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We consider a one-dimensional problem modeling two-phase flow inheterogeneous porous media made of two homogeneous subdomains, withdiscontinuous capillarity at the interface between them. We suppose that thecapillary forces vanish inside the domains, but not on the interface. Under theassumption that the gravity forces and the capillary forces are oriented inopposite directions, we show that the limit, for vanishing diffusion, is not ingeneral the optimal entropy solution of the hyperbolic scalar conservation lawas in the first paper of the series \cite{NPCX}. A non-classical shock canoccur at the interface, modeling oil-trapping.

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