Incompressible immiscible multiphase flows in porous media: a variational approach

TL;DR

This paper models the movement of multiple immiscible fluid phases in porous media using a variational gradient flow approach, proving convergence and existence results without relying on global pressure assumptions.

Contribution

It introduces a novel variational framework for incompressible multiphase flows in porous media, establishing convergence and existence results without the need for global pressure.

Findings

Proves convergence of a minimization scheme for the flow model.

Establishes a new existence result for the PDE system.

Models flows without global pressure assumptions.

Abstract

We describe the competitive motion of (N + 1) incompressible immiscible phases within a porous medium as the gradient flow of a singular energy in the space of non-negative measures with prescribed mass endowed with some tensorial Wasserstein distance. We show the convergence of the approximation obtained by a minimization schem\`e a la [R. Jordan, D. Kinder-lehrer \& F. Otto, SIAM J. Math. Anal, 29(1):1--17, 1998]. This allow to obtain a new existence result for a physically well-established system of PDEs consisting in the Darcy-Muskat law for each phase, N capillary pressure relations, and a constraint on the volume occupied by the fluid. Our study does not require the introduction of any global or complementary pressure.