On transport in formations of large heterogeneity scales

It has been recently suggested that in transport through heterogeneous aquifers, the effective dispersivity increases with the travel distance since plumes encounter heterogeneity of increasing scales. This conclusion is underlain, however, by the assumption of ergodicity. If the plume is viewed as made up from different particles, this means that these particles move independently from a statistical point of view. To satisfy ergodicity the solute body has to be of a much larger extent than heterogeneity scales. Thus, if the latter are increasing for ever and the solute body is finite, ergodicity cannot be obeyed. To demonstrate this thesis we relate to the two-dimensional heterogeneity associated with transmissivity variations in the horizontal plane. First, the effective dispersion coefficient is defined as half the rate of change of the expected value of the solute body second spatial moment relative to its centroid. Subsequently the asymptotic large time limit of dispersivity is evaluated in terms of the logtransmissivity integral scale and of the dimensions of the initial solute body in the direction of mean flow and normal to it. It is shown that for a thin plume aligned with the mean flow the effective dispersivity is zero and the effect of heterogeneity is a slight and finite expansion determined solely by the solute body size. In the case of a solute body transverse to the mean flow the effective dispersivity is different from zero, but has a maximal value which is again dependent on the solute body size and not on the heterogeneity scale. It is concluded that from a theoretical standpoint and for the definition of dispersivity adopted here for non-ergodic conditions, the claim of ever-increasing dispersivity with travel distance is not valid for the scale of heterogeneity analyzed here.