Numerical analysis of a two-phase flow discrete fracture matrix model

Abstract

We present a new model for two phase Darcy flows in fractured media, in which fractures are modelled as submanifolds of codimension one with respect to the surrounding domain (matrix). Fractures can act as drains or as barriers, since pressure discontinuities at the matrix-fracture interfaces are permitted. Additionally, a layer of damaged rock at the matrix-fracture interfaces is accounted for. The numerical analysis is carried out in the general framework of the Gradient Discretisation Method. Compactness techniques are used to establish convergence results for a wide range of possible numerical schemes; the existence of a solution for the two phase flow model is obtained as a byproduct of the convergence analysis. A series of numerical experiments conclude the paper, with a study of the influence of the damaged layer on the numerical solution.jump of the normal fluxes as well as additional transmission conditions at the matrix-fracture interfaces. These transmission conditions depend on the mathematical nature of the equidimensional model and on additional physical assumptions. They are typically derived for a single phase Darcy flow for which they specify either the continuity of the pressure in the case of fractures acting as drains [3,8] or Robin type conditions in order to take into account the discontinuity of the pressure for fractures acting either as drains or barriers [4,10,28,34].Fewer works deal with the extension of hybrid-dimensional models to two-phase Darcy flows. Most of them build directly the model at the discrete level as in [7,31,37] or are limited to the case of continuous pressures at the matrix-fracture interfaces as in [7,9,37]. In [32], an hybrid-dimensional two-phase flow model with discontinuous pressures at the matrix-fracture interfaces is proposed using a global pressure formulation. However, the transmission conditions at the interface do not take into account correctly the transport from the matrix to the fracture.In this paper, a new hybrid-dimensional two-phase Darcy flow model is proposed accounting for complex networks of fractures acting either as drains or barriers. The model takes into account discontinuous capillary pressure curves at the matrix-fracture interfaces. It also includes a layer of damaged rock at the matrix-fracture interface with its own mobility and capillary pressure functions. This additional layer is not only a modelling tool, it also plays a major role in the numerical analysis of the model and in the convergence of the non-linear Newton iterations required to solve the discrete equations.The discretisation of hybrid-dimensional Darcy flow models has been the object of many works using cell-centred Finite Volume schemes with either Two Point or Multi Point Flux Approximations (TPFA and MPFA) [1,2,4,30,33,38,40], Mixed or Mixed Hybrid Finite Element methods (MFE and MHFE) [3,31,34], Hybrid Mimetic Mixed Methods (HMM, which contains mixed-hybrid finite volume and mimetic finite difference schemes [19]) [5,8,10,27], Control Volume Finite Element Methods (...

A Hybrid High-Order Method for Darcy Flows in Fractured Porous Media

“…Using (75) for the consistency term in m T pp u T , v T q (see (20)), plugging the resulting relation into the expression of a ξ h pp u h , v h q (see (26)), and accounting for (70) in the fracture terms of a ξ h pp u h , v h q (where π k F can be cancelled using (9) after observing that λ ξ F rrv h ss F P P k pF q and λ F rrv h ss F P P k pF q for all F P F Γ h ) gives (66).…”

Uniform-in-Time Convergence of Numerical Schemes for a Two-Phase Discrete Fracture Model

“…We give here a brief overview of the notations, and refer to [3] for more details. Ω is a bounded polytopal domain of R d (d = 2, 3), partitioned into a fracture domain Γ and a matrix domain Ω \Γ .…”

“…Under the assumptions of Section 1, let (D l ) l∈N be a coercive, consistent, limit-conforming and compact sequence of space-time GD (see [3]), and let (u α,l ) l∈N be such that u α,l ∈ (X 0 l ) N l +1 is a sequence of solutions of (3) with D = D l . Then, there exists a weak solution (u α m , u α f ) α=1,2 of the model such that, for all µ ∈ {m, f } and a ∈ χ,…”

“…Before proving this theorem, we recall some convergence results established in [3], under the assumptions of Theorem 1. Here, if (w l ) l∈N is a sequence of functions in L 2 ((0, T ) × M) for some measured space M,…”

Numerical Analysis for the Pure Neumann Control Problem Using the Gradient Discretisation Method

“…for linear and non-linear elliptic and parabolic diffusion equations (including degenerate equations), the Navier-Stokes equations, variational inequalities, Darcy flows in fractured media, etc. See for example [16,17,18,1,22], and the monograph [15] for a complete presentation of the GDM. The contributions of this article are summarised now.…”