We develop a reduced-order modeling strategy aimed at providing numerical Monte Carlo simulations of groundwater flow in randomly heterogeneous transmissivity fields. We rely on moment equations for groundwater flow and conduct space reductions for both transmissivity, T, and hydraulic head, h. A truncated singular value decomposition (SVD) solver is employed to cope with the ill-conditioned stiffness matrix caused by (negative and thus) unphysical values of T that might arise due to possible low accuracy stemming from the order of model reduction. The performance of the approach is assessed through the analysis of various synthetic reference scenarios. These encompass diverse degrees of heterogeneity of the transmissivity field and various values of reduced-order dimensions, n and m, associated with h and T, respectively. Transmissivity is conceptualized as a composite (spatial) random field where there is uncertainty in the locations of regions associated with diverse geomaterials as well as in the heterogeneity of transmissivity therein. Our results are also compared against their counterparts that one could obtain upon performing a model reduction solely on the basis of hydraulic heads. Our findings show that: (i) resting on the truncated SVD solver is beneficial for coping with ill-conditioned stiffness matrices; (ii) the two model reduction strategies provide comparable solution accuracy for m ≥ 5n, while (iii) the computational cost associated with the reduced-order model based on space reduction for both T and h is always significantly smaller than that associated with space reduction based solely on h.

中文翻译：

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随机异质复合导水率场中地下水流的降阶蒙特卡洛模拟框架


我们开发了一种降阶建模策略，旨在提供随机异质导水率场中地下水流的数值蒙特卡洛模拟。我们依靠地下水流的弯矩方程，并对透射率 T 和水力水头 h 进行空间缩减。采用截断奇异值分解 （SVD） 求解器来处理由（负值，因此）T 的非物理值引起的病态刚度矩阵，该值可能由于模型缩减阶数可能产生的低精度而出现。通过分析各种合成参考情景来评估该方法的性能。这些包括透射率场的不同程度的异质性以及分别与 h 和 T 相关的降阶维度 n 和 m 的各种值。导水率被概念化为一个复合（空间）随机场，其中与不同土质材料相关的区域位置以及其中导水率的异质性存在不确定性。我们的结果还与仅基于液压水头进行模型缩减时可以获得的对应结果进行了比较。我们的研究结果表明：（i） 位于截断的 SVD 求解器上有利于应对病态刚度矩阵;（ii） 两种模型约简策略为 m ≥ 5n 提供了相当的求解精度，而 （iii） 与基于 T 和 h 空间缩减的降阶模型相关的计算成本总是明显小于仅基于 h 的空间缩减相关的计算成本。