques utilized in hierarchic of asymptotic models for approximating the water waves problem in the shallow water regime. Mathematically speaking, the asymptotic models manifest as nonlinear, non-local diffusion equations. We identify an energy dissipation law inherent to these models, thereby bolstering the physical validity and confidence in the proposed framework. A numerical strategy is proposed that preserved at the discrete level the energy dissipation.