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

Abstract

Highlights• Energy-stable model and scheme for coupled two-phase flow and contact mechanics in discrete fracture networks • Mechanics conforming discretization coupled with P 0 Lagrange multipliers to circumvent singularities, local contact equations • Flow discretization in the abstract gradient discretization framework accounting for a large family of schemes • Investigation of nonlinear algorithms both for the contact mechanics and for the fully coupled problem • Validation on benchmark 2D examples and application to a realistic axisymmetric case study

Numerical analysis of a mixed-dimensional poromechanical model with frictionless contact at matrix-fracture interfaces

“…Our convergence proof elaborates on previous works. In [17] we proved stability estimates and existence of a solution for the GD of a mixed-dimensional two-phase poromechanical model with Coulomb frictional contact. In [15,16] we obtained compactness estimates for a mixed-dimensional two-phase poromechanical model with no contact.…”

“…The modeling and numerical simulation of such mixed-dimensional poromechanical problems have been the object of many recent works [35,36,37,11,50,15,16,17]. In terms of discretization, they are mostly based on conservative finite volume schemes for the flow and use either a conforming finite element method [35,36,37,17] or a finite volume scheme for the mechanics [11,50].…”

“…The modeling and numerical simulation of such mixed-dimensional poromechanical problems have been the object of many recent works [35,36,37,11,50,15,16,17]. In terms of discretization, they are mostly based on conservative finite volume schemes for the flow and use either a conforming finite element method [35,36,37,17] or a finite volume scheme for the mechanics [11,50]. Note that the saddle point nature of the coupling between the matrix fluid pressure and the displacement field requires a compatibility condition between both discretizations, as in [11,50,15,16,17], or alternatively additional stabilization terms to get the stability of the pressure solution in the limit of low rock permeability, incompressible fluid, and small times.…”

“…In terms of discretization, they are mostly based on conservative finite volume schemes for the flow and use either a conforming finite element method [35,36,37,17] or a finite volume scheme for the mechanics [11,50]. Note that the saddle point nature of the coupling between the matrix fluid pressure and the displacement field requires a compatibility condition between both discretizations, as in [11,50,15,16,17], or alternatively additional stabilization terms to get the stability of the pressure solution in the limit of low rock permeability, incompressible fluid, and small times. The contact mechanics formulation is a key ingredient to efficiently handle the nonlinear variational inequalities of the contact fracture model.…”

“…The contact mechanics formulation is a key ingredient to efficiently handle the nonlinear variational inequalities of the contact fracture model. It is typically either based on a mixed formulation with Lagrange multipliers to impose the contact conditions as in [35,11,50,17], or on a consistent Nitsche penalization method as in [37]. Again, if a mixed formulation is used, a compatibility condition must be satisfied between the Lagrange multiplier and displacement field spaces [11,50,17], or a stabilization must be specifically designed [35].…”