Scaling of fracture systems in geological media

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.