An efficient discontinuous Galerkin - mixed finite element model for variable density flow in fractured porous media

Abstract

Modeling variable density flow (VDF) in fractured porous media is computationally challenging. The challenges are mainly related to: (i) the high permeability contrast between the matrix and the fractures and, (ii) the nonlinearity induced by density variations. Due to their local mass conservation property, cell-centered methods, such as finite volumes (FV), mixed finite elements (MFE) or Discontinuous Galerkin (DG) methods are well suited for modeling mass transfer in highly heterogeneous domains. When applied for fractured media, these methods require small grid cells next to fractures because of the cross-flow equilibrium assumption (i.e. the pressure and concentration in the fracture and in the adjacent matrix grid-cells are assumed the same). To avoid this constraint, an efficient model is developed in this work using advanced cell-centered numerical methods for VDF in fractured porous media with cross-flow equilibrium assumed only across the fractures. The new model uses the hybrid MFE method for the flow discretization in the matrix and in the fracture continua. Mass transport in the matrix is modeled using a monotonic upwind MFE scheme. Advection-dominated transport in fractures is discretized with the DG method, which is well adapted for hyperbolic equations. The new model ensures continuity of pressure, concentration, fluid mass flux, advective and dispersive contaminant fluxes at matrix-fracture interfaces as well as at the intersection of several fractures. The time discretization is performed using high-order adaptive time integration techniques via the method of lines (MOL). The developed model is first validated by comparison against a 2D-2D model for a test problem dealing with linear flow and transport in a 2D domain involving a “+”-shaped barrier/fracture network. Then, the MFE-DG model is used for the simulation of the Henry saltwater intrusion problem and the results are validated by comparison against the semi-analytical solution in the case of unfractured and fractured aquifers. Finally, the simulation of a transport problem with variable viscosity in a fractured heterogeneous domain points out the superiority of the new model in terms of accuracy and efficiency when compared to a standard finite element model.