Numerical analysis of coupled flow, deformation and frictional contact mechanics in a fractured porous medium

This paper studies the numerical approximation of a mathematical model of coupled flow and viscoelastic deformation in a fractured porous medium including fracture contact mechanics. We consider a mixed-dimensional model, where the fracture is modeled as a co-dimension one object within the porous medium. This model couples Darcy flow in the matrix and fracture, linear viscoelasticity of the matrix and frictional contact mechanics of the fracture surfaces; the contact conditions include normal compliance and a Coulomb friction law. We propose and analyze a fully discrete numerical scheme for this problem, using conformal finite elements in space and the implicit Euler method in time. Existence and uniqueness of the discrete solution is established, and stability and a priori error estimates are derived. A numerical experiment is performed in which numerical error estimates are computed and compared to the theoretical results.