# Coupling of multiscale and multi-continuum approaches

## TL;DR

This paper explores the relationship between multi-continuum methods and the Generalized Multiscale Finite Element Method (GMsFEM), proposing a coupled approach for efficient simulation of fractured media with multiscale features.

## Contribution

It establishes a connection between multi-continuum techniques and GMsFEM, and introduces a coupled multiscale modeling approach for complex fractured media.

## Key findings

- GMsFEM can automatically identify fracture networks.
- Simplified basis construction is effective for known, simple fracture geometries.
- Coupled approach efficiently models unresolved fractures using multi-continuum methods.

## Abstract

Simulating complex processes in fractured media requires some type of model reduction. Well-known approaches include multi-continuum techniques, which have been commonly used in approximating subgrid effects for flow and transport in fractured media. Our goal in this paper is to (1) show a relation between multi-continuum approaches and Generalized Multiscale Finite Element Method (GMsFEM) and (2) to discuss coupling these approaches for solving problems in complex multiscale fractured media. The GMsFEM, a systematic approach, constructs multiscale basis functions via local spectral decomposition in pre-computed snapshot spaces. We show that GMsFEM can automatically identify separate fracture networks via local spectral problems. We discuss the relation between these basis functions and continuums in multi-continuum methods. The GMsFEM can automatically detect each continuum and represent the interaction between the continuum and its surrounding (matrix). For problems with simplified fracture networks, we propose a simplified basis construction with the GMsFEM. This simplified approach is effective when the fracture networks are known and have simplified geometries. We show that this approach can achieve a similar result compared to the results using the GMsFEM with spectral basis functions. Further, we discuss the coupling between the GMsFEM and multi-continuum approaches. In this case, many fractures are resolved while for unresolved fractures, we use a multi-continuum approach with local Representative Volume Element (RVE) information. As a result, the method deals with a system of equations on a coarse grid, where each equation represents one of the continua on the fine grid. We present various basis construction mechanisms and numerical results.