Eulerian‐Lagrangian 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 ∇ · 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 + Q ν ( x , t ) and Q ν ( x , t ) is a dispersive flux. We show that Q ν , is given exactly by three space‐time convolution integrals of conditional Lagrangian kernels α ν with ∇· Q ν , β ν with ∇〈 c 〉 ν , and γ ν with 〈 c 〉 ν for a broad class of v ( x , t ) fields, including fractals. This implies that Q ν ( x , t ) is generally nonlocal and non‐Fickian, rendering 〈 c ( x , t )〉 ν non‐Gaussian. The direct contribution of random variations in f to Q ν depends on 〈 c 〉 ν rather than on ∇〈 c 〉 ν ,. We elucidate the nature of the above kernels; discuss conditions under which the convolution of β ν 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.