Discontinuous Galerkin methods for advective transport in single-continuum models of fractured media

Accurate simulation of flow and transport processes in fractured rocks requires that flow in fractures and shear zones to be coupled with flow in the porous rock matrix. To this end, we will herein consider a single-continuum approach in which both fractures and the porous rock are represented as volumetric objects, i.e., as cells in an unstructured triangular grid with a permeability and a porosity value associated with each cell. Hence, from a numerical point of view, there is no distinction between flow in the fractures and the rock matrix. This enables modelling of realistic cases with very complex structures. To compute single-phase advective transport in such a model, we propose to use a family of higher-order discontinuous Galerkin methods. Single-phase transport equations are hyperbolic and have an inherent causality in the sense that information propagates along streamlines. This causality is preserved in our discontinuous Galerkin discretization. We can therefore use a simple topological sort of the graph of discrete fluxes to reorder the degrees-of-freedom such that the discretized linear system gets a lower block-triangular form, from which the solution can be computed very efficiently using a single-pass forward block substitution. The accuracy and utility of the resulting transport solver is illustrated through several numerical experiments.