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 ${\mathbb{P}}_2$ finite elements for the mechanical deformation are also provided to illustrate the behavior of the solution to the model.

我们考虑裂缝多孔介质中的两相达西流，其中基质流与裂缝中的切向流相结合，被描述为平面网络。该流动模型还与基质的机械变形相结合，假设裂缝是打开的并被流体填充，以及小变形和线弹性本构定律。该模型使用梯度离散化方法 [30] 进行离散化，该方法涵盖了一大类符合和不符合方案。该框架允许使用离散功能工具的组合对耦合模型进行通用收敛分析。在这里，我们描述模型及其数值离散化，并使用离散紧凑性技术，我们证明了一个收敛结果（直到一个子序列），假设相迁移率是非简并的，并且离散解仍然是物理的，粗略地说，孔隙度没有消失，裂缝保持开放。据我们所知，这是考虑到裂缝性多孔介质中的两相流和非线性多孔机械耦合的此类模型的第一个收敛结果。以前的相关工作考虑了通过冻结裂缝导流能力获得的单相流的线性近似 [41]、[42]。数值测试采用两点通量近似 (TPFA) 有限体积方案进行流动和 据我们所知，这种模型的第一个收敛结果考虑了裂缝多孔介质中的两相流动和非线性多孔机械耦合。以前的相关工作考虑了通过冻结裂缝导流能力获得的单相流的线性近似 [41]、[42]。数值测试采用两点通量近似 (TPFA) 有限体积方案进行流动和 据我们所知，这种模型的第一个收敛结果考虑了裂缝多孔介质中的两相流动和非线性多孔机械耦合。以前的相关工作考虑了通过冻结裂缝导流能力获得的单相流的线性近似 [41]、[42]。数值测试采用两点通量近似 (TPFA) 有限体积方案进行流动和${\mathbb{磷}}_2$ 还提供了机械变形的有限元来说明模型解的行为。