Rock Anisotropy, Fracture and Earthquake Assessment 9783110432510, 9783110440706

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Yong-Gang Li (Ed.) Rock Anisotropy, Fracture and Earthquake Assessment

Rock Anisotropy, Fracture and Earthquake Assessment Edited by Yong-Gang Li

ISBN 978-3-11-044070-6 e-ISBN (PDF) 978-3-11-043251-0 e-ISBN (EPUB) 978-3-11-043253-4 Set-ISBN 978-3-11-043252-7 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2016 Higher Education Press and Walter de Gruyter GmbH, Berlin/Boston Cover image: argus456/iStock/thinkstock Printing and binding: CPI books GmbH, Leck ♾Printed on acid-free paper Printed in Germany www.degruyter.com

Preface This book is the third monograph of the earth science specializing in computational, observational and interpretational seismology and geophysics. It contains six chapters to describe: (1) the principle and theory of seismic wave propagation in anisotropic media, shear-wave splitting analysis, the ray series method for inhomogeneous anisotropic fractured rocks; (2) the bonded discrete element method using a new fracture criterion for reproducing the realistic compressivetensile strength ratio of rocks; (3) rock fracture mechanics in earthquake and its prediction for the earthquake nucleation and stress redistribution ; (4) the multiple linear regression analyses on the relationships between moment magnitude and fault measurements for earthquake hazard assessment; (5) the predictive model of earthquake physics using the pattern informatics algorithm based on statistical mechanics of complex systems; and (6) a fully probabilistic earthquake and tsuinami hazard assessment for Pacific Island Countries with a spatial resolution adequate for local seismic risk studies and building code applications. Each Chapter provides the comprehensive discussion of the state-of-the-art method and technique with their applications in case study. The editor approaches this as a broad interdisciplinary effort, with well-balanced observational, metrological and numerical modeling aspects. Linked with these topics, the book highlights the importance for characterizing the fractured crust that is closely related to earthquake physics. Researchers and graduate students in geosciences will broaden their horizons about advanced methodology and technique applied in seismology, geophysics and earthquake science. This book can be taken as an expand of previous two books in the series, covers multi-disciplinary topics to allow readers to grasp the various methods and skills used in data processing and analysis as well as numerical modeling for structural, physical and mechanical interpretation of geophysical problem and earthquake phenomena. Readers of this book can make full use of the present knowledge and techniques to understand the fractured crustal rocks and serve the reduction of earthquake disasters.

Contents

Rock Anisotropy, Fracture and Earthquake Assessment 1 1.1 1.2 1.2.1 1.2.2 1.2.3 1.3 1.3.1 1.3.2 1.3.3 1.4 1.4.1 1.4.2 1.5 1.5.1 1.5.2 1.5.3 1.5.4 1.5.5 1.6 1.6.1 1.6.2 1.6.3 1.7 1.7.1 1.7.2

Seismic Wave Propagation in Anisotropic Rocks with Applications to Defining Fractures in Earth Crust

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Introduction 12 Elastic Anisotropy of Crustal Rocks 14 Anisotropic Symmetry System 14 18 Transversely Isotropic Medium Anisotropy of Fractured Rock 20 28 Plane Wave Propagation in Homogeneous Anisotropic Medium Phase Velocities of Body Waves in Anisotropic Media 30 39 Group Velocities of Body Waves in Anisotropic Media Body Wave Polarizations 45 Reflection and Refraction of Plane Waves at a Planar Boundary between Anisotropic Media 52 53 Slowness Surface Method Reflection and Transmission Coefficients 67 Ray Tracing in Anisotropic Heterogeneous Media 72 73 Ray Series Method Body-Wave Polarization 78 79 Geometrical Spreading and Ray Amplitudes Specification of a Source and Ray Synthetic Seismogram 82 84 Least-Squares Inverse for Traveltime Ray Series Modeling of Seismic Wave Propagation in 3-D Heterogamous Anisotropic Media 88 The VSP Experiment at Hi Vista and Shear-Wave Splitting Observations 88 Theory 94 101 Traveltime and Amplitude Modeling Results Observation and Modeling of Fault-zone Fracture Seismic Anisotropy 110 110 The Experiment and Data Seismic Wave Traveltimes in a Heterogeneous Anisotropic Medium 114

viii 1.7.3 1.8 1.8.1 1.8.2 1.8.3 1.8.4

2 2.1 2.2 2.2.1 2.2.2 2.3 2.3.1 2.3.2 2.3.3 2.4 2.4.1 2.4.2 2.4.3 2.5 2.5.1 2.5.2 2.6

3 3.1 3.2 3.3 3.4 3.4.1 3.4.2 3.5 3.5.1 3.5.2 3.6 3.6.1

Contents

Polarization of Plane Waves in an Aligned Fracture Anisotropic Medium 118 Shear Wave Splitting Observations and Implications on Stress Regimes in the Los Angeles Basin, California 121 Tectonic Significance and Geological Setting 122 122 The Data and Method Implications from Shear Wave Splitting Observations 126 131 Ray Tracing Acknowledgements 134 134 References

Reproducing the Realistic Compressive-tensile Strength 142 Ratio of Rocks using Discrete Element Model Introduction 143 A Brief Introduction to the ESyS-Particle 146 146 The Equations of Particle Motion Force-displacement Laws and Calculation of Forces and Torques 150 The New Criterion for Bond Breakage Macroscopic Failure Criterion 150 150 Particle Scale Failure Criterion in DEM Model A New Failure Criterion for DEM 151 152 Calibration Procedures Input Microscopic Parameters and the Desired Macroscopic Parameters 153 Sample Generation 154 155 Numerical Set-ups Parametric Studies 157 Elastic Parameters 157 159 Fracture Parameters Discussions and Conclusions 169 170 Acknowledgements References 170

Rock Fracture under Static and Dynamic Stress Introduction 175 Stress Intensity Factor and Stress Field 177 186 Coulomb-Mohr Failure Criterion Energy Release and J-integral 189 189 Energy Release Rate J-integral 189 192 Crack Growth Maximum Hoop Stress Theory 192 196 Strain Energy Density Theory Crack Growth under Dynamic Loading 197 199 Dynamic Crack Propagation in Rock

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3.7 3.7.1 3.7.2 3.8 3.8.1 3.8.2 3.9

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4.1 4.2 4.3 4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.4.5 4.5

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5.1 5.1.1 5.1.2 5.1.3 5.2 5.2.1 5.2.2 5.2.3 5.3 5.3.1 5.3.2 5.3.3

Cohesive Model in Rock Fracture 204 Stress Change in Slip-weakening Model 205 Relationship between Energy Release Rate G and the Parameter in Slip-weakening Model 209 Numeric Method for Fracture Mechanics 210 211 Singularity Element Method Extended Finite Element Method 213 215 Discussion Acknowledgements 216 216 References

Multiple Linear Regression Analyses on the Relationships among Magnitude, Rupture Length, Rupture Width, 221 Rupture Area, and Surface Displacement Introduction 221 Data 223 226 Linear Models and Computational Approach Results 228 Simple Linear Regression Results 228 229 Multiple Linear Regression Results Model Diagnostics 233 Comparison between Multiple Models and Simple Models Model Fits on the Slip Factors 236 237 Concluding Remarks Acknowledgements 238 238 References

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PI Algorithm Applied to the Sichuan-Yunnan Region: A Statistical Physics Method for Intermediate-term Medium-range Earthquake Forecast 240 in Continental China PI Algorithm 241 241 Background and Basic Concepts Algorithm Validation 242 243 Test of the Algorithm: ROC and Beyond The Sichuan-Yunnan Region 244 Seismicity and Earthquake Catalogue 244 245 Tectonic Setting The 2008 Wenchuan Earthquake 245 PI Algorithm Applied to the Sichuan-Yunnan Region Parameter Setting 246 248 Sliding Window Retrospective Test Ergodicity 249

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x 5.4 5.4.1 5.4.2 5.4.3 5.5

6 6.1 6.2 6.2.1 6.2.2 6.3 6.4 6.5 6.6

Contents

Discussion and Development of PI Application 250 PI for Annual Estimate of Seismic Hazard: Useful, or Useless? 250 253 The 2008 Wenchuan Earthquake: Miss, or Hit? Sichuan-Yunnan versus Andaman-Sumatra: Separate, or Connected? 255 Concluding Remarks 258 259 Acknowledgements References 259

Probabilistic Seismic Hazard Assessment for Pacific Island 264 Countries Introduction 264 Data 266 266 Historical Earthquake Catalogs Subduction Segments, Crustal Faults and Geodetic GPS Data Kinematic Modeling Based on GPS and Active Faults Data 272 Modeling the Regional Seismicity Probabilistic Seismic Hazard Maps 276 278 Discussion Acknowledgements 279 279 References

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Rock Anisotropy, Fracture and Earthquake Assessment Yong-Gang Li

This book presents disciplines, methods and techniques for defining fractures in the earth crust, the bonded discrete element method using a new fracture criterion, rock fracture mechanics for analyzing the earthquake nucleation and stress redistribution, the multiple linear regression analyses on the relationships between earthquake moment magnitude and fault measurements for hazard assessment, the pattern informatics model based on the statistical mechanics of complex systems for evaluation of earthquake probability, and a fully probabilistic earthquake hazard assessment for local seismic risk studies and building code applications. Authors from global institutions illuminate multi-disciplinary topics with case studies. All topics in this book help further understanding earthquake physics and hazard assessment in global seismogenic regions. While a large variety of mechanisms may give rise to anisotropy, such as crystal alignment, grain alignment, stress-induced alignment of cracks, and thin sedimentary beds, seismic wave anisotropy is a wide spread phenomenon for fractured rocks in the earth crust. With the increasing resolution of seismic observations using three-component seismometers, rock anisotropy has been widely revealed in the crust and upper mantle. Many observations of seismic anisotropy is related to aligned cracks, since the upper brittle part of the earth crust is pervaded by distribution of cracks which are preferentially aligned by non-lithostatic stress. We are particularly interested in the seismic wave propagation in the anisotropic medium containing aligned cracks because the earthquake process and fluid transport in geothermal and hydrocarbon reservoirs, in nuclear waste repositories and in gas-rich shale sedimentary rocks are closely tied to the presence of fractures, particularly aligned fracture sets in the host rock. The search for fracture related earthquake precursors at crustal depths using seismic methods began since 1970s. Aftermath, shear-wave splitting (SWS) observation in three-component seismograms has been widely used as a direct means to char-

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acterize the population of fractures contained in the host rock. Studies of shearwave propagation through fractured media showed that a small degree of crustal fracture alignment along the wave propagation path would, in principle, induce separation or splitting between shear-waves polarized parallel to faces of the dominant fracture population and shear-waves polarized perpendicular to these fracture faces. Wave polarization anomalies are also found to be sensitive diagnostics of the degree of rock anisotropy, and its symmetry and orientation. In Chapter 1 of this book, we review the principle and theory of seismic wave propagation in an anisotropic medium induced by aligned cracks, introduce the 3-D ray series method with applications for inhomogeneous anisotropic medium, and illustrate observations and modeling of seismic anisotropy in fractured crustal rocks. The method and technique to defining the orientation and density of a population of crustal cracks described in this chapter are helpful for further understanding the damaged structure of fault zones in earth crust and also useful in exploration of geothermal reservoirs and hydro-fracturing shale gas. It is important to investigate fluid-solid interaction problems in engineering and scientific fields. These problems may include fluidized cracks and beds, liquefaction, particle suspension, fluvial erosion, transportation and sedimentation. One of such interaction problems deals with large deformation, or even fracturing of solids and the flow of fluids in the fractures of solids, for example, in petroleum industry where high pressure fluids are injected into boreholes to fracture reservoir rocks to enhance the flow of gas, oil or other fluids. In geophysical researches, there are interests to study the relations between earthquake occurrence and underground water flow or water injection in reservoirs. In the study of tsunami generation and inundation, there is a strong coupling between the movement of water and solid materials. In these problems, the movement of solid materials is accompanied, influenced or even driven by fluid flow of different forms, and flow patterns are strongly affected by the presence and movements of solids. Therefore the two-way solid and liquid coupling is critical in understanding the behavior of such interactions. For solid-fluid coupling problems, various combinations of models for the particle phase and fluid phase can be made depending on the type of problem. The current study introduced in Chapter 2 of this book is mainly focused on the impact of stiffness and fracture parameters on the macroscopic response. The bonded discrete element methods are used to model rock fracture, the compressive to tensile strength ratio and the internal frictional angle, in terms of a new fracture criterion, which is based not on forces in a single bond but on the average stress of neighboring particles and has the form of macroscopic Mohr-Coulomb criterion with explicit tensile cut-off. Elastic and fracture parameters for random packing of particles are then obtained by numerical simulations. It has been successfully utilized to the study of physical process such as rock fracture, stick-slip friction behavior, granular dynamics, heat-flow paradox, localization phenomena, Load-

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Unload Response Ratio (LURR) theory and Critical Point (CP) systems. It is also found in numerical models that the closest packing of particles in 3-D case generates anisotropic elasticity. Mature faults are planes of weakness in the earth crust. They facilitate slip under the prevailing stress orientation to initiate earthquakes. Extensive field and laboratory research, and numerical simulations indicate that the fault undergoes high, fluctuating stress and pervasive cracking during an earthquake. Quantitative studies of earthquakes based on the fault model in the past decades took kinematic and dynamic approaches. The kinematic approach is basically solving an inverse problem in which we determine the fault slip function in space and time from observed seismic records by means of the elasto-dynamic representation theorem; kinematic model parameters were also interpreted in terms of fracture mechanics. On the other hand, the dynamic approach attempts to predict the fault slip function based on a given distribution of rupture strength over the fault plane and the loading stress condition by means of the principle rupture mechanics. The two approaches are now combined to produce the distribution of stress drop and that of fracture strength over the fault plane. For all these models, knowledge of fracture mechanics is an important tool in improving our understanding of the mechanical performance during fault rupture in an earthquake. It applies the stress and strain on the microscopic crystallographic defects in real rocks to predict the macroscopic mechanical failure of the rock. The prediction of crack growth is a main task of the damage tolerance discipline. In Chapter 3 of this book, the propagation of cracks in rock materials during an earthquake is studied in detail. Fracture mechanics is introduced to earthquake and becomes an effective method in analyzing the earthquake development and nucleation and redistribution of stress induced by an earthquake. Other three Chapters (4, 5 and 6) of this book focus on the newly developed methods and technique for forecast and assessment of earthquakes in the fractured earth crust. Estimating maximum magnitudes of future earthquakes in a particular region is extremely important for evaluating potential earthquake hazard, as well as for disaster prevention and reduction. The relations between earthquake magnitude and observed fault measurements (the surface rupture length, subsurface rupture length, rupture width, rupture area, maximum displacement at surface and average displacement at depth) have been carefully analyzed using simple linear regression. However, there still remain issues unsolved, such as how many and which predictors are necessary for improving the estimation of the earthquake magnitude. A more reliable multiple linear regression analyses on the relationships between magnitude and fault measurements are presented in Chapter 4. Suffering from intense earthquake disasters, seismological community has being promoted the study of earthquake forecast and developed forecast schemes. The pattern informatics (PI) algorithm, a predictive algorithm for the analy-

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sis of seismic activity based on the statistical physics of complex systems, has been successfully applied to many seismogenic regions. The PI algorithm assumes that seismogenic dynamics can be regarded as a ‘threshold system’ driven by persistent forces or currents. By analyzing the fluctuations of seismicity, the PI algorithm estimates the increase of the probability of earthquakes at an intermediate-term time scale. Chapter 5 summarizes the results of this innovative method applied for seismogenic regions of Sichuan-Yunnan in China and Andaman-Sumatra in Indonesia. One component of the main objective for earthquake hazard assessment is to create a uniform and detailed regional seismic hazard model that captures the spatial variation of the hazard and could be used to support realistic estimation of the earthquake ground shaking risk across the region. A fully probabilistic earthquake hazard assessment carried out for fifteen Pacific Island Countries (PICs) is presented in Chapter 6. Based on historical and instrumental earthquake catalogs, subduction zone segmentation and plate motion information, geodetic data, and available data on crustal faults, a regional seismicity model was built by using different ground motion prediction equations for different types of earthquakes. Then the seismic hazard maps are developed and tested with a spatial resolution adequate for local seismic risk studies and building code applications. This study also includes modeling of earthquake-induced tsunami hazard. This book includes six Chapters. They are introduced as below. Chapter 1: “Seismic Wave Propagation in Anisotropic Rocks with Applications to Defining Fractures in Earth Crust” by Yong-Gang Li. In this Chapter, the author reviews explicit formulas derived from different elastic wave propagation theories and used in direct calculation of wave velocities in anisotropic medium, and introduces the ray theory for anisotropic media and the 3-D ray tracing system of equations applicable to the heterogeneous crackinduced anisotropic media. The most general form of rock anisotropy with 21 independent elastic constants is discussed. In this case, the Christoffel matrix is a 3 × 3 symmetric positive-definite matrix formed by elastic parameters and components of slowness vector. It has three real eigenvalues Gm with mutually orthogonal eigenvectors, corresponding to phase velocities and polarization vectors of three types of body-waves, respectively. Using matrix techniques to define elastic anisotropy of rocks containing aligned cracks greatly simplifies the analysis of anisotropic features of seismic wave propagation. In the computer procedure, only a subroutine is required to rotate the elastic tensor into the desired configuration while the main program remains independent of direction of wave propagation and type of anisotropy symmetry system. If there are two or more sets of cracks aligned in different directions, we can obtain the overall correction of elastic constants by calculating for each set and then adding them together.

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Velocities of body waves propagating in an anisotropic medium can be obtained from Kelvin-Christoffel equations and its characteristic equation associated with the Christoffel tensor. The computation of reflection and transmission coefficients at an interface between two anisotropic media is determined using a numerical technique called slowness surface method. The ray tracing system of equations applicable either to the isotropic or anisotropic medium is introduced. The ray tracing algorithm with inverse for computation of traveltimes and vector ray amplitudes in layered inhomogeneous anisotropic media is then developed and used for interpretation of the fracture structures in 3-D using the VSP data recorded across the fault zone associated with the M 5.7 earthquake at Oroville in Northern California and near the San Andreas fault at Hi Vista of Mojave Desert, and the data recorded at the seismic network in Los Angeles Basin in Southern California. Results from these investigations show that the ˇ Cerven´ y ray theoretic traveltime and amplitude computation is an accurate and robust technique for investigating the properties of low to moderate anisotropic and heterogeneous crustal rocks as well as the stress status in the region. Chapter 2: “Reproducing the Realistic Compressive-tensile Strength Ratio of Rocks using Discrete Element Model” by Yucang Wang and William W. Guo. Authors of this Chapter present the bonded discrete element methods used to model rock fracture, the compressive to tensile strength ratio and the internal frictional angle, in terms of a new fracture criterion with tensile cut-off. To overcome the difficulties in conventional DEM model, the new criterion based on averaging the stresses of a group of particles has been proposed: if the average stress of two particles satisfies the truncated Mohr-Coulomb criterion, the bond between particles can break. The new criterion is not based on forces in a single bond, but based on the stress obtained by averaging the stresses in the neighbor particles surrounding the particle in question. The results show that the new failure criterion leads to a better agreement with the known experimental results. In this Chapter, authors give a brief introduction to the ESyS-Particle, an open source DEM software developed at the University of Queensland, Australia. It was designed to provide a basis to study the physics of rocks and the nonlinear dynamics of earthquakes. The ESyS-Particle includes three fracture parameters: particle scale tensile strength, particle scale friction angle and tensile cut-off ratio. The major features that distinguish the ESyS-Particle from existing DEMs are the explicit representation of particle orientations using unit quaternion, complete interactions (six kinds of independent relative movements are transmitted between two 3-D interacting particles) and a new way of decomposing the relative rotations between two rigid bodies such that the torques and forces caused by such relative rotations can be uniquely determined. The current study is mainly focused on the impact of stiffness and fracture parameters on the macroscopic response. Once the samples are generated, uni-axial compressive tests are carried out to calculate the unconfined compressive strength,

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and the Brazilian and direct tensile tests are performed to evaluate the tensile strength. Elastic and fracture parameters for random packing of particles are then obtained by numerical simulations. The ESyS-Particle has been successfully utilized in study of physical process such as rock fracture, stick-slip friction behavior, granular dynamics, heat-flow paradox, localization phenomena, LoadUnload Response Ratio theory and Critical Point systems. It is also found in numerical models that the closest packing of particles in 3-D case generates anisotropic elasticity. For example, Face-Centered Cubic (FCC) lattice yields cubic elasticity, the simplest case for an orthotropic solid. Based on this research, the particle scale stiffness can be easily calibrated according to the given Young’s modulus and Poisson ratio in the case of regular packing. Chapter 3: “Rock Fracture under Static and Dynamic Stress” by Jiming Kang, Zheming Zhu, and Po Chen. This chapter presents the application of rock fracture mechanics in earthquake and its failure prediction. At first, a general theory is given, such as the three kinds of crack modes, the stress field near the crack tip and energy release rate, etc. Then, Authors introduce some recent applications of rock fracture mechanics and discuss some challenging issues. Because the stress distribution in the earth crust is complicated, it is still hard to get a whole picture of the energy distribution in an earthquake. Authors use the J-integral, a way to calculate the strain energy release rate, or work energy per unit fracture surface area, in the material.The J-integral is equal to the strain energy release rate for a crack in the rock subjected to monotonic loading. This is generally true only for linear elastic materials under quasi-static conditions. For rocks that experience small-scale yielding at the crack tip, J can be used to compute the energy release rate under special circumstances such as monotonic loading in Model III (antiplane shear). The strain energy release rate can also be computed from J for pure power-law hardening plastic materials that undergo small-scale yielding at the crack tip. The quantity J is path-dependent for monotonic mode I and mode II loading of elastic-plastic materials, so only a contour very close to the crack tip gives the energy release rate. J is path-independent in plastic materials when there is no non-proportional loading that is the reason for the path-dependence for the in-plane loading modes on elastic-plastic materials. Accordingly, authors provide detailed formulas used in maximum Hoop stress theory, strain energy density theory, crack growth under dynamic loading and dynamic crack propagation in rock. They further discuss the cohesive model in rock fracture, stress change in the slip-weakening model, and the relationship between energy release range and the parameter in slip-weakening model. Finally, the numeric methods for fracture mechanics, such as the singularity element method and extended finite element method, are developed to overcome difficulties in meshing and re-meshing within the crack tip region that contains the stress concentration. Therefore, these methods with fracture mechanics are

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applicable to understand the causes of failures and also verify the theoretical failure predictions with real earthquake failures. Chapter 4: “Multiple Linear Regression Analyses on the Relationships among Magnitude, Rupture Length, Rupture Width, Rupture Area, and Surface Displacement” by Annie Chu and Jiancang Zhuang. In this chapter, authors present the multiple linear regression analysis method to explain the relationship between earthquake moment magnitude and fault measurements (the surface rupture length, subsurface rupture length, rupture width, rupture area, maximum displacement at surface and average displacement at surface) reliably. Based on the highly linear correlations between pairs of the variables, the transformation method is applied to the six quantitative regressors, and then fit linear models between moment magnitude and the regressors. Akaike Information Criterion (AIC) is used as a model selection criterion. A model diagnosis is delivered by normal quantile-quantile plots and residual-onfit plots to verify the assumption that the errors follow normal distribution with mean 0 and constant variance. The normal QQ-plots appear approximately linear, and this phenomenon supports reasonably good fit to the model with no violation of normality. Other tests that they have implemented, such as examining Cook’s distance and leverage, show evidences that no outlier raises concerns. Through testing different models, authors show that these factors associated with a fault are neither independent nor equally important when the moment magnitude is estimated. Some of these factors may be eliminated when constructing a multiple linear regression model. The maximum displacement and rupture area provide adequate information to construct a very good model. When the maximum displacement is not available, the rupture surface and rupture width are the two predictors that provide the best model alternatively. Besides the quantitative predictors, their one-way analysis of variance (ANOVA) approach by adopting slip type and direction reveals that these factors are insignificant when predicting moment magnitude possibly due to two reasons: (1) The energy released by earthquakes is more concentrated on the asperity part or the locked part of the faults where the maximum displacement usually occur while the movement on the other part of the fault is more passive; (2) The errors for the estimate of the average displacement are much bigger than the maximum displacement. Chapter 5: “PI Algorithm Applied to the Sichuan-Yunnan Region: A Statistical Physics Method for Intermediate-term Medium-range Earthquake Forecast in Continental China” by Changsheng Jiang, John B. Rundle, Zhongliang Wu, and Yongxian Zhang. In the 2nd volume of this book series, Jiang and Wu have demonstrated some of useful tactics in analysis of earthquake catalog data and make notes on the existing methods used for analysis of seismicity, such as the Eclipse method and the Bayesian information criterion (BIC) and the de-clustered Benioff strain method.

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They use an ‘eclipse method’ based on the concept of modern astronomy for analyzing remote planets to screen out the seismicity in the neighboring active fault zones, for example, in analysis of seismicity for the 2008 MS 8.0 Wenchuan earthquake catalog data. The BIC consideration provides a useful aid to judge whether the apparent ‘accelerating’ trend is statistically significant. In this chapter, the authors introduce the pattern informatics (PI) model which is one of the recently developed predictive models of earthquake physics based on the statistical mechanics of complex systems. They demonstrate the retrospective forecast test of the PI model conducted for the earthquakes in Sichuan-Yunnan region and explore the possibility to apply this method to reverse tracing the increasing probability of strong earthquakes. For earthquakes in Sichuan-Yunnan region, they investigated the stability of the PI algorithm against the selection of model parameters. They adjusted the parameters systematically and tested the effect of such parameter variation. As a result, optimizing parameters were selected for the ‘target magnitude’ of MS 5.5 : a fifteen-year long ‘sliding time window’; the ‘anomaly training time window’ and ‘forecast time window’ both being 5 years; the spatial grid taken as D = 0.2◦ and only shallow earthquakes with depth ranging from 0 to 70 km are considered. Jiang and Wu (2010) also applied the same parameter settings to do retrospective case study for the 2008 Wenchuan MS 8.0 earthquake. Results from the ergodicity test for the SichuanYunnan region indicate that the PI algorithm is valid for this region in the time period under consideration. For different grid sizes and different cutoff magnitude values, the ergodicity test shows that the seismicity in Sichuan-Yunnan region has had a strong ergodicity since 1978, and the PI algorithm has been able to be used for the estimation of time-dependent earthquake rates. Chapter 6: “Probabilistic Seismic Hazard Assessment for Pacific Island Countries” by Y. Rong, J. Park, D. Duggan, M. Mahdyiar, and P. Bazzurro. In this Chapter, authors present a fully probabilistic earthquake hazard assessment study carried out for fifteen Pacific Island Countries (PICs). A regional seismicity model was built based on historical and instrumental earthquake catalogs, subduction zone segmentation and plate motion information, geodetic data, and available data on crustal faults. They used different ground motion prediction equations to account for different types of earthquakes. The effect of site conditions on ground motion was modeled based on shear wave velocities derived from microzonation studies and high-resolution topographic slope data. A comparison of their findings with those of earlier studies, such as the Globle Seismic Hazard Assessment Program (GSHAP), shows similarities, and in some cases, significant differences. The seismic hazard maps developed here have a spatial resolution that is adequate for local seismic risk studies and building code applications.

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The purpose of this book is to introduce the sophisticated approaches in solidearth geophysics research with case studies. The following methods and results presented in this book will be of particular interest to the readers: – Shear-wave slitting analysis, the 3-D ray series method for inhomogeneous anisotropic fractured rocks; – The bonded discrete element method using a new fracture criterion with tensile cut-off and non-spherical particles for rock fracture; – Rock fracture mechanics for analyzing the earthquake nucleation and stress redistribution; – The multiple linear regression analyses on the relationship between moment magnitude and fault measurements for hazard assessment; – The pattern informatics algorithm and reverse tracing earthquake probability; – A fully probabilistic earthquake hazard assessment for fifteen Pacific Island Countries. This book is a self-contained volume starting with an overview of the subject and then exploring each topic in depth detail. Extensive reference lists and cross references with other volumes could facilitate further research. The content is suitable for both the senior researchers and graduate students in geosciences who will broaden their horizons about observational, computational and applied seismology and geophysics. This book covers multi-disciplinary topics to allow readers to gasp the methods and techniques used in data analysis and numerical modeling for structural, physical and mechanical interpretation of geophysical and earthquake phenomena to aid the understanding of earthquake processes and hazards. The editor of this book series wishes to thank reviewers: C. Jayasundara, F. Alonso-Marroquin, B. Shen-Tu, K. Shabestari, S.Y. Zhou, C.C. Chen, J.C. Zhuang, R.S. Wu, Y.L. Chen, Z.M. Zhu, and X. Liu. We are grateful to many organizations and individuals, including HEP Director Bingxiang Li and Editor Yan Guan who help make our books possible. Key words: Anisotropic media, Shear-wave splitting, Ray-tracing and VSP in fractured rocks, Bonded discrete element model for rock fracture, Elastic and fracture parameters, Fracture mechanics, Strain energy density, Crack growth, Slip-weakening, Finite element method, Linear regression analyses, Moment magnitude, Fault measurements, Slip history and distribution. Pattern informatics, Earthquake probability, Probabilistic earthquake hazard assessment.

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Author Information Yong-Gang Li Research Professor at University of Southern California, Los Angeles, CA 90089, USA E-mail: [email protected]

Chapter 1

Seismic Wave Propagation in Anisotropic Rocks with Applications to Defining Fractures in Earth Crust Yong-Gang Li

Abundant evidence of rock anisotropy has been observed in most earth materials, at least in the upper crust, using three-component seismometers in microearthquake networks, borehole vertical seismic profiling (VSP), and reflection seismic exploration. A fundamental observation of seismic anisotropy is shearwave splitting, in which shear waves polarized in one direction travel faster than orthogonally polarized shear waves in the anisotropic medium. With the increasing resolution of seismometer in recent decades, wave polarization anomalies and directional velocity variations of three distinct body waves referred to as qP , qSV and qSH have been recorded in anisotropic media. Two shear waves also show the difference in attenuation when they travel in the anisotropic medium. These observations of seismic wave propagation anisotropy are becoming increasingly useful in stratigraphic and lithological interpretations. In this chapter, we review the principle and theory of seismic wave propagation in an anisotropic medium induced by aligned cracks, the 3-D ray series method with applications for inhomogeneous anisotropic medium, and observations and modeling of seismic anisotropy in fractured crustal rocks, part of which have been introduced in our previously published papers. The method and technique defining the orientation and density of a population of crustal cracks described in this chapter are helpful for further understanding the damaged structure in the earth crust and also useful in exploration of geothermal reservoirs and hydro-fracturing shale gas. Key words: Seismic wave polarization, Shear-wave splitting, Anisotropic medium, Crack-induced anisotropy, Ray tracing, Vertical seismic profiling.

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1.1

Yong-Gang Li

Introduction

Fractures in the crustal rock play an important role in various geophysical phenomena. The earthquake process and fluid transport in geothermal and hydrocarbon reservoirs, in nuclear waste repositories and in gas-rich shale sedimentary rocks are closely tied to the presence of fractures, particularly aligned fracture sets in the host rock. The search for fracture related earthquake precursors at crustal depths using seismic methods has begun since 1970s (e.g., Aki et al., 1970; Nur, 1971; Anderson and Whitcomb, 1975; Kanamori and Hadley, 1975; Bell and Nur, 1978; Crampin, 1978). Aftermath, shear-wave splitting (SWS) observation in three-component seismograms has been widely used as a direct means to characterize the population of fractures contained in the host rock (e.g., Steward et al., 1981; Crampin et al., 1985, 1986, 1991, 1994, 1999, 2003; Booth and Crampin, 1985; Peacock, 1985; Shearer and Orcutt, 1985, 1986; Leary et al., 1987; Li et al., 1987, 1988, 1990, 1994, 1996) Shih and Mejer, 1990; Daley and McEvilly, 1990; Aster and Shearer, 1992; Gao et al., 1995; Audoine et al., 2000; Cochran et al., 2003, 2006; Balfour et al., 2005). Studies of shear-wave propagation through fractured media showed that a small degree of crustal fracture alignment along the wave propagation path would, in principle, induce separation or splitting between shear-waves polarized parallel to faces of the dominant fracture population and shear-waves polarized perpendicular to these fracture faces (e.g., Hudson, 1980; Crampin, 1981, 1984a, 1986; Crampin and Booth, 1985; Petrashen and Kashtan, 1984). Wave polarization anomalies are also found to be sensitive diagnostics of the degree of rock anisotropy, and its symmetry and orientation (e.g., Long and Witherspoon, 1985; Majer et al., 1988). In this chapter, we begin with shear-wave splitting in the medium containing aligned fractures by a matrix presentation of elastic constants for some common anisotropic systems of seismological interest, with higher degrees of anisotropic symmetry, and give direct calculation of wave velocities in the transversely isotropic medium in terms of explicit formulas derived from different elastic wave propagation theories (e.g., Eshelby, 1957; Synge, 1957; Anderson et al., 1974; Garbin and Knopoff, 1973, 1975a, b; O’Connell and Budiansky, 1974; Crampin, 1978, 1984b, 1985; Levin, 1979; Hudson, 1981, 1982; Amadei, 1983). We compare the results from these formulas and use Hudson’s formulas with the first-order and second-order corrections of elastic constants due to aligned cracks to define elastic anisotropy for the medium containing aligned cracks. We then introduce the plane wave propagation in a homogeneous anisotropic medium and show distinct characteristics of seismic anisotropy including the directional phase velocities, the dispersion of group velocities and wave polarization deviations from the phase propagation direction or the wave front in anisotropic

1

Seismic Wave Propagation in Anisotropic Rocks. . .

13

media (e.g., Hanyga, 1984a, b, c; Crampin, 1984a, b; Fryer and Frazer, 1987). These characteristics are understood through examples for anisotropic medium with orthorhombic and hexagonal symmetry. In the calculation of phase velocities, group velocities and polarizations of three wave types qP, qSV and qSH, we use either analytic or numerical methods, which are associated with eigensolutions of the Christoffel elastic tensor Γ of the specific material. In the next step, we treat the discontinuity problem of elastic constants at the boundary between two anisotropic media. This formidable problem must be solved numerically for anisotropic media. We use the slowness surface method (e.g., Fedorov, 1968; Daley and Hron, 1977, 1979; Thomsen, 1988) to synthesize slowness surface of six types of waves: reflected P, SV, SH and transmitted P, SV and SH waves corresponding to a given incident wave at the plane interface between two anisotropic media. The directional phase velocities and emergent angles of six wave types are numerically obtained in a root-finding procedure. Six unknown reflection and transmission coefficients are then solved from a system of six simultaneous equations established by boundary conditions of displacement and stress. This numerical method for computation of reflection and transmission coefficients can be extended to the case of a locally smooth interface between two weak anisotropic media (Li, 1988). We then review the ray theory for anisotropic media and the ray tracing system of equations applicable to the heterogeneous anisotropic medium derived in ˇ ˇ ˇ published papers (e.g., Cerven´ y, 1972; Cerven´ y and Pˇsenˇcik, 1972, 1979; Cerven´ y ˇ ˇ et al., 1977; Cerven´ y and Horn, 1980; Cerven´ y and Firbas, 1984; Hanyga, 1982, 1984a, b, c, 1986; Shearer and Chapman, 1989; Chapman and Shearer, 1989; Babich, 1994). The first-order and second-order differential equations could be solved by Runge-Kutta integration with appropriate initial values associated with seismic source. Based on the knowledge of seismic anisotropy, we have developed a 3-D ray tracing computer procedure usable to calculate traveltimes, raypaths, wave polarizations and ray amplitudes of qP , qSV and qSH waves propagating in a layered anisotropic media with smooth interface (Li et al., 1990). The elastic constants of the host medium are vertically and laterally inhomogeneous, including a distribution of cracks aligned with their crack normal in an arbitrary direction. This computer program is useful to define the population of fractures in a particular region of the earth in terms of modeling observed traveltimes and amplitudes. The model parameters may be further improved by a least-squares inversion. Finally, we illustrate applications of the 3-D ray tracing computer procedure for anisotropic heterogeneous media to model the distribution of cracks in the basement rock near the San Andreas Fault at the Hi Vista borehole site in Mojave Desert, Southern California (Li et al., 1990). We quantitatively interpreted observations of shear-wave splitting and polarization anomalies and found a set of vertical cracks permeate in the crystalline basement rock at Hi Vista borehole

14

Yong-Gang Li

site. The crack density is ∼0.035 and cracks are preferentially aligned N30◦ W, consistent with the direction of the in-situ maximum horizontal principal stress measured by Zoback et al. (1980). We applied the ray tracing method to defining crack density and stress status in the upper crust at Los Angeles basin using three-component data recorded at network stations (Li et al., 1994; Li 1996). The method and technique introduced in this chapter are also useful for defining crustal fracture orientation and density of oil and gas traps and geothermal reservoirs.

1.2

Elastic Anisotropy of Crustal Rocks

Rock anisotropy is a rather common phenomenon caused by a variety of mechanisms, such as regular sequence of fine layers, crystal alignments, stress-induced lithological alignments, hydraulic and plastic flows, seafloor spreading, and most commonly aligned fractures and cracks in crustal rocks. Using matrix techniques to define elastic anisotropy of rock greatly simplifies the analysis of seismic wave propagation anisotropy in such rock. The principle which will be the basis for formulations of anisotropy described in this section is the restriction of wave propagation in a fixed coordinate direction in the full representation of the fourth-order tensor of elastic constants cijkl (i, j, k, l = 1, 2, 3). When a new direction is required, the tensor is rotated into another full fourth-order tensor. This treatment greatly simplifies both analytical expressions in terms of Kelvin-Christoffel equations (Christoffel, 1887; Musgrave, 1970; Auld, 1973) and numerical computations (e.g., Crampin, 1981; Fryer and Frazer, 1987) at no loss of generality. In the computer procedure, only a subroutine is required to rotate the elastic tensor into the desired configuration while the main program remains independent of direction of wave propagation and type of anisotropy symmetry system (e.g., Li, 1988).

1.2.1

Anisotropic Symmetry System

If we assume that the anisotropic rock can be described as linear, elastic, homogeneous and continuous, we obtain a general constitutive relationship between stress τ and strain e, ∂uk (1.1) τij = cijkl ekl = cijkl ∂xl

1

Seismic Wave Propagation in Anisotropic Rocks. . .

15

where uk is a component in k direction of the displacement vector u. In the general 3-D case, the fourth-order tensor of elastic constants has 81 components. However, due to the symmetry of both stress and strain tensors, only six of nine constitutive equations in (1.1) are independent. Therefore, only 36 distinct constants are required to describe an arbitrary anisotropy (Amadei, 1983). It allows replacing the fourth-order tensor of elastic constants cijkl by a 6×6 matrix crs (r, s = 1, 2, 3),   c11 c12 c13 c14 c15 c16  c  21 c22 c23 c24 c25 c26     c31 c32 c33 c34 c35 c36  (1.2)    c41 c42 c43 c44 c45 c46     c51 c52 c53 c54 c55 c56  c61 c62 c63 c64 c65 c66 If a strain energy function is presumed to exist, the number of independent elastic constants can be further reduced to 21 (Lekhnitskii, 1963). In this case, the constitutive equation remains unchanged after a symmetry transformation, cij = cji

(1.3)

21 distinct elastic constants are usually enough to describe an elastic system for the most general form of rock anisotropy. In fact, anisotropy systems with fewer than 21 elastic constants occur most commonly in the earth rock as described by Crampin (1984a, b). Such symmetry systems include the monoclinic with 13, orthorhombic with 9, tetragonal and trigon with 6, hexagonal with 5, and cubic anisotropy with 3 independent elastic constants. Most anisotropic rocks of interest to seismological study have some degree of elastic symmetry with, at least, a symmetry plane which is a plane if a point transformation, such as rotation, reflection, or inversion does not change the values of elastic constants. Such symmetry planes have two major effects on body waves when the wave propagation direction lies in these planes: (1) the polarization of Pwave and SV-wave is parallel to the symmetry plane but the polarization of SH-wave is perpendicular to the symmetry plane, and (2) P and SV motions are decoupled with SH motion. We shall limit our discussion to such classes of anisotropic systems with higher degrees of rock elastic symmetry and use the most convenient Cartesian coordinates in this study. One example of seismological interest is a structure with multiple populations of vertical cracks in different horizontal alignment orientations. This form of anisotropy is probably extremely common in the shallow crust where cracks pervade, spatially in the vicinity of fault zones (e.g., Crampin and Booth, 1985; Leary et al., 1987; Li et al., 1987, 1988, 1994, 1996; Cochran et al., 2003, 2006; Zhang et al., 2007). 13 independent non-zero elastic constants are required for defining the monoclinic anisotropic system with a single horizontal plane

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Yong-Gang Li

of elastic symmetry, for example, (x1 , x2 , x3 ),  c11 c12   c12 c22  c  13 c23   0 0   0  0 c16

c26

a x1 − x3 plane in the Cartesian coordinates c13

0

0

c23

0

0

c33

0

0

0

c44

c45

0

c45

c55

c36

0

0

c16



 c26   c36    0    0 

(1.4)

c66

Because most rocks of seismological interest have more than one plane of elastic symmetry with a higher symmetry than the monoclinic system, they may be described by fewer elastic constants than those of the monoclinic symmetry. For example, the orthorhombic symmetry with three mutually orthogonal planes of elastic symmetry is a common anisotropic system seen in the deep crust and upper mantle. Its tensor of elastic constants has nine independent elements and may be written as follows if the planes of elastic symmetry are parallel to the coordinate planes.    c11 c12 c13     c12 c22 c23   0     c    13 c23 c33   (1.5)    0 c44 0      0  0 c55 0    0 0 c66

This type of symmetry results from crystallization of the principal rock-forming minerals, such as olivine and orthopyroxene taking on a preferred orientation in response to tectonic stress in the deep crust and upper mantle (Ando et al., 1980; Christensen, 1984, 2002). Such symmetry is also found in ophiolites with a preferable direction parallel to seafloor spreading at the time of formation (Bibee and Shor, 1976; Stephen, 1981, 1985). Hexagonal symmetry is a more highly symmetric variant of orthorhombic system, with six-fold rotationable symmetry about a single axis, around which there are infinite planes of elastic symmetry. It is in fact difficult to distinguish the difference between the hexagonal and orthorhombic symmetry when an aggregate of the orthorhombic system occurs along one axis and the other two axes are more or less randomly distributed. When the symmetry axis is vertical along the x3 axis, a hexagonal system exhibits no azimuthal variation of elastic properties in the horizontal plane. Such a particular symmetry is also called transverse

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Seismic Wave Propagation in Anisotropic Rocks. . .

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isotropy and there are only five independent elastic constants in the 6 × 6 elastic tensor matrix,    c11 c12 c13  c  0 12 c11 c13       c13 c13 c33   (1.6)    0 c44 0    0 c44 0    0 0 0 c66

where c66 = 21 (c11 − c12 ). Transversely isotropic symmetry has been demonstrates for the earth rock and discussed by many authors (e.g., Levin, 1979; Crampin, 1986). This symmetry has received a great deal of attention in exploration seismology (Roberson and Corrigan, 1983; Helbig, 1984; Lyakhovitskiy, 1984) because this is the most common anisotropy encountered in the sedimentary rock and shale. For seismic wavelengths larger than the thicknesses of sedimentary beds, the regular sequences of thin layer are equivalent to a transversely isotropic solid, through which waves average the physical properties of thin layers. This is called the periodic thin-layered (PTL) anisotropy (Crampin, 1984a). The strong lithological transverse isotropy associated with calcareous shale and chalks was revealed in Tertiary and Cretaceous sediments using VSP (Gaiser et al., 1984). Phyllosilicates, calcite and quartz minerals also exhibit significant elastic anisotropy and grain alignments. These results suggest that mineralogy and grain orientation play a significant role in the degree of transverse isotropy (Daley and Horn, 1977). With a horizontal symmetry axis, hexagonal symmetry system has its elastic constant tensor by rotating (1.6) to a form in which the symmetry axis is aligned with the x1 coordinate    c33 c13 c13 c   0 13 c11 c12      c13 c12 c11    (1.7)    0 c66 0    0 c44 0    0 0 0 c44

Hexagonal symmetry is often seen in the shallow oceanic crust where thermal contraction of newly formed oceanic lithosphere results in a profound cracking in the vicinity of upwelling magma (Anderson et al., 1974). Cracks in the vicinity of the ridge have a preferred orientation parallel to the rise axis (Stephen, 1981, 1985). This form of symmetry also exists in the continental crust, with a preferable orientation controlled by the principal stresses (e.g., Nur, 1971; Li et al., 1994; Cochran et al., 2003, 2006). Igneous rocks containing micro-cracks

18

Yong-Gang Li

are frequently aligned by stress (e.g., Babuska and Pros, 1984). Sedimentary rocks with joints and fractures often possess consistent alignments over large areas (e.g., Engelder, 1982; Li, 1996). Stress-corrosion may cause subcritical crack growth in tensile fractures parallel to the maximum compressive stress (Crampin et al., 2003). The stress-related alignment of the average crustal fractures was detected in the crystalline rock of the flanks of the greater strike-slip stress province of the San Andreas Fault system in the VSP survey (Li et al., 1988, 1990). In the geothermal area, the aligned fractures are found to be associated with the overall hydrological characteristics, such as in the geysers geothermal field, northern California (Majer et al., 1988). In seismically active regimes, such as the active normal fault zone at Cleveland Hill, Oroville California, a dense distribution of cracks aligned parallel to the fault plane within the fault zone was quantitatively defined using tomographic VSP (Leary et al., 1987; Li et al., 1987).

1.2.2

Transversely Isotropic Medium

Elastic waves in an isotropic solid travel with velocities that are independent of direction of propagation. In the anisotropic medium, however, velocity depends on travel direction. Velocities of body waves in an arbitrary direction through an anisotropic rock cannot in general be written explicitly, but may be numerically solved as an eigenvalue problem for the slowness equation (1.23) (refer to Section 1.3 for details). For the transversely isotropic medium with higher degree of symmetry usually with a xs -axis normal to the free surface (this problem is often encountered in the exploration seismology), Levin (1979) gave convenient expressions for body-wave velocities based on the derivation of plane-wave surfaces for transversely isotropic media, 2 = Vp,sv

1 [2L + (A − L) sin2 θ + (C − L) cos2 θ ± 2ρ ([(A − L) sin2 θ + (C − L) cos2 θ]2 +

1

[(F + L)2 − (A − L)(C − L)] sin2 2θ) 2 ] 2 = Vsh

1 (N sin2 θ + L cos2 θ) ρ

(1.8) (1.8′ )

where A, C, F, L and N are five independent elastic constants in (1.6). A = c11 , C = c13 , F = c33 , L = c44 and N = c66 = 21 (c11 − c12 ). θ is the wave propagation direction with respect to the vertical symmetry axis of anisotropy. The plus sign in equation (1.8) corresponds to the P-wave and the minus sign to the SV-wave.

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Seismic Wave Propagation in Anisotropic Rocks. . .

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For example, we computed velocities of P, SV and SH waves for Model I, which has elastic constants as follows,    10.2 5.58 7.02   5.58 10.2 7.02   0      7.02 7.02 4.95      1.37 0.00 0.00     0.00 1.37 0.00    0 0.00 0.00 2.31

and the rock density 2.0 g/cm3 . Those values were estimated directly from the observation of shear wave traveltime data taken from the VSP experiment at Geary borehole near Oklahoma City (Roberson and Corrigan, 1983). The surface formation at the site is Dog Crack shale, which is a firm, reddish-brown shale of Permian age. Calculated velocity curves in Figure 1.1 are characteristic of anisotropy of wave propagation: (1) velocity variations dependent on the wave propagation direction and (2) the difference in velocity between two shear-waves. Three types of waves, P, SV and SH, travel along the vertical symmetry axis at their lowest velocities: 1875, 828, 828 m/s, respectively. In this case, � � c13 c44 ◦ , Vsv = Vsh = θ = 0 , Vp = ρ ρ

Fig. 1.1 Computed phase velocities of P-, SV- and SH-waves in Model I, Dog Creek shale of Permian age with transversely isotropic symmetry, using Levin’s equations (1.8) and (1.8′ ).

Two shear wave velocity surfaces come into tangential contact (called kiss singularity) in this direction. When the wave direction moves away from the vertical, they separate, but meets again with the same velocity 930 m/s as the

20

Yong-Gang Li

wave propagation direction is at an angle 40◦ to the vertical (called intersection singularity). When waves propagate horizontally,    c11 c44 c66 ◦ , Vsv = , Vsh = θ = 90 , Vp = ρ ρ ρ P-wave has fastest velocity of 2258 m/s in this direction. Two shear waves have the maximum difference between their velocities: 1075 m/s for the SHwave and 828 m/s for the SV-wave. In addition to the directional variations of phase velocities, seismic waves have abnormal particle motion direction and the group velocity dispersion in anisotropic media. We shall discuss the details in the following section.

1.2.3

Anisotropy of Fractured Rock

In this section, we review various methods for calculation of body-wave velocities in the anisotropic rock containing a population of cracks. We compare the results obtained by the methods developed by Garbin and Knopoff (1975a), Crampin et al. (1980) and Hudson (1981) for the elastic constants of rock permeated with cracks. The fractures, including both of microscopic and macroscopic cracks can be caused by tectonic stress and deformation, temperature gradient, cooling, recrystallization and layering. Cracks are usually aligned with a preferred orientation (Simmons and Nur, 1968; Nur, 1971; Simmons and Richter, 1976; Crampin and Booth, 1985). Earlier theoretical analyses have applied only to dilute concentration of cracks, in which cracks are assumed to be sufficiently far apart to permit the independent evaluation of the effect of each crack on properties of unfractured rock (Eshelby, 1957; Walsh, 1965; Wu, 1966). Nur and Simmons (1969) pointed out that flat cracks might be a common feature of rocks in the crust since a few kilo-bars are enough to close cracks with an aperture to diameter aspect ratio of the order of 10−3 . Garbin and Knopoff (1973) analyzed a random distribution of penny-shaped parallel cracks in the rock. Inserting the crack density ε defined by O’Connell and Budiansky (1974) into Garbin and Knopoff’s formulas (1973, 1975a, b), velocities of body-waves in dry cracks are Vp = 

8 1+ ε 3

Vs = 



Vp0

  8µs c ((λ + 2µ)c2 )2 + 3λ + 4µ 2µ(λ + µ) 2 2

16(λ + 2µ) ε 1+ 3



Vs0

(1.9)

′  2 2  (1.9 ) c2 sin2 ϕ (1 − 2c2 )2 cos2 ϕ s c cos2 ϕ + + 3λ + 4µ 3λ + 4µ 2(λ + µ)

1

where, Vp0 =



λ+2µ ρ

Seismic Wave Propagation in Anisotropic Rocks. . .

and Vs0 =



µ ρ

21

are P- and S-wave velocities in uncracked

isotropic rock, λ and µ are Lame constants for uncracked rock, s = sin θ and c = cos θ, θ is the incident angle with respect to the normal of the crack plane, and ϕ is the angle between the polarization of the S-wave and the plane containing the incident ray. For the horizontal crack plane, ϕ = 0◦ corresponding to SV-wave and ϕ = 90◦ corresponding to SH-wave which polarization parallel to the crack plane. ε = N a3 is the crack density where N is the number of cracks with radius a per unit volume. For example, ε = 0.125 means that a circular flat crack with the unit diameter is contained in a unit cube. Crack density depends only on the area and perimeter of the penny-shaped crack and not on the thickness, and hence it is independent of the crack void volume. This parameter is a useful parameter determining the elastic properties for a solid permeated with either dry or water saturated cracks. For water saturated cracks, the terms in brackets [ ] in equations (1.9) and (1.9′ ) are omitted. Crampin (1978) simplified the formulations (1.9) and (1.9′ ) for the Poisson solid, i.e. λ = µ, and computed velocities of body-waves in a cracked rock, but did not correct the factor of 2; the last term in equations (1.9′ ) and (1.10) should be multiplied by 2. The simplified equations are written as Vp = 



Vs = 



8 1+ ε 3

1 + 16ε

Vp0

(1.10)

  8 2 2 1 2 2 s c + (1 + 2c ) 7 4 Vs0 2

2

2 2



2

2 2

2

s c cos ϕ c sin ϕ (1 − 2c ) cos ϕ + + 7 7 4



(1.10′ )

where the terms in square brackets [ ] are omitted for the velocity variations due to water-saturated cracks. Putting ϕ = 90◦ for SH-wave and ϕ = 0◦ for SV-wave, and neglecting the second-order terms of ε in Garbin and Knopoff formulation, equations (1.10) and (1.10′ ) may be further simplified to Vqp = 

1−ε



Vqsh = 

1−ε



Vqsv = 



1−ε

Vp0 1 71 8 + cos 2θ − cos 4θ 21 3 21 Vs0 8 8 + cos 2θ 7 7

9 23 + cos 4θ 14 14

(1.11)

(1.11′ )



Vs0





(1.11′′ )

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Yong-Gang Li

for the solid containing dry cracks, and Vqp = 

1−ε



Vqsh = 

1−ε



Vqsv = 

1−ε



Vp0 8 8 − cos 4θ 21 21 Vs0 8 8 + cos 2θ 7 7 Vs0 8 8 + cos 4θ 7 7



(1.12)



(1.12′ )



(1.12′′ )

for the solid containing water-saturated cracks. In our second elastic Model II, in which the intact rock is a typical crustal granitic rock with elastic constants λ = µ = 26.25 × 1010 dynes/cm2 and rock density 2.5 g/cm3 . P-wave velocity is 5.61 km/s and S-wave velocity is 3.24 km/s. Assume a distribution of aligned cracks with crack density 0.075 contained in the granite rock, we computed wave velocities in the cracked rock using Equations (1.9) and (1.9′ ). The body-wave velocities are shown in Figure 1.2a for dry cracks and Figure 1.2b for water-saturated cracks. Velocity curves show the directional

Fig. 1.2 Computed phase velocities of P-, SV- and SH-waves in Model II, the granite rock containing aligned vertical (a) dry and (b) water-saturated cracks (crack density 0.075) with hexagonal anisotropic symmetry, using Garbin and Knopoff formulations for body-wave velocities (1.9) and (1.9′ ). Thick lines in (a) are obtained from their uncorrected formulations (1973, 1975a, b), missing the factor of 2. Thin lines are correct after multiplying a factor of 2.

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Seismic Wave Propagation in Anisotropic Rocks. . .

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variations as seen in Figure 1.1. For the rock with dry cracks, as waves propagate along the crack normal, three body-waves have their slowest velocities: P-wave travels at 4.66 km/s and two shear-waves travel at 2.99 km/s. As waves propagate parallel to crack planes, P-wave travels at its maximum velocity of 5.47 km/s while SV-wave travels at 2.99 km/s and SH-wave travels at 3.24 km/s. The SV-wave with polarization perpendicular to cracks can deform the rock easily due to the weakness zone (cracks). Hence it experienced a low effective rigidity, and thus has a lower velocity than SH-wave, which polarization is parallel to cracks and thus cannot sense the weakness zone so that it experiences a higher effective rigidity. In the rock containing saturated cracks, the P-wave velocity does not change much for comparison with its velocity in the uncracked rock. The minimum value of P-wave velocity is 5.45 km/s as P-wave propagates at an oblique angle 45◦ to the crack normal. SH-wave velocities are the same as those for the rock containing dry cracks while SV-wave velocities show a great deal directional variations. When SV-wave propagates at an incident angle θ below 62◦ , its velocity is faster than SH-wave velocity, but vice versa as θ is above 62◦ . The velocity curves of two shear-waves show the intersection singularity at this specific direction (θ = 62◦ in this case). We note that such a singularity also appears in Figure 1.2a for the rock containing dry cracks. The velocity curve of SH-wave intersects the velocity curve of SV-wave, but it is not correct. This problem with the intersection singularity of two shear waves for the rock containing dry cracks was first pointed out by Hudson (1981) when he derived formulations of body-wave velocities in the rock containing aligned cracks using a dynamic method. He noticed the discrepancy for SV velocities between his results and Garbin and Knopoff’s results (1975a) and found that the discrepancy was caused by a factor of 2 missed in the last term bracketed in equation (1.9′ ). Crampin (1984b) also corrected this erroneous result in his earlier papers (Crampin, 1978, 1980, 1981). Here, we confirm that a factor of 2 has been missed in the last term of equation (60) of Garbin and Knopoff (1975a) and here in equation (1.9′ ). After this correction, SV velocities shown by the thin line in Figure 1.2a always remain less than SH velocities as wave propagation direction changes between 0◦ and 90◦ . However, this missing factor of 2 does not affect results for water saturated cracks. If seismic wavelengths are longer than the size of cracks, observations will be made primarily of the overall properties of the cracked rock. Under such condition, Hudson (1981) derived explicit expressions of wave velocities and attenuation in cracked solids based on the mean taken over a statistical ensemble. The formulas may be valid to determine elastic properties of a single rock sample for small concentration of cracks because the first-order correction in the quantity ε = N a3 was used in his derivation (1981). The first-order correction of elastic

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Yong-Gang Li

constants for the cracked solid due to dilute cracks aligned in x1 direction is derived from ε 1 ¨kl (Ka) = − c0k1ip c0l1jq U (1.13) Cipjq µ where c0 is the elastic constant tensor for the uncracked solid, K is the wavenum¨kl (Ka) is the integral performed ber and a is the radius of cracks. The quantity U � over the points X of the face of a crack � ¨ (Ka) = 1 U Ukl eik·X;X e−ik·X dSx (1.14) a2 P ¨kl , the first-order correction of the elastic constants Since the symmetry of U for cracked solid with the crack density ε may be written as    (λ + 2µ)2 λ(λ + 2µ) λ(λ + 2µ)       λ(λ + 2µ) 0 λ2 λ2  U33     −ε  λ(λ + 2µ) λ2 λ2  1   Cij =    0 0 0 µρ       2 0 0  U11  0 µ   0

0

µ2

(1.15)

where U11 =

16 (λ + 2µ) , 3 (3λ + 4µ)

U33 =

4 (λ + 2µ) 3 (λ + µ)

For the solid containing water-saturated cracks, U33 = 0, λ and µ are Lame constants of the uncracked rock. This matrix form of the first-order correction of elastic tensor c1 has great advantages in computation of rock anisotropy caused by aligned cracks. If the crack normal is aligned in the x3 direction, we can easily obtain a new matrix (1.15′ ) for the correction of elastic constant tensor by rotating the axis x1 into x3 ,    λ2 λ2 λ(λ + 2µ)    λ2 λ2 λ(λ + µ)  U33   0      2 λ(λ + 2µ) λ(λ + µ) (λ + 2µ) −ε   1 Cij =    2   µρ  µ 0 0       0   0 µ2 0  U11  0

0

0

(1.15′ ) If there are two or more sets of cracks aligned in different directions, we can obtain the overall correction of elastic constants by calculating c1 for each set and then adding them together. The tensor of elastic constants of cracked rock is therefore obtained by c = c0 + c1 .

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Seismic Wave Propagation in Anisotropic Rocks. . .

25

Taking the mean over a statistical ensemble with the first-order in the crack density N a3 , Hudson (1981) gives the following expressions for velocities of P, SV and SH waves, Vp = 

4 1− ε 3

Vsh = 

Vp0

  16µs2 c2 ((λ + 2µ)c2 )2 + 3λ + 4µ 2µ(λ + µ) Vs0

16(λ + 2µ) ε 3



2



c 3λ + 4µ Vs0 =  2   2 2  16(λ + 2µ) (c − s2 )2 s c ε + 1− 3 3λ + 4µ λ+µ 1−

Vsv



(1.16)

(1.16′ )

(1.16′′ )

where s = sin θ and c = cos θ, θ is the angle with respect to the crack normal. The terms in the square brackets are omitted for the solid containing water-saturated cracks. Using the same parameters in Model II, we computed velocities of three type body-waves by equation (1.16) (see Fig. 1.3). Results from Hudson’s first-order equations are similar to those computed using corrected Garbin and Knopoff solutions (GKS). We multiply the last term in (1.9′ ) by 2 for correction of GKS. Velocity curves of SV- and SH-waves are no longer intersected for the solid containing dry cracks as seen in the uncorrected GKS (the thick lines in Fig. 1.2a). However, we find an obvious difference in

Fig. 1.3 Computed phase velocities of P-, SV- and SH-waves in Model II, the crystalline rock containing aligned vertical (a) dry and (b) water-saturated cracks (crack density 0.075) with hexagonal anisotropic symmetry, using the first-order formulation for body-wave velocities given in Equations (1.16), (1.16′ ) and (1.16′′ ) of Hudson (1981), are denoted by thick lines, compared with results from the corrected GKS (1.9) and (1.9′ ) denoted by thin lines.

26

Yong-Gang Li

P-wave velocities between two solutions is introduced by equation (1.16) because it neglects the second-order term in the crack density ε. We then introduce the second-order correction for elastic constants due to cracks aligned in the x1 direction (Hudson, 1982) as follows    (λ + 2µ)q λq λq    2   0 λq λ2 q/(λ + 2µ) λ2 q/(λ + 2µ)  U33     −ε2  λq λ2 q/(λ + 2µ) λ2 q/(λ + 2µ)  2   Cij =    0 0 0 15ρ     2   0  0 b 0  U11    0 0 0 (1.15′′ ) 2 where q = 15(λ/µ) + 28(λ/µ) + 28, b = 2µ(3λ + 2µ); and U11 and U33 are the same in (1.15). We write expressions of velocities for P, SV and SH waves including the second-order correction in ε below. � � � 2 2 2 − Vp2 ) − 4(Vs0 − Vsv ) 4 ′2 1 3(Vp0 ′ 2 2 (1.17) Vp = Vp − (Vsv − Vsv ) + 4 3 9 Vp0 � � �2 2 2 2 (3Vp0 + 2Vs0 ) Vsh 2 ′ 2 1− 2 Vsh = Vsh − (1.17′ ) 2 15 Vs0 Vp0 � � �2 2 2 2 (3Vp0 + 2Vs0 ) Vsv 2 ′ 2 1− 2 (1.17′′ ) Vsv = Vsv − 2 15 Vs0 Vp0

′ ′ where Vp′ , Vsv and Vsh are velocities of P, SV and SH waves including the secondorder correction for elastic constants due to cracks. Vp , Vsv and Vsh are velocities with the first-order correction obtained from (1.16). Vp0 and Vs0 are P- and Swave velocities of uncracked rock. We use Model II and compute P, SV and SH-wave velocities by (1.17). Figure 1.4 shows computed velocities with the second-order correction in crack density ε for comparison with those computed using corrected Garbin and Knopoff’s equations. The discrepancy between them is negligible. Finally, we multiplied the terms of c44 and c55 in Hudson’s first-order correction matrix of elastic constants (1.15) and the second-order correction matrix (1.15′ ) by a factor 2. We obtained the tensor of elastic constants for Model II shown below    78.25 26.25 26.25   26.25 78.75 26.25   0      26.25 26.25 78.75     × 1010 dyn/cm2   26.25 0.00 0.00     0.00 26.25 0.00    0 0.00 0.00 26.25 (1.18)

1

Seismic Wave Propagation in Anisotropic Rocks. . .

27

Fig. 1.4 Computed phase velocities of P-, SV- and SH-waves in Model II, the crystalline rock containing aligned vertical (a) dry and (b) water-saturated cracks (crack density 0.075) with hexagonal anisotropic symmetry, using the second-order formulation for body-wave velocities (1.15′′ ) of Hudson (1982) denoted by thick lines, compared with results from the corrected GKS (1.9) denoted by thin lines.

for the uncracked granite rock,    43.31 14.44 14.44   14.44 74.81 22.31   0      14.44 22.31 74.81     × 1010 dyn/cm2   26.25 0.00 0.00     0.00 21.75 0.00    0 0.00 0.00 21.75 (1.18′ ) for the cracked rock with the first-order correction in the crack density ε and    51.70 17.23 17.23   17.23 75.74 23.24   0    17.23 23.24 75.74       × 1010 dyn/cm2   26.25 0.00 0.00     0.00 22.04 0.00    0 0.00 0.00 22.04 (1.18′′ ) for the cracked rock with the first-order and second-order correction in crack density ε. We conclude that Hudson’s formulations of elastic constants for rocks containing aligned cracks may be with high accuracy even for the crack density as high as 0.1 if the second-order correction in the quantity of the crack density ε = N < a3 > is taken into account (refer to Fig. 1.4). When the wave propagation direction does not deviate far from the crack plane or the crack density

28

Yong-Gang Li

is low, Hudson’s formulation with the first-order correction for the elastic constants is still valid with appropriate accuracy. It may be wise to design a VSP survey with appropriate geometry so that the first-order theory can be applied to the fractured structures with high crack density (refer to later sections).

1.3

Plane Wave Propagation in Homogeneous Anisotropic Medium

Plane harmonic waves in a homogeneous anisotropic medium will propagate with slowness p, provided that the displacement u = U e−iω(t−p·x) satisfies the equation of motion, cijkl

∂ 2 Ui ∂Uk2 =ρ 2 , ∂xl ∂xj ∂t

i, j, k, l = 1, 2, 3

(1.19)

where Ui is the component of the polarization vector U , giving the direction of particle motion, cijkl is the fourth-order tensor of elastic constants and xi is the position coordinates in the Cartesian system. Taking the 3-D Fourier transform of (1.19), we obtain (1.20) kj kl cijkl Uk = ρω 2 Ui where ki (i = 1, 2, 3) is the component of vector wavenumber k. Replacing |k| = ωc where c is the phase velocity, and introducing the slowness vector p = k/|k|, equation (1.20) can be written in the form of Kelvin-Christoffel equation (cijkl pj pl − ρδik )Uk = 0

(1.21)

For non-trivial solutions, we require det |cijkl pj pl − ρδik | = 0

(1.22)

where δik is the Kronecker symbol. It forms a typical eigenvalue problem. Velocities of body-waves can be obtained from this equation. For a more convenient form, the slowness vector p is used instead of velocity c so that equation (1.22) is rewritten as Γ − Gm | = 0, (1.23) det |Γ where Gm (m = 1, 2, 3) denote three eigenvalues of the Christoffel matrix Γ which has its elements (1.24) Γik = aijkl pj pl where aijkl = cijkl /ρ is the fourth-order tensor of elastic constants divided by the material density, and pi (i = 1, 2, 3) are components of the slowness vector p.

1

29

Seismic Wave Propagation in Anisotropic Rocks. . .

As mentioned in Section 1.2.1, the symmetry of both stress and strain tensors allows a 6 × 6 matrix crs (r, s = 1, 2, 3) to replace the fourth-order tensor of elastic constants cijkl so that elastic parameters ars can be expressed as aij =cij /ρ. In isotropic media, equation (1.23) reduced to three separate second-order equations for slowness corresponding to P-, SV- and SH-waves. Three eigenvalues are G1 = α2 pi pi and G2 = G3 = β 2 pi pi , where α and β are velocities of P- and S-waves, respectively. For the general anisotropic medium, characteristic equation (1.23) is not possible to solve analytically for three slowness components pi (i = 1, 2, 3), directly in terms of ρ and components of the phase velocity c. In general, it must be solved numerically. Its three roots for c2 map three-sheeted non-spherical surface in the slowness space that gives the permissible values of slowness as a function of wave propagation direction (Synge 1957; Fedorov, 1968; Musgrave, 1970). For isotropic media, three sheets  are concentric ρ spheres, one with radius λ+2µ and two with the same radii µρ .

We then discuss the most general form of rock anisotropy with 21 independent elastic constants as described in Section 1.2. In this case, the Christoffel matrix Γ is a 3 × 3 symmetric positive-definite matrix formed by elastic parameters aij and components of slowness vector pi (i = 1, 2, 3). It has three real eigenvalues Gm with mutually orthogonal eigenvectors g m , corresponding to phase velocities and polarization vectors of three types of body-waves, respectively. In the most general anisotropic system with 21 independent elastic constants are expressed by Γ11 = a11 p2 + a66 r2 + a55 q 2 + 2a16 pr + 2a15 pq + 2a56 rq Γ12 = a16 p2 + a26 r2 + a45 q 2 + (a12 + a66 )pr + (a14 a56 )pq + (a46 + a25 )rq Γ13 = a15 p2 + a46 r2 + a55 q 2 + (a14 + a56 )pr + (a13 a55 )pq + (a45 + a36 )rq Γ22 = a66 p2 + a22 r2 + a44 q 2 + 2a26 pr + 2a46 pq + 2a24 rq Γ23 = a56 p2 + a24 r2 + a34 q 2 + (a25 + a46 )pr + (a36 a45 )pq + (a44 + a23 )rq Γ33 = a55 p2 + a44 r2 + a33 q 2 + 2a45 pr + 2a55 pq + 2a34 rq (1.25) where p = p1 , r = p2 and q = p3 are components of the slowness vector in the Cartesian coordinates. When an anisotropic system possesses the higher degree of elastic symmetry, such as monoclinic symmetry with its elastic tensor shown in equation (1.4), the elements in the Characteristic matrix Γ are simplified to Γ11 Γ12 Γ13 Γ22 Γ23 Γ33

= a11 p2 + a66 r2 + a55 q 2 + 2a16 pr = a16 p2 + a26 r2 + a45 q 2 + (a12 + a66 )pr = (a13 + a55 )pq + (a45 + a36 )rq = a66 p2 + a22 r2 + a44 q 2 + 2a26 pr = (a36 + a45 )pq + (a44 + a23 )rq = a55 p2 + a44 r2 + a33 q 2 + 2a45 pr

(1.26)

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Yong-Gang Li

For an orthorhombic anisotropic medium with its three planes of symmetry parallel to coordinate planes (refer to expression (1.5)), the Christoffel matrix Γ has its elements simplified to Γ11 = a11 p2 + a66 r2 + a55 q 2 Γ12 = (a12 + a66 )pr Γ13 = (a45 + a36 )rq Γ22 = a66 p2 + a22 r2 + a44 q 2

(1.27)

Γ23 = (a44 + a23 )rq Γ33 = a55 p2 + a44 r2 + a33 q 2 In the hexagonal medium with a horizontal symmetry axis, having its elastic tensor as shown in (1.7), the Christoffel matrix can be expressed as   a33 p2 + a44 r2 + a66 q 2 (a13 + a44 )pr (a13 + a44 )pq    Γ = a44 p2 + a11 r2 + a66 q 2 (a12 + a66 )rq (a13 + a44 )pr   (a12 + a66 )rq a44 p2 + a66 r2 + a11 q 2 (a13 + a44 )pq (1.28) If the hexagonal system has a vertical symmetry axis and this axis coincides with the x3 axis in the Cartesian coordinates (transversely isotropy), the matrix Γ has the simplest form with five non-zero elements only,   a11 p2 + a55 q 2 0 (a13 + a55 )pq    Γ = (1.29) 0 0 a66 p2 + a44 q 2   2 2 0 a55 p + a33 q (a13 + a55 )pq We shall discuss phase velocities, group velocities and polarizations of three distinct body-waves in anisotropic media with various degrees of elastic symmetry.

1.3.1

Phase Velocities of Body Waves in Anisotropic Media

We have shown that velocities of body waves propagating in an anisotropic medium can be obtained from Kelvin-Christoffel equation (1.21) and its characteristic equation (1.23) associated with the Christoffel tensor Γ . Eigenvalues and eigenvectors of matrix Γ may be computed numerically. For example, we used a numerical routine EIGRS in the scientific subroutine package IMSLS to obtain eigenvalues and eigenvectors directly in our 2-D ray tracing procedure

1

Seismic Wave Propagation in Anisotropic Rocks. . .

31

(refer to Leary et al., 1987; Li et al., 1987 for details). Finding eigenvalues Gm is then deduced to find roots of the characteristic equation � � � Γ 11 − Gm � Γ 12 Γ 13 � � � �=0 (1.30) det � Γ 12 Γ 22 − Gm Γ 23 � � Γ Γ 23 Γ 33 − Gm � 13 It can be rewritten in a cubic form,

G3m − P G2m + QGm − R = 0

(1.31)

For non-attenuating media, the polynomial has real coefficients so that it has either three real roots for wave propagation or three complex roots without physical meaning. The roots of the polynomial may be found by a computer numerical routine, for instance, the subroutine P OLRT in IBM-SSP scientific software package to compute the real and complex roots of a real polynomial as we used in our 3-D ray tracing program (refer to Li, 1988; Li et al., 1990 for details). Now, we choose the analytic solution because it is most rapid and suitable to be used in the anisotropy system with higher degrees of elastic symmetry. Coefficients P, Q and R in the cubic equation (1.31) are the trace, the second invariant and the determinant of the matrix Γ . They are written as P = Γ11 + Γ22 + Γ33 2 2 2 R = Γ11 Γ22 + Γ22 Γ33 + Γ11 Γ33 − Γ12 − Γ13 − Γ23

Q=

2 −(Γ11 Γ23

+

2 Γ22 Γ13

+

2 Γ33 Γ12 )

(1.32)

+ 2Γ12 Γ13 Γ23 + Γ11 Γ22 Γ33

Three real roots of the cubic equation (1.31) corresponding to three eigenvalues, are then obtained by the explicit algebraic expressions: G1 = A + B +

P 3

−1 (A + B) + G2 = 2 −1 (A + B) + G3 = 2 where

√ P 3 +i (A − B) 3 2 √ P 3 −i (A − B) 3 2

� �� � � � �2 � � −b � b 3  � +� A= + 2 2 � �� � � � �2 � � � −b b 3 −� + B = � 2 2 1 a = Q − P 2, 3

b=

� a  3 � a  3

2 2 (P Q − 2R) + P 3 3 27

(1.33)

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Yong-Gang Li

When the phase propagation vector n is given, inserting slowness pi = ni /v (i = 1, 2, 3) into the Christoffel tensor Γ of a specific anisotropic system possessing one form of equations (1.25) to (1.29), roots of polynomial (1.31) can be solved from (1.32) and (1.33). In the Cartesian coordinates (x1 , x2 , x3 ), n has its components, n1 = cos ϕ cos θ,

n2 = cos ϕ sin θ,

n3 = sin ϕ

(1.34)

where ϕ is the declination angle of the wave vector in the vertical plane with respect to the horizontal x1 − x2 plane; θ is the azimuthal angle between the projection of the wave vector on that horizontal plane and the x1 axis. For the transversely isotropic medium with a x3 symmetry, θ = 0◦ and so n1 = cos ϕ,

n2 = 0,

n3 = sin ϕ

(1.35)

Phase velocities of body waves are then computed straightforward by vi = √ √ √ √ Gm , i.e., v1 = G1 for P-wave, v2 = G2 and v3 = G3 for two shear waves. These equations demonstrate some of fundamental features of body-wave propagation in anisotropic media. The Kelvin-equation (1.21) shows that there are three types of body-waves for every phase propagation direction. One is called quasi P-wave (qP ) and the other two called quasi shear waves (qS1 and qS2 ). Two shear waves travel in general at different speeds except for some particular directions, for example, along the symmetry axis in the hexagonal medium as seen in Section 1.3. Phase velocities are not only a function of the elastic constants, but also depend on the propagation direction. Polarization vectors of three body waves remain orthogonal, but they are not in general parallel to the phase propagation direction for quasi P-wave and not perpendicular to the propagation for two quasi shear waves. We now use the Kelvin-Christoffel equation to solve the eigenvalue problem of our Model III which is the olivine crystalline rock with orthorhombic symmetry. The material density is 3.324 g/cm3 and its elastic constants are given by    32.4 59.0 79.0   59.0 19.8 78.0   0      79.0 78.0 24.9     × 1010 dyn/cm2 (1.36)   66.7 0.00 0.00     0.00 81.0 0.00    0 0.00 0.00 79.3 Figure 1.5 shows computed phase velocities of P, S1 and S2 waves in (a) the x2 = 0 plane, (b) the x2 = x1 plane and (c) the x1 = 0 plane. Phase velocities vary as a function of the angle of wave propagation direction with respect to the x3 = 0 plane. P-wave has its fastest speed 9.87 km/s when it propagates along the x1 axis and the lowest speed 7.72 km/s along the x2 axis. It travels

1

Seismic Wave Propagation in Anisotropic Rocks. . .

33

Fig. 1.5 Phase velocities of three quasi body-waves in the olivine crystalline rock with orthorhombic symmetry (Model III) when the phase propagation vector in (a) x2 = 0 plane, (b) x2 = x1 plane and (c) x1 = 0 plane. The angle denotes the phase propagation direction with respect to x3 = 0 plane.

at 8.66 km/s along the x3 axis. Two shear waves S1 and S2 are defined as the faster shear wave and slower shear wave, respectively. The S1 -wave (S2 -wave) propagates at 4.94 (4.88), 4.88 (4.48) and 4.94 (4.48) km/s along the x1 , x2 and x3 axis, respectively. Two shear waves have neither the kiss singularity (two shear velocity sheets tangentially touch) nor the intersection singularity (two sheets intersect). Figure 1.6 shows three velocity surfaces which are in general separated and not spherical in the orthorhombic anisotropic medium. In the second example, we use Model II in Section 1.2, and assume that the material density of granitic rock is 2.5 g/cm3 and lame constant λ = µ = 26.25 × 1010 dyn/cm2 . The rock contains homogeneous distribution of vertical cracks with the normal of crack plane along the x1 direction. The crack density is 0.075. Such cracked rock with aligned cracks forms a hexagonal anisotropic system with the symmetry axis in the x1 axis in the Cartesian coordinates. The fourth-order tensor of elastic constants of the uncracked granite is written as in equation (1.18). This elastic constant tensor of the cracked rock may be written as in (1.18′ ) having the first-order correction in the crack density ε (Hudson, 1981) or in (1.18′′ ) including the second-order correction in ε (Hudson, 1982). The Christoffel matrix Γ may be written in the form of equation (1.28). Then, we solve the cubic equation (1.31) for three real roots corresponding to velocities of P, S1 and S2 waves, taking the elastic constant tensor in equation (1.18′′ ) including the second-order correction due to the crack density ε. We still define S1 wave to be the faster shear wave and the S2 wave the slower shear wave. Computed phase velocities of P, S1 and S2 are plotted in Figure√1.7 for the phase propagation √ vector in (a) the x2 = 0 plane, (b) the x2 = 3/3x1 plane, (c) the x2 = 3x1 plane and (d) the x1 = 0 plane. Velocity curves on

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Yong-Gang Li

Fig. 1.6 Phase velocity sheets of three quasi body-waves in (a) x2 = 0 plane, (b) x2 = x1 plane and (c) x1 = 0 plane in Model III. The horizontal component is along x1 axis and the vertical component is along x3 axis. Solid lines are for P-wave, long dashed lines for S1 -wave and dash-dot lines for S2 -wave.

the left side are for the rock containing dry cracks and velocity curves on the right are for the rock containing water-saturated cracks. The angle denotes the direction of phase propagation with respect to the x3 = 0 plane in the Cartesian coordinates. Velocities of three quasi body-waves vary in general as a function of the direction of propagation except for the particular case as shown in Figure 1.7d. When the phase propagation direction in the x1 = 0 plane, i.e., the crack plane, three types of quasi waves remain constant for the rock containing either dry cracks or water-saturated cracks. They are 5.50 (5.61) km/s for P-wave, 3.24 (3.24) km/s for S1 -wave and 2.98 (2.95) km/s for S2 -wave (values in brackets for water-saturated cracks). In this case, S1 -wave has its particle motion in the x1 = 0 plane parallel to cracks so that it travels faster while S2 -wave has its particle motion perpendicular to the x1 = 0 plane so that it travels slower as we explained in Section 1.2.

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Seismic Wave Propagation in Anisotropic Rocks. . .

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When the phase propagation vector lies in the x2 = 0 plane, velocities shown in Figure 1.7a computed from the solution of eigenvalue problem are exactly the same as those shown in Figure 2.4 calculated by equation (1.17). In the case for the rock containing dry cracks, two shear waves have separate velocity curves denoted by dashed lines except for the symmetry direction. There is no intersection singularity between two shear-waves. This result verifies the computation we made in Section 1.2. The intersection singularity occurs in the rock containing water-saturated cracks only (Bush and Crampin, 1991). Two shear-waves have the same velocity 3.17 km/s as the phase propagation direction at an angle 62◦ with respect to the crack normal as shown in the right part of Figure 1.7a. When the phase propagation direction in along x1 axis, i.e. the crack normal, two shear-waves travel at the same speed 2.95 km/s, and then they separate from each other as the angle of phase propagation direction increases. We define that the faster shear wave is S1 -wave denoted by the long dashed line and the slower shear wave is S2 wave denoted by the dash-dot line. When the angle of phase propagation direction is below 62◦ , S1 -wave polarized in the

36

Yong-Gang Li

Fig. 1.7 Phase velocities of three quasi body-waves in the cracked granitic rock with hexagonal symmetry (Model II, the√crack density 0.075) when √ the phase propagation vector in (a) x2 = 0 plane, (b) x2 = 3/3x1 plane, (c) x2 = 3x1 plane and (d) x1 = 0 plan. The angle denotes the phase propagation direction with respect to x3 = 0 plane. The left parts are for the rock containing dry cracks and the right parts are for the rock containing water-saturated cracks.

vertical plane parallel to the crack normal travels faster than S2 -waves polarized transverse to the vertical plane. In this case, S1 -wave is called SV-wave and S2 -wave is called SH-wave. When the phase propagation direction is above 62◦ , S1 -wave polarized perpendicular to the vertical plane travels faster than S2 wave polarized in the vertical plane. Thus, SV-wave and SH-wave exchange the plotting styles in the figure. √ When the phase propagation direction lies in x2 = 3/3x1 plane, two shearwaves have different velocities for the rock containing dry cracks. They are completely separated from each other as shown in the left part of√Figure 1.7b. The shear-wave S1 with its particle motion perpendicular to x2 = 3/3x1 plane (usually called the sagittal plane on which the phase propagation vector lies)

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Seismic Wave Propagation in Anisotropic Rocks. . .

37

always travels faster than shear-wave S2 with its particle motion in the sagittal plane. However, for the rock containing water-saturated cracks, velocity sheets of two shear-waves have the intersection singularity although they do not have the kiss-type singularity (see √ the right part of Fig. 1.7b). When the phase propagation direction lies in x2 = 3x1 plane, shear-wave S1 with its particle motion in the sagittal plane travels faster than orthogonally polarized shear-wave S2 for the rock containing either dry cracks or water-saturated cracks as shown in Figure 1.7c. In summary, we plot three velocity surfaces of P, S1 and S2 waves as a function of the propagation direction lying in four vertical planes mentioned above (see Fig. 1.8a, b, c, d) for the rock containing dry cracks. The solid lines denote velocities of P-wave, the long dashed lines for the faster shear-wave S1 and the dash-dot lines for the slower shear-wave S2 . Three surfaces of body waves are

Fig. 1.8 Phase vector sheets of three quasi body-waves in Model II with dry cracks are plotted for the propagation vector lying in three vertical planes as same as in Figure 1.6.

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Yong-Gang Li

in general not spherical and not intersected except for the propagation direction parallel to the crack plane (x1 = 0 plane) in the hexagonal anisotropic medium containing aligned dry cracks. Figure 1.9 shows three phase velocity surfaces of P, S1 and S2 waves in the hexagonal anisotropic medium containing aligned water-saturated cracks. We notice that the intersection occurs between two shear-waves in Figure 1.9a and 1.9b. In the above examples, we illuminate the directional feature of phase velocities of body-waves traveling in the rock contain aligned cracks. We find that two shear-waves travel at different speeds in the anisotropic medium. The degree of difference in velocity between two shear waves not only depends on the elastic constants of the uncracked host rock, but also on the crack density and the wave propagation direction with respect to the crack plane. Results from these examples will help us to design an appropriate experiment to observe the meaningful shear-wave splitting in anisotropic media.

Fig. 1.9 Phase velocity sheets of three quasi body-waves in Model II with watersaturated cracks are plotted in the same format used in Figure 1.8.

1

1.3.2

Seismic Wave Propagation in Anisotropic Rocks. . .

39

Group Velocities of Body Waves in Anisotropic Media

In this section, we discuss body-wave group velocities which determine the wave energy transportation in the medium. It is well known that the kinematic group velocity Ug is the gradient of the frequency ω with respect to the wave number k. The classic expression for body wave group velocities in anisotropic media is written by T  ∂ω ∂ω ∂ω Ug = , , (1.37) ∂k1 ∂k2 ∂k3 Wave energy transport is in general no longer parallel to the wavefront normal, or the phase propagation direction in anisotropic media. It means that the direction of group velocity is not in the same direction of phase velocity when seismic waves propagate in an anisotropic medium. For example, when the wave phase travels in the x1 -direction with velocity c in a non-dispersive anisotropic medium, equation (1.37) can be written as below, Ug =



∂ω ∂ω c, , ∂k2 ∂k3

T

(1.38)

Therefore, wave energy transport not only has its principal component along the propagation vector at the phase velocity c, but also has its additional components along the wavefront. The group velocity Ug is in general larger than phase velocity c or equal to the phase velocity at some specific directions in anisotropic media. Synge (1957) described the surface of group velocity or called wave surface as being the envelope of wavefronts. The wave surface is spherical in the homogeneous isotropic medium, but it is no longer spherical in anisotropic media. The wavefronts of three body-waves are not consistent with the surfaces of phase velocities or wavefronts for the non-dispersive anisotropic medium. Components Ugi (i = 1, 2, 3) of the group velocity vector Ug in the Cartesian coordinates (x1 , x2 , x3 ) may be obtained from the expression (1.33) given by Crampin (1981) based on the derivation of Musgrave (1970). We describe briefly the derivation as follows. Let ξi be the coordinates of any point on the wave surface and the parametric equations for the envelope of the planes be expressed by (1.39) ni ξi = v where ni are directional cosines of the phase propagation vector and v is the magnitude of the phase velocity vector, whose extremity traces out velocity surfaces V which has been discussed in Section 1.3.1. V may be written as V = ak g k − 1 = 0

(1.40)

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Yong-Gang Li

Given by Kelvin (1904) where ak (i = 1, 2, 3) are the function of elastic constants and direction cosines of slownesses, gk are components of the polarization vector U . The required envelope is obtained by eliminating the differential dv from ξi dni − dv = 0, and

∂V ∂V dv = 0 dni + ∂ni ∂v

(1.41)

where ni are direction cosines of the phase propagation vector. Assuming N = ak gk be a normalization constant, so 2ρv ∂V =− ∂v N and hence ξi =

N ∂ak gk 2ρv ∂ni

From the energy flux principle, a single term of originally to Synge (1957),

(1.42) ∂(ak gk ) ∂ni

has the form due

N [(ρv 2 − Ak )(∂a2k /∂ni ) + a2k (∂Ak /∂ni )] (ρv 2 − Ak + a2k )2 In terms of the transportation of equation (1.22), we obtain the expression for the elements of the group velocity Ugi

� � 3 c � gk2 ∂a2k 2 2 ∂Ak (c − Ak ) = + ak 2 a2k ∂ni ∂ni

(1.43)

k=1

where Ak and ak are functions of elastic parameters aij and direction cosines ni of slowness vectors. A1 = Γ11 , A2 = Γ22 , A3 = Γ33 , a1 a2 = Γ12 , a1 a3 = Γ13 and a2 a3 = Γ23 . For the most general anisotropic medium, Ak and ak may be obtained from the matrix form of expression (1.44).   a11 a66 a55 a56 a15 a16  a24 a46 a26      a66 a22 a44  2 n A1 a  1 a34 a35 a45 55 a44 a33   A    n22   2      1 1 1    2  n   A3   a56 a24 a34 (a23 +a44 ) (a36 +a45 ) (a25 +a46 )  3    = 2 2 2  2n2 n3   a2 a3         2n n  1 1 1  a1 a3   3 1  a15 a46 a35 (a36 +a45 ) (a13 +a55 ) (a14 +a56 )    2 2 2 a1 a2   2n1 n2   1 1 1 a16 a26 a45 (a25 +a46 ) (a14 +a56 ) (a12 +a66 ) 2 2 2 (1.44)

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For a particular anisotropic system with higher degrees of symmetry discussed in Section 1.2, the number of non-zero elements in the first matrix on the right hand of equation (1.44) is reduced. In these cases, the computation is remarkably simplified. It is mentioned that two shear-wave surfaces will cause cusps in the vicinity of some particular directions, such as in symmetry axes because two phase velocity sheets of shear-waves have singularities due to their same phase velocity value. We then show computational results for group velocities in anisotropic media using previous models in Section 1.3.1. In the first example (Model III) with orthorhombic olivine rock having the elastic tensor as shown in equation (1.36), the Musgrave’s expression (1.44) is written as      n21 97.47 23.86 24.27 0 0 0 A1   A   23.86 59.67 20.07 2  0 0 0    n2   2        0 0 0   n23   A3   24.37 20.07 74.91   (1.44′ ) =   a2 a3   0 0 0 21.77 0 0   2n2 n3        a1 a3   0 0 0 0 24.07 0   2n3 n1  0 0 0 0 0 20.80 2n1 n2 a1 a2

where values in the first matrix on the right of the equation is the material normalized elastic constant with the unit of km2 /s2 . Computed group velocities for three types of body-waves are shown in Figure 1.10 for the cases in which phase propagation vector lies in (a) x2 = 0 plane, (b) x2 = x1 plane and (c) x1 = 0 plane, using the same format as in Figure 1.5. We plotted phase velocities by thin lines to compare with group velocities marked by thick lines. Group velocities are in general larger than phase velocities. In some specific cases, i.e. along the symmetry axes, group velocities

Fig. 1.10 Group velocities of three quasi body-waves in the orthorhombic olivine rock (Model III) using the same format in plot of Figure 1.5.

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Yong-Gang Li

are equal to phase velocities. However, group velocities are never slower than phase velocities in any cases. We see a big difference between group and phase velocities of P-wave occurring in the case of propagation direction in [1 1 0]. In this direction, the group velocity of P-wave increases to 8.95 km/s while the phase velocity is 8.61 km/s. We also note that one shear-wave S1 shows clearer separation of group velocities from phase velocities while the other shear-wave S2 has similar group and phase velocities. Figure 1.11 shows group velocity surfaces of three types of body-waves P, S1 and S2 , as a function of propagation direction in three vertical planes defined in Figure 1.11. The solid line denotes group velocity for P-wave, the long dashed line for the faster shear-wave S1 and the dash-dot line for the slower shear-wave S2 . Three types of wave surfaces are not spherical and do not have the cusp in the orthorhombic medium. In the next example, we use the same parameters in Model II for granitic rock containing vertical cracks with the crack normal parallel to x1 axis. The crack

Fig. 1.11 Group velocity sheets of three quasi body-waves in Model III are plotted in the same format as in Figure 1.6.

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Seismic Wave Propagation in Anisotropic Rocks. . .

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density is 0.075. The elastic tensor with the second-order correction due to the crack density ε has the hexagonal symmetry as shown in equation (1.18′′ ). In this example, Musgrave’s expression (1.44) is written as      n21 20.86 8.816 8.816 0 0 0 A1   A   8.816 30.30 10.50 2  0 0 0    n2   2        0 0 0   n23   A3   8.816 10.50 30.30  =  (1.44′′ )   a2 a3   0 0 0 9.712 0 0   2n2 n3        a1 a3   0 0 0 0 7.854 0   2n3 n1  0 0 0 0 0 7.854 a1 a2 2n1 n2

where the element in the first matrix on the right side of the equation has the unit km2 /s2 . Figure 1.12 shows computed group velocities for phase propagation direction √ √ vector lying in (a) x2 = 0 plane, (b) x2 = 3/3x1 plane, (c) x2 = 3x1 plane and (d) x1 = 0 plane. Calculated group velocities are plotted on the left for the rock containing dry cracks and on the right for the water-saturated cracks. Phase velocities are plotted too (denoted by thin lines) to be compared with group

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Fig. 1.12 Group velocities of three quasi body-waves in the hexagonal cracked granitic rock (Model II with crack density 0.075) using the same format as in Figure 1.10.

velocities (denoted by thick lines). Group velocities of P-wave are in general larger than its phase velocities in the rock containing dry cracks, except for the case in which the propagation direction is along the crack normal or parallel to the crack plane. However, there is no obvious difference between group and phase velocities of P-wave in the rock containing water-saturated cracks as shown on the right side of Figure 1.12. We note that two shear-waves have comparatively pronounced difference between group and phase velocities in the rock containing water-saturated cracks, but no obvious difference between them in the rock containing dry cracks. When the propagation vector lies in x1 = 0 plane, the group velocity is the same as the phase velocity for three quasi body-waves in the rock containing either dry or water-saturated cracks as shown in Figure 1.7d and Figure 1.12d. We then plot three velocity surfaces (i.e., wave surfaces) of P, S1 and S2 waves as a function of the propagation direction in x2 = 0 plane for the rock containing dry cracks

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in Figure 1.13a and for water-saturated cracks in Figure 1.13b. The surfaces of three quasi body-waves are in general not spherical and have no cusp for the hexagonal anisotropic medium due to aligned dry cracks, but are intersected for the medium with water-saturated cracks. Finally, the direction of the group velocity can be obtained from expressions (1.43) and (1.44) after determining three components of the group velocity vector Ug . The wave energy is transported along this direction. It is also called ray direction in the ray tracing procedure. In the homogeneous non-dispersive isotropic medium, the phase propagation direction is the same as the direction of wave energy transportation, or the ray direction. However, in anisotropic media, wave energy transports along a direction in general deviated from the phase propagation direction. We shall discuss this problem in detail in the next section.

Fig. 1.13 Group velocity sheets of three quasi body-waves in x2 = 0 plane (a) for the rock containing dry cracks and (b) for the rock containing water-saturated cracks. Solid lines are for P-wave, long dashed and dash-dot lines are for two shear-waves S1 and S2 , respectively.

1.3.3

Body Wave Polarizations

We have mentioned that three types of plane body-waves propagate in anisotropic media with three mutually orthogonal polarization directions which are not in general coincident with the phase propagation vector or the wavefront. The Pmotion is not along the slowness vector and two S-motions are not in the wavefront. Particle motion directions of three types of body-waves in anisotropic media can be obtained by solving eigenvectors of the Christoffel equation (1.21).

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Yong-Gang Li

Following Keith and Crampin (1977), and Fryer and Frazer (1987), components g1 , g2 and g3 , each of three eigenvectors Uk (k = 1, 2, 3), are related by g2 g3 g1  (1.45)  =  =    Γ12 Γ11 − G   Γ11 − G Γ22 − G  Γ12     det  det  Γ23  Γ13  Γ12 Γ22 − G  Γ23 g3 g2 g1  =  =  (1.46)      Γ13 Γ33 − G   Γ11 − G  Γ Γ 13 13 Γ11 − G       det  det  det  Γ23  Γ13 Γ33 − G  Γ12  Γ12 Γ23 g2 g3 g1 = =        Γ22 − G   Γ Γ Γ Γ − G − G Γ22 − G  23 23 12 12      det  det  det  Γ23 Γ33 − G  Γ13  Γ23  Γ33 − G Γ13   Γ12 det  Γ13

(1.47)

where terms in the denominators are given in Section 1.3.1 for specific anisotropic systems. Three unnormalized polarizations Uk (k = 1, 2, 3) can be in principle obtained by inserting three eigenvalues Gi (i = 1, 2, 3) into any of expressions (1.41)–(1.43). The choice of the suitable solution of eigenvectors from expressions (1.45)–(1.47) for three types of body-waves can be made as follows. Degeneracies of two shear waves occur in the vicinities of cusps where two S-wave surfaces touch together as described by Crampin (1981), and Crampin and Peacock (2005). Such phenomenon can be seen in previous demonstrations where two shear waves have the same velocity when wave propagate parallel to the symmetry of anisotropy or at some particular incident angles. In these cases, two shear-waves have the same vertical slowness so that three distinct eigenvectors are no longer obtained from a single expression. In general, we find the correct eigenvector for P-wave from (1.45), for the first shear-wave from (1.46) and for the second shear-wave from (1.47). If equation (1.46) gives the trivial solution, (1.47) is used for the first shear-wave S1 . If equation (1.47) gives the trivial solution, (1.46) is used for the second shear-wave S2 . We used Model III with the orthorhombic olivine having the elastic tensor as shown in (1.36) to illustrate wave polarizations. Figure 1.14 shows the deviation of the polarization vector from the slowness vector of P, S1 and S2 waves for the phase propagation vector lying in (a) x2 = 0 plane, (b) x2 = x1 plane and (c) x1 = 0 plane. The solid line denotes the deviation of P-wave polarization vector, the long dashed line for S1 -wave and the dash-dot line for S2 -wave. The deviation is measured by degrees of the angle of the polarization vector from the slowness vector (the normal of the wavefront) for P-wave and from the wavefront perpendicular to the slowness vector for two shear waves. For the propagation vector in x2 = 0 plane, the polarization vector of Pwave deviates from the slowness vector with the maximum angle 7.8◦ when the slowness vector is in the direction 35◦ with respect to x1 axis. Because S1 -wave has its polarization deviation with respect to the wavefront with the same value

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as that of P-wave polarization from the slowness vector, the long dashed curve for S1 -wave is overlapped by the solid curve for P-wave as shown in Figure 1.14a. The polarization deviation of S2 -wave from the wavefront is negligible in this specific case. For the propagation vector lying in x1 = x2 plane, polarization vectors of three types of body-waves are in general separated as shown in Figure 1.14b. For the propagation vector in x1 = 0 plane, S2 -wave has its polarization deviation from the wavefront with the same value as that of P-wave polarization deviation from the slowness vector as shown in Figure 1.14c. In this specific case, the polarization deviation of S1 -wave from the wavefront is negligible.

Fig. 1.14 Polarization deviation from slowness vectors of three quasi body-waves in Model III for the propagation vector lying (a) x2 = 0 plane, (b) x1 = x2 plane and (c) x1 = 0 plane. Solid lines are for P-wave, long dashed and dash-dot lines are for two shear-waves S1 and S2 , respectively.

For more details, we plot three components of the polarization vectors in Figure 1.15 (a) for P-wave, (b) for S1 -wave and (c) for S2 -wave, corresponding to the propagation vector lying in x2 = x1 plane. Three components are defined as the component (solid line) along the slowness vector direction, the component (long dashed line) perpendicular to the slowness vector and in the vertical plane, and the component (dash-dot line) perpendicular to the slowness vector but in the horizontal plane, respectively. In addition to the deviation of the polarization direction from the slowness vector or the wavefront, the energy transport direction (i.e. the group velocity vector or the ray vector) deviates in general from the slowness vector in anisotropic media as we discussed in Section 1.3.2. Here, we illustrate such deviations using Model IV. Figure 1.16 shows the group velocity vector deviation from the normal of wavefront for P-wave and from the wavefront for two S-waves when the propagation vector lies in x2 = x1 plane. The deviation is measured by degrees of the angle. The solid line is for P-wave, the long dashed line for S1 -wave and the dash-dot line for S2 -wave.

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Fig. 1.15 Three components of polarization vectors for (a) P-waves, (b) S1 -wave and (c) S2 -wave in x2 = x1 plane in Model III.

In the next example, we use Model II for the hexagonal granitic rock with aligned vertical cracks (the crack density ε = 0.075) having the elastic tensor shown in (1.18′′ ) including the second-order correction in ε. Figure 1.17 shows the deviation of polarization vector from the slowness vector of P, S1 and S2 waves for the phase propagation √ vector lying in (a) x2 = 0 plane, (b) x2 = √ 3/3x1 plane and (c) x2 = 3x1 plane. The deviation is measured by the angle of the polarization vector from the slowness vector for P-wave and from the wavefront for two S-waves. The solid line denotes the propagation deviation of P-wave. One shear-wave polarized in the plane including the slowness vector and perpendicular to the wavefront has the same deviation from the wavefront as the value of P-motion deviation from the slowness vector so that its deviation curve is overlaid by the solid line. The other shear-wave polarized orthogonally has its polarization direction vector in the wavefront in the hexagonal medium so that the deviation

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Fig. 1.16 Deviations of group velocity vectors from slowness vectors for P-wave (solid line), and two shear-waves S1 (long-dashed line) and S2 (dash-dot line) in x2 = x1 plane in Model IV.

value is zero. When the phase propagation vector is parallel to the crack plane, i.e., x2 = 0 plane, the deviation of polarization vector of three types of body waves is zero. The maximum value of the polarization deviation of P-wave for the rock containing dry cracks is 8.5◦ when the propagation direction is at 40◦ to x1 axis in the x2 = 0 plane as shown on the left side in Figure 1.17a. Such deviation of polarization direction is considerably reduced in the medium with water-saturated cracks (right plot). A specific case for the water-saturated cracks occurs when the propagation direction is at 45◦ with respect to x1 axis in x2 = 0 plane as shown on the right side in Figure 1.17a. The polarization vector of P-wave is coincident with the slowness vector and polarization vectors of two shear-waves lie on the wavefront. Another specific case is seen when the propagation direction is at 60◦ with respect to x2 axis in the same figure. Polarization deviation curves of two shear-waves are exchanged, corresponding to the intersection of slowness surfaces of two shear-waves. In this case, two shear-waves have the same phase velocity as we discussed in Section 1.3.1. We also computed deviations of group velocity vectors (the wave energy transport directions of three quasi body-waves) from the slowness vector using Model II. Figure√ 1.18 shows results for the propagation vector lying in (a) x2 = 0 plane, √ (b) x2 = 3/x1 plane and (c) x2 = 3x1 plane. The deviation is measured by degrees of the angle. The solid line denotes the deviation for P-wav, the long dashed line for the faster shear-wave S1 and the dash-dot line for the slower shear-wave S2 . Energy transport directions of three type waves are in general not along the phase propagation directions. The deviation of P-wave due to dry cracks (the crack density 0.075) is as large as 12◦ shown on the left side in Figure 1.18a. It is noted that deviations of P-wave due to water-saturated cracks reduced remarkably while deviations of two shear-waves due to saturated cracks

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Fig. 1.17 Polarization deviation from slowness vectors of three quasi body-waves in Model II √ with the crack density 0.075√for the propagation vector in (a) x2 = 0 plane, (b) x2 = 3/3x1 plane and (c) x2 = 3x1 plane. Left plot is for the dry crack. Right plot is for the water-saturated crack. Two specific cases occur at angle 45◦ and 60◦ , referring to the description in Section 1.3.3.

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Fig. 1.18 Deviations of group velocity vectors from slowness vectors of three type body-waves in Model II for the propagation vector lying on three vertical planes as used in Figure 1.18. Left plot is for the dry crack. Right plot is for the water-saturated crack. Two specific cases occur at angle 45◦ and 60◦ , referring to the description in Section 1.3.3.

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increase for comparison with those due to dry cracks. Two special cases are seen on the right side of Figure 1.18a, similar to those shown in Figure 1.17a. When the propagation direction is at 45◦ , the energy transportation is along the phase propagation direction. When the propagation direction is at 60◦ , wave surfaces of two shear-waves have the cusp phenomenon. We have discussed in detail the most important features of seismic wave propagation, including directional variation of phase velocities, traveltime separation between two shear-waves and deviation of polarizations and energy transportation directions apart from the wavefront or it’s normal in anisotropic media. These characteristics of seismic wave anisotropy have been observed in our field experiments (Leary et al., 1987; Li et al., 1987, 1990; Li, 1988).

1.4

Reflection and Refraction of Plane Waves at a Planar Boundary between Anisotropic Media

We have discussed propagation of plane waves in an unbounded homogeneous anisotropic medium in above sections. Now, we consider the problem of reflection and transmission of plane wave at the planar boundary between two anisotropic media. This problem had been treated primarily in study of the crystal by Synge (1957), Fedorov (1968) and Musgrave (1970), and then solved with a greater generality in study of seismic wave propagation by Keith and Crampin (1977), Daley and Horn (1977). Daley & Horn (1977, 1979) gave complete solutions of reflection and transmission coefficients foe transversely isotropic media. Since then, those solutions have been coded in quite commonly-used computer programs for wave propagation in azimuthally anisotropic media by Crampin et al. (1986), and Gajewski and Pˇsenˇcik (1987). We also coded a computer procedure for computation of reflection and a 3-D ray tracing modeling program for layered anisotropic heterogeneous media due to aligned cracks (Li et al., 1988; Li al., 1990). The computation of reflection and transmission coefficients at an interface between two anisotropic media is a formidable problem in seismic wave propagation because of the additional complications in algebraic handling of the pertinent equations. For the isotropic medium, complete solutions can be found ˇ in textbook (e.g., Cerven´ y et al., 1977; Aki and Richards, 1980; Kennett, 2002; Chapman, 2004). However, solutions for general anisotropic media have to be determined numerically. We discuss them in the following sections.

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Seismic Wave Propagation in Anisotropic Rocks. . .

53

Slowness Surface Method

When an elastic plane wave propagates through the boundary between two anisotropic media, all six types of reflected and transmitted P, SV and SH waves are generated at the interface. P, SV and SH waves are coupled with and converted to each other. The incident wave and its converted waves can no longer be thought as purely longitudinal or transverse waves with directionally constant wave velocities. Their velocities obtained from eigenvalues depend on the phase propagation direction, i.e., the wave normal. In addition, the direction of the energy transport does not coincident with the wave normal while the particle motion direction is also deviated from the wave normal or apart from the wavefront (refer to Section 1.3). In the anisotropic medium, Snell’s law can no longer simply be used. The reflection and transmission coefficients cannot be calculated analytically as they are usually calculated by explicit expressions in the isotropic medium. We introduce here a numerical technique called slowness surface method to solve this problem. In this method, the slowness surface is defined as the locus of the endpoint of the slowness vector s corresponding to different propagation directions. The slowness vector s is a vector having the same direction as the wavefront normal and the magnitude equal to the reciprocal of phase velocity in that direction. Let represent the slowness vector of the incident wave and s(v) (v = 1, 2, · · · , 6) represent slowness vectors of reflected and transmitted P, SV and SH waves, respectively. In isotropic media, the slowness surface consists of two concentric spherical sheets, the inner one representing P-wave and the outer one representing SV and SH waves. In anisotropic media, however, there are three shaped surfaces, one for each type of waves (refer to Section 1.4.2). Consider a plane boundary between two anisotropic media with unit normal n and set the origin of the coordinate system on the interface, the equation of the interface plane may be written as r·n= 0

(1.48)

where r is the position vector. The usual boundary conditions, i.e. the continuity of displacement and stress at the interface with the normal in the x3 axis, may be written as 6 

(−1)v uv = u0

(1.49)

v=1

6 

(−1)v σij (uv )nj = σi3 (u0 )nj ,

i, j = 1, 2, 3

(1.50)

v=1

where u0 is the displacement of incident wave and uv (v = 1, 2, · · · , 6) are displacements of reflected and transmitted waves. The factor (−1)v are used to

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denote the reflected or transmitted waves. nj are components of the normal of interface in the Cartesian coordinates.   1 m ∂ul ∂uk v σi3 (u ) = ci3kl + (1.51) 2 ∂xl ∂xk v where cm i3kl is the fourth-order tensor of elastic constants for the upper or lower (m = 1 or 2) layer. Equations (1.49) and (1.50) constitute six equations in six unknown values of displacements of reflected and transmitted waves. However, they cannot be solved in analytical forms of reflection and transmission coefficients in general, but have to be solved by numerical methods. Let us denote the plane wave by v (v) (1.52) Uj = Avj eiω(s ·r−t) In order to match boundary conditions in equations (1.49) and (1.50), the spatial variation of reflected and transmitted waves along the interface must be equal to that on the incident wave and for r on the interface, we have s(o) · r = s(1) · r = · · · = s(6) · r

(1.53)

This equation may be rewritten as (s(0) − s(1) ) · r = · · · = (s(0) − s(6) ) · r = · · ·

= (s(1) − s(2) ) · r = · · · = (s(5) − s(6) ) · r = 0

(1.54)

From expression (1.48) of the interface plane, any vector r contained in the interface n must be perpendicular to the difference of any two slowness vectors. In other words, the difference between the slowness of incident wave and the slowness of permissible reflected and transmitted waves must be parallel to the normal of the interface n. It can be written in the general vector product form [s(0) − s(1) , n] = · · · = [s(0) − s(6) , n] = · · ·

= [s(1) − s(2) , n] = · · · = [s(5) − s(6) , n] = 0

(1.55)

[s(0) , n] = [s(v) , n]

(1.56)

or Equations (1.55) and (1.56) give generalization of Snell’s law. It states that the vector product of the slowness vector of any type of waves by the normal, the interface must be equal to any other product of (1.56) a = [s(0) , n] = [s(v) , n]

(1.57)

so that the vector n normal to the interface and the slowness vector s(v) of all types of waves must lie in the incident plane defined by a·r = 0

(1.58)

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Then, forming the vector product b = n×a and using equation (1.57), we obtain n × a = n × s(v) × n = s(v) − s(v) n · n

(1.59)

s(v) = n × a + s(v) n · n = b + ξ (v) n

(1.60)

so that where the first vector b on the right side of equation (1.60) is the projection of the slowness vector on the interface and the second vector ξ (v) n is the projection of slowness vector on the n normal to the interface. Slowness vectors of all types of reflected and transmitted waves have a common projection vector b on the interface, but have their own projection vectors on the normal of the interface with different magnitudes ξ (v) (refer to Fig. 1.20). Thus, each slowness vector can be written as the sum of the common vector b and the vector ξ n for a permissible solution to the reflection and transmission problem at the interface. Corresponding to different wave normal directions ranging from 0◦ to π, the ends of slowness vector s(v) (v = 1, 2, · · · , 6) form six slowness surfaces on two sides of the boundary between two anisotropic media. When the normal of the interface is given and the common vector b may be known from the incident wave so that the problem is reduced to find the scalar value ξ (v) for each type of waves. The ξ (v) is simply the length of the perpendicular line drawn from the end point of b to the vth slowness surface. The value of ξ (v) for each type of waves may be numerically determined by the root-finding procedure. Its final value must satisfy the generalized Snell’s law (1.60). Slowness vector of six types of waves can be obtained from solutions of the eigenvalue problem as described in Section 1.3. Eigenvalues can be found from the characteristic equation (1.23). Γ − Gm | = 0, det |Γ

(1.61)

where Gm (m = 1, 2, 3) denote three eigenvalues of the Christoffel matrix Γ which has its elements (1.62) Γjk = aijkl si sl where aijkl si sl = cijkl /ρ is the fourth-order tensor of elastic constants divided by rock density. si (i = 1, 2, 3) are components of slowness vector s. There are six distinct eigenvalues on both sides of the interface between two anisotropic media. In the Cartesian coordinates, we define three components of the slowness vector s as s1 =

1 cos(i), v

s2 =

1 cos(j), v

s3 =

1 cos(k) v

(1.63)

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where cos(i), cos(j) and cos(k) are directional cosines of the slowness vector and v is the phase velocity for each wave. For a specific direction of phase propagation, the wave velocity is then determined by (1.61). In the root-finding procedure, we scan the angle between the slowness vector and the plane of the interface to produce a locus of the endpoint of the slowness vector for each wave, i.e. the slowness surface. This surface intersects the perpendicular line from the end of the common vector b on the interface between two anisotropic media. The required value of the scalar ξ (v) is simply the length between this intersection and the end of b (refer to Section 1.4.2). We have mentioned that the slowness surfaces are concentric spheres in the isotropic medium. Figure 1.19 shows an example of such case (Model V) with an interface between two isotropic media. The rock density in the bottom medium is 2.5 g/cm3 , and λ = µ = 26.25 × 1010 dyn/cm2 , with the elastic tensor as shown in equation (1.18). The top medium has the same rock density, but λ = µ = 18.75 × 1010 dyn/cm2 . The incident P-wave yields reflected and transmitted P- and SV-waves when it propagates from the top layer through the interface between the top and bottom layers. SH-wave is decoupled with P- and SVwaves in the isotropic case. The corresponding phase velocities of reflected and transmitted waves and their emergence angles with respect to the normal of interface are shown in the bottom-left plot in Figure 1.19. For the isotropic media, wave velocities are independent of propagation direction as shown at the right plot in Figure 1.19. Dashed lines denote P-wave velocities and solid lines denote S-wave velocities. Velocity curves above the interface are for the top layer while velocity curves below the interface are for the bottom layer. The next example is an anisotropic case (Model VI) including an interface between two anisotropic media with orthorhombic symmetry. The bottom medium is the olivine crystalline rock having the density 3.324 g/cm3 and the 6 × 6 elastic tensor as shown in equation (1.36). The 3 × 3 Christoffel matrix takes the form of equation (1.27). Assume that the top medium has the same material density and 32 values of elastic constants in the bottom layer. Figure 1.20 shows computed slowness surfaces when the slowness vector lies in (a) x2 = 0 plane, (b) x2 = x1 plane and (c) x1 = 0 plane. The incident P-wave with slowness vector s(0) propagates from the top layer at an angle 30◦ with respect to the normal of interface B in x3 = 0 plane with its normal n parallel to x3 axis. Six types of waves yield when the P-wave hits the interface. s(v) (v = 1, · · · , 6) are slowness vector of the incident P-wave, reflected P, S1 and S2 waves, and transmitted P, S1 and S2 waves. The common vector b lies on the interface plane. Six slowness vectors for reflected and transmitted waves have their ends intersecting the perpendicular line from the endpoint of b. Six slowness surfaces in the top and bottom anisotropic media are in general separated from each other and have non-spherical shape. Dash-dot lines denote P-waves; solid lines and dashed lines are for two shear-waves S1 and S2 , respectively. S1 -wave is the fast

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Fig. 1.19 Computed slowness surfaces of reflected and transmitted waves in Model V with isotropic media. The incident wave is P-wave. Four slowness surfaces of reflected and transmitted P-waves and S-waves are concentric spheres. Velocities are independent on the wave propagation direction. Solid lines are for S-waves and dashed lines for P-waves.

shear wave and S2 -wave is the slow shear wave. Phase velocities and emergence angles of reflected and transmitted waves corresponding to the incident P-wave are plotted at left bottom in Figure 1.20. The directional variation of velocities is shown at the right side in the figure. The angle denotes the propagation direction with respect to the normal of interface. Computed velocity curves in the bottom layer are exactly same as the results shown in Figure 1.19. For a comparison with the incident P-wave, Figure 1.21 shows computed results for the anisotropic case for (a) the incident S2 -wave and (b) the incident S1 -wave when the slowness vector lies in x2 = 0 plane. The further example is the case (Model VII) of granitic rock containing aligned dry cracks with hexagonal symmetry. The host rocks in the top and bottom layers have the same elastic constants as those in the isotropic Model V. The

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Fig. 1.20 Computed slowness surfaces of six reflected and transmitted waves in Model VI with the olive crystalline rocks have orthorhombic symmetry when slowness vector lying in (a) x2 = 0 plane, (b) x2 = x3 plane and (c) x1 = 0 plane. The angle denotes the phase propagation direction with respect to x3 = 0. The incident wave is P-wave. Six slowness surfaces are usually not spheres. Velocities depend on the propagation direction. Dashed-lines are for P-waves, solid lines and long dashed lines are for two shear-waves.

material density is 2.5 g/cm3 . We assume that a set of vertical cracks with the crack density 0.075 exists in the top and bottom layers. The normal of crack plane is in x1 axis. The 6 × 6 elastic tensor of the bottom medium may be obtained from equation (1.18′′ ) with the second-order correction due to aligned cracks. The elastic tensor of the top layer may be obtained in the same way. The 3 × 3 Christoffel matrix Γ has the form given in equation (1.28). The interface is assumed in x3 = 0 plane. The incident P-wave propagates from the top layer at the angle 30◦ to the normal of interface. For the rock containing dry cracks, Figure 1.22 shows computed slowness surfaces of six types of waves for the slowness vector lies in (a) x2 = 0 plane,

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Fig. 1.21 Six computed slowness surfaces of reflected and transmitted waves in Model VI with orthorhombic symmetry for (a) the incident S1 -wave and (b) the incident S2 wave in x2 = 0 plane.

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Fig. 1.22 Six computed slowness surfaces of reflected and transmitted waves in Model VII with the cracked granitic rock with dry cracks having when √ hexagonal symmetry √ slowness vectors lying in (a) x2 = 0 plane, (b) x2 = 3/3x1 , and (c) x1 = 3x1 plane and (d) x1 = 0 plane. Vertical cracks are aligned in x1 = 0 axis. Six slowness surfaces are in usually not spheres. Velocities vary with the wave propagation direction. Dashed-lines are for P-waves, solid lines and long dashed lines are for two shear-waves.

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√ √ (b) x2 = 33 x1 plane, (c) x2 = 3x1 plane and (d) x1 = 0 plane. The common vector b lies on the interface plane B. s(0) denotes the slowness vector of the incident P-wave and s(v) (v = 1, · · · , 6) denote slowness vectors of reflected and transmitted waves. The dash-dot line is for P-wave, the long dashed line and solid line are for two shear-waves SV and SH. When the slowness vector lies on the vertical plane parallel or perpendicular to the crack plane, SH-wave is not yielded by P-wave because they are decoupled from each other in the anisotropic symmetry plane. We still plot slowness surfaces in Figures 1.22a and 1.22d to illustrate the computation of phase velocities. When the wave normal in the vertical plane is perpendicular to the crack plane, SH-wave travels faster than SV-wave (Fig. 1.22a). Vice versa, when the wave normal in the vertical plane is parallel to the crack plane, SV-wave travels faster than SH-wave (Fig. 1.22d). Computed velocities of six types of waves corresponding to the incident P-wave are listed at bottom-left in this figure. Phase velocities plotted at right of the figure show directional variations. The angle is with respect to x3 = 0 plane. The computed results for the bottom layer are consistent with the results shown in Figure 1.12. Figure 1.23 shows computed slowness of six types of waves in Model VII with water-saturated cracks. The results are plotted using the same format as in Figure 1.22. In this case, slowness surfaces of two shear-waves are intersected in certain propagation directions as discussed earlier. For a comparison with the incident P-wave, Figure 1.24 shows computed results for the rock containing dry cracks when the incident waves are (a) S1 -wave and (b) S2 -wave as the slowness √ vector lies in x2 = 3/3x1 plane. These results from numerical computation using slowness surface method are consistent with the analytical results shown in Sections 1.1 and 1.3. Our rootfinding computer procedure for solving velocities and emergence angles of reflected and transmitted waves at the planar boundary between two anisotropic media may be extended to the general case with irregular interface as long as the interface is smooth enough in a wavelength, and the lateral variation of anisotropy and heterogeneity in not strong. We used this computer procedure in the 3-D ray tracing program for layered anisotropic inhomogeneous media (Li, 1988; Li et al., 1990). In the next section, we discuss the method used to compute reflection and transmission coefficients of six types of waves at the interface between two anisotropic media.

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Fig. 1.23 Six computed slowness surfaces of reflected and transmitted waves in Model VII with the rock containing water-saturated cracks.

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Fig. 1.24 Six computed slowness surfaces of reflected and transmitted waves in Model VII with the rock √ containing dry cracks. The incident wave is (a) S1 -wave and (b) S2 wave in x2 = 3/3x1 plane.

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Reflection and Transmission Coefficients

In the previous section, we have described the solutions for phase velocities and emergence angles of six reflected and transmitted waves at the boundary between two anisotropic media. We now discuss the solution of displacements of these waves. From boundary conditions (1.49) and (1.50), we have obtained a generalization of Snell’s law (1.56). Putting (1.56) into the characteristic equation (1.61), we constitute six simultaneous equations with six unknown displacements of reflected and transmitted waves. However, these equations become unmanageable for general anisotropic media. For example, the term in the scalar ξ 6 involves in a coefficient with summation over 12 indices and hence has 312 terms. However, for an anisotropic system with higher degrees of symmetry in seismological study, the number of terms is considerably reduced. We have mentioned in Section 1.3 that the orientation of the displacement vector Uk for a specific wave is determined by the eigenvector of the Christoffel tensor Γ . The eigenvector does not in general lie in the direction of the slowness vector for P-wave or in the wavefront for two shear-waves. The displacement of a plane wave may be written as U (γ) = A(γ) g (γ) e−ik

(γ)

(s1 x1 +s2 x2 +δ (γ) s3 x3 −v (γ) (ϕ(γ) θ (γ) )t)

(1.64)

where γ = 0, 1, · · · , 6; 0 corresponds to the incident wave, 1,3,5 to reflected P, S2 and S2 waves, and 2,4,6 to transmitted P, S1 and S2 waves. k (γ) and v (γ) are wavenumber and phase velocity of the γth wave, respectively. ϕ(γ) and θ(γ) are direction and azimuthal angle of the slowness vector corresponding to the γth wave. δ (γ) = −1 is for the incident and transmitted waves, and δ (γ) = 1 is for reflected waves. A(γ) is the amplitude of the γth wave, which has three (γ) (γ) (γ) components in directions determined by three components g1 , g2 and g3 (γ) of the eigenvector g . They have been solved in Section 1.3 and are used to determine the direction of particle motion. We now want to know the amplitude term A(γ) (xj ). Here, s1 , s2 and s3 are components of the slowness vector s, which may be expressed in the Cartesian coordinates [x1 x2 x3 ] as 1 (m1 i + m2 j + m3 k) (1.65) v where v is the phase velocity. Three components of the wave normal m may be written as s=

m1 = cos ϕ cos θ m2 = cos ϕ sin θ

(1.66)

m3 = sin ϕ where ϕ is the inclination angle of the wave normal with respect to x3 = 0 plane and θ is the azimuthal angle of the wave normal with respect to x1 axis. The

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continuity of displacement and stress on the boundary between two anisotropic media requires u0i + u1i + u3i + u5i = u2i + u4i + u6i

(1.67)

6 2 4 3 5 1 0 σi3 + σi3 = σi3 + σi3 + σi3 + σi3 + σi3

(1.68)

where i = 1, 2, 3 correspond to three components of displacement or stress in the Cartesian coordinates. Three components of stress are in general written as σ13 = c13kl

∂Uk = c15 s1 Ag1 + c25 s2 Ag2 + c35 s3 Ag3 + c56 (s2 Ag1 + s1 Ag2 ) + ∂xi

c55 (s3 Ag1 + s1 Ag3 ) + c45 (s3 Ag2 + s2 Ag3 ) σ23 = c23kl

(1.69)

∂Uk = c14 s1 Ag1 + c24 s2 Ag2 + c34 s3 Ag3 + c46 (s1 Ag2 + s2 Ag1 ) + ∂xi

c45 (s3 Ag1 + s1 Ag3 ) + c44 (s3 Ag2 + s2 Ag3 ) σ33 = c33kl

(1.70)

∂Uk = c13 s1 Ag1 + c23 s2 Ag2 + c33 s3 Ag3 + c36 (s3 Ag2 + s2 Ag1 ) + ∂xi

c35 (s3 Ag1 + s1 Ag3 ) + c34 (s3 Ag2 + s2 Ag3 )

(1.71)

where we have neglected exponential terms for brief. cij (i, j = 1, 2, 3) are elastic constants of anisotropic media. Components of the slowness vector si (i = 1, 2, 3) and components of the orientation of displacement gi (i = 1, 2, 3) may be solved as an eigen-problem as discussed before. Boundary conditions in equations (1.67) and (1.68) can be written in matrix form, 

g10





g12

 0  2  g2   g2     g0   g2  3   3  0 = 2  g13   g13     0   2  g23   g23 0 g33

2 g33

g14

g16

g24

g26

g34

g36

4 g13

6 g13

4 g23

6 g23

4 g33

6 g33

−g11

−g13

−g31

−g33

−g21

−g23

1 −g13

3 −g13

1 −g33

3 −g33

1 −g23

3 −g23

A  1    A2  −g25       5   A −g3  3     5  −g13 A  4    5  −g23   A5   5 −g33 A −g15

(1.72)

6

Six unknown reflection and transmission coefficients Ai (i = 1, · · · , 6) may be solved from six simultaneous equation (1.72) by a standard numerical routine, for example, LEQ2C in the software package IBMSSP as we used in our 3-D ray tracing program for the layered anisotropic heterogeneous media. Elements in the matrix on the left hand of (1.72) are terms for the incident wave propagating in the top medium. Even numbers of superscripts of elements in the first matrix on the right hand of (1.72) denote transmitted (down going) waves and odd numbers denote reflected (up going) waves. For the anisotropic medium with

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higher degrees of symmetry, such as the orthorhombic or hexagonal symmetry, stress components are simplified to   ∂U3i ∂U1i i = c55 (pi3 Ai g1i + pi1 Ai g3i ) (1.73) = c55 + σ13 ∂x3 ∂x1   ∂U3i ∂U2i i = c44 (pi3 Ai g2i + pi2 Ai g3i ) = c44 (1.74) + σ23 ∂x3 ∂x2 ∂U i ∂U i ∂U i i = c13 1 + c23 2 + c33 3 σ33 ∂x2 ∂x2 ∂x3 i i i i i i = c13 p1 A g1 + c23 p2 A g2 + c33 pi3 Ai g3i (1.75) For the transversely isotropic medium with the symmetry axis normal to the interface plane, reflection and transmission coefficients may be computed by the explicit formulation given by Daley and Horn (1977, 1979). A complete set of equations for computation of reflection and transmission coefficients at the interface between two transversely isotropic media is given in equation (1.6.1) of this chapter after correction of some typographic errors in D & H formulation. We now show examples in general cases below. The first example is an isotropic case using the same elastic parameters in Model V. Figure 1.25a shows computed reflection and transmission coefficients of P- and S-waves. The incident P-wave yields reflected P-wave (P` P´ ) and S´ and transmitted P-wave (P` P` ) and S-waves (P` S). ` The total reflection wave (P` S) of P-wave occurs when the incident angle is beyond the critical angle 52◦ with respect to the normal of the interface plane. The second example is the case of orthorhombic media having the same elastic parameters in Model VI. The incident P-wave yields six types of waves: reflected waves (P` P´ , P` S´1 , P` S´2 ) and transmitted waves (P` P` , P` S`1 , P` S`2 ). Computed re-

Fig. 1.25 (a) Computed reflection and transmission coefficients of reflected and transmitted P-waves and S-waves in the isotropic Model V corresponding to the incident P-wave. (b) Computed reflection and transmission coefficients of six types of reflected and transmitted P-waves and S-waves in the orthorhombic anisotropic Model VI corresponding to the incident P-wave.

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flection and transmission coefficients are plotted in Figure 1.25b, each coefficient curve marked by a symbol of wave type. The incident P-wave yields two types of orthogonal-polarized shear-waves S1 and S2 . The total reflection of P-wave occurs beyond the critical angle 55◦ with respect to the normal of the interface. Third example is the case of hexagonal anisotropic media due to aligned cracks. Elastic constants are same as in Model VII. When the slowness vector lies in the vertical plane parallel to the crack normal (i.e., x1 axis), the incident P-wave yields reflected and transmitted P-waves and S1 -waves (SV-waves) polarized in the vertical incident plane. The S2 -wave (SH-wave) decouples from them because the incident wave propagates in the anisotropic symmetry plane. Computed reflection and transmission coefficients are shown in Figure 1.26a for the rock containing dry cracks and Figure 1.26b for the rock containing water-saturated cracks with the crack density 0.075. The total reflection of P-wave occurs beyond the critical angle 65◦ with respect to the vertical (i.e., x3 axis) in the case of dry cracks and 55◦ in the case of saturated cracks. The critical angle depends on the increase of velocity contrast between the incident and transmitted P-waves,

Fig. 1.26 Computed reflection and transmission coefficients of reflected and transmitted P-waves and S1 -waves in the hexagonal anisotropic Model VII corresponding to the incident P-wave propagating in the vertical plane parallel to the normal of (a) dry cracks and (b) water-saturated cracks, and corresponding to the incident S1 wave for the rock containing (c) dry cracks and (d) water-saturated cracks.

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and is strongly affected by the presence or absence of fluid in crack. Figures 1.26c and d shows computed reflection and transmission coefficients of P-waves and S1 -waves in the rock containing dry cracks and water-saturated cracks when the incident S1 -wave travels in the vertical plane parallel to the crack normal. When√the incident P-wave travels in a non-symmetry plane, for example, in x2 = 3/3 x1 plane, it yields six types of reflected and transmitted waves at the interface. Figure 1.27 shows computed reflection and transmission coefficients in the case where six coefficient curves denoted by P` P´ , P` S´1 , P` S´2 for reflected waves and P` P` , P` S`1 , P` S`2 for transmitted waves. Two shear-waves have orthogonal polarization, one in the vertical plane and the other in the horizontal plane. Both planes contain the crack normal in x1 direction. The S1 -wave is defined as the faster shear-wave and the S2 -wave the slower shear-wave. The total reflection of P-wave occurs beyond the critical angle 60◦ with respect to the normal of the interface plane, i.e., x3 axis.

Fig. 1.27 Six computed reflection and transmission coefficient curves in Model VII √ corresponding to the incident P-wave propagating in x2 = 3/3 x1 plane. Six types of reflected and transmitted waves are yielded at the interface between two anisotropic media. Vertical dry cracks are aligned in x1 axis.

In this section, we have shown numerical solution for problems of reflection and transmission of the plane wave propagating through a planar boundary between two anisotropic media. The method discussed above has been used in our 3-D ray tracing procedure for multiple-layered anisotropic inhomogeneous media including aligned cracks (refer to later Sections).

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Ray Tracing in Anisotropic Heterogeneous Media

In the previous section, we have seen that wave propagation in anisotropic media is characterized by directional velocities and shear-wave splitting. Two orthogonally polarized shear-waves travel at different speeds in anisotropic media in general. Their particle motions no longer remain in the wavefront. Such characteristics of wave propagation are useful in stratigraphic and lithological interpretations because shear-wave splitting can be used to detect and quantify the orientation and density of cracks permeated in crustal rocks (e.g., Crampin, 1985; Malin and Waller, 1985; Kaneshima et al., 1987; Leary et al., 1987; Li et al., 1987, 1994; Peacock et al., 1988; Major et al., 1988; Shear and Chapman, 1989; Li 1996). In this section, we introduce the ray tracing method used in study of wave propagation in anisotropic heterogeneous media. We developed an algorithm for computation of traveltimes, ray paths, particle motion directions and ray amplitudes of P, SV and SH waves in 3-D layered heterogeneous anisotropic media with cracks aligned in an arbitrary symmetry axis. We use Hudson’s (1981) formula for elastic constants of rock matrix containing aligned cracks (refer to Sections 1.2 and 1.3) and use the slowness surface method (Synge, 1957; Fedorov, 1968) to compute reflection and transmission coefficients at a smooth boundary between two anisotropic layers (refer to Section 1.4). Our ˇ algorithm is similar to Cerven´ y (1972) ray tracing system and Gajewski and Pˇsenˇcik (1987) for anisotropic heterogeneous media to evaluate traveltimes, ray geometrical spreading and amplitudes along raypaths in 3-D space by means of standard numerical techniques. Ray synthetic seismograms may be constructed by elementary waves, which are simply computed from traveltimes and vector amplitudes obtained by ray tracing and computation of wave polarizations, reflection and transmission coefficients. Model parameters, including elastic constants of the uncracked rock, crack orientation and density are determined by the trial and error method first and then improved by an optimization procedure (Aki and Lee, 199; Aki et al., 1977). This computer program has been used to interpret the vertical seismic profiling (VSP) data at Oroville and Hi Vista in northern and southern California as well as the Los Angeles Basin Seismic Network data for detecting and quantifying the distribution of fractures in the crustal rocks, which are related to the tectonic stress distribution and hence useful in earthquake study in the interesting region (Leary et al., 1987; Li et al., 1987; Li et al., 1990, 1994; Li, 1996).

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Ray Series Method

In this section, we describe the ray tracing algorithm for computation of traveltimes, and vector ray amplitudes in layered inhomogeneous media used for interpretation of the fracture structure in a 3-D space (Li, 1988; Li et al., 1990). The ray method also known as the ray series method is a high-frequency asymptotic technique, which can only be applied if the velocity distribution (or elastic constants) in the model and at the interface under consideration are sufficiently smooth within a wavelength. The ray method is usually a fastest approach for computation when it is compared with other methods for study of wave propagation in the complex inhomogeneous anisotropic media with smoothly irregular interfaces. The equation of motion in an inhomogeneous anisotropic medium can be written in rectangular Cartesian coordinate xi as ∂ 2 Ui ∂Uk ∂ (cijkl )=ρ 2 ∂xj ∂xi ∂t

(1.76)

where Ui are components of the displacement vector U , cijkl = cijkl (xi ) is a fourth order tensor of elastic constants at xi , ρ(xi ) is the material density and t is the time. We seek the solution of equation (1.76) in the form of a ray series Uk (xi , t) =

∞ 

n=0

(n)

Uk (xi )fn (t − r(xi ))

(1.77)

where the function fn (θ), θ being the phase, satisfies the relation dfn+1 (θ) = fn (θ) (1.78) dθ Substitute (1.77) into (1.78) and compare coefficients of function fn (t − τ ) on the left-hand and right-hand sides of the equation, we obtain a basic system of equations (1.79) N (U (n) ) − M (U (n−1) ) + L(U (n−2) ) = 0 for n = 0, 1, 2, · · · , with U (−1) = U (−2) . Vector operators N , M , L are given by (n)

Nj (U (n) ) = Γjk Uk

(n)

− Uj (n)

∂Uk ∂ (n) + ρ−1 (ρaijkl pl Uk ) ∂xl ∂xi   (n) ∂Uk (n) −1 ∂ ρaijkl Lj (U ) = ρ ∂xi ∂xl

Mj (U (n) ) = pi aijkl

(1.80)

Γjk = pi pl aijkl aijkl = ρ−1 cijkl ∂τ pi = ∂xi.

(1.81)

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pi are components of the slowness vector p and the Christoffel tensor Γ is a 3 × 3 symmetric matrix. We have seen in previous sections that Γ plays an important role in study of wave propagation in anisotropic media. The basic function (1.79) may be used to derive the desired differential equations for phase function τ (xi ) in (1.87) and for amplitude coefficients U (xi ) in equation (1.105). Take the zeroth-order ray approximation, we put n = 0 in expression (1.79), N (U (n) ) = 0, f or n = 0

(1.82)

Putting (1.80) into (1.82), we obtain (0)

(Γjk − δjk )Uk

=0

(1.83)

where δjk is the Kronecker symbol. We have discussed this equation in Section (0) (0) (0) 1.3. It represents a system of three linear equations for U1 , U2 and U3 , and has a non-trival solution only if Γ − Gm | = 0 Det|Γ

(1.84)

By comparing (1.84) with (1.83), it is seen that equation (1.81) is satisfied if any of the three eigenvalues Gm (xi , pj ) is equal to 1, i.e., Gm (xi , pj ) = 1

(1.85)

This is a non-linear differential equation representing the slowness surface in eikonal equation for the anisotropic medium. For the isotropic medium, eigenvalues are simply written as G1 = αpi pi for P-wave and G1 = G2 = βpi pi for S-wave, where α and β are their velocities, respectively. Two shear waves have the same eigenvalue. For inhomogeneous anisotropic media, three separated wavefronts propagate independently. One of them corresponds to the quasiP (qP), the other two correspond to two different quasi-shear waves qSV and qSH. The eikonal equation (1.85) may be solved by means of characteristics (see Courant and Hibert, 1962), which employs the equivalence of a first-order partial differential equation with a system of first-order ordinary differential equations that is to be solved by the method of characteristics. Using Euler’s theorem on the characteristics equation (1.84), we obtain pi

∂Gm = 2Gm , (m = 1, 2, 3) ∂pi

(1.86)

In views of (1.86), the eikonal equation (1.85) is then equivalent to a system of the first-order differential equations written in the form 1 ∂Gm dxi = dτ 2 ∂pi −1 ∂Gm dpi = dτ 2 ∂xi

(1.87)

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Equation (1.87) represents rays propagating in an anisotropic medium under the zeroth-order approximation of the ray series. The system may be solved subject to the initial position xi (s0 ) and an initial direction of the wavefront defined by slowness pi (s0 ) at the initial location s = s0 . Then, the raypath with parametric representation xi (s) is determined and the normal of the wavefront is represented by the vector p. The parameter along the ray corresponds to the time of ray propagation along the raypath. This system of equations constitutes basic ray tracing system. Because of the difficulty in finding analytic solutions of eigenvalues Gm from the eikonal equation (1.85) for general anisotropic media, we seek analytical expressions for partial derivatives of the eigenvalue with respect to the slowness ∂Gm ∂Gm ∂pi and ∂xi instead of eigenvalues themselves. Using characteristic equation (1.84) and the theorem on implicit functions, the ray tracing system for inhoˇ mogeneous anisotropic media can be written as (Cerven´ y 1972) Djk dxi = αijkl pl dτ D −∂αijkl Djk dpi = pl ps dτ 2∂xi D

(1.88)

where i = 1, 2, 3; and D11 D22 D33 D12 D13 D23 D

2 = (Γ22 − 1)(Γ11 − 1) − Γ23 2 = (Γ11 − 1)(Γ33 − 1) − Γ13 2 = (Γ11 − 1)(Γ22 − 1) − Γ12 = D21 = Γ13 Γ23 − Γ12 (Γ33 − 1) = D31 = Γ12 Γ23 − Γ13 (Γ22 − 1) = D32 = Γ12 Γ23 − Γ23 (Γ11 − 1) = D11 + D22 + D33

(1.89)

We have discussed in detail the Christoffel elastic tensor Γ in Section 1.3. For a specific anisotropic medium, it may be defined by the corresponding expression. The ray tracing system of equations (1.88) and (1.89) is applicable either to the isotropic or anisotropic medium. It can be numerically solved subject to appropriate initial values xi (τ0 ) = xi0 (1.90) pi (τ0 ) = pi0 where xi0 are coordinates of the source location and are components of the slowness vector p of one of three quasi-waves at the source. Let p0 , r0 and q0 are components of the initial slowness p, written in the Cartesian coordinates as p0 = cos ϕ0 cos θ0 /v0 r0 = cos ϕ0 sin θ0 /v0 q0 = cos θ0 /v0

(1.91)

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where v0 is the phase velocity at the source location, ϕ and θ are two take-off angles as described by equation (1.34). In the case of rock containing aligned cracks, we set the normal of crack plane parallel to x1 axis in the Cartesian coordinates. ϕ0 is the inclination angle with respect to x3 = 0 plane and θ0 is the azimuthal angle to x1 axis. A standard fourth-order Rung-Kutta method is used in our computer program to solve the first-order ordinary differential equations of ray tracing system (5.88). The ray path represented by xi (τ ) and the normal direction of the wavefront are consequently obtained for each type of three quasi waves. If the normal to crack plane is parallel to the vertical plane including a survey line, the slowness vector remains in the anisotropic symmetry plane for a laterally homogeneous medium. The ray tracing procedure is thus reduced to a 2-D problem. The component of slowness vector p in the direction transverse to that plane, and hence the ray path remains in that plane. In this particular case, the Christoffel tensor Γ is written in the form of equation (1.29). Phase velocities of P, SV and SH waves can be analytically solved by the following explicit equations.  Vp = 0.5(a33 + a55 + (a11 − a33 ) sin2 ϕ + B)  Vsv = 0.5(a33 + a55 + (a11 − a33 ) sin2 ϕ − B) (1.92)  Vsh = 0.5(a44 sin2 ϕ + a66 cos2 ϕ) Where

B = [(a33 − a55 ) + 2A sin2 ϕ + A2 sin4 ϕ]

A1 = 2(a13 + a55 )2 − (a33 + a55 )(a11 + a33 − 2a55 )

(1.92′ )

A2 = (a11 + a33 − 2a55 ) − 4(a13 + a55 )2

where aij = cij /ρ are elastic parameters, and ϕ is the angle between the slowness vector and the crack plane. In this simple transversely isotropic case, SH-wave is decoupled with P and SV waves. The ray tracing system is separated into ˇ (Cerven´ y and Pˇsenˇcik, 1972) p(a11 + a55 − 2a11 a55 p2 + Aq 2 ) dx = dτ D q(a33 + a55 − 2a33 a55 q 2 + Ap2 ) dz = dτ D   ∂a12 2 2 ∂a33 2 ∂a55 ∂a11 2 p D2 + q D1 + F +2 p q (a13 + a55 ) − dp ∂x ∂x ∂x ∂x = dτ 2D   ∂a13 2 2 ∂a33 2 ∂a55 ∂a11 2 p D2 + q D1 + F +2 p q (a13 + a55 ) − dq ∂z ∂z ∂z ∂z = dτ 2D

(1.93)

1

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Seismic Wave Propagation in Anisotropic Rocks. . .

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D1 = 1 − a11 p2 − a55 q 2 D2 = 1 − a55 p2 − a33 q 2

(1.93′ )

A = a13 + 2a13 a55 − a11 a33 F = p2 + q 2 − a11 p4 − a33 q 4 + 2p2 q 2 a13 For quasi-P and quasi-SV waves, and dx = pa66 dτ dz = qa44 dτ  1 ∂a66 2 dp =− p + dτ 2 ∂x  1 ∂a66 2 dq =− p + dτ 2 ∂z

∂a44 2 q ∂x ∂a44 2 q ∂z



(1.94)



for quasi-SH wave. It should be stressed that the ray tracing system (1.88) enables the computation of rays and traveltime-position curve for an arbitrary anisotropic heterogeneous medium in general described by 21 elastic constants in all three coordinates. The medium may be vertically and laterally inhomogeneous and anisotropic. However, the variation of elastic constants in a wavelength cannot be too large for the ray method. If such variation is large in a local area, for example at the boundary between a fault zone and the wall rocks, we used a cubic spline procedure to smooth the variation as described in the later section. The laws of reflection and transmission described in Section 1.4 can be used to treat the boundary problem for an interface exists in the anisotropic media. The smoothly irregular interface may be treated as a locally planar boundary near the point where the ray strikes. The ray tracing procedure is performed in each layer from the source or from a new point at the interface where the last step of ray tracing has been terminated in the lower (or upper) layer. The initial values at that new point are obtained from the solution of the reflection and transmission laws. The fourth-order tensor of elastic constants of the medium are computed along the ray path. In the case of rock containing aligned cracks, Hudson’s formula (1981, 1982) can be used for computation of elastic constants of anisotropic rock with the first-order or second-order correction due to aligned cracks (refer to Section 1.2). The crack distribution may be vertically and laterally inhomogeneously in the rock. We used the ray tracing system to determine the ray path between a source and a receiver by an iterative procedure. In this approach, we search two takeoff angles ϕ and θ to make the ray tracing a small area including the receiver.

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Yong-Gang Li

This searching method is efficient for a receiver spread with a regular spacing, for example, locations of a set of borehole seismographs in the vertical seismic profiling (VSP) experiment. Once a ray strikes the receiver, the curve of xi (τ ) as a function of traveltime τ is registered as a successful ray path. The slowness vectors and eigenvalues of the Christoffel elastic tensor Γ along the successful ray path are stored in computer memory for the future use. Eigenvectors of Γ representing the wave polarization are computed simultaneously in the ray tracing procedure.

1.5.2

Body-Wave Polarization

We have discussed in Section 1.3 that the wave polarizations may be solved as an eigenvalue problem. The particle motion directions of three quasi-body waves qP, qSV and qSH are determined by the corresponding eigenvectors of the Christoffel elastic tensor Γ . Eigenvectors are found at the same time when the ray tracing is performed. Knowing particle motion directions is not only necessary for computation of reflection and transmission coefficients at the boundary but also useful for analysis of shear-wave splitting by means of polarization diagrams. In Section 1.3, we have discussed how to compute polarization of body-wave. For a plane harmonic wave with the slowness vector p and frequency ω propagating in the anisotropic medium, the displacement field is written as U = Ae−iωt(t−p·x)

(1.95)

where x is the position vector and A is the unit polarization vector giving the particle motion direction of the wave. Substitution of this solution into equation of motion (1.19) yields three simultaneous equations in the displacement vector U = Ui , which is written as Ui = pl pj aijkl Uk = Γik Uk

(1.96)

where pi , pj are components of the slowness vector p.aijkl = cijkl /ρ represents elastic parameters. Equation (1.96) may be rewritten as the Kelvin-Christoffel equation, (1.97) (aijkl pj pl − δjk )Uk = 0 It forms a typical eigenvalue problem. Eigenvalues of the Christoffel tensor Γ along ray paths have been determined in the ray tracing procedure so that the phase propagation directions of three types of quasi-waves are known at point xi on the ray. The particle motion directions represented by the corresponding eigenvectors may be obtained from Γ − IGm )gm = 0, m = 1, 2, 3 (Γ

(1.98)

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Seismic Wave Propagation in Anisotropic Rocks. . .

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where Gm are eigenvalues of Γ , I is the identity matrix and gm are three eigenvectors representing particle motion directions of qP, qSV and qSH, respectively. They are mutually orthogonal but are in general neither coincident with the coordinate system defined by the slowness vector p and the wavefront nor coincident with the coordinate system defined by the group velocity vector and the wave surface as described in Section 1.3. In the ray tracing procedure, the group velocity direction may be simply obtained by ray front special gradients dx, dy and dz in time increment dτ .   dx dy dz , , (1.99) Ug = dτ dτ dτ In our ray tracing program, the particle motion vector (PV), the slowness vector (SV) normal to the wavefront and the ray vector (RV) representing energy transition direction are computed simultaneously along the successful ray only (refer to the later section). Three types of vectors PV, SV and RV are in general apart from each other in the anisotropic medium (refer to Section 1.3). Synthetic polarizations are often used to constitute polarization diagrams for interpretation of shear-wave splitting. In addition to shear-waves, P-wave anisotropy has been observed and studied (e.g., Babuska and Pros, 1984; Gaiser et al., 1984; Hirahara and Ishikawa, 1984; Leary and Henyey, 1985). We also synthesize the P-wave particle motion deviation from the ray path for interpretation of our observed polarization anomalies of P-wave (Li et al., 1987).

1.5.3

Geometrical Spreading and Ray Amplitudes

As shown in previous sections, three types of wavefronts propagation in the anisotropic medium are determined by the eikonal equation (1.85). The polarization directions of particle motion direction vectors taken as the zero-order ray series coefficient vector U (0) for three types of quasi-waves may be also derived from this non-linear partial differential equation as discussed in Section 1.5.2. However, the amplitudes of the displacements are still unknown. We shall discuss them in this section. Assuming the higher order ray series coefficient vector to be written as (n) (n) (n) (1.100) U (n) = U1 g (1) + U2 g (2) + U3 g (3) (n)

(n)

(n)

where U1 , U2 and U3 are scalar coefficients of the high-order terms in ray series, and g (1) , g (2) and g (3) are three eigenvectors of the Christoffel elastic (n) tensor Γ . Three eigenvectors are mutually orthogonal. The component U1 g (1) has the same direction as the zeroth-order coefficient vector U (0) , and hence (n) called the principal component of U (n) . The other two components U2 g (2) (n) and U3 g (3) lie in the plane perpendicular to the eigenvector g (1) , having the

80

Yong-Gang Li

direction of g (2) and g (3) , respectively. They are called additional components of U (n) . (n) (n) (n) In order to obtain expressions of scalar coefficients U1 , U2 and U3 , we use the basic system equation (1.79) and introduce a vector Bn = M (U (n) ) − L(U (n−2) )

(1.101)

so that equation (1.79) may be written as (n)

(n)

(n)

N (U1 g (1) ) + N (U2 g (2) ) + N (U3 g (3) ) = Bn

(1.102)

Consider the additional components of U (n) and take into account equations (5.10) and (5.25), equation (1.102) yields (n)

(n)

U2 g (2) (G2 − 1)−1 + U3 g (3) (G3 − 1)−1 = Bn

(1.103)

where G2 and G3 are two eigenvalues of Γ . Because eigenvectors g (2) and g (3) are orthogonal, we can obtain expressions for additional components of the ray series coefficient vector U (n) as follows: (n)

U2

(n)

U3

= (G2 − 1)−1 (Bn · g (2) )

(1.104)

= (G3 − 1)−1 (Bn · g (3) )

where the vector Bn can be determined by equations (1.101) and (1.80). From equations (1.101) and (1.103), we then have (n−1) (1)

M (U1

g

(n−1) (2)

) · g (1) = L(U (n−2) ) − M (U2

g

(n−1) (3)

+ U3

g

)

(1.105)

Let (n)

ξn = M (U1 g (1) ) · g (1)

(1.106) (n)

and using (1.80), we obtain the expression for the principal component U1 the high-order coefficient vector U (n) in the Cartesian coordinates. (n)

dU1 dτ

of

(n)

+

U1 ∂(ρvi ) 1 = ξn 2ρ ∂xi 2

(1.107)

Expression (1.107) is called the transport equation where vi is the group velocity determined by equation (1.99) and ξi may be obtained from solutions of additional components. (n−1)

ξn = [L(U1

(n)

) − M (U2

(n)

+ U3 )] · g (1)

(1.108)

Introducing ray coordinates composed by the traveltime τ , and two ray parameters q1 and q2, the transport equation (1.107) can be simplified as (n)

dU1 dτ

(n)

+

U1 d(vρJ) 1 = ξn 2ρv dτ 2

(1.109)

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Seismic Wave Propagation in Anisotropic Rocks. . .

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where 1/v 2 = pi pi is the phase velocity, and J is proportional to the crosssection size of the wavefront within a ray tube. The function J is important in computation of ray amplitudes. It is written in the ray coordinates as |J| = |Xq1 × Xq2 |   ∂x1 ∂x2 ∂x3 Xq1 = , , ∂q1 ∂q1 ∂q1   ∂x1 ∂x2 ∂x3 , , Xq2 = ∂q2 ∂q2 ∂q2

(1.110)

The product of the phase velocity v and function J denotes the Jacobian transformation from the ray coordinates τ, q1 and q2 to the Cartesian coordinates xi (i = 1, 2, 3).    D(x1 , x2 , x3 )   (1.111) vJ =  D(τ, q1 , q2 ) 

The transport equation (1.107) is then integrated to yield the expression of the high-order principal coefficient   τ 1 (vρJ)τ0 (n) (n) U1 (τ ) = U1 (τ0 ) + (vρJ)τ (vρJ)τ ξn (t)dt (1.112) (vρJ)τ 2 τ0 where the subscription τ0 and τ are the traveltimes at any two successive points on the ray. If we consider the zeroth term only, for n = 0, equation (1.112) becomes  (vρJ)τ0 (n) (n) U1 (τ ) = U1 (τ0 ) (1.113) (vρJ)τ (0)

The zeroth-order principal coefficient U1 at an arbitrary point on the ray path may be determined by equation (1.113) if its value at an earlier point τ = τ0 on the ray path is known. The geometrical spreading is defined as the ratio of such two values. The initial value is usually taken as a unit. When ray paths are known for ray parameters q1 and q2 with small increments δq1 and ˇ y and Ravindra, 1971), the function J in δq2 , using the Laplace identity (Cerven´ equation (1.110) can be rewritten as √ J = EF − G E=

∂xi ∂xi ∂q1 ∂q1

F =

∂xi ∂xi ∂q2 ∂q2

G=

∂xi ∂xi ∂q1 ∂q2

(1.114)

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Yong-Gang Li

where xi are coordinates of the ray position. The first-order derivatives of xi with respect to parameters q1 and q2 are computed along the successful ray simultaneously with the basic ray tracing equations are constituted as below.     1 ∂ 2 Gm ∂xi ∂ 2 Gm ∂pj d ∂xi = + dτ ∂qτ 2 ∂pi ∂xj ∂qr ∂pi ∂pj ∂qr   2   1 ∂ Gm ∂xi ∂ 2 Gm ∂pj d ∂pi (1.115) =− + dτ ∂qτ 2 ∂xi xj ∂qr ∂xi pj ∂qr i, j, m = 1, 2, 3;

τ = 1, 2

This system of the first-order ordinary differential equations with the secondorder derivatives of the Christoffel elastic tensor Γ with respect to the position xi and the slowness pi are solved subject to appropriate initial values by a thirdorder Runge-Kutta method in our ray tracing program. The resulting first-order ∂xi ∂xi are used in equation (1.114). Although only ∂q needs for compuderivatives ∂q τ τ ∂p

tation of function J, ∂qrj must be computed simultaneously for solving equation (1.115). After knowing function J along a successful ray, the zeroth-order ray amplitude coefficient is straightforwardly computed by equation (1.113). For the layered medium, ray amplitude coefficients are discontinuous when rays strike the boundary between two layers. The system of differential equation (1.115) is then solved subject to new initial values at the striking points on the interface. The reflection or transmission coefficients at the boundary are then taken account into ray amplitudes. We shall discuss the initial values at the source location or at the interface below.

1.5.4

Specification of a Source and Ray Synthetic Seismogram

The appropriate initial values are necessary in solving a system of differential equations of ray tracing (1.87) or ray amplitudes (1.115). We have mentioned that those initial values are determined either at the source location or a point on the boundary between two layers where the ray strikes. A point source which may be saturated at any point in the media is considered. The initial values for computation of rays and traveltimes from such a source may be specified by equations (1.90) and (1.91). Two take-off angles ϕ0 and θ0 define shooting directions from the source. v0 gives the phase velocity in that direction at the source position. When a ray starts from a point on the boundary, those two angles and directional phase velocity at that point are determined by the reflection and transmission law. The initial values for computation of ray amplitudes from a point source in an anisotropic medium can be obtained in the way described by Hanyga (1984b).

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Seismic Wave Propagation in Anisotropic Rocks. . .

83

However, we consider a point source situated in the isotropic medium for simplification. This treatment may be reasonable in the case of weak anisotropy in the vicinity of the source. The initial value of the zeroth-order amplitude coefficient (0) U1 (τ0 ) may be written as  (0) (1.116) U1 (τ0 ) = ρ(τ0 )v(τ0 ) sin ϕ0 r(ϕ0 θ0 )

where ϕ0 θ0 is the radiation pattern as a function of take-off angles at the source location. Radiation patterns of various sources were well discussed by Aki and Richards (1980). We simply assume it constant in our test. Corresponding initial values of the system of differential equations (1.115) for a point source are specified by   ∂xi = 0, i = 1, 2, 3; τ = 1, 2 ∂qτ ∂p1 sin ϕ0 sin θ0 =− , ∂q2 v0

∂p1 cos ϕ0 cos θ0 =− ∂q2 v0

cos ϕ0 sin θ0 ∂p2 =− , ∂q1 v0

∂p2 sin ϕ0 sin θ0 =− ∂q2 v0

∂p3 = 0, ∂q1

(1.117)

∂p3 sin θ0 =− ∂q2 v0

These expressions of initial values are also used at the point as the boundary between two layers when the ray tracing stars from this point in the lower (or upper) layer. However, the constant value at that point becomes the amplitude coefficient multiplied by a factor of the reflection or transmission coefficient. Such treatment has some limits to the case when the elementary ray tube is expanded too much at that point or when the ray reaches the interface at a large grazing angle. However, this treatment is in general acceptable for the near-offset VSP. Let us now consider the source-time function f (t, ϕ) = f (t) cos ϕ + g(t) sin ϕ where ϕ is the phase shift and g(t) is the Hilbert transformation of f (t). f (t, ϕ) represents the signal of the wave under consideration at the starting point of the ray τ = τ0 . As the processes along the ray, the waveform remains the same only if the phase shift ϕ = 0 or ϕ =constant. When phase shifts occur due to complex reflection coefficients or the caustics, the waveform consequently changes. Computation of ray synthetic seismograms is associated with the original signal and the phase shift along the ray which can be described by a time phase function f (t, ϕ). The phase shift may be obtained in the ray tracing procedure. The original of the source is considered as a wavelet having the Gaussian envelope ˇ (Cerven´ y et al., 1977). f (t) = e(2πfm /τ ) cos(2πfm t + v)

(1.118)

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Yong-Gang Li

where fm , τ and v are shaping parameters of the signal. Optimal values are dependent on experiences. If γ large enough, the Hilbert transform of f (t) is approximately expressed by 2

g(t) = e(2πfm /τ ) sin(2πfm t + v)

(1.119)

This gives a very simple formula for f (t, ϕ), 2

f (t, ϕ) ∼ e(2πfm /τ ) cos(2πfm t + v − ϕ)

(1.120)

where ϕ is the phase shift caused by the complex reflection or transmission coefficients at interface. Finally, the ray synthetic seismograms can be constituted by the convolution of the amplitude coefficients with the source wavelet (refer to examples in the following section).

1.5.5

Least-Squares Inverse for Traveltime

The ray tracing and traveltime computation described in Section 1.5.1 can be used to solve direct kinematic problems in arbitrary inhomogeneous anisotropic 3-D layered media. However, it would be rather cumbersome to fit observed traveltime data by means of the trial and error method in modeling. A simpler procedure for the solution of inverse kinematic problems may be used to make the modeling procedure more efficient. This work is based on linearization, namely, the relation between the change in observable data and the model parameters is linear. The linearization procedure has been successfully applied for the inversion of traveltime data for isotropic media (e.g., Aki et al., 1977; Stewart, 1984; Ursin and Arntsen, 1985) and anisotropic media (e.g., Crampin and Bamfood, 1977; ˇ ˇ Cerven´ y and Jech, 1982; Cerven´ y and Firbas, 1984; Hirahara and Ishikawa, 1984; Li, 1988; Li et al., 1990). The limitation to the least-squares inverse procedure is that the perturbation of elastic parameters of media cannot be too great to remain linear. In order to improve the modeling procedure and refine model parameters, we used a least-squares inverse combined with the trial and error modeling. This updated inverse procedure is effective not only for saving a lot of computation time but also for reducing traveltime residual. For example, the residual traveltime after inverse modeling was reduced to the half of the residual obtained by the trial and error method (refer to the example) in the next section. The generalized inverse is nothing but the least-squares solution when zero eigenvalue exists in the data space U0 , but not in the model space V0 (Aki and Richards, 1980). The simplest inverse problem can be expressed by Gm = d

(1.121)

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Seismic Wave Propagation in Anisotropic Rocks. . .

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where d is the data vector, m is the model vector and G is called data kernel matrix. It can be solved in terms of the method of least-squares solution of an inverse problem by finding model parameters that minimize a particular measurement of the length of the estimated data dest , namely, its Euclidian distance from the observations. The length is usually quantified by L2 norm and expressed as (1.122) E = eT e = (d − Gm)T (d − Gm) where e is the prediction error vector and E is the overall error. The superscript T denotes transposing a matrix. (1.122) can be written explicitly in terms of components as  � � N M M � � � di − E= Gij mj  di − Gik mk i

=

j

M M � � j

mj mk

k

M � i

k

Gij Gik − 2

M � j

mj

N �

Gij dj +

i

N �

(1.123)

di di

i

where N is the number of data and M is the number of parameters. Then, we compute the derivative of the overall error E with respect to one of the model parameters, say mq , and set the result to zero. After some algebra, we have ∂E ∂ = ∂mq ∂mq

�

2

M � k

mk

N � i

Giq Gik − 2

N � i

Giq di

�

=0

(1.124)

Writing this expression in matrix form yields the normal equation GT Gm = GT d

(1.125)

Assuming that the inverse matrix (GT G)−1 exists, we have mest = (GT G)−1 GT d

(1.126)

which is the least-squares solution to the general linear inverse problem. Following Lanczos (1961), we can decompose matrix G using eigenvector corresponding to non-zero eigenvalue (1.127) GT = Up λp VpT where is a λp diagonal matrix, composed of non-zero eigenvalues, Up and λp are matrixes whose column vectors are the corresponding eigenvectors in data space and model space, respectively. Therefore, the generalized inverse solution may be written as T (1.128) mg = Vp λ−1 p Up d

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Yong-Gang Li

Where the superscript (−1) denotes the inverse matrix. When there is no V0 , but U0 exists, the solution is unique and we obtain GT G = Vp λ2p VpT

(1.129)

T (GT G)−1 = Vp λ−2 p Vp

It leads to T −1 T (GT G)−1 GT = Vp λ−2 p Vp · (Up λp Vp ) = Vp λp Up

(1.130)

Comparing equation (1.126) with equation (1.130), we show that the generalized inverse is equivalent to the least-squares solution for the unique inverse problem. In order to analyze the resolution and error of the least-squares solution, the data resolution matrix N , model resolution matrix R and the unit covariance matrix C are useful. They are expressed by N = G(GT G)−1 GT T

−1

G G

T

−1

T

R = (G G) |covu m| = (G G)

(1.131)

T

(1.132) T

−1

G G(G G)

T

−1

= (G G)

(1.133)

The data resolution matrix N tells how well the predictions match the observed data by the relation between them. dpre = N dobs

(1.134)

If N = I (unit matrix), then the prediction error is zero. The model resolution matrix R describes how closely a particular estimation of the model parameters is to the exact solution by the relation mest = Rdtrue

(1.135)

If R = I, then each model parameter is uniquely determined (in the overdetermined inverse problem). The unit data covariance (1.133) characterizes the degree of the error mapped from data space to model parameters by the expression (1.136) |covu m| = σd2 (GT G)−1

where σd2 is the data variance. Based on above knowledge, we can solve an actual inverse problem for seismic anisotropic model by means of the least-squares solution expressed in equation (1.126) and analyze the error and reliability of the results in terms of equations (1.131) to (1.136). The first step in the inverse procedure is to establish the data kernel matrix G, which has N × M elements. N is the number of data and M is the number of model parameters. The data are observed traveltimes. Let τ (x0i , xi ) = τ 0 (x0i ) + τ 1 (x0i , xi )

(1.137)

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Seismic Wave Propagation in Anisotropic Rocks. . .

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where τ 1 (x0i , xi ) is a small correction to the unperturbated traveltime τ 0 (x0i ) due to the perturbations in the elastic parameters, a1ijkl at the position xi in the medium. The basic linearization formula gives a linear relation between τ 1 (x0i , xi ) and aijkl .    1 ∂Gm τ 1 (x0i , xi ) = − a1 dτ 0 (1.138) 2 L0 ∂aijkl 0 ijkl The integral is taken along the unperturbed ray L0 , and dτ 0 is the infinitesimal time increment along L0 . Gm is eigenvalues of the Christoffel elastic tensor ˇ Γ in the unperturbed medium. Cerven´ y and Jesh (1982) gave a system of equations for traveltimes corrections τ 1 (x0i , xi ) of qP, qSV and qSH waves. However, these expressions seem complicated to use in practice. We adopt an indirect approach to obtain τ 1 (x0i , xi ). Beginning with initial values of the original model determined by the trial and error procedure, we perturb the model parameter one after another and run the ray tracing program to obtain the new traveltime τ (m0i , mi ) corresponding to the perturbed model parameter m0i + m1i . The perturbation of the model parameter has to be controlled in an appropriate range for the validity of linearization. The correction is given by τ 1 (m0i , mi ) = τ (m0i , m1i ) − τ 0 (m0i )

(1.139)

where τ (m0i ) is the travel time for the original medium. We then constitute the data kernel matrix G, each element in which contains the derivative of the travel time correction τ 1 (m0i , mi ) corresponding to the N th receiver with respect to the perturbation of the M th model parameter. We compute this value by Gij = τj1 (m0i , mi )/m1i The data vector d contains N perturbed travel time data τj1 (m0i , mi ) while the model vector m contains perturbations of M model parameters. Thus the least-squares solution may be written as mest = (GT W G)−1 GT W d

(1.140)

where W is the weighting function used to control weights of various model parameters. The improved model parameters are then obtained by m = minit + mest

(1.141)

In the inversion procedure, we compute equations (1.131)–(1.133) to monitor the quality of the solution. Then we perform the ray tracing again using improved model parameters and compute the root mean squares (RSM) of traveltime residuals. The inversion procedure may be repeated several times until the RSM of traveltime residuals procedure to a satisfactory value.

88

1.6

Yong-Gang Li

Ray Series Modeling of Seismic Wave Propagation in 3-D Heterogamous Anisotropic Media

In this section, we show the data recorded in a 600-m borehole VSP survey conducted in crystalline rock in the Mojave Block stress province, southern California, which yielded 100 oriented three-component seismograms from a fanshooting geometry with radius of 100 m using both the vibrator (provided by UC Berkeley) and an impact source (brought by Aki from MIT) (Li et al., 1990). The seismograms show up to 12 ms of shear wave splitting (SWS) and 5% lateral velocity heterogeneity. The first- and second-order ray tracings were used to model the travel times and amplitudes P-, SV- and SH-waves in a three dimensional mildly heterogeneous medium with elastic anisotropy induced by aligned fractures. A least squares inverse procedure was used to improve model parameters determined by forward modeling; the final root mean square travel time residual was ±2 ms. For purposes of amplitude computation the authors assume that the wavelets observed in a VSP section derive from a smooth intermediate interface (base of weathering layer or sedimentary/basement interface) that is illuminated by a narrow range of ray angles originating at a point source on the surface. Source-derived amplitude complications are thus minimized. The results strongly indicate a population of vertical cracks with an average crack density of 0.035 oriented in the N31.5◦ W direction. Such a fracture population is consistent with regional principal strain and stress determinations. Ray series geometric amplitudes were consistent with the observation that wave motion parallel to the aligned fractures decayed more slowly than wave motion transverse to the aligned fractures. Based on decay of amplitudes with travel distance, Q is estimated to be 75–100.

1.6.1

The VSP Experiment at Hi Vista and Shear-Wave Splitting Observations

The Hi Vista drillhole was part of US Geological Survey Mojave Desert borehole hydrofracture profile that ran normal to the San Andreas fault (SAF) ∼50 km southwest of Palmdale (Fig. 1.28), tending to confirm the existence of a maximum and a minimum horizontal principal stress consistent with the regional simple shear oriented 45◦ from the SAF trace (Zoback et al., 1980). With 3-D borehole VSP (Fig. 1.29), we explore aligned fracture populations in the crystalline rock off the SAF in Mojave Desert to study whether microcracks in the

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Fig. 1.28 Map of the western Mojave Desert in southern California, showing the locations of U.S. Geological Survey stress orientation boreholes at Hi Vista and Black Butte, and Cajon Pass scientific drillhole (Li et al., 1990).

crustal rock are notionally related to the current maximum horizontal principal stress field. A second aspect of this work was the incorporation of theoretic ray amplitudes. There was lack of in-situ observations and little connection between observations and model parameters such as crack density, crack aspect ratio and degree of fluid saturation. In this study, we computed VSP wavelet amplitudes with a minimum of source-related amplitude effects spatial amplitude variation of the source is correspondingly restricted. At Hi Vista, we assume that the source interface is a horizontal plane between an isotropic weathering zone and an anisotropic basement. Figure 1.29 shows the Hi Vista VSP survey layout and locations of the seismic source applied and borehole receivers deployed. In a pilot experiment we used a small repeatable vertical impact source moving along an east-south-west half circle with radius of 85 m about the wellhead, shooting at nine positions labeled by A through I. Source positions from A to G in the fan section were evenly spaced at 15◦ increments. Positions I and H were in the east (N70◦ E) and

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Fig. 1.29 Layout of VSP survey conducted at the Hi Vista 600-m borehole (Li et al., 1990). Left: Vertical impact source sites are denoted by asterisks labeled by A through I; vibrator sites and orientations shown as crossed arrows VP1, VP2 and VP3 100 m from the wellhead. σH and σh denote the maximum and minimum horizontal principal stress directions respectively from borehole hydrofracture measurements. Cracks (denoted by short lines) are aligned with the crack plane parallel to σH direction. Right: A three-component unoriented seismic sonde was deployed in the borehole at 18 depths from 90 m to 600 m. The impact source survey was conducted with a compass oriented sonde at depths 150 m, 300 m and 450 m. Straight-line ray paths give approximate incident angles of source rays.

west (S70◦ W) directions, respectively, from the borehole. A three-component compass-orientable digital borehole seismic sonde made by Leary et al. (1990) was deployed at depths of 150 m, 300 m and 450 m. In a subsequent run, the UCB vertical and horizontal vibrators were used. Three vibrator sites (VP1, VP2 and VP3) were located in the south (S20◦ E), southeast (S65◦ E) and east (N70◦ E) directions at 100-m offset from the borehole. 18 evenly spaced sonde levels were taken at depths between 90 m and 600 m. To profile the surface-most rock structure, a near-offset (33 m) VSP with 12 levels was conducted between depths 15 m and 90 m using an impact source. Such a three-dimensional configuration of VSP design was expected to define fracture-related heterogeneity and anisotropy as a function of depth, azimuth and possible relation to the regional stress. The P wave traveltimes picked from the vertical component seismograms obtained in the impact pilot survey are listed in Table 1.1. The digital sampling rate was 1,000 samples per second. A three-fold vertical stack was performed in the field to increase the signal to noise S/N ratio. The sharp onset of the first arrivals allowed traveltimes to be picked with an operational precision of

1

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Tab. 1.1 Hi Vista VSP Pilot Survey P wave Travel Times. Receiver Depth (m) 150 300 450

A 56 79 107

B 55 79 106

C 54 78 104

Travel Times D E F 55 56 56 77 79 80 105 107 109

G 57 81 110

H 57 81 109

I 57 81 108

1 ms although the presence of noise could make an uncertainty of several milliseconds. Thus, the accuracy of a phase pick might be 3 ms or 4 ms. Traveltime data showed a 5% variation as the source moved along the half circle shooting line from sites I to H. The polarization diagrams for source positions A-G are plotted in Figure 1.30 with the abscises corresponding to the direction of the H1 component (N65◦ E) and the ordinate to the H2 component (N25◦ W). The polarization pattern shows that S wave component polarized N25◦ W consistently travels faster than the S wave polarized N65◦ E, regardless of source-receiver orientation. The exceptional case for the source position C (S30◦ E) shows that the S wave in H2 (in the sagittal plane defined as a vertical plane including the borehole and the source) is much stronger than the orthogonal H1 . The particle motion remains linear in the direction of the sagittal plane. Here the vertical plane appears to be the symmetry plane which parallels crack faces.

Fig. 1.30 Horizontal motion diagrams for impact source positions A-G recorded at 450-m depth. Time proceeds from central dot in 1-ms steps. The first arrowhead marks the H2 component of S phase first motion; the second arrow marks the motion of the H1 component of this phase. H2 linear motion averages NNW-SSE.

Figure 1.31 exhibits example seismograms recorded at the 450-m sonde depth for the orthogonal source positions D and I. The S wave first arrival recorded by the horizontal geophone H1 in the N65◦ E direction is seen to be delayed relative to the S wave arrival recorded by the orthogonal horizontal geophone H2 at N25◦ W. The difference in traveltime between the two orthogonal horizontal components is 6-8 ms. The observed shear-wave splitting (SWS) is primary evidence for seismic evidence by aligned vertical fractures in the host rock. As is well known, S waves are retarded if shear displacement vector is normal to planes of weakness that exist for aligned microcracks. If the early arriving H2 component (N25◦ W) is polarized parallel to aligned microcracks, it travels at the same velocity as in the uncracked rock, while the orthogonal H1 component

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Fig. 1.31 Impact source seismograms recorded by the vertical component geophone V and two horizontal geophones H1 and H2 . (a) Seismograms are for the south source position D. (b) Seismograms for the east source position I. The H2 first arrival leads the H1 first arrival. The H2 component, compass oriented N31◦ W, is the same for both source positions, and corresponds to SV motion for source site D and to SH motion for source site I.

is retarded by a shear rigidity possibly weakened by aligned microcracks. This primary interpretation is supported by the Hi Vista hydrofracture data (Zoback et al., 1980; Hickman et al., 1988) for a regional maximum principal stress field orientation at N20◦ W. Impact P waves from the short-offset (33 m) VSP were used to obtain the near surface velocity and attenuation structure to a depth of 90 m. We found that the spectral sharply dropped at a depth of ∼50 m, which probably marks the bottom of the surface weathering layer overlying the granitic basement. It appears that a discontinuity exists at 50-m depth. We took this interface to be the source interface for ray amplitude computation. The P and S vibrator VSP data confirm shear wave splitting observations in the pilot survey. The vibrator signals were upsweeps (15–70 Hz for P waves and 10–50 Hz for S waves) recorded at 2-ms intervals. A total 54 three-component seismograms were returned from 18 depths between 100 m and 600 m for 3 azimuthal vibrator sites at 100-m offsets. Using the covariance matrix method (Majer et al., 1988), the three-component seismograms were rotated into three orthogonal components of waves: P wave motion, SV wave motion polarized in the sagittal plane, and SH wave motion polarized normal to the sagittal plane. Figure 1.32 shows seismograms of P, SV and SH waves for the south (VP1), southeast (VP2) and east (VP3) vibrator sites. The 18 traces in each panel correspond to the 18 borehole receiver depths between 100 m and 600 m. We analyze P, SV and SH components generated by the vertical, horizontal radial and transverse sources. The first arrivals of SV-waves propagating from VP1 south to borehole receivers are seen to be aligned in a steeped trend that precedes the trend of SH-wave first arrivals from the same source site. The reverse is observed for the arrivals from east vibrator site VP3.

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Fig. 1.32 Seismograms from oriented P, SV and SH vibrators recorded by P, SV and SH geophones at 18 borehole levels between 100 m and 600 m for south site VP1, southeast site VP2 and east site VP3 (Li et al., 1990).

Fig. 1.33 (a) Seismograms of SV and SH motions from SV and SH vibrators at southeast site VP2. The in-line SV motion is shown un-shaded. The SH motion is shown shaded. The SH polarization lags the SV polarization; on average the lag increases with depth. (b) Measured amplitudes of SV (void bars) and SH (solid black bars) motions in (a). The measurements are normalized to 1 at borehole receiver depth 100 m. (c) Traveltimes of P, SV and SH waves from three vibrator sites VP1 (triangles), VP2 (circles) and VP3 (pluses) to 18 borehole receivers at depths of 100– 600 m. Triangles denote traveltimes from the south siteVP1, circles the southeast site VP2, and pluses the east site VP3. The individual lines show observed basement heterogeneity; the splitting of the S wave lines shows observed basement anisotropy.

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SWS for the arrivals from southeast VP2 site is more clearly displayed in Figure 1.33a. The separation between SV and SH wavelets increases with depth. The SV-waves polarized parallel to the crack plane travel faster than SH-waves polarized transversely to the crack plane. The difference in traveltime between the two shear waves at 600 m depth is 12 ms. Figure 1.33b shows that the amplitudes of SH-waves decay faster than those of SV-waves. The traveltimes observed in the vibroseis VSP survey are plotted in Figure 1.33c. The measurements show 10-12 ms of shear wave splitting and ∼5% variation across the rock volume in the study area at Hi Vista.

1.6.2

Theory

In order to quantify the orientation and density of fractures contained in crustal rocks around Hi Vista borehole site in western Mojave Desert, we implemented the 3-D ray series equations for heterogeneous anisotropic media (see Section 1.5). The elastic anisotropy of a medium bearing aligned fractures is defined using formulas (see Sections 1.2 and 1.3). The elastic constants of a medium 0 1 +εCij + containing aligned cracks can be expressed by expression (1.13) Cij = Cij 3 2 2 ε Cij . For penny-shaped cracks, the crack density ε is defined as ε = N a /V , 0 where N is the number of cracks per unit volume V, and a is the crack radius. Cij 1 2 is the elastic constants of unfractured host rock; Cij and Cij are the first-order and second-order anisotropic perturbations, respectively, due to aligned cracks. Assuming that the normal to the crack plane is in x1 direction for all cracks, Hudson (1981) gives the first-order correction of elastic constants due to aligned dry cracks in equation (1.15), and then Hudson (1982) gives equation (1.15′′ ) as the second-order correction of elastic constants due to aligned cracks where λ and µ are Lame constants of the uncracked (isotropic) rock. The second-order correction of elastic constant due to aligned cracks is worth taking into account for the medium having strong anisotropy. For example, we consider the host rock has elastic constant λ = µ = 26.25 × 1010 dyn/cm2 and material density 2.5 g/cm3 , the P-wave velocity in the direction perpendicular to the crack plane is 4.18 km/s with the first-order correction and 4.66 km/s with the first-order and second-order corrections. As seismic waves propagate in an inhomogeneous anisotropic medium, the equation of motion is in general given by   ∂2U ∂Uk ∂ Cijkl = ρ 2 , i, j, k, l = 1, 2, 3 (1.142) ∂xi ∂xl ∂t where xi is the position coordinate, Uk is the component of the displacement vector U , and Cijkl is a fourth-order tensor of elastic constants given in our case

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by equations (1.15) and (1.15′′ ), and the plane wave is eiω(t−xi pi ) . The system of equations for the zero-order component of the displacement vector Uk is given by (0)

(Γjk − δjk )Uk Γjk = pi pj aijkl ,

=0

(1.143)

i, j, k, l = 1, 2, 3

(1.143′ )

where Γ is the Christoffel elastic tensor, a function of elastic constants and wave slowness pi , and is direction dependent. In a special case with aligned crackinduced anisotropic medium, Γ has a useful explicit form shown in (1.151). In equation (1.143′ ), aijkl = cijkl /ρ are elastic parameters normalized to unit (0) density, δ is the Kronecker symbol, and Uk is the zeroth-order component of the displacement vector Uk . For a non-trival solution, (1.143) requires that at least one of the eigenvalues Gm (xi , pi )(m = 1, 2, 3) of the Christoffel matrix Γ is equal to one. It leads to a general expression for the eikonal equation for anisotropic media, (1.144) Gm (xi pi ) = 1 dτ , equation (1.144) contains a condition on the time evolvSince slowness pi = dx i ing wave surface xi (τ ). This non-linear differential equation has three solutions (m = 1, 2, 3) corresponding to three independent wavefronts of qP , qSV and qSH waves propagating in the anisotropic medium. By means of the method of ˇ characteristics method (Cerven´ y and Firbas, 1984), a system of the first-order differential equations is obtained from equation (1.144).

1 ∂Gm dxi = , dτ 2 ∂pi

−1 ∂Gm dpi = , dτ 2 ∂xi

i = 1, 2, 3

(1.145)

where xi (i = 1, 2, 3) represents the position of the ray propagating in the anisotropic medium and the parameter τ corresponds to the traveltime along the ray. This is a basic ray tracing system of equations. Integrating (1.145) is equivalent to ray tracing. The eigenvalues Gm are in general found numerically for a specific anisotropic medium. However, they may be analytically determined in terms of a cubic form of the characteristic equation. The eigenvalues of the Christoffel elastic tensor Γ can be analytically determined in terms of the following cubic form of the characteristic equations: G3m − P G2m + QGm − R = 0

(1.146)

where coefficients P, Q and R are the trace, the second invariant and the determinant of the matrix Γ . They are expressed as P = (Γ11 + Γ22 + Γ33 ) 2 2 2 R = (Γ11 Γ22 + Γ22 Γ33 + Γ11 Γ33 − Γ12 − Γ13 − Γ23 )

Q=

2 −(Γ11 Γ23

+

2 Γ22 Γ13

+

2 Γ33 Γ12 )

+ 2Γ12 Γ13 Γ23 + Γ11 Γ22 Γ33

(1.147)

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Three roots of the cubic equation corresponding to three eigenvalues are then obtained by the following explicit algebraic expressions: G1 = A + B +

P 3

−1 (A + B) + G2 = 2 −1 G3 = (A + B) + 2

√ P 3 +i (A − B) 3 √2 P 3 −i (A − B) 3 2

(1.148)

where 

�� � � 12  13 2 1 � � −b b a 2  A= + + 2 2 3 

�� � � 12  13 2 1 � � −b b a 2  − B= + 2 2 3 1 a = Q − P 2, 3

b=

2 2 (P Q − 3R) + P 3 3 27

Using the theorem on implicit functions, Derived from equations (1.144) and (1.145), a set of coupled ordinary differential equations which can be directly integrated by numerical procedures such as the Runge-Kutta method: 1 Cijkl pl Djk dxi ρ = , dτ D

−1∂Cljks pl ps Djk dpi 2ρ∂xi = , dτ D

i, j, k, l, s = 1, 2, 3 (1.149)

where Djk and D are explicit function of p and Γ is given by equation (1.89). These ordinary differential equations with the first-order derivatives of elastic tensor with respect to the position xi are completely general and suitable for ray tracing when appropriate initial values for x and p are specified. Appropriate initial values for xi and pi , xi (τ0 ) = xi0 and pi (τ0 ) = pi0 can be easily specified in Cartesian coordinates as po = cos φo cos θo /vo ro = cos φo sin θo /vo

(1.150)

qo = sin φo /vo where vo is the phase velocity at the source position of the appropriate wave (P, SV or SH). The angles φo and θo are two take-off angles specifying the declination angle with respect to the borehole axis and the azimuthal angle with respect to the line connecting the wellhead and the source. The Christoffel elastic tensor Γ

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is computed on the ray in terms of the following general matrix for an anisotropic medium having the symmetry of aligned cracks.   C11 p2 + C66 r2 + C55 q 2 (C12 + C66 )pr (C13 + C55 )pq  Γ = C66 p2 + C22 r2 + C44 q 2 (C13 + C44 )rq (C12 + C66 )pr 2 2 2 (C13 + C44 )rq C55 p + C44 r + C33 q (C13 + C55 )pq (1.151) where p = p1 , r = p2 and q = p3 are three components of the slowness vector p in the Cartesian coordinate system (xi , i = 1, 2, 3), and C11 · · · C66 are the components of the matrix Cij in expression (1.13). As in equations (1.15)– (1.15′′ ), the x1 axis coincides with the normal to the crack plane and x2 and x3 axes lie in the crack plane. If the sagittal plane defined by the receiver spread in the borehole and the source position is horizontally rotated about the normal of crack plane by an angle θ, the slowness vector p can be transformed to new coordinates by the following transformation (Thomsen, 1988): P ′ = ΩP where



cos θ Ω =  sin θ 0

− sin θ cos θ 0

(1.152)  0 0 1

Once the appropriate ray parameters are determined for a given pair of source and receiver position, the ray amplitude may be computed. The ray amplitudes in an inhomogeneous anisotropic medium may be obtained from the general formulas for the higher order ray series given by equations (1.86), (1.88) and (1.92). However, when, considering only leading term of the ray series as in equation (1.143), the solution of the transport equation may be written as A(τ ) = A(τ0 )

�

(V ρJ)τ0 (V ρJ)τ

� 12

(1.153)

where τ , ρ, V and J are the ray traveltime, medium material density, phase velocity and ray tube cross-sectional area along the ray, with initial values τ0 , ρ0 , Vo and Jo . The ray amplitude A(τ ) along a ray is approximately computed relative to the initial value A(τ0 ) at the starting point of the ray. Unless there is good observational control on the source pulse amplitude or a wide range of source take-off angles, it is sufficient to set A(τ0 ) = 1. The function J measures the spreading of the ray tube along a ray. The spreading J in 3-D can be ˇ y, 1972), expressed in terms of dummy ray parameters q1 and q2 (Cerven´  �2  12 �2 � �2 �� 3 � 3 � 3 � ∂xi ∂xi  ∂xi ∂xi J = − ∂q ∂q ∂q1 ∂q2 1 2 i=1 i=1 i=1

(1.154)

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The first-order derivatives of the ray position coordinates xi (i = 1, 2, 3) with respect to ray parameter q1 and q2 may be solved from the following twelve second-order differential equations,     1 ∂ 2 Gm ∂xi ∂ 2 Gm ∂pj d ∂xi = + dτ ∂qr 2 ∂pi ∂xj ∂qr ∂pi ∂pj ∂qr   2   −1 ∂ Gm ∂xi ∂ 2 Gm ∂pj d ∂pi = (1.155) + dτ ∂qr 2 ∂xi ∂xj ∂qr ∂xi ∂pj ∂qr i, j, m = 1, 2, 3; r = 1, 2 where Gm is the eigenvalue of the Christoffel matrix Γ . These twelve firstorder differential equations with the second-order derivitives of Γ with respect to ray parameters and positions could be numerically solved by the third-order Runge-Kutta integration with appropriate initial values. Such amplitude related computations need only be performed along ray paths which were already determined by ray tracing. Among the twelve solutions of this equation system, only six first-order derivatives of the ray position xi are used in equation (1.154) for the computation of geometric spreading J along the ray. The above formulations of ray tracing system may be used only in the medium where the elastic constants Cijkl are continuous functions of coordinates. As the ray strikes a discontinuity in the medium, not only do the ray direction and amplitude change discontinuously, but also the incident wave converts to the orthogonal pair of displacements at the interface. Wave conversion at interfaces in anisotropic media may be treated using the slowness surface method (see Section 1.4.1). The method is based on a generalization of Snell’s law which can be expressed by (1.156) [pi n] = [piU n] = [piL n] where p is the slowness vector and n is the normal to the interface and [ ] denotes the vector product. The superscript I denotes the incident wave and i = 1, · · · , 6 correspond six types of reflected and refracted waves in the upper and lower layers denoted by subscript U and L. After some manipulation, equation (1.156) is expressed by (1.157) pi = b + ξ i n, i = 0, 1, · · · , 6 where the vector b is the projection of slowness vector pi on the interface and is the same for all type of waves, and ξ i is the projection of the ith slowness vector onto the normal n to the interface. The slowness vector p of six possible converted waves may be determined by the characteristic equation (1.143) as an eigenvalue problem. As the final value of ξ i must satisfy equation (1.156), the ξ i may be expressed as a root of equation (1.156) and numerically found by standard root-finding techniques. The wave velocities and their take-off angles from the interface are straightforwardly deduced from the corresponding slowness vectors. The reflection and transmission coefficients may be obtained

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by solving a six degree linear algebraic equation resulting from the boundary condition at the interface.     6 6   ∂us ∂us ∂uk ∂uk i i I i i I + = Cljks + (−1) u = u , (−1) Cljks ∂xs ∂xk i ∂xs ∂xk i i=1 i=1 (1.158) Writing the displacement of plane wave as U i = Ai g i e−ik

i

[pi1 x1 +pi2 x2 +(−1)i p3 x3 −v i (ϕi ;θ i )t]

(1.159)

Equation (1.158) may be conveniently expressed in matrix form (1.161). In equation (1.159), Ai and k i are the amplitude and wavenumber of the ith wave, respectively. The phase velocity v i is dependent on the azimuthal angle θi and declination angle ϕi of the slowness vector pi , which has been determined by the slowness surface method. The pi1 , pi2 and pi3 are components of pi in the Cartesian coordinates; the g1i , g2i and g3i are components of the eigenvector g i which may be solved as eigensolutions of equation (1.143) expressed as Γ − IGm )gm = 0, (Γ

m = 1, 2, 3

(1.160)

where Gm and gm are eigenvalues and eigenvectors of the Christoffel elastic tensor Γ , and I is the identity matrix. Boundary conditions of equation (1.158) appear as in matrix expression (1.72), where the superscript 0 denote the incident wave, odd numbers 1, 3, 5 the reflected waves and event numbers 2, 4, 0 0 0 , σ23 , σ33 of incident 6 the transmitted waves. When the values of g10 , g20 , g30 , σ13 wave (i = 0) are specified, six unknown reflected and transmission coefficients Ai (i = 1, · · · , 6) are then obtained from equation (1.72) by matrix inversion. For anisotropic media with hexagonal or orthorhombic symmetry, the stress i i i , σ23 and σ33 components may be written as equations (1.73), (1.74) and σ13 (1.75), in which we use 6×6 matrix convention for elastic tensor constants cij , (see Crampin, 1984b for details). When the seismic source is in the vertical plane coinciding with the anisotropic symmetry plane either parallel or perpendicular to crack plane, one component of the eigenvector g i is zero (for example g21 = 0). In this case, P and SV motions decouple with SH motion; reflection and transmission coefficients may be computed by explicit formulas. The following equations for computation of reflection and transmission coefficients between two transversely isotropic media are derived by Daley and Horn (1977, 1979). We have corrected some typographic errors in their published formulas and used them in our ray tracing procedure for two-dimensional case. The reflected/transmission coefficients Rij /Tij , where i designates the incident wave type and j the converted wave type, are

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2 2 Rp´ `p = (−t1 t2 x + t3 t4 P Q + t5 t6 P R + t7 t8 x P QRS − t9 t10 QS − t11 t12 RS)/D

Tp` `p = 2(t3 t10 P Q + t5 t11 P R)/D

Rp´ `s = −2x(t7 t11 P RS + t1 t3 P )/D

Tp` `s = −2x(t7 t10 P QS − t1 t5 P )/D

Rs`p´ = −2x(t8 t12 P Q + t2 t4 Q)/D Ts`p` = 2(t5 t8 P QR − t2 t10 Q)/D

Rs`s´ = (−t1 t2 x2 + t3 t4 P Q − t5 t6 P R + t7 t8 x2 P QRS + t9 t10 QS − t11 t12 RS)/D Ts`s` = 2(t4 t5 P Q + t10 t12 QS)/D

D = t1 t2 x2 + t3 t4 P Q + t5 t6 P R + t7 t8 x2 P QRS + t9 t10 QS + t11 t12 RS (1.161) where ω1 k2 l4 k2 x2 l4 , t3 = β2 ω2 + β1 , nm1 nl1 � � δ1 x2 l2 k1 x2 l3 m4 δ2 x2 l2 m3 , t6 =∈ t4 = ∈2 − , t5 = β1 ω1 + + , l1 nl1 m1 l1 nl1 � � k2 x2 ll4 m2 m3 m4 m2 , , t8 = δ2 − δ1 , t9 = β2 ω2 + t7 = β2 l − β1 m1 l1 l1 m1 nm1 t1 = ∈2 −

∈1 l2 , nl1

t2 = β2 ω2 − β1

m3 m4 k1 x2 ll3 ω 1 m2 + δ1 x2 , t11 =∈1 + δ2 x2 , t12 = β2 + β1 , l1 l1 m1 m1 v1 v3 v4 x = sin θ1 , n = , k1 = , k2 = , v2 v1 v2

t10 = ∈1

1

1

P = (1 − x2 ) 2 = cos θ1 , Q = (1 − k12 x2 ) 2 = cos θ3 , � �1 �1 � k 2 x2 2 x2 2 R = 1 − 12 = cos θ4 , S = 1 − 2 = cos θ2 , n n � � σ  12 {Q − aσ33 + aσ55 } σ σ σ + [a11 + a33 − 2a55 ]   sin2 θV  , lv =    2Qσ �

�  12 {Qσ − aσ33 + aσ55 } σ σ σ + [a11 + a33 − 2a55 ]   cos2 θV  , mv =    2Qσ 1

Qσ = ((aσ33 − aσ55 )2 + 2B1σ sin2 θV + B2σ sin4 θV ) 2 ,

B1σ = 2(aσ13 + aσ55 )2 − (aσ33 − aσ55 )(aσ11 + aσ33 − 2aσ55 ), B2σ = (aσ11 + aσ33 − 2aσ55 )2 − 4(aσ13 + 2aσ55 )2 ,

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l 1 + m2 u u u ) cos2 θ1 , + (m1 C33 − l1 C13 , ∈1 = l1 C13 l 1 + m1 V1 l l l ∈2 = ) cos2 θ2 ], − l2 C13 [l2 C13 + (m2 C33 V2 V1 (m3 cos2 θ3 − l3 sin2 θ3 ), ω1 = V2 (l1 + m1 ) V1 (m4 cos2 θ4 − l4 sin2 θ4 ), ω2 = V4 (l1 + m1 ) l=

u u − m3 C13 , δ1 = l3 C33

l l δ2 = l3 C33 − m3 C13 ,

u , for v = 0, 1, 3, β1 = C55

l β2 = C55 , for v = 2, 4

In above expressions V1 , V3 , V2 , V4 and θ1 , θ3 , θ2 , θ4 are velocities and emergent angles of reflected P, SV and transmitted P, SV waves. Angles are with respect to the normal of interface. They may be solved using the slowness surface method. If the incident wave is P-wave, θ0 = θ1 ; if the incident wave is SV-wave, θ0 = θ3 . σ = u denotes the upper layer and σ = l denotes the lower layer. v = 0, 2, 3 denote the incident wave, reflected P and SV waves. v = 2, 4 denote the transmitted P and SV waves. auij = cuij /ρ and alij = clij /ρ are elastic parameters of upper and lower media in the 6 × 6 matrix convention (see Section 1.4.1).

1.6.3

Traveltime and Amplitude Modeling Results

The velocity model used to fit the traveltime and amplitude data obtained from the vertical seismic profile (VSP) at Hi Vista, Mojave Desert in Southern California (Fig. 1.34a) is described in this section. A thin isotropic weathering layer overlies a half-space of granitic basement rock bearing fractures. The rigidity of the basement rock may vary vertically and laterally; vertical cracks may be aligned in an arbitrary horizontal direction; and the crack density may vary vertically and laterally. The traveltime data used for modeling are mainly taken from two orthogonal vibrator sites in the south (VP1) and east (VP3) directions, which yielded the ray paths shown in Figure 1.34a. The shorter offset (33 m) VSP data at site VP3 are used to determine the velocity structure to a depth of ∼90 m. P and S travel times and P wave spectral amplitudes are listed in Table 1.2. We find that the amplitude sharply dropped at a depth of ∼50 m and the average ratio of P travel times to S travel times above that depth is higher (1.65) than the average ratio (1.55) below that depth, suggesting that a discontinuity exists at 50 m depth which marks the bottom of the weathering layer overlying the granitic basement at Hi Vista. We took the interface at this depth as a source interface of ray amplitude computation in our ray tracing procedure.

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Fig. 1.34 (a) Ray-tracing model used to fit traveltimes and amplitudes observed at the Hi Vista borehole site. An isotropic weathering layer overlies the anisotropic basement rock. Vibrator source points are shown at south site VP1 and east site VP3 100 m from the wellhead. The impact source was located at 33 m offset for the near-surface velocity survey. (b) Best fit computed traveltime curves for (from left to right) P, SH and SV observed traveltimes for the vibrator source VP3. Observed data are plotted as boxes for P, circles for SH and crosses for SV waves at depths of 100-600 m. The rms error for the fit is ±2 ms for 48 observed three-component (P, SV and SH) arrivals (144 independent traveltime measurements).

1

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Seismic Wave Propagation in Anisotropic Rocks. . .

Tab. 1.2 Near-Offset VSP Travel Times and Amplitudes. Receiver Depth (m)

P wave (ms)

S Wave (ms)

7.5

29

15.0

P Wave Amplitudes Observed

Modeled

44

1.000

1.000

29

45

0.825

0.825

22.5

28

47

0.613

0.741

30.0

29

50

0.404

0.654

37.5

30

52

0.313

0.513

45.0

33

52

0.144

0.346

52.5

35

55

0.093

0.081

60.0

37

56

0.080

0.081

67.5

38

56

0.055

0.052

75.0

39

59

0.043

0.043

82.5

40

61

0.030

0.039

90.0

41

63

0.019

0.030

In our forward modeling, only the gradient in the vertical direction was used; two horizontal gradient parameters were held to zero. The interface between the weathering layer and the basement was modeled as horizontal with a constant depth of 50 m (as determined by the short-offset VSP). The P-wave velocity in the isotropic weathering layer increases rapidly from 1.2 km/s at the surface to 3.0 km/s at the bottom. At the top of the basement rock, VP abruptly increases to 5.2 km/s and then smoothly increases to 5.7 km/s at depth of 600 m. The crack density in the basement rock was fit to a linear increase from 0.03 to 0.035 in the same depth range. The initial crack orientation was set to N30◦ W (roughly parallel to the principal stress σH direction in the region). In this situation, rays shooting from the south site VP1 (S20◦ E) were not exactly parallel to the crack plane and rays coming from the east site VP3 (N70◦ E) were not exactly in the vertical plane because lateral gradients were omitted in the initial model. The polarization direction might be deviated out of the sagittal plane for P and SV waves, or not perpendicular to this plane for the SH wave, due to the aligned crack-induced anisotropy. In the laterally homogeneous model, we need to specify only the ray angle ϕ0 in the vertical sagittal plane at which the ray takes off from the source. We then evaluate initial values of slowness components for P, SV and SH waves corresponding to three source types at the ˇ vibrator site (Cerven´ y and Pˇsen˜cik, 1972; Li et al., 1990) P0 =

cos ϕ0 , VP0

P0 =

cos ϕ0 , 0 VSV

P0 =

cos ϕ0 , 0 VSH

q0 = tan ϕ0

(1.162)

For ray paths in the vertically inhomogeneous isotropic top layer, we use the same ray tracing procedure for the aligned crack-induced anisotropic heterogeneous medium but simply set a crack density to zero. As the ray strikes the

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weathering-basement interface, ray tracing was stopped. At this point, we treat the discontinuity problem at the interface as described before. For example, we assume that the upper medium has a reduced rigidity (rigidity/density) of 4 km2 /s2 and the lower medium has a reduced rigidity of 10 km2 /s2 while bearing vertical cracks with a crack density of 0.04. The media are Poisson solids and have the same density 2.5 km/cm3 . To illustrate the treatment of discontinuity at the interface (Fig. 1.35), consider the case that the incident P wave propagates in the vertical plane perpendicular to crack faces. The incident angle ϕ is 60◦ with respect to the interface. Calculated slowness surface of reflected and transmitted waves are shown in Figure 1.35a where the interface is denoted by B. the common wave vector projection on the interface is denoted by b and the interface is denoted by n. The slowness vectors of the incident wave is denoted by s(0) . reflected and P, SV waves in the isotropic medium are denoted by s(1) , s(3) , and the slowness vectors of transmitted P, SH and SV waves in the anisotropic medium are denoted by s(2) , s(4) , s(6) . In the particular case when the waves propagate in a plane either parallel or perpendicular to the anisotropic symmetry plane, there is no transmitted SH wave s(4) from conversion at the interface for the incident P wave. Figure 1.35 illustrates the SH slowness vector s(4) . Slowness vectors of reflected and transmitted waves have a common projection b on the interface; the vector b is defined as the projection of the incident slowness vector. The endpoints of slowness vectors lie on the vertical line drawn through the termination of the projection vector b. The loci of slowness vectors of reflected and transmitted waves are numerically computed using the slowness of P wave. Only one type of shear wave slowness surface, denoted by the solid line, appears in the isotropic top layer, while two separated shear wave slowness surfaces in the anisotropic bottom layer denoted by solid and dashed curves, respectively. The slowness surfaces are circular in the isotropic medium, but are ellipse-like in the anisotropic medium. The corresponding velocities of reflected P, SV and transmitted P, SH and SV waves are computed as a function of the phase propagation direction with respect to the interface and plotted in Figure 1.35b. Phase velocity curves are plotted with the same line types as those for the slowness surfaces. The velocities in the isotropic homogeneous medium remain constant while velocities vary as a function of angle for the anisotropic medium. Because Figure 1.35 illustrates the case for wave propagating in a plane perpendicular to the crack plane, the SH wave is faster than SV wave. For the incident P wave at 60◦ to the interface, the reflected P, SV and transmitted P, SH, SV waves have the phase velocities of 3.464, 2.00, 5.093, 3.135, 3.011 km/s, and emergent angles of 60.0◦, 73.22◦ , 42.68◦, 63.1◦ and 64.24◦, respectively. When waves propagate in the plane either parallel or perpendicular to the anisotropic symmetry plane, we check the reflection and transmission coefficients analytically using Daley and Hron’s formulas (in Eq. (1.161)). Figure 1.35c shows calculated coefficient curves of reflected P, SV and transmitted

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Fig. 1.35 (a) Slowness surfaces of reflected and transmitted waves derived from incident P wave in the vertical plane perpendicular to crack faces. Vertical s(0) , s(1) , s(3) , s(2) , s(4) and s(6) denote incident P, reflected P, SV and SH waves, respectively. B denotes the boundary between the upper and lower medium; n denotes the normal to the boundary, b denotes the common projection of slowness vector on the boundary B. (b) Directional velocities of reflected and transmitted P, SV and SH waves generated at the interface between two layers in (a). VP and VSV are in the top isotropic medium; VP` , VSV ` and VSH ` are in the lower anisotropic medium. The angle is with respect to the interface. (c) Calculated reflection and transmission coefficient curves of P and `P ´ and P ` SV ´ are reflected coefficients; P ` P, ` and P ` SV ` are transmission SV waves. P coefficients. SH waves are not excited. Angle is with respect to normal to interface B.

´ , P` P` and P` SV ` , respectively. The abscissa P, SV waves denoted by P` P´ , P` SV angle denotes the phase propagation direction with respect to the normal of the interface. The total reflection of P waves occurs at about 45◦ in this example. A second example, Figure 1.36, illustrates the general case computation for the interface discontinuity. The incident P wave now propagates in a plane different from the anisotropic symmetry plane. The normal of vertical cracks is at an angle of 30◦ from the vector b. In this case three types of transmitted waves P, SH and SV are generated in the anisotropic medium of the bottom layer in the model. The synthetic slowness surfaces are plotted in Figs. 6.10a and 6.10b, respectively. Corresponding to the incident P wave at angle of 60◦

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Yong-Gang Li

Fig. 1.36 (a) Slowness surfaces of reflected and transmitted waves derived from incident P wave in the vertical plane rotated from the normal of vertical cracks by 30◦ . (b) Directional velocities of reflected and transmitted P, SV and SH waves generated at the interface between two layers in (a). (c) Calculated reflection and transmission `P ´ and P ` SV ´ are reflected coefficients; coefficient curves of P, SH and SV waves in (a). P ` ` ` ` ` ` PP, PSH and PSV are transmission coefficients. Other notations are same in Figure 1.35.

with respect to the interface, velocities of reflected P, SV and transmitted P, SH, SV waves are 3.464, 2.000, 5.173, 3.142 and 3.012 km/s, respectively. Their corresponding emergent angles are 60.0◦ , 73.22◦, 41.7◦ , 63.03◦ and 64.23◦ with respect to the interface. Shear wave splitting is clearly seen in the anisotropic medium. Reflection and transmission coefficients in this case are computed using equation (1.161). Numerical results for this case are plotted in Figure 1.36c. As waves propagate in the anisotropic basement rock, all three slowness components are in general nonzero. We search for rays striking the borehole receivers by scanning the takeoff angles of the ray shooting from the source. The rays reaching the receivers are registered and their coordinates are stored for VP1, VP3, and the short-offset impact source position are plotted in Figure 1.34a. The traveltimes of P, SV, and SH waves are computed and compared with observations. In the initial model fitting, the rms traveltime residual error for a

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total of 144 observed values taken from 18 borehole seismograph depths for VP1 and VP3 and 12 depths for the short-offset VPs was 4.1 ms. In order to refine the model parameters, we followed the optimization procedure (Aki et al., 1977; see also Section 1.5.5) and implemented a least squares inversion to improve the model parameters. The perturbation was made for 14 initial parameters. Each model parameter was perturbed about 10% of its initial value to satisfy the requirement for the linear relation between the model parameter and the residual traveltime. The refined model parameters labeled by “inversion” are listed in Table 1.3. The rsm traveltime residual value fell to 2.15 ms after two passes of the inverse procedure, indicating a variance reduction of 75%. Tab. 1.3 Hi Vista Model Parameters. Parameter 1. rigidity/density at the top of weathering layer (km2 /s2 ) 2. rigidity/density at the bottom of weathering layer

(km2 /s2 ) (km2 /s2 )

Forward

Inverse

0.50

0.547

3.000

2.935

0.000

0.093

4. rigidity/density in WE direction in weathering layer (km2 /s2 )

0.000

−0.081

5. rigidity/density at the top of basement rock (km2 /s2 )

9.000

10.040

11.000

10.945

0.500

0.380

3.000

−0.257

3. rigidity/density in NS direction in weathering layer

6. rigidity/density at the bottom of basement rock

(km2 /s2 )

7. rigidity/density in NS direction in basement rock (km2 /s2 ) 8. rigidity/density in WE direction in basement rock

(km2 /s2 )

9. crack density at the top of basement rock

0.030

0.035

10. crack density at the bottom of basement rock

0.040

0.038

11. crack density gradient in NS direction (km−1 )

0.000

−0.002

12. crack density gradient in WE direction (km−1 )

0.000

0.004

13. crack orientation (degrees with respect to the north) 14. thickness of weathering layer (km)

30.00 0.050

31.50 0.052

Since the gradients of rock rigidity and crack densities in three coordinate directions are nonzero in the final model parameter inversion, the ray tracing have to be mode for a laterally heterogeneous anisotropic medium. Thus rays are curved not only in the sagittal plane but also horizontally. In this case, two takeoff angles are needed to specify the ray direction from the source. One angle out of the sagittal plane is the declination angle φ(0 < φ < π) and the other is the azimuthal angle (0 < θ < 2π). For example, the final travel time fitting results for source VP3 are shown in Figure 1.34b. The observed data are denoted by boxes for P waves, circles for SH waves, and crosses for SV waves. The computed travel times, denoted by three curves in successive panels, fit their corresponding observed travel time values to ∼2 ms. SH wave traveltimes precede SV wave traveltimes in this source direction. The observed anisotropy

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is best fit by a population of vertical dry cracks in the crustal rock at Hi Vista borehole site with the crack density 0.035 and crack alignment N31.5◦ W. This result is consistent with the in-situ stress field measured by Zoback et al. (1980). Ray amplitudes were also computed for those rays reaching borehole receivers. S wave anisotropic geometric spreading was computed along the ray via equations (1.87)–(1.89) with the second-order partial derivatives of eigenvalues of Christoffel elastic tensor with respect to the position coordinates and slowness components. Since the source was located on the isotropic medium, we used the ˇ initial values given by equation (3.30) in Cerven´ y et al. (1977). As discussed above, no attempt was made to establish a coherent model for SV and SH sources in a heterogeneous source layer. The wavelet amplitudes were computed for each ray path and reflect only the processes of spreading in the isotropic layer, plane wave transmission at the assumed plane source interface, and further spreading in the anisotropic study volume. The results of our computation and observations, shown in Figure 1.37 for the east and south vibrator source points, respectively, are evidence for anisotropic wavefronts. In Figure 1.37, observed S wave motion parallel to the crack faces (called SV wave denoted by solid line for the south source VP1 in left panel; called SH wave denoted by dashed line for the east source VP3 in right penal;) is greater than S wave motion transverse to the crack faces (SH, dashed line, left; SV, solid line, right). The trend of observed amplitudes was well modeled by the anisotropic geometric spreading computation. The top panels of Figure 1.37 show the scaled fits of the computed amplitude curves to the observed amplitudes. The bottom panels in Figure 1.37 show scaled fits of the interchanged model amplitude curves, i.e., scaling observed SH amplitudes with computed SV amplitudes and vice versa. The trends of computed SV and SH amplitudes better match the observed SV and SH trends than do the interchanged trends. The combined rms fitting error for the interchanged curves is triple the rms fitting error for the properly assigned curves. The difference in observed and computed amplitude decay rates does not suggest the presence of strong intrinsic attenuation. In particular, Figure 1.37 suggests that wave motion transverse to the crack faces decays slightly faster than wave motion parallel to the crack faces, similar to observations for southeast source site VP2 shown in Figure 1.33b. At a depth of 500 m, the waves have cycled about 5 times with negligible non-geometric loss in amplitude, suggesting the Q > 75–100. However, in practice, the scatter in the observed amplitude data is probably related to receiver coupling to the borehole and borehole coupling to the rock formation (Li et al., 1990). We have applied the ray tracing method in heterogeneous anisotropic media to study the fracture content in a test volume in Mojave Desert, southern California. The rock study volume may be heterogeneous and possess a non-uniform population of oriented fractures. By forward fitting observations of P and S wave traveltimes, a preliminary model of isotropic heterogeneity and fracture

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Fig. 1.37 Observed and computed amplitudes for 18 depths at Hi Vista for east (right) and south (left) vibrator source points. Solid lines denote SV motion, dashed line SH motion. In upper panels, computed SV and SH amplitude curves are scaled to best fit the observed SV and SH amplitudes between 200 m and 600 m. To test the significance of agreement between computed and observed amplitudes, the computed SV and SH curves are, in the lower panel, scaled to best fit the opposite component data. The lower panel cross-component fits have rms residual errors 3 times those of the upper panel fits, indicating that the fit difference between SV and SH amplitudes shown in the upper panel is significant.

content in the host rock is obtained. A standard inverse procedure is then used to refine the existing model parameters and/or to evaluate a more extensive list of model parameters. The three-component data define a population of microcracks in an alignment consist with local and regional stress measurements. The second-order ray theory geometric ray amplitudes for SV and SH waves in the fracture-induced anisotropic medium are consistent with observed amplitude trends for SV and SH waves: wave motion parallel to the aligned fracture direction fell off more slowly than wave motion transverse to the aligned fractures. The observed amplitude decay is consistent with a high Q(> 75–100) rock volume. We suggest that the ray theoretic traveltime and amplitude computation introduced in this chapter is an accurate and roust technique for investigating the properties of low to moderate anisotropic and heterogeneous crustal rock.

110

1.7

Yong-Gang Li

Observation and Modeling of Fault-zone Fracture Seismic Anisotropy

In this section, we demonstrated another VSP experiment conducted in the vicinity of an active normal fault at Oroville in northern California to investigate fault-zone related fracture structure in terms of shear-wave splitting observation and ray tracing modeling (Leary et al., 1987; Li et al., 1987 and Li, 1988). We show three-component VSP borehole seismograms and systematic shear-wave splitting that increases with proximity to the fault. Applying the ray tracing method in anisotropic heterogeneous media and Hudson’s formulation of elastic constants for media-bearing aligned fractures as described in previous sections, we have fitted a suite of P, SV and SH traveltimes with a simple model of aligned fractures flanking the fault zone. The dominant fracture set is best modeled as parallel to the fault plane and increasing in density with approach to the fault. The increase in fracture density is non-uniform (power law or Gaussian) with respect to distance to the fault. Although the hanging-wall and the foot-wall rock are petrologically the same unit, the fracture density is more intense and extensive in the hanging wall than in the foot wall. Upon approach to the fault core zone, the fracture density or fracture-density gradient becomes too great for the seismic response to be computed by ray tracing (the maximum fracture density that can be modeled is about 0.08). Within this 25 m fracture domain it appears more useful to model the fault zone and near field fractures as a lowvelocity waveguide. We have observed production of trapped waves within the confines of the intense fracture interval (Li et al., 1990; Li and Leary, 1990).

1.7.1

The Experiment and Data

The Cleveland Hill normal fault outcrops in a high-speed (Vp = 5.8 − 6.0 km/s) meta-sedirnentary stratum on the western flanks of the central Sierran uplift in California (Leary et al., 1987). The fault, dipping 60◦ to the west, sustained 15 cm of slip during a M5.7 earthquake in 1975 (Morrison et al., 1976; Lahr et al., 1976; Langston and Butler, 1976). Following the earthquake, the United States Geological Survey (USGS) drilled a 500 m borehole through the hanging wall into the foot wall. The borehole intersects the fault at a depth of 305 m (Fig. 1.38). USGS televiewer fracture logs of the borehole reveal that in the depth interval 220–320 m, crack faces are predominantly parallel to the fault plane. USGS borehole sonic-velocity log indicates that fracture density increases steadily as the fault zone is approached. Using a small repeatable, mobile, vertical impactsource and a three-component, orientable, 0.5 ms precise borehole seismic-sonde

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Fig. 1.38 The geometry of the borehole seismic experiment at Oroville. CA, showing the sagittal plane of borehole profile across the fault zone (Leary et al., 1987; Li et al., 1987). Orientated three-component geophones are located in 12 positions in the borehole while 20 source positions, denoted by vertical arrows, were deployed across the surface.

made in USC (Leary et al., 1987, 1990; Li et al., 1987; Li, 1988), we collected digital seismograms at 12 levels in the borehole at depths between 90 m and 305 m. At each borehole depth, 20 surface-source offsets were occupied between 60 m and 730 m from the borehole. The source points define a line through the borehole in a direction normal to the surface trace of the fault (Fig. 1.38). The orientation of the sonde at each depth in the borehole was determined by a digital magnetic compass aboard the sonde. The experiment was designed to offer a natural coordinate system of axes with which to describe seismic components of motion in relation to the fault plane. Since the rays are largely parallel to the fault zone (Fig. 1.38), two reference axes for seismic motion are chosen to be the ray longitudinal and transverse directions within the plane defined by the borehole and line of source points (the sagittal plane): the third reference axis is transverse to the sagittal plane. The selected source offsets and borehole depths provided ray-paths which sampled a full range of potential fracture populations in the hanging wall and, for purposes of contrast, a portion of the foot wall. In particular, source points near the borehole observed by shallow receivers define ray-paths largely avoiding the fault zone, while source offsets distant from the borehole observed at deep receivers define ray-paths restricted to the immediate fault zone environs.

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Figure 1.39a shows three-component seismograms from all source offsets at borehole depths of 90 m, 210 m and 290 m. This seismogram arrangement, termed the common receiver format, highlights variations in the properties of the source point. The sense of motion in the resultant seismograms are, from top to bottom, longitudinal motion (along the straight line ray-path), motion perpendicular to the ray-path in the vertical plane, and motion perpendicular to the ray-path in the horizontal plane. Motion in the sagittal plane transverse to the ray-vector is referred to as SV motion while the horizontal motion normal to the sagittal plane is referred to as SH motion. In terms of anisotropy created by fractures parallel to the fault plane, SV motion is normal to crack faces and SH motion is parallel to crack faces. For a fracture set that parallels the fault zone, the SH waves propagating at a greater velocity than the SV waves (Hudson, 1981: Crampin, 1984).

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Fig. 1.39 (a) Normalized amplitude seismograms of P, SV, and SH (from top to bottom) at three boreholc depths 90 m, 210 m and 290 m (from left to right) for a range of source offset. Note the clear velocity difference between arrivals of two shear waves (marked by solid lines with steeper slop for SH waves than SV waves), showing that the SH wave polarized parallel to fault plane travel faster than the SV wave polarized orthogonally. Note the transition from hanging wall to foot wall marked by early arriving energy at the bottom of each panel (brackets). (b) Observed P-wave seismograms recorded at the vertical component geophone and the horizontal geophone at depth of 305 m. Vertical P-first motions in the sagittal plane show a polarization change from downward to upward for the source offsets on the foot-wall: the horizontal P-first motions show a polarization change from westward to eastward for the source offsets near the wellhead.

For all source points the SH wavelet precedes the SV wavelet while the separation interval between SH and SV wavelets grows as the source point approaches the fault zone. For ray-paths in the hanging-wail rock of the fault zone, SVwaves are, on the average, delayed by 0.05 ms/m relative to SH-waves. In the intensely fractured fault zone (25 m effective width), the SV-SH delay rises to about 0.08 ms/m. The observed shear-wave splitting may be explained in terms of aligned cracks induced by faulting in the metamorphic crystalline host rock. These observations are agreeable with USGS televiewer data in the borehole depth interval 220–320 m.

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Rays originating in the hanging wall (source offset 3 /V, where N is the number of penny-shaped cracks per unit volume V and a is crack radius (O’Connell and Budiansky, 1974). The fourth-order tensor of the elastic constants for the cracked medium is given by Hudson (1980, 1981). The crack density e is determined by a trial and error

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Yong-Gang Li

Fig. 1.47 (a) Ray tracing diagram at station SCS for a M 2.2 earthquake (Event 32 in Li et al., 1994) occurring at latitude 33◦ 59.98, longitude 118◦ 11.47 and depth of 18-km on September 25 of 1990. Velocities of P and S waves in unfractured host rocks are plotted at the top and bottom of the Los Angeles basin and in crystalline basement. PP and SS denote ray paths of transmitted P and S waves. PS and SP denote P-to-S and S-to-P waves converted at the basin-basement boundary. (b) the plot of phase velocity difference of between two shear waves versus directions of the slowness vector from 0◦ to 360◦ with respect to north at the incident angle of 26◦ to station SCS for this local earthquake. The ordinate is N-S and the abscissa is E-W. The solid and dashed lines denote phase velocities of the fast and slow S waves, respectively. The arrow indicates the slowness vector direction from the source. Two asterisks denote the observed difference in velocities between the fast and slow S waves.

procedure of ray tracing to fit observed travel times. We obtained the best fit using a crack density of 0.04 for the crystalline basement and 0.02 for the sedimentary basin. Computed travel times of P, PS1 , S1 P, S2 P, S1 , and S2 waves are 3.94, 5.22, 5.85, 5.93, 6.93 and 7.02 second, respectively, generally agreeable with observations. The time difference between PS1 and PS2 , S1 and S1 S2 , and S2 and S2 S1 are neglected owing to the weak anisotropy of the sedimentary basin containing less vertical cracks than those in the crystalline basement. We further computed phase velocities of shear waves in the crystalline basement to confirm the results from raying. The slowness surfaces of body waves propagating in the anisotropic medium are defined by the equation G(x, p) = 1, Γ − Gm | = 0, m = where G are eigenvalues of the characteristic equation det |Γ 1, 2, 3. The Christoffel matrix Γ a = aijkl pj pl , where pi (i = 1, 2, 3) are three components of the slowness vector p. The term aijkl = cijkl /ρ is the fourthorder tensor of the elastic constants for the anisotropic medium divided by the material density ρ. For a medium containing aligned cracks, the tensor of elastic constants is given in Section 1.2. Eigenvalues Gm may be found as roots of the characteristic equation (see Section 1.5). Phase velocities of body waves √ can be computed in a straight-forward manner from vi = Gm ; i = 1, 2, 3 for

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the quasi-P and two quasi-S waves. In the cracked medium, the phase velocity varies with the direction of the slowness vector with respect to the crack plane. We assume that the elastic constants of the crystalline basement are: λ = µ = 33 × 1010 dyne/cm2 and ρ = 2.7 gm/cm3 . The crack density e is 0.04. The vertical crack plane is in the N-S direction. Figure 1.47 shows the differences in computed phase velocities between two split quasi-S waves versus the direction of the slowness vector at station SCS for the M 2.2 earthquake in the above example. The computer differences in computed phase velocities between two split quasi-S waves are consistent with observations. We have showed such results for 24 local earthquakes occurring at different depths and azimuthal angles within the shear-wave window of station SCS (see Fig. 14 of Li et al., 1994). Based on these results, we determine that the apparent density of vertical cracks in the crystalline basement beneath station SCS in the Los Angeles basin to be 0.04. Using the same method, we determined the apparent crack density within the shear wave window of other stations to be 0.044 (DHB), 0.08 (IPC), 0.052 (LCL), 0.051 (LNA), 0.055 (RCP), 0.05 (HTB), 0.05 (SAT), 0.04 (GVR), 0.045 (FLA), 0.06 (VPD), 0.026 (GFP), 0.023 (PAS), 0.035 (PVP), and 0.04 (PVR) (refer to Fig.1.43). The mean value of apparent crack densities at stations located in the strike-slip regions is 0.052±0.011, while the mean value of apparent crack densities at stations in the reverse-thrusting regions is 0.031±0.008. In this section, we have illustrated shear wave splitting recorded at 15 stations for earthquakes occurring beneath the Los Angeles basin and adjacent areas, showing that polarization directions of the fast shear waves are in general nearly N-S. Most events in the Los Angeles basin show strike-slip faulting mechanisms with steeply dipping nodal planes and the N-S P axis (Hauksson, 1990); there is no dominant polarization of shear waves in the N-S direction due to the source radiation effect. Therefore we conclude that the observed shear wave polarizations are controlled by the rock anisotropy in the medium between source and station. We find that the separation time between the split shear waves increases as the travel distance within the crustal basement rock increases. The increasing rate ranges from 2.8 ms/km to 7.8 ms/km with the greater value in the vicinity of the Newport-Inglewood fault and Whittier fault where the strike-slip faulting is dominant. We interpret that observed shear-wave splitting is mainly caused by stress-aligned vertical microcracks in the seismogenic layer beneath the Los Angeles basin. The shear wave splitting data yield a maximum principal stress direction of N-S±15◦ beneath the Los Angeles basin and adjacent areas, consistent with the results from geological mapping (Yerkes et al., 1965; Wright, 1987; Davis et al., 1989) and focal mechanism (Hauksson, 1990). While at stations in the vicinity of local faults, the maximum principal stress direction is affected by rock fractures and follows the direction of fault strikes as observed at other active faults (e.g., Leary et al., 1997; Cochran et al., 2003, 2006). Based on shear

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wave splitting data, we determine the apparent crack density of vertical crustal microcracks to be 0.03 to 0.08 beneath the Los Angeles basin with the higher crack density in regions with strike-slip faulting.

Acknowledgements This research was supported by the National Science Foundation Grants EAR8319254, EAR-8519938, DOE Contract DE-FG03-85ER-1336, U.S. Geological Survey Cooperative Agreement 53-4831-8375, and the Southern California Earthquake Center (SCEC) through USGS Cooperative Agreement 14-08-0001-A0899 and National Science Foundation Cooperative Agreement EAR-8920136. The author especially thanks the late Professor Keiiti Aki, Peter Leary, Stuart Crampin, Tom Henyey, Ta-Liang Teng, Peter Shearer, Martha Savage, Rick Aster and Peter Malin for beneficial discussions in his study and research on seismic anisotropy to defining crustal fractures. The author acknowledge the cooperation of Peter Leary, Derek Manov and John Scott of USC in acquiring the data in the field VSP experiments, and Egell Hauksson, John McRaney, Lily Hsu and Michael Roberson of USC in acquiring the data recorded at Los Angeles basin seismic network.

References Aki K., Richards P.G., 1980. Quantitative Seismology: Theory and Methods. W.H. Freeman and Co. Aki K., Defazkio T., Resenberg, P., Nur A., 1970. An active experiment with earthquake fault for an estimation of the in situ stress. Bull. Seism. Soc. Am., 60: 1315–1336. Aki K., Lee W.H.K., 1976. Determination of three-dimensional velocity anomalies under a seismic array using first P arrival time from local earthquakes: 1. A homogeneous initial model. J. Geophys. Res., 81 (23): 4381–4399. Aki K., Christoffersson A., Husebye E.S., 1977. Determination of the three-dimensional seismic structure of the lithosphere. J. Geophys. Res., 82: 277–296. Amadei, B., 1983. Rock Anisotropy and the Theory of Stress Measurements. SpringerVerlag. Anderson D. L., Minster B., Cole D. 1974. The effect of oriented cracks on seismic velocities. J. Geophys. Res., 79: 4011–4015. Anderson D.L., Whitcomb J. H., 1975. Time-dependent seismology. J. Geophys. Res., 80: 1497–1503.

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Author Information Yong-Gang Li Department of Earth Sciences, University of Southern California, Los Angeles, CA 90089, USA E-mail: [email protected]

Chapter 2

Reproducing the Realistic Compressivetensile Strength Ratio of Rocks using Discrete Element Model Yucang Wang and William W. Guo

When the bonded discrete element models are used to model rock fracture, the compressive-tensile strength ratio and the internal frictional angle obtained from the conventional discrete element models are found to be much lower than the laboratory results. The methods to improve this include the use of non-spherical particles, and some modifications to the methods. However, these methods require either the huge computational time or the use of some specific parameters. In this chapter we propose a new fracture criterion with tensile cut-off. The new criterion is not based on forces in a single bond, but based on the stress obtained by averaging the stresses in the neighbouring particles surrounding the particle in question. The impact of the fracture criterion on macroscopic strength is discussed. The results show that the new failure criterion, while keeping a simple particle shape and linear contact laws without any loss of computational efficiency for contact detection and force calculation, leads to a better agreement with the known experimental results. The compressive-tensile strength ratio of up to 45 and internal frictional angle of 60 degree are achieved. Key words: Discrete element method, Rock fracture, UCS, Tensile strength, Compressive-tensile strength ratio, Parameter calibration.

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Reproducing the Realistic Compressive-tensile Strength Ratio of Rocks. . .

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Introduction

The Discrete Element Method (DEM) has received considerable attention over the past three decades for its ability to simulate the behavior of granular materials (Cundall and Strack, 1979). Some DEM models allow the neighbouring particles to be bonded through springs or elastic bonds. Bonds can break when certain criterion is reached, explicitly modelling microscopic fracture events. When bonded DEM models are used to model rock fracture, the crack formation and coalescence appear naturally as a part of simulation process without the need for the complex constitutive laws and re-meshing techniques due to its discrete nature. These advantages allow DEMs to study with great simplicity highly non-linear dynamic problems with large deformations and complex geometries, especially with a large number of frequently changing contacts, such as wave propagation (Toomey and Bean, 2000), fracture of intact materials such as rocks (Mora and Place, 1993, 1994, 1998; Place and Mona, 1999, 2001; Place et al., 2002; Young et al., 2000; Wang et al., 2000; Chang et al., 2002; Hazzard et al., 2000; Hzzzard and Young, 2000; Potyyondy and Cundall, 2004; Potyondy et al., 1996; Donze et al., 1997; Boutt and Mcpherson, 2002; Hunt et al., 2003; Hentz et al., 2004a, b ) and blast and impact of brittle materials at high strain rates (Donze et al., 1997; Hentz et al., 2004a, b; Magnier and Donze, 1998). While the method is versatile and attractive, the use of DEM has been limited by its inherent difficulties. The first limitation lies in the huge computational time and a small time step required for time integration of equations of motion, constrained by the physical laws and numerical accuracy. One way to speed up the simulation may be parallelizing the code and using multiple CPUs in super-computers. The second challenge in DEM analyses is that it requires extensive calibration of the input parameters. In the DEM simulations, microscopic properties, such as contact stiffness and bond strength, are usually assigned as input parameters. However, the target macroscopic parameters such as Young’s modulus, Poisson ratio, uni-axial compressive strength (UCS), tensile strength and internal frictional angle, are not known a priori. The relation between these two parametric sets are not directly related and far from being fully understood. Therefore a careful calibration approach is often required to select the input parameters so that the satisfactory quantitative predictions are produced. In most DEM simulations, the trial-and-error calibration of input parameter is generally conducted (Chang et al., 2002; Hazzard et al., 2000; Hazzard and Young, 2000; Boutt and Mcpherson, 2002; Matsuda and Iwase, 2002). This simple approach typically involves conducting a series of numerical tests such as direct tensile test, Brazilian test, uni-axial compressive test and tri-axial test, and varying the input parameters until the desired macroscopic responses are

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reached. A dimensional approach is established and some dimensionless parameters are introduced to improve the calibration (Huang, 1999; Fakhimi and Villegas, 2007; Moon et al., 2007; Rojek et al., 2011). Yoon (2007) proposed a statistical approach for calibrating micro-parameters to fit material macro-properties. During the calibration process, it is found that matching the macroscopic elastic constants is relatively easy while reproducing the realistic fracturing parameters is quite difficult. Three disconcerting problems have been identified when calibrating DEM models consisting of spherical particles to match the macroscopic fracturing of real rocks: (1) the ratio of compressive to tensile strength (UCS/T) generated from DEM models for cohesive rock is considerably low (typically 3–5) compared with that measured in natural rocks, which is typically around 10–20, or even higher; (2) the macroscopic internal friction angle obtained with the conventional DEM models is much lower than the known hard rock experimental values; (3) a linear failure envelope without a tension cut-off is usually obtained in DEM simulations. This clearly does not match laboratory results. Since the strength ratio and friction angle are two critical parameters in rock mechanics and geotechnical engineering design, it is important to reconcile this inconsistency between rock and their DEM analogues if DEM is to be applied to the practical problems where the stress path may lead to either tensile or compressive failure. The extensive sensitivity studies have indicated that adjusting contact parameters in classic DEMs have little effect on these deficiencies. Previous attempts to solve these problems involve some modifications to the conventional DEM which are briefly summarised here. Some researchers suggested that the problems might be caused by the use of circular or spherical particles (Potyondy and Cundall, 2004; Moon et al., 2007), therefore DEMs with the irregular shaped elements are proposed. There are three different logics along this line. Boutt and McPherson (2002) and Potyondy and Cundall (2004) suggested that the macroscopic friction angle could be successfully increased and adjusted to match the failure envelope of granite in 2D DEM by using particle clusters of appropriate size. Cho et al. (2007) showed that the introduction of rigid particle clumps of suitable size yielded a significant reduction of the tensile strength, and the correct ratio of UCS/T were reproduced. Here the difference between the “cluster logic” and “clump logic” is that in clusters the particles are bonded with a finite stiffness and bond breakage may or may not be permitted (Potyondy and Cundall, 2004; Boutt and Mcpherson, 2002), while particle clumps move as a single unbreakable particle of complex shape. The third option is direct use of a polygonal particles instead of circular shape (Kazerani and Zhao, 2010; Lan et al., 2010). Each particle is calculated as one entity, as compared to sphere clumps which are calculated as many individual spheres glued together. With these approaches, the realistic tri-axial failure envelope as well as UCS/T ratio are shown to be reproducible.

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145

In spite of more realistic shapes in accordance with the true microstructure of rock and some improvement in reproducing the realistic rock strength, the particle models based on the non-circular or non-spherical particles are computationally more demanding, limiting their applications mainly to small scale 2-D problems. Besides, it is observed that the post-peak response in the cluster model is like perfectly plastic type without an abrupt stress drop and damage does not seem to localize into discrete macroscopic planes as in granite (Potyondy and Cundall, 2004). For circular or spherical particles, the contact detection and force calculation are much simpler and faster compared to the other particle shapes. Therefore there are some models still employing the simple particle shapes, but having certain modifications to the conventional DEM algorithms. Fakhimi (2004), Fakhimi and Villegas (2007) reported that by slightly overlapping circular particles during the sample genesis and adopting a double normal interaction forcedisplacement law the UCS/T ratio could be increased to 22–25. Scholtes and Donze (2013) showed that the increase of the degree of interlocking by increasing the interaction range between spherical particles would result in a high ratio of UCS/T and produce non-linear failure envelopes as well. More recently, Azevedo and Lemos (2013) obtained a better agreement with the experimental behaviour by presenting a 3-D generalized rigid particle contact model based on a multiple contact formulation. In these methods, the extra empirical parameters, which are linked to the specific model, such as genesis pressure (Fakhimi, 2004; Fakhimi and Villegas, 2007), interaction range coefficient (Scholtes and Donze, 2013) and the number of layers and the total number of point per layer (Azevedo and Lemos, 2013), have to be introduced. Schopfer et al. (2009) suggested that the low UCS/T ratio might be caused by the packing method [34]. They reported that internal frictional angle and the UCS/T ratios decrease with the increasing porosity. The UCS/T ratio increases with the increasing density of pre-existing crack. Although the value of UCS/T > 10 can possibly be achieved by using a greater proportion of non-bonded contacts, it is found that the stress and volumetric stain behaviour becomes less similar to that observed in brittle rock (Schopfer et al., 2009). While different factors affecting the strength parameters are discussed, no discussion on the impact of fracture criterion has been made. In this chapter, we present a new failure criterion with tensile cut-off and discuss the impact of fracture criterion on macroscopic strength. The new criterion, which is not based on forces in a single bond, but on the stress averaged over the neighboring particles, is implemented into the ESyS-Particle, an open source DEM code. We show that by simply changing the degree of tensile cut-off strength introduced at particle scale, the macroscopic response of the particle assembly can be adjusted to the desired behaviour. The adoption of the new failure criterion, while still keeping a simple particle shape and linear contact law and thus maintaining the

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computational efficiency for contact detection and force calculation, leads to a better agreement with the known experimental results, e.g. the nature of failure envelope and the UCS/T ratio. The rest of this chapter is arranged as follows. In Section 2.2, the open source DEM software, ESyS-Particle, is briefly introduced; the new criterion for bond breakage is proposed in Section 2.3; the numerical set-up, including sample generation and simulation of a number of numerical tests, are detailed in Section 2.4; and the numerical simulation results are given in Section 2.5, followed by the discussions and conclusions in Section 2.6.

2.2

A Brief Introduction to the ESyS-Particle

The ESyS-Particle is an open source DEM software developed at the University of Queensland, Australia. It was designed to provide a basis to study the physics of rocks and the nonlinear dynamics of earthquakes. The ESyS-Particle has been extended to include single particle rotation and a full set of interactions between particles. The major features that distinguish the ESyS-Particle from existing DEMs are the explicit representation of particle orientations using unit quaternion, complete interactions (six kinds of independent relative movements are transmitted between two 3-D interacting particles) and a new way of decomposing the relative rotations between two rigid bodies such that the torques and forces caused by such relative rotations can be uniquely determined (Wang et al., 2006, 2009; Wang, 2009). It has been successfully utilised to the study of physical process such as rock fracture (Mora and Place, 1997; Place et al., 2002; Wang and Mora, 2008), stick-slip friction behaviour (Mora and Place, 1994, 1999), granular dynamics (Mora and Place, 1998; Place and Mora, 1999), heat-flow paradox (Mora and Place, 1998; Place and Mora, 1999), localization phenomena (Place and Mora, 2000), Load-Unload Response Ratio theory (Mora et al., 2002; Wang et al., 2004) and Critical Point systems (More and Place, 2002). The basic equations and contact law is presented below.

2.2.1

The Equations of Particle Motion

The particle motion can be decomposed into two completely independent parts, translational motion of the centre of mass and rotation about the centre of mass. The former is governed by the Newtonian equation r¨(t) = f (t)/M

(2.1)

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where r¨(t) and M are the particle position and the particle mass respectively. f (t) is the total forces acting on the particle. The equation above can be integrated using the velocity Verlet scheme (Mora and Place, 1994, 2002; Place and Mora, 1999). The particle rotation depends on the total applied torque and usually involves two coordinate frames. One is fixed in space called space-fixed frame, in which equation (2.1) is applied and the other is attached to the principal axes of the rotation body, referred to as body-fixed frame. The particle rotation is governed by the Euler’s equations (in the body-fixed frame) (Goldstein, 1980): τxb = Ixx ω˙ xb − ωyb ωzb (Iyy − Izz ) τyb = Iyy ω˙ yb − ωzb ωxb (Izz − Ixx ) τzb

=

Izz ω˙ zb

−

ωxb ωyb (Ixx

(2.2)

− Iyy )

where τxb , τyb and τzb are components of total torque τ b expressed in body-fixed frame, ωxb , ωyb and ωzb are components of angular velocities ω b measured in body-fixed frame, and Ixx , Iyy and Izz are three principle moments of inertia in body-fixed frame. In case of 3-D spheres, I = Ixx = Iyy = Izz . In our model, the unit quaternion q = q0 + q1 i + q2 j + q3 k is used to explicitly describe the orientation of each particle (Evans, 1977; Evans and Murad, 1977). The physical meaning of a quaternion is that it represents a one-step rotation around the vector q1ˆi + q2 ˆj + q3 kˆ with a rotation angle of 2arccos(q0 ) (Kuipers, 1998). A quaternion for each particle satisfies the following equation (Evans, 1977; Evans and Murad, 1977], 1 Ω Q˙ = Q0 (q)Ω 2

(2.3)

where 

 q˙0  q˙1   Q˙ =   q˙2  , q˙3



q0  q1 Q0 (q) =   q2 q3

−q1 q0 q3 −q2

−q2 −q3 q0 q1

 −q3 q2  , −q1  q0



 0  ωxb   Ω =  ωb  y

ωzb

Detailed algorithms to solve equations (2.2) and (2.3) can be found in Wang et al. (2006, 2009).

2.2.2

Force-displacement Laws and Calculation of Forces and Torques

The force-displacement law plays an important role in DEM simulations. Three kinds of interactions exist between contact particles in the current ESyS-Particle

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model: bonded, solely normal repulsive and cohesionless frictional interaction. Bonded interactions between particles in contact allow transmission of tensile forces, which can be used to model behaviours of continuum or intact materials. The bonds can break if a criterion is reached. The breakage of bonds provides an explicit mechanism for microscopic fracture. The solely normal repulsive and cohesionless frictional interaction do not allow tensile forces to be transmitted between particles, and therefore are suitable for modelling the motions and behaviours of “real discrete” or granular materials. The normal force exists only when two particles contact, i.e., d < R1 + R2 , here d is the distance between two particles, R1 and R2 are radii of two particles. The normal force can be calculated according the first formula in Equation (2.4). In the case of the frictional interaction, forces are transmitted in both normal and tangential directions when d < R1 + R2 . The normal force is dealt with exactly the same way as in the case of the elastic interaction. In tangential direction, static-dynamic frictional force is employed in the current ESyS-Particle model (Wang and Mora, 2009). For bonded interactions, the three important issues that need to be specified are types of interactions being transmitted between each particle pair, the algorithm to calculate the interactions between bonded particles due to the relative motion and the criterion for a bond to break.

2.2.2.1

Force-displacement law and interactions transmitted

Theoretically, six independent parameters are required to represent a 3-D particle, thus six kinds of relative motions exist between two bonded particles (Fig. 2.1). We hope that the relationship between interactions and relative displacements between two bonded particles can be written in the linear form fn = Kn ∆un , τt = Kt ∆αt ,

fs1 = Ks1 ∆us1 , τb1 = Kb1 ∆αb1 ,

fs2 = Ks2 ∆us2 τb2 = Kb2 ∆αb2

(2.4)

where ∆un , ∆us1 (∆us2 ) are the relative displacements in normal and tangent directions. ∆αt and ∆αb1 (∆αb2 ) are the relative angular displacements caused by torsion and rolling. fn , fs1 , fs2 , τt , τb1 and τb2 are forces and torques, Kn , Ks1 , Ks2 , Kt , Kb1 and Kb2 are relevant stiffness. Assuming that the bonds are identical in every direction, Ks = Ks1 = Ks2 and Kb = Kb1 = Kb2 . It should be pointed out that equation (2.4) is valid only for infinitesimal small deformation. In case of finite deformation, rotation series are non-commutative, or order dependent. Actually the two finite rolling motions, ∆αb1 and ∆αb2 , cannot be completely decoupled. Therefore in practice they are treated as one rolling motion, controlled by an orientation angle (Wang, 2009).

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Fig. 2.1 Six kinds of interactions between bonded particles. fn is normal force pulling and pressing in the radial direction, fs1 and fs2 are shearing forces in two tangential directions, τt is twisting torque, τb1 and τb2 are bending torques.

2.2.2.2

Calculation of interactions due to relative motion

Before solving equations (2.1), (2.2) and (2.3), forces and torques due to the relative motions between bonded particles have to be calculated first based on the position and orientation of each particle. We developed a Finite Deformation Method (FDM) (Wang, 2009) as an alternative to the incremental method (Potyondy and Cundall, 2004; Sakaguchi and Muhlhaus, 2000). The FDM scheme requires a complete decomposition of the total relative (translational and rotational) displacements between two bonded particles. A new algorithm was developed to decompose the relative displacements such that twisting (or torsion) and bending (or rolling) are fully decoupled and order-independent. Thus the torques and forces caused by such relative rotations can be uniquely determined. Such decomposition is physically reliable and numerically accurate since it is consistent with the non-commutativity of finite rotations. The details of the decomposition can be found in Wang et al. (2006, 2009).

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2.3

The New Criterion for Bond Breakage

2.3.1

Macroscopic Failure Criterion

The most commonly used macroscopic failure criterion for rocks is the Coulomb criterion. It can be expressed as (Scholz, 1990) τ = τ0 + µσn

(2.5)

where τ and σn are shear and normal stresses respectively, τ0 is cohesion, µ is the coefficient of internal friction, which is usually expressed as µ = tan ϕm , where ϕm is the internal frictional angle. Expressed in terms of the principle stress σ1 and σ3 , the Coulomb criterion can also be written as (Scholz, 1990)     σ1 µ2 + 1 − µ − σ3 µ2 + 1 + µ = 2τ0 (2.6) This is a straight line in the (σ1 , σ3 ) space, which has a slope of

k = tan2 (π/4 + ϕm /2) = (1 + sin ϕm )/(1 − sin ϕm )

(2.7)

and it intersects the σ1 -axis at   µ2 + 1 + µ = 2τ0 cos ϕm /(1 − sin ϕm ) σu = 2τ0

(2.8)

where σu is the macroscopic uni-axial compression strength. Equation (2.6) can also be written as (2.9) σ1 = σu + σ3 tan2 (π/4 + ϕm /2)

2.3.2

Particle Scale Failure Criterion in DEM Model

In the current DEM models, equations (2.5) and (2.9) are not used directly. Instead, single-bond failure criteria are generally adopted, that is, the failure is controlled completely by the forces and torques on that bond. This is conceptually and numerically simple. There are mainly two kinds of single-bond failure criteria. In the first criterion, when either an extensional normal force or a tangent force limit is reached, the bond breaks (Potyondy et al., 1996). In the second criterion, the shear strength of a bond depends not only on the tangent force, but also on the normal force component (Hentz et al., 2004a, b; Wang et al., 2006; Wang and Tonon, 2009a, b). Although this criterion has the similar form as the Mohr-Coulomb criterion, the “local” normal and shear stress obtained by the normal force and the tangent force in a bond divided

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by the contact area between two particles forming the bond are not necessarily the macroscopic stress which should be determined by all of the neighbouring particles around the particle pair involved in this bond. Although widely used, the single-bond failure criteria would cause strong crystalline or direction dependent failure patterns. In some directions, cracks extent very fast, while in other directions, the breaks do not occur spatially continuously and no unstable cracks are observed (Wang, 2016). As will be pointed out later, the single-bond failure criteria may be partly responsible for the low UCS/T ratio and low friction angle obtained from the conventional DEM simulations.

2.3.3

A New Failure Criterion for DEM

To overcome the above mentioned difficulties, a new criterion based on averaging the stresses of a group of particles has been proposed (Wang, 2016). In DEM this average stress can be calculated based on the contact forces. Then, a criterion with a similar form of macroscopic Mohr-Coulomb criterion as noted in equation (2.9) can be used. In practical DEM simulations, it is not convenient to calculate the average stresses for a group of particles. Rather the stress for each particle can be easily obtained. We can image that if the stress in a particle is large enough, the particle breaks. However in the conventional DEM models, a single particle is a non-breakable rigid body. A bond connecting two particles is permitted to break, explicitly representing a microscopic fracturing event. This reminds us to the following idea: if the average stress of two particles is large enough, the bond connecting the two particles breaks. The average stress of two particles is   1 2 /2 (2.10) σ ij = σij + σij 1 2 where σij and σij are stress of particle 1 and 2, which can be calculated using

σij =

1  Fi nj V

(2.11)

where V , Fi and nj are volume of the particle, contact force and the branch vector respectively, and the summation is taken to all of its neighbours. The new criterion can be expressed as: if the average stress of two particles satisfies the truncated Mohr-Coulomb criterion below (the red line in Fig. 2.2), the bond breaks, where σ1 and σ2 are the maximum and minimum principle stress of σ ij . It can be mathematically expressed as below: when σ1 > σpu (1 − α): σ1 = σpu + σ3 tan2 (π/4 + ϕp /2)

(2.12)

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when σ1  σpu (1 − α):

σ3 = −ασpt

(2.13)

The new criterion has three independent input parameters: σpt : particle scale tensile strength; α = σptc /σpt : tension cut-off ratio; ϕp : particle scale friction angle. where the ratio α(0  α  1) is a factor describing the degree of how much tensile strength is reduced. If α = 1, the fracture criterion is not truncated under tension. If α = 0, bonds have no tensile strength at all. Therefore large tension cut-off corresponds to small value of α. The slope of the envelope is (1 + sin ϕp )/(1 − sin ϕp ) (Fig. 2.2). The particle scale uni-axial compression strength σpu is not an independent parameter, and is determined by σpu = σpt (1 + sin ϕp ) / (1 − sin ϕp )

Fig. 2.2 (See color insert.) Particle scale Mohr-Coulomb failure criterion with tensile cut-off. This criterion has three independent input parameters: particle scale tensile strength σpt , tensile cut-off ratio σptc /σpt and particle scale friction angle ϕp .

2.4

Calibration Procedures

The calibration process of DEM involves recognition of the main input parameters, sample generation and running of a number of numerical simulations by varying the main input parameters until the desired macroscopic behaviour is reproduced.

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Input Microscopic Parameters and the Desired Macroscopic Parameters

In modelling of solid deformation and fractures, the task of the calibration is to determine whether micro-mechanical parameters can produce the desired macroscopic parameters. These macroscopic parameters may include Young’s modulus, Poisson ratio, UCS, tensile strength and internal frictional angle. According to the literatures, the factors affecting the macroscopic properties include stiffness parameters (Fakhimi and Villegas, 2007; Cho et al., 2007; Kazerani and Zhao, 2010; Yang et al., 2006; Plassiard et al., 2009); fracture parameters (Fakhimi and Villegas, 2007; Cho et al., 2007; Kazerani and Zhao, 2010); particle shape (Potyondy and Cundall, 1996; Boutt and Mcpherson, 2002; Cho et al., 2007; Kazerani and Zhao, 2010); particle size (Potyondy and Cundall, 1996; Cho et al., 2007; Yang et al., 2006; Plassiard et al., 2009; Akram and Belheine, 2009); particle size distribution (Schoper et al., 2009); coordinate number (Scholtes and Donze, 2013); porosity (Schoper et al., 2009) and proportion of pre-existing cracks (Schoper et al., 2009). However, different DEM models may have different contact laws and therefore have different numbers of stiffness parameters. There are four stiffness parameters in the ESyS-Particle model: Kn , Ks , Kb and Kt . The two extra parameters compared with some DEM models are related to rotation: Kb and Kt (Eq. 2.4), which introduce extra complexity to the calibration process. Fortunately these two parameters are not independent and can be related to normal and shear stiffness by Kb = k1 R2 Kn and Kt = k1 R2 Ks (Wang et al., 2006). Preliminary results suggest that k1 and k2 have minor influence on the macroscopic elasticity. In this study, the fixed values of k1 = 1/3 and k2 = 1/3 are used. Similarly different models have different failure criteria, thus different fracture parameters. In general, bonded DEM models employ single-bond failure criteria which involve normal bond strength and shear bond strength at particle scale (Potyondy and Cundall, 1996; Fakhimi and Villegas, 2007; Cho et al., 2007; Scholtes and Donze, 2013). As described in Section 3.3, the ESyS-Particle adopts a new failure criterion, and thus includes three fracture parameters: particle scale tensile strength σpt , particle scale friction angle ϕp and tensile cut-off ratio σptc /σpt . The current study is mainly focused on the impact of stiffness and fracture parameters on the macroscopic response. Other parameters will be kept constant.

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Sample Generation

The densely packed random particle arrangement is generated using the particle insertion algorithm developed by the ESyS-Particle group (Place and Mora, 2001). In this method, the “seed” particles are first generated randomly within the specimen domain. Then particles are iteratively inserted into the model space so that each new particle touches four neighbours or walls. The filling-up of the specimen terminates if no further particles within the chosen size range can be inserted. The advantage of this algorithm is that it provides a relatively fast way to generate dense, random particle packing. The total number of particles and the final porosity that can be achieved are determined by the predefined particle size range (Rmin –Rmax ). It was found that samples generated with the particle insertion method had a power-law particle size distribution (PSD) and a porosity of 23% if the maximum to minimum particle radius ratio (Rmax /Rmin ) is 10 (Schoper et al., 2009). Three kinds of samples are generated in this study. Long cylindrical sample composed of 254,509 particles ranging from 0.2 mm to 2.0 mm (Fig. 2.3a) is used for unconfined compression tests and direct tensile tests. Shorter cylindrical sample composed of 60,544 particles ranging from 0.2 mm to 2.0 mm (Fig. 2.3b) is used for the Brazilian tests, and rectangular parallelepiped sample is used for tri-axial tests and Young’s modulus and Poisson ratio measurement (Fig. 2.3c). Although the samples are not exactly the same, they have the same particle size range (Rmax /Rmin = 10), the same particle size distribution and thus the same statistical properties. Numerical tests have been carried out to calculate the macroscopic strength for different particle arrangements with the same macroscopic statistical distribution, only small difference between them has been found (within 5%–10%).

Fig. 2.3 (See color insert.) Samples used in fracture simulations. (a) cylinder sample used in UCS tests and direct tensile tests; (b) cylinder sample used in Brazilian tests; (c) rectangular parallelepiped used in tri-axial tests and Young’s modulus and Poisson ratio measurement.

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Numerical Set-ups

Once the samples are generated, uni-axial compressive tests are carried out to calculate the unconfined compressive strength and the Brazilian and direct tensile tests are performed to evaluate the tensile strength. The difference between tensile strength obtained from two methods is also discussed. Tri-axial tests are used to study the failure envelopes and obtain the macroscopic internal frictional angle.

2.4.3.1

Uni-axial compressive tests

Uni-axial compression simulations are performed by moving slowly the two rigid platens towards one another. To avoid longitudinal splitting of the specimen in case of frictionless platens, stick-slip style friction condition is introduced between the particles and the loading platens, with the static and dynamic friction coefficients µs = 0.6 and µd = 0.4. The macroscopic UCS σu is then directly calculated using the force recorded in the platens and the circular area of the sample. The typical stress-strain curve and fracture patterns are shown in Figures 2.4 and 2.5a.

Fig. 2.4 Typical stress-strain curve obtained from uni-axial compression test.

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Fig. 2.5 (See color insert.) Typical fracture patterns under different loading conditions: (a) uni-axial compressive loading; (b) direct tensile test; (c) Brazilian test; (d) tri-axial compression.

2.4.3.2

The Brazilian tests

The cylindrical specimen of diameter D = 50 mm and length L = 25 mm has been generated (Fig. 2.3b) with the same distribution of particle radius as the rock model used in UCS tests. To load the sample, two flat walls are moved towards each other along the vertical direction. The typical failure mode is shown in Figure 2.5c. Compression induced tensile failures can be seen along the loading diameter direction and two crush zones are observed in two regions near the loading flat ends, splitting the sample into two halves. The failure modes correspond very well to the experimental observations. The force-time

Fig. 2.6 Typical force- time step curve obtained from Brazilian test.

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step curve is plotted in Figure 2.6. Taking the maximum force P, we find the Brazilian tensile strength as 2P (2.14) σbt = πLD

2.4.3.3

Direct tensile test

In this test, particles forming the top and bottom boundaries of the sample in Figure 2.3a are identified, tagged and bonded to two rigid platens moving slowly away from each other. In order to avoid stress concentration at the boundaries, two buffer zones consisting particles of the upper and lower 15% of the sample are formed within which bonds are set to be non-breakable. In this way, the macroscopic failure of the sample occurs roughly in the central part of the sample, rather than along the boundaries (Fig. 2.5b). The macroscopic tensile strength σdt is then directly calculated using the force recorded in the platens and the circular area of the sample.

2.4.3.4

Tri-axial test

Confined tri-axial compression tests (σ1 > σ2 = σ3 ) are performed by moving the top and bottom platens towards each other with a constant velocity, while maintaining a constant confining pressure between 5 to 30 MPa using four servocontrolled lateral walls. The failure envelopes are obtained using the peak stress σ1 value at a given confining pressure (σ3 ). The internal friction angle ϕm is calculated from the slope of the failure envelopes in σ1 − σ3 space (Eq. 2.9). The typical failure mode is shown in Figure 2.5d.

2.5

Parametric Studies

2.5.1

Elastic Parameters

For regular packing of the same sized particles, the analytical relation between the particle scale stiffness and the macroscopic elastic constants of materials can be derived (Wang and Mora, 2006). It is found that the 2-D triangular lattice of equal-sized particles yields isotropic elasticity, while in 3-D case, the closest packing generates anisotropic elasticity. For example, Face-Centered Cubic

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(FCC) lattice yields cubic elasticity, the simplest case for an orthotropic solid. Based on this research, the particle scale stiffness can be easily calibrated according to the given Young’s modulus and Poisson ratio in the case of regular packing. For random packing, however, the elasticity of lattice has to be studied numerically. Besides Kn and Ks , the macroscopic elastic parameters may also be influenced by the detailed packing geometry such as particle size distribution for random packing. This needs to be carefully investigated. The studies show that for dense random packing, particles size only has a weak effect on the macroscopic elasticity. Uni-axial compression tests with different combination of Kn and Ks are carried out for random packing under non-failure condition. The forces on the platens and lateral and axial strains are recorded to calculate Young’s modulus and Poisson ratio. Figures 2.7 and 2.8 show the dependence of Young’s modulus and Poisson ratio on bond stiffness in the case of random packing. As can be seen from these

Fig. 2.7 Macro-scopic Poisson ratio vs Ks /Kn for the random packing.

Fig. 2.8 Macro-scopic Young ’s modulus vs Ks /Kn for the random packing.

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plots, Young’s modulus is related to both Kn and Kn /Ks , while Poisson ratio is only related to the ratio Kn /Ks , which is similar to the regular packing (Wang and Mora, 2008).

2.5.2

Fracture Parameters

Even for the regular packing, it is very difficult to have a theoretical analysis on how to choose input parameters to match the desired macro-mechanical strength, like UCS σu and tensile strength σt . This problem can only be investigated numerically. The numerical simulation results are presented in the following sections.

2.5.2.1

Effect of particle scale tensile strength σpt

Figure 2.9 shows the dependences of UCS σu , direct tensile strength σdt and Brazilian tensile strength σbt on particle scale tensile strength σpt value for Ks /Kn = 0.5, ϕp = 36◦ and α = σptc /σpt = 1.0.

Fig. 2.9 The dependences of UCS σu , direct tensile strength σdt and the Brazilian tensile strength σbt on particle scale tensile strength σpt .

It can be seen that σu , σdt and σbt all linearly increase with the increase of σpt . This suggests that in order to increase UCS, direct tensile strength and Brazilian tensile strength, a larger particle scale tensile strength σpt should be used. However, the linear relations in Figure 2.9 also implies that the UCS/T ratio is independent of σpt . It is noted that σbt is nearly 1.5 times of σdt for this group of parameters, and this will be discussed in Section 2.5.2.3.

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2.5.2.2

Effect of particle scale friction angle ϕp

Figure 2.10 shows the variation of UCS with particle scale friction angle ϕp for σpt = 20 MPa, Ks /Kn = 0.5 and different values of σptc /σpt . UCS increases with the increasing of ϕp for all σptc /σpt values. The reason is that increasing ϕp implies that shear fracture is more suppressed than tensile fracture. Figures 2.11 and 2.12 show the variation of the direct tensile strength σdt and Brazilian tensile strength σbt with particle scale friction angle ϕp for the same group of parameters. It is found that the direct tensile strength is nearly independent of the particle scale friction angle. This is because in this test, the stress state is close to the pure tensile (σ1 is close to 0 and σ3 < 0, see Fig. 2.2) where the failure is mostly controlled by σptc and σpt . However, the Brazilian tensile strength increases with the increasing of ϕp without tensile cut-off, but is independent of

Fig. 2.10 The variation of UCS σu with particle scale friction angle ϕp for different degree of tensile cut-offs.

Fig. 2.11 The variation of the direct tensile strength σdt with particle scale friction angle ϕp for different degree of tensile cut-offs.

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Fig. 2.12 The variation of the Brazilian tensile strength σbt with particle scale friction angle ϕp for different degree of tensile cut-offs.

Fig. 2.13 The variation of the compressive-tensile strength ratio with particle scale friction angle ϕp for different degree of tensile cut-offs. The direct tensile strength σdt is used.

ϕp in case of large degree of tensile cut-off. In this test in the top and bottom areas close to the platens, there are large compressive and shear stress, while in the central part of the sample, tensile stress dominates. When there is larger tensile cut-off, tensile cracks appear earlier, quickly extending to the two ends, and separating the sample into two halves. In this case, ϕp nearly plays no role to the strength. When there is no tensile cut-off, tensile cracks appear later, and shear fracture may develop at the top and bottom end, therefore ϕp affects the Brazilian tensile strength. For different values of tensile cut-off, Figure 2.13 gives the variation of the UCS/T ratios with particle scale friction angle ϕp . In this plot, the direct tensile

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strength is used to calculate the strength ratio. The strength ratio based on the Brazilian tensile strength exhibits the similar results. It is seen that the strength ratios increase with the increasing of particle scale friction angle for all σptc /σpt values. This is different from the results of Scholtes and Donze (2013) and Cho et al. (2007). They report that the macroscopic strength ratio is not influenced by the particle scale friction angle at contact (ϕ) (Scholtes and Donze, 2013) and the bond shear to normal strength ratio (Cho et al., 2007) which is a representative of particle scale friction angle (there is no tensile cut-off in this model).

2.5.2.3

Effects of the tensile cut-off ratio σptc /σpt

The introduction of tensile cut-off in the failure criterion reduces UCS, the direct tensile strength and the Brazilian tensile strength. Figures 2.14–2.16 show the dependence of the macroscopic strength on the tensile cut-off ratio σptc /σpt for σpt = 20 MPa, Ks /Kn = 0.5, and different particle scale friction angle ϕp .

Fig. 2.14 The dependence of UCS σu on the tensile cut-off ratio σptc /σpt for different particle scale friction angle values of ϕp .

It can be seen that the direct tensile strength drops linearly with the decreasing of σptc /σpt for all ϕp values, implying that it is very sensitive to the tensile cut-off ratio σptc /σpt but independent of ϕp . However, UCS and the Brazilian tensile strength are less sensitive to the tensile cut-off ratio in case of smaller ϕp values compared to the case with larger ϕp values. It is interesting to note that when there is no tensile cut-off, the direct tensile strength is larger than the Brazilian tensile strength, especially for small ϕp , but with the increasing degree of tensile cut-off, the differences between them

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Fig. 2.15 The dependence of the Brazilian tensile strength σbt on the tensile cut-off ratio σptc /σpt for different particle scale friction angle values of ϕp .

Fig. 2.16 The dependence of the direct tensile strength σdt on the tensile cut-off ratio σptc /σpt for different particle scale friction angle values of ϕp .

become smaller and smaller. This can be best shown in Figure 2.17, where the direct tensile and Brazilian tensile strength ratios are plotted against σptc /σpt . For ϕp = 26◦ , the ratio σdt /σbt is about 2.11 in case of σptc /σpt = 1.0, and this ratio decreases to unity when σptc /σpt drops to 0.1–0.3. For larger ϕp values, even for the smaller degree of tensile cut-off (σptc /σpt = 0.8), ratio σdt /σbt is close to 1.0. Although some laboratory measurements suggest that the Brazilian tensile strength is slightly larger than the direct tensile strength (Fahimifar and Malekpour, 2012), we would argue that the σdt /σbt ratio close to 1.0 is acceptable and desirable, since they both describe the capability of the material to resist the tensile load. The parameter range to ensure σdt /σbt ≈ 1.0 is listed in Table 2.1.

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Fig. 2.17 The variation of the ratio of direct tensile strength to the Brazilian tensile strength with degree of tensile cut-off ratio σptc /σpt for different particle scale friction angle ϕp . Tab. 2.1 The parameter range which ensures that the direct tensile strength and Brazilian tensile strength are roughly equal (or σdt /σbt ≈ 1.0). ϕp (degree)

26

36

46

56

66

σptc /σpt

0.1 − 0.4

0.1 − 0.6

0.1 − 0.8

0.1 − 0.9

0.1 − 1.0

Figure 2.18 shows that the dependence of the UCS/T value determined using the direct tensile strength on tensile cut-off ratio σptc /σpt . The strength ratio based on the Brazilian tensile strength exhibits the similar results (Fig. 2.19). By decreasing values of σptc /σpt , the UCS/T ratio increases significantly for all ϕp values. The strength ratio of more than 40 can be achieved. This is a big improvement since in the conventional DEM simulations, the highest value of this ratio is about 5–8. It is interesting to note that the same parameters to generate the nearly equalled direct or indirect tensile strength also produce desired higher UCS/T ratios. Is this a mere coincidence, or is there underlying physics behind this phenomenon? The answer to this question is not clear to authors at present and it deserves further investigations. Probably the work of Scholtes and Donze (2013) is the only one to investigate the impact of the tensile cut-off on the UCS/T ratio. The tensile cut-off ratio σptc /σpt can be related to c/t ratio in their paper by σptc /σpt = t·tan ϕ/c, where t, c and ϕ are bond tensile strength, bond cohesion and the local frictional angle. They found that when c/t ratios increase from 1 to 20 while keeping ϕ = 25◦ , the macroscopic UCS/T ratio only changed from 3 to 5. This corresponds to the change of σptc /σpt value from 0.0233 to 0.466. They conclude that the increasing c/t ratio has very little influence on the compressive-direct tensile strength ratio

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Fig. 2.18 The variation of the compressive-tensile strength ratio with degree of tensile cut-off ratio σptc /σpt for different particle scale friction angle ϕp . The direct tensile strength σdt is used.

Fig. 2.19 The variation of the compressive-tensile strength ratio with degree of tensile cut-off ratio σptc /σpt for different particle scale friction angle ϕp . The Brazilian tensile strength σbt is used.

if it is not associated with an increase of the number of bonds in the medium, implying that if the single bond failure criterion is adopted, the degree of tensile cut-off has very little influence on the UCS/T ratio.

2.5.2.4

Effect of contact stiffness ratio

As was described in Section 2.4.1, for a fixed normal stiffness Kn , the higher the contact shear to normal stiffness ratio (Ks /Kn ) corresponds to the higher macroscopic Young’s modulus and the lower Poisson ratio. The Ks /Kn ratio also has impact on sample strength, which is shown in Figures 2.20 and 2.21.

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Fig. 2.20 The dependence of UCS σu on the stiffness ratio Ks /Kn for two different values of tensile cut-off ratio σptc /σpt .

Fig. 2.21 The dependence of the Brazilian tensile strength σbt and the direct tensile strength σdt on the stiffness ratio Ks /Kn for two different values of tensile cut-off ratio σptc /σpt .

Generally when this ratio is increased, UCS σu , the direct tensile strength σdt and the Brazilian tensile strength σbt values also increase, and these strength increase much faster when Ks /Kn < 0.125. It is found that the sample tends to be more ductile when this ratio is below 0.125. However, as shown in Figure 2.22, when Ks /Kn > 0.125, the macroscopic UCS/T ratios nearly stay constant between 14 and 16 in case of larger tensile cut-off and between 4 and 6 without tensile cut-off (i.e. as in a conventional approach). This implies that beyond 0.125 this ratio has little effect on the UCS/T ratios in terms of both the direct tensile strength and the Brazilian tensile strength. This result is similar to Cho et al. (Yoon, 2007) who found that the contact shear to normal stiffness ratio has little effect on the macroscopic UCS/T ratio.

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Fig. 2.22 The variation of the compressive-tensile strength ratio with the stiffness ratio Ks /Kn for two different values of tensile cut-off ratio σptc /σpt . Both the Brazilian tensile strength σbt and the direct tensile strength σdt are used.

2.5.2.5

Determination of macroscopic friction angle

The simulations in the previous section suggest that increasing the particle scale tensile cut-off and friction angle significantly increases the macroscopic UCS/T ratios. In order to evaluate the effect of these two parameters to the failure envelope and the macroscopic friction angle, a series of tri-axial tests with different confining pressures are carried out for two cases: σptc /σpt = 1.0 (no tensile cutoff) and σptc /σpt = 0.2 (large tensile cut-off). The failure envelopes for these two cases are presented in Figures 2.23 and 2.24.

Fig. 2.23 Failure envelope for different particle scale friction angle ϕp in tri-axial tests, σptc /σpt = 1.0 (no tensile cut-off).

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Fig. 2.24 Failure envelope for different particle scale friction angle ϕp in tri-axial tests, with tensile cut-off ratio σptc /σpt = 0.2.

Fig. 2.25 The variation of macroscopic friction angle ϕm with particle scale friction angle ϕp for two different σptc /σpt ratios.

It can be seen that without the particle scale tensile cut-off, the linear failure envelopes are obtained (as usually obtained using the conventional DEM approach). However, large tensile cut-off tends to produce non-linear failure envelopes which are similar to the laboratory test data, further confirming the necessity to introduce the particle scale degree of tensile cut-off. Figures 2.23 and 2.24 also reveal that in the case of larger particle scale friction angle, the failure envelope has higher curvature. The reason for this is that σptc /σpt ratio mainly affects the axial strength at lower confining pressure which favours tensile failure, since the tensile strength is very sensitive to σptc /σpt ratio (Fig. 2.16), while the compressive strength is not, especially in case of small particle scale friction angle (Fig. 2.14). Figure 2.25 shows the variation of macroscopic friction angle with particle scale friction angle for two different σptc /σpt ratios. The macroscopic friction angles are found to increase with the increase of particle scale friction angle, and the introduction of tensile cut-off at particle scale boosts this value. The macroscopic friction angle close to 60 degree is obtained.

2

2.6

Reproducing the Realistic Compressive-tensile Strength Ratio of Rocks. . .

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Discussions and Conclusions

The bonded discrete element models have been used to model brittle failure of rock-like materials over the past two decades. This kind of simulation usually requires a careful and time-consuming calibration of the input parameters. It is found that the conventional DEM models produce much lower UCS/T ratio and lower internal friction angles compared with measured laboratory values. The previous methods to reduce these deficiencies include the use of non-spherical particle shapes or modification of the conventional DEM models by introducing some specific, extra parameters. The present work proposed a new failure criterion for bonded discrete element model and investigated the impact of particle scale fracturing parameters on the macroscopic strength. The new criterion, which is not based on forces in a single bond but on the average stress of neighbouring particles, has the form of macroscopic Mohr-Coulomb criterion with explicit tensile cut-off. It has three independent particle-scale parameters: tensile strength, internal friction angle and degree of tensile cut-off. The impact of stiffness and fracture parameters to the macroscopic responses are numerically investigated. The results show that UCS, direct tensile strength and Brazilian tensile strength are found to be dependent upon particle scale tensile strength, particle scale friction angle, the tensile cut-off ratio and stiffness ratio. However the UCS/T ratio is only controlled by the tensile cut-off ratio and particle scale friction angle. For small particle scale friction angle without internal tensile cut-off, there is a big difference between the direct tensile strength and the Brazilian tensile strength, and only when larger degree of the tensile cutoff is used, the two kinds of the tensile strength are approximately equal. It is interesting to note that the same combination of parameters to generate the nearly equalled direct or indirect tensile strength also reproduces higher UCS/T ratios. In tri-axial tests, non-linear strength envelops are reproduced, and the macroscopic friction angle is found to be related to particle scale friction angle and the tensile cut-off ratio. The new failure criterion has several advantages over the widely used singlebond failure criteria: (1) In the single-bond failure criteria, adjusting of contact parameters has very little effect on the UCS/T ratios and frictional angle, even though the particle scale tensile cut-off is used. While in the new criterion the UCS/T ratios and frictional angle are quite sensitive to the tuning parameters; (2) Each of the three parameters in the new criterion has a clear physical meaning; (3) Direction dependent failure patterns is removed (Wang, 2016); (4) Unstable cracks are observed and the extension of cracks occur in a spatially continuous manner (Wang, 2016).

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The findings from the current research indicate that even for the spherical particles, if the new fracture criterion is adopted, by simply changing the degree of tensile cut-off and particle scale internal friction angle, it is possible to reproduce most of the basic features of hard brittle rocks: high UCS/T ratio (up to 40) and high macroscopic internal friction angle (up to 60◦ ). This suggests that the proposed criterion can be used to realistically simulate the rock fracture for a wide range of materials, from weak to hard rocks. The results in this chapter have some theoretical implications to DEM parameter calibration. For example, in order to increase the UCS/T ratio and macroscopic friction angle, higher particle scale friction angle ϕp and lower ratio of σptc /σpt should be used. It should be pointed out that the results presented here (Figs. 2.7–2.25) are obtained only for a given packing assembly. For a random packing with different particle size distribution, the whole calibration procedure must be repeated. It is impossible to give a general formula to relate the micro-mechanical parameters to the macro-mechanical properties. However the basic and qualitative shapes are expected to be similar to those presented in this chapter.

Acknowledgements This work was financially supported by CSIRO Geothermal energy program and Huainan Coal Mining Group in China. It is also supported by the (NCI) National Facility at the Australia National University (ANU) and Interactive Virtual Environments Centre (iVEC) at the University of Western Australia (UWA) through the use of advanced computing resources.

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Authors Information Yucang Wang, William W. Guo School of Engineering and Technology, Central Queensland University, Australia E-mail: [email protected]

Chapter 3

Rock Fracture under Static and Dynamic Stress Jiming Kang, Zheming Zhu, and Po Chen

Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material’s resistance to fracture. As is well known, earthquake happens when energy is released at the fault. Detailed study of active fault shows that this model—elastic-rebound theory—applies to most major earthquake. But the stress distribution in the earth crust is so complex that it is still hard for us to get a whole picture of the distribution of earthquake. Recently, fracture mechanics is introduced to earthquake, and becomes an effective method in analyzing the earthquake development and nucleation and redistribution of stress induced by an earthquake. This chapter presents the application of rock fracture mechanics in earthquake and its prediction. A general theory will be given at first, such as three kinds of modes of crack, the stress field near the crack tip and energy release rate, etc. Then we will introduce some recent application of rock fracture mechanics and discuss some challenging issues. Key words: Fracture mechanics, Strain energy density, Crack growth, Slipweakening, Finite element method, Earthquake nucleation, Stress redistribution.

3.1

Introduction

Fracture mechanics is an important tool in improving the performance of mechanical components. It applies the physics of stress and strain, in particular the theories of elasticity and plasticity, to the microscopic crystallographic de-

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fects found in real materials for predicting the macroscopic mechanical failure of bodies. Fractography is widely used with fracture mechanics to understand the causes of failures and also verify the theoretical failure predictions with real life failures. The prediction of crack growth is a main task of the damage tolerance discipline. There are three ways of applying a force to enable a crack to propagate (Fig. 3.1): – Mode I fracture – Opening mode – Mode II fracture – Sliding mode – Mode III fracture – Tearing mode

Fig. 3.1 The three modes of fracture.

A fault is a planar fracture or discontinuity in a volume of rock, across which there has been a significant displacement along the fractures as a result of the earth movement. Large faults within the Earth’s crust result from the action of plate tectonic forces, forming the boundaries between the plates, such as subduction zones or transform faults. Energy release associated with rapid movement on active faults is the cause of most earthquakes. Some seismic faults can be considered as a group of collinear cracks and subjected to a compressed stress from environment around it. A simple model is shown in Figure 3.2 (the mixed model of Mode I and Mode II):

Fig. 3.2 An infinite plane containing a group of collinear cracks under compression, where σN is normal stress, σS is shear stress.

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3.2

Rock Fracture under Static and Dynamic Stress

177

Stress Intensity Factor and Stress Field

The stress intensity factor K is used in fracture mechanics to predict the stress state around the tip of a crack caused by a remote load or residual stresses (Griffith, 1921; Irwin, 1957; Erdogan, 2000). It is a theoretical construct usually applied to a homogeneous, linear elastic material, and is useful for providing a failure criterion for brittle materials and a critical technique in the discipline of damage tolerance. The concept can also be applied to materials that exhibit small-scale yielding at a crack tip (Orowan, 1948). The magnitude of K depends on sample geometry, the size and location of the crack, and the magnitude and the modal distribution of loads on the material. Linear elastic theory predicts that the stress distribution (σij ) near a crack tip, in polar coordinates (r, θ) with origin at the crack tip, has the form (Dugdale, 1960; Barenblatt, 1962; Willis, 1967; Anderson, 1995; Camacho and Ortiz, 1997; Buckley, 2005): K σij (r, θ) = √ fij (θ) + O(r, θ) 2πr where K is the stress intensity factor and fij is a dimensionless quantity that varies with the load and geometry. This relation breaks down very close to the tip (small r) because as r approaches to 0, the stress σij goes to ∞. Thus, plastic distortion occurs at high stresses and the linear elastic solution is no longer applicable close to the crack tip. However, if the crack-tip plastic zone is small, it can be assumed that the stress distribution near the crack is still given by the above relation. In this chapter, the stress intensity factors for an infinite plane containing collinear cracks (Newman, 1971; Nemat-Nasser and Horii, 1982; Steif, 1984; Ashby and Hallam, 1986; Li and Nordlund, 1993; Germanovich et al., 1994; Baud et al., 1996; Zhu, 1999; Rice et al., 2001; Zhu et al., 1997, 2006) under a far field principal stress σ1 and σ3 shown in Figure 3.2 will be studied by using complex stress functions. According to Zhu et al. (1996), for the elastic plane problem, the stress can be expressed in terms of the complex stress functions, Φ(z) and Ω (z), as σx + σy = 2[Φ(z) + Φ(z)] σy − iτxy = Φ(z) + Ω (¯ z ) + (z −

(3.1) z¯)Φ ′ (z)

(3.2)

where z = x + iy. The equilibrium equations and the compatibility conditions are automatically satisfied, and the resultant force boundary condition can be expressed as i



B

A

(X + iY )ds = [ϕ(z) + ω(¯ z ) + (z − z¯)Φ(z)]B A

(3.3)

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where ϕ(z) =



Φ(z)dz and ω(z) =



Ω (z)dz, and X, Y are the surface forces

along the boundary in x and y directions, respectively, and A is an arbitrary point on boundary. The positive direction of the integral is assumed that the region always lies to the left. The displacement boundary condition can be expressed as 2G(u + iv) =

3−µ ϕ(z) − ω(¯ z ) − (z − z¯)Φ(z) 1+µ

(3.4)

where G = E/2(1 + µ), E is elastic modules and µ is Poisson’s ratio. The stress boundary condition can be written as σx l + τxy m = Xn τxy l + σy m = Yn

(3.5)

where l and m are the direction cosines of the normal line of the boundary, and Xn , Yn are the force components on the boundary. From equation (3.5) and Figure 3.2, on the crack surfaces (σy )+ = −σN , (σy )− = −σN ,

(τxy )+ = −σf (τxy )− = −σf

(upper surface) (lower surface)

(3.6)

where σf and σN are the crack surface friction and normal stress, respectively (see Fig. 3.1). Substituting equation (3.6) into (3.2), one can have (σy )+ − i(τxy )+ = Φ + (t) + Ω − (t) = −σN + iσf (σy )− − i(τxy )− = Φ − (t) + Ω + (t) = −σN + iσf or

(3.7)

+

[Φ(t) + Ω (t)] + [Φ(t) + Ω (t)]− = −2σN + 2iσf = Q + [Φ(t) − Ω (t)] − [Φ(t) − Ω (t)]− = 0

(3.8)

where Q is a constant. This is a simple Hilbert problem. 3 If the shear stress σs (σs = σ1 −σ sin 2a), which is parallel to the crack sur2 3 + face, is less than the crack surface fiction, i.e., σs < f σN , where σN = σ1 +σ 2 σ1 −σ3 cos 2a, then the crack tip stress intensity factor is zero because large crack 2 surface friction will result in a crack tip without stress concentration, only when σs  f σN , the crack tip SIF exists, and in this case σf can be written as:   σ1 − σ3 σ1 + σ3 + cos 2a f (3.9) σf = σN f = 2 2 For the problem of multiple cracks shown in Figure 3.2, the Plemelj function can be written as  X(z) = 1/ (z − a1 )(z − a2 )(z − a3 ) · · · (z − an ) (3.10)

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It can be seen that as z → ∞, z n X(z) → 1, where n is the number of crack tips. Suppose t0 is a point on crack surfaces, and if X(z) has a unique limit as z → t0 along any path in the right neighborhood of t0 , we denote this limit by X + (t0 ). Similarly, if a unique limit exists on approach to t0 from the left neighborhood, it is denoted by X − (t0 ). In order to investigate the relation between X + (t) and X − (t) as X(z) crosses the crack, let’s trace a path beginning at a point D on a crack surface and leading from the right side of the crack around the crack tip A1 to approach D from the left as shown in Figure 3.3. It can be seen that the modulus of all the vectors will not change except for z − a8 , and all vectors will increase by 2π in argument whereas the vector z − a8 will return to its original value in argument. Therefore, from equation (3.10), one can have X − (t) = X + (t) · e0.5×(n−1)×(−2π)i = −X + (t)

(3.11)

where n is crack tip number, and for the case in Figure 3.3, n = 8.

Fig. 3.3 Sketch of a path.

In equation (3.8), Φ(t) and Ω (t) are stress complex functions and according to the stress single-valued condition in multi-connected zones, they can be written as 1 a1 a2 1+ν (X + iY ) + 2 + 3 + · · · 8π z z z 1 b1 3−ν b2 (X + iY ) + 2 + 3 + · · · Ω (z) = B + C − iD + 8π z z z Φ(z) = B −

(3.12) (3.13)

where ν is Poisson’s ratio, B, C, and D are constants, and they can be determined by the far field principal stresses, i.e. 1 (σ1 + σ3 ) 4 π  σ −σ σ1 − σ3 1 3 cos 2 −α = cos 2α C=− 2 2 2   σ1 − σ3 σ1 − σ3 π sin 2 −α = sin 2α D = τxy = 2 2 2 B=

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As z → ∞, from equations (3.12) and (3.13), one can have Φ(∞) = B and Ω (∞) = B + C − iD. Let E(t) = Φ(t) + Ω (t) and F (t) = Φ(t) − Ω (t), equations (3.3) and (3.4) can be rewritten as E + (t) + E − (t) = Q(t) +

−

F (t) − F (t) = 0

(3.14) (3.15)

Equations (3.14) and (3.15) are simple Hilbert problem, and dividing through by X + (t) and X − (t), from equation (10), one can have E − (t) Q(t) E + (t) − = + + − X (t) X (t) X (t) Let G(t) =

(3.16)

Q(t) E(t) and g(t) = + , then equation (3.16) can be rewritten as X(t) X (t) G+ (t) − G− (t) = g(t)

(3.17)

According to the theory of Cauchy integral on an arc (England, 1971), the general solution of equation (3.17) can be written as  1 g(t)dt + P2 (z) G(z) = (3.18) 2πi L t − z According to the function’s relationship defined above, from equation (3.18), one can have  X(z) Q(t)dt + X(z)P2 (z) (3.19) Φ(z) + Ω (z) = 2πi L X + (t)(t − z) Equation (3.10) can be rewritten as F + (t) = F − (t)

(3.20)

This means that the function F (z) is analytic in the entire plane except possibly at infinite where it can at most have a pole. Hence, by Laurent’s theorem, F (z) is a polynomial, and considering the condition that as z → ∞, F (z) is limited, then the general solution is F (z) = Φ(z) − Ω (z) = Φ(∞) − Ω (∞) = −C + iD

(3.21)

From equations (3.19) and (3.21), the two complex stress functions for the case of collinear cracks under compression can be formulated as (see Erdogan, 1962; England, 1971; Horiand Nemat-Nasser, 1982; Gong, 1994; Wang, 1997; Chen et al., 2003).  1 Q(t)dt X(z) + X(z)Pn (z) − (C − iD) (3.22) Φ(z) = + 4πi L X (t)(t − z) 2  X(z) 1 Q(t)dt Ω (z) = + X(z)Pn (z) + (C − iD) (3.23) + 4πi L X (t)(t − z) 2

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Fig. 3.4 An arc and a lacet surrounding it.

where Pn (z) = k0 + k1 z + k2 z 2 + · · · + kn z n ; 1 C − iD = (σ1 − σ3 )e−2iα 2 1 1 1 kn = B + (C − iD) = (σ1 + σ3 ) + (σ1 − σ3 )e−2iα 2 4 4 where k0 → kn−1 are constants. They will be determined. In order to get the solution of Φ(z) and Ω (z) from equations (3.22) and (3.23), first we have to solve the following equation  Q(t)dt (3.24) I(z) = + L X (t)(t − z) where L = L1 + L2 is the union of the two cracks. For each crack, the integrals along Lk in equation (3.24) can be expressed in terms of integral along a lacet Ck surrounding it as shows in Figure 3.4, and the integral over the lacet can be evaluated by the residue theory. Considering the contour integral around the lacet Ck , one can have    f (ζ)dζ f (t)dt f (ζ)dζ = + lim − + ρ→0 Ck X(ζ)(ζ − z) Lk X (t)(t − z) |z−bk |=ρ X(ζ)(ζ − z) (3.25)   f (t)dt f (ζ)dζ + lim − ρ→0 |z−a |=ρ X(ζ)(ζ − z) Lk X (t)(t − z) k It can be seen that the second and the fourth integrals in equation (3.25) tend to zero as ρ → 0, and combining equation (3.25), one can have      1 f (ζ)dζ f (t) f (t)dt 1 = − − dt = 2 + + X (t) Ck X(ζ)(ζ − z) Lk t − z X (t) Lk X (t)(t − z) (3.26) From equation (3.26), the relationship between the integrals along a crack Lk and along its lacet Ck can be obtained   1 f (t)dt f (ζ)dζ = (3.27) I(z) = + 2 C X(ζ)(ζ − z) L X (t)(t − z)

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By using equations (3.27), (3.22) and (3.23) can be rewritten as  X(z) 1 Q(t)dt + X(z)Pn (z) − (C − iD) Φ(z) = 8πi C X(t)(t − z) 2  X(z) 1 Q(t)dt Ω (z) = + X(z)Pn (z) + (C − iD) 8πi C X(t)(t − z) 2

(3.28) (3.29)

From the residue theory, the integral in equation (3.27) along contour C can be expressed in terms of the sum of the residues at point z and at infinity, thus equation (3.28) can be rewritten as:    QX(z) 1 1 1 Φ(z) = + Res , ∞ + X(z)Pn (z) − (C − iD) 4 X(z) X(t)(t − z) 2 (3.30) In order to determine the coefficient k0 → kn−1 , the conditions of single-valued displacement have to be applied, which can be expressed as   Φ(z)dz − Ω (¯ z )d¯ z = 0 (k = 1, 2, · · · , n) (3.31) κ Ck

Ck

where κ = (3 − ν)/(1 + ν) for plane stress and κ = 3 − 4ν for plane strain, ν is Poisson’s ratio, and Ck is the contour of the kth crack. If the contour Ck shrinks to a lacet around the crack, equation (3.31) can be rewritten as   + − κ [Φ (t) − Φ (t)]dt − [Ω − (t) − Ω + (t)]dt = 0 (k = 1, 2, · · · , n) (3.32) Lk

Lk

where t is the coordinates of a point on the crack surface. Applying equation (3.32) for k = 1, 2, · · · , n, one can yield n linear equations, from which the n coefficients can be obtained. Therefore, for an infinite plate containing finite collinear cracks under tension, one may obtain the stress function Φ(z) and Ω (z). However, because the stress functions contain the Plemelj function X(z), the process of solving n coefficients maybe difficult due to the high-order integrals involved. In order to qualitatively investigate the effect of confining stress on SIFs, the corresponding photo-elastic experiment using poly-carbonate (PC) plates is conducted, and the test results are shown in Figure 3.4. The PC plate size is 15cm × 15cm × 0.6cm; the crack length is 1.0 cm; the crack width is 0.1 cm; and the crack orientation is 60◦ . The vertical stress varies from 0 to 2 MPa. From the photo-elastic results in Figure 3.5, it can be clearly seen that the density of fringes on crack tips increase as the stress increase, indicating that SIF increases as confining stress increases, and therefore, the photo-elastic results qualitatively are in line with the theoretical results. First, consider a single crack under compression as Figure 3.6.

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Fig. 3.5 (See color insert.) The photo-elastic results of the order of fringes for PC plates with confining stress; the parameters: crack lengths are 1.0 cm and 2.0 cm, crack tip distance is 1.0 cm, α = 60◦ . (a) σ1 = 0.5 MPa; (b) σ1 = 1.0 MPa; (c) σ1 = 2.0 MPa.

Fig. 3.6 Single crack in infinite plane under compression.

Based on the previous result, for an infinite plane as shown in Figure 3.6, the corresponding function can be simplified as:  (3.33) X(z) = 1/ (z 2 − a2 ) and the corresponding polynomial Pn (z) = k0 + k1 z where k1 = kn . In this case, Q = −2σN + 2iσf = (−1 + f )2σN and Φ(z) can be written as:  X(z) 1 Q(t)dt Φ(z) = + X(z)Pn (z) − (C − iD) (3.34) + 4πi L X (t)(t − z) 2  X(z) 1 Q(t)dt Ω (z) = + X(z)Pn (z) + (C − iD) (3.35) + 4πi L X (t)(t − z) 2

Substituting equations (3.34) and (3.35) into equation (3.32), one can have:  a k0 + k1 z (2κ + 1) =0 (3.36) X(z) −a

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Because the crack and load is symmetric, k0 = 0 where 1 1 1 k1 = B + (B ′ − iC ′ ) = − (σ1 + σ3 ) − (σ1 − σ3 )e2iα 2 4 4 Thus Φ(z) can be expressed as:

where S0 = −

Φ(z) = X(z)S0 z +

Q 1 − (C − iD) 4 2

Ω(z) = X(z)S0 z +

Q 1 + (C − iD) 4 2

Q + k1 4

The stress intensity factors (SIFs) can be calculated by    √ Q KI − iKII = lim 2 2π(z − ct)Φ(z) = 2 πa − + k1 z→a 4          √ σ1 − σ3 σ1 − σ3 σ1 + σ3 + sin 2α − sin 2α = i πa f 2 2 2 √ = −i πa(−f σN + σs ) and then KI = 0,

KII =

√ πa(−f σN + σs )

(3.37)

where KI , KII are the SIFs of model I and model II respectively. Because the right part of the equation (3.37) is pure imaginary function, the model I factor KI is zero, which means that the KI has no impact on the crack or fault under compression. One also can get the stress field at the crack tip as follows:    θ 3θ θ KII 2 + cos cos − sin σx = √ 2 2 2 2πr θ 3θ KII θ σy = √ sin cos cos 2 2 2 2πr   θ 3θ θ KII 1 − sin sin cos τxy = √ 2 2 2 2πr or

  θ KII θ 1 − 3 sin2 sin σrr = √ 2 2 2πr   θ KII θ −3 sin cos2 σθθ = √ 2 2 2πr   KII θ 1 − 3 sin2 σrθ = √ 2 2πr

In order to get more precise stress field, a new coordinate system (see Fig.3.7) should be introduced as below

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Fig. 3.7 Polar coordinate with r1 , r2 and θ1 , θ2 .

The Westergaard Stress Function is given for the sake of simplicity:   z √ ZII = − 1 τ∞ z 2 − a2

(3.38)

where z = reiθ , z − a = r1 eiθ1 , z + a = r2 eiθ2 , and then equation (3.38) can be written as   θ1 +θ2 r ei(θ− 2 ) − 1 τ ∞ (3.39) ZII = √ r1 r2 The derivative of ZII with respect to z is ′ ZII =

−a2 τ ∞ −i 32 ( θ1 +θ −a2 τ ∞ 2 ) 2 = e 2 3/2 3/2 −a ) (r1 r2 )

(z 2

(3.40)

and the calculus of ZII is     1 θ1 +θ2 II = τ ∞ z 2 − a2 − z = τ ∞ (r1 r2 )ei 2 ( 2 ) − re−iθ Z So the stress components can be given:

σx + σy = ImZII 2 σx − σy ′ = ImZII + yReZII 2 ′ τxy = ReZII − yImZII

(3.41)

Substituting equations (3.39) and (3.40) into equation (3.41), one can have   rτ ∞ θ1 + θ2 σx + σy = √ sin θ − 2 r1 r2 2     a2 r θ1 + θ2 3(θ1 + θ2 ) r σx − σy ∞ =τ − sin θ − sin θ cos √ 2 r1 r2 2 2 (r1 r2 )3/2     θ1 + θ 2 r −1 − τxy = τ ∞ cos θ − √ r1 r2 2  a2 r 3(θ1 + θ2 ) (3.42) sin θ sin 3/2 2 (r1 r2 )

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Also the displacement field can be given by 2µ  ImZII + yReZII E′ 2µ(1 − ν ′ )  2µν = − ReZII − yImZII E′

2µu =

Substituting equations (3.29) and (3.40) into equation (3.42), one can have:          θ 1 + θ2 4µ √ θ 1 + θ2 2µu = −r sin θ +r sin θ cos θ− −1 τ ∞ r r sin 1 2 E′ 2 2         θ1 +θ2 2µ(1−ν ′) √ θ1 +θ2 −r sin θ −r sin θ sin θ− τ∞ 2µν = r1 r2 cos E′ 2 2 (3.43) When y = 0 and |x| < a, the displacements on the upper and bottom crack surfaces can be written as: 2τ ∞  2 a − x2 u+ = E′ (1 − ν ′ )τ ∞ ν+ = x E′ 2τ ∞  2 u− = − ′ a − x2 E (1 − ν ′ )τ ∞ ν− = x E′ Thus,

4τ ∞ √ 2 a − x2 E′ ∆ν = 0

∆u =

3.3

(3.44)

Coulomb-Mohr Failure Criterion

In the above analyses, we show that the crack surface frictions, confining stresses and distances between two crack tips can affect the stress intensity factors significantly, and thus they affect fault fracture behavior. It is well known that the crack propagation criterion for mode II crack can be written as (Muskelishvili, 1953; Nemat-Nasser and Horii, 1982; Kachanov, 1987; Ballarini and Plesha, 1987; Wang, 1997; Eberhardt and Kim, 1998; Lauterbach and Gross, 1998; Chen, 2003; Fan, 2003; Li et al., 2003; Zhu et al., 2006; Millwater, 2010; Jin, 2013) (3.45) KII  KIIC

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where KIIC is material fracture toughness. The crack propagation criterion in equation (3.45) is difficult to apply because the fracture toughness is generally difficult to measure. The technique used in measuring material fracture toughness is not flawless, and the measurement results are usually scattered in a large range, especially for brittle materials. In order to avoid such difficulties in measuring material fracture toughness, equation (3.29) will be transformed into a new form expressed in terms of principal stresses, without KII and KIIC involved. Considering a specimen containing two cracks under uni-axial compression as shown in Figure 3.8a, its critical stress is σU , and the corresponding KII can be obtained from equation (3.37) by substituting σf from equation (3.9) and taking σ3 = 0:   √ σU (1 + cos 2α) σU sin 2α U + (3.46) KII = πa −f 2 2 Under the critical condition, the relation between KII and KIIC can be written as KII = KIIC Substituting equation (3.46) into equation (3.47), we have   √ σU (1 + cos 2α) σU sin 2α U KII = πa −f + = KIIC 2 2

(3.47)

(3.48)

When the confining stress is applied on this specimen as shown in Figure 3.8b, its critical stress will increase from σU to σ1 . The corresponding KII can be written as     √ (σ1 + σ3 ) (σ1 + σ3 ) (σ1 − σ3 ) cos 2α + + sin 2α (3.49) KII = πa −f 2 2 2

Fig. 3.8 A specimen containing two inclined cracks under (a) uni-axial compression and (b) bi-axial Compression.

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Substituting equations (3.48) and (3.49) into equation (3.47), one can have f (σ1 + σ3 ) + (σ1 − σ3 )(f cos 2α − sin 2α)  σU (f + f cos 2α − sin 2α) (3.50) From equation (3.50), the crack propagation criterion for single crack under compression is obtained (Steif, 1984; Ashby and Hallam, 1986; Basista and Gross, 2000) 1 + f tan α σ3 σ1  σU + (3.51) 1 − f tan α

Equation (3.51), known as Coulomb-Mohr failure criterion, is an alternative form of crack propagation criterion for single crack under compression, derived from the general crack propagation criterion, equation (3.35). This criterion is expressed in terms of principal stresses, without KII and KIIC being involved, which is more convenient in application. The critical stress σ1 in equation (3.41) is related to four parameters: confining stress σ3 , uni-axial compression strength σU , crack orientation α, and crack surface friction coefficient f . Under the critical state, the relation between σ1 and σ3 from equation (3.51) can be written as (Erdogan, 1962; England, 1971; Olver et al., 2010) σ1 = σU +

1 + f tan α σ3 1 − f tan α

It should be noted that the shear stress σS in Figure 3.6, which is parallel to the crack surface, must be larger than the crack surface friction f σN , i.e.   (σ1 + σ3 ) (σ1 − σ3 ) cos 2α (σ1 − σ3 ) sin 2α > f + 2 2 2 Otherwise, the SIF is zero due to a large crack surface friction and then the above equation can be rewritten as 1 − f tan α σ3 = σ1 1 + f tan α

(3.52)

Therefore, before applying equations (3.49), one should use equation (3.52) to examine if the crack SIF is zero, and if equation (3.52) is not satisfied, the crack tip SIF is zero. σ3 Because > 0, from equation (3.52), one can have 1 − f tan α > 0, or σ1 tan α > f . This means that tan α should be larger than crack surface friction coefficient f . Otherwise, the corresponding crack SIF must be zero.

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3.4

Energy Release and J-integral

3.4.1

Energy Release Rate

189

The strain energy release rate is the energy dissipated during fracture per unit of newly created fracture surface area. This quantity is central to fracture mechanics because the energy that must be supplied to a crack tip for it to grow must be balanced by the amount of energy dissipated due to the formation of new surfaces and other dissipative processes such as plasticity. For the purposes of calculation, the energy release rate is defined as G=−

∂(U − W ) ∂A

(3.53)

where U is the potential energy available for crack growth, W is the work associated with any external forces acting, and A is the crack area (crack length for two-dimensional problems). The unit of G is J/m2 . The energy release rate failure criterion states that a crack will grow when the available energy release rate G is greater than or equal to a critical value Gc , G  Gc where Gc is the fracture energy and is considered to be a material property which is independent of the applied loads and the geometry of the body. For two-dimensional problems (plane stress, plane strain, anti-plane shear) involving cracks that move in a straight path, the mode I stress intensity factor K2 KI is related to the energy release rate (G) by G = I′ where E is the Young’s E E ′ ′ for plane strain. modulus and E = E for plane stress and E = (1 − µ2 )

3.4.2

J-integral

The J-integral represents a way to calculate the strain energy release rate, or work (energy) per unit fracture surface area, in a material (Van Vliet, 2007). The theoretical concept of J-integral was developed by Cherepanovn (1967) and by Rice (1968) independently, who showed that an energetic contour path integral was independent of the path. Later, experimental methods were developed to measure the critical fracture properties using laboratory-scale specimens for materials in which sample sizes are too small and for which the assumptions of Linear Elastic Fracture Mechanics (LEFM) do not hold (Lee and Donovan,. 1987), and to infer a critical value of fracture energy JIC . The quantity JIC defines the point at which large-scale plastic yielding during propagation takes place under mode one loading (Rice, 1968).

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The J-integral is equal to the strain energy release rate for a crack in a body subjected to monotonic loading (Lee and Donovan, 1987). This is generally true, under quasi-static conditions, only for linear elastic materials. For materials that experience small-scale yielding at the crack tip, J can be used to compute the energy release rate under special circumstances such as monotonic loading in mode III (anti-plane shear). The strain energy release rate can also be computed from J for pure power-law hardening plastic materials that undergo small-scale yielding at the crack tip. The quantity J is not path-independent for monotonic mode I and mode II loadings of elastic-plastic materials, so only a contour very close to the crack tip gives the energy release rate. Also, Rice shows that J is path-independent in plastic materials when there is no non-proportional loading. Unloading is a special case of this, but non-proportional plastic loading also invalidates the path-independence. Such non-proportional loading is the reason for the pathdependence for the in-plane loading modes on elastic-plastic materials. The two-dimensional J-integral was originally defined as (see Fig. 3.9)       → → ∂ u ∂ui wdx2 − t wdx2 − ti ds = ds (3.54) J= ∂x1 ∂x1 Γ Γ where w(x1 , x2 ) is the strain energy density, x1 , x2 are the coordinate directions, t = n · σ is the surface traction vector, n is the normal to the curve Γ , σ is the Cauchy stress tensor, and u is the displacement vector.

Fig. 3.9 Line J-integral around a notch.

For Mode II crack under shear stress τ on the surface, we can have the conservative expression (see Fig. 3.10) JQ + JQ+ P + − JP + JP − Q− = 0

(3.55)

where JQ is a J-integral along the closure from the bottom surface point Q− to the upper surface point Q+ . JQ+ P + is a J-integral along the line from the upper surface point Q+ to the surface point P + (see Fig. 3.10).

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Fig. 3.10 J-integral for the Mode II crack.

Equation (3.55) can be rewritten as JQ +



0

σ

Q



∂δ ∂x1



dx1 = JP +



0

σ

P



∂δ ∂x1



dx1

(3.56)

When P is close to the crack tip O, and equation (3.56) has JQ +



0

Q

σ



∂δ ∂x1



dx1 = lim JP = G P →O

(3.57)

For the crack surface under uniform shear stress τs , equation (3.57) can be simplified as (3.58) G = JQ − τs δQ Rice (1968) also showed that the value of the J-integral represents the energy release rate for planar crack growth. The J-integral was developed because of the difficulties involved in computing the stress close to a crack in a nonlinear elastic or elastic-plastic material. Rice showed that if monotonic loading was assumed (without any plastic unloading) then the J-integral could be used to compute the energy release rate of plastic materials too. For isotropic, perfectly brittle and linear elastic materials, the J-integral can be directly related to the fracture toughness if the crack extends straight ahead with respect to its original orientation (Ramberg et al., 1943; Hutchinson, 1968; Rice, 1968; Rosengren, 1968; Yoda, 1980; Meyers and Chawla; 1999). Under Mode I loading conditions, this relation is JIC = GIC =

2 KIC E′

where GIC is the critical strain energy release rate, JIC is the fracture toughness, and E is the Young’s modulus and E ′ = E for plane stress and E ′ = E/(1 − µ2 ) for plane strain.

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For Mode II loading, the relation between the J-integral and the mode II fracture toughness KIIC is JIIC = GIIC =

2 KIIC E′

For Mode III loading, the relation is JIIIC = GIIIC =

3.5

2 KIIIC



1+µ E



Crack Growth

Crack criterion is an important problem, which is needed to research from micro, sub-micro and macro views. As an integral crack criterion, it should be able to solve two problems: – In which condition will crack initial and continue? – In which direction will crack extend? The research on the rock fracturing mechanism can be traced back to the theory of Griffith brittle fracture model which means that the crack can’t extend until the release rate of the strain energy is larger than the crack surface energy. Later, Irwin introduced SIF—known as K criterion—as crack criterion based on Griffith brittle fracture model. But K criterion cannot answer the problem of the crack with mixed model containing Model I and Model II, so a series of new crack criterion are developed.

3.5.1

Maximum Hoop Stress Theory

Erdogan and Sih (1968) raised the theory when they found that the mixed-model crack extended along the surface that was perpendicular to the maximum hoop stress. In the 2-D model, which means KIII is zero, the stress field on the crack tip is shown below:   1 θ θ KI cos (3 − cos θ) + KII sin (3 cos θ − 1) σrr = √ 2 2 2 2πr θ 1 σθθ = √ cos [KI (1 + cos θ) − 3KII sin θ] (3.59) 2 2 2πr θ 1 σrθ = √ cos [KI sin θ − KII (3 cos θ − 1)] 2 2 2πr

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According to the following theory: Crack extends along the angle which have the max hoop stress σθθ , and the direction satisfies ∂ 2 σθθ ∂σθθ = 0, 0 ∂θ ∂ θ

(3.66)

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For mode II, KI = KIII = 0 and 2 KII [4(1 − µ)(1 − cos θ) + (1 + cos θ)(3 cos θ − 1)] 16πµ 2 KII ∂S = [(1 − 2µ − 3 cos θ) sin θ] ∂θ 16πµ 2 KII ∂2S = [(1 − 2µ) cos θ − 3 sin 2θ] 2 ∂ θ 16πµ

S=

From equation (3.66), one can get the solution:  and the second θ satisfies

θ=0 cos θ = (1 − 2µ)/3

∂2S > 0, so θ0 is: ∂2θ θ = arccos[(1 − 2µ)/3]

and 2 Smin = KII

3.6

(3.67)

2(1 − µ) − µ2 12πµ

(3.68)

Crack Growth under Dynamic Loading

The basic equations of plane elastic-dynamics are as below. ∂τxy ∂σx + = ρ¨ u ∂x ∂y

(3.69)

∂σy ∂τxy + = ρ¨ v ∂x ∂y The strain-displacement relations are εx =

∂u , ∂x

εy =

∂v , ∂y

εxy =

1 2



∂u ∂v + ∂y ∂x



(3.70)

and constitutive relations are: σx = λ(εx + εy ) + 2Gεx σy = λ(εx + εy ) + 2Gεy τxy = 2Gεxy

(3.71)

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and λ is given by λ= Where λ∗ =

 ∗   λ , planes − strain ∗   2λ G , λ∗ + 2G

plane − stress

Eµ , (1 + µ)(1 − 2µ)

G=

E 2(1 + µ)

To analyze crack growth under a dynamic loading, we first analyze the stress and displacement fields near a stationary crack tip. Because the displacements are finite, their derivatives with respect to time are also bounded for stationary cracks. At the same time, the stresses and their derivatives with respect to spatial coordinates are singular at the crack tip. The left side terms in the equations of motion equation (3.54) thus dominate the inertial terms on the right side. Hence in the near-tip region, the forms of the equations of motion are simplified as follows: ∂τxy ∂σx + =0 ∂x ∂y (3.72) ∂σy ∂τxy + =0 ∂x ∂y Because the material is still considered to be linear elastic, the stress and displacement fields near the crack tip will be the same as those in the quasi-static linear elastic fracture mechanics. Hence, the inertial effect does not change the singularity structure of stress and displacement fields near the tip of a stationary crack under dynamic loading. The crack tip stress and displacement fields are given by � � � � θ 3θ KII (t) θ 3θ θ θ KI (t) 1 − sin sin − √ 2 + cos cos cos sin σx = √ 2 2 2 2 2 2 2πr 2πr � � θ 3θ KII (t) θ 3θ θ θ KI (t) 1 + sin sin + √ (3.73) cos sin cos cos σy = √ 2 2 2 2 2 2 2πr 2πr � � θ 3θ KII (t) θ 3θ θ θ KI (t) τxy = √ + √ 1 − sin sin cos sin cos cos 2 2 2 2 2 2 2πr 2πr and

� � 3θ KI (t) √ θ + u= 2πr (2κ − 1) cos − cos 8µπ 2 2 � � 3θ θ KII (t) √ 2πr (2κ + 3) sin + sin 8µπ 2 2 � � √ KI (t) 3θ θ v=u= + 2πr (2κ + 1) sin − sin 8µπ 2 2 � � 3θ θ KII (t) √ 2πr (2κ − 3) cos + cos 8µπ 2 2

(3.74)

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where KI (t) and KII (t) are the dynamic stress intensity factors and (r, θ) are the polar coordinates at the crack tip with crack surfaces at θ = ±π · KI (t). and KII (t) depend not only on the magnitude of the dynamic load and crack configuration, but also time t. In general, the peak values of the dynamic stress intensity factors for a given crack are higher than the corresponding SIF under the quasi-static loading of the same magnitude. The monographs by Sih et al. (1972) and Freund (1990) provide the dynamic stress intensity factor solutions for various cracks under crack face impact loads and wave loads. Closed-form solutions of dynamic stress intensity factors are available for only a few problems of dynamically loaded stationary cracks. Maue (1954) considered a semi-infinite crack under a sudden crack face pressure σ at time t = 0 and gave the dynamic stress intensity factor as follows: √ 2 2 cR √ σ0 πct (3.75) KI (t) = π c1 where cR is the Rayleigh surface wave speed, which is the smallest real root of the following equation:  2 1  1  c2 c2 2 c2 2 2 − R2 1 − R2 − 4 1 − R2 =0 c2 c1 c2

(3.76)

Note that cR < c2 < c1 . Equation (3.75) shows that the dynamic stress intensity factor depends on both dilatational and Rayleigh surface wave speeds. Moreover, the dynamic stress intensity factor is zero at the beginning of loading and increases with time. The crack tip stress field equation (3.73) shows that similar to the quasistatic cracks, the dynamic stress intensity factor determines the intensity of the singular stresses around the crack tip. The crack initiation thus can be predicted based on the SIF criterion, that is, the crack initiation occurs when the dynamic stress intensity factor reaches the dynamic fracture toughness of the material KId . (3.77) KI (t) = KId This fracture criterion is applicable to brittle materials such as engineering ceramics and rocks, as well as metals when small-scale yielding conditions prevail.

3.6.1

Dynamic Crack Propagation in Rock

In a linear elastic material, a crack will propagate unstably once it has initiated. The crack propagation will lead to dynamic failure of the material unless it is arrested. Studies of the stress and displacement fields near a propagating crack

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tip usually use a moving coordinate system (x1 , y1 ) centered at the crack tip, as shown in Figure 3.16, where (x, y) is a fixed coordinate system and a(t) is a current crack propagation distance. The two systems are related by x1 = x − a(t),

y1 = y

(3.78)

The material derivative in the moving coordinates becomes d( )/dt = ∂( )/∂t − V · ∂( )/∂x1

(3.79)

where V = a˙ = da/dt is the crack propagation speed. Using equations (3.78) and (3.79), the wave equation (3.69) in the moving coordinates can be written as follows:   2 1 ∂2ϕ ∂2ϕ ∂2ϕ ∂2ϕ ˙ ∂ϕ + V 2 ∂ ϕ − V + = − 2V ∂x21 ∂y12 c21 ∂t2 ∂x1 ∂t ∂x1 ∂x21 (3.80)   2 ∂ψ 1 ∂2ψ ∂2ψ ∂2ψ ∂2ψ 2∂ ψ ˙ −V + = 2 − 2V +V ∂x21 ∂y12 c2 ∂t2 ∂x1 ∂t ∂x1 ∂x21 These equations can be simplified as   1 ∂2ϕ ∂ 2ϕ ∂ 2 ϕ ∂ 2ϕ ˙ ∂ϕ − V = − 2V α2 2 + ∂x1 ∂y12 c21 ∂t2 ∂x1 ∂t ∂x1  2  2 2 2 ∂ψ ∂ ψ 1 ∂ ψ ∂ ψ 2∂ ψ ˙ −V + = 2 − 2V β ∂x21 ∂y12 c2 ∂t2 ∂x1 ∂t ∂x1 where α and β are given by  α = 1 − (V /c1 )2 ,

β=

 1 − (V /c2 )2

(3.81)

(3.82)

Note that α > β when c1 > c2 . When the stresses are singular at the moving crack tip, the terms on the left part of equation (3.81) dominate because the terms on the right part have weaker singularities. In the crack tip, equation (3.81) can be written as α2

∂ 2 ϕ ∂ 2ϕ + =0 ∂x21 ∂y12

∂2ψ ψ β + =0 ∂x21 ∂y12 2∂

2

(3.83)

Now we introduce two polar coordinate systems, (r1 , θ) and (r2 , θ), at the moving crack tip as show in Figure 3.16. x1 = r1 cos θ1 , x1 = r2 cos θ2 ,

r1 sin θ1 α r2 sin θ2 y1 = β y1 =

(3.84)

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201

Fig. 3.16 Coordinate system on the moving crack tip.

thus, ∂( ) ∂( ) sin θ1 ∂( ) = cos θ1 − ∂x1 ∂r1 ∂θ1 ∂r1 ∂( ) ∂( ) cos θ1 ∂( ) = α sin θ1 + α ∂y1 ∂r1 ∂θ1 r1

(3.85)

in the (r1 , θ1 ) system, and ∂( ) ∂( ) sin θ2 ∂( ) = cos θ2 − ∂x2 ∂r2 ∂θ2 r2 ∂( ) ∂( ) cos θ2 ∂( ) = β sin θ2 + β ∂y2 ∂r2 ∂θ2 r2

(3.86)

in the (r2 , θ2 ) system. Also we can have  r1 = x21 + (αy1 )2 ,  r2 = x21 + (βy1 )2 ,

 αy1 θ1 = arctan x1   βy1 θ2 = arctan x1 

(3.87)

From these two equations, we can get: x1 ∂r1 = = cos θ, ∂x1 r1

∂r1 α2 y1 = = α sin θ1 ∂y1 r1

αy1 sin θ1 ∂θ1 =− 2 =− , ∂x1 r1 r1 x1 ∂r2 = = cos θ, ∂x1 r2

∂θ1 αx1 cos θ1 = 2 =α ∂y1 r1 r1

∂r2 β 2 y1 = = β sin θ2 ∂y1 r1

αy1 sin θ2 ∂θ2 =− 2 =− , ∂x1 r2 r2

∂θ2 βx1 cos θ2 = 2 =β ∂y1 r2 r2

(3.88)

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Since ϕ = ϕ(r1 , θ1 ) and ψ = ψ(r2 , θ2 ), equation (3.80) can be written as follows: 1 ∂2ϕ ∂2ϕ ∂ϕ + + r =0 1 ∂r12 ∂r1 r12 ∂θ12 1 ∂2ψ ∂ϕ ∂2ψ + 2 2 =0 + r2 2 ∂r2 ∂r2 r2 ∂θ2

(3.89)

And the displacement potential functions are supposed to have the following form:  1 ), ϕ = r1s ϕ(θ  2 ), ψ = r2s ψ(θ

r1 → 0 r2 → 0

(3.90)

where s is the eigenvalue to be determined by the boundary conditions. Substituting equation (3.90) into equation (3.89), we have d2 ϕ  + s2 ϕ =0 dθ12

d2 ψ + s2 ψ = 0 dθ22

(3.91)

The general solutions of these equations are

ϕ  = C1 cos(sθ1 ) + C2 sin(sθ1 )

(3.92)

ψ = C3 cos(sθ2 ) + C4 sin(sθ2 )

In the following we consider Mode II crack propagation only. Mode II symmetry consideration gives C1 = C4 = 0. Hence equation (3.92) is simplified to ϕ  = C2 sin(sθ1 )

ψ = C3 cos(sθ2 )

(3.93)

Substituting equation (3.90) into equation (3.86), and through the equations (3.84)–(3.88), we obtain the following asymptotic expressions for the displacement potential, displacement, and stressed at the moving crack tip: ϕ = A2 r1s sin(sθ1 ) ψ = A3 r2s cos(sθ2 ) u = sA2 r1s−1 sin(s − 1)θ1 − βsA3 r2s−1 sin(s − 1)θ2

ν = αsA2 r1s−1 cos(s − 1)θ1 − sA3 r2s−1 cos(s − 1)θ2

(3.94)

3

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203

∂v ∂u +λ ∂x ∂y = (λ + 2G)[A2 s(s − 1)r1s−2 sin(s − 2)θ1 −

σxx = (λ + 2G)

βA3 s(s − 1)r2s−2 sin(s − 2)θ2 ]+

λ[−α2 A2 s(s − 1)r1s−2 sin(s − 2)θ1 + βA3 s(s − 1)r2s−2 sin(s − 2)θ2 ]

= ρc22 (2α2 + 1 − β 2 )A2 s(s − 1)r1s−2 sin(s − 2)θ1 − 2ρc22 βA3 s(s − 1)r2s−2 sin(s − 2)θ2

∂u ∂v +λ ∂y ∂x = (λ + 2G)[−α2 A2 s(s − 1)r1s−2 sin(s − 2)θ1 +

σyy = (λ + 2G)

(3.95)

βA3 s(s − 1)r2s−2 sin(s − 2)θ2 ]+

λ[A2 s(s − 1)r1s−2 sin(s − 2)θ1 − βA3 s(s − 1)r2s−2 sin(s − 2)θ2 ]

= ρc22 (−1 − β 2 )A2 s(s − 1)r1s−2 sin(s − 2)θ1 +

τxy

2ρc22 βA3 s(s − 1)r2s−2 sin(s − 2)θ2   ∂u ∂v + =G ∂y ∂x = G[2αA2 s(s − 1)r1s−2 cos(s − 2)θ1 −

(1 + β 2 )A3 s(s − 1)r2s−2 cos(s − 2)θ2 ]

Because of the square root singularity, we can get s = 3/2

(3.96)

And for Model II, the displacements of crack surface should satisfy σyy = 0,

θ1 = θ2 = π

(3.97)

Substituting equation (3.97) into equation (3.94), we can have the relation between A2 and A3 as follows A2 =

2β A3 1 + β2

(3.98)

A3 can be determined by the Model II dynamic stress intensity factor (SIF) KII (t) 1 + β2 4 KII (t) √ f, f = (3.99) A3 = 3 G 2π 4αβ − (1 + β 2 )2

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with KII (t) is defined by KII (t) = lim

r→0

 √  2πrτxy (r1 , r2 , θ1 , θ2 )

θ1 =θ2 =r1 =r2 =r=0

The crack tip stress and displacement fields can be expressed   θ1 θ2 2KIIf 2β √ √ √ − β r2 sin r1 sin u= 2 2 G 2π 1 + β 2   2KIIf θ1 √ θ2 2αβ √ v= √ − r2 cos r1 cos 2 2 G 2π 1 + β 2

(3.100)

σxx = ρc22 (2α2 + 1 − β 2 )A2 s(s − 1)r1s−2 sin(s − 2)θ1 −

2ρc22 βA3 s(s − 1)r2s−2 sin(s − 2)θ2   β 2β(2α2 + 1 − β 2 ) 1 KII f θ1 θ2 + − = √ sin sin √ √ 1 + β2 r1 2 r2 2 2π

σyy = ρc22 (−1 − β 2 )A2 s(s − 1)r1s−2 sin(s − 2)θ1 + 2ρc22 βA3 s(s − 1)r2s−2 sin(s − 2)θ2   β KII f θ1 θ2 1 − √ sin = √ 2β √ sin r1 2 r2 2 2π

(3.101)

τxy = G[2αA2 s(s − 1)r1s−2 cos(s − 2)θ1 −

(1 + β 2 )A3 s(s − 1)r2s−2 cos(s − 2)θ2 ]   KII f θ1 1 θ2 4αβ 1 2 − (1 + β = √ cos ) cos √ √ 2 r2 2 2π 1 + β 2 r1

The crack tip velocity field can also be obtained by equation (3.80) (∂u/∂t = 0)   β β θ1 θ2 2V KII f 1 + (3.102) − sin sin u˙ = − √ √ √ 1 + β 2 r1 2 2 r2 2 G 2π

3.7

Cohesive Model in Rock Fracture

Linear elastic fracture mechanics (LFEM) is valid only as long as nonlinear material deformation is confined to a small region around the crack tip. In many materials (such as iron, concrete, and rock), it is impossible to characterize the fracture behavior with LEFM. Elastic-plastic fracture mechanics applies to materials that is related to timeindependent, nonlinear behavior. In order to remove the singularity near the crack tip, Barenblatt (1962) gave a cohesive model, which considers the cohesive stress g(x) near the crack tip, as shown in Figure 3.17.

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Fig. 3.17 Barenblatt cohesive model. ′ Suppose KII is the SIF which is related to the cohesive stress g(x) √  −2 a a g(x)dx ′ √ KII = √ π c a2 − x2 ′ In order to remove the singularity near the crack tip, KII should be equal to KII , √  2 a a g(x)dx ′ √ = √ (3.103) KII = −KII π c a2 − x2

3.7.1

Stress Change in Slip-weakening Model

The seismic fault failure can cause additional static stress around the crack tip. It is believed that the Coulomb failure stress change generated by earthquakes will have impact on the other remote earthquakes (Andrews, 1976; Aki and Richards, 1980; Chen and Knopoff, 1986; Harris, 1998, 2000; He et al., 2011). The Slip-weakening model (the extension of Barenblatt Cohesive Model) is used to examine the calculations of the Coulomb failure stress change. A seismic fault under compression is shown in Figure 3.18, where σs is the shear strength; σd is the sliding friction force; σf is the static friction force; ∆σe = σs − σd is the effective stress drop. Suppose the shear stress at the fault surface increases to σs and then drops to σd , the effective stress drop ∆τe = σs − σd will be determined in the earthquake wave data (Fig. 3.19). The Coulomb stress change reflects the static stress drop ∆τ = σf − σd , 0 < |x| < a so a coefficient γ can be written as (Andreev, 1995; He, 1995). ∆τ γ= (3.104) ∆τe

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Fig. 3.18 Seismic fault under compression.

Fig. 3.19 Shear stress variation with time at a point on the fault surface (Yamashita 1976; Knopoff and Chen 2009).

Then ∆τ = γ∆τe

(3.105)

For most of the time, the σd is not a constant, so ∆τ = σf − σd (x), and the distribution in the fault surface is as below.  |x| < c σd , σd (x) = σ(x), c < |x| < a Then the static stress drop ∆τ = σf − σd (x) can be written as ∆τ =



σf − σd , σf − σd (x),

|x| < c c < |x| < a

(3.106)

3

Rock Fracture under Static and Dynamic Stress

and the stress function of the additional stress field can be written as � a� 2 2z a − ξ2 ∆τ (ξ)dξ ZII = √ 2 2 π z − a 0 z2 − ξ2 � � � a 2 a2 − ξ 2 2z √ Z�II = ∆τ (ξ)dξ π π z 2 − a2 0 z 2 − ξ 2

207

(3.107) (3.108)

and the SIF on the both crack tips under the cohesive stress are the same because of the stress symmetry. √ � 2 a a ∆τ (ξ)dξ + − � KII = KII = √ (3.109) π 0 a2 − ξ 2 Substituting equation (3.108) into equation (3.25), the shear stress outside the fault surface can be shown: � a� 2 2x a − ξ 2 ∆τ (ξ)dξ , |x| > a, y = 0 (3.110) τxy = √ x2 − ξ 2 π x2 − a2 0 By utilizing equation � 1 x2 − a2 a2 − ξ 2 � � = − z2 − ξ2 a2 − ξ 2 (x2 − ξ 2 ) a2 − ξ 2

the shear stress can be expressed as below. √ � � a 2x x2 − a2 a 2x ∆τ (ξ)dξ ∆τ (ξ)dξ c � � − , τxy = √ 2 2 π π x −a 0 a2 − ξ 2 0 (x2 − ξ 2 ) a2 − ξ 2 |x| > a, y = 0 (3.111) The first part in equation (3.111) should be zero because there is no singularity c will be: after the cohesive stress being introduced, so the shear stress τxy √ � 2x x2 − a2 a ∆τ (ξ)dξ c � τxy = , |x| > a, y = 0 (3.112) 2 π (x − ξ 2 ) a2 − ξ 2 0

For a real seismic fault, the crack surface is covered by all kinds of uneven cleavages, which makes ∆τ (x) nonlinear and more complicated. From equation (3.112), one can find that the different distribution of ∆τ (x) has no impact on c . It is the final result of the calculus that has the final impact the stress field τxy c on τxy . For the linear distribution of the shear stress σd (x) along the crack surface (see Fig. 3.20),   σd , |x| < c a−x σd (x) = (3.113)  σs − (σs − σd ) , c < |x| < a a−c

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Fig. 3.20 Linear distribution of σd (x) along a fault surface.

The corresponding static stress drop can be written as   |x| < c  γ(σs − σd ), � � ∆τ = x−c  (σs − σd ), c < |x| < a  γ− a−c

(3.114)

Substituting equation (3.114) into equation (3.107) and (3.108), one can have the shear stress (|x| > a, y = 0)    � � 2 2 2 2 2 x a − c a − c c  x arctan = (σs − σd ) −γ + − c arctan τxy π(a − c) x2 − a2 c x2 − a2

(3.115)

At the crack surface (|x| < a, y = 0), the shear stress is � � x−c c , |x| < a, y = 0 = −∆τ = −(σs − σd ) γ − τxy a−c

(3.116)

Substituting equation (3.74) into equation (3.71), one have � 2 √ �II = 2(σs − σd ) − πc − πγ(a − c)z + A z 2 − a2 + Z π(a − c) 4 2 � √ √ √ √ cz c z 2 − a2 + iz a2 − c2 z 2 − a 2 + i a 2 − c2 z 2 + c2 √ √ ln √ ln √ − 4i 2i c z 2 − a2 − iz a2 − c2 z 2 − a 2 − i a 2 − c2 (3.117) At the crack surface y = 0,

θ1 = π, θ2 = 0, r1 = a − x, √ √ z 2 − a2 = i a2 − x2

r2 = a + x,

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Rock Fracture under Static and Dynamic Stress

209

the displacement (plain strain) on the crack upper surface is  √ √  2(1 − µ) x2 + c2 − σ ) a2 − x2 + a2 − c2 (σ s d c+ 2 2 √ √ A a −x + ln − u = πG (a − c) 4 a2 − x2 − a2 − c2  √  √ cx  x a2 − c2 + c a2 − x2  √ ln  √   x a2 − c2 − c a2 − x2  2 √ a 2 − c2 A= 2   (1 − 2µ) (σs − σd ) c2 c+ + γ(a − c)x (3.118) v = πG (a − c) 2 and the displacement on the crack bottom surface uc− = −uc+ v c− = v c+

(3.119)

The displacement change between the upper and bottom surfaces are ∆uc− = 2uc+ ∆v = 0

3.7.2

(3.120)

Relationship between Energy Release Rate G and the Parameter in Slip-weakening Model

The stress distribution near the crack tip is shown in Fig. 3.21, the stress drops from σs to σd (slip-weakening zone from c to a), and the displacement reaches to its maximum δ ∗ . If we apply J-integral along the crack tip from bottom surface to upper surface and consider the disappearance of singularity at the crack tip, we have the result from Section 3.4.2   0 ∂δ σ (3.121) 0 = JQ + dx1 ∂x1 Q According to the equation (3.73), the second part of the equation above can be written as   0  δ∗ ∂δ ∂δ σ(δ) dx1 = − [σ(δ) − σ d ] dx1 − σ d δQ (3.122) ∂x1 ∂x1 0 Q where Q is the beginning and end point of the closure. From equation (3.122), the equation (3.121) can be converted to  δ∗ ∂δ d JQ − σ δQ = [σ(δ) − σ d ] dx1 ∂x1 0

(3.123)

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Fig. 3.21 The stress distribution near the crack.

The left part is independent of Q and can be considered as the driving force. The right part can be reckoned as the resisting force, so equation (3.108) can be thought of being a criterion of crack growth. From equation (3.58), the equation (3.123) can be converted to G = JQ − σ d δQ = [σ s − σ d ]δ where δ¯ is the average displacement of the slip-weakening zone (from c to a), which means G can be got from multiplier of the stress drop and the average displacement δ.  δ∗ ∂δ d JQ − σ δQ = [σ(δ) − σ d ] dx1 ∂x1 0

3.8

Numeric Method for Fracture Mechanics

The numeric method, especially for finite element method (FEM), is an effective tool to solve some complex problems, such as objects with complicated surfaces, or fluid-solid coupling problems. Compared to other numeric methods, FEM has its own advantages so that it is wildly used in fracture mechanics. However, the conventional finite element method considers the crack surface as the border of elements and treats the crack tip as a node of an element. These treatments increase the complex of computation. In order to solve this problem, some new thoughts are introduced to deal with the crack surface problems. Among them, the singularity element and the extended finite element method (XFEM) are reckoned as the commonly used strategies.

3

3.8.1

Rock Fracture under Static and Dynamic Stress

211

Singularity Element Method

A significant advancement in FEM was the simultaneous and independent “quarterpoint” elements developed by Henshell and Shawn (1975), Barsoum (1976), Hibbitt (1977), and Hoenig (1982). In their method, the proper crack-tip displacement, stress and strain fields are modeled by the standard quadratic order iso-parametric finite elements if one moves the element’s mid-side node to the position by one quarter of the way from the crack tip to the far end of the element. This procedure introduces a singularity into the mapping between the element’s parametric coordinate space and the Cartesian space. Henshell and Shaw (1975) described a quadrilateral quarter-point element, illustrated in Figure 3.22a. Barsoum (1977) proposed collapsing one edge of the element at the crack tip. This is shown in Figure 3.22b, where the crack-tip nodes (1, 4 and 8) are constrained to move together. The discovery of the quarter-point elements was a significant milestone in the development of FEM. With these elements standards, FEM can be used to model the crack tip fields accurately with only minimal modification. The effect of moving the side node of a quadratic element to the quarterpoint position can be best illustrated with a one-dimensional element. A onedimensional quadratic order element is illustrated in Figure 3.23. The displace-

Fig. 3.22 (a) Quadrilateral and (b) collapsed quadrilateral quarter-point elements.

Fig. 3.23 (a) the parametric space and (b) the Cartesian space of the element.

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ment u at any point within the element is determined by interpolating the nodal displacements, ui using the standard lagrange second order shape functions, u=

3 

Ni u i =

i=1

or

1 1 ξ(ξ − 1)u1 + (1 − ξ 2 )u2 + ξ(ξ + 1)u3 2 2

  1 1 u = u2 + (u3 − u1 )ξ + (u3 + u1 ) − u2 ξ 2 2 2

(3.124)

Using an iso-parametric formulation, the same shape functions are used to interpolate the geometry of the element: r=

3 

1 Ni ri = αl + lξ + 2 i=1



 1 l − αl ξ 2 2

(3.125)

First, consider the case where the center node is located at the mid-point of the element. That is α = 1/2 and r = l(1 + ξ)/2. Substituting this expression for ξ into equation (3.124) yields the expected quadratic expression in r for the displacements r r2 u = u1 + (4u2 − u3 − 3u1 ) + 2(u3 + u1 − 2u2 ) 2 l l

(3.126)

Differentiating this expression yields the expected linear expression in r for the strains in the element ε=

du 1 r = (4u2 − u3 − 3u1 ) + 4(u3 + u1 − 2u2 ) 2 . dr l l

Then consider the case where the middle√ node is moved to the quarter-point position. For this case, α = 1/4 and ξ = 2 l lr − 1. Substituting this expression for ξ into equation (3.109) and differentiating yields the following expressions for the displacements and strains in the element √ r rl u = u1 + 2(u3 + u1 − 2u2 ) + (u3 − 3u1 + 4u2 ) l l   1 3 1 du 1 = 2(u3 + u1 − 2u2 ) + 2u2 − u1 − u3 √ ε= (3.127) dr l 2 2 lr One can see that the three terms in the displacement expression model a constant value, a linear variation in r, and the square root variation in r. This corresponds to the leading terms in the LFEM expressions for the near crack-tip displacement. The expression for the strains contains a constant term and a singular term that varies as r1/2 , the form of lead term in the LFEM stress and strain expansions.

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213

For two dimension (see Fig. 3.24), the displacements along the crack face for the quarter-point element interpolation are √ r rl vupper = va + (2va + 2vc − 4vb ) + (4vb − vc − 3va ) l √l r rl vlower = va + (2va + 2ve − 4vd ) + (4vd − ve − 3va ) l l

Fig. 3.24 Two dimensional singularity element at the crack tip.

The FEM crack opening displacement (COD) is vupper − vlower

√ r rl = [4(vb − vd ) + 2(vc − ve )] + [4(vb − vd ) − vc + ve ] l l

The square root term of the COD can then be substituted into the analytical crack-tip displacement field to yield √ G 2π √ [4(vb − vd ) + ve − vc ], KI = (2 − 2µ) r (3.128) √ G 2π √ [4(ub − ud ) + ue − uc ], KII = (2 − 2µ) r

3.8.2

Extended Finite Element Method

The extended finite element method (XFEM) retains all advantages of the common finite element method (CFEM), and overcomes difficulties in meshing and re-meshing within the crack tip region that contains the stress concentration. XFEM uses the partition of unity framework (Moes et al., 1999, 2003; Babuska and Melenk, 1997) to model strong and weak discontinuities independent of the finite element mesh (Fig. 3.25). This allows discontinuous functions to be implemented into a traditional finite element framework through the use of enrichment functions and additional degrees of freedom.

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Fig. 3.25 Enriched nodes for crack.

Cracks are modeled using a combination of two enrichment functions. One for the complicated behavior at the crack tip and another is a Heaviside step function to represent the discontinuity across the body of the crack. The Heaviside function takes a value of 1 above the crack and −1 below the crack, thus putting a displacement discontinuity across the body of the crack in elements whose support is cut by the crack. For the crack tip the enrichment functions originally introduced by Fleming et al. (1997) for use in the element free Galerkin method. They were later adopted by Belytschko et al. (2001) for use in XFEM. These four functions span the crack tip displacement field. Also note that the first function is discontinuous across the crack within the element containing the crack tip. For information on modeling bi-material or branching cracks please refer to the papers by Sukumar et al. (2001) and Daux (2000). The XFEM displacement approximation for a domain with a crack takes the following form.   4   (3.129) NI (x) uI + H(x)aI + FJ (x)bJI uh (x) = I∈N

J=1

where uI is Nodal DOF; NI is Shape Function; aI is Nodal enriched DOF; bJI is Nodal DOF at the crack tip enrichment;  1, if (x − x∗ ) · n  0 H(x) = −1, otherwise  √ √ √ √ r sin θ2 , r cos θ2 , r sin θ sin θ2 , r sin θ cos 2θ is Heaviside function and FJ (x) = is the crack tip asymptotic functions. Predicting where a crack will initiate is a challenging area of computational mechanics. The most common approach is to place a crack at the location of

3

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215

Fig. 3.26 (See color insert.) Edge crack under bi-axial tension. (a) σx ; (b) σxy ; (c) σy .

maximum stress (Edke and Chang, 2010). However, it is well known that the stress fields from finite element simulations converge at a rate which is much slower than displacements. Therefore, it may be difficult, if not impossible, to identify the exact location of maximum stress. It is possible for an optimization problem to be formulated to identify the initial crack location. While in general this optimization problem may be too expensive to consider with regards to a finite element simulation with a large number of degrees of freedom, the use of the proposed reanalysis algorithm (Paris, 2010) makes the solution of the optimization problem possible.

3.9

Discussion

In this chapter, a crack model is established by using complex functions, and an analytical solution of the crack tip stress intensity factors for three cracks under compression has been presented. Coulomb–Mohr failure criterion, energy release and J-integral, as well as the numeric method have been analyzed. Generally, this chapter has presented some novel methods and techniques for the study of fault stability and earthquake. But the rock environment is so complex that many other factors need to be involved. First, the properties of rocks will change dramatically when the depth increases because the temperature increases with it. The mechanism of the rock will be similar to that of the liquid, and the strength of it decrease sharply, and thus the strain and stress will change with the time, which is named as a creep. Compared to the linear or plastic theory, the creep is more complicated and there is no universal theory until now because of its non-linear character. Second, the chemical effect of the rock coupled with water is another important factor. The character of the rock is reshaped when under water, so it is the

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mechanic property. A better understanding of it depends on the research of geochemistry.

Acknowledgements On the completion of this chapter, we should like to express our deepest gratitude to all those whose kindness and advice have made this work possible.

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Authors Information Jiming Kang, Po Chen Department of Geology and Geophysics, University of Wyoming, Laramie, WY 82071, USA E-mail: [email protected] Zheming Zhu College of Architecture and Environment, Sichuan University, Chengdu, China

Chapter 4

Multiple Linear Regression Analyses on the Relationships among Magnitude, Rupture Length, Rupture Width, Rupture Area, and Surface Displacement Annie Chu and Jiancang Zhuang

Using fault data, Wells and Coppersmith (1994) have fit simple linear regression (SLR) models to explain linear relations between magnitude and logarithms of fault measurements such as rupture length, rupture width, rupture area and surface displacement. This chapter extends their analyses to multiple linear regression (MLR) models by considering two or more predictors. We have discovered that fitting moment magnitude on logarithms of rupture area and maximum displacement provides the best-fit model of two predictors. And when maximum displacement is unavailable, the best alternative predictors are surface length and rupture area. We have also verified that neither slip type nor slip direction is a significant predictor. Key words: Earthquakes, Earthquake forecasting, Global seismicity, Linear models, Linear regression.

4.1

Introduction

The first formula for determining the magnitude of an earthquake was given by Richter and Gutenberg (1935), now known as the local magnitude (ML ) or the Richter scale. This scale is based on ground motion measured by a particular type of seismometer (a Wood-Anderson seismograph) at a distance of 100

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kilometers from the earthquake epicenter. Subsequently, many other magnitude scales were proposed, such as surface wave magnitude, duration magnitude, and body wave magnitude. All these magnitude scales have a saturation problem for very high magnitudes or other shortcomings. Nowadays, the moment magnitude (Mw ) scale, which was introduced by Hanks and Kanamori (1979) to maintain consistency between magnitude and energy released by an earthquake, is commonly used. The moment magnitude is defined by Mw =

2 log10 (M0 ) − 6 3

where the earthquake moment M0 = µAD µ is the shear modulus, usually taking a value of 32 GPa in crust and 75 GPa in mantle; A is the area of the earthquake fault and D is the average displacement of rupture. The above formula for the earthquake moment is given under ideal situation and cannot be used directly to determine the moment magnitude since the crustal shear modulus and the average displacement cannot be measured directly and the geometry of an earthquake fault is complex. In practice, the seismic moment is calculated from the amplitude spectra of seismic waveforms. On the other hand, after a large earthquake, we can always obtain some partial information related to the earthquake source, by geophysical inversion or field trip observation. A careful investigation between the earthquake magnitude and these quantities is useful, especially in the estimation of magnitudes of paleoearthquakes in the field of observations, where only few quantities of the earthquake fault can be observed. As described in Wells and Coppersmith (1994), estimating maximum magnitudes of future earthquakes in a particular region, which also relies on the knowledge of paleoearthquakes that occurred in this region, is of importance for evaluating potential earthquake hazard, as well as for disaster prevention and reduction. Wells and Coppersmith (1994) carefully analyzed the relations between earthquake magnitude and observed fault measurements using simple linear regression (SLR) models. Similar studies have been conducted by many researchers (e.g., Bonilla et al. 1984; Mark 1977; Singh et al. 1980). However, there still remain several important issues unsolved: (1) How can we improve estimation when the model includes two or more predictors? (2) How many and which predictors are necessary for estimating the magnitude? That is to ask, among the candidate predictors, which may be eliminated? In this chapter, we perform multiple linear regression (MLR) analyses on the data adopted from Wells and Coppersmith (1994) to answer the above-stated questions.

4

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Multiple Linear Regression Analyses on the Relationships among Magnitude. . .

4.2

Data

The data of Wells and Coppersmith (1994) are delineated in Table 4.1. The data have 244 seismic events, and each event is recorded with several variables. To construct a prediction model for moment magnitude, we consider the following quantitative variables: moment magnitude (M ), rupture length of surface (SRL), rupture length of sub-surface (RLD), rupture width (RW), rupture area (RA), maximum displacement of surface (MD), and average displacement of surface (AD). It is noticed that the rupture area is the product of rupture sub-surface length and rupture width. Moreover, when we multiply rupture surface length and rupture width, we perceive an extremely strong correlation with correlation coefficient 0.95 between the result product and the rupture area. Tab. 4.1 Correlation coefficients between variables, calculated based on the subset of data that the seven variables, M , SRL, RLD, RW, RA, MD and AD are all observed. The sample size n is 28. Variable

M

log(SRL)

0.88

log(RLD)

0.90

log(SRL)

log(RLD)

log(RW)

log(RA)

log(MD)

0.90

log(RW)

0.71

0.54

0.66

log(RA)

0.91

0.84

0.96

0.85

log(MD)

0.73

0.64

0.53

0.32

0.49

log(AD)

0.72

0.62

0.54

0.34

0.51

0.95

Although the total number of earthquakes is 244 in the provided data, missing and unreliable observations occur frequently. Before we proceed to statistical modeling, we depict the frequency histograms of the quantitative variables in Figure 4.1. Due to the absence of an event under one or more variables, the events used to produce each variable’s frequency histogram are a portion of the 244 events. For example, 22 out of 244 events have missing records on moment magnitude. Figure 4.1a shows the frequency histogram of the 222 events with known moment magnitude. The minimum and maximum of magnitude are 4.57 and 8.14; the mean and median are 6.38 and 6.36, respectively. The histogram approximately resembles a normal distribution with majority of the data lying between 6 and 7. Besides that moment magnitude is missing for some events, other variables are missing for some other events. For example, there are only 77 events with their SLR values available and its right-skewed histogram is displayed in Figure 4.1b. The frequency histograms of RLD, RW, RA, MD, and AD are shown in Figure 4.1c–g; the histogram of RW resembles a normal curve, while all other variables’ histograms are right-skewed.

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Fig. 4.1 Histograms of frequencies of the variables M , SLR, RLD, RW, RA, MD and AD. The sample size for each variable is (a) M , 77 events, (b) SLR, 77 events, (c) RLD, 168 events, (d) RW, 154 events, (e) RA, 148 events, (f) MD, 79 events, (g) AD, 55 events.

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Fig. 4.2 Scattersplots of original data together with the fitted simple linear regression models and 95% confidence intervals. Positive correlations are depicted in all plots. (a) M on log(SLR), n = 77, (b) M on log(RLD), n = 168, (c) M on log(RW), n = 154, (d) M on log (RA), n = 148, (e) M on log(MD), n = 79, (f) M on log(AD), n = 55. A 95% confidence interval of the predicted values is provided for each plot. As expected, the models with smaller correlation coefficients, for example, (e) and (f), have a larger portion of data that are unable to be explained by the linear model.

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To unify the standard in comparing different regression models, we extract the data by taking the intersection of all non-missing, reliable portion and obtain 28 useable events. Before carrying out the regression analyses, it is useful and important to understand how magnitude correlates with the predictors. We construct a correlation matrix of moment magnitude and logarithms (logarithms of base 10) of the quantitative predictors (Table 4.1). Among the predictors themselves, the scatter plots are found to be linear or close-to-linear: log(MD) and log(AD) are strongly correlated linearly, and the rest four variables appear to have weak correlations with respect to log(MD) and log(AD). However, log(SRL), log(RLD), log(RW) and log(RA) all appear to have strong linear correlations when any pair is considered. For the case of RW, logarithm transformation does not improve linearity: the correlation coefficients are both approximately 0.7 between M and RW, and between M and log(RW). As shown in Table 1, the correlation coefficients are calculated using the maximal possible portion of non-missing, reliable data while all but M have logarithms applied to their values. The reader may also observe the correlations between M and the logarithms of SLR, RLD, RW, RA, MD and AD by the scatter plots in Figure 4.2.

4.3

Linear Models and Computational Approach

Based on the high linear correlations between some pairs of the variables, we expect that SRL, RLD, and RW are redundant when RA is known. However, in actual fieldwork, measurements may vary in reliability and obtainability. Therefore, in model fitting we consider all four of SRL, RLD, RW and RA, so that when RA is absent in data, SRL or RW may still be useful. The main task of this study is to fit linear models between moment magnitude and multiple regressors. (An independent variable is also known as a regressor or predictor, and we will use these terminologies throughout this chapter.) Logarithms of base 10 are applied to the six quantitative independent variables according to the correlations obtained in previous section. Same logarithmic transformations are implemented in Wells and Coppersmith (1994). From the six possible regressors, two or more are selected to fit multiple linear regression models.  (j) βj xi + errori (4.1) Mi = α + j

(j)

where Mi is the moment magnitude, and xi is the i-th observation of one variable from log(SRL), log(RLD), log(RW), log(RA), log(MD) and log(AD), and βj (j) is the coefficient. When there is only one term βj xi on the right hand side, the regression model is simplified to a simple linear regression model, and they are

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227

presented in Wells and Copperfield (1994) for the cases of log(SRL), log(RLD), log(RW), log(RA), log(MD) and log(AD). For the reader’s convenience, we firstly reproduce the six simple linear models by using M as the dependent variable and one of SRL, RLD, RW, RA, MD and AD as the regressor. Then, we fit multiple linear regression models to verify whether the prediction of M can be improved when we use two or more predictors simultaneously. In addition to fitting multiple linear regression models using quantitative regressors, we construct a one-way analysis of variance (ANOVA) model Mi = α + τi + errori and an analysis of covariance (ANCOVA) model  (j) βj xi + τi + errori Mi = α +

(4.2)

(4.3)

j

where τi denotes a categorical factor, slip type or slip direction, and xi denotes a quantitative regressor, log(SRL), log(RLD), log(RW), log(RA), log(MD) or log(AD). We use Akaike Information Criterion (AIC; see Akaike, 1974) as a model selection criterion to determine which model fits better than the others. The statistic AIC = −2maxθ log L(θ) + 2k

(4.4)

is computed for each model fitted to the data. θ is a vector consisting of k parameters. log L(θ) stands for the model’s log-likelihood function. AIC is computed when maximum log-likelihood estimate (MLE) is attained by having an optimal θ. In comparing models with different numbers of parameters, the addition of quantity k roughly compensates for the additional flexibility that the extra parameters provide; i.e., large k is penalized to prevent over-fitting. The model with the lowest AIC value is taken as an optimal choice for forward prediction purposes. As a rule of thumb, in testing a model with k + d parameters against a null model with k parameters, we regard a difference of 2 in AIC values as a rough estimate of significance at 5% level (Ogata and Zhuang, 2006, Zhuang et al., 2014). In the case of linear regression, if the model assumes normality in residuals, AIC may be computed for model selection using the formula (Gorβ, 2003): AIC = N log(SSE/N ) + 2k

(4.5)

where SSE stands for sum of square errors, computed from the residuals of the fit model. Burnham and Anderson (2002) suggest that when the sample size is small compared to k, an alternative form, a second-order correction for finite sample sizes should be used: AICC = AIC + 2k(k + 1)/(N − k − 1)

(4.6)

In this study, judging that the sample size of each analysis ranges mostly between 30 and 60, we choose to use AICC .

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4.4

Results

4.4.1

Simple Linear Regression Results

In Table 4.2 we list the sample sizes, estimated coefficients and their standard errors, correlation coefficients, and statistics such as maximum and minimum of a variable like in Wells and Copperfield (1994). The sample sizes of available events are denoted by n and it slightly differs in the models for RLD, RW, MD and AD due to Wells and Coppersmith’s excluding event #89 for the simTab. 4.2 Results from simple linear regression models following the procedures in Wells and Coppersmith (1994). For each model, the top part shows our reproduced work, and the bottom part gives the results of Wells and Coppersmith. Coefficients and their standard errors

Correlation Coefficient

Minimum and maximum of magnitude

Minimum and maximum of the regressor

(Intercept) 5.0837 (0.0998) (Variable) 1.1612 (0.0668)

0.894

5.12, 8.14

1.3, 432

(Intercept) 5.08 (0.1) (Variable) 1.16 (0.07)

0.89

(Intercept) 4.3477 (0.0578) (Variable) 1.5176 (0.0440)

0.938

4.57, 8.144

1.1, 350

167

(Intercept) 4.38 (0.06) (Variable) 1.49 (0.04)

0.94

4.8, 8.1

154

(Intercept) 4.042 (0.117) (Variable) 2.261 (0.120)

0.836

4.57, 7.9

153

(Intercept) 4.06 (0.11) (Variable) 2.25 (0.12)

0.84

4.8, 8.1

148

(Intercept) 4.0338 (0.0660) (Variable) 0.9956 (0.0294)

0.836

4.57, 7.9

2.2, 5184

(Intercept) 4.07 (0.06) (Variable) 1.02 (0.10)

0.84

79

(Intercept) 6.6895 (0.0444) (Variable) 0.7698 (0.0720)

0.77

5.22, 8.14

0.02, 14.6

80

(Intercept) 6.69 (0.04) (Variable) 0.74 (0.07)

0.78

55

(Intercept) 6.9331 (0.0536) (Variable) 0.8224 (0.0990)

0.752

56

(Intercept) 4.38 (0.06) (Variable) 1.49 (0.04)

0.75

Model

n

M on log(SLR)

77

M on 168 log(RLD)

M on log(RW)

M on log(RA)

M on log(MD)

M on log(AD)

1.1, 350

0.01, 14.6 5.55, 8.14

0.05, 8.0

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Multiple Linear Regression Analyses on the Relationships among Magnitude. . .

229

ple linear regression models of fitting M on log(RLD), log(RW) and log(RA). The moment magnitude of this event is 4.57 and it causes the difference in the minimum magnitude between our work and Wells and Coppersmith’s. We have observed presence on this event under RLD. All of the regression coefficients are highly significant, with p-values less than 0.001 (0.1% significance level). Not surprisingly, our reproduced parameter estimates and their standard errors coincide with the results listed in Wells and Copperfield (1994), only different in some decimal places that might be due to rounding. In Table 4.2, the two rightmost columns provide the minimum and maximum of a variable, and our result coincide with Wells and Coppersmith’s; the only difference in our work is due to the exclusion of event #169 with its DM 0.01. This event is excluded in the reproduced model because of its unreliable DM.

4.4.2

Multiple Linear Regression Results

Since rupture lengths of surface and subsurface, rupture width, and rupture area are geophysically meaningful related (classified as group A), and maximum and average displacements are also geophysically related (classified as Group B), we expect that one variable from Group A and one from Group B would yield a good model. Observing that the log(MD) and log(AD) are strongly linearly correlated, we select exact one regressor between them when building a model. Assuming the case of all six regressors being available, we use the following selection rules to include two, three or four regressors: – Two-regressor models: one from Group A and one from Group B. – Three-regressor models: two from Group A and one from Group B. – Four-regressor models: three from Group A and one from Group B. Among all the two-regressor multiple linear regression models that we have implemented, Table 4.3 selectively presents those that fit better than the others by employing the criteria of R2 and residual analysis. We had examined models that have two, three, four regressors, and additional terms of interaction of two or three regressors, for example, interaction of log(SLR) and log(RA). To discover the proportion of the data that a regression model can explain, the statistic R2 , coefficient of determination, is used. R2 is between 0 and 1; a high R 2 -value indicates that the model can explain most of the variation of the data. It is not surprising that the presented models have similarities in R2 (approximately 0.8 to 0.9 for most models) and p-value, since the predictors are linearly correlated to a substantial level. Furthermore, we also computed adjusted R2 . Adjusted R2 is useful when a model is to be compared with another model that has a different number of parameters, which is not the case for the models in Table

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4.3. We provide this adjusted R2 as a future reference. The regression results suggested the models in Table 4.3 fit the data well. Tab. 4.3 Results from selective multiple linear regressions of using two predictors from SRL, RLD, RW, RA, and MD. n is the number of events used in a model. Predictors and number of events

Modeling results

Model (4.7) log(RA)

Estimate Std. Err. t-value

p-value

intercept

4.42

0.192

23.0

< 2e − 16

log(MD)

log(RA)

0.842

0.0736

11.4

1.75e − 14

n = 45

log(MD)

0.312

0.0590

5.28 4.23e − 06

R2 = 0.892, adjusted R2 = 0.887 Model (4.8)

Estimate Std. Err. t-value 4.14

0.194

21.3

p-value

log(SLR)

intercept

< 2e − 16

log(RA)

log(SLR)

0.455

0.1380

3.30

n = 45

log(RA)

0.719

0.1245

5.77 8.43e − 07

0.00198

R2 = 0.892, adjusted R2 = 0.887 Estimate Std. Err. t-value 38.5

p-value

log(SRL)

intercept

5.37

0.140

< 2e − 16

log(MD)

log(SRL)

0.944

0.0975

9.69 1.11e − 14

n = 75

log(MD)

0.252

0.0728

3.46

9e − 04

R2 = 0.829, adjusted R2 = 0.824 Estimate Std. Err. t-value log(RLD)

intercept

log(MD)

log(RLD)

n = 56

log(MD)

4.86

p-value

0.160

30.4

< 2e − 16

1.18

0.103

11.5

4.89e − 16

0.263

0.0598

4.39 5.43e − 05

R2 = 0.873, adjusted R2 = 0.838 Estimate Std. Err. t-value log(RW)

intercept

5.14

0.294

17.5

p-value < 2e − 16

log(MD)

log(RW)

1.34

0.268

5.01 7.70e − 06

n = 51

log(MD)

0.563

0.0761

7.41 1.74e − 09

R2 = 0.718, adjusted R2 = 0.706

With all six regressors being processed, our analyses reveal that the best model contains RA and MD: M = 4.42 + 0.842 log(RA) + 0.312 log(MD)

(4.7)

The sample size n of this model is 45. It is not surprising that SRL and RW are insignificant predictors in the multiple linear regression analyses, since RA is the product of RLD and RW. To visualize the linear model, which is a plane in

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a three-dimensional coordinate system, we show the plane along with the events in Figure 4.3. Figures 4.3a and 4.3b indicate that the data points are close to the prediction plane. Figures 4.3c and 4.3d give additional different perspectives showing how events are scattered. We observe from the plots that the data points are closely located to the plane, indicating that the model provides a good fit to the data.

Fig. 4.3 Three dimentional plots of model (4.7) and related data points showing in different perspectives.

By assuming the absences of MD and AD, we also have n = 45. The best prediction model contains predictors SRL and RA: M = 4.14 + 0.455 log (SRL) + 0.719 log (RA)

(4.8)

Figure 4.4 provides four different perspectives of the events along with the prediction plane. Similar to Figure 4.3, the plots also show that the model provides a good fit to the data. When three or four predictors are employed to improve models (4.7) and (4.8), we find that either at least one predictor becomes insignificant, implying that additional one or two predictors are not useful. In general, using three or four predictors tends to reduce the sample size and possibly introduces more bias; and the inclusion of MD tends to have better

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Fig. 4.4 3.D plots of model (4.8) and related data points showing in different perspectives.

fit than the inclusion of AD. The only three-predictor models significant at 5% level are: M = 4.59 + 0.697 log (SRL) + 0.990 log (RW) + 0.245 log (MD)

(4.9)

M = 4.77 + 0.779 log (SRL) + 0.787 log (RW) + 0.327 log (AD)

(4.10)

M = 4.51 + 0.929 log (RLD) + 0.644 log (RW) + 0.305 log (MD)

(4.11)

These three models have similar goodness-of-fit according to their diagnostics plots and correlation of determination, R2 (0.866, 0.893 and 0.891 for models (4.9) to (4.11), respectively). When MD or AD is not in presence, the only model with three significant predictors at 5% level is: M = 4.10 + 0.503 log (SRL) + 0.652 log (RLD) + 0.794 log (RW)

(4.12)

Note that SRL and RLD have a very high linear correlation of 0.9, and that RA is the product of RLD and RW. Therefore, for the cases with and without fault displacement, it is reasonable that two predictors suffice a good fit. Indeed, when we consider the maximal possible reliable data with n = 28, AICC = −17.1 for model (4.7); and for models (4.9) to (4.11), AICC = −5.85, −4.57

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and −15.6, respectively. For models (4.8) and (4.12), AICC = 1.77 and 5.90, respectively. It is noted that a decrement of 2 or more in AIC suggests significant model improvement. Apparently, the parsimonious models (4.7) and (4.8) with only two predictors, are sufficiently good, and an additional predictor does not improve the fit. Since we use different samples extracted from the original data, concerns might be raised whether such comparisons are reasonable. As a work of validation, we use the common reliable data where n = 28 and apply the same modeling scheme of two predictors, it is realized that the model using log(RA) and log(MD), and the one using log(SRL) and log(RA) still excel, with their coefficient estimates and p-values similar to those of models (4.7) and (4.8). We prefer the coefficient estimates in models (4.7) and (4.8) since they are estimated with relatively larger sample sizes.

4.4.3

Model Diagnostics

Model diagnostics are carried out by normal quantile-quantile plots (a.k.a. normal QQ-plot or normal probability plots) and residual-on-fit plots, in order to verify the assumption that the errors follow normal distribution with a zero mean and a constant variance. This assumption is the standard normal assumption and it is a requirement for inference and prediction to be made. The normal QQ-plots of the fitted residuals appear approximately linear, and this phenomenon supports that model (4.1) fits the data reasonably well without violating the normality. Other tests that we have implemented, such as examining Cook’s distance and leverage, show evidences that no outlier raises concerns. The residual-on-fit and normal quantile-quantile plots for models (4.7) and (4.8) are presented in Figures 4.5 and 4.6. In many cases, a slight deviation from linearity in the normal QQ-plot implies that the errors do not follow closely to a normal distribution. We have observed in the residual-on-fit plots that the residuals have a slight fan-shape pattern for some simple linear models that we have examined. Further mathematical transformations may be applied to reduce the minor heteroscedasticity. To keep the models simple, we do not implement such transformations. It is worth mentioned that when we use square root transformation on MD and obtain a competitive model (4.7), the heteroscedasticity is reduced and the model yields R2 = 0.92. Nevertheless, we think that the commonly accepted logarithmic transformations on rupture surface length and rupture area provide an adequately decent model, due to its simplicity. Consequently, this issue is negligible. Besides the heteroscedasticity issue, missing and unreliable data raise a different question that cannot be answered easily.

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Fig. 4.5 Residual-on-fit and normal QQ-plots. (a) and (b) are for models (4.7). (c) and (d) are for models (4.8). Both residual-on-fit plots show randomness of the residuals. The normal QQ-plots show the points are close to a line with y-intercept zero and slope one, suggesting that the residuals follow a standard normal distribution.

4

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Fig. 4.6 Residual-on-fit and normal QQ-plots. (a) and (b) are for model (4.9). (c) and (d) are for model (4.10). (e) and (f) are for model (4.11).

4.4.4

Comparison between Multiple Models and Simple Models

The fits of two-predictor models (4.7) and (4.8) are found to be manifestly improved compared to the simple linear regression method, suggested by AIC. To test whether the multiple linear regression models (4.7) and (4.8) are better fit than the simple linear regression models, we use the maximal possible data to fit simple linear regression models. The simple linear regression models that uses RA and MD as predictors, M = 3.83 + 1.07 log(RA)

(4.13)

M = 6.59 + 0.71 log(MD)

(4.14)

and have AICC of 12.4 and 53.1, respectively. Note that the numbers of events used for models (4.13), (4.14) and (4.15) below are not exactly equivalent to those used to compute the coefficients in Table 4.2. So the estimates of the coefficients are not equal to those in models (4.7) and (4.8). The multiple linear regression model in (4.7) yields an AICC value of −8.23. As for model (4.8), we compare its AICC with the simple linear regression models in (4.13), and M = 5.09 + 1.13 log (SRL)

(4.15)

using the same data. We obtain 26.0 and 10.1 for models (4.13) and (4.15) respectively, while the multiple linear regression model (4.8) yields an AICC of 2.02. Thus we can conclude that the multiple linear regression models (4.7) and (4.8) have better fit that any of (4.13), (4.14), and (4.15).

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4.4.5

Model Fits on the Slip Factors

In this section, we implement a one-way analysis of variance model and analysis of covariance model, as described in equations (4.2) and (4.3) by regarding the slip type and direction as the categorical factors. Slip direction may be easily grouped into three groups (center, left and right). However, it is not straightforward to categorize the slip type from the given data. Instead, our approach adopts five groups: (a) normal, not strike, (b) reverse, not strike, (c) strike, not normal and not reverse, (d) strike and normal, and (e) strike and reverse. In the actual analyses fitting moment magnitude on slip type and direction, we apply the fit on maximal possible reliable events where n = 192. This yields the sizes of the sub-categories: 27, 36, 83, 14, and 32 for groups (a) to (e), respectively. And for the sub-categories classified by slip direction, the sizes are 63, 46, and 83 for center, left and right, respectively. As seen in Table 4.4, analysis results support the null hypothesis that moment magnitude does not depend on slip type and direction. Figure 4.7 presents the modified box-and-whisker plots of moment magnitude on slip type and direction. Modified box-and-whisker plots have the ability to show outliers beyond Q1−1.5IQR (interquartile range) and Q3+1.5IQR. As we have observed that all the five groups of slip types have very similar means, it is not surprising that this categorical factor is insignificant in predicting the moment magnitude. Similarly, slip direction is found to be an insignificant factor in the one-way ANOVA model. Apparently, no outliers is observed under this criterion. Tab. 4.4 Results from fitting one-way ANOVA model of M against slip type and slip direction. The columns are degrees of freedom (DF), sum of squares (SS), mean square (MS), F -value and p-value. The p-values indicate neither factor is significant at 5% level. Fitting M on slip type DF

SS

MS

F -value

p-value

Model

4

0.924

0.231

0.365

0.833

Residuals

187

118

0.632

Fitting M on slip direction DF

SS

MS

F -value

p-value

Model

2

0. 899

0.450

0.718

0.489

Residuals

189

118

0.626

Furthermore, we combine the quantitative variables and two slip factors, and implement analysis of covariance (ANCOVA). When including the quantitative predictors in both models (4.7) and (4.8), the data yield n = 45 with the sizes of the five groups (a) to (e) reduced to 9, 8, 18, 3 and 7. Such ANCOVA models

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Fig. 4.7 Modified box-and-whisker plots for the moment magnitude on two categorical variables. (a) shows the moment magnitude on slip type with: a. normal, not strike, b. reverse, not strike, c. strike, not normal and not reverse, d. strike and normal, and e. strike and reverse. (b) shows the moment magnitude on slip direction. In both plots, the similar values of the means (depicted by black squares) explain why the one-way ANOVA models show insignificance between groups.

are also implemented using two or three quantitative predictors and one or two categorical predictors. Although some models yield p-value less than 0.05, we do not consider those models because there are so few events in some groups and this causes substantial bias in prediction. In summary, we conclude that models (4.7) and (4.8) suffice good prediction models.

4.5

Concluding Remarks

Extending the study of Wells and Coppersmith (1994), we apply multiple regression analyses to explain the relationships between moment magnitude and fault measurements. Our study shows that, when we estimate the moment magnitude, the factors of rupture lengths, rupture width, rupture area, and surface displacements, are neither independent nor of equally importance. Some of these factors may be eliminated when we build a multiple linear regression model; maximum displacement and rupture area already provide adequate information to construct a very good model. And when maximum displacement is not available, using rupture surface and rupture width as the two predictors provides the best alternative. Besides the quantitative predictors, our ANOVA approach by adopting slip type and direction has revealed that these factors are insignificant when predicting moment magnitude. Geophysically, why do the models with maximum displacement surpass those with average displacement? Two possible explanations are: 1. The energy released by earthquakes is more concentrated

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on the asperity part or the locked part of the faults, where the maximum displacement usually occur, while the movement on the other part of the fault is more passive; 2. The errors for the estimate of the average displacement are much bigger than the maximum displacement. A remaining issue is the missing data. In this study, only the reliable events are analyzed while the pattern of missingness or unreliability is unknown. We need to be aware that this may introduce bias in model prediction.

Acknowledgements We thank Prof. Yosihiko Ogata for his support, and Dr. Rodolfo Console and Dr. Yong-Gang Li for helpful discussions.

References Akaike H., 1974. A new look at the statistical model identification. IEEE Transactions on Automatic Control, AC-19: 716–723. Bonilla M.G., Mark R.K., Lienkaemper J. J., 1984. Statistical relations among earthquake magnitude, surface rupture length, and surface fault displacement. Bull. Seismol. Soc. Amer., 74(6): 2379–2411. Burnham K.P., Anderson D.R., 2002. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd ed. Springer-Verlag. Gorβ, J., 2003. Linear Regression. Springer-Verlag. Hanks T.C., Kanamori H., 1979. Moment magnitude scale. Journal of Geophysical Research, 84 (B5): 2348–2350. Jennrich R.I., 1995. An Introduction to Computational Statistics. Prentice-Hall, New Jersey. Mark R.K., 1977. Application of linear statistical models of earthquake magnitude versus fault length in estimating maximum expectable earthquakes. Geology, 5(8): 464–466. Ogata Y., Zhuang J., 2006. Space-time ETAS models and an improved extension. Tectonophysics, 413(1-2): 13–23. Richter, C.F., 1935. An instrumental earthquake magnitude scale. Bull. Seismol. Soc. Amer., 25 (1-2): 1–32. Singh S.K., Bazan E., Esteva L., 1980. Expected earthquake magnitude from a fault. Bull. Seismol. Soc. Amer., 70(3): 903–914. Wells D.L., Coppersmith K.J., 1994. New empirical relationships among magnitude, rupture length, rupture width, rupture area, and surface displacement. Bull. Seismol. Soc. Amer., 84(4): 974–1002.

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Wyss M., 1979. Estimating maximum expectable magnitude of earthquakes from fault dimensions. Geology, 7(7): 336–340. Zhuang J., Vere-Jones D., Guan H., Ogata Y., Ma L., 2005. Preliminary analysis of observations on the ultra-low frequency electric field in a region around Beijing. Pure and Applied Geophysics, 162: 1367–1396, doi:10.1007/s00024-004-2674-3.

Authors Information Annie Chu Department of Mathematics, Woodbury University, 7500 N. Glenoaks Boulevard, Burbank CA 91504, USA E-mail: [email protected] Jiancang Zhuang The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan E-mail: [email protected]

Chapter 5

PI Algorithm Applied to the SichuanYunnan Region: A Statistical Physics Method for Intermediate-term Mediumrange Earthquake Forecast in Continental China Changsheng Jiang, John B. Rundle, Zhongliang Wu, and Yongxian Zhang

In the 2nd volume of this book series, Jiang and Wu (2014) have demonstrated some of the useful tactics in the analysis of earthquake catalog data. In connection to the accelerating moment release (AMR) analysis for the Sichuan-Yunnan region of southwest China, they proposed an “eclipse method” to screen out the seismicity in the neighboring active fault zones. As a development of the “curvature parameter” discriminating linear increase and accelerating trend, they proposed to use the Bayesian information criterion (BIC) to evaluate the statistical significance of the acceleration. One of the issues worth pointing out is that when an algorithm is applied to a specific region, something new will come out and has to be taken into account. The same thing happened to the same region, for another algorithm. In this chapter, the pattern informatics (PI) algorithm, which is one of the recently developed predictive models of earthquake physics based on the statistical mechanics of complex systems, is applied to the Sichuan-Yunnan region. Retrospective forecast test was conducted, obtaining the “optimal” parameters suitable for this region. Moreover, three issues were discussed, extending the application of the algorithm. The first issue is whether PI algorithm, which is mainly for 5 years or longer time scales, is suitable for the annual forecast. Retrospective test shows that simple application of PI method to annual forecast is not necessarily a good idea, but PI algorithm can be applied to the annual forecast in a way similar to the reverse tracing of precursors

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(RTP) approach. The second issue is whether PI algorithm could be able to provide some indications of the approaching of the 2008 Wenchuan earthquake. The conclusion is shown to be dependent on the input data: If “traditional” parameter settings are used, then there is no precursory “hotspot” seen associated with the Wenchuan earthquake. However, if a lower cutoff magnitude is used, then the PI “hotspots” could be seen not only near the epicenter but also spanning the whole rupture zone. The third issue is about the reported correlation of seismicity in the Sichuan-Yunnan region and the Andaman-Sumatra region. The application of PI algorithm confirmed such a correlation. Key words: Pattern informatics, Earthquake probability, Seismicity precursor, ergodicity, Sichuan-Yunnan region.

5.1

PI Algorithm

5.1.1

Background and Basic Concepts

Since last three decades, the application of concepts and methodologies of physics in nonlinear systems and complexity to the forecast of earthquakes has attracted much attention among seismological and physical communities. The Abdus Salam International Centre for Theoretical Physics (ICTP) organized the serial workshops on nonlinear dynamics and earthquake prediction from the 1980s to the 1990s. The APEC Cooperation on Earthquake Simulation (ACES) project was focusing on the simulation of earthquake preparation and earthquake occurrence as a nonlinear process. Since the 21st century, the StatSeis series of international symposia has provided another platform for the exchange of the statistics and statistical physics of earthquakes. Among these concepts and methodologies, pattern informatics (PI) algorithm (Rundle et al., 2000; 2003; Tiampo et al., 2002), a predictive algorithm for the analysis of seismic activity based on the statistical physics of complex systems, has been successfully applied to California (Rundle et al., 2000, 2003; Tiampo et al., 2002), central Japan (Nanjo et al., 2006), and Taiwan (Chen et al., 2005), as well as other regions. In many aspects, the algorithm has also been discussed (e.g., Zechar and Jordan, 2008), evaluated (e.g., Cho and Tiampo, 2012), improved (e.g., Wu et al., 2008), and extended (e.g., Wu et al., 2011; Jiang and Wu, 2011). PI algorithm assumes that seismogenetic dynamics can be regarded as a “threshold system” driven by persistent forces or currents. By analyzing the fluctuations of seismicity, PI algorithm estimates the increase of the probability of earthquakes at an intermediate-term time scale.

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PI algorithm has been introduced to continental China since 2006, in connection to the Western Pacific Geophysics Meeting (WPGM) in Beijing. Institute of Geophysics of China Earthquake Administration (IGPCEA) applied the MatLab code of Rundle and Chen, and China Earthquake Networks Center (CENC) wrote the MatLab code independently. PI algorithm has been tested retrospectively against the earthquakes in continental China, with discussions and developments based on the situation of seismic activity. This chapter summarizes the results of the test since the last decade.

5.1.2

Algorithm Validation

The input data for PI calculation are regional earthquake catalogues; the output of PI calculation is the relative increase of the probability of earthquakes, or the distribution of “hotspots”, described by the top 30% of the normalized probability increase. PI algorithm uses counts of earthquakes within spatial-temporal cells to describe the averaged seismic activity and its fluctuation. In the algorithm, the whole region under the study is binned into boxes or “pixels” with size D × D centered at a point xi . Each point xi is associated with a time series Ni (t), where Ni (t) is the time-dependent average rate of earthquakes with magnitude greater than the cutoff magnitude MC in box i and its Moore neighborhood, as seen in Figure 5.1a. Ni (t) is calculated for box i within a period staring from time tb to time t(t > tb ), as seen in Figure 5.1b. The “seismic activity intensity” function of box i is defined as the average rate of occurrence of earthquakes: Ii (tb , t) =

t 1  Ni (t′ ) t − tb ′

(5.1)

t =tb

The probability of a future strong earthquake in box i, Pi (t0 , t1 , t2 ), is defined as the square of the average intensity fluctuation: 2

Pi (t0 , t1 , t2 ) = ∆Ii (t0 , t1 , t2 )

(5.2)

in which t1 is the starting time of the “anomaly identification window”, t2 is the ending time of the “anomaly identification window” and the starting time of the “forecast window”, and t3 is the ending time of the “forecast window”. The “sliding window” for PI calculation is selected to start from t0 . Subtracting the mean probability over all boxes and denoting this change as the probabilityincrease of future earthquakes via ∆Pi (t0 , t1 , t2 ) = Pi (t0 , t1 , t2 ) − �Pi (t0 , t1 , t2 )�

(5.3)

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The “hotspots” are defined to be the boxes where ∆Pi (t0 , t1 , t2 ) is positive, or the probability function Pi (t0 , t1 , t2 ) is larger than the background level. Physically, since the probability function has quadratic form, either “seismic activation” or seismic quiescence can be reflected by the PI “hotspot” map, which is one of the reasons why PI forecasts outperform the “relative intensity” (RI) forecasts.

Fig. 5.1 Sketch map showing the spatial and temporal range selection of earthquake time series. (a) Sketch map of spatial range selection: The spatial range of the time series for site xi includes the box centered at xi and the eight nearest neighbor boxes (the Moore neighborhood) shown by gray squares; (b) the time tags and time windows related to earthquake time series of point xi .

5.1.3

Test of the Algorithm: ROC and Beyond

In the evaluation of the performance of earthquake forecasts, receiver operating characteristic (ROC) test (Swets, 1973; Molchan, 1997) was used by systematically changing the “alarm threshold” of the “forecast region” and counting the “hit rate” and “false alarm rate” relative to real earthquake activity. Here, “hit rate” means the number of “forecasted” events divided by the total number of “target” events. “Hit” or “forecasted” is defined as the case that the “target event” occurs within any “alarmed cell” or one of the nearest neighbors. For the ROC test, the PI forecast is compared with a random guess. The larger the area under the ROC curve is when compared to the area of the triangle for random forecast, the better the performance of the forecast will be. In addition, ROC test also compared the PI forecast with the “relative intensity” (RI) forecast (Chen et al., 2005; Holliday et al., 2006). Here the “relative intensity” forecast is used as a null hypothesis to test the necessity of using the PI algorithm. In the RI algorithm, it is assumed that strong earthquakes will occur in the regions where earthquakes occurred before, being similar to the “clustering” argument (Kagan and Jackson, 2000). In the comparison, RI index is computed simply by the box-counting of the number of events. Similar to the

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PI algorithm, the RI statistics were also normalized, and the “alarm threshold” was determined by the relative value of the RI statistics ranging from 0 to 1. For visual comparison with the PI forecast, only top 30% of the RI increases are regarded as “hotspots” and are shown on the map.

5.2

The Sichuan-Yunnan Region

5.2.1

Seismicity and Earthquake Catalogue

Sichuan-Yunnan region is located between latitudes 20.8◦ N–34.0◦N and longitudes 97.2◦ E–107.0◦E, including Sichuan and Yunnan Provinces as well as their surrounding regions. There are 43 major earthquakes with the magnitude larger than MS 7.0 occurred in the last five centuries in this region, with 3 of them larger than MS 8.0: the 1833 Songming-Yanglin, Yunnan MS 8.0 earthquake, the 1879 Wudu, Gansu MS 8.0 earthquake, and the 2008 Wenchuan, Sichuan MS 8.0 earthquake. The recent major earthquake was the April 20, 2013, Lushan, Sichuan earthquake (Wu et al., 2014). Yi et al. (2002) and Xu et al. (2005) observed that strong earthquakes in this region have a significant component of stochastic clustering, with a recurrence process that does not demonstrate simple periodicity and does not seem to be well-described by the time-predictable model or magnitude-predictable model. Earthquake catalogue available for this region is the Monthly Earthquake Catalogue provided by the China Earthquake Networks Center (CENC), compiled based on the catalogues provided by regional/local seismic networks, with magnitude unified as ML . Su et al. (2003) used the regional earthquake catalogue of Sichuan and Yunnan from 1970 to 2001 to discuss the time-dependent and regionalized completeness of the catalogue, indicating that since the early time of earthquake monitoring in the late 1970s, the earthquake catalogues of the Sichuan-Yunnan region have been complete down to ML 3.0. The surface wave magnitude is used for the definition of “target earthquake” to avoid the bias caused by the difference between different local magnitudes. For this purpose, the “target earthquakes” are selected from the catalogue of the CENC (http://www.csndmc.ac.cn/newweb/data.htm#).

5

5.2.2

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245

Tectonic Setting

The Sichuan-Yunnan region is located in the east margin of the Qinghai-Xizang (Tibetean) Plateau, being the transitional zone between the rapidly upheaving Tibetean Plateau and relatively steady Yangze Platform. Under the pressure exerted by the northward movement of the Indian plate, the Sichuan-Yunnan region has been undergone strong deformation, becoming one of the regions with most intensive seismicity in continental China. The Sichuan-Yunnan diamond and its boundary fault zones, with the Xianshuihe, Anninghe, Zemuhe, Xiaojiang, Honghe, Lancangjiang, and Longmenshan fault systems, located in this region. These fault systems as well as other neighboring fault systems cut the whole region into different “tectonic blocks” and control the distribution of earthquakes (Xu and Deng, 1996). Figure 5.2 shows the earthquakes larger than MS 5.5 since 1970 in the Sichuan-Yunnan region, with tectonic faults shown by gray lines. In the subplot showing the temporal distribution of earthquakes, the three vertical dash lines to the right represent t1 the starting time of the “anomaly training window”, t2 the ending time of the “anomaly training window” and the starting time of the “forecast window”, and t3 the ending time of the “forecast window”. For the sliding window considered in this example, the catalogue is selected to start from t0 .

5.2.3

The 2008 Wenchuan Earthquake

Remarkably, in the Sichuan-Yunnan region, there occurred the great Wenchuan earthquake on May 12, 2008 (Chen and Booth, 2010). The earthquake stroke at 14 : 28 pm local time, with magnitudes MS 8.0 and Mw 7.9, having a thrustdominated focal mechanism with its aftershock belt being 330-km long, and causing fatalities of up to 69, 227 and near 18, 000 missing (by September 25, 2008, from the Ministry of Civil Affairs of China). The earthquake is one of the largest in China in recent decades and caused much attention not only within scientific community but also in the public 1 . Figure 5.2 shows the epicenters of the Wenchuan mainshock and its aftershocks. After the 2008 Wenchuan earthquake, there was a debate on whether the earthquake could have been “forecasted” if observational data were carefully examined. This question is important not only for this earthquake but also for the study of earthquake predictability in the future.

1 See: Elsevier. Virtual Special Issue about the 2008 Wenchuan Earthquake. http://www.journals.elsevier. com/tectonophysics/virtual-special-issues/virtual-special-issue-on-the-2008-wenchuan-earthquake/.

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Fig. 5.2 (See color insert.) (a) Earthquakes larger than MS 5.5 since 1970 in the Sichuan-Yunnan region, with tectonic faults shown by gray lines (reproduced from Jiang and Wu, 2010). Yellow star and yellow dots indicate the epicenters of the Wenchuan mainshock and its aftershocks, respectively. The region under the study is shown in the index to the bottom right, in which the Wenchuan earthquake is shown by yellow star. (b) Frequency-magnitude distribution showing the selection of the magnitude of completeness. (c) Temporal distribution, with the three vertical dash lines to the right representing t1 the starting time of the “anomaly training window”, t2 the ending time of the “anomaly training window” and the starting time of the “forecast window”, and t3 the ending time of the “forecast window”. For the sliding window considered in this example, the catalogue is selected to start from t0 .

5.3

PI Algorithm Applied to the SichuanYunnan Region

5.3.1

Parameter Setting

To select proper parameters of predictive models is one of the critical issues in statistical seismology. PI algorithm suffers from the same difficulty and complexity. Previous works provided experiences for the selection of PI parameters. Nanjo et al. (2006) used a 35 year catalogue, 27 year “anomaly training window” and more than 9 years “forecasting window” to forecast the M  5 target events

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by setting the box size as 0.1◦ × 0.1◦ . For earthquake forecasting verification of PI method, Holiday et al. (2005) used a 68 year catalogue and took the “anomaly training window” and the “forecast window” to be both 10 years, forecasting the M  5 target events of Southern California and central Japan by setting box size as 0.1◦ × 0.1◦ , and for world-wide application with magnitudes greater than 7.0 by setting box size as 1.0◦ × 1.0◦ . To detect the precursory seismic activity preceding the 1999 Chi-Chi, Taiwan, earthquake by using PI analysis, Chen et al. (2005) used a near 12 year catalogue, and an approximately 6 year “anomaly training window”. Holliday et al. (2006) applied the modified PI method to a five-year short-term forecast, and partitioned the area by using square bins with edge length 0.1◦ , corresponding to the linear size of a magnitude 6 earthquake. As a contributed model for RELM testing of Southern California M  5 target events forecasting, Holliday et al. (2007) used a catalogue of more than 55 years, took the “anomaly training window” and the “forecast window” to be more than 20 years and 5 years, respectively, and the research area was divided

Fig. 5.3 (See color insert.) Retrospective test of PI algorithm: the Sichuan-Yunnan region, period of 01/01/1992–01/01/1997. Color-coded hotspots highlight the relative probability increase for earthquakes above MS 5.5, with spatial grid size of 0.2◦ . Blue circles stand for the earthquakes above MS 5.5 occurring within the “forecast window”, while gray reverse triangles show the earthquakes above MS 5.5 occurring within the “anomaly training window”. Difference between the blue circles and the gray reverse triangles shows the reason why PI algorithm outperforms RI algorithm.

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into 0.1◦ ×0.1◦ pixels. Wu et al. (2008) discussed the migration of PI hotspots of the December 26, 2006, Pingtung doublets by setting the box size as 0.1◦ × 0.1◦, and using a 13 year catalogue; in their work t2 was fixed, with t1 shifting from 5 years to 2 years before t2 using the sliding step of three months. For earthquakes in the Sichuan-Yunnan region, Jiang and Wu (2008) conducted retrospective forecast test of PI algorithm to investigate the stability of the algorithm against the selection of model parameters. They adjusted the parameters systematically and investigated the effect of such parameter variation. As a result, the “optimal” parameters were selected for the “target magnitude” of MS 5.5: a fifteen-year long “sliding time window”; the “anomaly training time window” and “forecast time window” both being 5 years; the spatial grid taken as D = 0.2◦ and only shallow earthquakes with depth ranging from 0 to 70 km are considered. Figures 5.3 and 5.4 shows an example of the application.

Fig. 5.4 ROC test for the Sichuan-Yunnan region: period of 01/01/1992–01/01/1997. Thick solid line represents the ROC result for PI forecast, thin solid line the ROC result for RI forecast, and black broken line the result for random forecast. Gray broken line shows the difference between the “hit rate” of PI algorithm and that of RI algorithm. H-hit rate; F-false alarm rate; G-difference between the hit rate of PI and that of RI.

5.3.2

Sliding Window Retrospective Test

To evaluate the performance of PI algorithm for a long time duration, the test firstly considers the “anomaly training window” and the “forecast window” to be 5 years, with a “sliding window” of 15 years, and slides t2 from 01/01/1988 to 01/01/2003, with sliding step being 0.5 year. Figure 5.5a gives the ROC test result, in which gray zone delimitates the range of all ROC curves, with the gray

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Fig. 5.5 ROC test of PI algorithm (reproduced from Jiang and Wu, 2008). “Anomaly training window” and “forecast window” are taken as 5 years. The sliding window is taken as 15 years. “Forecast window” slides from t2 = 01/01/1988 to t2 = 01/01/2003, with sliding step being 0.5 year. (a) ROC curves for PI forecasts. Gray zone delimitates the range of all ROC curves, with gray line and black line representing the results of the first and the last sliding, respectively. (b) PI forecast versus RI forecast — difference between the hit rate of PI algorithm and RI algorithm changing with false alarm rate. Gray zone delimitates the range of all curves, with gray line and black line representing the results of the first and the last sliding, respectively.

line and the black line representing the results of the first and the last sliding, respectively. From the figure, it can be seen that despite the variation of the performance, generally the PI forecast shows a much better performance than the random forecast. Figure 5.5b compares the results of PI algorithm and RI algorithm. It can also be seen that PI algorithm also outperforms RI algorithm.

5.3.3

Ergodicity

Figure 5.6 shows the ergodicity of seismicity for the Sichuan-Yunnan region, indicating that PI algorithm is valid for this region and for the time period under consideration. The ergodicity is evaluated following Tiampo et al. (2004; 2007) by plotting the Thirumalai-Mountain (TM) metric (Thirumalai et al., 1989) versus time. For different grid sizes and different cutoff magnitude values, TM metric plots show that since 1978 seismicity in the Sichuan-Yunnan region has had a strong ergodicity, and PI algorithm has been able to be used for the estimation of time-dependent earthquake rates.

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Fig. 5.6 Plots of the inverse TM metric for the Sichuan-Yunnan region from 01/01/1971 to 15/06/2008 (reproduced from Jiang and Wu, 2010). In each subplot, D is the spatial grid size, and Mc is the cutoff magnitude.

5.4

Discussion and Development of PI Application

5.4.1

PI for Annual Estimate of Seismic Hazard: Useful, or Useless?

Motivated by the needs of three-year and one-year forecasts, we change the “forecast window” to 3 years and 1 year, respectively. Figures 5.7 and 5.8 give the test of such forecasts, using the same parameters as those shown in Figure 5.5. It is seen that although the capability of forecast decreases, even for the one year forecast, PI algorithm still outperforms the random forecast. On the other hand, for the annual time scale, PI algorithm has no more significant advantage. This phenomenon was also observed by Zhou et al. (2006) who found that the probability for strong earthquakes increases significantly in the regions near the earthquakes larger than magnitude 5 occurring in the previous year; they even used this observation as one of the criteria to identify the alarm regions of strong earthquakes at an annual scale. When the “forecast window” is taken

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Fig. 5.7 ROC test of PI algorithm (reproduced from Jiang and Wu, 2008). “Anomaly training window” and “forecast window” are taken as 5 years and 3 years, respectively. “Sliding window” is taken as 15 years. Captions are the same as Figure 5.5.

Fig. 5.8 ROC test of PI algorithm (reproduced from Jiang and Wu, 2008). “Anomaly training window” and “forecast window” are taken as 5 years and 1 year, respectively. “Sliding window” is taken as 15 years. Captions are the same as Figure 5.5.

as one year, there is the case of “complete false alarm”, i.e., there is no “target earthquake” occurring in the “forecast window”. The ROC test fails to reflect this case. Physically, the phenomenon that PI fails to outperform RI at annual scale implies that at the one year time scale, clustering plays a predominant role in determining the behavior of seismicity. But the test shows that simple application of PI algorithm to the annual forecast is not a good idea. Among the intermediate-term and medium-range earthquake forecast with variable definitions of their spatio-temporal ranges, the annual earthquake fore-

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cast plays a special role by balancing the capability of the time-dependent seismic hazard estimation and the annual plan for the earthquake hazard preparedness. In China, such an approach is implemented by the Annual Consultation Meeting on the Likelihood of Earthquakes organized by China Earthquake Administration (CEA). By combining tectonic, seismic, and other geophysical information, the group of experts draws conclusions about the seismic tendency in the next year and identifies the areas with a higher seismic risk (Shi et al., 2000, 2001; Zheng et al., 2000; Zhang et al., 2002; Ma et al., 2004; Wu et al., 2007a, b; Zhuang and Jiang, 2012). One of the problems of the Annual Consultation is its false-alarm rate, which has been apparently higher than expected (Wang, 2005). To solve this problem, we use the reverse tracing of precursors (RTP) approach proposed by KeilisBorok et al. (2004). The idea is to trace long-term anomalies within the regions identified by the Annual Consultation as the “alarm regions in the coming year”. There are several options for choosing the long-term anomalies (e.g., Field, 2007) in the RTP consideration. Our choice is PI and RI algorithms. This approach is, to some extent, a development of the idea of Keilis-Borok et al. (2004), because RTP is originally based on the analysis of a set of intermediate-term precursors in an area of short-term anomalies, such as the region of long-range activation of seismicity detected by earthquake chains (Shebalin et al., 2004, 2006). In our approach, RTP is based on the analysis of PI/RI anomalies with a five-year time scale in an “alarm region” of the increase of probability of earthquakes for one year time scale. We have intentionally chosen the testing period of 1990 to 2003 as the time when PI/RI algorithms were still “independent” of the methods used in the Annual Consultation to analyze seismicity. Figure 5.9 depicts the application of PI result to RTP of the Annual Consultation for the Sichuan-Yunnan region. The results are obtained by applying the following three steps. Step 1: plot on the same map of the alarm regions produced by the Annual Consultation and PI “hotspots”. Step 2: calculate the proportion of overlapped area between each alarm region and PI “hotspots” to empirically determine a threshold proportion for the detection of the false alarm regions. Step 3: eliminate the regions which have a proportion of “overlapped area” lower than the threshold, thus eliminate the false-alarm regions. The selecting of the threshold proportion is done empirically, because in practice one has to balance the correct removal of the false alarm regions and to avoid the “incorrect removal” (i.e., removal of the alarm regions in mistake causing miss-to-hit). For PI algorithm, the threshold is empirically selected as 7%, and for RI algorithm, the threshold is 13%.

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Fig. 5.9 (See color insert.) PI applied for RTP (reproduced from from Zhao et al., 2010). (a) The “hotspot” distribution using PI method for the forecast period 01/01/1995– 01/01/2000 and the result of the Annual Consultation for the year 01/01/1995– 01/01/1996 (delimitated by black bold lines). Blue circles show the “target earthquakes” occurring in the period 01/01/1995–01/01/1996, and gray reverse triangles show the earthquakes occurring in the “anomaly training window”. (b) Proportion of overlapping area between PI “hotspots” and the alarm region from the Annual Consultation. Text shows the proportion of the overlapped area and the decision for the removal or the confirmation of each alarm region when the threshold proportion is 7%.

5.4.2

The 2008 Wenchuan Earthquake: Miss, or Hit?

We investigated whether PI algorithm, which shows a good performance for the Sichuan-Yunnan region, could have provided some clues to the approaching of the Wenchuan earthquake. Figure 5.10 shows the “forward forecast” for different time ranges from t2 = 01/01/2004 to t2 = 01/01/2008 with a one-year step, covering the time of the Wenchuan earthquake. In the figure we mark the segment of the Longmenshan

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Fig. 5.10 (See color insert.) Hotspot maps of PI algorithm for different “forecasting time windows” (reproduced from Jiang and Wu, 2010). (a) t2 = 01/01/2004; (b) t2 = 01/01/2005; (c) t2 = 01/01/2006; (d) t2 = 01/01/2007; (e) t2 = 01/01/2008. “Forecast time window” t3 − t2 = 5 years. Green box delimitates the mid-to-north segment of the Longmenshan fault which accommodated the Wenchuan earthquake. Star shows the epicenter (or, nucleation point) of the great earthquake.

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fault which accommodated the main rupture and epicenter (or, initiation point) of the Wenchuan earthquake. This area also covers the aftershock zone. The hotspots likely exist in the “forecast time window”, and the “hotspot” cluster looks spanning along the whole rupture area. Note that the original objective of PI algorithm is to forecast the position of the epicenter of earthquakes, rather than the rupture area. On the other hand, this earthquake is too large to be treated as a “point source” so that such a correlation between “hotspots” and the rupture area seems to make sense in the physics of earthquakes. One noticeable problem is the selection of the grid size. As pointed out by Kossobokov and Soloviev (2008) for the 2004 Sumatra earthquake, a larger grid size may be more appropriate for forecasting larger events. To explore the possibility of this case, we tested bigger grid size, but in this case the “hotspots” near the mid-and-north Longmenshan fault disappeared (Jiang and Wu, 2010). This result is consistent with those of Holliday et al. (2005), which were published before the earthquake occurred, using a 1◦ grid size. In their paper, a prominent PI anomaly was located about 200 km away from the Wenchuan earthquake. This suggests that the convention, in which the cutoff magnitude is adopted 2 units less than that of the “target earthquake”, is problematic in dealing with inland earthquakes. Physically, the population of small events may contain information about the increase in probability of strong earthquakes. If only larger events and larger grids are used in analysis, other patterns, such as the migration of PI “hotspots” (Wu et al., 2008), should be taken into account.

5.4.3

Sichuan-Yunnan versus Andaman-Sumatra: Separate, or Connected?

The relation between Sichuan-Yunnan and Andaman-Sumatra regions located approximately 2000 km to the south is interesting, but has not been fully explored yet. Tectonically, these two regions are interconnected by the boundaries between Eurasia-Burma-Sunda and India-Australia plates. Seismicity in these two regions seems apparently correlated with each other (Zhang et al., 2005). Aiming at the forecast of “target earthquakes” no less than Mw 7.0, we treat these two regions as a unified one in the “learning” and “forecasting” process, and conduct retrospective test of the forecast at biennial time scale. Figure 5.11 shows the distribution of earthquakes larger than 7.0 since 1995 in SichuanYunnan and Andaman-Sumatra. We use NEIC catalogue from 1973 to 2007 in PI calculation. The earthquake magnitude is unified to Mw by empirical relations between magnitudes: log M0 = 0.14m2b + 0.36mb + 10.76 (Johnston, 1996), log M0 = 1.5MS + 9.1 (Kanamori, 1977), MS = 1.27(ML − 1)− 0.016ML2 (Gutenberg and Richter, 1956), and Mw =

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Fig. 5.11 (a) Earthquakes larger than 7.0 since 1995 in Sichuan-Yunnan and Andaman-Sumatra (reproduced from Jiang et al., 2009). (b) and (c) Temporal distribution for the two regions, respectively. (d) Frequency-magnitude distribution of the catalogue, indicating the completeness down to Mw 5.0.

2/3lgM0 − 6.06 (Kanamori, 1977). The catalogue is filtered by the completeness cutoff magnitude Mw 5.0, and only shallow events are used. Considering previous works (Rundle et al., 2002; Tiampo et al., 2002), the spatial grid is taken as D = 1.0◦ The “anomaly training window” from t1 to t2 is taken as 10 years, same as those used by others. Starting time of the catalogue t0 = 01/01/1973. The “forecast window”, from t2 to t3 , is taken as 2 years, considering the needs of earthquake forecast study in China. Results of the retrospective forecast test are shown in Figure 5.12. The ROC test (Jiang et al., 2009) indicates that both PI and RI forecasts outperform random guess, but the performances of PI and RI are comparable, indicating that at the biennial time scale, clustering plays a predominant role. Or in another word, in PI calculation, the cause to changes of seismicity is mainly due to an activation mechanism. Based on the data from September 1997 to September 2007, the experimental forward forecast was conducted with parameter settings identical to that of the retrospective test. Figure 5.13 shows the forward forecast.

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Fig. 5.12 (See color insert.) PI map of the retrospective forecast test (reproduced from Jiang et al., 2009). Circles represent the “target” earthquake occurring between t2 and t3 . Colored pixels (“hotspots”) represent areas with large seismicity change caused by either seismic activation or quiescence, indicating higher probability for future large events. (a) t1 = 07/11/1985, t2 = 07/11/1995, and t3 = 07/11/1997; (b) t1 = 04/01/1988, t2 = 04/01/1998, and t3 = 04/01/2000; (c) t1 = 02/13/1991, t2 = 02/13/2001, and t3 = 02/13/2003; (d) t1 = 12/26/1994, t2 = 12/26/2004, and t3 = 12/26/2006.

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Fig. 5.13 (See color insert.) Experimental forward forecast for the period with t1 = 09/12/1997, t2 = 09/12/2007, and t3 = 09/12/2009 (reproduced from Jiang et al., 2009). Circles represent the “target” earthquakes occurred just at the margin between the “learning period” and the “forecast period” (from north to south, they are earthquakes of 09/13/2007 Mw 7.0, 09/12/2007 Mw 7.9, and 09/12/2007 MS 8.5). As a forward forecast test, the green star indicates the earthquake of 10/25/2007 Mw 7.1 which occurred after the submission of the manuscript to Acta Seismologica Sinica (on October 22, 2007), locating near one of the “hotspot” clusters. However, the 2008 Wenchuan earthquake fell within a weak hotspot cluster to the north in the figure.

5.5

Concluding Remarks

It has been nearly ten years since the starting of the application of PI algorithm to the Sichuan-Yunnan region. During these years there occurred the 2008 Wenchuan earthquake. Test of the algorithm has learnt a lot as per how to apply PI algorithm to inland earthquakes and the practical estimation of the time-dependent seismic hazard in the Annual Consultation. Yet more questions remained, from the performance of the forecast versus the deformation rate, to how to make use of PI “hotspot” maps to the decision-making-oriented earthquake forecast.

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Suffering from intense seismic disasters and having been striving for promoting the study of earthquake forecast since the 1960s, the seismological community in China, while independently developing forecast schemes, is enthusiastic in applying and testing all the concepts and methods for earthquake forecast. Continental China also plays the role of the natural laboratory for the experiments of earthquake forecast approaches. As pointed out in one of our reports introducing the Annual Consultation on the Likelihood of Earthquakes (Wu, 2007a): If you love your model, come to China; If you hate your model, come to China.

Acknowledgements We are grateful to Prof. Y. G. Li, the editor-in-chief of this volume, for kind invitation and valuable suggestions for improving the manuscript. Thanks are also due to the students joining in the study, including Hui Jiang, Yingchun Li, Shengfeng Zhang and Yizhe Zhao. This work is supported by the China National Special Fund for Earthquake Scientific Research in Public Interest (grants 201308011) and the international science and technology cooperation project (Grant No. 2012DFG20510), from the Ministry of Science and Technology of China.

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Authors Information Changsheng Jiang, Zhongliang Wu Institute of Geophysics, China Earthquake Administration, Beijing 100081, China E-mail: jiang [email protected] John B. Rundle Department of Physics, University of California at Davis, Davis, California 95616, USA Yongxian Zhang China Earthquake Networks Center, Beijing 100045, China

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Chapter 6

Probabilistic Seismic Hazard Assessment for Pacific Island Countries Y. Rong, J. Park, D. Duggan, M. Mahdyiar, and P. Bazzurro

A fully probabilistic earthquake hazard assessment study was carried out for fifteen Pacific Island Countries (PICs): Cook Islands, Fiji, Kiribati, Republic of Marshall Islands, Federated States of Micronesia, Nauru, Niue, Palau, Papua New Guinea, Samoa, Solomon Islands, Timor-Leste, Tonga, Tuvalu and Vanuatu. A regional seismicity model was built based on historical and instrumental earthquake catalogs, subduction zone segmentation and plate motion information, geodetic data, and available data on crustal faults. We used different ground motion prediction equations to account for different types of earthquakes. The effect of site conditions on ground motion was modeled based on shear wave velocities derived from microzonation studies and high-resolution topographic slope data. A comparison of our findings with those of earlier studies, such as GSHAP (Global Seismic Hazard Assessment Program), shows similarities, and in some cases, significant differences. The seismic hazard maps developed here have a spatial resolution that is adequate for local seismic risk studies and building code applications. Key words: Probabilistic seismic hazard assessment (PSHA), Pacific Islands, Seismicity model, Ground motion.

6.1

Introduction

Many of the PICs are located close to one of the most active subduction zones in the world and are prone to a high seismic risk. Since the year 2000, fifteen earthquakes of moment magnitude (Mw ) greater than or equal to 7.5 have oc-

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curred in the region, with four having Mw > 8.0. The complicated tectonics and the high seismicity of the region are due mainly to the interaction of four major plates (Fig. 6.1), the Pacific, Philippine Sea, Sunda, and Australia plates. The Philippine Sea plate subducts to the west under the Sunda plate at a rate of about 100 mm/yr, and the Australia plate subducts to the north beneath the Sunda plate at a rate of about 70–80 mm/yr. The convergence between the Australia and Pacific plates results in a shortening at the subduction plate boundaries along Papua New Guinea, the Solomon Islands, Vanuatu, Fiji and Tonga. The convergence rate is about 60–70 mm/yr at the Tonga trench, and about 100 mm/yr at other trenches.

Fig. 6.1 (See color insert.) Regional tectonic setting and seismic source zones in the South Pacific region. Thick red lines indicate the major plate boundaries between the four major plates: Pacific (PA), Philippine Sea (PS), Sunda (SU), and Australia (AU) plates. Thick black arrows illustrate the movement of the PA, PS, and AU plates relative to the SU plate (Bird, 2003). The source zones are illustrated by blue polygons, and the numbers are the zone IDs. The area in the green polygon was covered by the AIR Southeast Asia earthquake model.

Until this study, a consistent and up-to-date view of the seismic hazard for this region was not available in the open literature. The Global Seismic Hazard Assessment Program (GSHAP) conducted a uniform but coarse seismic hazard study for the entire region (McCue, 1999). In the present study, however, we include spatial resolution sufficient for local level applications, more recent ground motion prediction equations (GMPEs), and site effects. The seismic hazard of some countries such as Fiji, Papua New Guinea (PNG), and Vanuatu has been studied in more detail (e.g., Jones, 1998; Suckale and Gr¨ uthal, 2009), but the methodologies used were not the same for all regions and the resulting seismic hazard was not expressed in a uniform way across studies. Detailed seismic

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hazard studies for countries such as Tonga, Tuvalu, Samoa and the Solomon Islands are even rarer or non-existent in the open literature. As part of the Pacific Catastrophe Risk Assessment and Financing Initiative project funded by the World Bank and supported by other agencies, including the Asian Development Bank and the Applied Geoscience and Technology Division (SOPAC) of the Secretariat of the Pacific Community, we developed country-specific seismic risk assessment models for fifteen PICs: Cook Islands, Fiji, Kiribati, the Republic of Marshall Islands, the Federated States of Micronesia, Nauru, Niue, Palau, Papua New Guinea, Samoa, Solomon Islands, Timor-Leste, Tonga, Tuvalu, and Vanuatu. The main objective of the hazard assessment component of this study was to create a uniform and detailed regional seismic hazard model that captures the spatial variation of the hazard and could be used to support realistic estimation of the earthquake ground shaking risk across the region. The study included modeling of earthquake-induced tsunami hazard, but that component is not covered here.

6.2

Data

6.2.1

Historical Earthquake Catalogs

The availability of historical earthquake data for the Pacific Ocean region is limited compared to other parts of the world. To obtain a relatively complete, single catalog for the entire region, we selected, processed, and merged the following three historical earthquake catalogs: – PAGER-CAT by Allen et al. (2009) for the period 1900–2009, – Engdahl et al. (1998) relocated global earthquake catalog for the period 1964–2004, and – Earthquake database assembled by Geoscience Australia (http://www.ga. gov.au/earthquakes/searchQuake.do) for the period 1958–2009. Before merging the catalogs, the magnitude values reported in the Engdahl et al. catalog (1998) were converted to moment magnitude, Mw , using the regression relationships established by Suckale and Gr¨ uthal (2009). The magnitudes other than Mw (the surface magnitude, MS ; the body wave magnitude, Mb ; the local magnitude, ML ; the duration magnitude, MD ; and unknown magnitudes, MUK ) reported in PAGER-CAT were converted to Mw using the magnitude regression relationships derived in this study:

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Mw = 0.8252MS + 1.2188

(6.1)

Mw = 0.7546mb + 1.7809

(6.2)

Mw = 1.3210ML − 1.8631

(6.3)

Mw = 0.8110MUK + 1.2640

(6.4)

Mw = 0.9644MD + 0.4677

(6.5)

Equations (6.1) through (6.4) were derived based on the PAGER-CAT data from the South Pacific region, while Equation (6.5) was derived from all the PAGER-CAT data. For events in PAGER-CAT with only the teleseismic bodywave magnitude, mB , available, we used the relationship by Bormann and Saul (2008): (6.6) Mw = 1.33mB − 2.36 The events contained in the three historical earthquake catalogs were also augmented with an additional 42 large (Mw  7.0) historical events that were identified from various publications. Some of them were recorded pre-1900. Due to the remoteness of the region and poor coverage by seismographs, about 20% of the earthquakes in the merged historical earthquake catalog either lacked a hypocentral depth or had a default depth value (e.g., 33 km). The hypocentral depth for these events was probabilistically assigned based on empirically-based depth distributions derived from other historical events in this region. Finally, we declustered the merged catalog from foreshocks and aftershocks using the method of Reasenberg (1985). Figure 6.2 shows the epicenters of the historical

Fig. 6.2 (See color insert.) Epicenters of earthquakes in the Pacific Island Countries region contained in the merged historical catalog (down to Mw 5.0). Circles are colorcoded by depth and sized by magnitude.

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earthquakes included in the final combined catalog (down to Mw 5.0; smaller events were not included in the plot). The completeness time in the catalog varies by region. Generally, the catalog is considered to be complete for events of Mw 5.3 or greater that have occurred since 1964. For Fiji and its vicinity, events of Mw 6.5 and greater are complete since 1910. For other regions, events of Mw 7.0 and greater are complete since 1900.

6.2.2

Subduction Segments, Crustal Faults and Geodetic GPS Data

Among the fifteen PICs studied in this project, Palau, Papua New Guinea, Samoa, Solomon Islands, Vanuatu, Timor-Leste, and Tonga are located on or close to subduction zones. Segmentation models of subduction zones (e.g., Nishenko, 1991), have been customarily considered as a guide to the location and extent of mega-thrust earthquakes. However, the validity of segmentation models in seismic hazard studies has been recently challenged by the most recent mega-thrust earthquakes: the 2004 Sumatra, Indonesia, the 2010 Maule, Chile, and the 2011 Tohoku, Japan earthquakes. Here we used segmentations as a model framework, but allowed some earthquakes to rupture multiple segments. In addition to the subduction of plates, active crustal faults are potential sources of large earthquakes. For example, the 1953 Suva earthquake, the most damaging earthquake of Fiji in recent times, took place on a near-shore crustal fault. Potentially active faults have been identified in Viti Levu of the Fiji Islands (Rahiman, 2006), in PNG (Puntodewo et al., 1994; Tregoning et al., 1998), and in the Tonga-New Hebrides region (Pelletier et al., 1998). The crustal faults and subduction trenches are displayed in Figure 6.3. We used a total of 254 GPS velocity vectors from various sources: Simons et al. (2007), Wallace et al. (2004), Calmant et al. (2003), Tregoning et al. (1998), Bevis et al. (1995), Puntodewo et al. (1994), Dawson et al. (2010, personal communications, IAG data), and Wallace et al. (2010, personal communications, GNS data). The spatial distribution of these GPS stations is uneven (Fig. 6.3). Along the subduction zone, PNG and Vanuatu have the best coverage, whereas we could not locate any GPS measurements in the Solomon Islands. The GPS stations on the Sunda plate (Fig. 6.1) are not included here since that region was already covered in the AIR Southeast Asia earthquake model (2004), which is not discussed here.

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Fig. 6.3 (See color insert.) Location of GPS stations (triangles) used in this study. The pink lines illustrate curvilinear grids used in our regional kinematic model. Subduction trenches and geological faults are displayed in thick blue lines.

6.3

Kinematic Modeling Based on GPS and Active Faults Data

We used the 254 GPS velocity vectors and the active faults data shown in Figure 6.3 to invert the regional strain rate field using the well-established methodology by Haines and Holts (1993) and Haines et al. (1998). We also adopted the relative plate motion between the Australia and Pacific plates as a constraint in the inversion procedure to supplement the GPS and the active faults data. The relative plate motion was derived based on Global Strain Rate Map by Kreemer et al. (2003). The reference frames of the data include ITRF94, ITRF2000, ITRF2005, No-Net-Rotation (NNR), stable Pacific Plate, and stable Australia

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Plate. In the inversion, the original reference frame of each individual study was left undefined a priori, and then it was solved upon fitting geodetic velocities to one self-consistent velocity gradient tensor field.

Fig. 6.4 (See color insert.) Modeled (red arrows) and observed (black and white arrows) horizontal GPS velocities relative to the fixed Australia plate for Papua New Guinea and Solomon Islands region (top panel), and Vanuatu, Fiji, and Tonga region (bottom panel).

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Based on the regional tectonic, geologic, and seismic information, and the spatial distribution of GPS data, the region was divided into 3648 curvilinear grid cells, which cover both rigid and deforming zones (Fig. 6.3). (The grid cells in Figure 6.5 below will show only the deforming zones, which are allowed to deform in the inversion procedure.) Figure 6.4 shows the modeled and observed horizontal velocities in the region, relative to the fixed Australia plate. The modeled velocities match the observed GPS vectors at most of the sites, but significant differences exist at some stations in the Tonga region, in the New Britain Island of Papua New Guinea, and in West Papua, Indonesia. These discrepancies may be due to the

Fig. 6.5 (See color insert.) Predicted strain rate field (black arrows) for Papua New Guinea and Solomon Islands region (top panel), and for the Vanuatu, Fiji, and Tonga region (bottom panel). The beach-balls show the focal mechanisms of historical earthquakes: green refers to earthquakes with 5.3  Mw  6.0, which are shown only for non-subduction areas; blue refers to earthquakes with Mw > 6.0. The red lines illustrate curvilinear grids and the thin pink lines illustrate the subduction zones and faults.

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incompatibility between these GPS data and others. Figure 6.5 illustrates the predicted strain rate field for a large part of the South Pacific region. We also plotted the moment tensors of large earthquakes in the global CMT catalog (http://www.globalcmt.org/) as a verification of the modeled strain rate field. The predicted strain rate field is consistent with the plate motions, regional seismic moment tensor solutions, and observed GPS velocities. For example, the large strain rate at the southern New Hebrides subduction zone segment is in line with the large converging rate observed by GPS data. We converted the predicted strain rate field to seismic moment rate budget for each source zone and used the budget for modeling regional seismicity.

6.4

Modeling the Regional Seismicity

Our regional seismicity model comprises 58 source zones, which were delineated based on the regional seismotectonics and historical seismicity (Fig. 6.1). These source zones capture subduction zones, fore arc and back arc regions, transform zones, and background area. For the regions around Vanuatu, the source zones are based on Suckale and Gr¨ uthal (2009), while for the region around Fiji the zones are based on Jones (1998). We used the same methodology to model the regional seismicity as in Rong et al. (2010) except that here we employed the results from the geodetic modeling to constrain the upper bound seismic moment rate for each source zone. For those subduction segments with possible cascading ruptures, various cascading scenarios were considered using a stochastic modeling scheme (Mahdyiar and Rong, 2006). The rate of subduction mega earthquakes of Mw 8.5–9.0 was modeled by a moment conservation principle as in Rong et al. (2010). The seismicity model can be partially verified by comparing the modeled seismicity rates vs. historical seismicity rates. This has been done for each of the seismic source zones. For example, Figure 6.6 shows the cumulative magnitudefrequency distributions from the model (curve) and historical catalog (dots) for some zones around Vanuatu. The long tails of the curves derive from the moment rate based on the geodetic model that is considerably higher than the historical moment rate. The modeled and historical seismicity rates are also compared at a larger scale. Figure 6.7 shows the excellent agreement between the magnitude-frequency distributions of modeled and historical earthquakes in areas around PNG and Fiji, respectively.

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Fig. 6.6 Cumulative magnitude-frequency distributions from the model (curve) and from the historical catalog (dots) for the source zones 16 (a), 19(b), and 38 (c) around Vanuatu.

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Fig. 6.7 (See color insert.) Cumulative magnitude-frequency distributions from the model (red curve) and historical catalog (blue dots) for five large areas covering PNG, Solomon Islands, Vanuatu, Fiji and Tonga. (a) five large areas; and (b)–(f) magnitudefrequency distributions for five areas.

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Probabilistic Seismic Hazard Maps

The hazard is computed using an earthquake catalog of simulated events consistent with the magnitude-frequency curves of the source zones. The synthetic catalog includes 10,000 realizations of next year seismicity and contains about 7.6 million events with Mw  5.0. We applied a three-dimensional model to capture the spatial distribution of subduction interface, intraplate, and deep earthquakes. We used Benioff zone depth contours to define the subduction interface. Large subduction zone earthquakes are distributed along the subduction slab. Figure 6.8 illustrates the distribution of a sample 50-year simulated catalog. Both the spatial and the depth distributions are consistent with the historical earthquake catalog shown in Figure 6.2.

Fig. 6.8 (See color insert.) A sample 50-year synthetic catalog. Circles are colorcoded by depth and sized by magnitude.

Seven ground motion prediction equations (GMPEs) were used to calculate ground motions. For crustal earthquakes, four Next Generation Attenuation (NGA) relation models of equal weight were used: Boore and Atkinson (2008), Campbell and Bozorgnia (2008), Chiou and Youngs (2008), and Abrahamson and Silva (2008). We adopted Youngs et al. (1997), Atkinson and Boore (2003), and Zhao et al. (2006) to estimate ground motion for subduction and deep earthquakes. Half the weight was given to Zhao et al. (2006), and the other half was split equally between Youngs et al. (1997) and Atkinson and Boore (2003). Due to the lack of GMPEs for oceanic earthquakes, the ground motion for this type of events was calculated using both crustal GMPEs and subduction GMPEs, with each type of GMPEs having half the weight. Given the size of the stochastic catalog, for each type of earthquake, we applied a single weighted average

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GMPE in the PSHA study. Strictly speaking, averaging GMPEs is probabilistically incorrect. However, this method provides in a manageable amount of time a close approximation to the average of the hazard results computed using each GMPE separately. In our model, the geometry of all sources is formulated by a three-dimensional rupture area which assures that the distance calculation in the GMPEs is accurate and realistic. Site conditions have a strong influence on earthquake ground motions and, therefore, on seismic hazard and risk estimates. Shorten et al. (2001) have derived seismic microzonation maps of four capital cities (Suva, Fiji; Port Vila, Vanuatu; Honiara, Solomon Islands; and Nuku’alofa, Tonga) by carrying out microtremor surveys. Given the lack of similar studies for all the other regions in the 15 countries, we derived site-condition maps based on high-resolution SRTM topography data (Jarvis et al., 2008) using the method of Allen and Wald (2009). Estimates of the shear wave velocity in the top 30 m of soil derived from the topography-based approach and the microzonation studies were embedded in the ground motion calculations. The probabilistic seismic hazard analysis (PSHA) included the ground motion intensity measures peak ground acceleration (PGA),

Fig. 6.9 (See color insert.) Map of free surface PGA, including site conditions, with 10% probability of exceedance in 50 years (475-year mean return period) for some of the Pacific Island Countries.

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and 5%-damped linear spectral accelerations at different oscillator periods. As an example of the results obtained, Figure 6.9 shows free surface PGA, including site effects, with 10% probability of exceedance in 50 years (i.e., 475-year mean return period) for some of the PICs.

6.6

Discussion

Figure 6.10 compares the hazard results from this study and the corresponding results from earlier studies. It shows that: – In general, our estimates of the 475-year mean return period rock PGA values are closer to the results from the detailed local studies than to those from the GSHAP study. – Historical seismicity in the neighborhood of the PICs does not explain the very similar 475-year rock PGA estimates for Kiribati, Marshall Islands, Micronesia, Nauru, Tuvalu, Samoa, Timor-Leste, and Tonga in the GSHAP study (Fig. 6.10b) because: (1) Tonga is on the seismically active Tonga subduction zone, while Tuvalu, Kiribati, Marshall Islands, and Nauru are not only far from the regional subduction zones but are also located in seismically inactive areas; (2) Samoa is located at about 130 km north of the northern tip of the Tonga subduction zone; and (3) Timor-Leste is in the middle of the seismically active Timor and Banda Sea trenches. The relative difference in the seismic hazard is correctly reflected in the PGA estimates from this study. – It is important to account for site conditions in seismic hazard analysis. Based on the results from this study, the hazard with site conditions considered is about 20%–30% higher than the rock hazard at the capitals of Vanuatu, Tonga, Timor-Leste and Micronesia, about 50% higher at the capital of Samoa, and more than 80% higher at the capitals of Tuvalu and Kiribati. The seismic hazard maps presented here have sufficient details to be used in local seismic risk studies and were developed using the current state of practice in probabilistic seismic hazard assessment. The large differences between this study and GSHAP can be attributed to the combined impact of differences in earthquake source models, GMPEs, site conditions, and the details of the hazard calculations.

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Fig. 6.10 (See color insert.) Expected PGA (g), with a 10% probability of exceedance in 50 years (475-year mean return period), at the capitals of some of the PICs. The blue and purple bars are PGA values from this study with and without site conditions accounted for, respectively. The red bars are PGA values by GSHAP. The green bars in (a) are the results of local studies (Jones 1998 for Fiji; Suckale and Gr¨ uthal 2009 for Vanuatu).

Acknowledgements The authors would like to thank Dr. Bingming Shen-Tu and Dr. Khosrow Shabestari of AIR Worldwide for constructive discussions. The financial support from World Bank and AIR Worldwide is appreciated.

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Authors Information Y. Rong FM Global, Norwood, MA, USA (formerly at AIR Worldwide, Boston, MA, USA) E-mail: [email protected] J. Park, D. Duggan AIR Worldwide, San Francisco, CA, USA M. Mahdyiar AIR Worldwide, Boston, MA, USA P. Bazzurro I.U.S.S., Pavia, Italy (formerly at AIR Worldwide, San Francisco, CA, USA)