The Richards equation with hysteresis and degenerate capillary pressure

Abstract

We study the Richards equation with a dynamic capillary pressure, including hysteresis. We provide existence and approximation results for degenerate capillary pressure curves pc, treating two cases. In the first case, the permeability function k can be degenerate, but the initial saturation does not take the critical values. In the second case, the permeability function k is strictly positive, but the capillary pressure function can be multi-valued. In both cases, the degenerate behavior of pc leads to the physically desired uniform bounds for the saturation variable. Our approach exploits maximum principles and relies on the corresponding uniform bounds for pressure and saturation. A new compactness result for the saturation variable allows to take limits in nonlinear terms. The solution concept uses tools of convex analysis.