Analysis of macrodispersion through volume averaging: comparison with stochastic theory

Abstract

Velocity variability at scales smaller than the size of a solute plume enhances the rate of spreading of the plume around its center of mass. Macroscopically, the rate of spreading can be quantified through macrodispersion coefficients, the determination of which has been the subject of stochastic theories. This work compares the results of a volume-averaging approach with those of the advection dominated large-time small-perturbation theory of Dagan [1982] and Gelhar and Axness [1983]. Consider transport of an ideal tracer in a porous medium with deterministic periodic velocity. Using the Taylor-Aris-Brenner method of moments, it has been previously demonstrated [Kitanidis, 1992] that when the plume spreads over an area much larger than the period, the volume-averaged concentration satisfies the advection-dispersion equation with constant coefficients that can be computed. Here, the volume-averaging analysis is extended to the case of stationary random velocities. Additionally, a perturbation method is applied to obtain explicit solutions for small-fluctuation cases, and the results are compared with those of the stochastic macrodispersion theory. It is shown that the method of moments, which uses spatial averaging, for sufficiently large volumes of averaging yields the same result as the stochastic theory, which is based on ensemble averaging. The result is of theoretical but also practical significance because the volume-averaging approach provides a potentially efficient way to compute macrodispersion coefficients. The method is applied to a simplified representation of the Borden aquifer.