Polytopic Discontinuous Galerkin methods for the numerical modelling of flow in porous media with networks of intersecting fractures
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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.
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