The gradient flow structure for incompressible immiscible two-phase flows in porous media

TL;DR

This paper reveals that the classical model for two-phase flow in porous media has a formal gradient flow structure, deriving key laws without algebraic transformations, thus providing a new theoretical perspective.

Contribution

It introduces a gradient flow framework for two-phase flow models, avoiding traditional algebraic transformations and extending to multiple phases.

Findings

Derives Darcy-Muskat law from energy and dissipation principles

Establishes a gradient flow structure for two-phase flow models

Can be extended to more than two phases

Abstract

We show that the widely used model governing the motion of two incompressible immiscible fluids in a possibly heterogeneous porous medium has a formal gradient flow structure. More precisely, the fluid composition is governed by the gradient flow of some non-smooth energy. Starting from this energy together with a dissipation potential, we recover the celebrated Darcy-Muskat law and the capillary pressure law governing the flow thanks to the principle of least action. Our interpretation does not require the introduction of any algebraic transformation like, e.g., the global pressure or the Kirchhoff transform, and can be transposed to the case of more phases.