Effective dispersion in temporally fluctuating flow through a heterogeneous medium

In this paper we investigate the effective transport of a passive solute in temporally fluctuating flow through a spatially heterogeneous medium. The Darcy equation for an incompressible fluid in the heterogeneous medium is solved with temporally fluctuating boundary conditions using perturbation theory. We distinguish between a spatial random process reflecting the medium heterogeneities and a temporal random process which models the fluctuations of the boundary conditions. By appropriately averaging over the corresponding random fields, we evaluate the second-order perturbation approximation to the time evolution of the "effective" and "ensemble" dispersion coefficients. Both quantities consist of three terms reflecting: (1) local dispersion; (2) dispersion caused by spatial heterogeneities (identical to the corresponding dispersion in steady random flow); (3) dispersion linked to the enhanced solute spreading caused by the interactions between temporal fluctuations, local dispersion, and spatial heterogeneity. The behavior of this latter contribution is complex due to the interplay of three different time scales (set by fluctuating boundary conditions, local dispersion, and advection). Temporal fluctuations of the velocity field lead to effective transverse dispersion coefficients that evolve in time to macroscopic values (i.e., independent of the local dispersion), which is consistent with observations in the field but was not predicted by theories based on steady flow. Due to their perturbative nature, the derived results are intrinsically limited to moderately fluctuating random velocity fields. However, numerical transport simulations indicate a wide range of applicability. The reported results support remediation techniques that attempt to enhance the mixing of injected reactants with contaminated ground water by temporal variations of injection and pumping rates.