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We present a numerical approximation of Darcy's flow through a porous mediumthat incorporates networks of fractures with non empty intersection. Our schemeemploys PolyDG methods, i.e. discontinuous Galerkin methods on generalpolygonal and polyhedral (polytopic, for short) grids, featuring elements withedges/faces that may be in arbitrary number (potentially unlimited) and whosemeasure may be arbitrarily small. Our approach is then very well suited to tamethe geometrical complexity featured by most of applications in thecomputational geoscience field. From the modelling point of view, we adopt areduction strategy that treats fractures as manifolds of codimension one and weemploy the primal version of Darcy's law to describe the flow in both the bulkand the fracture network. In addition, some physically consistent conditionscouple the two problems, allowing for jump of pressure at their interface, andthey as well prescribe the behaviour of the fluid along the intersections,imposing pressure continuity and flux conservation. Both the bulk and fracturediscretizations are obtained employing the Symmetric Interior Penalty DG methodextended to the polytopic setting. The key instrument to obtain a polyDGapproximation of the problem in the fracture network is the generalization ofthe concepts of jump and average at the intersection, so that the contributionfrom all the fractures is taken into account. We prove the well-posedness ofthe discrete formulation and perform an error analysis obtaining a priorihp-error estimates. All our theoretical results are validated performingpreliminary numerical tests with known analytical solution.

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