Generalized Multiscale Inversion for Heterogeneous Problems

TL;DR

This paper introduces a generalized multiscale inversion algorithm based on GMsFEM for heterogeneous problems, effectively capturing complex features like fractures without fine-grid resolution, and demonstrating its effectiveness through numerical experiments.

Contribution

It develops a new inversion method that identifies coarse-grid parameters in highly heterogeneous media using GMsFEM, bypassing the need for fine-scale details.

Findings

Accurately captures channels and fractures in heterogeneous media.

Reduces degrees of freedom compared to fine-grid models.

Demonstrates effectiveness through numerical experiments.

Abstract

In this work, we propose a generalized multiscale inversion algorithm for heterogeneous problems that aims at solving an inverse problem on a computational coarse grid. Previous inversion techniques for multiscale problems seek a coarse-grid media properties, e.g., permeability and conductivity, and by doing so, they assume that there exists a homogenized representation of the underlying fine-scale permeability field on a coarse grid. Generally such assumptions do not hold for highly heterogeneous fields, e.g., fracture media or channelized fields, where the width of channels are very small compared to the coarse-grid sizes. In these cases, grid refinement can lead to many degrees of freedom, and thus unattractive to apply. The proposed algorithm is based on the Generalized Multiscale Finite Element Method (GMsFEM), which uses local spectral problems to identify non-localized features, i.e., channels (high-conductivity inclusions that connect the boundaries of the coarse-grid block). The inclusion of these features in the coarse space enables one to achieve a good accuracy. The approach is valid under the assumption that the solution can be well represented in a reduced-dimensional space by multiscale basis functions. In practice, these basis functions are non-observable as we do not identify the fine-scale features of the permeability field. Our inversion algorithm finds the discretization parameters of the resulting system. By doing so, we identify the appropriate coarse-grid parameters representing the permeability field instead of fine-grid permeability field. We illustrate the approach by numerical results for fractured media.