Asymptotic dispersion for two-dimensional highly heterogeneous permeability fields under temporally fluctuating flow

Temporal fluctuations of water flux have been investigated as a mechanism that strongly enhances transverse dispersion in heterogeneous media. Unfortunately, most results have been obtained by linear stochastic theories on permeability fields of limited variability. Worse, results are inconsistent regarding the impact of fluctuations on longitudinal dispersion, which motivates our work to find the effect of temporal velocity fluctuations on macrodispersion. We perform numerical Monte Carlo simulations for highly variable permeability fields of up to 800 correlation lengths. We find that fluctuations longitudinal to the main flow direction hardly modify macrodispersion because they do not alter the flow lines. Fluctuations transverse to the main flow direction not only increase transverse dispersion, which is well known, but also reduce the longitudinal macrodispersion in a significant and consistent way, which contradicts previous findings. The reduction of the longitudinal dispersion is comparable to the increase of transverse dispersion. Most surprisingly, for high heterogeneity, temporal fluctuations cause total (longitudinal plus transverse) macrodispersion to drop with respect to the steady state one. Enhancement of the transverse macrodispersion comes from both the increase of the transverse velocity variability and Lagrangian correlation. Reduction of the longitudinal macrodispersion results from the reduction of the Lagrangian correlation of the longitudinal velocity. That is, temporal fluctuations reduce longitudinal spreading both by breaking the fastest velocity paths on the plume front and by letting solute bypass the low-permeability zones that tend to block or trap the solute in steady state flow conditions.