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

Abstract

International audienceFlow and transport in fracturedporous media are of paramount importance for many applications such as petroleum exploration and production, geological storage of carbon dioxide, hydrogeology, or geothermal energy. We consider here the two-phase discrete fracture model introduced in [3] which represents explicitly the fractures as codimension one surfaces immersed in the surrounding matrix domain. Then, the two-phase Darcy flow in the matrix is coupled with the two-phase Darcy flow in the fractures using transmission conditions accounting for fractures acting either as drains or barriers. The model takes into account complex networks of fractures, discontinuous capillary pressure curves at the matrix fracture interfaces and can be easily extended to account for gravity including in the width of the fractures. It also includes a layer of damaged rock at the matrix fracture interface with its own mobility and capillary pressure functions. In this work, the convergence analysis carried out in [3] in the framework of gradient discretizations [2] is extended to obtain the uniform-in-time convergence of the discrete solutions to a weak solution of the model

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

“…Based on compactness arguments, we showed in Theorem 4.1 the strong L 2 convergence of the saturations and the weak L 2 convergence for the pressures to a solution of Model (1). In Theorem 4.11, we established uniform-in-time, weak L 2 in space convergence for the saturations, a result that is extended to uniform-in-time, strong L 2 in space convergence in [23].…”

“…Remark 4.2 It is additionally proved in [23] that the saturations converge uniformly-in-time strongly in L 2 (that is, in L ∞ (0, T ; L 2 )).…”