Recursive Conditional Moment Equations for Advective Transport in Randomly Heterogeneous Velocity Fields

Flow and transport parameters such as hydraulic conductivity, seepage velocity, and dispersivity have been traditionally viewed as well-defined local quantities that can be assigned unique values at each point in space-time. Yet in practice these parameters can be deduced from measurements only at selected locations where their values depend on the scale (support volume) and mode (instruments and procedure) of measurement. Quite often, the support of the measurements is uncertain and the data are corrupted by experimental and interpretive errors. Estimating the parameters at points where measurements arc not available entails an additional random error. These errors and uncertainties render the parameters random and the corresponding How and transport equations stochastic. The stochastic flow and transport equations can be solved numerically by conditional Monte Carlo simulation. However, this procedure is computationally demanding and lacks well-established convergence criteria. An alternative to such simulation is provided by conditional moment equations, which yield corresponding predictions of flow and transport deterministically. These equations are typically integro-differential and include nonlocal parameters that depend on more than one point in space-time. The traditional concept of a REV (representative elementary volume) is neither necessary nor relevant for their validity or application. The parameters are nonunique in that they depend not only on local medium properties but also on the information one has about these properties (scale, location, quantity, and quality of data). Darcy's law and Fick's analogy are generally not obeyed by the flow and transport predictors except in special cases or as localized approximations. Such approximations yield familiar-looking differential equations which, however, acquire a non-traditional meaning in that their parameters (hydraulic conductivity, seepage velocity, dispersivity) and state variables (hydraulic head, concentration) are information-dependent and therefore, inherently nonunique. Nonlocal equations contain information about predictive uncertainty, localized equations do not. We have shown previously (Guadagnini and Neuman, 1997, 1998, 1999a, b) how to solve conditional moment equations of steady-state flow numerically on the basis of recursive approximations similar to those developed for transient flow by Tartakovsky and Neuman (1998, 1999). Our solution yields conditional moments of velocity, which are required for the numerical computation of conditional moments associated with transport. In this paper, we lay the theoretical groundwork for such computations by developing exact integro-differential expressions for second conditional moments, and recursive approximations for all conditional moments, of advective transport in a manner that complements earlier work along these lines by Neuman (1993).