A multiscale flux basis for mortar mixed discretizations of reduced Darcy–Forchheimer fracture models

Abstract

In this paper, a multiscale flux basis algorithm is developed to efficiently solve a flow problem in fractured porous media. Here, we take into account a mixed-dimensional setting of the discrete fracture matrix model, where the fracture network is represented as lower-dimensional object. We assume the linear Darcy model in the rock matrix and the non-linear Forchheimer model in the fractures. In our formulation, we are able to reformulate the matrix-fracture problem to only the fracture network problem and, therefore, significantly reduce the computational cost. The resulting problem is then a non-linear interface problem that can be solved using a fixed-point or Newton-Krylov methods, which in each iteration require several solves of Robin problems in the surrounding rock matrices. To achieve this, the flux exchange (a linear Robin-to-Neumann co-dimensional mapping) between the porous medium and the fracture network is done offline by pre-computing a multiscale flux basis that consists of the flux response from each degree of freedom on the fracture network. This delivers a conserve for the basis that handles the solutions in the rock matrices for each degree of freedom in the fractures pressure space. Then, any Robin sub-domain problems are replaced by linear combinations of the multiscale flux basis during the interface iteration. The proposed approach is, thus, agnostic to the physical model in the fracture network. Numerical experiments demonstrate the computational gains of pre-computing the flux exchange between the porous medium and the fracture network against standard non-linear domain decomposition approaches.

Splitting-based domain decomposition methods for two-phase flow with different rock types

“…We reformulate next the multi-domain pressure problem (4.2)-(4.3) as a reduced problem posed on the interface between the rocks [3]. The reduced problem is then solved by an iterative procedure, which requires solving subdomain pressure problems at each iteration.…”

“…The problem is then decomposed into a set of subproblems in different subdomains, and solved in each time step in a fixed point type iteration based on the L-scheme linearization method [32]. Other related works can be found in [3,21,23,31].…”

Intersecting fractures in porous media: mathematical and numerical analysis

“…However, the simulation of such models remains challenging: one needs to choose between a conforming or a non-conforming mesh, and in the latter case, how to express the transmission conditions for pressure and flux across the fractures. A large number of numerical methods have been used [52,53,45,1,59,32], among which one finds finite volume methods [4,3,27], extended finite elements (XFEM) [22,29], and mixed finite elements [49,7,16,13,9]. Models that specifically take into account the behavior of the flow at fracture intersection include [7,16,13,60].…”

“…We note that a fairly general, and quite elegant, formulation for hierarchical problem has been proposed and analyzed in [16] (see also [52,53]). The models have been extended to deal with non-linear flow in the fractures (Forchheimer model), see [33,1] and multiphase flow, see [17,37,38,43,35,58,41]. A posteriori error estimates were studied in [51,39].…”

“…The fractures define the subdomains between them, and the physical transmission conditions across the fractures give transmission conditions for the DD method. DD was first proposed as a possible formulation in [49,5], was studied in more details in [28] and more recently in [2,1]. The original mixed-dimensional problem is reduced to one posed only on the fractures, which is somewhat non-standard from the DD viewpoint, as it has to account both for the subdomain solutions, and also for the flow inside the fractures.…”