Stress transfer in earthquakes, hazard estimation and ensemble forecasting: Inferences from numerical simulations

Observations indicate that earthquake faults occur in topologically complex, multi-scale networks driven by plate tectonic forces. We present realistic numerical simulations, involving data-mining, pattern recognition, theoretical analyses and ensemble forecasting techniques, to understand how the observable space–time earthquake patterns are related to the fundamentally inaccessible and unobservable dynamics. Numerical simulations can also help us to understand how the different scales involved in earthquake physics interact and influence the resulting dynamics. Our simulations indicate that elastic interactions (stress transfer) combined with the nonlinearity in the frictional failure threshold law lead to the self-organization of the statistical dynamics, producing 1) statistical distributions for magnitudes and frequencies of earthquakes that have characteristics similar to those possessed by the Gutenberg–Richter magnitude–frequency distributions observed in nature; and 2) clear examples of stress transfer among fault activity described by stress shadows, in which an earthquake on one group of faults reduces the Coulomb failure stress on other faults, thereby delaying activity on those faults. In this paper, we describe the current state of modeling and simulation efforts for Virtual California, a model for all the major active strike slip faults in California. Noting that the Working Group on California Earthquake Probabilities (WGCEP) uses statistical distributions to produce earthquake forecast probabilities, we demonstrate that Virtual California provides a powerful tool for testing the applicability and reliability of the WGCEP statistical methods. Furthermore, we show how the simulations can be used to develop statistical earthquake forecasting techniques that are complementary to the methods used by the WGCEP, but improve upon those methods in a number of important ways. In doing so, we distinguish between the “official” forecasts of the WGCEP, and the “research-quality” forecasts that we discuss here. Finally, we provide a brief discussion of future problems and issues related to the development of ensemble earthquake hazard estimation and forecasting techniques.     

山东阜山金矿区构造应力场及其转化

在估算了阜山金矿区成岩成矿期的构造差应力值后,获得约2km2区域内的主压应力和差异应力的空间分布规律。结果表明,矿体部位往往对应于应力或应变强烈变化的区段,而非应力或应变的极高值或极低值处。依据应变测量和磁组构分析,研究了阜山金矿成岩期和成矿期的构造应变场特征,指出:阜山金矿在成岩和成矿期间构造应力场经历了由NE向挤压转为NW向挤压,构造应力场的这种转化对该区的成矿意义重大。      

深切河谷区地应力场测试数据的统计分析

以某深切峡谷区应力解除法和水压致裂法实测数据为基础资料，运用统计分析方法对该区河谷和岸坡中的地应力场的特征进行分析，分别获得了整个坝区、河谷区和岸坡中最大水平应力的方向，以及河谷区最大水平应力量值与埋深的拟合方程。并根据岸坡中地应力场的变化特征，对空间地应力测试数据进行了筛选，最终获得了代表岸坡相对稳定区的地应力场的量值和方向。     

Statistical Analysis of Fracture-Length Distribution Sampled Under the Truncation and Censoring Effects

The present paper addresses statistical analysis and estimation of fracture-length distributions at scales influenced by the truncation and censoring effects. The computational method employed here uses fracture-length distributions of a given set of measurements and information about observational constraints (i.e., window of observation) to estimate the probability density of the truncated and censored parts of fracture data sets. The results are benchmarked against power-law based maximal likelihood estimations commonly used for the same purpose. The relationship between the accuracy of estimates and size of the window of observation is studied. The utility of employing statistical models with arbitrary probability distributions of fracture lengths in order to provide a valid statistical model approximation is also considered. A verification of the suggested approximation using the Kolmogorov–Smirnov test applied to truncated and censored data is proposed. Numerical computations show that the proposed method can represent an essential improvement compared to other commonly employed techniques.     

A generalized principal components model in petrology

Principal components analysis of elemental distributions among minerals elucidate chemical, thermal, and equilibrium effects. Previous approaches in applying this multivariate statistical method to petrologic data are united into a single model, so-called three-mode principal components analysis. The major advantage of the model lies in the separate analysis of each of the three modes: element, mineral, and sample. The three sets of principal components are related through a “core matrix”. The model accommodates the ordination of pair-wise combination of the modes, such as samples and minerals in visualization of multidimensional tie-lines. Because of its generality, the model permits great flexibility in the study of petrologic data.     

Linking dynamic elastic parameters to static state of stress: toward an integrated approach to subsurface stress analysis

Stress is the most important parameter to understand basin dynamics and the evolution of hydrocarbon systems. The state of stress can be quantified by numerical geo-mechanical modelling techniques. These techniques require static elastic parameters of the rocks as input, while tectonic and gravitational forces are given as explicit boundary conditions to compute the local state of stress at different scales. We developed a technique to determine the density and elastic constants at seismic frequencies using full Zoeppritz inversion on angle-dependent seismic reflection data. The dynamic elastic parameters as obtained from seismic data differ from their static equivalents, which are necessary to determine the static state of stress. The dynamic elastic parameters are related to their static equivalents through experimentally obtained relations. In these rock-physics experiments, the static and dynamic elastic parameters are measured simultaneously during different external loading conditions. The experiments used here are all carried out in a tri-axial pressure machine under equal axial stresses. Then pre-stack seismic data analysis in combination with the relation between the static and dynamic elastic parameters, from the rock-physics experiments, provides the input parameters for geo-mechanical modelling.     

Probabilistic Horizontal Stress Ratios in Rock

Prior to excavation, any point in a rock mass is subjected to stress, which has to be considered in the overall stress distribution after the excavation work is completed. Over the past three decades, empirical, analytical, and field studies have shown that the ratio of average horizontal stress to vertical stress plays a dominant role in engineering structure design. Notationally, this ratio is referred to as the coefficient of lateral stress K, which is a dimensionless positive quantity. The intent of this paper was to bring out the intrinsic characteristic of K with the use of probabilistic and statistical techniques, which have not been considered extensively in any previous context.     

地道桥结构与土相互作用的有限元分析

借助有限元软件ANSYS来分析地道桥结构与土的相互作用,采用三维实体建模方法模拟空间范围内的地道桥结构及其周围土体.分析有无列车荷载等两种不同工况下地道桥结构的位移、应力分布规律以及由其引起的地表沉降和土体的应力变化规律.通过有限元分析可以求出结构和土体的位移、应力峰值及其空间位置,为地道桥结构设计提供可靠的分析依据.     

重载铁路路基面动应力峰值随机分布特征研究

掌握列车移动荷载作用下路基的动应力响应特性可为路基沉降预测、状态评估提供依据。建立重载铁路车辆-轨道-路基三维动力有限元模型,并在钢轨轨面引入中国三大干线不平顺轨道谱以实现轨道不平顺的模拟。通过数值计算,分析了轨道高低不平顺对路基面动应力分布的影响,统计了4种列车轴重下路基面3个位置（钢轨下方,线路中心）处的动应力峰值沿线路纵向的分布,对路基面动应力峰值的随机分布形式进行了正态性检验,对动应力峰值的最大值进行了预测。结果表明：轨道不平顺导致路基面动应力沿路基面中心不对称,路基面动应力峰值沿线路纵向服从正态分布。随着轴重的增大,动应力峰值离散程度增大,动应力峰值分布曲线变陡。动应力峰值的预测值和现场实测值相吻合,证明了有限元模型及统计分析方法的正确性。研究结果对路基动变形及累积沉降的可靠性分析提供依据。     

Statistical analysis of stress transmission in wet granular materials

The micromechanics of wet granular materials encompasses complex microstructural and capillary interconnects that can be readily described through a formal derivation of stress transmission in such a 3‐phase medium. In the quest for defining an appropriate stress measure, the stress tensor expression that results from homogenization [Duriez et al. J Mech Phys Solids 99 (2017): 495‐511] of such a medium provides theoretical insights necessary to extract useful information on the relationship between capillary effects and microforce interactions via several small‐scale parameters whose evaluation can be challenging. Using instead a statistical approach where microvariable distributions are described by probability density functions, the current study provides simple estimates of stress components in terms of only a few tractable microvariables such as coordination number and fabric anisotropy. In particular, the latter recognizes details of contacts such as force interactions being either mechanical or capillary, including interactions with and without mechanical contact. The developed expressions are in a good agreement with discrete element method simulation results of the triaxial loading of a wet granular assembly, notably for hydrostatic (mean) pressure. A new set of dimensionless groups is also identified to characterize the significance of mechanical and capillary physics, which facilitates a better understanding of the contribution of dominating elements to stress, while also providing the opportunity to incorporate important capillary effects in micromechanically based constitutive formulations.     

三维初始地应力场反分析的径向基函数法

通过对初始地应力场计算方法的回顾,提出了基于径向基函数的人工神经网络初始地应力场的三维反分析方法。径向基网络的可以避免在反分析过程中出现过拟合现象,从而保证反分析结果更加可靠.此外,该方法综合了目前各种初始地应力场模拟方法的优点,并考虑了多种因素对地应力场分布的影响。实例证明,由作者提出方法所模拟的初始地应力场是合理的,精确度也较高。此外,由于考虑了断层的作用对初始地应力场的影响,可以较好地解释实测地应力值分散的原因。     

含不同半径孔洞的颗粒体模型的力学行为数值模拟

本文通过编程建立了非连续介质(颗粒体材料)模型,采用FLAC软件模拟了静水压力条件下不同半径的巷道围岩中的剪切应变增量、最小主应力及最大主应力的分布规律。研究表明,随着孔洞半径的增大,呈圆环形的剪切应变增量与最小主应力的高值区的圈数、呈辐射状的最大主应力的高值区的延伸范围及剪切应变增量的最大值都呈先慢后快的增长趋势。模型中最大的拉应力接近于在模型四周所施加的压应力,而最大的压应力约为所施加的压应力的5～10倍。模型内部的剪切应变增量、最小主应力及最大主应力的分布是高度不均匀的。具有较高的差应力的位置与具有较高的剪切应变增量的位置具有很好的一致性。     

测定岩石经历的最高古应力状态实验研究

当前国内外尚无测定三维古构造应力场的理想方法.提出了一种新方法: 从待测地点取大岩心或新鲜石块, 从6个以上方向取岩石试样, 每方向至少10个试样, 以声发射的广义抹录不净现象, 测定岩石经历的各主要构造运动期的最大主应力值.而后, 统计出岩石记忆的最高古应力状态的最大主应力值σ_mt.由于岩石对最高古应力状态在各取样方向的正应力σ_mn尚有记忆, 只不过隐藏于各期较低古应力状态的最大主应力值之间, 不易区分, 故以σ_mt为目标, 通过“AE反演搜索法”, 则可将隐藏的各σ_mn搜索出来, 从而计算出岩石经历的最高古应力状态.以准平面应力状态模拟实验检验了该新方法的可行性.      

深埋泥岩地层的最小主应力测定方法探讨

结合理论分析与实际资料统计,根据泥岩地层在水力压裂垂直裂缝剖面中所占的比例大小和瞬时停泵压力的变化趋势,获得了鄂尔多斯盆地西峰地区长8层(2 100 m)和靖边地区盒8地层(3 000 m)的泥岩最小主应力,这可为油气钻采、压裂设计以及测井、地震资料解释提供参考依据。间接证据表明,利用该方法求取的泥岩地层地应力更符合实际情况。      

黄骅坳陷新生代构造应力场演化的光弹模拟与石油地质条件分析

在构造演化分析的基础上对黄骅坳陷的新生代构造应力场演化进行了光弹实验模型,认为黄骅坳陷新生代最大主应力方向具有从NNE向挤压向NE、NEE向转变的右旋特征,构造应力场方向的右旋转变控制着断层的活动与坳陷的形成,使得早期(孔店期、沙河街期)以引张形式活动的断裂到晚期(东营期及其以后)逐渐转变为以走滑形式活动,促成了多期构造叠加,形成了多种圈闭类型,为油气成藏富集创造了有利条件。      

Incorporation of incomplete fault-slip data into stress tensor inversion

A fault-slip datum used in the stress tensor inversion techniques is composed of fault orientation, slip orientation and the sense of slip. Sometimes the latter two items are not available. This paper proposes a method to deal with such incomplete fault-slip data. Firstly, the constraints on stress state from full and incomplete data are theoretically specified. The admissible stresses are expressed by the probability distributions in the parameter space. The objective function of inversion is given by superposing the probability distributions for all data including full and incomplete ones. Finally, the peaks of the objective function are detected to enumerate possible solutions. The validity of the present method is tested by simulated data. The incorporation of incomplete data can enhance the detectability of stress states and expands the applicability of stress tensor inversion.     

Probabilistic risk assessment of major accidents: application to offshore blowouts in the Gulf of Mexico

Major accidents are low-frequency, high-consequence accidents which are not well supported by conventional statistical methods due to the scarcity of directly relevant data. Modeling and decomposition techniques such as event tree have been proved as robust alternatives as they facilitate incorporation of partially relevant near accident data–accident precursor data—in probability estimation and risk analysis of major accidents. In this study, we developed a methodology based on event tree and hierarchical Bayesian analysis to establish informative distributions for offshore blowouts using data of near accidents, such as kicks, leaks, and failure of blowout preventers collected from a variety of offshore drilling rigs. These informative distributions can be used as predictive tools to estimate relevant failure probabilities in the future. Further, having a set of near accident data of a drilling rig of interest, the informative distributions can be updated to render case-specific posterior distributions which are of great importance in quantitative risk analysis. To cope with uncertainties, we implemented the methodology in a Markov Chain Monte Carlo framework and applied it to risk assessment of offshore blowouts in the Gulf of Mexico.     

P波作用下饱和土中输流管道动力响应

根据Biot波动理论,采用复变函数与多级坐标法,求解了P波作用下饱和土体中地下输流管道的波动散射方程,分析了管道中流体介质性质、入射波角度及管道埋深等对地下输流管道周边动应力集中系数及孔压集中系数分布的影响。计算结果表明：在低频弹性波入射时,管道周边动应力集中系数及孔压集中系数分布相对均匀,而随入射频率的增加,其分布变得复杂化,但其峰值有所减小;中低频波入射作用下,管道内为水、石油介质时,动应力集中系数和孔压集中系数较空气介质时小,而在高频波作用时情况相反;对于管道内流体介质为水时,随着入射角度的变化,应力集中分布也沿一定角度的方向发生转动,入射波自管道下方垂直入射时,管道周边动应力集中系数峰值相对较大;随埋深的增加,动应力集中系数和孔压集中系数均呈震荡减小的趋势。     

Field stresses around an inclined shaft

In this study, the vertical overburden pressure in the vicinity of an inclined circular underground opening is defined by relation to the geometry of the medium. The stresses around the opening consisting of six components are induced by geostatic field stress. In this regard, the inclined circular opening i.e. shaft changes the original stress condition and an interrupted region develops beneath it. Three zones are defined on a plane which is perpendicular to the axis of shaft. The mathematical expressions of vertical overburden stresses at these three zones are established, respectively. The example given in appendix demonstrates the variation of radial and tangential stresses around the inclined shaft. It is seen that the stress distributions in the third zone, which includes the interrupted region, beneith the shaft display different configurations than that of those obtained by undisturbed field stresses. In the interrupted region the stresses around the shaft linearly grow up due to increasing overburden pressure by radial distance from the periphery of the shaft. At the boundary of interrupted region stresses jump to the original values induced by field stresses. © 1998 John Wiley & Sons, Ltd.