Upscaling of solute transport in heterogeneous media with non-uniform flow and dispersion fields

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 V-e and an effective dispersion coefficient D-e. It is shown that both V-e and D-e are scale-dependent (dependent on the length scale of the microscopic heterogeneity, epsilon), dependent on the Peclet number P-e, 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 V-e and dispersion coefficient D-e 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. (C) 2013 Elsevier Inc. All rights reserved.