On the Longitudinal Dispersion in Conservative Transport Through Heterogeneous Porous Formations at Finite Peclet Numbers

We consider transport of a conservative solute through an aquifer as determined: (i) by the advective velocity, which depends upon the hydraulic conductivity K and (ii) by the local spreading due to the pore-scale dispersion (PSD). The flow is steady, and it takes place in a porous formation where, owing to its erratic spatial variations, the hydraulic log conductivity Y=In K is modeled as a stationary Gaussian random field. The relative effect of the above mechanisms (i)–(ii) is quantified by the Peclet number (Pe) which, in most of the previous studies, was considered infinite (i.e., no PSD) due to the overtake of advective heterogeneities upon the PSD. Here we aim at generalizing such studies by accounting for the impact of finite Pe on conservative transport. Previous studies on the topic required extensive numerical computations. In the present note, we remove the computational burden by adopting the rational approximate expression of Dagan and Cvetkovic (1993) for the covariance of the velocity field. This allows one to obtain closed form expressions for the quantities characterizing the longitudinal plume's dispersion. Transport can be straightforwardly investigated by dealing with a modified Peclet number (Pe) incorporating both the PSD and the aquifer's anisotropy. The satisfactory match to Cape Cod field data suggests that the present theoretical results lend themselves as a useful tool to assess the impact of the PSD upon conservative transport through heterogeneous porous formations.