Motion of fronts in porous media and laws for the capillary pressure

The motion of two non-mixing fluids in a porous material is commonly described by the two-phase-flow or Leverett equations. Capillarity induces a pressure difference between the two fluids, the capillary pressure. We propose and analyze a model for the motion of fronts in a porous material, aiming at the justification of this pressure jump. We work on a meso-scale in which the single front is still resolved, and homogenize the effects of the single pores. Stationary methods and the methods of periodic homogenization both turn out not to be suited for this problem. Using the language of micro-local-patterns we can formulate a conditional result for upscaled equations; in the stochastic case the upscaled system is satisfied almost surely.