Complexity Analysis of Three-dimensional Stochastic Discrete Fracture
Networks with Fractal and Multifractal Techniques
Abstract
The fractal dimension and multifractal spectrum can characterize the
complexity of fracture sets. However, studies of impacts of fracture
geometries on their fractal and multifractal characteristics are largely
insufficient, especially for three-dimensional (3-D) fracture networks
(natural fractures are always 3-D instead of 2-D). In this work, we
construct 3-D stochastic discrete fracture networks with an open-source
DFN software, HatchFrac. Systematical investigations are then conducted
to study the impact of geometrical fracture properties and system sizes
on the fractal and multifractal characteristics. The box-counting method
is adopted to calculate the fractal dimension and multi-fractal
descriptors. The fractal dimension, D, and the difference of the
singularity exponent, ∆α, represent the fractal and multifractal
patterns, respectively. Two critical (percolative and over-percolative)
stages of fracture networks are considered. 3-D fracture networks share
similar characteristics with 2-D fracture networks at percolation.
However, results at an over-percolative stage are systematically
different. At the first stage, fracture orientations (κ), lengths (a)
and system sizes (L) have positive correlations with D and ∆α. D is
weakly correlated with fracture positions (FD), meaning that the fractal
dimension is insensitive to clustering effects. However, ∆α is strongly
correlated with FD, implying that ∆α can characterize the heterogeneity
caused by clustering effects. a and L are positively correlated with ∆α,
and κ and FD have negative correlations. At stage two, the sensitivity
results on D are similar to stage one, but a and L become negatively
correlated with ∆α. Impacts of κ and FD become more significant.