Three-Dimensional Monte Carlo Simulations of Movement of a Contaminant Plume in Heterogeneous Aquifers

Monte Carlo (MC) approach has been applied in studying solute transport in heterogeneous porous media. Most studies with MC simulations were either conducted in two dimensions or in three dimensions with no more than a hundred realizations or addressed only the longitudinal macrodispersion even though our understanding about the transverse macrodispersion is less satisfactory than the longitudinal macrodispersion in both ergodic and non-ergodic transport. In this study Monte Carlo approach was adopted to simulate transport of a finite solute body by steady-state groundwater flow with a uniform mean velocity in a three-dimensional heterogeneous and statistically isotropic aquifer with more than a thousand realizations. The hydraulic conductivity (K) is modeled as a random field which is assumed to be log-normally distributed with an exponential covariance. The ensemble averages of the second spatial moments of the plume about its center of mass, < S_ii(t)> , and the plume centroid covariance, R_ii(t) (i=1,2,3), were simulated for three values of the variance of log K, 0.1, 0.25, and 0.5, and a line source of different lengths normal to the mean flow. It is shown that for the small variance of 0.1 all simulated < S_ii(t)> and R_ii(t) agree well with the first-order theoretical values. As the variance increases to 0.25 and 0.5, the simulated longitudinal moments, < S₁₁(t)> and R₁₁(t), still agree well with the first-order theoretical results but the simulated transverse moments are significantly different from the first-order theoretical results, contrary to the assumption that the first-order theory is generally valid for the variance of log K less than one. The simulated one-particle longitudinal displacement variance, X₁₁(t), is in good agreement with the first-order theory. The simulated transverse moment, X₂₂(t) and X₃₃(t), are larger than the first-order values but in excellent agreement with the quasi-linear theory based on the Corrsin­_s conjecture. We also found that more than 1000 realizations are needed in order to obtain stable second spatial moments for the variance of log K as small as 0.5, indicating the existing MC results for a much large variance with much less realizations may not valid.