Ian G. Main

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, hy- drocarbon reservoir management, and earthquake haz- ard assessment. Relevant publications are therefore spread widely through the literature. Although it is rec- ognized 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 expo- nents and fractal dimensions from observations, al- though outwardly simple, is problematic, and uncritical use of analysis techniques has resulted in inaccurate and even meaningless exponents. We review these tech- niques and suggest guidelines for the accurate and ob- jective estimation of exponents and fractal dimensions. Syntheses of length, displacement, aperture power law exponents, and fractal dimensions are found, after crit- ical appraisal of published studies, to show a wide vari- ation, 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 phys- ical causes of power law scaling and variation in expo- nents and fractal dimensions are still poorly understood.

Statistical physics, seismogenesis, and seismic hazard

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 strong local interactions, and the impact of zoning a potentially multifractal distribution of epicentres with smooth polygons. The results of some models show premonitory patterns of seismicity which could in principle be used as mainshock precursors. However, there remains no consensus, on both theoretical and practical grounds, on the possibility or otherwise of reliable intermediate-term earthquake prediction.

Sequential growth of deformation bands in the laboratory

We investigate the formation and evolution of localised faulting in high porosity sandstone by laboratory triaxial compression of intact 100-mm-diameter core samples. Experiments were carried out dry, at constant confining pressure (34 MPa), constant axial strain rate (5 10 ˇ6 s ˇ1 ) and increasing axial strain (1.5‐11.2%). Tests generated fault zones consisting of sets of distinct pale granulated strands, separated by lenses of apparently undamaged host rock. The sets of strands were sub-parallel to the shear direction but showed complex anastamosing geometry in perpendicular section. The individual strands had reduced grain size, porosity and sorting compared to undeformed rock. A strong correlation was found between the number of strands occurring in a fault zone and the applied axial strain. Mean grain size, however, reached a steady value irrespective of axial strain. This implies that a limited amount of strain is accommodated on each strand with further strain requiring new strands to form. However, no direct evidence for strain hardening was observed in the post-failure macroscopic stress‐strain curves. Our laboratory induced deformation zones strongly resemble the key characteristics of natural deformation bands. We show the first laboratory evidence for the sequential development of increasing numbers of discrete deformation bands with increasing strain. # 1999 Elsevier Science Ltd. All rights reserved.

Temporal variations in seismicity during quasi-static and dynamic rock failure

Abstract A comprehensive model is presented which can explain temporal fluctuations in seismic b-values in the period leading to mechanical failure in terms of the underlying physical processes of time-varying applied stress and stress corrosion-enhanced crack growth under conditions of constant strain rate. The form of the b-value anomaly in the period leading to failure depends on the form of the stress/time relationship. For the case where dynamic failure occurs at peak stress after a period of strain hardening, the model predicts a single cusp-like b-value anomaly, reaching a critically low value of 0.5 at failure. For the physically most realistic case where dynamic failure is preceded by a period of precursory strain energy release during strain softening, the model predicts two minima in the b-value. separated by a temporary maximum or inflection point. These fluctuations in the b-value are consistent with reported “intermediate-term” and “short-term” earthquake precursors separated by a period of seismic quiescence. For the case of quasi-static cataclastic flow, the b-value mirrors the stress and never falls to the critical value because there is no critical rupture. New results from a series of controlled laboratory experiments are presented in which microseismic event rates and b-value were monitored contemporaneously with stress/time data for all variants of the stress/time relationship. Recent field observations of temporal changes in seismicity rates and b-value preceding major earthquakes are also reported. Both data sets exhibit b-value anomalies which are consistent with the model predictions.

Apparent Breaks in Scaling in the Earthquake Cumulative Frequency-Magnitude Distribution: Fact or Artifact?

It has been suggested that the finite width of the seismogenic lithosphere can have a strong effect on the frequency-moment relation for large earthquakes. Theories have been proposed in which large earthquakes have either a shallower or a steeper power-law slope in the incremental frequency-moment distribution, at characteristic seismic moments corresponding to earthquakes that rupture the entire seismogenic depth. However, many authors have applied the predicted double-slope distribution, which requires five independent parameters, to cumulative frequency data, and used the location of the break of slope to make inferences on characteristic size effects in the earthquake source in different seismotectonic regions. Here we examine the problem in a forward modeling mode by adding a realistic degree of statistical scatter to ideal incremental frequency-moment distributions of various commonly used forms. Adopting a priori the assumption of a piecewise linear distribution, we find in each case apparently statistically distinct breaks of slope that are not present in the parent distribution. These breaks of slope are artifacts produced by a combination of (a) high-frequency noise introduced by the random statistical scatter, (b) the more gradual natural roll-over in the cumulative frequency data near the maximum seismic moment, and (c) a systematic increase in the apparent regression coefficient due to the natural smoothing effect of the use of cumulative-frequency data. Therefore, if there is no apparent break of slope in the incremental distribution, it is unwise to interpret the cumulative-frequency data uniquely in terms of a break in slope. Until such breaks of slope can be distinguished in incremental-frequency data, we conclude that alternative methods (inversion of fault length and width from individual source mechanisms/aftershock sequences etc.) should be preferred for the examination of finite-depth effects, and that simpler solutions to the frequency-moment problem should be adopted for seismic-hazard applications.

One slope or two? Detecting statistically significant breaks of slope in geophysical data, with application to fracture scaling relationships

The scaling of displacement as a function of length is important for a variety of applications which depend on the mechanical and hydraulic properties of faults and fractures. Recently it has been suggested that the power-law exponent ν which has been found to characterise this relationship may change significantly at a characteristic length for a variety of reasons, for example when cracks begin to interact, or when faults grow to a length comparable to a characteristic size in the brittle layer. Such a break of slope requires a second straight line, requiring two extra model parameters. Here we present a new method for analysing such data, which penalises the extra parameters using a modified form of Schwarzs Information Criterion, and a Bayesian approach which represents uncertainty in the unknown parameters. We apply the method to data from the Krafla fissure zone in the north of Iceland, and find a significant break of slope, from ν ≈ 3/2 to ν ≈ 2/3, at a characteristic length of 12m.

Scattering attenuation and the fractal geometry of fracture systems

Scattering of seismic waves can be shown to have a frequency dependenceQ−1 ∝ ω3−v if scattering is produced by arrays of inhomogeneities with a 3D power spectrumW3D(k) ∝k−v. In the earths crust and upper mantle the total attenuation is often dominated by scattering rather than intrinsic absorption, and is found to be frequency dependent according toQ−1 ∝ ωγ, where −1<γ≤−0.5. IfD1 is the fractal dimension of the surface of the 3D inhomogeneities measured on a 2D section, then this corresponds respectively to 1.5<D1≤1.75, since it can be shown that γ=2(D1−2). Laboratory results show that such a distribution of inhomogeneities, if due to microcracking, can be produced only at low stress intensities and slow crack velocities controlled by stress corrosion reactions. Thus it is likely that the earths brittle crust is pervaded by tensile microcracks, at least partially filled by a chemically active fluid, and preferentially aligned parallel to the maximum principal compressive stress. The possibility of stress corrosion implies that microcracks may grow under conditions which are very sensitive to pre-existing heterogeneities in material constants, and hence it may be difficult in practice to separate the relative contribution of crack-induced heterogeneity from more permanent geological heterogeneities.By constrast, shear faults formed by dynamic rupture at critical stress intensities produceD1=1, consistent with a dynamic rupture criterion for a power law distribution of fault lengths with negative exponentD. The results presented here suggest empirically thatD1∼-1/2(D+1), thereby providing the basis for a possible framework to unify the interpretation of temporal variations in seismicb-value (b∼-D/2) and the frequency dependence of scattering attenuation (γ).

Spatial variations of the fractal properties of seismicity in the Anatolian fault zones

Abstract The Anatolian fault zones are seismically active strike-slip fault zones transcending the Anatolian plate in E-W and N-S directions. We investigate the spatial variations of seismicity along these zones in an attempt to investigate fault complexity along strike, quantified by the Gutenberg-Richter b -value and the fractal (correlation) dimension of earthquake epicentres, using the maximum likelihood method and the correlation integral, respectively. The investigation covers instrumentally recorded carthquakes of magnitude M > 4.5 occurring between 1900 and 1992. We find systematic spatial variations which may be related to structural or mechanical variability along strike. In particular the large change in strike at the northern apex of the North Anatolian Fault Zone is associated with the highest correlation dimension and lowest b -value for seismicity this century. The correlation dimension and b -value show a negative correlation with respect to each other, similar to results reported in other regional studies of Japan and southern California. This statistical correlation is stronger when more objective seismic zoning is carried out (based on number of events) rather than more subjective seismotectonic zoning in common use in seismic hazard analysis.

Temporal variations in seismic event rate and b-values from stress corrosion constitutive laws

Abstract Main, I.G., Meredith, P.G. and Sammonds, P.R., 1992. Temporal variations in seismic event rate and b-values from stress corrosion constitutive laws. In: T. Mikumo, K. Aki, M. Ohnaka, L.J. Ruff and P.K.P. Spudich (Editors), Earthquake Source Physics and Earthquake Precursors. Tectonophysics, 211: 233–246. A model is developed to characterise the state of damage for a fractal population of cracks in terms of a mean energy release rate 〈G〉which depends on the stress σ, the event rate N and the seismic b-value. N is assumed to be proportional to the total number of potentially active cracks, and b to be proportional to the exponent D of the crack length distribution. 〈G〉is then positively correlated with N and negatively correlated with b, consistent with experimental observation based on stress intensity as a constitutive variable. The model predicts that higher b-values are associated either with lower stress intensity or greater material heterogeneity. Both experiment and theory predict intermediate-term seismic quiescence of similar magnitude to that observed in the field (45–90%), for only a small simultaneous reduction in stress and stress intensity (2–7%) or the equivalent mean energy release rate. The theoretical model predicts three different types of quiescence associated with increasing, constant or decreasing b-value. The first two are associated with stable intermediate-term quiescence due to a reduction in 〈G〉, and the third is associated with unstable short-term quiescence associated with constant or increasing 〈G〉. The theory can be used to infer relative changes in 〈G〉 from acoustic emissions in the laboratory. Experiments to date show that quiescence of the order 10% can be seen in the laboratory at strain rates of 10−5 s−1, but that this has the characteristics of short-term rather than intermediate-term quiescence.

INFLUENCE OF FRACTAL FLAW DISTRIBUTIONS ON ROCK DEFORMATION IN THE BRITTLE FIELD

Abstract The geometrical distribution of flaws plays a crucial role in the physical behaviour of geological materials under stress. Flaws are present in the earth on all scales, from microcracks to plate-rupturing faults. They may be distributed on one characteristic length scale (e.g. joints, ‘characteristic’ earthquakes), or more commonly exhibit scale-invariance over a specified range of sizes. Scale-invariance implies that the discrete length distribution in a finite range is a power law of negative exponent D, where 1 ≤ D < 3. Fault systems where motion is concentrated on a dominant fault (e.g. San Andreas) have D ≈ 1, but more diffuse fault systems have D near 2. D is one of the fractal dimensions of the fracture system. The length distribution of faults or microcracks may be inferred from the slope b of the log-linear frequency—magnitude distribution of earthquakes, or laboratory-scale acoustic emissions, since it can be shown that D = 3b/c. The scaling factor c depends on the relative time constants of the seismic event and the recording instrument, and is usually equal to 3/2. b is found experimentally to be negatively correlated with the stress intensity on the dominant flaw, which depends in turn on the applied stress and the flaw length. Thus a fracture mechanics model of rock failure which includes a range of flaw sizes can be tested by seismic monitoring. We describe a fracture mechanics model of rock failure for a variety of styles of deformation, ranging from elastic failure to quasi-static cataclastic flow, and predict the time-dependence of D and the seismic b-value at different times up to and including failure. Critical coalescence of microcracks during dynamic failure (e.g. earthquake foreshocks) occurs when D = 1 (b = 0.5); random processes (e.g. cataclastic flow, background seismicity) are associated with D = 2 (b = 1); positive feedback in the concentration of stress on the dominant flaw (e.g. during strain softening and shear localisation) occurs when D < 2 (b < 1); negative feedback in stress concentration (e.g. during the early stages of dilatancy), and where a highly diffuse fracture system is produced, occurs at low stress intensities and is associated with D > 2 (b > 1). It has long been a goal of structural geologists to measure stress on rocks, since most geometrical signatures of deformation are strain-related. We show that stress is not usually as significant in rock fracture as stress intensity, and furthermore that the geometric signature of the length distribution of microcracks is well-correlated with the stress intensity.