Eulerian‐Lagrangian Theory of transport in space‐time nonstationary velocity fields: Exact nonlocal formalism by conditional moments and weak approximation

Abstract

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)〉ν and 〈J(x, t)〉ν conditioned (as implied by the subscript) on local hydraulic measurements through the use of the latter in obtaining a relatively smooth unbiased estimate ν(x, t) of v(x, t). These predictors satisfy ∂〈c(x, t)〉v/∂t + ∇ · 〈J(x, t)〉ν = 〈g(x, t)〉ν, where 〈J(x, t)〉ν = ν(x, t)〈c(x, t)〉ν + Qν(x, t) and Qν(x, t) is a dispersive flux. We show that Qν, is given exactly by three space‐time convolution integrals of conditional Lagrangian kernels αν with ∇·Qν, βν with ∇〈c〉ν, and γν with 〈c〉ν for a broad class of v(x, t) fields, including fractals. This implies that Qν(x, t) is generally nonlocal and non‐Fickian, rendering 〈c(x, t)〉ν non‐Gaussian. The direct contribution of random variations in f to Qν depends on 〈c〉ν rather than on ∇〈c〉ν,. We elucidate the nature of the above kernels; discuss conditions under which the convolution of βν 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〉ν at early time; use the latter to conclude that linearizations which predict that 〈c〉ν 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〉ν 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.