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

摘要

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 ε 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, ε), dependent on the Peclet number P_e, and on a dimensionless parameter a that represents the effects of microscopic heterogeneity. The parameter α, 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 ε. 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.