Laws for the Capillary Pressure in a Deterministic Model for Fronts in Porous Media

We propose and analyze a model for sharp fronts in porous media, aiming at an investigation of the capillary pressure. Using the notion of microlocal patterns we analyze the local behavior of the system. Depending on the structure of the local patterns we can derive upscaled equations that characterize the capillary pressure and include the hysteresis effect that is known from the physical system

Instability of gravity wetting fronts for Richards equations with hysteresis

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

Interfaces determined by capillarity and gravity in a two-dimensional porous medium

We consider a two-dimensional model of a porous medium where circular grains are uniformly distributed in a squared container. We assume that such medium is partially filled with water and that the stationary interface separating the water phase from the air phase is described by the balance of capillarity and gravity. Taking the unity as the average distance between grains, we identify four asymptotic regimes that depend on the Bond number and the size of the container. We analyse, in probabilistic terms, the possible global interfaces that can form in each of these regimes. In summary, we show that in the regimes where gravity dominates the probability of configurations of grains allowing solutions close to the horizontal solution is close to one. Moreover, in such regimes where the size of the container is sufficiently large we can describe deviations from the horizontal in probabilistic terms. On the other hand, when capillarity dominates while the size of the container is sufficiently large, we find that the probability of finding interfaces close to the graph of a given smooth curve without self-intersections is close to one.The authors acknowledge support of the Hausdorff Center for Mathematics of the University of Bonn as well as the project CRC 1060 The Mathematics of Emergent Effects, that is funded through the German Science Foundation (DFG). This work was also partially supported by the Spanish Government projects DGES MTM2011-24109, MTM2011-22612 and MTM2014-53145-P and the Basque Government project IT641-13