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We consider an immiscible two-phase flow in a heterogeneous one-dimensionalporous medium. We suppose particularly that the capillary pressure field isdiscontinuous with respect to the space variable. The dependence of thecapillary pressure with respect to the oil saturation is supposed to be weak,at least for saturations which are not too close to 0 or 1. We study theasymptotic behavior when the capillary pressure tends to a function which doesnot depend on the saturation. In this paper, we show that if the capillaryforces at the spacial discontinuities are oriented in the same direction thatthe gravity forces, or if the two phases move in the same direction, then thesaturation profile with capillary diffusion converges toward the unique optimalentropy solution to the hyperbolic scalar conservation law with discontinuousflux functions.

Asymptotic behavior of two-phase flows in heterogeneous porous media for  capillarity depending only on space. I. Convergence to the optimal entropy  solution