Scaling of fracture systems in geological media

Abstract

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. INTRODUCTIONThe study of fracture systems (terms in italic are defined in the glossary, after the main text) has been an active area of research for the last 25 years motivated to a large extent by the siting of hazardous waste disposal sites in crystalline rocks, by the problems of multiphase flow in fractured hydrocarbon reservoirs, and by earthquake hazards and the possibility of prediction. Here we define a fracture as any discontinuity within a rock mass that developed as a response to stress. This comprises primarily mode I and mode II fractures. In mode I fracturing, fractures are in tensile or opening mode in which displacements are normal to the discontinuity walls (joints and many veins). Faults correspond to mode II fractures, i.e., an in-plane shear mode, in which the displacements are in the plane of the discontinuity. Fractures exist on a wide range of scales from microns to hundreds of kilometers, and it is known that throughout this scale range they have a sign...

The impact of different aperture distribution models and critical stress criteria on equivalent permeability in fractured rocks

“…In sandstones, aperture scaling may be described by a universal exponent [Hooker et al, 2014], whereas for carbonates variable exponents are inferred [Hooker et al, 2012, and references therein]. These different values are likely related to trends in lithology or mechanical units, which may constrain the growth of fractures, but no physical relation between these outcrop properties and scaling exponents has been observed [Odling et al, 1999;Bonnet et al, 2001;Hooker et al, 2012].…”

“…where F is the frequency for aperture X (in m), b is the power law exponent, and a is a fracture network density constant [Bonnet et al, 2001;Ortega et al, 2006;Hooker et al, 2012Hooker et al, , 2013Hooker et al, , 2014. In sandstones, aperture scaling may be described by a universal exponent [Hooker et al, 2014], whereas for carbonates variable exponents are inferred [Hooker et al, 2012, and references therein].…”

Joint Inversion of Fracture Model Properties for CO2Storage Monitoring or Oil Recovery History Matching

“…This map is generally equal to the initial density map D 0 . Then for each cell the number of points N loc to draw is computed as a function of the total number of points N tot : (14) Then for each cell of D 0 the N loc points are drawn. With this method deformation trajectories are restricted to the cells of D 0 .…”

“…Then, a multifractal method is proposed to analyze and to constrain the sub-seismic fault model to the visible seismic ones. The fault length distribution, [14], is modeled using a power law distributions; n(l)∝l -a where n(l) is the number of faults shorter than l and a is the power law exponent. This choice is justified from outcrop observations and geomechanical models.…”

“…Stochastic methods based on fractal geometry have been extensively used to generate sub-seismic fault models [11][12][13][14]. These methods assume that a fracture network is similar at different observation scales.…”