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

An analytical approach is developed for describing the ensemble averag e of the second moment of a solute plume in three-dimensional heteroge neous porous media. While existing approaches describe scale-dependent dispersion in terms of a single scale, the plume displacement, the ap proach 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 descript ion of scale-dependent dispersion is thus achieved. The growth of the ensemble average second moment is related to the product of concentrat ion values at two points. The concept of the separation distribution f unction related to the latter is introduced. The separation distributi on function physically describes the fraction of solute particles whic h have another solute particle at a given separation. An Eulerian part ial differential equation based on a small perturbation approach is de veloped to describe the evolution of the separation distribution funct ion. Simple analytical expressions for the second moment growth rates are presented. These expressions incorporate the influence of the plum e size through a low wavenumber filter depending of the plume second m oment. Asymptotic expressions for the second moment growth rate are pr esented which apply at large displacement. These expressions indicate that the longitudinal second moment growth rate depends on the transve rse second moments of the plume. Comparison of predicted second moment evolution with results from earlier numerical simulations indicates e xcellent agreement. Application to the Borden tracer test indicates a significant reduction in the longitudinal second moment from that pred icted by existing three-dimensional theories and better agreement with the measured second moments.