Asymptotic Behavior of Two-Phase Flows in Heterogeneous Porous Media for Capillarity Depending Only on Space. I. Convergence to the Optimal Entropy Solution

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. key words. entropy solution, scalar conservation law, discontinuous porous media, capillarity AMS subject classification. 35L65, 76S05 1 Presentation of the problem The resolution of multi-phase flows in porous media are widely used in oil engineering to predict the motions of oil in subsoil. Their mathematical study is however difficult, then some physical assumptions have to be done, in order to get simpler problems (see e.g. [6, 11, 30]). A classical simplified model, so-called dead-oil approximation, consists in assuming that there is no gas, i.e. that the fluid is composed of two immiscible and incompressible phases and in neglecting all the different chemical species. The oil-phase and the water-phase are then both made of only one component. 1.1 the dead-oil problem in the one dimensional case Suppose that R represents a one dimensional homogeneous porous medium, with porosity φ (which is supposed to be constant for the sake of simplicity). If u denotes the saturation of the water phase, and so (1− u) the saturation of the oil phase thanks to the dead-oil approximation, writing the volume conservation of each phase leads to: φ∂tu+ ∂xVw = 0, (1) −φ∂tu+ ∂xVo = 0, (2) where Vo (resp. Vw) is the filtration speed of the oil phase (resp. water phase). Using the empirical diphasic Darcy law, we claim that Vβ = −K kr,β(u) μβ (∂xPβ − ρβg) , β = o, w, (3) UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France ([email protected]) The author is partially supported by GNR MoMaS 1 ha l-0 03 60 29 7, v er si on 4 6 N ov 2 00 9 whereK is the global permeability, only depending on the porous media, μβ , Pβ , ρβ are respectively the dynamical viscosity, the pressure and the density of the phase β, g represents the effect of gravity, kr,β denotes the relative permeability of the phase β. This last term comes from the interference of the two phases in the porous media. There exists s⋆ ∈ [0, 1) such that the function kr,w is non-decreasing, with kr,w(u) = 0 if 0 ≤ u ≤ s⋆ < 1, and kr,w is increasing on [s⋆, 1]. The function kr,o is supposed to be nonincreasing, with kr,o(1) = 0. We suppose that there exists s ⋆ ∈ (s⋆, 1] such that kr,o(s) = 0 for s ∈ [s, 1), and kr,o is decreasing on [0, s). The pressures are supposed to be linked by the relation Pcap(u) = Pw − Po, (4) where Pcap is a smooth non-decreasing function called capillary pressure. Adding (1) and (2), and using (3) and (4) yields −∂x 