Is transverse macrodispersivity in three-dimensional groundwater transport equal to zero? A counterexample

In advective transport through weakly heterogeneous aquifers of random stationary and isotropic three-dimensional permeability distribution, transverse macrodispersivity alpha(T) is found to be zero. This was determined in the past by solving the transport equation at first order in the log conductivity variance sigma(2)(Y). However, field findings indicate the presence of small but finite alpha(T). The aim of the paper is to determine alpha(T) for highly heterogeneous formations using a model that contains inclusions of conductivity K, submerged in a matrix of conductivity K(0), for large kappa = K/ K(0). In the dilute medium approximation, valid for small volume fraction n, but arbitrary kappa, and for spherical inclusions, it is found that alpha(T) = 0 because of the axisymmetry of flow past a sphere. A medium made up of rotational ellipsoids of arbitrary random orientation, macroscopically isotropic, and of the same k and n is devised as a counterexample. It is found that because of the intertwining of streamlines alpha(T) > 0, being of order (kappa -> 1) 4 for kappa -> 1. These findings are confirmed by accurate numerical simulations of flow through a large number of interacting inclusions; for kappa = 10 and n = 0.2 (jamming limit), the large value alpha(T)/alpha(L) similar or equal to 0.15 is attained. The numerical simulations display the strong permanent deformation of stream tubes responsible for this phenomenon, coined as "advective mixing.'' The two-point covariance, used in practice in order to characterize the aquifer structure, is not able to detect the structures that produce advective mixing. Nevertheless, the presence of high-conductivity lenses inclined with respect to the mean flow may explain the occurrence of finite alpha(T) in field applications.