Upscaling of Solute Transport in Heterogeneous Media with Non-uniform Flow and Dispersion Fields

TL;DR

This paper develops an analytical and computational model to upscale solute transport in heterogeneous media with non-uniform flow and dispersion, providing scale-dependent effective parameters and efficient solutions.

Contribution

The paper introduces a novel homogenization approach that accounts for non-uniform flow and dispersion fields, deriving scale-dependent effective parameters for macroscopic modeling.

Findings

Effective advection velocity and dispersion coefficient depend on heterogeneity scale and Péclet number.

The model achieves good agreement with direct numerical solutions while reducing computational cost.

Homogenized solutions can accurately describe macroscopic solute transport in complex media.

Abstract

An analytical and computational model for non-reactive solute transport in periodic heterogeneous media with arbitrary non-uniform flow and dispersion fields within the unit cell of length {\epsilon} is described. The model lumps the effect of non-uniform flow and dispersion into an effective advection velocity Ve and an effective dispersion coefficient De. It is shown that both Ve and De are scale-dependent (dependent on the length scale of the microscopic heterogeneity, {\epsilon}), dependent on the P\'eclet number Pe, and on a dimensionless parameter {\alpha} that represents the effects of microscopic heterogeneity. The parameter {\alpha}, confined to the range of [-0.5, 0.5] for the numerical example presented, depends on the flow direction and non-uniform flow and dispersion fields. Effective advection velocity Ve and dispersion coefficient De can be derived for any given flow and dispersion fields, and {\epsilon}. Homogenized solutions describing the macroscopic variations can be obtained from the effective model. Solutions with sub-unit-cell accuracy can be constructed by homogenized solutions and its spatial derivatives. A numerical implementation of the model compared with direct numerical solutions using a fine grid, demonstrated that the new method was in good agreement with direct solutions, but with significant computational savings.