Scale-dependent dispersivity: a velocity fluctuation model

Abstract

In the previous paper, it was shown that the cumulative effect of multiple one-dimensional velocity fluctuations can explain qualitative features of the observed scale dependent dispersivity in natural aquifers, but not the magnitude of the effect. It is plausible that in real systems the enhancement of dispersion caused by a single fluctuation may be larger than that derived for the 1-dimensional stepped fluctuation, because for example there are additional enhancement mechanisms in 2- and 3-dimensional systems. However this paper shows that to achieve the observed magnitude, it is not enough to increase the size of enhancement factor but in addition the rate at which the effect of a single fluctuation changes with fluctuation length and with position along the fluctuation sequence need to be modified. Several variations are explored. Simple assumptions are shown to lead to dispersivity formulas in terms of purely algebraic power laws, while more elaborate assumptions yield expressions that are still analytic but contain non-elementary functions. In either case it is possible to find the required variation of the dispersivity over 3 or more orders of magnitude and with curve shapes that are consistent with historical observations Moreover, this is achieved with plausible parameter values, leading for example to the conjecture that in the observed systems the porous medium could not have been homogeneous on a scale of more than centimeters. The model presented is schematic in the sense that it contains some detail assumptions not derived from first principles, but is believed to capture the essentials of the mechanism that causes scale dependent dispersivity. It sets some boundaries for viable detail models, but within those boundaries the final predictions are not very sensitive to the detail assumptions. A key merit of the treatment is that it identifies crucial variables that need to be measured or controlled in experimental studies.