On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types

We consider a simplified model for two-phase flows in one-dimensional heterogeneous porous media made of two different rocks. We focus on the effects induced by the discontinuity of the capillarity field at interface. We first consider a model with capillarity forces within the rocks, stating an existence/uniqueness result. Then we look for the asymptotic problem for vanishing capillarity within the rocks, remaining only on the interface. We show that either the solution to the asymptotic problem is the entropy solution to a scalar conservation law with discontinuous flux, or it admits a non-classical shock at the interface.