Hg. Borgos et al., Practicalities of extrapolating one-dimensional fault and fracture size-frequency distributions to higher-dimensional samples, J GEO R-SOL, 105(B12), 2000, pp. 28377-28391
Previously published theory, which extrapolates fault and fracture populati
on statistics observed in a one-dimensional sample to two- and three-dimens
ional populations, is found to be of limited value in practical application
s. We demonstrate how significant the discrepancies may be and how they ari
se. There are two main sources for the discrepancies: (1) deviations from i
deal spatial uniformity (spatial Poisson process) of a fault or fracture pa
ttern and (2) non-power law scaling of the size frequency distributions of
the population. We show that even small fluctuations in spatial density, co
mbined 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 ch
aracteristic of natural data sets, and we demonstrate how this can affect t
he 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 s
egments and we show how this property of natural patterns can be quantified
and included in the theory. These results are relevant to building more re
alistic three-dimensional models of the physical properties of fractured ro
cks, such as fracture permeability and seismic anisotropy.