Abstract In large scale heterogeneous aquifer simulations, determining the appropriate coarsening scale λ to define an effective hydraulic conductivity Keff is a challenging task, that involves a trade-off between accuracy and cost. Efficiently adjusting the scale λ is then key, in particular for uncertainty quantification. In this paper, we obtain improved analytical results for the variance of Keff, valid at any scale, in the context of energy dissipation formulation. Using this formulation, we then derive an efficient Keff numerical estimator, and compare it with those of the potential-flow average and permeameter formulations in 2D, for lognormal and binary media, over a wide range of λ and of heterogeneity. We analyze the probability density function (pdf), mean, and variance, of these estimators, comparing them with the analytical results. In the lognormal case, the pdf’s are rather similar for the three estimators, and remain lognormal at all scales. In the binary case, slow convergence to an asymptotic regime is observed close to the percolation threshold.

中文翻译：

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关于随机非均质介质中有效水力传导率的多尺度表征：历史调查和一些新观点。

摘要 在大规模非均质含水层模拟中，确定适当的粗化尺度 λ 以定义有效水力传导率 Keff 是一项具有挑战性的任务，涉及精度和成本之间的权衡。有效地调整尺度 λ 是关键，特别是对于不确定性量化。在本文中，我们在能量耗散公式的背景下获得了 Keff 方差的改进分析结果，在任何尺度下都有效。使用这个公式，我们然后推导出一个有效的 Keff 数值估计器，并将其与二维中的势流平均值和渗透计公式的那些进行比较，对于对数正态和二元介质，在广泛的 λ 和非均质性范围内。我们分析这些估计量的概率密度函数 (pdf)、均值和方差，并将它们与分析结果进行比较。在对数正态情况下，三个估计量的 pdf 相当相似，并且在所有尺度上都保持对数正态。在二元情况下，在接近渗透阈值时观察到缓慢收敛到渐近状态。