Dispersion tensor evaluation in heterogeneous media for finite Peclet values

In natural formations the transport process at the local scale is characterized by the spatial heterogeneity of hydraulic conductivity and by the pore-scale dispersion. Usually, theoretical investigations consider only the effect due to the spatial variation of hydraulic conductivity, because the first prevails over the latter by some order of magnitude. Nevertheless, the pore-scale dispersion has a noteworthy impact on the concentration variance evaluation and on the ergodicity condition, so that its influence on the overall dispersion processes may be relevant. The note illustrates a theoretical procedure that allows one to define the global dispersion tensor at the local scale by taking into account the pore-scale effects as evaluated in laboratory columns. To reach this goal, the pore-scale and the local-scale velocity fluctuations are coupled via a rigorous analytical procedure in the three-dimensional (3-D) domain, and it is demonstrated that the porous media heterogeneity modifies the pore-scale dispersion effects as measured in laboratory columns. The impact of the hydraulic conductivity heterogeneity on the pore-scale dispersion at the local scale is due to (1) the path line sinuosity and (2) the module of the local velocity variation. A quantitative evaluation of these effects is made according to a first-order analysis, i.e., assuming that the spatial fluctuations of the hydraulic conductivity are small, and, in the 2-D case, a comparison with nonlinear results is made also. The results demonstrate that the only path line sinuosity coupled with the pore-scale anisotropy enhances the transversal mixing and reduces the longitudinal one, but the global heterogeneity effect, considering also the velocity module fluctuations, gives a generalized increase of the local dispersion tensor components.