On the asymptotic behavior of dilution parameters for gaussian and hole-gaussian log-conductivity covariance functions

Using a Lagrangian approach, the authors have previously shown that the kinetics of concentration variance and dilution of passive solutes in heterogeneous aquifers depend on the ratio between two- and one-particle covariances. Extending this approach, it is here demonstrated that, for point injections, the two-particle covariance coincides with the variance of the centroid location. Then, based on an Eulerian formulation, analytical first-order approximations are obtained for the two-particle covariance, for two types of log-conductivity covariance. For Gaussian covariance functions, whereas the transverse moments are asymptotically constant, the longitudinal moment tends to increase logarithmically over time. For hole–Gaussian covariances, both transverse and longitudinal moments tend to constant values. In both cases, the longitudinal two-particle correlation, which plays a crucial role in determining the dynamics of the concentration fluctuations, is controlled by the magnitude of the local dispersivity. The theoretical predictions are compared to the data collected at the Cape Cod site in terms of time derivative of the longitudinal two-particle moment, for Gaussian logconductivity covariance and resorting to maximum likelihood estimates. Generally, good agreement is there, between experimental data and analytical expressions.