Energy-stable discretization of two-phase flows in deformable porous media with frictional contact at matrix-fracture interfaces

TL;DR

This paper develops an energy-stable numerical method for simulating two-phase flows in deformable fractured porous media, incorporating frictional contact at interfaces, and demonstrates its effectiveness through benchmark and realistic tests.

Contribution

It introduces a novel discretization framework combining gradient discretization, TPFA, and finite elements to handle complex fracture-matrix interactions with energy stability.

Findings

Energy estimates and existence of solutions are established.

The method effectively handles singularities at fracture tips and intersections.

Numerical simulations validate the approach on benchmark and realistic scenarios.

Abstract

We address the discretization of two-phase Darcy flows in a fractured and deformable porous medium, including frictional contact between the matrix-fracture interfaces. Fractures are described as a network of planar surfaces leading to the so-called mixed- or hybrid-dimensional models. Small displacements and a linear elastic behavior are considered for the matrix. Phase pressures are supposed to be discontinuous at matrix-fracture interfaces, as they provide a better accuracy than continuous pressure models even for high fracture permeabilities. The general gradient discretization framework is employed for the numerical analysis, allowing for a generic stability analysis and including several conforming and nonconforming discretizations. We establish energy estimates for the discretization, and prove existence of a solution. To simulate the coupled model, we employ a Two-Point Flux Approximation (TPFA) finite volume scheme for the flow and second-order ($\mathbb P_2$) finite elements for the mechanical displacement coupled with face-wise constant ($\mathbb P_0$) Lagrange multipliers on fractures, representing normal and tangential stresses, to discretize the frictional contact conditions. This choice allows to circumvent possible singularities at tips, corners, and intersections between fractures, and provides a local expression of the contact conditions. We present numerical simulations of two benchmark examples and one realistic test case based on a drying model in a radioactive waste geological storage structure.