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.
翻译:我们认为,在一个支离破碎和变形的多孔介质中存在两阶段的达西流,骨折被描述为形成所谓的混合式模型的平板表面网络,骨折假设是开放的,由流体和小变形以线性弹性成份法填充,在矩阵中则考虑的是线性弹性成份法。与[10]相反,在矩阵骨折接口中,阶段压力假设不是连续的,这给与流动中额外的跨方方方方程式和未知方程式有关的趋同分析带来了新的挑战。如[16,2]号文件所示,与单阶段流不同,两阶段流动的不连续压力模型比连续压力模型的准确性强,甚至与高渗透性骨折的连续压力模型更强。 由一个阶段完全填补的骨折可成为另一阶段的障碍,导致矩阵骨折接口的压力不连续不连续。 该模型采用渐离散的离散分解法方法[22],包括大量符合和不兼容的系统计划。正如[16,这一框架允许使用离层-TP流动的美元压力模型进行一般性的趋同化分析。