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 scaling properties of earthquake populations show remarkable similarities to those observed at or near the critical point of other composite systems in statistical physics. This has led to the development of a variety of different physical models of seismogenesis as a critical phenomenon, involving locally nonlinear dynamics, with simplified rheologies exhibiting instability or avalanche‐type behavior, in a material composed of a large number of discrete elements. In particular, it has been suggested that earthquakes are an example of a “self‐organized critical phenomenon” analogous to a sandpile that spontaneously evolves to a critical angle of repose in response to the steady supply of new grains at the summit. In this stationary state of marginal stability the distribution of avalanche energies is a power law, equivalent to the Gutenberg‐Richter frequency‐magnitude law, and the behavior is relatively insensitive to the details of the dynamics. Here we review the results of some of the composite physical models that have been developed to simulate seismogenesis on different scales during (1) dynamic slip on a preexisting fault, (2) fault growth, and (3) fault nucleation. The individual physical models share some generic features, such as a dynamic energy flux applied by tectonic loading at a constant strain rate, strong local interactions, and fluctuations generated either dynamically or by fixed material heterogeneity, but they differ significantly in the details of the assumed dynamics and in the methods of numerical solution. However, all exhibit critical or near‐critical behavior, with behavior quantitatively consistent with many of the observed fractal or multifractal scaling laws of brittle faulting and earthquakes, including the Gutenberg‐Richter law. Some of the results are sensitive to the details of the dynamics and hence are not strict examples of self‐organized criticality. Nevertheless, the results of these different physical models share some generic statistical properties similar to the “universal” behavior seen in a wide variety of critical phenomena, with significant implications for practical problems in probabilistic seismic hazard evaluation. In particular, the notion of self‐organized criticality (or near‐criticality) gives a scientific rationale for the a priori assumption of “stationarity” used as a first step in the prediction of the future level of hazard. The Gutenberg‐Richter law (a power law in energy or seismic moment) is found to apply only within a finite scale range, both in model and natural seismicity. Accordingly, the frequency‐magnitude distribution can be generalized to a gamma distribution in energy or seismic moment (a power law, with an exponential tail). This allows extrapolations of the frequency‐magnitude distribution and the maximum credible magnitude to be constrained by observed seismic or tectonic moment release rates. The answers to other questions raised are less clear, for example, the effect of the a priori assumption of a Poisson process in a system with s...
[1] The characterization of time-dependent brittle rock deformation is fundamental to understanding the long-term evolution and dynamics of the Earth's crust. The chemical influence of pore water promotes time-dependent deformation through stress corrosion cracking that allows rocks to deform at stresses far below their short-term failure strength. Here, we report results from a study of time-dependent brittle creep in water-saturated samples of Darley Dale sandstone (initial porosity, 13%) under triaxial stress conditions. Results from conventional creep experiments show that axial strain rate is heavily dependent on the applied differential stress. A reduction of only 10% in differential stress results in a decrease in strain rate of more than two orders of magnitude. However, natural sample variability means that multiple experiments must be performed to yield consistent results. Hence we also demonstrate that the use of stress-stepping creep experiments can successfully overcome this issue. We have used the stress-stepping technique to investigate the influence of confining pressure at effective confining pressures of 10, 30, and 50 MPa (while maintaining a constant 20 MPa pore fluid pressure). Our results demonstrate that the stress corrosion process appears to be significantly inhibited at higher effective pressures, with the creep strain rate reduced by multiple orders of magnitude. The influence of doubling the pore fluid pressure, however, while maintaining a constant effective confining pressure, is shown to influence the rate of stress corrosion within the range expected from sample variability. We discuss these results in the context of microstructural analysis, acoustic emission hypocenter locations, and fits to proposed macroscopic creep laws.
Concrete bridges in the United Kingdom represent a major legacy that is starting to show signs of distress. Therefore, the need for monitoring them is an urgent task. The acoustic emission ͑AE͒ technique was proposed as a valid method for monitoring these bridges but more study is needed to develop methods of analyzing the data recorded during the monitoring. The writers would like to propose a b-value analysis as a possible way to process AE data obtained during a local monitoring. The b-value is defined as the log-linear slope of the frequency-magnitude distribution of acoustic emissions. This paper presents the results of a b-value analysis carried out on data recorded during a laboratory test on a reinforced concrete beam designed as representative of a bridge beam. During the experiment, the specimen was loaded cyclically and it was continuously monitored with an AE system. The data obtained were processed and a b-value analysis was carried out. The b-value was compared with the applied load, with a damage parameter, and with the cracks appearing on the beam. The damage parameter represents the cumulative damage in terms of total sum of acoustic emissions. The results showed a good agreement with the development of the fracture process of the concrete. From a study of the b-value calculated for a whole loading cycle and for each channel, some quantitative conclusions were also drawn. Further development work is needed to make the b-value technique suitable for practical use on a real bridge.
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