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

TL;DR

This paper analyzes the asymptotic behavior of two-phase flows in heterogeneous porous media with space-dependent capillarity, showing convergence to the unique entropy solution under certain flow conditions.

Contribution

It demonstrates the convergence of saturation profiles with capillary diffusion to the entropy solution in media with discontinuous capillary pressure fields, considering space-dependent effects.

Findings

Convergence to the entropy solution when capillary and gravity forces align.

Capillary diffusion effects vanish in the limit, leading to a hyperbolic conservation law.

Results apply to media with discontinuous capillary pressure functions.

Abstract

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.