A continuum model of unstable infiltration in porous media endowed with an entropy function

We propose a thermodynamic approach to modeling unsaturated flow in porous media, where the liquid saturation is understood as the state variable. The free energy functional is designed as a symmetric expansion of the traditional capillary energy density in Richards equation, therefore removing ambiguities on the interpretation of the higher-order term in the model equation. The proposed definition renders a formulation that leads naturally to an entropy function of the system, and we show that the model describes an entropy-increasing process for an isolated system. The new formulation reproduces gravity fingering during infiltration in soil. We show that the nonlinear and singular structure of the capillary pinning function in the fourth-order term plays a fundamental role in the behavior and stability of infiltration fronts, promoting front pinning and the persistence of fingered infiltration at relatively large flux ratios.