Geological media are inherently fractal over multiple length scales, exhibiting a wide range of fractal dimensions . We study the size dependent conductivity of models of geological media based on so called regular random fractals. Specifically, we consider generalizations of the classic Menger sponges where the impermeable cells can take random locations. This results, for a given random fractal generator and a given construction order, in multiple realizations of the target media, each one maintaining the same . We use direct numerical simulation to estimate the conductivity, over a number of patterns, with orders . The conductivity bounds and estimates for regular fractal media are, in general, not applicable to these random fractal models. The conductivity of a specific realization may differ from the ensemble average by a few percents, which suggests a lack of self-averaging; however, the self-averaging is regained if the size of the domain of interest is beyond the maximal fractality length scale. We propose a low computational cost approach for estimating the ensemble average conductivity of a given random fractal at a given order, which matches the order of magnitude predicted from more intensive calculations. Through tabulation of conductivities of generators of random fractals with small scaling factors and fitting models, we explain how these fractals can model geological media in which fractal dimensions and conductivities (or effective diffusion coefficients) are measured.

中文翻译：

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地质介质分形模型的电导率估计

地质介质在多个长度尺度上本质上是分形的，表现出广泛的分形维数。我们研究了基于所谓的规则随机分形的地质介质模型的大小相关电导率。具体来说，我们考虑了经典 Menger 海绵的推广，其中不可渗透的细胞可以随机放置。对于给定的随机分形生成器和给定的构造顺序，这会导致目标媒体的多个实现，每个实现都保持相同。我们使用直接数值模拟来估计电导率，在许多模式上，订单. 常规分形介质的电导率界限和估计值一般不适用于这些随机分形模型。特定实现的电导率可能与整体平均值相差几个百分点，这表明缺乏自平均；然而，如果感兴趣域的大小超出最大分形长度尺度，则重新获得自平均。我们提出了一种低计算成本的方法，用于估计给定顺序的给定随机分形的集合平均电导率，这与从更密集的计算预测的数量级相匹配。通过小比例因子和拟合模型的随机分形发生器的电导率列表，