Renormalization Group theory outperforms other approaches in statistical comparison between upscaling techniques for porous media

TL;DR

This study demonstrates that Mode-Elimination Renormalization-Group (MG) theory provides a more accurate and computationally efficient method for upscaling permeability in porous media, outperforming traditional approaches especially near the percolation threshold.

Contribution

The paper introduces a novel MG-based upscaling technique that effectively captures tensorial permeability properties and improves accuracy over existing methods.

Findings

MG method outperforms other upscaling techniques in accuracy.

MG maintains low computational cost.

MG provides reliable flow estimates near the percolation threshold.

Abstract

Determining the pressure differential required to achieve a desired flow rate in a porous medium requires solving Darcy's law, a Laplace-like equation, with a spatially varying tensor permeability. In various scenarios, the permeability coefficient is sampled at high spatial resolution, which makes solving Darcy's equation numerically prohibitively expensive. As a consequence, much effort has gone into creating upscaled or low-resolution effective models of the coefficient while ensuring that the estimated flow rate is well reproduced, bringing to fore the classic tradeoff between computational cost and numerical accuracy. Here we perform a statistical study to characterize the relative success of upscaling methods on a large sample of permeability coefficients that are above the percolation threshold. We introduce a new technique based on Mode-Elimination Renormalization-Group theory (MG) to build coarse-scale permeability coefficients. Comparing the results with coefficients upscaled using other methods, we find that MG is consistently more accurate, particularly so due to its ability to address the tensorial nature of the coefficients. MG places a low computational demand, in the manner that we have implemented it, and accurate flow-rate estimates are obtained when using MG-upscaled permeabilities that approach or are beyond the percolation threshold.