Fractal and Multifractal Characterization of Stochastic Fracture
Networks and Real Outcrops
Abstract
The fractal dimension and multifractal spectrum are widely used to
characterize the complexity of natural fractures. However, a systematic
investigation on the impact of different fracture properties (fracture
lengths, orientations, center positions, system sizes) on the fractal
and multifractal characterization of complex fracture networks is
missing. We utilize an in-house developed DFN modeling software,
HatchFrac, to construct stochastic fracture networks with prescribed
distributions and systematically study the impact of four geometrical
properties of fractures on the fractal and multifractal
characterization. We calculate the single fractal dimension and
multifractal spectrum with the box-counting method. The single fractal
dimension, D, and the difference of singularity exponent, ∆α, are used
to represent the fractal and multifractal patterns, respectively. We
find that fracture lengths, orientations and system sizes have positive
correlations with D and ∆α, while the system size has the most
significant impact among the four parameters. D is uncorrelated with
fracture positions (FD), which means that a single fractal dimension
cannot capture the complexity caused by clustering effects. However, ∆α
has a strong negative correlation with FD, which implies that clustering
effects make fracture networks more complex, and ∆α can capture the
difference. We also digitize 60 outcrop maps with a novel fracture
detection algorithm and calculate their fractal dimension and
multifractal spectrum. We find wide variations of D and ∆α on those
outcrop maps, even for outcrops at similar scales. It means that a
universal indicator for characterizing fracture networks at different
scales or the same scale is almost impossible.