Temporal behaviour of a solute cloud in a chemically heterogeneous porous medium

In this paper we investigate the temporal behaviour of a solute cloud in a heterogeneous porous medium using a stochastic modelling approach. The behaviour of the plume evolving from a point-like instantaneous injection is characterized by the velocity of its centre-of-mass and by its dispersion as a function of time. In a stochastic approach, these quantities are expressed as appropriate averages over the ensemble of all possible realizations of the medium. We develop a general perturbation approach which allows one to calculate the various quantities in a systematic and unified way. We demonstrate this approach on a simplified aquifer model where only the retardation factor R(x) due to linear instantaneous chemical adsorption varies stochastically in space. We analyse the resulting centre-of-mass velocity and two conceptually different definitions for the dispersion coefficient: the ‘effective’ dispersion coefficient which is derived from the average over the centred second moments of the spatial concentration distributions in every realization, and the ‘ensemble’ dispersion coefficient which follows from the second moment of the averaged concentration distribution. The first quantity characterizes the dispersion in a typical realization of the medium as a function of time, whereas the second one describes the (formal) dispersion properties of the ensemble as a whole. We show that for finite times the two quantities are not equivalent whereas they become identical for t→∞ and spatial dimensions d[ges ]2. The ensemble dispersion coefficient which is usually evaluated in the literature considerably overestimates the dispersion typically found in one given realization of the medium. We derive for the first time explicit analytical expressions for both quantities as functions of time. From these, we identify two relevant time scales separating regimes of qualitatively and quantitatively different temporal behaviour: 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, and spatial dimensions d[ges ]2, do the effective and the ensemble dispersion coefficients become equivalent due to mixing caused by the local transversal dispersion. Finally, the formalism is generalized to an extended source. With growing source size the convergence of the effective dispersion coefficient to the ensemble dispersion coefficient happens faster as the extended source already represents an ensemble of point sources. In the limit of a very large source size, convergence occurs on the time scale of advective transport over one disorder length. We derive explicit results for the temporal behaviour in the different time regimes for both point and extended sources.