Hierarchical multiscale modeling for flows in fractured media using Generalized Multiscale Finite Element Method

TL;DR

This paper introduces a multiscale finite element method based on GMsFEM for simulating flows in fractured media, efficiently capturing fracture effects through adaptive basis functions and reduced computational cost.

Contribution

The paper develops a GMsFEM-based multiscale approach that effectively models fractures using local spectral problems, combining DFM and EFM, with adaptive basis functions and randomized snapshots.

Findings

Accurate flow simulation in fractured media demonstrated.

Adaptive basis functions improve computational efficiency.

Randomized snapshots reduce offline computational cost.

Abstract

In this paper, we develop a multiscale finite element method for solving flows in fractured media. Our approach is based on Generalized Multiscale Finite Element Method (GMsFEM), where we represent the fracture effects on a coarse grid via multiscale basis functions. These multiscale basis functions are constructed in the offline stage via local spectral problems following GMsFEM. To represent the fractures on the fine grid, we consider two approaches (1) Discrete Fracture Model (DFM) (2) Embedded Fracture Model (EFM) and their combination. In DFM, the fractures are resolved via the fine grid, while in EFM the fracture and the fine grid block interaction is represented as a source term. In the proposed multiscale method, additional multiscale basis functions are used to represent the long fractures, while short-size fractures are collectively represented by a single basis functions. The procedure is automatically done via local spectral problems. In this regard, our approach shares common concepts with several approaches proposed in the literature as we discuss. Numerical results are presented where we demonstrate how one can adaptively add basis functions in the regions of interest based on error indicators. We also discuss the use of randomized snapshots (\cite{randomized2014}) which reduces the offline computational cost.