PLUME SCALE-DEPENDENT DISPERSION IN HETEROGENEOUS AQUIFERS .2. EULERIAN ANALYSIS AND 3-DIMENSIONAL AQUIFERS

An analytical approach is developed for describing the ensemble average of the second moment of a solute plume in three-dimensional heterogeneous porous media. While existing approaches describe scale-dependent dispersion in terms of a single scale, the plume displacement, the approach developed here presents an enhanced picture of scale-dependent dispersion involving two scales: the plume displacement and the plume scale. The plume scale arises naturally in the formulation, permitting a distinction between the dispersive role of heterogeneity at scales smaller than the plume size and the variability in the plume location caused by larger scale heterogeneity. A physically consistent description of scale-dependent dispersion is thus achieved. The growth of the ensemble average second moment is related to the product of concentration values at two points. The concept of the separation distribution function related to the latter is introduced. The separation distribution function physically describes the fraction of solute particles which have another solute particle at a given separation. An Eulerian partial differential equation based on a small perturbation approach is developed to describe the evolution of the separation distribution function. Simple analytical expressions for the second moment growth rates are presented. These expressions incorporate the influence of the plume size through a low wavenumber filter depending of the plume second moment. Asymptotic expressions for the second moment growth rate are presented which apply at large displacement. These expressions indicate that the longitudinal second moment growth rate depends on the transverse second moments of the plume. Comparison of predicted second moment evolution with results from earlier numerical simulations indicates excellent agreement. Application to the Borden tracer test indicates a significant reduction in the longitudinal second moment from that predicted by existing three-dimensional theories and better agreement with the measured second moments.