Temporal behavior of a solute cloud in a heterogeneous porous medium 1. Point-like injection

We investigate the temporal behavior of transport coefficients in a model for transport of a solute through a spatially heterogeneous saturated aquifer. In the framework of a stochastic approach we derive explicit expressions for the temporal behavior of the center-of-mass velocity and the dispersion of the concentration distribution after a point-like injection of solute at time t = 0, using a second-order perturbation expansion. The model takes into account local variations in the hydraulic conductivity (which, in turn, induce local fluctuations in the groundwater flow velocities) and in the chemical adsorption properties of the medium (which lead to a spatially varying local retardation factor). In the given perturbation theory approach the various heterogeneity-induced contributions can be systematically traced back to fluctuations in these quantities and to cross correlations between them. We analyze two conceptually different definitions for the resulting dispersion coefficient: the effective dispersion coefficient which is derived from the average over the centered second moments of the spatial concentration distributions in every realization and the ensemble dispersion coefficient which follows from the second moment of the ensemble-averaged concentration distribution. The first quantity characterizes the dispersion in a typical realization of the medium, whereas the second one describes the (formal) dispersion properties of the ensemble as a whole. We give explicit analytic expressions for both quantities as functions of time and show that for finite times their temporal behavior is remarkably different. The ensemble dispersion coefficient which is usually evaluated in the literature considerably overestimates the dispersion typically found in one given realization of the medium. From our explicit results we identify two relevant timescales separating regimes of qualitatively and quantitatively different temporal behavior: The shorter of the two scales is set by the advective transport of the solute cloud over one disorder correlation length, whereas the second, much larger one, is related to the dispersive spreading over the same distance. Only for times much larger than this second scale, do the effective and the ensemble dispersion coefficient become equivalent because of mixing caused by the local transversal dispersion. The formulae are applied to the Borden experiment data. It is concluded that the observed dispersion coefficient matches the effective dispersion coefficient at finite times proposed in this paper very well.