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 α_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 σ²_γ. However, field findings indicate the presence of small but finite α_T . The aim of the paper is to determine α_T for highly heterogeneous formations using a model that contains inclusions of conductivity K, submerged in a matrix of conductivity K₀, for large κ = K/K₀. In the dilute medium approximation, valid for small volume fraction n, but arbitrary κ, and for spherical inclusions, it is found that α_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 κ and n is devised as a counterexample. It is found that because of the intertwining of streamlines α_T > 0, being of order (κ - 1)⁴ for κ » 1. These findings are confirmed by accurate numerical simulations of flow through a large number of interacting inclusions; for κ = 10 and n = 0.2 (jamming limit), the large value α_T/α _T≃ 0-15 is attained. The numerical simulations display the strong permanent deformation of stream tubes responsible for this phenomenon, corned 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 α_T in field applications.