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 f or transport of a solute through a spatially heterogeneous saturated aquife r. In the framework of a stochastic approach we derive explicit expressions for the temporal behavior of the center-of-mass velocity and the dispersio n 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 tur n, induce local fluctuations in the groundwater flow velocities) and in the chemical adsorption properties of the medium (which lead to a spatially va rying local retardation factor). In the given perturbation theory approach the various heterogeneity-induced contributions can be systematically trace d back to fluctuations in these quantities and to cross correlations betwee n them. We analyze two conceptually different definitions for the resulting dispersion coefficient: the "effective" dispersion coefficient which is de rived from the average over the centered second moments of the spatial conc entration distributions in every realization and the "ensemble" dispersion coefficient which follows from the second moment of the ensemble-averaged c oncentration distribution. The first quantity characterizes the dispersion in a typical realization of the medium, whereas the second one describes th e (formal) dispersion properties of the ensemble as a whole. We give explic it analytic expressions for both quantities as functions of time and show t hat for finite times their temporal behavior is remarkably different. The e nsemble dispersion coefficient which is usually evaluated in the literature considerably overestimates the dispersion typically found in one given rea lization 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 t ransport of the solute cloud over one disorder correlation length, whereas the second, much larger one, is related to the dispersive spreading over th e 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 app lied to the Borden experiment data. It is concluded that the observed dispe rsion coefficient matches the effective dispersion coefficient at finite ti mes proposed in this paper very well.