Asymptotic and numerical modelling of flows in fractured porous media

Abstract

Abstract. This study concerns some asymptotic models used to compute the flow outside and inside fractures in a bidimensional porous medium. The flow is governed by the Darcy law both in the fractures and in the porous matrix with large discontinuities in the permeability tensor. These fractures are supposed to have a small thickness with respect to the macroscopic length scale, so that we can asymptotically reduce them to immersed polygonal fault interfaces and the model finally consists in a coupling between a 2D elliptic problem and a 1D equation on the sharp interfaces modelling the fractures. A cell-centered finite volume scheme on general polygonal meshes fitting the interfaces is derived to solve the set of equations with the additional differential transmission conditions linking both pressure and normal velocity jumps through the interfaces. We prove the convergence of the FV scheme for any set of data and parameters of the models and derive existence and uniqueness of the solution to the asymptotic models proposed. The models are then numerically experimented for highly or partially immersed fractures. Some numerical results are reported showing different kinds of flows in the case of impermeable or partially/highly permeable fractures. The influence of the variation of the aperture of the fractures is also investigated. The numerical solutions of the asymptotic models are validated by comparing them to the solutions of the global Darcy model or to some analytic solutions.Mathematics Subject Classification. 76S05, 74S10, 35J25, 35J20, 65N15.

A mixed finite element method for Darcy flow in fractured porous media with non-matching grids

“…This is not a strict limitation of the proposed method; however, handling the case of a "partially immersed fracture" is more complex from the analytical point of view (see [2] for a thorough investigation concerning the proper analytical setting of the continuous problem).…”

“…This requires the derivation of a reduced model for the fracture flow, for which we refer to [1,2,12,15]. Following the cited references, we assume that the flow within the fracture is governed by a reduced Darcy's equation.…”

“…Thanks to the algebraic identity ab = a {b} + {a} b and to the interface conditions (2.4) we can write 3 In the case of a "partially immersed" fracture this step would be more complex, see [2], Section 3.…”

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

“…γ − ), see Figure 1. For a sufficiently regular function ϕ in Ω \ γ, we consider two traces ϕ + and ϕ − on γ, one for each side; see [3] for a more rigorous definition. We also introduce the tangential gradient ∇ τ and the tangential divergence div τ operators along the fracture.…”

“…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.…”