Simulating groundwater flows in heterogeneous aquifers is one of the most widely studied problems. The heterogeneity is modeled through random hydraulic conductivity fields log-normally distributed. In this paper, we aim to generate the realization of the log-normal hydraulic conductivity by summing up a finite number of random periodic modes with the Kraichnan algorithm. To address Neumann conditions, we propose a change of variables that results in Dirichlet boundary conditions along all boundaries of the computational domain. Then, we suggest using the radial basis function partition of unity method (RBFPUM) and RBFPUM with QR factorization (RBFPUM-QR) to analyze groundwater flow in the heterogeneous porous medium. We evaluate the performance of the proposed methods by increasing the number of random modes and the variance of the log-hydraulic conductivity fields using Gaussian and exponential correlation. In addition, we verify the effectiveness of the presented approaches in one and two dimensions using the method of manufactured solutions. Furthermore, we provide statistical inferences through Monte Carlo simulations. Numerical results demonstrate that RBFPUM-QR performs well in handling higher heterogeneity in both single realizations and Monte Carlo simulations.

中文翻译：

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使用统一方法的径向基函数分区模拟具有随机导水率的地下水流

模拟非均质含水层中的地下水流是研究最广泛的问题之一。通过对数正态分布的随机水力传导率场对非均质性进行建模。在本文中，我们的目标是通过用 Kraichnan 算法对有限数量的随机周期模式求和来生成对数正态水力传导率的实现。为了解决诺依曼条件，我们提出了变量的变化，从而导致沿着计算域的所有边界的狄利克雷边界条件。然后，我们建议使用统一径向基函数划分方法（RBFPUM）和带有QR分解的RBFPUM（RBFPUM-QR）来分析非均质多孔介质中的地下水流。我们通过使用高斯和指数相关性增加随机模式的数量和对数水力电导率场的方差来评估所提出方法的性能。此外，我们使用制造解决方案的方法验证了所提出的方法在一维和二维方面的有效性。此外，我们通过蒙特卡罗模拟提供统计推论。数值结果表明，RBFPUM-QR 在单一实现和蒙特卡罗模拟中处理更高的异质性方面表现良好。