We present a numerical approximation of Darcy's flow through a porous medium that incorporates networks of fractures with non empty intersection. Our scheme employs PolyDG methods, i.e. discontinuous Galerkin methods on general polygonal and polyhedral (polytopic, for short) grids, featuring elements with edges/faces that may be in arbitrary number (potentially unlimited) and whose measure may be arbitrarily small. Our approach is then very well suited to tame the geometrical complexity featured by most of applications in the computational geoscience field. From the modelling point of view, we adopt a reduction strategy that treats fractures as manifolds of codimension one and we employ the primal version of Darcy's law to describe the flow in both the bulk and the fracture network. In addition, some physically consistent conditions couple the two problems, allowing for jump of pressure at their interface, and they as well prescribe the behaviour of the fluid along the intersections, imposing pressure continuity and flux conservation. Both the bulk and fracture discretizations are obtained employing the Symmetric Interior Penalty DG method extended to the polytopic setting. The key instrument to obtain a polyDG approximation of the problem in the fracture network is the generalization of the concepts of jump and average at the intersection, so that the contribution from all the fractures is taken into account. We prove the well-posedness of the discrete formulation and perform an error analysis obtaining a priori hp-error estimates. All our theoretical results are validated performing preliminary numerical tests with known analytical solution.
翻译:我们通过一个包含断裂网络和非空十字路口的多孔介质,展示达西流动的数值近似值。我们的计划采用多维DG方法,即对一般多边形和多面网格(光谱,短)采用不连续的Galerkin方法,其中的边缘/面元素可能是任意数字(可能不受限制),其测量量可能任意小一些。我们的方法非常适合于驯服计算地球科学领域大多数应用所呈现的几何复杂性。从建模角度看,我们采用了一种削减战略,将骨折作为共振一的柱子处理,我们采用达西法律的原始版本来描述大体和断裂网络的流量。此外,有些实际条件将两者相近于任意数字(可能不受限制)的边框/面元素,允许其界面的压力跳跃,并同时规定在交叉点上流体的行为,使压力的连续性和通量保存。所有大和断裂分解都是使用Symica DG方法,将骨质断分解为多端测试的元,我们用原始分析结果的初版法来分析结果。我们所认识的分解的分解系统的分解系统是用来分析的分解的分解的分解的分解的分解系统,在先分解的分解的分解的分解模型的分解。我们所学的分解的分解的分解的分解点,在总的分解的分解的分解的分解法是用来的分解的分解的分解的分解的分解的分解的分解的分解的分解。我们的分解法是先分解的分解的分解的分解的分解的分解的分解的分解的分解点的分解点的分解。我们的分解的分解的分解的分解的分解的分解法,在先的分解法是分解。的分解的分解。的分解的分解的分解的分解的分解。我们的分解的分解的分解的分解的分解的分解的分解的分解的分解的分解的分解的分解的分解的分解的分解的分解的分解点的分解点,