Fractures result from complex mechanical processes producing irregular, hierarchical, and correlated networks. The statistical analysis of such networks is an important step toward characterizing and modeling fractures. However, established exploratory statistics for the investigation and quantification of fracture networks use only first-order or mean-value characteristics such as the density or the length and orientation distributions of fractures, leaving much to be desired. Here, we present a second-order statistical theory to characterize the inner variability of fracture networks in 2D. We use ideas from marked point process theory treating the barycenters of fractures or fracture branches as “points” and fracture lengths and orientation as “marks.” The statistics are based on oriented distances between object centers, which are represented by pair-correlation and mark-correlation functions describing fracture network variability. The forms of the corresponding plots give information on the degree of randomness, the most frequent center-to-center distances, and possible local order, all with respect to fracture orientations. We demonstrate the application of these ideas by analyzing three fracture networks. First, we study a synthetic structure as a benchmark test of the methods under ideal conditions. Then, we focus on two previously characterized field exposures: a well-developed and highly connected network and a very irregular network with many small and isolated fractures. The correlation functions successfully characterize the different spatial arrangements of fractures in all three cases.

中文翻译：

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裂缝网络的方向对相关分析

裂缝是由复杂的机械过程产生的，产生不规则的、分层的和相关的网络。这种网络的统计分析是对裂缝进行表征和建模的重要一步。然而，用于研究和量化裂缝网络的已建立的探索性统计仅使用一阶或平均值特征，例如裂缝的密度或长度和方向分布，还有很多不足之处。在这里，我们提出了一个二阶统计理论来描述二维裂缝网络的内部变异性。我们使用标记点过程理论的思想，将裂缝或裂缝分支的重心视为“点”，将裂缝长度和方向视为“标记”。统计数据基于对象中心之间的定向距离，它们由描述裂缝网络变异性的对相关和标记相关函数表示。相应图表的形式给出了关于随机性程度、最常见的中心到中心距离和可能的局部顺序的信息，所有这些信息都与裂缝方向有关。我们通过分析三个裂缝网络来展示这些想法的应用。首先，我们研究合成结构作为理想条件下方法的基准测试。然后，我们专注于两个先前表征的现场暴露：一个发达且高度连接的网络和一个非常不规则的网络，具有许多小的和孤立的裂缝。相关函数成功地表征了所有三种情况下裂缝的不同空间排列。