Dispersion of a passive solute in non-ergodic transport by steady velocity fields in heterogeneous formations

Abstract

An inert solute is convected by a steady random velocity field, which is associated with flow through a heterogeneous porous formation. The log conductivity and the velocity are stationary random space functions. The log conductivity Y is assumed to be normal, with an isotropic two-point correlation of variance σ_Y² and of finite integral scale I. The solute cloud is of a finite input zone of lengthscale l. The transport is characterized with the aid of the spatial moments of the solute body. The effective dispersion coefficient is defined as half of the rate of change with time of the second spatial moment with respect to the centroid. Under the ergodic hypothesis, which is bound to be satisfied for l/I [Gt ] 1, the centroid moves with the mean velocity U and the longitudinal dispersion coefficient [dscr ]_L tends to its constant, Fickian, limit. Under a Lagrangian first-order analysis in σ_Y² it has been found that [dscr ]_L = σ_Y²UI.This study addresses the computation of the effective longitudinal dispersion coefficient for a finite input zone, for which ergodic conditions may not be satisfied. In this case the centroid trajectory and the second spatial moments are random variables. In line with a previous work (Dagan 1990) the effective dispersion coefficient D_L is defined as half the rate of change of the expected value of the second spatial moment for large transport time. The aim of the study is to derive D_L and its dependence upon l/I and in particular to determine the conditions under which it tends to the ergodic limit [dscr ]_L. The computation is carried out separately for a thin body aligned with the mean flow and one transverse to it. In the first case it is found that DL is equal to zero, i.e. the streamlined body does not disperse in the mean. This result is explained by the correlation between the trajectories of the leading and trailing edges, respectively, once the latter reaches the position of the first. The relatively modest increase of the mean second spatial moment is effectively computed. In the case of a thin body initially transverse to the mean flow, D_L may reach the ergodic limit [dscr ]_L for a ratio l/I of the order 10². For smaller values, D_L is found to be bounded from above, and its maximum depends on l but not on I. The uncertainty caused by the randomness of the velocity field is manifested in the trajectory of the centroid rather than in the effective dispersion.