Instability of gravity wetting fronts for Richards equations with hysteresis

Abstract

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 an appropriate geometry and with adequate 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.

Hysteresis models and gravity fingering in porous media

“…Our study extends previous results of [7,8,15]. In [8], we have shown analytically that the full hysteresis model is well-posed for the Richards equation, and we have shown the well-posedness for the two-phase flow model in [7].…”

“…For the configuration of Figure 1, right, the Richards equation with static hysteresis and switching in the boundary condition, we have shown an instability result in [15]. With the gravity term, the system possesses unstable front solutions.…”

“…In [8], we have shown analytically that the full hysteresis model is well-posed for the Richards equation, and we have shown the well-posedness for the two-phase flow model in [7]. In [15], we have shown rigorously that the Richards equation with static hysteresis does not define an L 1 -contraction. In particular, we have shown that an instability occurs for these equations, making fingering possible.…”

Travelling wave solutions for the Richards equation incorporating non-equilibrium effects in the capillarity pressure

“…Such regularisation has been used in [39,42] for proving the existence of weak solutions to such models, and for developing appropriate numerical schemes. It is an intriguing question whether this approach is motivated physically, and in particular how the parameter ε can be interpreted from physical point of view.…”

“…In general, one cannot expect that solutions exist in a classical sense. We refer to [5,6,12,13,14,25,28,27,41,42,43] for results concerning the existence and uniqueness of weak solutions for hysteresis models, dynamic capillarity models, or for models including both effects. In particular we refer to [41,42,13] where, as suggested in [3,4], (2.12) is used to express ∂ t S as a function of S and p. We rely on the same idea for the TW analysis below.…”