Transport of a passive scalar in a stratified porous medium

A uniform and horizontal head gradient J is applied to a stratified formation whose given random conductivity K is function of the vertical coordinate x₃ only. K is assumed to be stationary and of finite integral scale I_v. By Darcy's law, the velocity field V₁ (x₃) = J K depicts a fluctuating shear flow. A solute body is injected instantaneously in the formation. In a Lagrangean framework, the second spatial moment of the mean concentration <C(x,t)> can be related to the one-particle trajectories variance X₁₁ (t, Pe) where Pe = <V₁> I_v/ D_dT and D_dT is the transverse pore-scale dispersion coefficient. X₁₁ was determined in the past by Matheron and de Marsily (1980). The present study is concerned with determining the local concentration variance σ_C², that depends on the two-particles trajectories covariance Z₁₁ (t). The latter is derived exactly and <C> and σ_C² are determined by assuming normal or lognormal probability distribution of trajectories. The results are illustrated for small and very large (ergodic) solute plumes. For large travel time the concentration coefficient of variation at the center of the plume tends asymptotically to a constant value, unlike formations with finite horizontal correlation length of the hydraulic conductivity. The results may serve for benchmarking of numerical codes and in applications for short travel distances in highly anisotropic formations.