Algebraic multiscale method for flow in heterogeneous porous media with embedded discrete fractures (F-AMS)

Abstract

This paper introduces an Algebraic MultiScale method for simulation of flow in heterogeneous porous media with embedded discrete Fractures (F-AMS). First, multiscale coarse grids are independently constructed for both porous matrix and fracture networks. Then, a map between coarse-and fine-scale is obtained by algebraically computing basis functions with local support. In order to extend the localization assumption to the fractured media, four types of basis functions are investigated:(1) Decoupled-AMS, in which the two media are completely decoupled, (2) Frac-AMS and (3) Rock-AMS, which take into account only one-way transmissibilities, and (4) Coupled-AMS, in which the matrix and fracture interpolators are fully coupled. In order to ensure scalability, the F-AMS framework permits full flexibility in terms of the resolution of the fracture coarse grids. Numerical results are presented for two-and three-dimensional heterogeneous test cases. During these experiments, the performance of F-AMS, paired with ILU(0) as second-stage smoother in a convergent iterative procedure, is studied by monitoring CPU times and convergence rates. Finally, in order to investigate the scalability of the method, an extensive benchmark study is conducted, where a commercial algebraic multigrid solver is used as reference. The results show that, given an appropriate coarsening strategy, F-AMS is insensitive to severe fracture and matrix conductivity contrasts, as well as the length of the fracture networks. Its unique feature is that a fine-scale mass conservative flux field can be reconstructed after any iteration, providing efficient approximate solutions in time-dependent simulations.

Upscaling of the single-phase flow and heat transport in fractured geothermal reservoirs using nonlocal multicontinuum method

“…For the unstructured meshes where fractures are gridded explicitly, the discrete fracture models are used. In another approach, called the embedded fracture model (EFM) [21,24,45], the fractures are not resolved by grid but are considered as overlaying continua. In EFM, matrix and fracture are viewed as two porosity types co-existing at the same spatial location, thus simple structured meshes can be used for the domain discretization.…”

“…For the fine-grid approximations, we will use the finite volume method that is widely used for reservoir simulations. Specifically, we employ the cell-centered finite-volume method with two-point flux approximation together with the use of the embedded fracture model [8,24,45,46]. Let T h = ∪ i K i be a structured fine grid with rectangular cells of the domain and E γ = ∪ l γ l be the fracture mesh.…”

Algebraic dynamic multilevel method for embedded discrete fracture model (F-ADM)

“…The sub-block (P p ) i i−1 in the prolongation operator interpolates the pressure from resolution i to i − 1. Here, the twolevel multiscale strategy [36] is extended to multiple levels. Even though the fine-scale pressure equation is a TPFA-based system, starting from coarse level 1, all the coarse-scale systems develop MPFA-based connectivities, due to the local basis functions.…”

“…where the tp f a(•) extracts the TPFA components of the matrix (heptadiagonal matrix for 3D Cartesian grids), by considering the transmissibility between cells i and j equal to entry A ij of the original matrix. The algebraic multiscale procedure for the calculation of basis functions is now performed on this TPFA system at level i − 1 [36], in order to obtain the basis functions at level i, i.e., (P p ) i i−1 . Note that the TPFA reduction of the MPFA multilevel system is done independently for each sub-domain of matrix, and individual fractures.…”

“…In this work, an ADM method for simulation of multiphase flow in heterogeneous fractured porous media is developed (F-ADM). This is achieved by devising multilevel multiscale basis functions for embedded discrete fracture modelling (EDFM) approach [33][34][35][36][37][38]. More precisely, first, an EDFM fine-scale discrete system is obtained [39][40][41]34,[42][43][44][45], which allows for independent grids for fractures and matrix.…”