A New Paradigm for the Analysis of Transient Saturated/Unsaturated Flow and Transport in Randomly Heterogeneous Soils

小行星9628133 本项目的目标是：（a）建立一个精确的概念/数学框架，用于最佳地预测和分析随机非均匀地下环境中的瞬态饱和和非饱和水流，这些水流受任意初始条件、源项和边界条件的影响;（B）相关概念/在饱和和部分饱和介质中瞬态流动下非反应性溶质运移的预测和分析的数学框架;（c）这种流动和溶质输运的相关低阶和高阶计算方案。 理论框架和计算方法将依赖于随机推导的确定性流和输运方程，其中包含本地和非本地信息依赖的参数。 与通常难以在理论上证明或与测量值进行比较的放大量相比，进入我们方程的所有量将在与潜在可用的现场数据（其支持尺度）兼容的尺度上定义。 理论框架将包含明确的表达式来评估预测的不确定性。 当土的性质在空间上随机变化时，相应的流动和输运方程是随机的。 要解析求解这些方程，方程必须是线性的，并且土壤性质受到严格的限制。 通过（条件）Monte Carlo模拟数值求解它们是计算密集型的，并且通常不能保证收敛。 理想情况下，蒙特卡罗方法收敛到一个（条件）平均解，它构成了不确定性下系统行为的最佳无偏预测器，以及一个（条件）方差，它测量预测不确定性。 我们的目标是规避蒙特卡罗模拟的需要，通过评估这些（条件）的时刻确定性。 为此，我们建议首先推导出精确的流动和输运方程，这些时刻的时空演化，然后求解它们的数值近似。 我们期望这样的流动和输运方程是积分-微分方程，因此是非局部的、非达西的和非菲克的。 我们把这种新的解决方案范例称为“平滑”。 与其中网格必须基于特设标准先验地定义的尺度放大相反，在这里，网格基于期望的矩函数的平滑程度来后验地定义。 它们的平滑度部分取决于时空中条件（测量）点的质量和分布。 在大多数情况下，这样的点是稀疏的，足以使矩函数比它们的随机对应物更平滑。 因此，解决前者所需的网格通常比解决后者所需的网格粗糙得多。 与蒙特卡罗方法相比，最终结果应该可以大大节省计算机时间和存储空间。 Neuman和Orr（1993）已经为稳态饱和流建立了精确的条件矩方程，Neuman（1993）和Zhang和Neuman（1995 e）为非反应性溶质运移建立了精确的条件矩方程。 这些非局部方程涉及局部和非局部的流动和运输参数的数据条件，因此非唯一的。 虽然方程是精确的，但它们的非局部参数不能在没有高分辨率蒙特卡罗模拟或近似的情况下直接计算;第三种选择是通过逆方法间接估计它们。 我们建议将上述非局部流动和运输理论扩展到瞬态饱和和不饱和流动条件;为相应的参数开发合适的近似;并开发相应的非局部平均流动和运输方程的解决方案的计算方案，以及相应的预测误差的评估。

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.