Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Non-classical shocks to model oil-trapping

We consider a one-dimensional problem modeling two-phase flow in heterogeneous porous media made of two homogeneous subdomains, with discontinuous capillarity at the interface between them. We suppose that the capillary forces vanish inside the domains, but not on the interface. Under the assumption that the gravity forces and the capillary forces are oriented in opposite directions, we show that the limit, for vanishing diffusion, is not in general the optimal entropy solution of the hyperbolic scalar conservation law as in the first paper of the series \cite{NPCX}. A non-classical shock can occur at the interface, modeling oil-trapping.