9628133 Neuman The objectives of this project are to develop (a) an exact conceptual/mathematical framework for the optimum prediction and analysis of transient saturated and unsaturated water flows in randomly heterogeneous subsurface environments, subject to arbitrary initial conditions, source terms, and boundary conditions; (b) a related conceptual/mathematical framework for the prediction and analysis of nonreactive solute transport under transient flow in saturated and partially saturated media; (c) associated low- and higher-order computational schemes for such flows and solute transport. The theoretical framework and computational methodology will rely on stochastically derived deterministic flow and transport equations which contain both local and nonlocal information-dependent parameters. In contrast to upscaled quantities which are often difficult to justify theoretically, or compare with measurements, all quantities entering into our equations will be defined on a scale compatible with potentially available field data (their support scale). The theoretical framework will contain explicit expressions to assess prediction uncertainty. When the properties of a soil vary randomly in space, the corresponding flow and transport equations are stochastic. To solve them analytically, the equations must be linear(ized) and the soil properties severely restricted. To solve them numerically by (conditional) Monte Carlo simulation is computationally intensive and does not usually guarantee convergence. Ideally, the Monte Carlo method converges to a (conditional) mean solution which constitutes an optimum unbiased predictor of system behavior under uncertainly, and a (contional) variance which measures prediction uncertainly. Our aim is to circumvent the need for Monte Carlo simulations by evaluating these (conditional ) moments deterministically. For this, we propose to first derive exact flow and transport equations that govern the space-time evolution of these moments, then solv e them numerically by approximation. We expect such flow and transport equations to be integro-differential, hence nonlocal, non-Darcian, and non-Fickian. We refer to this new solution paradigm as "smoothing". Contrary to upscaling in which a grid must be defined a priori on the basis of ad hoc criteria, here the grid is defined a posteriori based on how smooth the moment functions are expected to be. Their smoothness is controlled in part by the quanlity and distribution of conditioning (measurement) points in space-time. In most cases such points are sparse enough to render the moment functions much smoother than are their random counterparts. Hence a grid required to resolve the former is usually much coarser than that required to resolve the latter. The net result should be a considerable saving in computer time and storage when compared to the Monte Carlo method. Exact conditional moment equations have been developed for steady state saturated flow by Neuman and Orr (1993), and for nonreactive solute transport by Neuman (1993) and Zhang and Neuman (1995e). These nonlocal equations involve local and nonlocal flow and transport parameters that are conditional on data and thus nonunique. Though the equations are exact, their nonlocal parameters cannot be evaluated directly without either high-resolution Monte Carlo simulation or approximation; a third option is to estimate them indirectly by inverse methods. We propose to extend the above nonlocal flow and transport theories to transient saturated and unsaturated flow conditions; to develop suitable approximations for the corresponding parameters; and to develop computational schemes for the solution of the corresponding nonlocal mean flow and transport equations, as well as for the assessment of the corresponding prediction errors.
