Dual virtual element methods for discrete fracture matrix models

Abstract

The accurate description of fluid flow and transport in fractured porous media is of paramount importance to capture the macroscopic behavior of an oil reservoir, a geothermal system, or a CO2 sequestration site, to name few applications. The construction of accurate simulation models for flow in fractures is challenging due to the high ratio between a fracture’s length and width. In this paper, we present a mixed-dimensional Darcy problem which can represent the pressure and Darcy velocity in all the dimensions, i.e. in the rock matrix, in the fractures, and in their intersections. Moreover, we present a mixed-dimensional transport problem which, given the Darcy velocity, describes advection of a passive scalar into the fractured porous media. The approach can handle both conducting and blocking fractures. Our computational grids are created by coarsening of simplex tessellations that conform to the fracture’s surfaces. A suitable choice of the discrete approximation of the previous model, by virtual finite element and finite volume methods, allows us to simulate complex problems with a good balance of accuracy and computational cost. We illustrate the performance of our method by comparing to benchmark studies for two-dimensional fractured porous media, as well as a complex three-dimensional fracture geometry.