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

## TL;DR

This paper presents an algebraic multiscale method (F-AMS) for simulating flow in heterogeneous porous media with embedded fractures, offering scalable, accurate, and mass-conservative solutions adaptable to complex fracture networks.

## Contribution

The novel F-AMS framework constructs independent multiscale coarse grids for matrix and fractures, with flexible coupling strategies and proven scalability for complex heterogeneous media.

## Key findings

- F-AMS is insensitive to high contrast in conductivities.
- The method achieves efficient convergence with reduced CPU times.
- It provides accurate, mass-conservative flux reconstructions after iterations.

## 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.