Instability of gravity wetting fronts for Richards equations with hysteresis

Abstract

We study the evolution of saturation profiles in a porous medium. When there is a more saturated medium on top of a less saturated medium, the effect of gravity is a downward motion of the liquid. While in experiments the effect of fingering can be observed, i.e., an instability of the planar front solution, it has been verified in different settings that the Richards equation with gravity has stable planar fronts. In the present work we analyze the Richards equation coupled to a play-type hysteresis model in the capillary pressure relation. Our result is that, in a homogeneous medium, imposing appropriate initial and boundary conditions, the planar front solution is unstable. In particular, we find that the Richards equation with gravity and hysteresis does not define an $L^1$-contraction.