Stochastic modeling of mass transport in a random velocity field

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Abstract
The purpose of this paper is to develop and solve a stochastic transport equation which is based directly on a stochastic representation of the velocity field. This equation, which describes the spatial and temporal variation in ensemble mean concentration, is similar in form to the advection‐dispersion equation. Of particular note are the coefficients of the second‐order terms, which are termed ensemble dispersion coefficients. For a simple system where neighboring velocities are uncorrelated, these coefficients vary as a function of the mean velocity, the variance in velocity, and travel distance of the tracer. For cases where the spatial correlation in velocity can be approximated by an exponentially decaying function, ensemble dispersion coefficients ultimately reach a constant value. If ensemble averages and spatial averages for a realization are equivalent, the ensemble dispersion coefficients can be related to the process of dispersion in a single realization. A hybrid deterministic‐probabilistic technique used to solve the stochastic equation has been verified with a one‐dimensional analytic solution. The results of trials with the numerical model show how variability in ensemble mean concentration is controlled by the coefficients in the stochastic transport equation for two different types of loading functions. The generality of the model makes it suitable for application to media with very different spatial structure.