Abstract We illustrate and test an approach grounded on embedding moment equations (MEs) of groundwater flow within a Monte Carlo based modeling strategy to yield a Reduced-Order Model (ROM) that enables the efficient and accurate evaluation of probability distributions of hydraulic heads in randomly heterogeneous transmissivity fields. The projection space determining the accuracy of the ROM solution is typically computed through the principal component analysis of a selected number of full system model solutions (the so-called snapshots). Computationally expensive sensitivity analyses are then required to assess the independence of the ROM from the snapshots. Here, we propose to compute the projection vectors upon relying on the hydraulic head covariance evaluated from the solution of corresponding MEs of groundwater flow. Our workflow to compute hydraulic head distributions is organized according to the following steps: (i) approximation of mean hydraulic head and head covariance matrix through (second-order accurate) solutions of MEs; (ii) computation of the leading eigenvectors of the head covariance matrix to form the basis set for the ROM projection space; and (iii) construction of the ROM. Sample probability density functions of hydraulic heads are then efficiently obtained via Monte Carlo simulations relying on the developed ROM. The proposed methodology is compared against snapshot-based ROMs and the full system model in a two- and a three-dimensional steady-state groundwater flow setting where pumping from a point source is superimposed to a mean uniform flow. Our results show that the projection space computed by relying on MEs provides a more accurate ROM solution than the one resulting from reliance on snapshots.

中文翻译：

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用于地下水流蒙特卡罗模拟的降阶建模策略中矩方程的整合

摘要 我们说明并测试了一种基于在基于蒙特卡洛的建模策略中嵌入地下水流的矩方程 (ME) 的方法，以产生降阶模型 (ROM)，该模型能够高效准确地评估随机水头的概率分布。异质透射场。确定 ROM 解决方案精度的投影空间通常通过对选定数量的完整系统模型解决方案（所谓的快照）进行主成分分析来计算。然后需要进行计算成本高昂的敏感性分析来评估 ROM 与快照的独立性。在这里，我们建议依靠从地下水流的相应 ME 的解中评估的水头协方差来计算投影向量。我们计算水头分布的工作流程是根据以下步骤组织的：（i）通过 ME 的（二阶精确）解决方案近似平均水头和水头协方差矩阵；(ii) 计算头部协方差矩阵的主要特征向量以形成 ROM 投影空间的基组；(iii) ROM 的构建。然后依靠开发的 ROM 通过蒙特卡罗模拟有效地获得液压头的样本概率密度函数。在二维和三维稳态地下水流设置中，将所提出的方法与基于快照的 ROM 和完整系统模型进行比较，其中从点源抽水叠加到平均均匀流。