Scaling of fracture systems in geological media

Scaling in fracture systems has become an active field of research in the l ast 25 years motivated by practical applications in hazardous waste disposa l, hydrocarbon reservoir management, and earthquake hazard assessment. Rele vant publications are therefore spread widely through the literature. Altho ugh it is recognized that some fracture systems are best described by scale -limited laws (lognormal, exponential), it is now recognized that power law s and fractal geometry provide widely applicable descriptive tools for frac ture system characterization. A key argument for power law and fractal scal ing is the absence of characteristic length scales in the fracture growth p rocess. All power law and fractal characteristics in nature must have upper and lower bounds. This topic has been largely neglected, but recent studie s emphasize the importance of layering on all scales in limiting the scalin g characteristics of natural fracture systems. The determination of power l aw exponents and fractal dimensions from observations, although outwardly s imple, is problematic, and uncritical use of analysis techniques has result ed in inaccurate and even meaningless exponents. We review these techniques and suggest guidelines for the accurate and objective estimation of expone nts and fractal dimensions. Syntheses of length, displacement, aperture pow er law exponents, and fractal dimensions are found, after critical appraisa l of published studies, to show a wide variation, frequently spanning the t heoretically possible range. Extrapolations from one dimension to two and f rom two dimensions to three are found to be nontrivial, and simple laws mus t be used with caution. Directions for future research include improved tec hniques for gathering data sets over great scale ranges and more rigorous a pplication 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.