Self-organised criticality and fluid-rock interactions in the brittle field

The concept of self-organised criticality (SOC) has recently been suggested as a paradigm for the long-term behaviour of earthquakes, even though many of the currently-proposed models require some tuning of the state variables or local conservation rules to produce the universally-observed Gutenberg-Richter frequency-magnitude distribution witha b value near 1. For example, a systematic negative correlation is predicted between modelb values and the degree of conservation of local force after the slip of a single element in an elastic spring/block/frictional slider model. A similar relation is described here for a cellular automaton model with constitutive laws based on fracture mechanics. Such systems, although critical phenomena in the sense of producing order on all scales, are clearly not universal, and may not in general even be true examples of SOC. Nevertheless they adequately reproduce both the observed power-law (fractal or multifractal) scaling and its reported short-term fluctuation.We also present experimental and field evidence for similar systematic variations inb value with the degree of force conservation (expressed in terms of a normalised crack extension force) during subcritical crack growth involving the physical and chemical influence of pore fluids during a single cycle of failure both in tension and compression. We find that the level of conservation is strongly influenced by fluid-rock interaction under stress, allowing energy partition into processes such as: physico-chemical stress corrosion reactions; the dissolution and precipitation of mineral species on crack surfaces; and the purely mechnical phenomenon of dilatant hardening. All of these are known to occur in the Earth on a local scale, but few have been explicitly included in automaton models of seismicity. The implication is that over long time periods pore fluids may exert a strong physical and chemical influence on the universal state of SOC which the system evolves in a complex interplay of local feedback mechanisms keeping the system near criticality, perhaps most strikingly due to the valve action of faults. In the short term, crustal fluids might nevertheless be responsible for systematic local fluctuations about this average state.     

Foreshocks and b Value: Bridging Macroscopic Observations to Source Mechanical Considerations

Spring-block models, such as the Olami-Feder-Christensen (OFC) model, were introduced several years ago to describe earthquake dynamics in the context of self-organized criticality. With the aim to address the dependency of the seismicity style on source’s material properties, we present an analytical enrichment of a 2D OFC model. We conclude with an analytical expression which introduces, through an appropriate constitutive equation, an effective dissipation parameter a ^eff related analytically not only with the elastic properties of the fault plane, but also with stochastic structural heterogeneities and structural processes of the source through a gradient coefficient. Moreover, within the proposed formulation, the low b values experimentally observed in foreshock sequences can be modeled by a process of material softening in the seismogenic volume. To check our analytical findings, a cellular automaton was built-up whereas simulation results have verified the model’s predictions for the evolution of b in macroscopic records.     

Recent seismicity in Northeast India and its adjoining region

Recent seismicity in the northeast India and its adjoining region exhibits different earthquake mechanisms – predominantly thrust faulting on the eastern boundary, normal faulting in the upper Himalaya, and strike slip in the remaining areas. A homogenized catalogue in moment magnitude, M _W, covering a period from 1906 to 2006 is derived from International Seismological Center (ISC) catalogue, and Global Centroid Moment Tensor (GCMT) database. Owing to significant and stable earthquake recordings as seen from 1964 onwards, the seismicity in the region is analyzed for the period with spatial distribution of magnitude of completeness m _t, b value, a value, and correlation fractal dimension D _C. The estimated value of m _t is found to vary between 4.0 and 4.8. The a value is seen to vary from 4.47 to 8.59 while b value ranges from 0.61 to 1.36. Thrust zones are seen to exhibit predominantly lower b value distribution while strike-slip and normal faulting regimes are associated with moderate to higher b value distribution. D _C is found to vary from 0.70 to 1.66. Although the correlation between spatial distribution of b value and D _C is seen predominantly negative, positive correlations can also be observed in some parts of this territory. A major observation is the strikingly negative correlation with low b value in the eastern boundary thrust region implying a possible case of extending asperity. Incidentally, application of box counting method on fault segments of the study region indicates comparatively higher fractal dimension, D, suggesting an inclination towards a planar geometrical coverage in the 2D spatial extent. Finally, four broad seismic source zones are demarcated based on the estimated spatial seismicity patterns in collaboration with the underlying active fault networks. The present work appraises the seismicity scenario in fulfillment of a basic groundwork for seismic hazard assessment in this earthquake province of the country.     

基于微震监测的页岩气开采区b值特征研究

四川盆地南部的长宁页岩气开发区附近地震频发,近年来已发生近10次ML＞4.0的中型地震和万余次ML1.0～3.0的小微地震,灾害风险持续增高.由于国家地震台网的固定台站较为分散,难以捕捉到1级以下微震事件的精确信息,通过近场微震监测数据来分析页岩气开发区的地震风险演化趋势,已经成为势在必行的科学问题.本文基于专门布设的...     

On the physical essence of the b value for AE of rock tests and natural earthquakes in terms of fracture mechanics

The problem of physical essence of theb value forAE of rock tests and natural earthquakes has been a controversial topic during the past two decades. In the present paper the order and energy of each microfracturing of the microcrack system existing in a rock specimen is discussed from the viewpoint of fracture mechanics, then the whole series ofAE and subsequently the value ofb are determined. The order of microfracturing depends on the parametersKei/Kci whereKei is the effective stress intensity factor of thei-th microcrack andKci is the fracture toughness in the site of thei-th crack. The energy of eachAE can be expressed asηi ∫ loili Gids whereG is the energy release rate of microcarck,loi andli are the original and final crack lengths respectively andηi is the emanating efficiency for thei-th crack. If we assume that the distribution density function of microcrack lengthl isp(l)=Bl−v whereB andγ are constants, then the expressionb=3γ/2 can be deduced. Hence, we have come to the conclusion that the value ofb mirrors essentially the “crack-system-configuration” of the material, which means the distribution of the sizes, shapes and orientations of microcracks over space as well as the distribution of the other relevant physical parameters such as fracture toughness, friction coefficient, etc. Our conclusion is somewhat similar in character to Mogi’s. He concluded thet the heterogeneity of the material plays the most important role in determining the value ofb. The term heterogeneity of course covers the idea of “crack-system-configuration, but we think that our view has a little bit deeper insight to the problem than that of Mogi. This subject is supported by Chinese Joint Seismological Science Foundation.     

The experimental research on b value of AE for the rock specimens with pre-existing crack or notch under uniaxial compression

A lot of researches onb value have been made in seismology. Since the 1960’s Mogi, Scholz and others have studied AE of rock specimens in laboratory and discovered that it is related to natural earthquakes. All former researchers used integral specimens to studyb value in the laboratory. However a major earthquake is usually related to a existing seismic-fault in that area. For this reason, a series of fracture experiments with rock and glass specimens having pre-existing crack or notch is performed in order to examine the effect of preexisting crack tob value. The experimental results show that theb value begins to decrease as soon as the initiation of the crack and finally drop to a very low value when the specimen breaks unstably. Based on these, a brief discussion on the possible mechanism ofb value change for natural earthquakes is given. The Chinese version of this paper appeared in the Chinese edition ofActa Seismologica Sinica,9, 393–400, 1987. Projects sponsored by the Chinese Joint Seismological Science Foundation.     

Evaluation of seismicity and seismic hazard parameters in Turkey and surrounding area using a new approach to the Gutenberg–Richter relation

In this study, the spatial distributions of seismicity and seismic hazard were assessed for Turkey and its surrounding area. For this purpose, earthquakes that occurred between 1964 and 2004 with magnitudes of M ≥ 4 were used in the region (30–42°N and 20–45°E). For the estimation of seismicity parameters and its mapping, Turkey and surrounding area are divided into 1,275 circular subregions. The b-value from the Gutenberg–Richter frequency–magnitude distributions is calculated by the classic way and the new alternative method both using the least-squares approach. The a-value in the Gutenberg–Richter frequency–magnitude distributions is taken as a constant value in the new alternative method. The b-values calculated by the new method were mapped. These results obtained from both methods are compared. The b-value shows different distributions along Turkey for both techniques. The b-values map prepared with new technique presents a better consistency with regional tectonics, earthquake activities, and epicenter distributions. Finally, the return period and occurrence hazard probability of M ≥ 6.5 earthquakes in 75 years were calculated by using the Poisson model for both techniques. The return period and occurrence hazard probability maps determined from both techniques showed a better consistency with each other. Moreover, maps of the occurrence hazard probability and return period showed better consistency with the b-parameter seismicity maps calculated from the new method. The occurrence hazard probability and return period of M ≥ 6.5 earthquakes were calculated as 90–99% and 5–10 years, respectively, from the Poisson model in the western part of the studying region.     

Fracture Toughness Evaluation Based on Tension-softening Model and its Application to Hydraulic Fracturing

This paper discusses the applicability of the tension-softening model in the determination of the fracture toughness of rocks, where the fracture toughness evaluated based on the tension-softening model is compared with the crack growth resistance deduced from laboratory-scale hydraulic fracturing tests. It is generally accepted that the fracture process is dominated by the growth of a fracture process zone for most types of rocks. In this study, the J-integral based technique is employed to determine the fracture toughness of Iidate granite on the basis of the tension-softening model, where compact tension specimens of different dimensions were tested in order to examine the specimen size effect on the measured fracture toughness. It was shown that the tension-softening relation deduced from the J-integral based technique allowed us to determine the specimen size independent fracture toughness K_c of Iidate granite. Laboratory-scale hydraulic fracturing tests were performed on cubic specimens (up to a 10 m sized specimen), where cyclic pressurization was conducted using a rubber-made straddle packer to observe the extent of the hydraulically induced crack. The experimental results of pressure and crack length were then used to construct the crack growth resistance curve based on the stress intensity factor K. The crack growth resistance obtained from the hydraulic fracturing tests was observed to initially increase and then level off, giving a constant K value for a long crack extension stage. The plateau K value in the crack growth resistance curve was found to be in reasonable agreement with the fracture toughness K_c deduced from the tension-softening relation. It was demonstrated that the tension-softening model provides a useful tool to determine the appropriate fracture toughness of rocks, which may be applicable for the analysis of the process of large-scale crack extension in rock masses.     

Anomalously high b-values in the South Flank of Kilauea volcano, Hawaii: evidence for the distribution of magma below Kilauea's East rift zone

The pattern of b-value of the frequency–magnitude relation, or mean magnitude, varies little in the Kaoiki-Hilea area of Hawaii, and the b-values are normal, with b=0.8 in the top 10 km and somewhat lower values below that depth. We interpret the Kaoiki-Hilea area as relatively stable, normal Hawaiian crust. In contrast, the b-values beneath Kilauea's South Flank are anomalously high (b=1.3–1.7) at depths between 4 and 8 km, with the highest values near the East Rift zone, but extending 5–8 km away from the rift. Also, the anomalously high b-values vary along strike, parallel to the rift zone. The highest b-values are observed near Hiiaka and Pauahi craters at the bend in the rift, the next highest are near Makaopuhi and also near Puu Kaliu. The mildest anomalies occur adjacent to the central section of the rift. The locations of the three major and two minor b-value anomalies correspond to places where shallow magma reservoirs have been proposed based on analyses of seismicity, geodetic data and differentiated lava chemistry. The existence of the magma reservoirs is also supported by magnetic anomalies, which may be areas of dike concentration, and self-potential anomalies, which are areas of thermal upwelling above a hot source. The simplest explanation of these anomalously high b-values is that they are due to the presence of active magma bodies beneath the East Rift zone at depths down to 8 km. In other volcanoes, anomalously high b-values correlate with volumes adjacent to active magma chambers. This supports a model of a magma body beneath the East Rift zone, which may widen and thin along strike, and which may reach 8 km depth and extend from Kilauea's summit to a distance of at least 40 km down rift. The anomalously high b-values at the center of the South Flank, several kilometers away from the rift, may be explained by unusually high pore pressure throughout the South Flank, or by anomalously strong heterogeneity due to extensive cracking, or by both phenomena. The major b-value anomalies are located SSE of their parent reservoirs, in the direction of motion of the flank, suggesting that magma reservoirs leave an imprint in the mobile flank. We hypothesize that the extensive cracking may have been acquired when the anomalous parts of the South Flank, now several kilometers distant from the rift zone, were generated at the rift zone near persistent reservoirs. Since their generation, these volumes may have moved seaward, away from the rift, but earthquakes occurring in them still use the preexisting complex crack distribution. Along the decollement plane at 10 km depth, the b-values are exceptionally low (b=0.5), suggesting faulting in a more homogeneous medium.     

Physical background on estimating b value

The G-R relation lgN=a-bM (1954) is an empirical formula used widely in the seismicity research. But the linearity of b curves has great difference in different time and space domains. An interested question in this paper is that in how large a space-time-strength domain the b value has certain physical connotation. This study told us that we can get optimal statistical results of b value in those space-time domains which can develop correspondent strong shocks with magnitude interval (M _s≥8.5, 8.0≤M _s<8.5, 7.0≤M _s<8.0). Thus, the possible seismogenic areas in which strong shocks with different magnitude intervals develop can be inferred in different regions of the mainland of China. Finally, some new problems are proposed, such as the delimitation of seismic province, the seismicity parameter determination in seismic hazard analysis and in earthquake predictions by using b value. Contribution No. 96A-0074, Institute of Geophysics, SSB, China.     

Space-time correlations ofb values with stress release

A total of 1503 events for a 2-month period associated with am _N 2.6 rockburst is investigated for possible space-time correlations between low magnitude (–1.1 to –0.4)b values and several estimates of stress (static stress drop, apparent stress, and dynamic stress). Spatial variations of decreasingb values were found to be well correlated with increasing stress release estimates for time intervals prior to the rockburst and following the aftershock sequence. The strongest correlation tob value was with the dynamic stress drop, having correlation coefficients of 0.87 and 0.79 for the two intervals, respectively. The rockburst was found to actually occur at the intersection of the spatial coordinates corresponding to the largest gradient inb value. Based on these correlations, we conclude that the low magnitude seismicity is an indicator of the stress state within the rock mass, and can be used to study and forecast stress patterns in the vicinity of an impending major event. Time variations, however, did not show the same clear correlations and these are discussed in terms of departure from steady state conditions. Regardless, our results favour the use ofb values in a spatial, context rather than in a time analysis approach, and we consider thatb values provide valuable information regarding the changing stress conditions within the seismogenic volume.     

An experimental study on variation of b value during deformation of rock samples with different structural models

A preliminary study of b value of rocks with two kinds of structural models has been made on the base of a new acoustic emission recording system. It shows that b value of the sample decreases obviously when the sample with compressive en echelon faults changes into a tensile one after interchange occurs between stress axis σ ₁ and σ ₂. A similar experiment is observed when the sample with tensile en echelon faults changes into that with a bend fault after two segments of the en echelon fault linking up. These facts indicate that the variation of b value may contain the information of the regional dominant structural model. Therefore, b-value analyses could be a new method for studying regional dominant structural models. Foundation item: National Natural Science Foundation of China (Grant No. 40072067) and Minister of Science and Technology of China (2004BA601B01).     

建筑物出口的人员疏散研究

根据建筑物出口疏散的特点,提出了一种改进型的元胞自动机模型。研究了影响人员疏散速度的两个因素:人员之间的相互拥挤产生的冲突和人员移动到出口时由于转弯导致的移动速度减慢。使用数学方法推导得出了摩擦冲突函数和转弯函数。通过计算机仿真与实际疏散实验对比发现,在不考虑转弯因素的情况下,仿真结果与实验结果偏差较大,而在同时考虑相互冲突和转弯因素时的仿真结果与实验结果偏差很小,验证了这两个因素是疏散仿真不可忽略的因素。通过分析仿真结果,验证了本文所采用的模型和函数能够准确地反映实际疏散过程,具有较高的实用价值。     

2016年新疆呼图壁6.2级地震前b值异常特征研究

利用2009年以来中国地震台网中心地震编目资料,以2016年新疆呼图壁6.2级地震作为研究对象,采用最大似然法进行b值空间扫描计算,获取震前震中及其邻区地震b值的空间图像。研究结果显示,2016年呼图壁6.2级地震发生在震前显著低b值区域,该异常特征可能反映了地震孕育的应力积累过程,印证了b值的物理意义。     

Calculation of b value and its application in earthquake prediction

The improved calculation method ofb value is presented in this paper. The method can enlarge the role of earthquake occurrence frequency inb value calculation and thus increase theb value variation amplitude. In this case, the combination structure variation between earthquake magnitudes and corresponding frequencies could be shown clearly. According to the calculation and analysis for limited mainshocks in the complete seismicity data of selected monitored area with assigned consistent lowest magnitude, the precursor anomaly features, quantitative indexes and the calculation formula of relative subject function ofb value variation have been preliminarily worked out. The prediction in short period (from 1 to 3 months) for damage earthquakes in the monitored area mentioned above can be put forward on the basis of the results of quantitative calculation and analysis.     

Ito equations out of domino cellular automaton with efficiency parameters

Ito equations are derived for simple stochastic cellular automaton with parameters describing efficiencies for avalanche triggering and cell occupation. Analytical results are compared with the numerical one obtained from the histogram method. Good agreement for various parameters supports the wide applicability of the Ito equation as a macroscopic model of some cellular automata and complex natural phenomena which manifest random energy release. Also, the paper is an example of effectiveness of histogram procedure as an adequate method of nonlinear modeling of time series.     

The experimental study of acoustic emission during creep of rocks

Under normal temperature, the creep experiments with complete samples of Gabbro and Marble rocks are made under uniaxial compression. It is found that at the instantaneous creeping stage, AE activities increase with loading; at the steady creeping stage, large AE signals may appear at lower background of AE action, andm—value which shows the relationship between AE amplitude and frequency keeps stable on the low value or decreases; at the accelerate creeping stage, AE activities increase andm—value decreases quickly or decreases again after recovery. These experimental results are related to the quality of the samples. In this paper, AE activities during three stages of creeping process are connected with the seismic patterns (for exampleb—value, foreshocks). Finally, it is pointed out the possibility that the foreshock—mainshock—aftershock earthquake sequence has been formed by the mechanism of creep fracture of crustal rocks. The Chinese version of this paper appeared in the Chinese edition ofActa Seismologica Sinica,13, 104–112, 1991. This subject is sponsored by the Chinese Joint Seismological Science Foundation.     

古登堡-里希特定律中的b值统计样本量研究

b值是研究地震活动的重要指标,其广泛应用于地震危险性分析和地震预测研究之中,与实际资料的完整性、样本量的大小、计算方法等因素有着重要的关系。常见的b值计算方法有最小二乘法和最大似然法,样本量的大小对这2种方法影响很大。本文利用蒙特卡罗模拟地震目录和汾渭地震带实际目录作为样本,从中抽取不同大小的样本量进行计算,研究不同样本量下这2种方法计算得到的b值与设定值或真实值之间的差别。结果表明,最小二乘法需要的最低样本量为1000,最大似然法为200;当样本量达不到要求时,计算出的b值是不可靠的;由于对样本量的要求不同,前者适用于计算区域的整体b值,而后者在研究某区域b值在时间轴上的变化方面更有优势。本研究为确定2种b值计算方法对样本量的最低要求提供了参考依据。     

Tidal Stress/Strain and the b-values of Acoustic Emissions at the Underground Research Laboratory, Canada

The correlation between the b-values of acoustic emissions (AEs) and the phase of the moon was investigated at the Underground Research Laboratory (URL) in Canada. The same data as those used in Iwata (2002) were examined, which showed that the occurrence of AEs is correlated with the phase of the moon. It was expected, therefore, that the b-value of the AEs would also be sensitive to tidal stress/strain fluctuations. We investigated the variation of the b-values as a function of the phase of the moon. Results show that b-values immediately following the times of full/new moon are higher than those at other times. Using AIC (Akaike Information Criterion) and random (Monte Carlo) simulations, it was confirmed that this feature is statistically significant. We also investigated whether or not there was a change in the b-values immediately before the times of full/new moon, but no statistically significant change was observed. The results suggest that the effect of stress/strain fluctuations on AE occurrences at the URL is asymmetric to the times of full/new moon.All authors are members of the Academic Robotics Group. In listing The Academic Robotics Group, the authors are endeavoring to place each of the participant institutions on an equal footing in terms of effort and authorship. M. A. Talamini is serving as presenter.