2020/02/15 by Paola F. Antonietti, Antonietti, Paola Francesca, Chiara Facciolà +3 · 2 citations
Computer Science · Earth and Planetary Sciences · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Seismic Imaging and Inversion Techniques
paper · pdf · doi:10.48550/arxiv.2002.06420
openalex publication_date 2020/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a numerical approximation of Darcy's flow through a porous medium\nthat incorporates networks of fractures with non empty intersection. Our scheme\nemploys PolyDG methods, i.e. discontinuous Galerkin methods on general\npolygonal and polyhedral (polytopic, for short) grids, featuring elements with\nedges/faces that may be in arbitrary number (potentially unlimited) and whose\nmeasure may be arbitrarily small. Our approach is then very well suited to tame\nthe geometrical complexity featured by most of applications in the\ncomputational geoscience field. From the modelling point of view, we adopt a\nreduction strategy that treats fractures as manifolds of codimension one and we\nemploy the primal version of Darcy's law to describe the flow in both the bulk\nand the fracture network. In addition, some physically consistent conditions\ncouple the two problems, allowing for jump of pressure at their interface, and\nthey as well prescribe the behaviour of the fluid along the intersections,\nimposing pressure continuity and flux conservation. Both the bulk and fracture\ndiscretizations are obtained employing the Symmetric Interior Penalty DG method\nextended to the polytopic setting. The key instrument to obtain a polyDG\napproximation of the problem in the fracture network is the generalization of\nthe concepts of jump and average at the intersection, so that the contribution\nfrom all the fractures is taken into account. We prove the well-posedness of\nthe discrete formulation and perform an error analysis obtaining a priori\nhp-error estimates. All our theoretical results are validated performing\npreliminary numerical tests with known analytical solution.\n