EULERIAN-LANGRANGIAN THEORY OF TRANSPORT IN SPACE-TIME NONSTATIONARY VELOCITY-FIELDS - EXACT NONLOCAL FORMALISM BY CONDITIONAL MOMENTS AND WEAK APPROXIMATION

A unified Eulerian-Lagrangian theory is presented for the transport of a conservative solute in a random velocity field that satisfies locally del . 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 partial derivative c(x, t)/partial derivative t + del . 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)]nu, and [J(x, t)]nu conditioned (as implied by the subscript) on local hydraulic measurements through the use of the latter in obtaining a relatively smooth unbiased estimate nu(x, t) of v(x, t). These predictors satisfy partial derivative [c(x, t)]nu/partial derivative t + del.[J(x, t)]nu = [g(x, t)]nu, where [J(x, t)]nu = nu(x, t)[c(x, t)]nu + Q(nu)(x, t) and Q(nu)(x, t) is a dispersive flux. We show that Q, is given exactly by three space-time convolution integrals of conditional Lagrangian kernels alpha(nu) with del.Q(nu) beta, with del[c]nu and gamma(nu) with [c]nu for a broad class of v(x, t) fields, including fractals. This implies that Q(nu)(x, t) is generally nonlocal and non-Fickian, rendering [c(x, t)]nu non-Gaussian. The direct contribution of random variations in f to Q(nu) depends on [c]nu rather than on del[c]nu. We elucidate the nature of the above kernels; discuss conditions under which the convolution of beta(nu) and del[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), at early time; use the latter to conclude that linearizations which predict that (c), 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), 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 approximations for the second conditional moment of the concentration prediction error; demonstrate that it yields the so-called ''two-particle covariance'' as a special case; conclude that the (conditional) variance of c does not become infinite merely as a consequence of disregarding local dispersion; and discuss how to estimate explicitly the cumulative release of a contaminant across a ''compliance surface'' together with the associated estimation error.