Practicalities of extrapolating one‐dimensional fault and fracture size‐frequency distributions to higher‐dimensional samples

Previously published theory, which extrapolates fault and fracture population statistics observed in a one‐dimensional sample to two‐ and three‐dimensional populations, is found to be of limited value in practical applications. We demonstrate how significant the discrepancies may be and how they arise. There are two main sources for the discrepancies: (1) deviations from ideal spatial uniformity (spatial Poisson process) of a fault or fracture pattern and (2) non‐power law scaling of the size frequency distributions of the population. We show that even small fluctuations in spatial density, combined with variance in the estimator of population statistics, can lead to considerable deviations from the theoretical predictions. Ambiguity about power law scaling or otherwise of the underlying population is a typical characteristic of natural data sets, and we demonstrate how this can affect the extrapolation of one‐dimensional data to higher dimensions. In addition, we present new theoretical approaches to the problem of extrapolation when clustering of faults and fractures is explicitly considered. Clustering is commonly observed in the field as en echelon arrays of fault or fracture segments and we show how this property of natural patterns can be quantified and included in the theory. These results are relevant to building more realistic three‐dimensional models of the physical properties of fractured rocks, such as fracture permeability and seismic anisotropy.