Gradient discretization of two-phase flows coupled with mechanical deformation in fractured porous media

Abstract

We consider a two-phase Darcy flow in a fractured porous medium consisting in a matrix flow coupled with a tangential flow in the fractures, described as a network of planar surfaces. This flow model is also coupled with the mechanical deformation of the matrix assuming that the fractures are open and filled by the fluids, as well as small deformations and a linear elastic constitutive law. The model is discretized using the gradient discretization method [30], 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. Here, we describe the model together with its numerical discretization and, using discrete compactness techniques, we prove a convergence result (up to a subsequence) assuming the non-degeneracy of the phase mobilities and that the discrete solutions remain physical in the sense that, roughly speaking, the porosity does not vanish and the fractures remain open. This is, to our knowledge, the first convergence result for this type of model taking into account two-phase flows in fractured porous media and the non-linear poromechanical coupling. Previous related works consider a linear approximation obtained for a single phase flow by freezing the fracture conductivity [41,42]. Numerical tests employing the Two-Point Flux Approximation (TPFA) finite volume scheme for the flows and P2 finite elements for the mechanical deformation are also provided to illustrate the behavior of the solution to the model.

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

“…As in [10], this work focuses on a hybrid-dimensional two-phase Darcy flow model coupled with a linear poroelastic deformation of the matrix. The fractures are assumed to remain open and fully filled by the fluids.…”

“…In [10], the continuity of both phase pressures is used as a transmission condition at matrix fracture interfaces. This is a classical assumption in the case of highly permeable fractures such as open fractures.…”

“…Let us refer to [2] for striking examples including the desaturation by suction at the interface between the atmosphere and a low permeable and fractured storage rock. This potential inaccuracy of continuous pressure models motivates us to consider the extension of the analysis carried out in [10] to hybrid-dimensional discontinuous pressure flow models [34,23,16,2]. For such flow models, the Darcy fluxes between the matrix fracture interface and the fracture are modelled using a two-point flux approximation combined with an upwind approximation of the mobilities [23,16,2].…”

“…It must be kept sufficiently small to maintain the accuracy of the solution (see [23]). Following [10] and [23], this new hybrid-dimensional poro-mechanical model is discretized using the gradient discretization method [22]. This framework is based on abstract vector spaces of discrete unknowns combined with reconstruction operators.…”

“…Then, we prove the convergence of the discrete solution to a weak solution of the model. Compared with [10], new difficulties arise from the interfacial additional nonlinear flux and accumulation terms including the damaged rock type. Assuming that the fracture normal transmissivity is fixed in the interfacial two-point fluxes, i.e.…”