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

Clément Cancès (1)

1 LJLL - Laboratoire Jacques-Louis Lions

We consider an immiscible two-phase flow in a heterogeneous one-dimensional porous medium. We suppose particularly that the capillary pressure field is discontinuous with respect to the space variable. The dependence of the capillary 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 the asymptotic behavior when the capillary pressure tends to a function which does not depend on the saturation. In this paper, we show that if the capillary forces at the spacial discontinuities are oriented in the same direction that the gravity forces, or if the two phases move in the same direction, then the saturation profile with capillary diffusion converges toward the unique optimal entropy solution to the hyperbolic scalar conservation law with discontinuous flux functions.

Mots clés: entropy solution; scalar conservation law; discontinuous porous media; capillarity

Domaines: Equations aux dérivées partielles [math.AP]

Clément Cancès. Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. I. Convergence to the optimal entropy solution.. SIAM Journal on Mathematical Analysis, 2010, 42 (2), pp.946-971. ⟨hal-00360297v4⟩