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 in 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.

我们提出了通过多孔介质的达西流的数值近似，该介质包含具有非空交叉点的裂缝网络。我们的方案采用 PolyDG 方法，即一般多边形和多面体（简称为多面体）网格上的不连续 Galerkin 方法，其特征是边/面可以是任意数量（可能是无限的）并且其度量可以任意小。我们的方法非常适合用于控制计算地球科学领域的大多数应用所具有的几何复杂性。从建模的角度来看，我们采用了一种减少策略，将裂缝视为一维的流形，我们采用达西定律的原始版本来描述整体和裂缝网络中的流动。此外，一些物理上一致的条件将这两个问题结合在一起，允许它们界面处的压力跳跃，并且它们还规定了流体沿交叉点的行为，强加了压力连续性和通量守恒。使用扩展到多面体设置的对称内部惩罚 DG 方法获得体积和断裂离散化。获得裂缝网络中问题的 polyDG 近似的关键工具是在交叉点处对跳跃和平均概念的推广，以便考虑所有裂缝的贡献。我们证明了离散公式的适定性，并进行了误差分析以获得先验 它们还规定了流体沿交叉点的行为，强加了压力连续性和通量守恒。使用扩展到多面体设置的对称内部惩罚 DG 方法获得体积和断裂离散化。获得裂缝网络中问题的 polyDG 近似的关键工具是在交叉点处对跳跃和平均概念的推广，以便考虑所有裂缝的贡献。我们证明了离散公式的适定性，并进行了误差分析以获得先验 它们还规定了流体沿交叉点的行为，强加了压力连续性和通量守恒。使用扩展到多面体设置的对称内部惩罚 DG 方法获得体积和断裂离散化。获得裂缝网络中问题的 polyDG 近似的关键工具是在交叉点处对跳跃和平均概念的推广，以便考虑所有裂缝的贡献。我们证明了离散公式的适定性，并进行了误差分析以获得先验 获得裂缝网络中问题的 polyDG 近似的关键工具是在交叉点处对跳跃和平均概念的推广，以便考虑所有裂缝的贡献。我们证明了离散公式的适定性，并进行了误差分析以获得先验 获得裂缝网络中问题的 polyDG 近似的关键工具是在交叉点处对跳跃和平均概念的推广，以便考虑所有裂缝的贡献。我们证明了离散公式的适定性，并进行了误差分析以获得先验hp -错误估计。我们所有的理论结果都通过使用已知解析解进行初步数值测试得到验证。