Temporal behavior of a solute cloud in a heterogeneous porous medium: 2. Spatially extended injection

We investigate the temporal behavior of transport coefficients in a stochastic model for transport of a solute through a spatially heterogeneous saturated aquifer. While the first of these two companion papers [Dentz et al., this issue] investigated a situation characterized by a point‐like solute injection, we now focus on the case of spatially extended solute sources. The analysis of the finite time behavior of the transport coefficients makes it necessary to distinguish between two fundamentally different quantities characterizing the solute dispersion. We define an “effective” dispersion coefficient which is derived from the average over the centered second moments of the spatial concentration distributions in every realization and an “ensemble” dispersion coefficient which follows from the second moment of the ensemble‐averaged concentration distribution. While the two quantities are equivalent in the asymptotic limit of infinite times or infinitely extended sources, they are qualitatively and quantitatively different for the more realistic situation of finite times and finite source extent. We demonstrate that in this case the ensemble quantity, used more or less implicitly in most of the previous studies, overestimates the true dispersion of the plume. Using a second‐order perturbation theory approach, we derive explicit solutions for the temporal behavior of the dispersion coefficients for various types of isotropic and anisotropic initial conditions. We identify the relevant timescales which separate regimes of different temporal behavior and apply our formulae to the Borden experiment data. We find a good agreement between theory and experiment if we compare the observed dispersion with the appropriate effective dispersion coefficient (including the leading effects of the local dispersion), whereas the ensemble dispersion coefficient commonly used in the literature to analyze these data overestimates the experimental results considerably.