Vertex Approximate Gradient Scheme for Hybrid Dimensional Two-Phase Darcy Flows in Fractured Porous Media

Abstract

This paper presents a finite volume discretization of two-phase Darcy flows in discrete fracture networks taking into account the mass exchange between the matrix and the fracture. We consider the asymptotic model for which the fractures are represented as interfaces of codimension one immersed in the matrix domain, leading to the so called hybrid dimensional Darcy flow model. The pressures at the interfaces between the matrix and the fracture network are continuous corresponding to a ratio between the normal permeability of the fracture and the width of the fracture assumed to be large compared with the ratio between the permeability of the matrix and the size of the domain. The discretization is an extension of the Vertex Approximate Gradient (VAG) scheme to the case of hybrid dimensional Darcy flow models. Compared with Control Volume Finite Element (CVFE) approaches, the VAG scheme has the advantage to avoid the mixing of the fracture and matrix rocktypes at the interfaces between the matrix and the fractures, while keeping the low cost of a nodal discretization on unstructured meshes. The convergence of the scheme is proved under the assumption that the relative permeabilities are bounded from below by a strictly positive constant. This assumption is needed in the convergence proof in order to take into account discontinuous capillary pressures in particular at the matrix fracture interfaces. The efficiency of our approach compared with CVFE discretizations is shown on two numerical examples of fracture networks in 2D and 3D.

Gradient Discretization of Hybrid Dimensional Darcy Flows in Fractured Porous Media

“…located at the boundary of K with Using these unified notations and following [6] Section 4 for the VAG scheme and [13] Remark 2.7 for the HFV scheme, we can define for both schemes the matrix and fracture fluxes which are exhibited in Figure 4 inside a cell K with a fracture face σ. The matrix fluxes connect a cell K ∈ M to its boundary d.o.f.…”

“…Following [6], when applying the VAG or HFV discretization to two phase Darcy flow models or to the coupling of the Darcy flow equation with a tracer equation, the choice of the cells and fracture faces partitioning defining the control volumes is done in order to avoid the mixture of heterogeneous properties inside each control volume. In particular, at the matrix fracture interfaces, one simply need to set ω K,ν = ∅ for all ν such that x ν ∈ Γ.…”

“…It leads to a sparse discretization on tetrahedral or mainly tetrahedral meshes. It has the advantage, compared with CVFE approaches to avoid the mixing of the control volumes at the fracture matrix interfaces, which is a key feature for its application to multiphase Darcy flows (see [6]). It will be compared to the HFV discretization using face and fracture edge unknowns in addition to the cell unknowns which can be as well eliminated without any fill-in.…”

A multi-layer reduced model for flow in porous media with a fault and surrounding damage zones

“…This approach has been successfully applied to different kind of physics, ranging from advection of a passive scalar, heat transport, multi-phase flow. An incomplete list of references is the following [31,35,23,3,32,26,12,19,18,33,22,37,30,38,13,1,10,27,8,4,39,17,7,9,34].…”

First-order indicators for the estimation of discrete fractures in porous media

“…see [4,25,24,32,26,22], as well as in the mathematical literature, e.g. see [30,3,29,16,20,34,27,12,11], to name just a few. We use here the model developed in [30], where flow in the fracture as well as in the rock is governed by Darcy's law.…”