### 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} + 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.

Original language | English (US) |
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Pages (from-to) | 633-645 |

Number of pages | 13 |

Journal | Water Resources Research |

Volume | 29 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1993 |

### ASJC Scopus subject areas

- Water Science and Technology