Solute transport in three‐dimensional heterogeneous media with a Gaussian covariance of log hydraulic conductivity

Abstract

The analytical solutions for the velocity covariance, uij one particle displacement covariance Xij, and the macrodispersivity tensor αij defined as (0.5/μ)(dXij/dt), were derived in three‐dimensional heterogeneous media. A Gaussian covariance function (GCF) of logarithmic hydraulic conductivity, log K, was used, assuming uniform mean flow and first‐order approximation in log‐conductivity variance, where μ is the magnitude of the mean flow velocity μ. Based on these solutions, the time‐dependent ensemble averages of the second spatial moments, Zij ≡ 〈Aij〉 − Aij∥0) = Xij − Rij and the effective dispersivity tensor γij defined as (0.5/μ)(d〈Aij〉/dt, were evaluated for a finite line source either normal or parallel to μ, where Aij(0) is the initial value of the second spatial moment of a plume, Aij and Rij is the plume centroid covariance. The results obtained in this study were compared with previous results for an exponential covariance function (ECF). It was found that in a stationary log K field the spreading of a solute plume depends not only on the variance and integral scale of the log K field but also on the shape of its covariance function. The more strongly correlated the hydraulic conductivities at short separation distances are, the faster Zii and γii grow at early time. Also, the earlier that γii approaches its asymptote or peak, and the higher the peak is, the larger the negative transverse dispersivity. More importantly, the ergodic limits for GCF are reached faster than those for ECF, as the initial size of a plume increases. The ergodic limit X11 for GCF is slightly larger than that for ECF, but X22 and X33 are significantly smaller than those for ECF even though the asymptotic αii for GCF is the same with that for ECF.