Free energy diminishing discretization of Darcy-Forchheimer flow in poroelastic media

TL;DR

This paper introduces a novel discretization method for the nonlinear Darcy-Forchheimer flow in deformable porous media, preserving the system's gradient flow structure and enabling efficient, stable numerical solutions.

Contribution

It develops a structure-preserving mixed finite element discretization for the coupled flow and deformation model, ensuring existence, uniqueness, and stability, with potential for computational cost reduction.

Findings

Discretization preserves the gradient flow structure.

Ensures existence, uniqueness, and stability of solutions.

Enables finite volume discretizations for efficiency.

Abstract

In this paper, we develop a discretization for the non-linear coupled model of classical Darcy-Forchheimer flow in deformable porous media, an extension of the quasi-static Biot equations. The continuous model exhibits a generalized gradient flow structure, identifying the dissipative character of the physical system. The considered mixed finite element discretization is compatible with this structure, which gives access to a simple proof for the existence, uniqueness, and stability of discrete approximations. Moreover, still within the framework, the discretization allows for the development of finite volume type discretizations by lumping or numerical quadrature, reducing the computational cost of the numerical solution.