Plume scale‐dependent dispersion in heterogeneous aquifers: 1. Lagrangian analysis in a stratified aquifer

Abstract

An analytical approach is developed for describing the ensemble average of the second moment of a solute plume in heterogeneous porous media. The growth of the ensemble average second moment is related to the growth of the mean square separation of a pair of particles. Exact Lagrangian expressions are developed for the growth of the mean square separation in a perfectly stratified aquifer. These exact expressions are made possible by an exact relation between the Lagrangian and Eulerian velocity covariances in the perfectly stratified aquifer. The ensemble average second moment is shown to depend on the initial vertical dimensions of the concentration distribution. As the vertical size increases, the second moment growth rate is larger. The second moment expressions are contrasted with the expressions for the mean square displacement of a single particle. While all scales of heterogeneity contribute to the mean square displacement of a single particle, only scales of heterogeneity smaller than the plume size contribute to the second moment. Asymptotic large displacement expressions for the case of a Gaussian covariance function describing the hydraulic conductivity variations are derived. These expressions indicate that at large displacement, the second moment grows as the 3/2 power of displacement. However, for any finite source size no matter how large, the second moment growth rate approaches a different asymptote from the rate of growth of the mean square displacement of a single particle. This is in conflict with the traditional notion of ergodicity which leads to the expectation that for large source sizes, the dispersivity approaches the rate of growth of the mean square displacement of a single particle.