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

Abstract

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 \cite{bonaldi:hal-02549111}, 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 \cite{BHMS2018,gem.aghili}, 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 \cite{gdm}, 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 \cite{bonaldi:hal-02549111} 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.

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

“…l Having established the existence of a solution to the scheme and estimates on this solution, the natural question would be to analyze its convergence as the mesh size and time steps tend to zero. This convergence requires to establish complex compactness results on sequences of approximate solutions; see, e.g., the convergence analysis for the model without contact in [16], which assumes that the fracture width and porosity remain suitably bounded from below. For the models with Coulomb friction considered here (either with or without the small porosity assumption of Remark 3.1), a very challenging element is the variational inequality (11c): either the Lagrange multiplier or the jump of displacement would need to converge strongly in appropriate spaces.…”

“…where 𝐶 depends only on the data in Assumptions (H), (𝑏, 𝑀, d𝔞 , φ𝔞 ) excepted. From here, the same arguments as in [16] provide the estimates on ( 𝑝, u) stated in the theorem. Using these estimates, the definition (11d) of ( 𝑝 𝐸 𝜔 ) 𝜔=𝑚, 𝑓 , the coercivity bound (14) and the fact that 0 ≤ 𝑈 𝜔 ( 𝑝) ≤ 2| 𝑝|, we also obtain a bound on Π 𝑓 D 𝑝 𝑝 𝐸 𝑓 𝐿 2 (0,𝑇 ;𝐿 2 (Γ)) and…”

“…We assume here that the gradient discretizations we consider for the flow are coercive [16], that is: there exists…”

“…We then follow the arguments in [16,Lemma 4.3]. Taking 𝜑 𝛼 = 𝑝 𝛼 in (11a), summing over the phases and adding (18), and accounting for the fact that (11d) and (12a) ensure that 𝑑 𝑓 , D u ≥ 𝑑 0 , we obtain [16, Eq.…”

“…This test case mostly presents the same geometry, boundary conditions, and data set as in [16]. Unlike here, however, the model and simulations in this reference assume that all fractures remain open during the process.…”