Abstract
Scaling in fracture systems has become an active field of research
in the last 25 years motivated by practical applications in hazardous
waste disposal, hydrocarbon reservoir management, and earthquake
hazard assessment. Relevant publications are therefore spread widely
through the literature. Although it is recognized that some fracture
systems are best described by scale-limited laws (lognormal, exponential),
it is now recognized that power laws and fractal geometry provide
widely applicable descriptive tools for fracture system characterization.
A key argument for power law and fractal scaling is the absence of
characteristic length scales in the fracture growth process. All
power law and fractal characteristics in nature must have upper and
lower bounds. This topic has been largely neglected, but recent studies
emphasize the importance of layering on all scales in limiting the
scaling characteristics of natural fracture systems. The determination
of power law exponents and fractal dimensions from observations,
although outwardly simple, is problematic, and uncritical use of
analysis techniques has resulted in inaccurate and even meaningless
exponents. We review these techniques and suggest guidelines for
the accurate and objective estimation of exponents and fractal dimensions.
Syntheses of length, displacement, aperture power law exponents,
and fractal dimensions are found, after critical appraisal of published
studies, to show a wide variation, frequently spanning the theoretically
possible range. Extrapolations from one dimension to two and from
two dimensions to three are found to be nontrivial, and simple laws
must be used with caution. Directions for future research include
improved techniques for gathering data sets over great scale ranges
and more rigorous application of existing analysis methods. More
data are needed on joints and veins to illuminate the differences
between different fracture modes. The physical causes of power law
scaling and variation in exponents and fractal dimensions are still
poorly understood.
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