Effect of disorder on the spatial structure of damage in slowly compressed porous rocks.

Faults and damage zone properties control a range of important phenomena, from the hydraulic properties of underground reservoirs to the physics of earthquakes on a larger scale. Here, we investigate the effect of disorder of porous rocks on the spatial structure of damage emerging under compression. Model rock samples are numerically generated by sedimenting particles where the amount of disorder is controlled by the particle size distribution. To obtain damage bands with a sufficiently large length along axis, we performed simulations of 'Brazilian'-type compression tests of cylindrical samples. As failure is approached, damage localization leads to the formation of two conjugate shear bands. The orientation angle of bands to the loading direction increases with disorder, implying a decrease in the internal coefficient of friction. The width of the damage band scales as a power law of the degree of disorder. Inside the damage band, the sample is crushed into a large number of pieces with a power law mass distribution. The shape of fragments undergoes a crossover at a disorder-dependent size from the isotropy of small pieces to the anisotropic flattened form of the large ones. The results provide important constraints in understanding the role of disorder in geological fractures.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Rupture cascades in a discrete element model of a porous sedimentary rock.

We investigate the scaling properties of the sources of crackling noise in a fully dynamic numerical model of sedimentary rocks subject to uniaxial compression. The model is initiated by filling a cylindrical container with randomly sized spherical particles that are then connected by breakable beams. Loading at a constant strain rate the cohesive elements fail, and the resulting stress transfer produces sudden bursts of correlated failures, directly analogous to the sources of acoustic emissions in real experiments. The source size, energy, and duration can all be quantified for an individual event, and the population can be analyzed for its scaling properties, including the distribution of waiting times between consecutive events. Despite the nonstationary loading, the results are all characterized by power-law distributions over a broad range of scales in agreement with experiments. As failure is approached, temporal correlation of events emerges accompanied by spatial clustering.

Effect of the loading condition on the statistics of crackling noise accompanying the failure of porous rocks.

We test the hypothesis that loading conditions affect the statistical features of crackling noise accompanying the failure of porous rocks by performing discrete element simulations of the tensile failure of model rocks and comparing the results to those of compressive simulations of the same samples. Cylindrical samples are constructed by sedimenting randomly sized spherical particles connected by beam elements representing the cementation of granules. Under a slowly increasing external tensile load, the cohesive contacts between particles break in bursts whose size fluctuates over a broad range. Close to failure breaking avalanches are found to localize on a highly stressed region where the catastrophic avalanche is triggered and the specimen breaks apart along a spanning crack. The fracture plane has a random position and orientation falling most likely close to the centre of the specimen perpendicular to the load direction. In spite of the strongly different strengths, degrees of 'brittleness' and spatial structure of damage of tensile and compressive failure of model rocks, our calculations revealed that the size, energy and duration of avalanches, and the waiting time between consecutive events all obey scale-free statistics with power law exponents which agree within their error bars in the two loading cases.

Seismic events miss important kinematically governed grain scale mechanisms during shear failure of porous rock.

Catastrophic failure in brittle, porous materials initiates when smaller-scale fractures localise along an emergent fault zone in a transition from stable crack growth to dynamic rupture. Due to the rapid nature of this critical transition, the precise micro-mechanisms involved are poorly understood and difficult to image directly. Here, we observe these micro-mechanisms directly by controlling the microcracking rate to slow down the transition in a unique rock deformation experiment that combines acoustic monitoring (sound) with contemporaneous in-situ x-ray imaging (vision) of the microstructure. We find seismic amplitude is not always correlated with local imaged strain; large local strain often occurs with small acoustic emissions, and vice versa. Local strain is predominantly aseismic, explained in part by grain/crack rotation along an emergent shear zone, and the shear fracture energy calculated from local dilation and shear strain on the fault is half of that inferred from the bulk deformation.

Predicting mining collapse: Superjerks and the appearance of record-breaking events in coal as collapse precursors.

The quest for predictive indicators for the collapse of coal mines has led to a robust criterion from scale-model tests in the laboratory. Mechanical collapse under uniaxial stress forms avalanches with a power-law probability distribution function of radiated energy P∼E^{-}^{ɛ}, with exponent ɛ=1.5. Impending major collapse is preceded by a reduction of the energy exponent to the mean-field value ɛ=1.32. Concurrently, the crackling noise increases in intensity and the waiting time between avalanches is reduced when the major collapse is approaching. These latter criteria were so-far deemed too unreliable for safety assessments in coal mines. We report a reassessment of previously collected extensive collapse data sets using "record-breaking analysis," based on the statistical appearance of "superjerks" within a smaller spectrum of collapse events. Superjerks are defined as avalanche signals with energies that surpass those of all previous events. The final major collapse is one such superjerk but other "near collapse" events equally qualify. In this way a very large data set of events is reduced to a sparse sequence of superjerks (21 in our coal sample). The main collapse can be anticipated from the sequence of energies and waiting times of superjerks, ignoring all weaker events. Superjerks are excellent indicators for the temporal evolution, and reveal clear nonstationarity of the crackling noise at constant loading rate, as well as self-similarity in the energy distribution of superjerks as a function of the number of events so far in the sequence E_{sj}∼n^{δ} with δ=1.79. They are less robust in identifying the precise time of the final collapse, however, than the shift of the energy exponents in the whole data set which occurs only over a short time interval just before the major event. Nevertheless, they provide additional diagnostics that may increase the reliability of such forecasts.

Fragmentation and shear band formation by slow compression of brittle porous media.

Localized fragmentation is an important phenomenon associated with the formation of shear bands and faults in granular media. It can be studied by empirical observation, by laboratory experiment, or by numerical simulation. Here we investigate the spatial structure and statistics of fragmentation using discrete element simulations of the strain-controlled uniaxial compression of cylindrical samples of different finite size. As the system approaches failure, damage localizes in a narrow shear band or synthetic fault "gouge" containing a large number of poorly sorted noncohesive fragments on a broad bandwidth of scales, with properties similar to those of natural and experimental faults. We determine the position and orientation of the central fault plane, the width of the shear band, and the spatial and mass distribution of fragments. The relative width of the shear band decreases as a power law of the system size, and the probability distribution of the angle of the central fault plane converges to around 30 degrees, representing an internal coefficient of friction of 0.7 or so. The mass of fragments is power law distributed, with an exponent that does not depend on scale, and is near that inferred for experimental and natural fault gouges. The fragments are in general angular, with a clear self-affine geometry. The consistency of this model with experimental and field results confirms the critical roles of preexisting heterogeneity, elastic interactions, and finite system size to grain size ratio on the development of shear bands and faults in porous media.

Record-breaking events during the compressive failure of porous materials.

An accurate understanding of the interplay between random and deterministic processes in generating extreme events is of critical importance in many fields, from forecasting extreme meteorological events to the catastrophic failure of materials and in the Earth. Here we investigate the statistics of record-breaking events in the time series of crackling noise generated by local rupture events during the compressive failure of porous materials. The events are generated by computer simulations of the uniaxial compression of cylindrical samples in a discrete element model of sedimentary rocks that closely resemble those of real experiments. The number of records grows initially as a decelerating power law of the number of events, followed by an acceleration immediately prior to failure. The distribution of the size and lifetime of records are power laws with relatively low exponents. We demonstrate the existence of a characteristic record rank k(*), which separates the two regimes of the time evolution. Up to this rank deceleration occurs due to the effect of random disorder. Record breaking then accelerates towards macroscopic failure, when physical interactions leading to spatial and temporal correlations dominate the location and timing of local ruptures. The size distribution of records of different ranks has a universal form independent of the record rank. Subsequences of events that occur between consecutive records are characterized by a power-law size distribution, with an exponent which decreases as failure is approached. High-rank records are preceded by smaller events of increasing size and waiting time between consecutive events and they are followed by a relaxation process. As a reference, surrogate time series are generated by reshuffling the event times. The record statistics of the uncorrelated surrogates agrees very well with the corresponding predictions of independent identically distributed random variables, which confirms that temporal and spatial correlation in the crackling noise is responsible for the observed unique behavior. In principle the results could be used to improve forecasting of catastrophic failure events, if they can be observed reliably in real time.

Heterogeneity: The key to failure forecasting.

Elastic waves are generated when brittle materials are subjected to increasing strain. Their number and energy increase non-linearly, ending in a system-sized catastrophic failure event. Accelerating rates of geophysical signals (e.g., seismicity and deformation) preceding large-scale dynamic failure can serve as proxies for damage accumulation in the Failure Forecast Method (FFM). Here we test the hypothesis that the style and mechanisms of deformation, and the accuracy of the FFM, are both tightly controlled by the degree of microstructural heterogeneity of the material under stress. We generate a suite of synthetic samples with variable heterogeneity, controlled by the gas volume fraction. We experimentally demonstrate that the accuracy of failure prediction increases drastically with the degree of material heterogeneity. These results have significant implications in a broad range of material-based disciplines for which failure forecasting is of central importance. In particular, the FFM has been used with only variable success to forecast failure scenarios both in the field (volcanic eruptions and landslides) and in the laboratory (rock and magma failure). Our results show that this variability may be explained, and the reliability and accuracy of forecast quantified significantly improved, by accounting for material heterogeneity as a first-order control on forecasting power.

Approach to failure in porous granular materials under compression.

We investigate the approach to catastrophic failure in a model porous granular material undergoing uniaxial compression. A discrete element computational model is used to simulate both the microstructure of the material and the complex dynamics and feedbacks involved in local fracturing and the production of crackling noise. Under strain-controlled loading, microcracks initially nucleate in an uncorrelated way all over the sample. As loading proceeds the damage localizes into a narrow damage band inclined at 30°-45° to the load direction. Inside the damage band the material is crushed into a poorly sorted mixture of mainly fine powder hosting some larger fragments. The mass probability density distribution of particles in the damage zone is a power law of exponent 2.1, similar to a value of 1.87 inferred from observations of the length distribution of wear products (gouge) in natural and laboratory faults. Dynamic bursts of radiated energy, analogous to acoustic emissions observed in laboratory experiments on porous sedimentary rocks, are identified as correlated trails or cascades of local ruptures that emerge from the stress redistribution process. As the system approaches macroscopic failure consecutive bursts become progressively more correlated. Their size distribution is also a power law, with an equivalent Gutenberg-Richter b value of 1.22 averaged over the whole test, ranging from 3 to 0.5 at the time of failure, all similar to those observed in laboratory tests on granular sandstone samples. The formation of the damage band itself is marked by a decrease in the average distance between consecutive bursts and an emergent power-law correlation integral of event locations with a correlation dimension of 2.55, also similar to those observed in the laboratory (between 2.75 and 2.25).

Entropy production and self-organized (sub)criticality in earthquake dynamics.

We derive an analytical expression for entropy production in earthquake populations based on Dewar's formulation, including flux (tectonic forcing) and source (earthquake population) terms, and apply it to the Olami-Feder-Christensen numerical model for earthquake dynamics. Assuming the commonly observed power-law rheology between driving stress and remote strain rate, we test the hypothesis that maximum entropy production (MEP) is a thermodynamic driver for self-organized 'criticality' (SOC) in the model. MEP occurs when the global elastic strain is near-critical, with small relative fluctuations in macroscopic strain energy expressed by a low seismic efficiency, and broad-bandwidth power-law scaling of frequency and rupture area. These phenomena, all as observed in natural earthquake populations, are hallmarks of the broad conceptual definition of SOC (which has, to date, often included self-organizing systems in a near but strictly subcritical state). In the MEP state, the strain field retains some memory of past events, expressed as coherent 'domains', implying a degree of predictability, albeit strongly limited in practice by the proximity to criticality and our inability to map the natural stress field at an equivalent resolution to the numerical model.

Origin and nonuniversality of the earthquake interevent time distribution.

Many authors have modeled regional earthquake interevent times using a gamma distribution, whereby data collapse occurs under a simple rescaling of the data from different regions or time periods. We show, using earthquake data and simulations, that the distribution is fundamentally a bimodal mixture distribution dominated by correlated aftershocks at short waiting times and independent events at longer times. The much-discussed power-law segment often arises as a crossover between these two. We explain the variation of the distribution with region size and show that it is not universal.