Transport of a passive scalar in a stratified porous medium RID A-2321-2010

A uniform and horizontal head gradient J is applied to a stratified formation whose given random conductivity K is function of the vertical coordinate x(3) only. K is assumed to be stationary and of finite integral scale I. By Darcy's law, the velocity field V-1(x(3)) = JK 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 X-11(t, Pe) where Pe=< V-1>I-v/D-dT and D-dT is the transverse pore-scale dispersion coefficient. X-11 was determined in the past by Matheron and de Marsily (1980). The present study is concerned with determining the local concentration variance sigma(C)(2), that depends on the two-particles trajectories covariance Z(11)(t). The latter is derived exactly and (C) and sigma(C)(2) 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.