  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.
