  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.
