Algebraic Multiscale Solver for Flow in Heterogeneous Fractured Porous Media

Abstract

A general algebraic multiscale framework is presented for fractured porous media, which enables the treatment of fractures with multiple length scales and wide ranges of conductivity contrasts. To this end, fully integrated local basis functions for both matrix and fractures are constructed. These basis functions are employed to construct the multiscale coarse system for both matrix and fractures, and then interpolate the coarse solutions back to the fine-scale reference system. Combined with a second stage fine-scale solver, here, ILU(0), our development leads to an iterative multiscale strategy for heterogeneous fractured media, allowing for error reduction to any arbitrary level, while honoring mass conservation after any multiscale finite volume stage. In order to maintain generality, it is shown that when each fracture network is modeled using a single coarse grid cell, our formulation automatically reduces to that proposed by Hajibeygi et al. (2011). However, in order to facilitate the treatment of general ranges of conductivity contrasts and deviation term norms, we introduce the ability to refine the coarse grid to more than one cell per network. Our experiments show that this flexibility results in a significant improvement on convergence properties. This added degree of flexibility is made possible through an algebraic formulation, which leads to a multi-stage multiscale conservative linear solver for multiphase flow in fractured media.

The multiscale restriction smoothed basis method for fractured porous media (F-MsRSB)

“…To resolve this computational challenge, a multiscale restriction smoothed basis method for fractured media (F-MsRSB) is developed. The F-MsRSB benefits from the previously developed multiscale methods for fractured media [1,2] and the MsRSB formulation for non-fractured systems [3].…”

“…The F-MsRSB method is devised on the idea of introducing basis functions for both matrix and fracture domains, similar to F-AMS [2] where n f = (n m f + n f f ) and n c = (n cm + n cf ) are total degrees of freedom (matrix and fractures) at fine and coarse scales.…”

“…This leads to higher iterative convergence rates, as shown in Fig. 5.4, if F-MsRSB is combined with ILU(0) in an iterative multiscale procedure [2]. Convergence is determined on the basis of setting a threshold value for the scaled residual norm, i.e., ||r b || 2 = ||Ap q|| 2 /||q|| 2 .…”

“…Later, Sandve et al [51] used the MSFV method to develop e↵ective coarse-scale MINC-type model for fracture networks. Very recently, the treatment of fracture network was extended in a general formulation by proposing an algebraic multiscale solver for fractured media (F-AMS) [2,52]. In the F-AMS, fracture basis functions were introduced on the basis of a coarsening ratio inside fracture domain, similar as in the matrix rock.…”

“…Following F-AMS [2,52], F-MsRSB constructs basis functions for fractures and matrix in a general way, allowing for di↵erent level of coupling between them. In addition to F-AMS, though, F-MsRSB constructs its multiscale formulation on the basis of the MsRSB approach.…”