The concept of the Dilution Index

TL;DR: In this paper, the authors introduced a new macroscopic measure of dilution, the dilution index E, which measures the overall rate at which a tracer plume spreads about its centroid and depend critically on the heterogeneity of the formation.

Abstract: In many applications, it is important to make the distinction between spreading and dilution of a plume in groundwater. Spreading is associated with the stretching and deformation of a contaminant plume, whereas dilution is associated with the increase in volume of the fluid occupied by the solute. The dilution and spreading of a Gaussian plume in a homogeneous porous medium with constant velocity are related in a simple fashion and are both characterized by the same parameters, the dispersion coefficients. However, the geological formations of interest in field applications are heterogeneous, and the plumes are irregular in shape. The dispersion coefficients that are deduced from tracer tests usually measure an overall rate at which a tracer plume spreads about its centroid and depend critically on the heterogeneity of the formation. These macroscopic dispersion coefficients are not reliable measures of the rate at which the maximum concentration is reduced because in heterogeneous formations the rates of dilution and spreading can be quite different. The main objective of this work is to introduce a new macroscopic measure of dilution, the dilution index E. Examples serve to demonstrate the usefulness of the measure. A general expression for the rate of dilution of a tracer plume is derived. The exact rate of increase of the dilution index under the idealized conditions of constant dispersion coefficients and a Gaussian plume is computed, and a lower bound is found to the same quantity for non-Gaussian plumes. For the general heterogeneous case the analysis demonstrates that the instantaneous rate of increase of ln E is proportional to the small-scale dispersion coefficients, everything else being the same. The rate of increase of ln E depends also on the degree of irregularity in the shape of the plume. Thus, in the long term, geologic heterogeneity should increase the rate of dilution because spatial variability in the flow velocity tends to deform plumes and make them less regular.