A two dimensional semi-continuum model to explain wetting front instability in porous media

Modelling fluid flow in an unsaturated porous medium is a complex problem with many practical applications. There is enough experimental and theoretical evidence that the standard continuum mechanics based modelling approach is unable to capture many important features of porous media flow. In this paper, a two-dimensional semi-continuum model is presented that combines ideas from continuum mechanics with invasion percolation models. The medium is divided into blocks of finite size that retain the nature of a porous medium. Each block is characterized by its porosity, permeability, and a retention curve. The saturation and pressure of the fluids are assumed to be uniform throughout each block. It is demonstrated that the resulting semi-continuum model is able to reproduce (1) gravity induced preferential flow with a spatially rich system of rivulets (fingers) characterized by saturation overshoot, (2) diffusion-like flow with a monotonic saturation profile, (3) the transition between the two. The model helps to explain the formation of the preferential pathways and their persistence and structure (the core and fringe of the fingers), the effect of the initial saturation of the matrix, and the saturation overshoot phenomenon.

Traveling wave solutions for the Richards equation with hysteresis

We investigate the one-dimensional non-equilibrium Richards equation with play-type hysteresis. It is known that regularized versions of this equation permit traveling wave solutions that show oscillations and, in particular, the physically relevant effect of a saturation overshoot. We investigate here the non-regularized hysteresis operator and combine it with a positive $\tau $-term. Our result is that the model has monotone traveling wave solutions. These traveling waves describe the behavior of fronts in a bounded domain. In a two-dimensional interpretation, the result characterizes the speed of fingers in non-homogeneous solutions.