  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.
