Gradient discretization of two-phase poro-mechanical models with discontinuous pressures at matrix fracture interfaces

TL;DR

This paper develops a gradient discretization framework for two-phase poro-mechanical models with discontinuous pressures at matrix-fracture interfaces, proving convergence and comparing numerical solutions.

Contribution

It extends the gradient discretization method to handle discontinuous pressures in two-phase flow models, providing a rigorous convergence analysis.

Findings

Discontinuous pressure models offer higher accuracy than continuous ones in fractured media.

The extended gradient discretization scheme converges to a weak solution.

Numerical tests compare continuous and discontinuous pressure models using TPFA and P2 finite elements.

Abstract

We consider a two-phase Darcy flow in a fractured and deformable porous medium for which the fractures are described as a network of planar surfaces leading to so-called hybrid-dimensional models. The fractures are assumed open and filled by the fluids and small deformations with a linear elastic constitutive law are considered in the matrix. As opposed to [10], the phase pressures are not assumed continuous at matrix fracture interfaces, which raises new challenges in the convergence analysis related to the additional interfacial equations and unknowns for the flow. As shown in [16, 2], unlike single phase flow, discontinuous pressure models for two-phase flows provide a better accuracy than continuous pressure models even for highly permeable fractures. This is due to the fact that fractures fully filled by one phase can act as barriers for the other phase, resulting in a pressure discontinuity at the matrix fracture interface. The model is discretized using the gradient discretization method [22], which covers a large class of conforming and non conforming schemes. This framework allows for a generic convergence analysis of the coupled model using a combination of discrete functional tools. In this work, the gradient discretization of [10] is extended to the discontinuous pressure model and the convergence to a weak solution is proved. Numerical solutions provided by the continuous and discontinuous pressure models are compared on gas injection and suction test cases using a Two-Point Flux Approximation (TPFA) finite volume scheme for the flows and $P_2$ finite elements for the mechanics.