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.
翻译:我们处理分两阶段流动的分解问题,即分解介质,分解介质介质分解,分解介质介质介质的分解介介质,包括矩阵-裂裂变之间的摩擦接触。裂痕被描述为导致所谓混合或混合的模型的平面表面网络。考虑为矩阵考虑小规模的流离和线性弹性行为。阶段压力在矩阵-裂变界面界面中应该是不连续的,因为它们提供比持续压力模型更好的准确性,即使对于高度骨折多变性者来说也是如此。一般梯度离散化框架用于数字分析,允许进行一般性的稳定分析,包括若干符合和不兼容的地质离异分析。我们为离散或混合的模型建立能源估计,并证明存在一种解决办法。为了模拟混合模型,我们采用了双点通性通性通性通感调整和线弹性弹性弹性动作(TPFA)在流动和第二顺序(M2美元)下提供机械迁移的限定要素,同时加上面基常数($mathbbbbip P_0$)在数值分析中采用一般稳定分析,并包括一些符合和不兼容的离散和不相兼容的离异化的地质分解分解分解的分解的分解的分解和不兼容的分解的分解的分解框架。我们为我们为,我们为分解的能源的能源的能源的能源估算的能源的能源估算,并存的分解和分解结构的分解结构的分解的分解和分解的分解的分解的分解的分解的分解的分解的分解和分解的分解的分解的分解的分解和分解的分解的分解情况。