This article presents numerical investigations on accuracy and convergence properties of several numerical approaches for simulating steady state flows in heterogeneous aquifers. Finite difference, finite element, discontinuous Galerkin, spectral, and random walk methods are tested on one- and two-dimensional benchmark flow problems. Realizations of log-normal hydraulic conductivity fields are generated by Kraichnan algorithms in closed form as finite sums of random periodic modes, which allow direct code verification by comparisons with manufactured reference solutions. The quality of the methods is assessed for increasing number of random modes and for increasing variance of the log-hydraulic conductivity fields with Gaussian and exponential correlation. Experimental orders of convergence are calculated from successive refinements of the grid. The numerical methods are further validated by comparisons between statistical inferences obtained from Monte Carlo ensembles of numerical solutions and theoretical first-order perturbation results. It is found that while for Gaussian correlation of the log-conductivity field all the methods perform well, in the exponential case their accuracy deteriorates and, for large variance and number of modes, the benchmark problems are practically not tractable with reasonably large computing resources, for all the methods considered in this study.

中文翻译：

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非均质地下水地层中流动数值解的基准

本文介绍了对模拟非均质含水层稳态流动的几种数值方法的准确性和收敛特性的数值研究。在一维和二维基准流问题上测试了有限差分、有限元、不连续伽辽金、谱和随机游走方法。对数正态水力传导率场的实现由 Kraichnan 算法以封闭形式生成为随机周期模式的有限和，允许通过与制造的参考解决方案进行比较来直接验证代码。评估方法的质量是为了增加随机模式的数量和增加具有高斯和指数相关性的对数水力电导率场的方差。从网格的连续细化计算收敛的实验阶数。通过比较从数值解的蒙特卡罗集合获得的统计推论与理论一阶微扰结果之间的比较，进一步验证了数值方法。发现虽然对于对数-电导场的高斯相关，所有方法都表现良好，但在指数情况下，它们的准确性会下降，并且对于大方差和模式数，基准问题实际上无法通过合理的大量计算资源来解决，对于本研究中考虑的所有方法。