A unified Eulerian-Lagrangian theory is presented for the transport of a conservative solute in a random velocity field that satisfies locally ∇⋅v(x,t)=f(x,t), where f(x,t) is a random function including sources and/or the time derivative of head. Solute concentration satisfies locally the Eulerian equation ∂c(x,t)/∂t+∇⋅J(x,t)=g(x,t), where J(x,t) is advective solute flux and g(x,t) is a random source independent of f(x,t). We consider the prediction of c(x,t) and J(x,t) by means of their unbiased ensemble moments ⟨c(x,t)⟩&ngr; and ⟨J(x,t)⟩&ngr; conditioned (as implied by the subscript) on local hydraulic measurements through the use of the latter in obtaining a relatively smooth, unbiased estimate &ngr;(x,t) of v(x,t). These predictors satisfy ∂⟨c(x,t)(⟩&ngr;/∂t+∇⋅⟨J(x,t)&ngr;= ⟨g(x,t)⟩&ngr;, where ⟨J(x,t)⟩&ngr;=&ngr;(x,t)⟨c(x, t)⟩&ngr;+Q&ngr;(x,t) and Q&ngr;(x,t) is a dispersive flux. We show that Q&ngr; is given exactly by three space-time convolution integrals of conditional Lagrangian kernels &agr;&ngr; with ∇⋅Q&ngr;, &bgr;&ngr; with ∇⟨c⟩&ngr;, and &ggr;&ngr; with ⟨c⟩&ngr; for a broad class of v(x,t) fields, including fractals. This implies that Q&ngr;(x,t) is generally nonlocal and non-Fickian, rendering ⟨c(x,t)⟩&ngr; non-Gaussian.

The direct contribution of random variations in f to Q&ngr; depends on ⟨c⟩&ngr; rather than on ∇⟨c⟩&ngr;. We elucidate the nature of the above kernels; discuss conditions under which the convolution of &bgr;&ngr; and ∇⟨c⟩ becomes pseudo-Fickian, with a Lagrangian dispersion tensor similar to that derived in 1921 by Taylor; recall a 1952 result by Batchelor which yields an exact expression for ⟨c⟩&ngr; at early time; use the latter to conclude that linearizations which predict that ⟨c⟩&ngr; bifurcates at early time when the probability density function of v is unimodal cannot be correct; propose instead a weak approximation which leads to a nonlinear integro-differential equation for ⟨c⟩&ngr; due to an instantaneous point source and which improves with the quantity and quality of hydraulic data; demonstrate that the weak approximation is analogous to the ''direct interaction'' closure of turbulence theory; offer non-Fickian and pseudo-Fickian weak ap