Two-phase flow in porous media with hysteresis

Two-phase flow through a porous medium with hysteresis effects is considered. The model consists of a system of two coupled nonlinear equations: a transport equation for the water saturation and an evolution equation for the hysteresis variable. The latter is not in conservation form and contains discontinuous functions of the two unknown variables as coefficients. Some qualitative properties of piecewise smooth solutions of the system are proved. In particular, the hysteresis variable is shown to satisfy a maximum principle, and that its total variation is bounded by the total variation of its initial value. The traveling waves are investigated under the assumption that the convective term is convex. Riemann solvers for the inviscid system are constructed. Non-uniqueness due to hysteresis loops is finally discussed; several solutions are discarded by the maximum principle for the hysteresis variable.