Modelling and inversion of microearthquakes

Modelling and inversion of microearthquakes

Hom Nath Gharti

Dissertation for the degree Doctor of Philosophy

DEPARTMENT OF GEOSCIENCES FACULTY OF MATHEMATICS AND NATURAL SCIENCES UNIVERSITY OF OSLO

July 01, 2011

To my Parents and Grandparents

Contents

Preface

iii

Acknowledgments

iv

List of Publications

vi

1

Introduction

1

1.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Possibilities and challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.4

Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2

Source location and moment-tensor inversion

7

3

Wave propagation and travel times

33

4

Elastoplastic failure

64

5

Multistage excavation

99

6

Concluding remarks

136

6.1

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.2

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

A Softwares packages

140

A.1 Major contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

i

A.1.1 MIGLOC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 A.1.2 SPECFEM3D_SLOPE . . . . . . . . . . . . . . . . . . . . . . . . . . 141 A.1.3 SPECFEM3D_EXCAVATION . . . . . . . . . . . . . . . . . . . . . . 142 A.1.4 TREVERSAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 A.2 Minor contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 A.2.1 SPECFEM3D_SESAME / SPECFEM2D . . . . . . . . . . . . . . . . 142 A.2.2 E3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 A.2.3 FDTIMES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

ii

Preface This PhD project was conducted within the framework of the ACUPS (Active Use of Passive Seismics) project — a competence building project (KMB) of the Research Council of Norway (NFR). This project was mainly funded by the NFR, and supported by the industry partners BP, Statoil, and Total. It was executed in cooperation between NORSAR, the Norwegian Geotechnical Institute (NGI), and the University of Oslo. The PhD fellowship was jointly funded by the NFR (75%) and NORSAR (25%). The development of a 3D spectral-element software package for elastoplastic problems in geomechanics started up when the author was at Princeton University, USA, for a three months external research stay. During that time, the author was partially supported by the Department of Geosciences, Princeton University.

iii

Acknowledgments I would like to thank my supervisors Volker Oye and Michael Roth for their comprehensive guidance, inspiration, and support during this work. They not only guided me through my research but also helped and supported in every possible way to find my path ahead. I am indebted to my supervisor Valérie Maupin for her regular advice. She was greatly interested in my work, and her insightful remarks were very helpful to come up with new ideas. I thank the members of the microseismic group at NORSAR — Daniela Kühn, Peng Zhao, Julie Albaric, and Kamran Iranpour for helpful discussions, especially in our weekly meetings. This work would not have been possible without the friendly and exciting atmosphere at NORSAR. I am grateful to all NORSAR staff. Special thanks to Winnie Lindvik for her great support on practical matters; Hilmar Bungum, Tormod Kværna, Conrad Lindholm, and Johannes Schweitzer for discussions and regular inspiration; Emrah Erduran, Steven Gibbons, Jürg Hauser, Tina Kaschwich, Dominik Lang, and Isabelle Lecomte for helpful discussions. I am also very thankful to Vidar Døhli who always helped me installing and fixing softwares. Many thanks to Eyvind Aker, Fabrice Cuisiat, and the industry partners for helpful feedback on my work during every project meeting. I thank Dani Schmid and his group at PGP (UiO) for helpful discussions on my work. I am grateful to Jeroen Tromp for accepting my request for a three months research stay in his group at Princeton University. Even after this period he provided regular support and important feedback on my work. I greatly benefited from the exciting research atmosphere of his group — Yang Luo, Tarje Nissen-Meyer, Christina Morency, Daniel Peter, and Hejun Zhu. During my visit, I had the opportunity to meet Albert Tarantola, and to attend his lecture on inversion. I am grateful to his important remarks on my work. I am also very thankful to

iv

Dimitri Komatitsch for regular discussions and critical remarks on my work on the spectralelement method. Special thanks to Roland Martin for his feedback on my work. I thank Shawn Larsen for great help. His quick and detailed answers to my questions greatly helped me to understand the finite-difference method. I am also grateful to Jean-Paul Montagner and Vaclav Vavryˇcuk for helpful discussions on time-reversal and moment-tensor inversion. I gratefully acknowledge the access I got to the Titan cluster owned by the University of Oslo and the Norwegian metacenter for High Performance Computing (NOTUR), as well as the Princeton Institute for Computational Science and Engineering (PICSciE), USA. I also want to acknowledge the help received from the Research Computing Services group at USIT, the University of Oslo IT-department. I am also very thankful to Thorvald, Neni, and their families with whom my family and I enjoyed wonderful Norwegian life including cabin tour, hiking to beautiful mountains, and boating at breathtaking fjords. I remain grateful to my families forever for their unconditional support and motivation. My wife Basundhara and son Deven are the sources of my strength that helped me to accomplish this work.

v

List of Publications This thesis consists of an introduction, four reviewed articles and two extended abstracts.

Chapter 2 1. Gharti, H. N., V. Oye, M. Roth, and D. Kühn, 2010, Automated microearthquake location using envelope stacking and robust global optimization: Geophysics, 75, MA27MA46. doi:10.1190/1.3432784. 2. Gharti, H. N., V. Oye, D. Kühn, and P. Zhao, 2011, Simultaneous microearthquake location and moment-tensor estimation using time-reversal imaging: SEG Technical Program Expanded Abstracts, accepted.

Chapter 3 1. Gharti, H. N., V. Oye, and M. Roth, 2008, Travel times and waveforms of microseismic data in heterogeneous media: SEG Technical Program Expanded Abstracts, 27, 13371341. doi:10.1190/1.3059162. 2. Gharti, H. N., V. Oye, M. Roth, and D. Kühn, 2011, Wave propagation modeling based on the spectral-element method – application to microearthquakes and acoustic emissions, to be submitted.

vi

Chapter 4 1. Gharti, H. N., D. Komatitsch, V. Oye, R. Martin, and J. Tromp, 2011, Application of an elastoplastic spectral-element method to 3D slope stability analysis: International Journal for Numerical Methods in Engineering, in revision.

Chapter 5 1. Gharti, H. N., V. Oye, D. Komatitsch, and J. Tromp, 2011, Simulation of multistage excavation based on a 3D spectral-element method: Computers & Structures, in review.

Other publications 1. V. Oye, H. N. Gharti, E. Aker, and D. Kühn, 2010, Moment tensor analysis and comparison of acoustic emission data with synthetic data from spectral element method, SEG Technical Program Expanded Abstracts, 29, 2105-2109. 2. Kühn, D., H. N. Gharti, V. Oye, and M. Roth, 2009, Analysis of mining-induced seismicity using focal mechanism and waveform computations in heterogeneous media. EAGE extended abstracts, Amsterdam.

Presentations 1. Gharti, H. N., V. Oye, M. Roth, and D. Kühn, 2010, Microearthquake location using envelope stacking and robust global optimization – application to mine and rock slope monitoring, Doctoral Student Congress, IPGP, France. 2. Gharti, H. N., J. Tromp, V. Oye, M. Roth, and D. Kühn, 2010, Seismic wave propagation modeling based on the spectral-element method – applications to microearthquakes and acoustic emission, Petroleum Geology and Geophysics seminar, Oslo University.

vii

3. Gharti, H. N., V. Oye, and M. Roth, 2009, Automated event location for noisy microearthquake data using envelope stacking and robust global optimization, Eos Trans. AGU, 90(52), Fall Meet. Suppl., Abstract S32B-04 4. Gharti, H. N., V. Oye, M. Roth, and D. Kühn, 2009, Automated microearthquake location using envelope stacking and robust global optimization, Petroleum Geology and Geophysics seminar, Oslo University. 5. Oye, V. and H. N. Gharti, 2009, Location of microearthquakes in various noisy environments using envelope stacking, Eos Trans. AGU, 90(52), Fall Meet. Suppl., Abstract S32B-1751 6. Kühn, D., H. N. Gharti, V. Oye, and M. Roth, 2009, Automatic determination of full moment tensor solutions from P-wave first motion amplitudes, Passive seismic workshop, 22-23 March 2009, Cypros. 7. Kühn, D., H. N. Gharti, V. Oye, and M. Roth, 2008, Automatic determination of focal mechanisms from P-wave first-motions applied to mining-induced seismicity, presentation at ESC Crete, Greece.

viii

Chapter 1 Introduction 1.1

Background

The seismic wavefield contains information on its source and on the properties of the propagation media. The analysis of seismic data therefore allows us, in principle, to determine source parameters and to image the subsurface. Figure 1.1 shows an example of seismic wavefield propagation in a mine. The seismic energy emitted by the source interacts with the structural and material heterogeneities of the media resulting in a complex wavefield. The wavefield is recorded by a limited number of receivers placed near the top. Since the data recorded by the receivers carries information on source and propagation media, they can be analyzed and interpreted in order to obtain the source location and possibly also the source mechanism. If the spatial distribution of receiver network and seismic sources provides good ray-coverage, even tomographic methods can be applied to obtain properties of the media. Such a process is often referred to as inversion or imaging. On the other hand, if information on the source and the media are known, we can simulate wave propagation in the media and generate synthetic data. Such a process is often referred to as forward modeling. Both the inversion and the forward modeling are important to study subsurface structures or processes. Active or controlled-source seismics is probably the most widely used method to image subsurface structures. In active seismics, sources are artificially generated, for example, by blasts, vibrators, hammer blows. Therefore, the receivers as well as the seismic sources are controlled, and they can be distributed in a geometry best suited to the specific purposes. Since 1

Receivers

12 11 10 9 8 7 6 5 4 3 2 1 0

0.04

0.08 0.12 Time (s)

0.16

0.2

Figure 1.1: An example of seismic wavefield propagation in a mine model. Left: Snapshot of the wavefield. The model consists of mined-out volumes (blue) and ore body (brown) surrounded by the host rock. The source is shown in red. Black, numbered dots represent the receivers. Right: Synthetic seismograms recorded at the receivers. detailed information on the sources (e.g., origin time and source signals) is available, recording parameters for the receivers (e.g., trigger time, sampling interval, and duration of the record) can be precisely tuned to optimize the data acquisition. Additionally, the knowledge of the seismic source signal allows for the removal of its imprint from the recorded data and simplifies the imaging process. Controlled sources have limited mechanisms, and their position is usually restricted to the surface or shallow depths. Hence, the information obtained from active seismics is not always sufficient to illuminate the subsurface. The term ‘passive seismics’ is commonly used as a synonym for small-scale high-resolution seismology. In passive seismics, sources are small earthquakes generated by natural processes, for example, earthquake faulting, volcanoes, and landslides (e.g., Bulut et al., 2009), or induced by human activities, for example, mining and tunneling (e.g., Mendecki, 1997; Shapiro, 2011). Small earthquakes with magnitudes less than 3 are often referred to as microearthquakes (e.g., Lee and Stewart, 1981; Bohnhoff et al., 2010). The type, occurrence, and location of these microearthquakes are unknown beforehand. Hence, the main goal of passive seismic monitoring is to determine the source parameters, providing information on the process that triggered the microearthquake, and to deduce the structural properties of the subsurface. The locations of the microearthquakes are indications of potential weakness zones or the volumes where sudden

2

release of energy occurs. Similarly, the magnitudes and mechanisms of the microearthquakes provide information on local dynamic changes including in-situ stress state, stress drop, and stress evolution. Consequently, the interpretation of these events often yields important information in several problems, for example, location of potentially unstable zones in mines (e.g., Mendecki, 1997; Young et al., 1992), monitoring CO2 sequestration (e.g., Maxwell et al., 2004), and imaging the internal structure of geothermal (e.g., Phillips et al., 1997) and hydrocarbon reservoirs (e.g., Rutledge et al., 1998; Oye and Roth, 2003).

1.2

Possibilities and challenges

Passive seismic methods offer a wide range of possibilities but also pose several challenges. Seismic events typically exhibit a power law distribution between frequency of occurrence and magnitude, which means that small earthquakes occur in large numbers. In addition, the relation between magnitude (M0 ) and corner frequency (fc ) is usually given by M0 ∼ fc−3 (e.g., Aki, 1967; Hanks, 1977), which implies that smaller earthquakes have higher corner frequencies and consequently higher frequency content. Therefore, small-scale structural or material heterogeneities will have a strong influence on the wavefield. In working environments with high ambient noise and natural or man-made structural heterogeneities, many of the seismic signals have low signal-to-noise ratios (SNR) and complicated waveforms (e.g., Oye and Roth, 2003). A large amount of data with complicated signals is a challenge for processing. Therefore, automated and robust data-processing methods are desirable. Simulation of wave propagation provides valuable information on the interaction of the wavefield with structures and media including wave scattering, wave conversion, etc. Synthetic data are also important to assess applicability and reliability of source-location and inversion algorithms. Furthermore, we can compute Green’s functions which are required for momenttensor inversion. However, the simulation of wave propagation in such environments is often challenging. Due to the natural or man-made structural complexities, accurate geometry mapping is very difficult, and the presence of strong velocity contrasts causes problems for the numerical stability. Microearthquake monitoring can be used to determine potentially unstable or failure-prone 3

zones in the subsurface. Moreover, the modeling of a failure process itself can provide complimentary information on the unstable zones. However, the failure phenomenon is an inherently nonlinear process, which is complicated and demands high computational costs. Another class of problems in geomechanics in which the model geometry changes with time is represented, for example, by mining, (de)glaciation, volcanic eruptions, and dike intrusions. In such problems, stress redistribution due to the geometry modification with time has a considerable consequence on response and stability of the models. Therefore, a multistage simulation of such problems provides progressive information during the process. All of these methods are computationally intensive. Hence, high-performance computing tools are often necessary.

1.3

Objectives

The main objective of this work is to develop robust and efficient methods and tools to process microearthquake data, and to study microearthquake mechanisms and failure processes. More specifically, the following four objectives are identified: 1. to develop a robust and efficient source-location method for microearthquake data, and to investigate the possibility of a moment-tensor inversion simultaneously, 2. to compute travel times of seismic phases and full waveforms, and study the wavefield interaction in complex heterogeneous models of various scales, using finite-difference and spectral-element methods, 3. to implement a 3D spectral-element method to problems of elastoplastic failure in geomechanics, and apply this method to slope stability analysis, and 4. to implement a 3D spectral-element method for multistage excavation, and demonstrate an application to a mine.

4

1.4

Structure of the thesis

This thesis consists of six chapters including this introductory chapter. Chapter 2 contains two articles related to the work on source location and moment-tensor inversion. The first article introduces a new migration-based source location method for microearthquake data with low signal-to-noise ratio (SNR). The second article presents the time-reversal imaging for simultaneous source location and moment-tensor inversion. Two articles related to travel time and full waveform computation are included in Chapter 3. One article presents the results of first-arrival time and full waveform computations in an underground ore mine, namely the Pyhäsalmi mine in Finland. The first-arrival times are computed using a finite-difference eikonal solver, and the full waveforms are computed using a viscoelastic finite-difference code. The other article presents the results of full waveform computations in various environments: the Pyhäsalmi mine, a rock slope at Åknes in Norway, and a weakly anisotropic sample used in acoustic emission experiment. These latter computations are based on the spectral-element method. Chapter 4 comprises an article related to elastoplastic failure in geomechanics. This article describes the implementation of the 3D spectral-element method for elastoplastic problems, and demonstrates several examples of slope failure modeling including an earthen dam and a rock slope. Chapter 5 contains a paper on the implementation of the 3D spectral-element method to multistage excavation. It illustrates several examples of multistage excavations in both elastic as well as elastoplastic materials. This article also demonstrates the multistage excavation in the Pyhäsalmi mine. Chapter 6 concludes the thesis with final remarks and an outlook for future work. Finally, a list of all software packages is presented, and briefly described in the Appendix.

References Aki, K., 1967, Scaling law of seismic spectrum: Journal of Geophysical Research, 72, 1217– 1231. 5

Bohnhoff, M., G. Dresen, W. Ellsworth, and H. Ito, 2010, Passive seismic monitoring of natural and induced earthquakes: Case studies, future directions and socio-economic relevance, in New Frontiers in Integrated Solid Earth Sciences, (International Year of Planet Earth): Springer, 261–285. Bulut, F., M. Bohnhoff, W. L. Ellsworth, M. Aktar, and G. Dresen, 2009, Microseismicity at the North Anatolian Fault in the Sea of Marmara offshore Istanbul, NW Turkey: Journal of Geophysical Research, 114, B09302. Hanks, T. C., 1977, Earthquake stress drops, ambient tectonic stresses and stresses that drive plate motions: Pure and Applied Geophysics, 115, 441–458. Lee, W. H. K., and S. W. Stewart, 1981, Principles and applications of microearthquake networks: Academic Press, New York. Maxwell, S. C., D. J. White, and H. Fabriol, 2004, Passive seismic imaging of CO2 sequestration at Weyburn: SEG Technical Program Expanded Abstracts, 23, 568–571. Mendecki, A. J., 1997, Seismic monitoring in mines: Chapman and Hall. Oye, V., and M. Roth, 2003, Automated seismic event location for hydrocarbon reservoirs: Computers & Geosciences, 29, 851–863. Phillips, W. S., L. S. House, and M. C. Fehler, 1997, Detailed joint structure in a geothermal reservoir from studies of induced microearthquake clusters: Journal of Geophysical Research, 102, 11745–11763. Rutledge, J. T., W. S. Phillips, and B. K. Schuessler, 1998, Reservoir characterization using oilproduction-induced microseismicity, Clinton county, Kentucky: Tectonophysics, 289, 129– 152. Shapiro, S. A., 2011, Microseismicity: a tool for reservoir characterization: EAGE Publications BV. Young, R. P., S. C. Maxwell, T. I. Urbancic, and B. Feignier, 1992, Mining-induced microseismicity: Monitoring and applications of imaging and source mechanism techniques: Pure and Applied Geophysics, 139, 697–719.

6

Chapter 2 Source location and moment-tensor inversion

7

GEOPHYSICS, VOL. 75, NO. 4 共JULY-AUGUST 2010兲; P. MA27–MA46, 24 FIGS., 3 TABLES. 10.1190/1.3432784

Automated microearthquake location using envelope stacking and robust global optimization

Hom Nath Gharti1, Volker Oye1, Michael Roth1, and Daniela Kühn1

arrival times 共computed using an eikonal solver兲 and stacked for a predefined time window centered on the arrival time of the corresponding phase. This was done for each component and phase individually, and the squared sum of the stacks was defined as the objective function. We applied a robust global optimization routine called differential evolution to maximize the objective function and thereby locate the seismic event. Our source location method provides a complete algorithm with only a few control parameters, making it suitable for automatic processing. We applied this method to single and multicomponent data using P and/ or S phases. We conducted controlled tests using synthetic seismograms contaminated with a minimum of 30% white noise. The synthetic data were computed for a complex and heterogeneous model of the Pyhäsalmi ore mine in Finland. We also successfully applied the method to real seismic data recorded with the in-mine seismic network of the Pyhäsalmi mine.

ABSTRACT Most earthquake location methods require phase identification and arrival-time measurements. These methods are generally fast and efficient but not always applicable to microearthquake data with low signal-to-noise ratios because the phase identification might be very difficult. The migration-based source location methods, which do not require an explicit phase identification, are often more suitable for such noisy data. Whereas some existing migration-based methods are computationally intensive, others are limited to a certain type of data or make use of only a particular phase of the signal. We have developed a migration-based source location method especially applicable to data with relatively low signal-to-noise ratios. We projected seismograms onto the ray coordinate system for each potential source-receiver configuration and subsequently computed their envelopes. The envelopes were time shifted according to synthetic P- and S-wave

INTRODUCTION Microseismic monitoring is well established in many fields, and its application helps to mitigate risk and optimize operation or production. For example, it is used to assess potentially unstable zones in mines 共Young et al., 1992; Mendecki, 1997兲, to image CO2 sequestration 共Maxwell et al., 2004兲, and to study the internal structure of geothermal 共Phillips et al., 1997兲 and hydrocarbon reservoirs 共Rutledge et al., 1998兲. Seismic events typically exhibit a power law distribution between the frequency of occurrence and magnitude. That means microseismic events occur in large numbers. In working environments with high ambient noise and natural or man-made structural heterogeneity, many of the seismic signals have a low signalto-noise ratio and complicated waveforms. Therefore, an automatic and robust source location algorithm is desirable. Standard seismic source location methods generally exploit only the phase arrival times. Therefore, they require explicit phase identi-

fication and measurement of arrival times. These methods usually are based on Geiger’s scheme 共Geiger, 1912兲, which minimizes the arrival-time residual of seismic phases using an iterative procedure, e.g., the Gauss-Newton iteration. Several deterministic and probabilistic variants of these methods are illustrated in Thurber and Rabinowitz 共2000兲. Probabilistic methods also take into account possible errors in measurement and provide the source location with uncertainty 共Tarantola and Valette, 1982; Sambridge and Kennett, 1986; Lomax et al., 2000; Husen et al., 2003兲. The location error due to uncertainties in phase picks and velocity structure is significantly reduced by waveform crosscorrelation, double differencing, or joint hypocenter determination 共Pujol, 2000; Waldhauser and Ellsworth, 2000; Schaff et al., 2004兲. However, the phase identification and accurate determination of the arrival time can be difficult for data with a relatively low signal-to-noise ratio. Alternative location methods, which exploit the waveforms, are based mainly on Kirchhoff migration and do not require the phase

Manuscript received by the Editor 25 September 2009; revised manuscript received 15 January 2010; published online 2 August 2010. 1 NORSAR, Kjeller, Norway. E-mail: [email protected]; [email protected]; [email protected]; [email protected]. © 2010 Society of Exploration Geophysicists. All rights reserved.

MA27

Downloaded 03 Aug 2010 to 193.156.134.55. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

MA28

Gharti et al.

identification. The objective function is generally formulated as a superposition of signals from all receivers, and it is optimized toward its maximum value to locate the source. Backward extrapolation methods 共e.g., McMechan, 1982; Chang and McMechan, 1991; Gajewski and Tessmer, 2005; Fink, 2006; Lu, 2007兲 use the finitedifference method to compute the wavefield considering time-reversed seismograms as sources. The actual source location is estimated from the best energy focus. This method is computationally intensive, and the energy focusing can be ambiguous for noisy data and very heterogeneous models. Another class of migration-based methods is based on the idea of delay and sum of the traces to enhance the signal-to-noise ratio, for example, beam forming 共e.g., Arnold 1977; Ringdal and Kværna, 1989兲. Similar to this idea, Kao and Shan 共2004兲 propose the sourcescanning algorithm 共SSA兲 to map the distribution of seismic sources in space and time. The SSA, however, sums only absolute amplitudes at computed arrival times, instead of summing the entire trace. An image is constructed computing the brightness function at each point in space and time for certain regions of interest. The brightness function is obtained by stacking the absolute amplitudes of normalized seismograms recorded at all stations across a predefined time window. The position of the time window is constrained by the arrival time computed for a certain phase having the largest amplitude 共usually the S phase兲. The brightest point in the image represents the source location. Baker et al. 共2005兲 suggest a similar method, especially designed to run continuously in real time. Unlike the SSA, this method uses the envelopes of seismograms, and the arrival times are computed using a P-wave velocity model. An image at a certain time is constructed summing the amplitudes of expected P-wave signals for each trial source. The point showing the maximum value represents the coherent summation of the signals and hence the source location. The algorithm of Baker et al. 共2005兲 is designed so that the recordings before and after the current time step do not need to be stored. Because such an algorithm exploits only the P-wave signals, this method is favorable for early warning and rapid response systems. Kao and Shan 共2007兲 also implement P-wave envelopes in the SSA to identify the earthquake rupture plane. Drew et al. 共2005兲 propose another method for continuous microseismic mapping in real time and apply it to a vertical array of three-component seismic sensors. This method stacks the product of

Q

L T

P and S signal-to-noise ratios at computed arrival times 共corresponding to a trial source兲 to obtain a so-called coalescence map in 4D space. The event is said to be detected if the maximum value of the coalescence function exceeds a certain threshold. Subsequently, the events are located considering the polarization of P-wave arrivals. Rentsch et al. 共2007兲 propose a location method for three-component recordings. This method requires the preliminary identification of the time interval, which contains the P-wave signal. Subsequently, the P-wave polarization is determined to perform the initial-value ray tracing. Each ray is represented by a corresponding Gaussian beam. The energy contribution of all the beams at a specific point 共a trial source兲 is stacked. The summation of the stacked energy over the previously determined time interval gives the total stacked energy independent of the origin time. The point with the maximum value of total stacked energy corresponds to the source location. Because of the stacking, an accurate identification of the P phase is not required. Due to the usage of Gaussian beams, stacking is confined only to physically relevant regions allowing a fast computation. However, the application of this method is limited to three-component data. In addition, the back tracing of rays through a large and heterogeneous model involves high computational cost, and the accuracy of the back tracing strongly depends on the quality of the polarization analysis. Migration-based methods are often computationally expensive. The objective function is generally not varying smoothly and often possesses multiple peaks. Therefore, gradient-based algorithms are not useful to identify the global peak. Hence, in most migrationbased methods, the source usually is located by computing the objective function in the entire region of interest. In our study, we present a new method of source location, especially targeting low-quality signals. The method is migration-based, thereby avoiding explicit phase identifications and arrival-time measurements. This method uses the idea of delay and sum of the traces and stacks a certain portion of the traces similar to the SSA 共Kao and Shan, 2004兲. We use seismogram envelopes and take advantage of Pand S-wave energy. Original traces are rotated onto the local ray coordinate system to optimize the signal-to-noise ratio. In addition, we implement a robust global search algorithm, namely, the differential evolution 共DE兲 algorithm, to locate the source. By using both rotation and stacking, we enhance the signal-to-noise ratio as much as possible, without the requirement of three-component data and relatively high P-wave signal quality 共needed for polarization analysis, e.g., in the Gaussian beam method兲共Rentsch et al., 2007兲, and using the DE enables us to locate the source in a robust manner, even in the case of highly nonlinear objective functions. In the following, we explain the formulation of the objective function, summarize the DE algorithm, and demonstrate the application using synthetic and observed data sets in a mine, which represents a very heterogeneous and complex environment.

METHODOLOGY Formulation of the objective function z Source y x

Figure 1. The LTQ system for a particular source-receiver configuration.

We denote the east, north, and vertical component of the seismograms recorded at the ith receiver as f xi 共t兲, f iy共t兲, and f zi 共t兲, respectively. To increase the signal-to-noise ratio, we project the seismograms onto the local ray coordinate system denoted by LTQ 共Figure 1兲. The L-axis is oriented along the ray direction. The T- and Q-axes are mutually perpendicular and situated on a plane perpendicular to the ray direction. For simplicity, we orient the T-axis horizontally.


Automated microearthquake location Therefore, the vertical component of the seismogram has no contribution to the T-component. The P-wave signal is polarized along the L-axis, and the S-wave signal is polarized on the TQ-plane. The LTQ system is formed for every trial source-receiver configuration based on the geometric ray direction. The seismograms obtained by the rotation of f ix共t兲, f iy共t兲, and f zi 共t兲 onto the LTQ system are represented by f iL共t兲, f iT共t兲, and f Qi 共t兲, respectively. For the case of single-component data, we project the seismogram onto the LTQ system as well. The vertical component can contribute to only the L- and Q-components, and the horizontal components can contribute to the L-, T-, and Q-components. Due to the radiation pattern of the source, the amplitudes at different receivers might cancel out during stacking of the seismograms. To avoid this, we use the envelopes of the rotated seismograms, which are computed via Hilbert transform. By using envelopes, we obtain a more stable objective function, despite losing some resolution. If we compute the Hilbert transform after the LTQ rotation, the computational cost is significantly increased because we need to compute the Hilbert transform every time we make the LTQ rotation. Due to linearity of the Hilbert transform, however, we can compute the Hilbert transform only once at the beginning and make the LTQ rotation afterward, saving huge computational costs. We stack the area under the envelope for the L-, T-, and Qcomponents individually over a predefined time window. Subsequently, the objective function is computed as the sum of the squares of the stacked components expressed as

S共x兲 ⳱

冢

toⳭtiPⳭwP

N

冕

兺

i⳱1

Ⳮ

冢

FLi 共t兲dt

toⳭtPi N

兺

i⳱1

toⳭtiSⳭwS

冕

冣冢冣 2

toⳭtSi

Ⳮ

N

兺

i⳱1

toⳭtSi ⳭwS

冕

toⳭtSi

FTi 共t兲dt

冣

2

2

FQ i 共t兲dt

,

共1兲

where FLi 共t兲, FTi 共t兲, and FQi 共t兲 are the envelopes of the rotated seismograms for the ith receiver; N is the total number of receivers; tPi and tSi are traveltimes for P and S phases computed at the ith receiver for a trial source located at x ⳱ 共x,y,z,0兲; to denotes the origin time measured relative to the start time of the seismogram; and wP and wS are the durations of the P and S time windows, respectively. To obtain the objective function independent of the origin time, we have to integrate the objective function S共x兲 in equation 1 within the range of the certain origin time. The first term on the right-hand side represents the contribution from the L-component 共i.e., mainly P-wave contribution兲, and the second and third terms are contributions from the T- and Q-components 共i.e., mainly S-wave contribution兲. We do not scale the envelopes with respect to geometric spreading or other measures and keep the amplitudes as they are. For homogeneous models, we compute P- and S-phase arrival times and ray directions analytically. For heterogeneous and complex models, we compute the traveltimes numerically using an eikonal solver based on Huygens’ principle 共Podvin and Lecomte, 1991兲. For isotropic media, the ray direction at a particular receiver is the outward normal to the wavefront at the receiver. Both traveltimes and ray directions can be stored in lookup tables. Alternatively, one can use ray tracing to generate the lookup tables 共e.g., Vinje et al., 1996兲.

MA29

Global search algorithm A brute-force method to determine the event location is to conduct a full grid search, i.e., to compute the objective function for all grid points in the volume of interest. This is too time-consuming to be used in a routine localization, but it reveals properties of the objective function. The number and magnitude of local extrema and the roughness of the objective function depend on the receiver distribution, complexity of the velocity model, noise contamination of the data, and presence of secondary phases. Gradient-based methods require a start model that is relatively close to the global extremum, which might be difficult to estimate beforehand. Therefore, we decided to use a global optimization algorithm, and found the differential evolution 共DE兲 method to be suitable for our purpose. The DE, which was introduced by Storn and Price 共1995兲, is an evolutionary algorithm for global optimization. The key difference between the DE and other evolutionary algorithms is its differential mutation scheme. The method is simple and robust, and can be adapted efficiently to parallel processors. Storn and Price 共1997兲 demonstrate that the DE converges faster and with more certainty than many other acclaimed global optimization methods. It has been applied to several optimization problems 共Price et al., 2005兲. Ruzek and Kvasnicka 共2001兲 apply the DE for the nonlinear problem of earthquake hypocenter location based on the minimization of time residuals, and show that the method is highly reliable. In the following, we briefly summarize the DE algorithm. For the general explanation, we denote the size of the parameter vector as M 共in our case, M ⳱ 4, because we estimate x, y, z, and to兲. The DE algorithm consists of four major steps: initialization, mutation, crossover, and selection.

Step one: Initialization The initial population consists of a set of parameter vectors randomly distributed over the given parameter ranges. The parameter ranges depend on the region of interest. The size of the population is normally taken as 5 to 10 times the size of a parameter vector, and it remains unchanged for all generations. The initial population represents the first generation of the evolution.

Step two: Mutation A new set of trial parameter vectors is generated using a differential mutation from the current population. Three vectors are randomly chosen from the existing population, and a mutant vector is computed perturbing one vector with a scaled difference of the other two:

v p ⳱ x p1 Ⳮ F共x p2 ⳮ x p3兲,

共2兲

where v p represents the pth mutant vector; and x p1, x p2, and x p3 represent the three randomly chosen vectors so that p, p1, p2, and p3 are distinct. The scaling factor F denotes the mutation factor, and its value is taken between 0 and 2. This factor controls the rate at which the population evolves.

Step three: Crossover To increase the diversity of the parameter vectors, each parameter vector of the current population is crossed with a corresponding mutant vector. The resulting population is called the competing population. The elements of a parameter vector in the competing population


MA30

Gharti et al.

are randomly inherited from both the mutant vector and current parameter vector based on the predefined crossover probability as expressed below:

um ⳱

再

vm If randm ⱕ C or m ⳱ mrand

xm otherwise

冎

,

共3兲

where um, vm, and xm represent the mth element of a parameter vector in the competing population, mutant population, and current population, respectively. Each mth element of a mutant vector is assigned a probability randm randomly chosen between 0 and 1, and an index mrand, also randomly chosen between 1 and M. Those elements of a mutant vector whose probability 共randm兲 do not exceed the crossover probability C, or have the random index 共mrand兲 equal to the running index 共m兲, are inherited to the competing population. Other elements of a competing vector are inherited from the parameter vector of the current population. The crossover probability controls the fraction of the elements in the mutant vectors to be carried over into the competing population. The condition m ⳱ mrand ensures that a potential duplicate of the current population does not occur within the competing population.

the competing population and the current population one by one. The parameter vector with a higher value of the objective function becomes a member of the next generation. Steps two through four are repeated until a stop criterion is reached. Different stop criteria can be used according to the problem setting, e.g., maximum number of generations or variation 共i.e., standard deviation兲 of the population. When the variation becomes insignificantly small, the inversion has converged. It is important to choose appropriate values of DE control parameters 共i.e., size of population, mutation factor, and crossover probability兲 for the optimum convergence-speed trade-off. The convergence is more likely with a larger size of the population and a smaller value of the mutation factor, but this increases the computational cost. A higher value of crossover probability usually increases the convergence speed. Therefore, some parameter tuning is necessary depending on the nature of the objective function. Recommended values of control parameters are a population size of 10 times the size of a parameter vector, a mutation factor of 0.5, and a crossover probability of 0.9 共Price et al., 2005兲. If the objective function contains a very large number of local peaks, a larger population size and a smaller mutation factor are recommended.

CONTROLLED TESTS

Step four: Selection Finally, the parameter vectors for the next generation are selected by comparing the objective functions for each parameter vector in

a)

b)

x y z

To verify the described methodology, we conducted several tests using a simple homogeneous velocity model. Because the tests were successful, we applied the method to synthetic data computed for the Pyhäsalmi ore mine. Located in central Finland, this active mine consists of a zinc-copper ore body extending down as low as 1.4 km 共Figure 2a兲. Due to strong heterogeneity and structural complexity, this model poses a challenging task for a source location algorithm. The in-mine seismic network comprises 18 geophones, out of which 6 共geophone numbers 1, 5, 9, 13, 17, and 18兲 are three-component instruments 共Puustjärvi, 1999兲. For this mine, we have a detailed 3D velocity model and observed microseismic data from the in-mine network at our disposal 共for more details on the mine and the microseismic event characteristics, see Oye et al., 2005兲. For the computation of synthetic data, we used the idealized model as shown in Figure 2b and Table 1. The detailed model of the mine allowed us to compute relatively realistic synthetic seismograms. We computed two sets of synthetic data. First we computed synthetic seismograms using the 3D viscoelastic finite-difference code E3D from Larsen and Grieger 共1998兲. These data will be inverted for the source location. Next we generated traveltime lookup tables using a 3D finite-difference eikonal solver 共Podvin and Lecomte, 1991兲. These lookup tables will be used to estimate the traveltimes for a trial source during the inversion. We computed synthetic seismograms for a strike-slip point source at x ⳱ 261 m, y ⳱ 330 m, and z ⳱ 380 m 共Figure 2b兲. For the source, we chose a moment magnitude of ⳮ0.59 Mw 共corresponding to a seismic moment of 1.62e Table 1. Seismic properties of the mine model.

Figure 2. 共a兲 Structural model of Pyhäsalmi ore mine with surrounding infrastructure: the zink/copper ore body is shown in pink/brown, access tunnels are shown in yellow, the elevator shaft is shown in dark blue, and seismic stations are numbered. Passage for the quarried ore is marked by KN1. 共b兲 Idealized velocity model 共see Table 1兲 used to generate the synthetic data: stopes 共i.e., mined-out cavities兲 are shown in blue, the ore body is in brown, the host rock is in gray, and geophones are shown in green and numbered.

Air Rock Ore

VP 共m/s兲

VS 共m/s兲

300 6000 6300

0 3460 3700


␳ 共kg/ m3兲 1.25 2000 4400

Automated microearthquake location Ⳮ 8 Nm兲 and a dominant frequency of 250 Hz 共a Ricker wavelet with an 0.004-s dominant period兲 to reproduce waveforms of a particular event that occurred in the mine, and used a sampling time interval of 0.0001 s. We clearly observed complicated wavefield interactions with the strong heterogeneities caused by voids 共Figure 3a and b兲. Nevertheless, the traveltimes computed by the eikonal solver reasonably matched with the respective arrivals in the wavefield computed by E3D for this mine model 共for more details, see Gharti et al., 2008兲. Further, we observed a complex waveform and a strong coda 共Figure

MA31

4a-c兲. Evidently, the voids strongly interact with the wavefield, thereby generating complex secondary arrivals. To save computational costs while generating the lookup tables, we applied the principle of reciprocity, i.e., we considered the receivers to be the sources and stored the traveltimes at all grid points 共which will serve as potential source locations in the inversion兲. The ray direction for a trial source is computed assuming undistorted wavefronts 共i.e., straight raypaths兲. This assumption is acceptable for the Pyhäsalmi mine model. The size of the heterogeneities in the

a)

z (m)

a)

b)

y (m) x (m)

z (m)

b)

c)

y (m) x (m)

Figure 3. Wavefield snapshot at 0.05 s. Superimposed is the first-arrival isochron of 0.05 s shown in green. Solid objects represent the ore body in brown and mined-out voids in blue. 共a兲 P-wave potential and P-wave first-arrival isochron. 共b兲 S-wave potential and S-wave first-arrival isochron. The wavefield is computed using E3D, and the first-arrival isochron is computed using the eikonal solver.

Figure 4. Synthetic seismograms computed for the idealized model of the Pyhäsalmi ore mine: 共a兲 east component, 共b兲 north component, and 共c兲 vertical component. Superimposed are P onsets shown in green and S onsets shown in red for the estimated source location. Seismograms are normalized to their maximum amplitudes.


MA32

Gharti et al.

mine 共stopes, access tunnels, and so on兲 is in the same order or smaller than the wavelength of the typical microseismic signals we observed. The scattering effect of the heterogeneities is therefore more pronounced on the amplitudes and complexity of the wavefield. It is less pronounced on the traveltimes because the wavefront heals quickly after passing a void due to contributions of diffracted waves. For the computation of the objective function, we considered a time window length of 0.008 s, which is equal to twice the dominant period. For simplicity, we used the same time window length for P and S signals. To visualize the properties of the objective function, we computed it for the entire volume on a 1-m sampled grid for a true origin time. The global maximum is clearly visible as a red sharp peak and accurately coincides with the true source location 共Figure 5a-c兲. To display the variation of the objective function with respect to the origin time, we calculated in addition the objective function at the true source location for a time window of Ⳳ0.04 s around the true origin time 共Figure 5d兲. The global maximum is accurately located at the true origin time 共i.e., to ⳱ 0 s兲.

Some trial runs were necessary to choose the optimum control parameters for the DE inversion. We chose a population size of 60, a mutation factor of 0.4, and a crossover probability of 0.9. Compared with the recommended values of DE control parameters, we prefer a larger population size and a smaller mutation factor to make the algorithm more robust. We defined the parameter range for the spatial coordinates from 100 to 600 m and for the origin time to from ⳮ0.2 to 0.1 s 共relative to the start time of the seismogram section兲. Intentionally, we opted for a very large parameter range to demonstrate the reliability of the search algorithm. We discretized the space and time with 1 m and 0.0001 s, respectively, yielding a total of more than 3.7e Ⳮ 11 potential parameter vectors in the 4D space. As stop criteria, we chose a maximum of 50 generations, and a standard deviation of the last generation of 2 m and 0.0002 s for location and origin time, respectively. It took only 23 generations to achieve convergence, computing the objective function for only 1408 parameter vectors. Both P- and S-wave onsets corresponding to the best source location accurately

b)

a)

y (m)

z (m) y (m)

x (m)

c)

d)

z (m) x (m)

to (s)

Figure 5. Objective function computed on cross sections through the true source location: 共a兲 xy-plane, 共b兲 yz-plane, and 共c兲 zx-plane. The objective function was computed at the true origin time. The color scale represents the objective function value. The spot with the maximum value marks the best source location. The true source location is shown in green. 共d兲 Objective function computed at the true source location for a time window of Ⳳ0.04 s around the true origin time. The objective functions are normalized to their maximum values.


Automated microearthquake location matched with the onsets of the waveforms 共Figure 4a-c兲. Figure 6a-d shows the distribution of all trial locations. The parameters are uncorrelated, and the source region marked by the highest values is focused. Such a distribution indicates a reliable source location with a good resolution. An unreliable source location would result in a randomly scattered distribution having no clear maximum. These snapshots also replicate the major features of the objective function 共Figure 5a-c兲 with significantly fewer sampling points. Furthermore, closer to the final source location, the model space is sampled more closely. In other words, the DE provides both the source location and image of the potential source region with high resolution in the proximity of the source, while reducing the compu-

0 100 200 300 400 500 600 600 500 400 300 y (m) 200100

Figure 6. All trial locations visualized in 共a兲 3D space, 共b兲 xy-plane, 共c兲 yz-plane, and 共d兲 zx-plane. Colors represent the value of the objective function 共high values in red and low values in blue兲. The spot with the maximum value represents the best source location.

500 400 300 200

0 0

600 500 300400 100 200 x (m)

100 100

200

200

300

300

z (m)

d) 100

400

400

500

500

600 100

200

300

400

500

200

300

600 100

600

200

300

a)

1

b)

1

5

Generation number

5

10 15 20

600

400

500

600

Figure 7. All generations plotted against 共a兲 parameter x, 共b兲 parameter y, 共c兲 parameter z, and 共d兲 parameter to. Colors represent the value of the objective function 共high values in red and low values in blue兲.

10 15 20

200

300

400

500

23 100

600

200

300

400

500

600

c)

1

d)

1

5

Generation number

y (m)

Generation number

x (m)

5

10 15 20 23 100

500

x (m)

Generation number

y (m)

23 100

400

x (m)

c) 100

z (m)

tational cost significantly. Initially, the DE algorithm samples the complete parameter ranges. Therefore, the population range is large and similar to the parameter range in the first generation. In later generations, the population range decreases 共Figure 7a-d兲. The narrow population range toward newer generations indicates a finer sampling within the vicinity of the best source location. If the population range remains relatively wide from generation to generation, it indicates either that no source can be located with the given set of input data 共seismograms兲, or that the number of generations needs to be increased. The normalized standard deviation of the population generally decreases 共Figure 8a兲, and the higher the decreasing rate is, the faster

b) 600

y (m)

z (m)

a)

10 15 20

200

MA33

300

400

z (m)

500

600

23 –0.2

–0.1

0

0.1

to (s)


MA34

Gharti et al.

the convergence is. The convergence rate depends not only on the objective function but also on the DE control parameters. All parameter vectors tend to increase their fitness 共objective function兲 in the next generation. The mean objective function shows a smooth increasing trend, unlike the irregular trend of the best objective function 共Figure 8b兲. At the convergence, all parameter vectors eventually attain a fitness close enough to that of the best parameter vector. Figure 9 shows the envelopes of the LTQ seismograms corresponding to the best source location. The LTQ transformation clearly separated the P and S components as indicated by the filled areas. Variations in the relative strength of the P- and S-wave envelopes are due also to the radiation pattern of the strike-slip source. The remaining signal energy, mainly concentrating in the S-wave coda, represents primarily scattered energy from the cavity walls 共see also Figure 3兲. The final result of the DE inversion is listed in Table 2. The result correctly matches with the true source location. To examine the influence of P versus S stacks in the location result, we also applied the DE inversion using only the S stack 共last two terms in equation 1兲 and only the P stack 共first term in equation 1兲, and listed the results in Table 2. The result from S stacking is correct and almost identical to the result from the combined stacking, whereas stacking only P phases provides a completely wrong location. In our seismograms, the S signal is stronger than the P signal. As we move the time windows through the entire traces, some S signals might contribute to the P stack, and vice versa. The contribution of the weaker phase to the stronger phase stacking is insignificant. On the contrary, the contribution of the stronger phase to the weaker

phase stacking can be significant, which might result in a false position of the maximum. This is confirmed by the objective functions computed for three cases 共Figure 10a-c兲. The P stack results in the wrong maximum attributed to the undesirable contribution from the stronger S signals. The S stack provides the correct location; the undesirable contribution from the weaker P signals does not have a sig-

a)

b)

a) x y z to

c) Generation number

S(x)

b)

Generation number

Figure 8. 共a兲 Normalized standard deviations and 共b兲 objective function S共x兲 plotted against the generation number.

Figure 9. Time-shifted envelopes 共for P and S arrival times兲 of rotated seismograms corresponding to the estimated source location: 共a兲 L-component, 共b兲 T-component, and 共c兲 Q-component. Stacked areas are shown as filled regions 共L-component in green, and T- and Q-components in red兲. Superimposed are P onsets shown in green and S onsets shown in red. Envelopes are normalized to their maximum amplitudes.


Automated microearthquake location

MA35

computed as 30% of the peak amplitude observed in all seismograms. That means we add an equal noise level to all seismograms, which is what we generally observe in the mine. Hence geophones that are close to the source will be disturbed with somewhat more than 30% noise 共relative to the traces’ peak amplitude兲, whereas faraway geophones will be superimposed with noise that is several times larger than their peak amplitudes. The noises on the individual geophones are uncorrelated. These noise-contaminated seismograms are band-pass filtered from 20 to 400 Hz 共Figure 11a-c兲. The phase identification is very difficult in these seismograms. Even with this high noise level, the global maximum is clearly visible in the objective function 共Figure 12a-c兲. The variation of the objective function as computed for a time window of Ⳳ0.04 s around the true origin time also shows a clear global maximum that coincides with the true origin time 共Figure 12d兲. Compared with the previous case 共Figure 5兲, more local peaks are visible due to the added noise, and only a robust global search algorithm can recover this maximum. Some distinct dark blue spots near the source are associated with the voids within the mine model 共Figure 12b and c兲. These

nificant effect. On the other hand, the combined stacking also gives the correct location because the objective function is dominated by the stronger S phases. To have a crude idea of how much the location is influenced by errors in the velocity model, we performed the DE inversion using an equivalent homogeneous velocity model of VP ⳱ 5500 m / s and VS ⳱ 3500 m / s 共Oye et al., 2005兲, and obtained a reasonably good location 共Table 2兲. A fairly good agreement between the location results of the homogeneous and the heterogeneous model can be explained as follows. As mentioned earlier, for this particular mine, the scattering effect of the heterogeneities is more pronounced on the amplitudes and complexity of the wavefield 共see Figures 3 and 4兲, and it is less pronounced on the traveltimes. As our method avoids the phase identification, the complexity of the signals does not significantly affect the location accuracy. The main strength of our inversion procedure is to locate microearthquakes even using the data that are strongly disturbed with noise. To prove this point, we take the same synthetic data as above and add white noise to the original seismograms. The white noise is

Table 2. Location results and comparison with true source location: Ng and Nf represent numbers of generations and computed functions, respectively. Actual error is calculated as the absolute difference of the computed location and the true location.

True 0% noise

30% noise

Phase/s

x 共m兲

Source location y z 共m兲共m兲

— PS S P a PS PS S P a PS

261 259 259 182 259 258 257 124 256

330 331 331 363 328 331 329 134 331

380 380 380 382 381 378 381 100 386

to 共s兲

⌬x 共m兲

Actual error ⌬y ⌬z 共m兲共m兲

0 0.0003 0.0003 0.0137 ⳮ0.0011 0.0004 0.0002 ⳮ0.0362 ⳮ0.0006

— 2 2 79 2 3 4 137 5

— 1 1 33 2 1 1 196 1

— 0 0 2 1 2 1 280 6

Cost ⌬t 共s兲

Ng

Nf

— 0.0003 0.0003 0.0137 0.0011 0.0004 0.0002 0.0362 0.0006

— 23 29 41 21 28 36 46 27

— 1408 1698 2460 1307 1950 1996 2146 1648

a

Computed using homogeneous velocity model with VP ⳱ 5500 m / s and VS ⳱ 3500 m / s.

a)

b)

0

c)

z (m)

100

1.00

200

0.75

300

0.50

400

0.25 0.00

500 600 0

100

200

300

y (m)

400

500

600

0

100

200

300

y (m)

400

500

600

0

100

200

300

400

500

600

y (m)

Figure 10. Objective functions plotted on the yz-plane through the true source location: 共a兲 combined P and S stacks, 共b兲 only the S stack, and 共c兲 only the P stack. The color scale represents the objective function value. The objective functions are normalized to their maximum values. The true source location is shown in green.


MA36

Gharti et al.

are not artifacts, but instead trial sources within the voids are associated with very late phase arrivals 共due to the low velocity in the voids兲, and therefore result in low values of the objective function. The global maximum in Figure 12a-d is not as sharp as in the noisefree case 共Figure 5兲, but it does not deviate too much from the original position, demonstrating the reliability of the formulation of the objective function for such noisy data. We performed the inversion using the same parameters as defined

a)

b)

c)

Figure 11. Synthetic seismograms computed for the idealized model of the Pyhäsalmi ore mine, contaminated with 30% white noise, and band-pass filtered from 20 to 400 Hz: 共a兲 east component, 共b兲 north component, and 共c兲 vertical component. Superimposed are P onsets shown in green and S onsets shown in red for the estimated source location. Seismograms are normalized to their maximum amplitudes.

earlier. The result was achieved after 28 generations, by computing the objective function for only 1950 parameter vectors. Although P and S signals are almost buried in most of the seismograms, we observed, after careful examination, that the estimated onsets reasonably match with the P and S signals 共Figure 11a-c兲. The spatial distribution of all trial sources 共Figure 13a and b兲 demonstrates that the entire parameter space was thoroughly searched, and that the source location was reliably recovered. Compared with the previous case, however, the searched parameter ranges are narrowing more gradually from generation to generation toward the final location 共Figure 14a and b兲. This indicates that the objective function is more complex, possessing several local maxima. Figure 15a-c shows the rotated envelopes and the stacked sections for the estimated source location. We observed clear signals on four seismograms of the L-component, at least six seismograms of the T-component, and at least 14 seismograms of the Q-component. Primarily, these clear signals contribute to the reliable source location. The signals outside the stacked sections represent either the scattered energy or noise. The final result of the DE is appended to Table 2. The source was reasonably located. Due to the roughness of the objective function 共i.e., quite a large number of local maxima兲, the convergence is slower, and the number of generations required for the convergence is larger than in the previous case. Using only the S stack, we obtained as well a reliable source location, but using only the P stack we got a completely wrong location, which is also very different from the previous noise-free case in which only the P stack was used. Expectedly, the noise contaminates the weaker phase 共P phase, in this example兲 significantly. The DE inversion using the equivalent homogeneous velocity model gives a reasonably good location also for this case 共Table 2兲. Due to strong secondary phases in the seismograms and the significant noise level, we could not judge the reliability of the method for noisy data by testing just one particular case of random noise. Therefore, a Monte Carlo type of computation was necessary. We repeated the computation of the source location 100 times with the same minimum noise level of 30%, each time computing different realizations of the white noise. The location error computed for all realizations is plotted in Figure 16. It ranges from 1 to 11 m, the error for most locations being below 6 m. This shows that the new location method is reliable for this kind of noisy data. As with all kinds of migration-based location methods, the estimation of the location error, especially in combination with the DE inversion, is not straightforward. As an approximation, we can interpret the spatiotemporal distribution of all DE trial locations. One alternative is to look at all trial locations, whose objective function is larger than a certain threshold, say, 95% of the maximum of the objective function. If we take the absolute value of the range of these locations 共which is equivalent to taking a 95% contour of the objective function兲, the error is largely overestimated 共Figure 16兲. On the other hand, twice the standard deviation 共also called expanded uncertainty兲 of these locations provides a reasonable estimate of the inversion error. Strictly speaking, this uncertainty estimate is valid only for the current source-receiver configuration, but we will assume in the following that it is relevant for events that occur within the mining network. Further, we conducted a jackknife test for both noise-free and noise-contaminated data, i.e., removing one geophone at a time for all 18 geophones. For the noise-free case, we obtained the mean source location at x ⳱ 259, y ⳱ 331, z ⳱ 379 m, and an origin time



a)

MA37

b)

z (m)

y (m) x (m)

y (m)

c) d)

to (s)

z (m) x (m)

Figure 12. Objective function computed on cross sections through the true source location: 共a兲 xy-plane, 共b兲 yz-plane, and 共c兲 zx-plane. The objective function was computed at the true origin time. The color scale represents the objective function value. The spot with the maximum value marks the best source location. The true source location is shown in green. 共d兲 Objective function computed at the true source location for a time window of Ⳳ0.04 s around the true origin time. The objective functions are normalized to their maximum values.


MA38

Gharti et al.

of to ⳱ 0.0003 s with standard deviations of 1, 1, 1 m, and 0.0001 s, respectively. For the noise-contaminated data, the mean source was located at x ⳱ 261, y ⳱ 331, z ⳱ 381 m, and an origin time of to ⳱ 0.0003 s with standard deviations of 1, 2, 2 m, and 0.0002 s, respectively. These tests also demonstrate the stability of our method.

APPLICATION TO OBSERVED DATA Finally, we demonstrated the application of our method to a typical microseismic event that occurred in the Pyhäsalmi ore mine on February 10, 2006. This particular event was processed by the microseismic monitoring software MIMO 共Oye and Roth, 2003兲 using a homogeneous velocity model. Due to the complexity of the signals, we could not identify the S phase reliably; therefore, only the P phases were used. This particular source was characterized by a moment magnitude of ⳮ0.68 Mw 共seismic moment of 1.21e Ⳮ 8 Nm兲 and a corner frequency of 210 Hz. During this particular event, out of a total of 18 geophones 共12 one-component and 6 three-component兲, geophone numbers 11, 12, 17, and the north component of geophone number 13 were down 共Figure 17a-c兲. We projected all seismograms onto the LTQ system to be able to deal with single and multicomponent data in a straightforward manner, but the full advantage of the rotation will be effective only for three-component data.

a)

We searched for the best location of the microseismic event using the entire range of the velocity model and an origin time range from ⳮ1 to 1 s, again with the same DE control parameters. In practice, such a large search domain is not necessary if one is interested only in a certain volume of the mine, but we wanted to keep the restrictions low to test the methodology. We sampled this 4D space with 1 m and 0.001 s, respectively, yielding a total search space of more than 4.8e Ⳮ 11 parameter vectors. The time window length for both P and S was taken as 0.012 s 共approximated to cover the main P- and S-wave energy兲. We ran the program for a predefined maximum standard deviation of 2 m and 0.002 s, and a maximum number of generations of 50. It took only 31 generations to achieve the convergence, by computing the objective function for only 1763 parameter vectors. The P- and S-wave onsets based on the estimated best source location fit very well with the observed P- and S-wave traveltimes 共Figure 17a-c兲. The best source was reliably located, as shown by the distribution of all trial locations 共Figure 18a and b兲. The highest values are focused around the best source location. The image of the potential source region is reliably reproduced. In the evolution of the parameters 共Figure 19a and b兲, we observed a long and sharp tip toward the best source location, suggesting a good convergence and high resolution of the location. The even sharper and longer tip of the origin time is due to the smaller standard deviation defined for its last generation, but also due to faster convergence.

z (m)

a)

x (m)

y (m)

x (m)

b)

z (m)

b)

y (m) to (s)

Figure 13. All trial locations visualized in 共a兲 3D space and 共b兲 yz-plane. Similar features are observed in xy- and zx-planes. Colors represent the value of the objective function 共high values in red and low values in blue兲. The spot with the maximum value represents the best source location.

Figure 14. All generations plotted against: 共a兲 parameter x and 共b兲 parameter to. Similar features are observed for parameters y and z. Colors represent the value of the objective function 共high values in red and low values in blue兲.


Automated microearthquake location To look at the results in more detail, we computed the objective function corresponding to the estimated origin time 共Figure 20a-c兲. We also computed the objective function at the estimated source location for a time window 共0.24 to 0.32 s兲 around the estimated origin time 共Figure 20d兲. Although the objective function is complex, the global maximum is clearly visible and the number of local peaks is relatively low. The convergence is faster in the case of fewer local

a)

b)

MA39

peaks. The DE inversion has accurately located the global maximum, as indicated by the estimated source location. The shape of the objective function is elongated along the z-axis, implying a larger uncertainty in depth 共Figure 20c兲. Figure 21a-c displays the rotated envelopes and stacked sections for the estimated source location. The signals are reasonably polarized in the LTQ directions on at least eight seismograms of the L-component, all five seismograms of the T-component, and at least nine seismograms of the Q-component. The remaining parts of the signals that are mostly visible around the S-wave coda represent mainly the scattered energy due to the minedout voids. The final result of the inversion is listed in Table 3. Although the MIMO location was obtained using the homogeneous velocity model and only the P phase, it is in close agreement with the computed result. As noted earlier, the estimated error in the z-direction is larger than for the horizontal direction and is probably due to the source-receiver distribution. The S signal is stronger than the P signal; therefore, only the S stack gives approximately the same result as the P and S stacks combined. Again, using only the P stack provides a wrong location. In addition, the location using only the P stack is not similar to the MIMO location, which also uses only the P phase 共in this example兲, and uses the phase-identification-based method. Further, we performed the DE inversion using the equivalent homogeneous velocity model, and obtained a fairly good location 共Table 3兲. The jackknife computation for this case gives a mean location of x ⳱ 358, y ⳱ 382, z ⳱ 388 m, and to ⳱ 0.2761 s with standard deviations of 1, 1, 3 m, and 0.00028 s, respectively. Due to the larger uncertainty in the z-location, the jackknife computation also shows the largest discrepancy in the z-location.

DISCUSSION In most stacking methods, trace normalization, automatic gain control 共AGC兲, and the correction for geometric spreading are the standard steps to obtain a better stacking result. In our approach, however, we did not apply any normalization because the source-toreceiver distances are in a range of about 50 to 500 m and the noise conditions are more or less equal throughout the mine. Hence, we di-

c)

Iteration number

Figure 15. Time-shifted envelopes 共for P and S arrival times兲 of rotated seismograms corresponding to the estimated source location: 共a兲 L-component, 共b兲 T-component, and 共c兲 Q-component. Stacked areas are shown as filled regions 共L-component in green, and T- and Q-components in red兲. Superimposed are P onsets shown in green and S onsets shown in red. Envelopes are normalized to their maximum amplitude.

Figure 16. Actual location error and two estimates for location error computed for 100 Monte Carlo iterations. The actual error is the absolute error of the estimated location with respect to the true source location. The error estimate 1 corresponds to the absolute value of the range of all trial locations that have a value of the objective function larger than 95% of the maximum value, and the error estimate 2 is twice the standard deviation of the same trial locations.


MA40

Gharti et al.

rectly stacked the true amplitudes of the envelopes. This implies that the traces with the largest amplitudes, which usually are the ones recorded on the geophones closer to the source, dominate the stacking result. Nevertheless, we favor this stacking approach because we are more confident in the seismograms recorded closer to the source. In addition, AGC or trace normalization would significantly increase the contribution of noisy traces to the stacking result. In other environments, where noise levels differ significantly, an appropriate weighting scheme might improve the stacking result.

a)

b)

Our method requires only a few control parameters. For the objective function, the only control parameter that we have to define in advance is the length of the time window used for stacking. Ideally, we want to have a time window length just long enough to cover the main P- and S-wave energy, thereby cumulating the major contributions of the signals. In other words, the time window length should approximately match the dominant period of the signals. For the automatic processing, one can take a time window length depending on the characteristic frequency observed in the region. Our practice shows that the time window length does not significantly influence the accuracy of the location, when being within the range of about one to three times the period of the dominant signal. However, some improvement might be achieved by coupling the time window length to the traveltime along the source-receiver path and the associated buildup of a coda. All control parameters are generally constant for the DE algorithm. The parameter ranges are determined based on the region of interest. The parameter quantization is limited by the grid interval of the model used for the traveltime computations and the sampling rate of recordings. Other parameters, such as the mutation factor and crossover probability, are usually taken as recommended values or approximated after some trial runs. The rotation into the LTQ system significantly improves the objective function and consequently the source image. For instance, in

z (m)

a)

y (m)

c)

x (m)

z (m)

b)

y (m)

Figure 17. Seismograms observed in the Pyhäsalmi ore mine: 共a兲 east component, 共b兲 north component, and 共c兲 vertical component. Superimposed are P onsets shown in green and S onsets shown in red for the estimated source location. Seismograms are normalized to their maximum amplitude.

Figure 18. All trial locations visualized in 共a兲 3D space and 共b兲 yz-plane. Similar features are observed in xy- and zx-planes. Colors represent the value of the objective function 共high values in red and low values in blue兲. The spot with the maximum value represents the best source location.



MA41

an example shown in Figure 22a and b, the source image is clear in the case of noise-free data and noticeably blurred for noisy data. After the LTQ rotation, however, both images are significantly improved 共Figure 22c and d兲. The location of the real microearthquake was successfully determined and was in agreement with the standard phase-picking results. The noise level of the data example was such that P-wave picking was possible, but reliable S-wave picking was not 共Figure 17兲. However, in comparison with the noisy synthetic data 共Figure 11兲, the signal-to-noise level was still relatively large. Hence, the objective function obtained for the real data is smoother than the one for the noisy synthetic data. Although this could be expected, it is worth noticing that the real microearthquake data consist of only 24 channels 共5 E-, 4 N-, and 15 Z-components兲 as compared to 54 channels 共18 three-components兲 in the synthetic case, decreasing the effectiveness of the LTQ rotation. Furthermore, the waveforms of the real data are more complicated than the synthetics, containing more scattered energy expressed in longer codas and secondary phases. The real P- and S-wave velocity model certainly is more complicated than the synthetic 3D model 共Figures 2 and 3兲; moreover, it is constantly changing due to mining operations. The stacking method presented in this study takes into account the combination of P- and S-wave envelopes; however, we also investi-

gated the cases of using only P- or S-wave envelopes, respectively. The larger part of the signal energy in the synthetic as well as in the real data seismograms is concentrated in the S-wave and the S-wave coda. Hence, we obtain a similar result using only the S-wave envelopes compared to the combination of P- and S-wave envelopes. On the contrary, if we use only the P-wave envelopes, sometimes the P-wave onsets related to the maximum objective function can match with some of the S-wave onsets, thus providing a wrong location. Other stacking/migration studies that use only P-waves therefore need to ensure that the P phase on the channels selected for inversion is the dominating seismic phase, or implement some sort of phase identification 共e.g., Rentsch et al., 2007兲 to avoid wrong locations. For our method, it would be favorable also to constrain the stacking of each component within the approximately determined time window of the corresponding phase. This would result in a more accurate location, especially when stacking the weaker phase 共P, in our case兲. However, for most low signal-to-noise ratio data, the data quality is not sufficient to conduct these necessary steps. We therefore recommend using a combined stacking of P- and S-wave envelopes for the automated processing. The use of the combined stacking of P- and S-wave envelopes enabled us to obtain a reasonable location, even when some of the geophones were situated on the nodal plane. 共Figures 23 and 24.兲

a)

Figure 19. All generations plotted against: 共a兲 x and 共b兲 to. Similar features are observed for parameters y and z. Colors represent the value of the objective function 共high values in red and low values in blue兲.

x (m)

b)

to (s)


MA42

Gharti et al.

b)

a)

z (m)

y (m) x (m)

y (m)

c)

d)

to (s) z (m) x (m)

Figure 20. Objective function computed on cross sections through the estimated source location: 共a兲 xy-plane, 共b兲 yz-plane, and 共c兲 zx-plane. The objective function was computed at the estimated origin time. The color scale represents the objective function value. The spot with the maximum value marks the best source location. The estimated source location is shown in green. 共d兲 Objective function computed at the estimated source location for a time window 共0.24 to 0.32 s兲 around the true origin time. The objective functions are normalized to their maximum values.



MA43

Figure 21. Time-shifted envelopes 共for P and S arrival times兲 of rotated seismograms corresponding to the estimated source location: 共a兲 L-component, 共b兲 T-component, and 共c兲 Q-component. Stacked areas are shown as filled regions 共L-component in green, and T- and Q-components in red兲. Superimposed are P onsets shown in green and S onsets shown in red. Envelopes are normalized to their maximum amplitudes.

a)

b)

c)

Table 3. Location results and comparison with MIMO: Ng and Nf represent numbers of generations and computed functions, respectively. Estimated error for the new method is approximated as described in the ⴖControlled Testsⴖ section.

Phase/s MIMO New method

a

P PS S P a PS

x 共m兲

Source location y z 共m兲共m兲

356 358 358 320 354

372 383 382 341 379

380 387 386 621 383

to 共s兲

⌬x 共m兲

Estimated error ⌬y ⌬z 共m兲共m兲

0.2760 0.2760 0.2760 0.0003 0.2760

14 5 5 11 5

11 6 5 10 7

14 7 9 11 8

Cost ⌬t 共s兲

Ng

Nf

0.0027 0.0011 0.0011 0.0019 0.0012

— 31 45 36 28

— 1763 2486 1971 1574

a

Computed using homogeneous velocity model with VP ⳱ 5500 m / s and VS ⳱ 3500 m / s.


MA44

Gharti et al.

b)

z (m)

a)

y (m)

y (m)

d)

z (m)

c)

y (m)

y (m)

Figure 22. Typical objective function in a cross section of the Pyhäsalmi ore mine model without LTQ rotation: 共a兲 computed from synthetic seismograms, and 共b兲 computed from seismograms contaminated with 30% white noise and band-pass filtered from 40 to 400 Hz. 共c, d兲 Same as 共a, b兲 but with LTQ rotation. The color scale represents the objective function value. The true source location is shown in green. The objective functions are normalized to their maximum values.



MA45

Figure 23. Evolution of the objective function computed on a cross section through the estimated source location for the synthetic data without noise 共left兲, and for the synthetic data contaminated with 30% white noise and band-pass filtered from 20 to 400 Hz 共right兲. The color scale represents the objective function value. The spot and the time at which the objective function attains the maximum value mark the best source location and origin time. The objective functions are normalized to their maximum values 共figure enhanced online兲. 关DOI: http://dx.doi.org/10.1190/

1.3432784.1兴

Figure 24. Evolution of the objective function computed on a cross section through the estimated source location for the real data. Color scale represents the objective function value. The spot and the time at which the objective function attains the maximum value mark the best source location and origin time. The objective functions are normalized to their maximum values 共figure enhanced online兲. 关DOI:

http://dx.doi.org/10.1190/1.3432784.2兴


MA46

Gharti et al.

CONCLUSIONS The proposed method is fully automatic, flexible, and complete. The application of the LTQ rotation significantly improves the objective function. Further, the DE global search algorithm is robust, and drastically reduces the amount of calculations compared with a full grid search. Hence, the method is applicable and reliable for the processing of low-quality data. We tested the algorithm on synthetic and real data with different formulations of the objective function, e.g., different time window lengths, using P or S envelopes and combined envelopes, and with and without the LTQ rotation. We conclude that a combined use of P and S envelopes on rotated traces is the most reliable way to construct the objective function for noisy data. In future work, we will apply the method to different data sets, e.g., rock slope monitoring and large ocean-bottom networks. We also intend to investigate further how source-receiver distributions, different weighting schemes, and errors in the velocity model influence the stability of the objective function.

ACKNOWLEDGMENTS We thank Shawn Larsen for his E3D code, Pascal Podvin for his eikonal solver, and Markus Bühren for the differential evolution code. We thank Valerie Maupin for helpful discussions and Isabelle Lecomte for suggestions on the manuscript. We thank Katja Sahala and Integrated Seismic System International for access to the mine model and the in-mine data. This work was funded by the Norwegian Research Council and supported by industry partners BP, StatoilHydro, and Total. We thank an anonymous reviewer, E. Gaucher, M. van der Baan, and the associate editor for their thoughtful comments and suggestions.

REFERENCES Arnold, M. E., 1977, Beam forming with vibrator arrays: Geophysics, 42, 1321–1338. Baker, T., R. Garant, and R. W. Clayton, 2005, Real-time earthquake location using Kirchhoff reconstruction: Bulletin of the Seismological Society of America, 95, 699–707. Chang, W. F., and G. A. McMechan, 1991, Wavefield extrapolation of body waves for 3-D imaging of earthquake sources: Geophysical Journal International, 106, 85–98. Drew, J., D. Leslie, P. Armstrong, and G. Michaud, 2005, Automated microseismic event detection and location by continuous spatial mapping: Presented at the Annual Technical Conference and Exhibition, SPE, Paper 95513. Fink, M., 2006, Time-reversal acoustics in complex environments: Geophysics, 71, no. 4, SI151–SI164. Gajewski, D., and E. Tessmer, 2005, Reverse modeling for seismic event characterization: Geophysical Journal International, 163, 276–284. Geiger, L., 1912, Probability method for the determination of earthquake epicenters from the arrival time only: Bulletin of St. Louis University, 8, 60–71. Gharti, H. N., V. Oye, and M. Roth, 2008, Traveltimes and waveforms of microseismic data in heterogeneous media: 78th Annual International Meeting, SEG, Expanded Abstracts, 1337–1341. Husen, S., E. Kissling, N. Deichmann, S. Wiemer, D. Giardini, and M. Baer, 2003, Probabilistic earthquake location in complex three-dimensional velocity models: Application to Switzerland: Journal of Geophysical Research, 108, 1–12. Kao, H., and S. J. Shan, 2004, The source-scanning algorithm: Mapping the distribution of seismic sources in time and space: Geophysical Journal International, 157, 589–594. ——–, 2007, Rapid identification of earthquake rupture plane using source-

scanning algorithm: Geophysical Journal International, 168, 1011–1020. Larsen, S., and J. Grieger, 1998, Elastic modeling initiative: Part 3 — 3-D computational modeling: 68th Annual International Meeting, SEG, Expanded Abstracts, 1803–1806. Lomax, A., J. Virieux, P. Volant, and C. Brege-Thierry, 2000, Probabilistic earthquake location in 3D and layered models, in C. Thurber, and N. Rabinowitz, eds., Advances in seismic event location: Kluwer Academic Publishers, 101–134. Lu, R., 2007, Time-reversed acoustics and applications to earthquake location and salt dome flank imaging: Ph.D. thesis, Massachusetts Institute of Technology. Maxwell, S. C., D. J. White, and H. Fabriol, 2004, Passive seismic imaging of CO2 sequestration at Weyburn: 74th Annual International Meeting, SEG, Expanded Abstracts, 568–571. McMechan, G. A., 1982, Determination of source parameters by wavefield extrapolation: Geophysical Journal of the Royal Astronomical Society, 71, 613–628. Mendecki, A. J., ed., 1997, Seismic monitoring in mines: Chapman and Hall. Oye, V., H. Bungum, and M. Roth, 2005, Source parameters and scaling relations for mining related seismicity within the Pyhäsalmi ore mine, Finland: Bulletin of the Seismological Society of America, 95, 1011–1026. Oye, V., and M. Roth, 2003, Automated seismic event location for hydrocarbon reservoirs: Computers and Geoscience, 29, 851–863. Phillips, W. S., L. S. House, and M. C. Fehler, 1997, Detailed joint structure in a geothermal reservoir from studies of induced microearthquake clusters: Journal of Geophysical Research, 102, 11745–11763. Podvin, P., and I. Lecomte, 1991, Finite difference computation of traveltimes in very contrasted velocity models: A massively parallel approach and its associated tools: Geophysical Journal International, 105, 271–284. Price, K. V., R. M. Storn, and J. A. Lampinen, 2005, Differential evolution: A practical approach to global optimization: Springer. Pujol, J., 2000, Joint event location — The JHD technique and application to data from local seismic networks, in C. Thurber, and N. Rabinowitz, eds., Advances in seismic event location: Kluwer Academic Publishers, 163–204. Puustjärvi, H., ed., 1999, Pyhäsalmi modeling project: Technical report 13.5.1997-12.5.1999: Geological Survey of FinlandArchive Report M 19/ 3321/99/1/10. Rentsch, S., S. Buske, S. Lüth, and S. A. Shapiro, 2007, Fast location of seismicity: Migration-type approach with application to hydraulic-fracturing data: Geophysics, 72, no. 1, S33–S40. Ringdal, F., and T. Kværna, 1989, A multi-channel processing approach to real time network detection, phase association, and threshold monitoring: Bulletin of the Seismological Society of America, 79, 1927–1940. Rutledge, J. T., W. S. Phillips, and B. K. Schuessler, 1998, Reservoir characterization using oil-production-induced microseismicity, Clinton county, Kentucky: Tectonophysics, 289, 129–152. Ruzek, B., and M. Kvasnicka, 2001, Differential evolution algorithm in the earthquake hypocenter location: Pure and Applied Geophysics, 158, 667–693. Sambridge, M. S., and B. L. N. Kennett, 1986, A novel method of hypocenter location: Geophysical Journal International, 87, 679–697. Schaff, D. P., G. H. R. Bokelmann, W. L. Ellsworth, E. Zanzerkia, F. Waldhauser, and G. C. Beroza, 2004, Optimizing correlation techniques for improved earthquake location: Bulletin of the Seismological Society of America, 94, 705–721. Storn, R., and K. Price, 1995, Differential evolution — A simple and efficient adaptive scheme for global optimization over continuous spaces: Technical Report TR-95-012, International Computer Science Institute, Berkeley, ftp://ftp.icsi.berkeley.edu/pub/techreports/1995/tr-95-012.pdf, accessed 5 January 2009. ——–, 1997, Differential evolution — A simple and efficient heuristic for global optimization over continuous spaces: Journal of Global Optimization, 11, 341–359. Tarantola, A., and B. Valette, 1982, Inverse problems⳱ quest for information: Journal of Geophysics, 50, 159–170. Thurber, C. H., and N. Rabinowitz, 2000, Advances in seismic event location: Kluwer Academic Publishers. Vinje, V., E. Iversen, K. Åstebøl, and H. Gjøystdal, 1996, Estimation of multivalued arrivals in 3D models using wavefront construction — Part 1: Geophysical Prospecting, 44, 819–842. Waldhauser, F., and W. L. Ellsworth, 2000, A double-difference earthquake location algorithm: Method and application to the Northern Hayward fault, California: Bulletin of the Seismological Society of America, 90, 353–1368. Young, R. P., S. C. Maxwell, T. I. Urbancic, and B. Feignier, 1992, Mininginduced microseismicity: Monitoring and applications of imaging and source mechanism techniques: Pure and Applied Geophysics, 139, 697–719.


Simultaneous microearthquake location and moment-tensor estimation using time-reversal imaging Hom Nath Gharti∗ , Volker Oye, Daniela Kühn, Peng Zhao, NORSAR SUMMARY We use time-reversal imaging for microearthquake location and investigate the possibility of a simultaneous qualitative moment-tensor estimation. We cross-correlate the data with the synthetic strain Green’s tensor and stack individually for each moment-tensor component. The objective function for the source location is then formulated as the squared sum of those stacked components. The maximum value of the objective function corresponds to the estimated source location and origin time. Similarly, the corresponding stacked components at the estimated source location give the entire time history of the qualitative estimation of the moment tensor. We apply the method to synthetic data of various types of moment-tensor sources computed for a complex and heterogeneous model, namely the Pyhäsalmi ore mine in Finland. We also test the method with the same data adding white noise up to 40% of the absolute maximum. Although the method is computationally intensive, it is fully automatic and can easily be adapted to parallel processing. The preliminary results show that the method is robust and reliable.

INTRODUCTION Microearthquake monitoring has become an important tool in many fields, such as mining, CO2 -storage, geothermal, and oil and gas industry (e.g., Mendecki (1997); Oye and Roth (2003); Shapiro (2011)). Microearthquakes usually occur in large numbers. The accurate source location and the moment tensor are important for a reliable interpretation of these events. In many cases, microearthquake data have very low signalto-noise ratio (SNR) making the use of phase-identificationbased methods difficult. Therefore the migration-based approach that does not require phase picking is gaining interest recently (e.g., Drew et al. (2005); Gajewski and Tessmer (2005); Rentsch et al. (2007); Gharti et al. (2010)). Timereversal is also a migration-based approach, in which the observed seismograms are back propagated in time. The resulting wavefield focuses at the correct source location and origin time (e.g., McMechan (1982); Fink (1999)). Therefore, the time-reversal can be used to approximate the inversion (e.g., Claerbout (2001); Kawakatsu and Montagner (2008)). The time-reversal has been successfully applied for global-scale model (e.g., Larmat et al. (2006)) using spectral-element method to compute the time-reversal wavefield. From the computational point of view, the time-reversal for the source location can be realized with two equivalent approaches. In the first approach, entire wavefield is computed similar to forward propagation but using time-reversed seismograms as the sources (e.g., Tromp et al. (2008); Artman et al. (2010)). In this approach, the entire wavefield has to be computed on the entire domain for all sources. In the second approach, the Green’s function database is computed solving

the forward wave equation. Then the time-reversed data are convolved with the Green’s functions (equivalently the crosscorrelation with the data). This approach requires the large storage for the Green’s function database. Once the Green’s function database is prepared, however, we can constrain the region of potential-source locations as required. Kawakatsu and Montagner (2008) have discussed the theoretical framework for an application of the time-reversal to the source location and moment-tensor inversion. We use the same concept to locate the microearthquake sources and investigate the possibility of simultaneous moment-tensor estimation. We apply the method to synthetic data computed for a complex and heterogeneous mine in Finland. We consider various types of moment-tensor sources. We also test the reliability of the method for noisy data adding a minimum of 40% white noise to the synthetic data. In this article, we briefly describe the method and present our preliminary results.

METHODOLOGY For a general moment-tensor source located at ξ , the displacement field at any point x can be expressed in terms of the Green’s function (Aki and Richards (2002)) as un (x, t) = Gni, j (x, t; ξ , 0) ∗ Mi j (ξ , t) ,

(1)

where, the symbol (*) denotes the convolution, un the nth component of the displacement field, Gni the Green’s function at x for a unit force applied at ξ , and Mi j the moment tensor. Summation convention applies to repeated indices. The index ( j) after comma (,) represents the partial differentiation with respect to the source coordinates (ξ j ). Using the spatial reciprocity of the Green’s function, Equation 1 can be reformulated as un (x, t) = Ei jn (ξ , t; x, 0) ∗ Mi j (ξ , t) ,

(2)

where, Ei jn is also called the strain Green’s tensor, which represents the strain components computed for a unit force applied in nth direction (e.g., Zhao et al. (2006)). Due to the symmetry, the strain Green’s tensor has 18 independent components. It is given by Ei jn (ξ , t; x, 0) =

) 1( Gin, j (ξ , t; x, 0) + G jn,i (ξ , t; x, 0) 2

. (3)

The strain Green’s tensor can be computed numerically (e.g., Eisner and Clayton (2001); Okamoto (2002); Zhao et al. (2006)) using the finite-difference method (e.g., Larsen and Schultz (1995); Saenger and Bohlen (2004)) or the spectral-element method (e.g., Komatitsch and Tromp (1999)). For the given data un (x, t), the time-reversal operation can be approximated (Kawakatsu and Montagner (2008)) as ∑ Ei jn (ξ , t; x, 0) ∗ un (x, t0 − t) , (4) Mˆ i j (ξ , t) ≈ Receivers

Microearthquake location and moment tensor where, un (x, t0 − t) is the time-reversed version of the data un (x, t). Alternatively, the time-reversal can also be expressed as the cross-correlation operation with the original data un (x, t). ∑ Mˆ i j (ξ , t) ≈ Ei jn (ξ , t; x, 0) ⋆ un (x, t) , (5) Receivers

where, the symbol (⋆) denotes the cross-correlation. In this formulation, Mˆ i j is not the actual moment tensor. However, this quantity focuses at the correct source location and should give a qualitative estimation of the moment tensor. It is interesting to note that the time-reversal operation is also closely related to a matched-filter processing or a waveform crosscorrelation, in which the empirical Green’s function (masterevent template) is used for cross-correlation (e.g., Gibbons and Ringdal (2006); Kummerow (2010)). The cross-correlation and the stacking approach usually enhance the SNR of the data. Finally, the objective function at the possible source location ξ and origin time t0 is formulated as ∫ t0 +tw S(ξ , t0 ) = Mˆ i j (ξ , t)Mˆ i j (ξ , t) dt , (6)

(a)

t0

where, tw is certain time window. The objective function can be normalized by the total waveform energy of the data. The maximum value of the objective function corresponds to the estimated location and origin time of the source. Similarly the Mˆ i j at the estimated source location gives the qualitative approximation of the moment tensor. The time-reversal of the moment tensor Mˆ i j at a grid point (potential-source location) can be computed independently. Hence the processing can be parallelized easily and efficiently, because there is virtually no parallel communication overhead. We have parallelized our code based on the SPMD (Single Program Multiple Data) model using the Matlab parallel toolkit. (b)

NUMERICAL RESULTS We test the method with the synthetic data computed for the Pyhäsalmi mine. This mine is located in central Finland. It consists of a volcanogenic massive sulphide (VMS) deposit. Figure 1(a) shows the ore body and the infrastructure of the mine. The in-mine seismic network comprises 18 geophones including six 3-component instruments (Puustjärvi (1999)). A large number of microearthquake events are frequently observed in the mine (e.g., Oye et al. (2005)). The 3D velocity model of the mine consists of an ore body, a host rock, and stopes (Figure 1(b) and Table 1). Due to structural complexities and strong velocity contrasts, this model poses a challenging task for any source location or moment tensor inversion algorithm (Gharti et al. (2010); Kühn and Vavryˇcuk (2011)). We compute the strain Green’s tensor using the reciprocity principal (e.g., Zhao et al. (2006)) requiring only three simulations per receiver. We use the 3D viscoelastic finite-difference code by Larsen and Schultz (1995). The model is discretized with 2 m grid interval, and a smooth Gaussian source is used. For the computation of synthetic data, we position the source at x = 350 m, y = 300 m, and z = 480 m with the origin time taken as 0. The source is placed near the cavities so that all possible

Figure 1: a) Pyhäsalmi ore mine with surrounding infrastructure: the Copper/Zink ore body (brown/pink), access tunnels (yellow), the elevator shaft (dark blue), and seismic stations (numbered). Passage for the quarried ore is marked by KN1. b) 3D velocity model (see Table 1) of the mine: stopes, i.e., mined-out cavities (blue) and the ore body (brown). Remaining part is the host rock. Geophones are shown in black and are numbered. Similarly, x-, y- and z-axes correspond to East, North, and vertical direction respectively.

Air Rock Ore

Vp (m/s) 300 6000 6300

Vs (m/s) 0 3460 3700

ρ (kg/m3 ) 1.25 2000 4400

Table 1: Material properties of the Pyhäsalmi mine model.

Channels

18Z 18N 18E 17Z 17N 17E 16Z 16N 16E 15Z 15N 15E 14Z 14N 14E 13Z 13N 13E 12Z 12N 12E 11Z 11N 11E 10Z 10N 10E 9Z 9N 9E 8Z 8N 8E 7Z 7N 7E 6Z 6N 6E 5Z 5N 5E 4Z 4N 4E 3Z 3N 3E 2Z 2N 2E 1Z 1N 1E 0

0.1 0.15 Time (s)

0.2

0.25

18Z 18N 18E 17Z 17N 17E 16Z 16N 16E 15Z 15N 15E 14Z 14N 14E 13Z 13N 13E 12Z 12N 12E 11Z 11N 11E 10Z 10N 10E 9Z 9N 9E 8Z 8N 8E 7Z 7N 7E 6Z 6N 6E 5Z 5N 5E 4Z 4N 4E 3Z 3N 3E 2Z 2N 2E 1Z 1N 1E 0

0.05

0.1 0.15 Time (s)

(a)

(a)

(b)

(b)

0.2

0.25

Moment tensor

0.05

Moment tensor

Channels

Microearthquake location and moment tensor

−0.2

−0.1

0 0.1 Origin time (s)

0.2

(c)

Figure 2: a) 3-component synthetic waveforms computed for the Pyhäsalmi mine model. Seismograms are trace normalized. b) Objective function computed at the correct origin time. Color scale represents the value of the normalized objective function. The exact source location is shown in black. c) Moment-tensor traces obtained from the time-reversal. The moment-tensor traces are normalized to the absolute maximum.

−0.2

−0.1


0.2

(c)

Figure 3: a) Synthetic waveforms contaminated with the white noise of 40% of the absolute maximum and band-pass filtered (40 - 400 Hz). Seismograms are trace normalized. b) Objective function computed at the correct origin time. Color scale represents the value of the normalized objective function. The exact source location is shown in black. c) Moment-tensor traces obtained from the time-reversal. The moment-tensor traces are normalized to the absolute maximum.

Microearthquake location and moment tensor complexities can be reproduced. We use a Ricker wavelet with a central frequency of 250 Hz. For the first test, we take a pure double-couple source with moment-tensor components Myz = Mzy = 1.6e8 Nm (≈ -0.46 Mw). The seismograms are recorded with the sampling interval of 0.1 ms for a total of 0.25 s duration. Figure 2(a) shows the resulting seismograms. Due to the complicated interaction of the wavefield with the cavities, we observe very complex waveforms (e.g., Gharti et al. (2008)).

absolute maximum to the previous data. Figure 3(a) shows the band-pass filtered (10 - 400 Hz) data. In the presence of high noise level, seismic phases are barely visible. We use these data to compute the time-reversal. Even with these noisy data the source is clearly visible at the correct location (Figure 3(b)). Although we observe the presence of more noise on the moment-tensor traces, the nature of the actual momenttensor component is correctly recovered (Figure 3(c)). We made another test with an explosion source, and found that the method works reliably. Therefore, we conduct a final test with a general moment-tensor source with fictitious components. We take Mxx = 1.6e8 Nm, Mxy = Myx = 1.6e8 Nm, and Myz = Mzy = -1.6e8 Nm. For this case also, the source is located accurately with high resolution (Figure 4(a)). The nature of the moment tensor is also correctly recovered. The sign of the Myz component is flipped as expected (Figure 4(b)).

DISCUSSION

Moment tensor

(a)

−0.2

−0.1


0.2

(b)

Figure 4: The general fictitious moment-tensor source. a) Objective function computed at the correct origin time. Color scale represents the value of the normalized objective function. The exact source location is shown in black. b) Momenttensor traces obtained from the time-reversal. The momenttensor traces are normalized to the absolute maximum. In order to compute the time-reversal of the moment tensor (Equation 5) and the objective function (Equation 6), we take 48 × 48 grid on yz-plane through the true source location with a 4 m spacing. Figure 2(b) shows the objective function computed at the correct origin time. This image can also be interpreted as the time-reversal image of the source. The point with the highest value gives the estimated source location, which accurately coincides with the correct source location. Although the data are fairly complicated, the source is sharply and accurately located giving a high resolution. Time-reversal traces of the moment tensor also correctly recovers the nature of the correct moment-tensor component (Myz ) (Figure 2(c)). We also observe some fluctuations on the other components, but these are not significant. The moment tensor has focused at the correct origin time, and the actual source-time function (Ricker wavelet) has been retrieved. In the next experiment, we add white noise of 40% of the

The time-reversal technique can be a reliable tool for the processing and analysis of microearthquake data . The synthetic tests show that the method may be used for the source location and the qualitative moment-tensor estimation simultaneously. Although the method is computationally intensive, it can efficiently be adapted to parallel processing. In an usual stacking procedure, we have to use the envelope to avoid the cancellation of the amplitudes at different receivers (due to the radiation pattern). That reduces the resolution of the location. On the other hand, the time-reversal approach considers the actual seismograms for the cross-correlation, and utilizes the entire waveforms. Additionally, moment-tensor correction is taken into account automatically during the stacking. Therefore, we obtain high-resolution locations even in the presence of high noise level. We plan to investigate the influence of error in the velocity model, receiver configurations, and different normalizations with the application to real data. Although, this approach only gives the qualitative estimation of the moment tensor, it is not difficult to invert for the actual moment tensor once we have the accurate source location and the Green’s functions. Simultaneously locating source and determining moment tensor requires a large computational effort. In practice, we are interested in a specific region. Once the Green’s function database is prepared, we can compute the time-reversal only in the specified region. Additionally, the optimization methods (e.g., Differential Evolution (Price et al. (2005)) can be used for the source location (e.g., Gharti et al. (2010)), further reducing the computational cost.

ACKNOWLEDGMENTS We thank Shawn Larsen, Valérie Maupin, Jean-Paul Montagner, and Vaclav Vavryˇcuk for the helpful discussion, Michael Roth for his comments on the manuscript, and Katja Sahala and ISS for access to the mine model. The programs were run on the Titan cluster of the University of Oslo, Norway. This work was funded by the Norwegian Research Council, and supported by BP, Statoil and Total.

Microearthquake location and moment tensor REFERENCES Aki, K., and P. G. Richards, 2002, Quantitative seismology, second ed.: University Science Books. Artman, B., I. Podladtchikov, and B. Witten, 2010, Source location using time-reverse imaging: Geophysical Prospecting, 58, 861–873. Claerbout, J. F., 2001, Basic earth imaging: Stanford University. Drew, J., D. Leslie, P. Armstrong, and G. Michaud, 2005, Automated microseismic event detection and location by continuous spatial mapping: SPE Annual Technical Conference and Exhibition. Eisner, L., and R. W. Clayton, 2001, A reciprocity method for multiple-source simulations: Bulletin of the Seismological Society of America, 91, 553–560. Fink, M., 1999, Time-reversed acoustics: Scientific American, 281, 9197. Gajewski, D., and E. Tessmer, 2005, Reverse modelling for seismic event characterization: Geophysical Journal International, 163, 276–284. Gharti, H. N., V. Oye, and M. Roth, 2008, Travel times and waveforms of microseismic data in heterogeneous media: SEG Technical Program Expanded Abstracts, 27, 1337– 1341. Gharti, H. N., V. Oye, M. Roth, and D. Kühn, 2010, Automated microearthquake location using envelope stacking and robust global optimization: Geophysics, 75, MA27–MA46. Gibbons, S. J., and F. Ringdal, 2006, The detection of low magnitude seismic events using array-based waveform correlation: Geophysical Journal International, 165, 149–166. Kawakatsu, H., and J.-P. Montagner, 2008, Time-reversal seismic-source imaging and moment-tensor inversion: Geophysical Journal International, 175, 686–688. Komatitsch, D., and J. Tromp, 1999, Introduction to the spectral element method for three-dimensional seismic wave propagation: Geophysical Journal International, 139, 806– 822. Kühn, D., and V. Vavryˇcuk, 2011, Computation of full waveform moment tensors in a very heterogeneous mining environment: EAGE 3rd Passive Seismic Workshop, 120–124. Kummerow, J., 2010, Using the value of the crosscorrelation coefficient to locate microseismic events: Geophysics, 75, MA47–52. Larmat, C., J.-P. Montagner, M. Fink, Y. Capdeville, A. Tourin, and E. Clévédé, 2006, Time-reversal imaging of seismic sources and application to the great Sumatra earthquake: Geophysical Research Letters, 33. Larsen, S., and C. A. Schultz, 1995, ELAS3D: 2D/3D elastic finite difference wave propagation code: Technical Report No. UCRL-MA-121792: Technical report. McMechan, G. A., 1982, Determination of source parameters by wavefield extrapolation: Geophysical Journal of the Royal Astronomical Society, 71, 613–628. Mendecki, A. J., 1997, Seismic monitoring in mines: Chapman and Hall. Okamoto, T., 2002, Full waveform moment tensor inversion by reciprocal finite difference Green’s function: Earth Planets Space, 54, 715–720.

Oye, V., H. Bungum, and M. Roth, 2005, Source parameters and scaling relations for mining related seismicity within the pyhaesalmi ore mine, finland: Bulletin of the Seismological Society of America, 95, 1011–1026. Oye, V., and M. Roth, 2003, Automated seismic event location for hydrocarbon reservoirs: Computers & Geosciences, 29, 851–863. Price, K. V., R. M. Storn, and J. A. Lampinen, 2005, Differential evolution: a practical approach to global optimization: Springer. Puustjärvi, H., 1999, Pyhaesalmi modeling project, section b. geology: Technical report: Technical report, Geological Survey of Finland, and Outokumpu Mining Oy. Rentsch, S., S. Buske, S. L¨th, and S. A. Shapiro, 2007, Fast location of seismicity: A migration-type approach with application to hydraulic-fracturing data: Geophysics, 72, S33– S40. Saenger, E. H., and T. Bohlen, 2004, Finite-difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid: Geophysics, 69, 583–591. Shapiro, S. A., 2011, Microseismicity: a tool for reservoir characterization: EAGE Publications BV. Tromp, J., D. Komatitsch, and Q. Liu, 2008, Spectral-element and adjoint methods in seismology: Communications in Computational Physics, 3, 1–32. Zhao, L., P. Chen, and T. H. Jordan, 2006, Strain Green’s tensors, reciprocity, and their applications to seismic source and structure studies: Bulletin of the Seismological Society of America, 96, 1753–1763.

Chapter 3 Wave propagation and travel times

33

Travel times and waveforms of microseismic data in heterogeneous media Hom Nath Gharti*, Volker Oye and Michael Roth, NORSAR Summary

Receiver line Z Vp Vs Rock 6.00 3.46 Ore 6.30 3.70 Void 0.30 0.00

ρ 4.40 2.00 0.00125 622 m

We compute P- and S-wave first arrival times for a heterogeneous model of the Pyhäsalmi ore mine using a finite-difference Eikonal code (Podvin and Lecomte, 1991). For the very same model we are computing the complete wavefield using a viscoelastic finite difference scheme (Larsen and Schultz, 1995). We compare the synthetic results amongst each other and with real microseismic data recorded with an in-mine seismic network. The resulting first arrival times generally agree with the onset of the synthetic wavefield, whereas the amplitude distribution is strongly affected by the heterogeneities in the model. Such amplitude variations at the different in-mine geophones as well as P- and S-wave coda lengths generally agree with the observed microseismic data.

Y

Void Rock Ore Source 622 m

Introduction Microseismic monitoring in mines is a well-established tool used to minimize risk for personnel and to optimize production (e.g. Mendecki, 1997). The seismic wave propagation in a mining environment is generally very complex due to heterogeneities such as excavations, access tunnels and stopes (voids) as well as the geological structural changes in rock types. In the following we are going to assess wave propagation effects in the Pyhäsalmi (Puustjaervi, 1999) underground ore mine, Finland. For this mine we have been provided with a detailed underground model as well as microseismic data recorded with an inmine seismic network. We are modeling first arrival times using the Eikonal finite-difference code of Podvin and Lecomte (1991), which is based on Huygens’ Principle and valid in the high-frequency limit. It is very robust and, since it takes refractions and diffractions into account, it also delivers correct arrival times in shadow zones. For the full waveform modeling we are using Shawn Larsen’s E3D code (Larsen and Schulz, 1995). Finally we compare observed microseismic data from the ore mine with synthetic waveforms. Pyhäsalmi ore mine model The Pyhäsalmi ore mine model consists of an ore body with mined-out stopes surrounded by host rock. Figure 1 shows the S-N cross section that we used for 2D Eikonal and full waveform modeling. For the 2D case we place a line of 36 receivers arbitrarily at the depth of 80 m and position a source at y = 310 m and z = 500 m; y and z being northwards and downwards, respectively.

Figure 1: S-N cross section of the Pyhäsalmi ore mine model (Vp, Vss and ρ are in km/s and g/cm3, respectively) and the sourcereceiver configuration.

X Y

1 4 3

2

Z 6

5

622 m

9 15 16

10

17 13

7 Source

11 8 12 18

14 622 m

622 m Figure 2: Pyhäsalmi ore mine model with in-mine seismic network and source location

Figure 2 shows the geometry of the ore mine model in 3D and the position of the in-mine seismic network. The inmine network consists of 18 geophones out of which 6 are 3-component instruments (for more details on the mine, the network and the microseismic event characteristics see Oye et al. 2005). We positioned a seismic source at x = 489 m, y = 274 m and z = 414 m; x, y and z pointing East, North and downwards, respectively. This particular source location corresponds to a real microseimic event with a moment magnitude MW -0.6 that occurred on May 03, 2005 in the Pyhäsalmi ore mine (see section 3D modeling).

Travel times and waveforms

2D modeling For the 2D numerical experiments we discretized the model in Figure 1 onto a 1 m equispaced grid. We performed travel time computations using Podvin’s Eikonal method for three different cases: a) a homogeneous model with host rock alone, b) a heterogeneous model with host rock and the ore body, and c) the complete heterogeneous model with host rock, ore body and mined-out stopes. For the full waveform modeling using Larsen’s E3D code we considered only the complete heterogeneous model. As source we used a strike-slip source with a central frequency of 250 Hz. Figure 3 shows the results for the travel time computations. The travel time field is represented in terms of 0.01 s isochrones, i.e. the wavefront at 0.01 s time steps. For the homogeneous model (case a)) the wavefronts are concentric rings as expected. For case b) the part of the wavefront that passes through the ore body advances faster and overtakes the wavefront for the homogeneous case. More complicated interaction is evident for the complete heterogeneous model of case c). The voids represent obstacles for the wavefronts due to the extremely low velocity. The first arrivals behind the stopes are not from a wavefront transmitted through the void, but from the wavefronts diffracted around the obstacle. The wavefronts have kinks in the shadow zones which straighten out at further distances due to the effect of wavefront healing.

Figure 3: Isochrones, i.e. wavefronts at 0.01 s time steps. Case a) host rock only (in red), case b) host rock and ore body (in black), and case c) host rock, ore body and mined-out stopes (in green). Complete heterogeneous mine model is shown in magenta.

In Figure 4 we display snapshots of the compression potential (i.e. P-wave component) computed with E3D and the first arrival time curve computed with Podvin’s code. The voids interact strongly with the wavefield generating reflections, conversions and diffractions. The 0.02 s snapshot shows clearly an additional secondary phase propagating downwards, which was generated when the direct wave hit the host-rock/ore interface. At 0.03 s snapshot, reflection of direct wave from the ore/void interface is clearly exposed. At 0.04 s snapshot, diffraction around the void is distinctly visible and at 0.05 s snapshot, very strong S-to-P conversion has been generated at the lower boundary of the void. The wavefronts obtained by full waveform computation exhibit low amplitudes behind the voids, because in these shadow zones the wavefield consists solely of diffracted energy. For the chosen source location much of the seismic energy is reflected downwards from the voids and never reaches the upper part of the model. Behind the voids the wavefronts converge and heal out with increasing distance. The first arrival time computed with the Eikonal code accurately describes the wavefront of the full waveform results. t = 0.02 s

t = 0.03 s

t = 0.04 s

t = 0.05 s

t = 0.06 s

t = 0.07

Figure 4: P-wave potential superimposed with first arrival isochrones (in green). Complete heterogeneous mine model is shown in magenta.


The synthetic seismograms for the given receiver line (Figure 1) marked with the Eikonal results for the P- and Swave first arrivals are shown in Figure 5. Also in this display the Eikonal first arrivals nicely match with the onsets of the computed waveforms. Due to the orientation of the strike-slip source the amplitudes of the first arrival signals are strong on the vertical component and weak on the horizontal components.

400 Hz) around the central signal frequency to match the frequency range of the synthetic data. Secondary phases and multiple scattering due to complicated interaction of the wavefield with the interfaces and voids are clearly visible and can similarly be explained as in the 2D case. First arrival isochrones obtained by the Eikonal computation accurately match with the earliest onsets obtained by the full waveform computation. Again, the amplitude of the P-wave first arrivals behind the voids has been reduced significantly.

t = 0.03 s

t = 0.04 s

t = 0.05 s

t = 0.06 s

Figure 6: P wave potential and superimposed P-wave first arrival wavefront (in green).

Figure 5: Trace-normalized seismograms (top: horizontal component, bottom: vertical component) recorded at the receiver line (Figure 1). Superimposed are the P- (in magenta) and S-wave (in green) first arrival times computed with the Eikonal code.

3D modeling For the 3D experiments we discretized the real model in Figure 2 onto a 2 m grid. The seismic source was a strikeslip source oriented in the x-z plane with a central frequency of 200 Hz and at the same location as the observed event. The synthetic results are shown as P-wave potential superimposed with the first arrival isochrone in snapshots (Figure 6) and as seismograms overlapped with the first arrival P and S onsets at the locations of the inmine seismic network in Figure 7-9. In addition we display the observed seismograms of the May 03, 2005 microseismic event that have been bandpass filtered (20 to

Computed onsets and waveforms reflect the general features of the observed data. The arrival times of the regular P and S phases reasonably match with the observed ones. The discrepancies between synthetic and observed arrivals for the z-components of geophones 8 and 16 are justifiable. These are 1-component geophones in reality and the observed first arrival signals can be reconstructed in superposing all three components of the synthetic data. The coda of the observed data exhibits more complexity as that of the synthetic seismograms, which is understandable, because any model is a simplification of the real structural conditions. However, the coda duration and amplitudes of the observed data are represented reasonably. Our choice of the fault plane mechanism was arbitrary and therefore the P-and S-wave amplitude distribution onto the three geophone components differs from the observed data.


Figure 7: Trace-normalized seismograms and superimposed P- (in magenta) and S-wave (in green) onsets. Top: synthetic Eastcomponents with Eikonal onsets. Bottom: observed Eastcomponents with manually picked onsets. Note that only six of the 18 in-mine geophones are 3-component instruments and that some of the channels did not record the event.

Figure 9: Same as Figure 7 for the vertical components.

Discussion The Eikonal method provides accurate first arrival P- and S-wave onset times even in very heterogeneous media, where ray-tracing might be problematic. The comparison with finite difference waveforms illustrates that the Eikonal method models the wavefront correctly and that it can deal with diffractions and shadow zones. Lookup tables computed with the Eikonal code therefore, can be used in an inversion routine to localize microseismic events. However, full waveform modeling illustrates the complexity of the wave propagation, and with respect to the determination of, for instance, focal mechanisms (e.g. Kuehn et al., 2008) or full moment tensor inversions, it is important to assess the propagation effects. Acknowledgement We thank S. Larsen for his E3D code and his helpful comments on the waveform modeling. We thank K. Sahala and ISS for access to the mine model and the in-mine data. This work was funded by the Norwegian Research Council and industry partners BP, StatoilHydro and Total. Most of the figures were created using the open-source visualization application ParaView.

Figure 8: Same as Figure 7 for the North components.

References Kuehn, D., V. Oye, M. Roth, H. N. Gharti, 2008, Automatic determination of focal mechanisms from P-wave first-motions applied to mining-induced seismicity: 78th Annual International Meeting, SEG, Expanded Abstracts, submitted. Larsen, S., and C. A. Schultz , 1995, ELAS3D: 2D/3D elastic finite difference wave propagation code: Technical Report No. UCRL-MA-121792, 19 pp. Mendecki, A. J., 1997, Seismic Monitoring in Mines: Chapman & Hall Oye, V., H. Bungum, and M. Roth, 2005, Source parameters and scaling relations for mining related seismicity within the Pyhaesalmi ore mine, Finland: Bulletin of Seismological Society of America, 95, 1011-1026. Podvin, P., and I. Lecomte, 1991, Finite difference computation of traveltimes in very contrasted velocity models: a massively parallel approach and its associated tools: Geophysical Journal International, 105, 271-284. Puustjärvi, H., 1999, Pyhaesalmi modeling project, section B. Geology: Technical report, Geological Survey of Finland, and Outokumpu Mining Oy.

Wave propagation modeling based on the spectral-element method – application to microearthquakes and acoustic emissions Hom Nath Gharti, Volker Oye, Michael Roth, and Daniela K¨ uhn July 1, 2011 Abstract We simulate wave propagation for microearthquakes and acoustic emissions based on the spectral-element method. In the first example, we compute the full wavefield in 2D and 3D models of an underground ore mine, namely the Pyh¨ asalmi mine in Finland. In the second example, we simulate wave propagation in a homogeneous velocity model including the actual topography of an unstable rock slope at ˚ Aknes, western Norway. Finally, we compute the full wavefield for a weakly anisotropic cylindrical sample at laboratory scale, which was used for an acoustic emission experiment under triaxial loading. We investigate the characteristic features of wave propagation in those models and compare synthetic waveforms with observed waveforms wherever possible. We illustrate the challenges associated with the spectral-element simulation in those models.

1

1

Introduction

Microearthquakes and acoustic emissions are small events having a magnitude less than 3 (e.g., [27, 4]). Small earthquakes have a high-frequency content (e.g., [2, 17]), and therefore small-scale structural or material heterogeneities may strongly influence the wavefield. Due to this reason, observed waveforms of microearthquakes are often complicated (e.g., [35]). The simulation of wave propagation helps to understand complexities of the waveforms and provides valuable information on the wavefield interaction. In addition, synthetic data are important to assess the applicability and reliability of data processing algorithms. Similarly, synthetic Green’s functions are necessary for moment-tensor inversion. However, wave propagation modelling is a challenging task for microearthquakes. An accurate geometry mapping is very difficult due to the small-scale natural or man-made structural complexities. In addition, the presence of strong velocity contrasts causes problems for the numerical stability. The finite-difference method (FDM) is probably the most widely used method for wave propagation modelling (e.g., [51, 8, 3, 28]). Direct discretization of the governing equation on structured grid (i.e., regular grid) makes the FDM simple and efficient, and easily adaptable to parallel processing (e.g., [26, 43]). However, it is difficult to accurately model complex boundaries and interfaces using the structured grid. In the presence of surface topography, the FDM is less accurate due to the approximation of boundary conditions (e.g., [19, 32]). The spectral-element method (SEM) is a higher order finite-element method (FEM) that uses a nodal quadrature, namely the Gauss-Lobatto-Legendre quadrature for numerical integration over an element. While the nodal quadrature results in a diagonal mass matrix enabling an efficient time-marching scheme, the higher-order elements give a high degree of spatial accuracy. Further, since the SEM solves a weak form of the wave equation, freesurface boundary conditions are automatically satisfied. It is also possible to model complex boundaries and interfaces accurately, using the unstructured mesh (e.g., [45, 23]). The nodal quadrature was originally limited to specific types of elements, e.g., quadrilaterals in 2D and hexahedra in 3D. Hexahedral meshing is a challenging task and an area of active research (e.g., [30, 37, 47]). Only a few hexahedral meshing tools are currently available, e.g., CUBIT [44], Gmsh [14], and TrueGrid [41]. The hexahedral meshing is usually not fully automated, and careful mesh design is necessary. Recently, there have been several attempts to implement the SEM using other types of elements (e.g., triangles in 2D, tetrahedra in 3D) using socalled Fekete points (e.g., [18, 48, 20, 30]). Since the nodal quadrature includes end points of the numerical integration interval, the order of integration may not always be sufficiently high [11]. Due to the high degree of spatial accuracy, however, the influence of low-order integration on accuracy and convergence may not be significant, depending on the specifics of the problem (e.g., [31, 46, 9]). Hence, the SEM is a versatile tool due to an efficient timemarching scheme and a high degree of spatial accuracy. The SEM was originally developed to solve fluid dynamics problems (e.g., [36, 5, 7, 10]). Recently, it has been widely used to simulate seismic wave propagation in various scales (e.g., [12, 21, 22, 50, 34, 38]). In this paper, we present the results of full wavefield simulations in an underground ore mine, an unstable rock slope, and a weakly anisotropic cylindrical sample used during an acoustic emission laboratory experiment. We discuss the characteristic features of the wavefields, and compare the waveforms with observed data wherever possible. We illustrate the challenges associated with the application of the SEM in these problems. We use the SEM code developed by Komatitsch and Tromp [21] for our simulations. 2

Air Rock Ore

Vp (m/s) 300 6000 6300

Vs (m/s) 0 3460 3700

ρ (kg/m3 ) 1.25 2000 4400

Table 1: Material properties of the Pyhäsalmi mine model.

2

Pyh¨ asalmi ore mine

The Pyhäsalmi mine is an underground ore mine located in central Finland. This mine consists of a volcanogeneic massive sulphide (VMS) deposit, and produces mainly copper, zinc, and pyrite. The copper-zinc ore body in the mine extends down to a depth of ∼ 1.4 km (Figure 1a). The in-mine seismic network consists of 18 geophones including 6 three-component instruments (geophones 1, 5, 9, 13, 17, and 18) [40]. The geophones have a sampling rate of up to 3 kHz. For this mine, we have a detailed 3D velocity model and observed microseismic data at our disposal (for more details on the mine and the microseismic event characteristics, see [33]). The 3D velocity model of the mine is shown in Figure 1b and described in Table 1. Even though the model is a simplification of the original structure (Figure 1a), it still poses a challenging task for the modelling of wave propagation in terms of geometrical discretization, numerical stability, and accuracy. Gharti et. al. [15] used this model to compute full waveforms and first-arrival times using a 3D visco-elastic finite-difference code [26] and a 3D finite-difference eikonal solver [39], respectively, and used these data to locate the microearthquakes in the mine [16].

2.1

2D model

In this test, we consider a 2D model of the Pyhäsalmi mine taking a North-South (i.e., yz-plane at x = 311 m in Figure 1b) cross-section, and perform two simulations. In the first simulation, we mesh the entire model including the air in the stopes (i.e., mined-out cavities). Since the S-wave velocity is zero in air, we cannot satisfy the dispersion condition in this region. The P-wave velocity for the air is also very low, therefore we need a very fine mesh within this region. In the second simulation, we mesh the model excluding the air in mined-out stopes. The main purposes of these simulations are to assess the stability of the SEM in the presence of a strong velocity contrast, and to investigate whether there are any discrepancies in the waveforms when including or excluding the air during meshing of the model. For the computation purpose, we place a receiver line with 36 equispaced geophones near the top surface (at z = -80 m) as shown in Figure 2. We take a source represented by a Ricker wavelet having a central frequency of 250 Hz, which is located at y = 310 m and z = −500 m. This source is characterized by a strike-slip mechanism on the yz-plane and a seismic moment (M0 ) of −1010 Nm. We select a sampling interval of 1 µs for the seismogram recordings. A quadrilateral mesh is required for the SEM simulation in 2D models. For the quadrilateral meshing, we use the mesh generation tool CUBIT [44]. For the case with air included, we use an average element size of 2 m resulting in a total of 53,071 spectral elements (Figure 3a). In order to preserve the same Courant number for numerical stability outside the stopes, we have 3

(a)

(b)

Figure 1: a) Pyhäsalmi ore mine with surrounding infrastructure: the copper/zink ore body is shown in brown/pink, access tunnels are shown in yellow, the elevator shaft is shown in dark blue, and seismic stations are numbered. The passage for the quarried ore is marked by KN1. b) 3D velocity model (see Table 1) of the mine used to generate the synthetic data: stopes (i.e., mined-out cavities) are shown in blue and the ore body is in brown. Remainder is the host rock. Geophone locations are shown in black and are numbered. The x, y, and z axes represent East, North, and vertical directions, respectively. y

Figure 2: 2D model of the Pyhäsalmi mine (North-South cross-section of the 3D model, i.e., yz-plane at x = 311 m in Figure 1b). The model consists of host rock (light gray), ore body (dark gray), mined-out stopes (white), and source (sphere). The receiver line of 36 geophones (black) is used for the computation purpose. 4

(a)

(b)

Figure 3: Spectral element mesh for the 2D model of the Pyhäsalmi mine. a) Including air. b) Excluding air.

to maintain the same degree of mesh fineness outside the stopes in both cases. Therefore, we remove the mesh on the stopes to obtain the mesh for the case with air excluded, resulting in a total of 40,594 spectral elements (Figure 3b). Excluding the stopes for the meshing reduces the number of elements by about 30%, which is significant regarding the computational cost. Practically, a coarser mesh may be used when excluding the air. Both simulations give stable results. For visualization, we calculate P-wave and S-wave potentials. The P-wave potential is computed as the divergence of the displacement field, and the S-wave potential is derived from the sum of the components of the curl of the displacement field [26]. Figure 4 and 5 show snapshots of P-wave and S-wave potentials, respectively. Wavefields are very similar in both cases of air included and air excluded. In case of air included, only P waves travel with a very slow speed within the voids. We observe the diffracted waves around the stopes. The converted waves, i.e., P to S and S to P are also visible. Due to multiple reflections and conversions at the strong-contrast interfaces, the wavefield is very complicated. It is severely distorted by the voids, but tends to heal further away from the voids. For both cases (i.e., with and without air), the computed waveforms are very similar (Figure 6), and discrepancies are negligible. The P and S first arrival times computed by an eikonal solver [39] are in good agreement with the respective arrivals in computed waveforms. The strong signals observed between first P and S arrivals as well as after the S arrivals are the secondary waves generated by multiple reflections and conversions at the interfaces with strong velocity contrast. Because of source radiation pattern, these secondary waves are stronger on vertical components than on horizontal components. These results show that the SEM is stable even in the presence of high velocity contrasts. For this particular mine model, we may safely exclude the air during meshing to compute the full waveforms unless we are interested in the acoustic waves in the stopes. In some cases, excluding the air drastically reduces the computational cost and avoids possible numerical instability due to the high velocity contrast. 5

y

y

y

y

y

y

y

y

Figure 4: P-wave potential for the model with air included (left column) and with air excluded (right column). The lines in cyan represent the outlines of the ore body and the stopes. High values of the potential are shown in red and low values are shown in blue. 6

y

y

y

y

y

y

y

y

Figure 5: Same as Figure 4, but for the S-wave potential.

7

35

30

30

25

25 Receivers

Receivers

35

20 15

20 15

10

10

5

5

0

0.04

0.08 0.12 Time (s)

0.16

0.2

(a)

0

0.04

0.08 0.12 Time (s)

0.16

0.2

(b)

Figure 6: Synthetic waveforms computed for the 2D model of the Pyhäsalmi mine without air (black) and with air (red). (a) Horizontal components. (b) Vertical components. Superimposed are the P- (black) and S-wave (blue) first-arrival times computed with a finitedifference eikonal solver. Seismograms are normalized to trace maximum.

In case where a regularly gridded (i.e., structured grid) model is already available, it is not always practicable to discretize a model with an unstructured mesh, in particular, if the model is complicated. In such cases, there is the possibility of using the spectral-element method considering each grid cell as a quadrilateral element. In the next example, we simulate wave propagation considering the structured grid as spectral-element mesh. We have the gridded velocity model with 2 m grid interval for this 2D model of the mine. We use these 2 m grid cells as spectral elements and compute full waveforms. First-arrival times and general features of the waveforms computed using structured and unstructured mesh match reasonably (Figure 7). It is usually difficult to perfectly honor the complicated interfaces with a structured grid. We observe some discrepancies, in particular, in the secondary arrivals and coda. Using grid cells as spectral elements may not always be feasible. Particularly, if the structured grid is very fine, direct conversion of the grid cells into spectral elements will result in a unnecessarily large number of spectral nodes, which requires increased computational cost. Alternatively, one can extract larger grid cells interpolating the velocity model, and convert them into spectral elements. In the next example, we extract larger grid cells of 4 m and use them as the spectral elements for the simulation of wave propagation. With larger grid cells, we observe a significant discrepancy on the amplitudes of the secondary arrivals and the coda (Figure 8), although general features of waveforms and firstarrival times are preserved. On coarser grid, discrepancy of the velocity model obtained by interpolation may be significantly large depending on the velocity model and the degree of grid coarseness, and therefore, we may observe the significant discrepancy on the waveforms. 8

35

30

30

25

25 Receivers

Receivers

35

20 15

20 15

10

10

5

5

0

0.04

0.08 0.12 Time (s)

0.16

0.2

0

0.04

(a)

0.08 0.12 Time (s)

0.16

0.2

(b)

35

35

30

30

25

25 Receivers

Receivers

Figure 7: Synthetic waveforms computed for unstructured (black) and structured (red) mesh (2 m grid) of the 2D model of the Pyhäsalmi mine. (a) Horizontal components. (b) Vertical components. Superimposed are the P- (black) and S-wave (blue) first-arrival times computed with a finite-difference eikonal solver. Seismograms are normalized to trace maximum.

20 15

20 15

10

10

5

5

0

0.04

0.08 0.12 Time (s)

0.16

0.2

(a)

0

0.04

0.08 0.12 Time (s)

0.16

0.2

(b)

Figure 8: Synthetic waveforms computed for unstructured (black) and structured (red) mesh (4 m grid) of the 2D model of the Pyhäsalmi mine. (a) Horizontal components. (b) Vertical components. Superimposed are the P- (black) and S-wave (blue) first-arrival times computed with a finite-difference eikonal solver. Seismograms are normalized to trace maximum.

9

2.2

3D model

Now, we simulate wave propagation in a 3D model of the mine. We use an observed event for this simulation. The source location was estimated using a microseismic monitoring software (MIMO see [35]), and the full moment-tensor inversion was preformed using first motion polarities and P-wave amplitudes [29]. The source is located at x = 368 m, y = 371 m, and z = −392 m. It is characterized by a Ricker wavelet with a central frequency of 200 Hz and moment-tensor components Mxx = −1.0158 × 108 Nm, Myy = 0.0858 × 108 Nm, Mzz = 0.3540×108 Nm, Mxy = 0.4394×108 Nm, Myz = 0.1025×108 Nm, and Mzx = 0.0731×108 Nm. Due to the complicated waveforms and very heterogeneous velocity model, the uncertainty in the moment tensor was large, nevertheless we will use it for our simulation. The estimated moment tensor consists of both isotropic and deviatoric components implying a complex source mechanism. The sampling interval of the seismogram recordings is set to 1 µs.

Figure 9: 3D model of the Pyhäsalmi mine including ore body (solid), two major stopes, and 18 geophones (numbered). Although, there are several tools for 2D quadrilateral meshing, only a few tools are available for 3D hexahedral meshing, and the functionalities of such tools are limited. Including all small cavities and inclusions is very difficult for the hexahedral meshing. It is also unfeasible in view of computational cost, because these small inclusions will result unnecessarily in a very large number of elements. Therefore, we simplify the original 3D model including only two major stopes (Figure 9). We exclude the stopes from the meshing. Automatic hexahedral meshing is currently not possible with the CUBIT for such a complex 3D model like the Pyhäsalmi mine. The complex model has to be decomposed into several volumes which can be meshed with the functionalities available within the CUBIT. We decompose the 3D model into 78 volumes (Figure 10a). We use average element sizes of 9.5 m for rock and 10 m for ore body resulting in a total of 107,712 spectral elements and a total of 7,161,572 spectral nodes (Figures 10a-b). We partition the mesh into 24 domains for parallel processing (Figure 11). In order to balance the load among the processors, we use an open-source graph partitioning tool SCOTCH [6] for the mesh partition. Figure 12 shows snapshots of P-wave and S-wave potentials. We again observe reflected and converted waves. The wavefield is distorted by the stopes but gradually heals away from the 10

(a)

(b)

Figure 10: a) Spectral-element mesh for the 3D model of the Pyhäsalmi mine. b) Interior section of the mesh visualizing the ore body and stopes. Colors represent different volumes created for meshing.

Figure 11: Spectral-element mesh for the 3D model of the Pyhaesalmi mine partitioned into 24 domains for parallel processing.

11

Figure 12: P-wave potential (left) and S-wave potential (right). High values are shown in red and low values are shown in blue. 12

stopes. Figure 13 illustrates the synthetic and observed seismograms. During this particular event, geophones 8, 11, 12, 17, and the North channel of geophone 13 were not functioning. The first-arrival times and coda durations for computed and observed waveforms generally match. Due to the very complicated velocity model and an uncertain source mechanism, we do not expect an accurate match. From the synthetic data, e.g., geophones 5-10, we observe that it would be difficult to identify and pick the P and S phases if the geophones were only single component. Due to the shadow zones created by the voids or the source radiation pattern, first arrivals have small amplitudes at some of the receivers. Therefore, 3-component data are important for reliable data processing. First-arrival times computed by a finite-difference eikonal solver using structured grid are in reasonable agreement with respective arrivals in the computed waveforms. However, there is always a discrepancy in velocity model when converting unstructured mesh into structured grids.

3

Unstable rock slope at ˚ Aknes

In this example, we simulate wave propagation in a rock slope at ˚ Aknes, Western Norway, which is monitored because of its instability (Figure 14). The slope mainly consists of gneiss, and it is densely jointed along the foliation. The mass of the sliding volume has been estimated to be ∼ 35-40 million m3 (e.g., [13, 25]). Several monitoring instruments including a seismic network (e.g. [42]) are in place to monitor the slope movement. The seismic network consists of 8 threecomponent geophones covering an area of about 250 × 150 m2 . In addition, a high-sensitive broadband seismometer has been installed in the middle of the slope. Microearthquakes are frequently observed at the site (see [42]). Due to the rough topography and unavailability of a realistic velocity model, we cannot locate the microearthquakes. It is nevertheless important to identify the characteristic features of the synthetic wavefield. Once a realistic velocity model is available, we can utilize these synthetic results to locate and characterize microearthquakes. We have access to Digital Elevation Map (DEM) of the slope. Therefore, we build a 3D model including a realistic topography (Figure 15a). Since we do not have detailed seismic properties of the slope, we use a homogeneous model with a P-wave velocity (Vp ) of 3000 m/s, a S-wave velocity (Vs ) of 1732 m/s, and a mass density (ρ) of 2000 kg/m3 . We position the source on the free surface with x = 171 m, y = 470 m, and z = 461 m so that all complexities due to the free surface topography can be captured. The source is represented by an explosion mechanism given by the moment-tensor components as Mxx = Myy = Mzz = 1015 Nm, and is characterized by a Ricker wavelet having a central frequency of 120 Hz. The sampling interval for the seismogram recordings is set to 10 µs. Using the CUBIT, we mesh the model with an average element size of 10 m resulting in a total of 107,712 spectral elements and a total of 7,161,572 spectral nodes. The mesh is partitioned into 8 domains for the parallel processing (Figure 15b). For visualization of the wavefield, we plot P-wave and S-wave potentials (Figure 16). Multiple reflections and conversions on the free surface result in a complicated wavefield, despite the simple source mechanism. Figure 17 shows the synthetic waveforms. Although, we have used an explosion source, S waves are generated by free-surface reflections and conversions, which are stronger than the P waves. First-arrival times computed by a finite-difference eikonal solver are generally in 13

18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

Receivers

16 15 14 13

10 9 7 6 5 4 3 2 1

0.02

0.04

0.06

0.08 0.1 Time (s)

0.12

0.14

0.16

0.18

0.25

0.3

0.35 Time (s)

0.4

0.25

0.3

0.35 Time (s)

0.4

0.25

0.3

0.35 Time (s)

0.4

18 16 15 14 13 Receivers

18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

18

10 9 7 6 5 4 3 2 1

0.02

0.04

0.06

0.08 0.1 Time (s)

0.12

0.14

0.16

0.18

18 16 15 14 13 Receivers

Receivers Receivers Receivers

18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

10 9 7 6 5 4 3 2 1

0.02

0.04

0.06

0.08 0.1 Time (s)

0.12

0.14

0.16

0.18

Figure 13: Synthetic (left column) and observed (right column) waveforms for the Pyhäsalmi mine. Top to bottom are East, North, and vertical components, respectively. Superimposed are the P- (black) and S-wave (blue) first-arrival times computed with a finite-difference eikonal solver. Seismograms are normalized to trace maximum.

14

Figure 14: ˚ Asknes rock slope. Dotted line represents the approximate outline of the ˚ Aknes model used for the simulation.

600

500

1

4

8 400

7

3

2

6 5

300

z (m) 200

100

600 500

0

400

500 400

300 300

y (m)

200

200

x (m)

100

100 0 0

(a)

(b)

Figure 15: a) 3D model of the ˚ Aknes rock slope. Geophones are numbered (solid black). The x, y, and z axes represent the East, North, and vertical directions respectively. b) Spectralelement mesh for the ˚ Aknes slope model. The mesh is partitioned into 8 domains.

15

Figure 16: P-wave potential (left) and S-wave potential (right). High values are shown in red and low values are shown in blue. The side plane in each plot represents a yz-slice through the source location (red sphere). 16

Channels

8Z 8N 8E 7Z 7N 7E 6Z 6N 6E 5Z 5N 5E 4Z 4N 4E 3Z 3N 3E 2Z 2N 2E 1Z 1N 1E 0

0.05

0.1

0.15

0.2

0.25

Time (s)

Figure 17: Synthetic waveforms computed for the ˚ Aknes slope model. Superimposed are the P- (black) and S-wave (blue) first-arrival times computed with a finite-difference eikonal solver. Seismograms are normalized to geophone maximum.

good agreement with the respective arrivals in the computed waveforms, but we observe some discrepancies in a few geophones, namely 1, 7, and 8. It is difficult to accurately compute arrival times using the finite-difference eikonal solver due to the rough topography, especially when the receivers are situated on the surface.

4

Weakly anisotropic cylinder at laboratory scale

Finally, we simulate wave propagation for acoustic emissions during a laboratory experiment (Figure 18a). The laboratory sample consists of a Vosges sandstone cylinder with a diameter of 25.4 mm and a height of 63.5 mm. A small cylindrical hole of 5.2 mm diameter is drilled through the center of the sample mimicking a bore-hole in a real field problems. The triaxial experiment was performed to fracture the sample thereby inducing the acoustic emissions. Twelve piezoelectric sensors are mounted on the surface of the cylindrical sample [1] to record the radial displacement. Full waveforms are recorded with a sampling rate of 10 MHz. About 2500 events were detected during a triaxial experiment (due to limitations of the data acquisition system, this does not represent all events) with automatic processing (e.g., [24, 34]). We use a homogeneous velocity model with a P-wave velocity (Vp ) of 3660 m/s and a Swave velocity (Vs ) of 2286 m/s estimated from experimental data. We take a mass density (ρ) of 2000 kg/m3 . For anisotropy, two Thomsen’s parameters ϵ and γ are estimated to be about -0.0869 and 0.07613, respectively, from the experimental data; and the third Thomsen’s parameter δ is assumed to be -0.1 (e.g., [49]). We again use the CUBIT to mesh the model. The mesh consists of 82,240 spectral elements with an average element size of 1.5 mm resulting in a total of 5,462,768 spectral nodes. The mesh is partitioned into 8 domains using SCOTCH for parallel processing (Figure 19). We select a source located near the bore hole with x = −9.0 mm, y = −5.1 mm, and z = 1.8 mm. Full moment-tensor inversion was performed for this source considering a homogeneous isotropic model using first motion polarities and amplitudes [29, 24]. The moment tensor components for this source were estimated as Mxx = −0.0673 × 106 Nm, Myy = 1.4297 × 106 Nm, Mzz = 1.9070 × 106 Nm, Mxy = −0.8306 × 106 Nm, Myz = −0.4332 × 106 Nm, and Mzx = −0.5052 × 106 Nm. Due to the complicated waveforms and the discrepancy between the homogeneous model and the real 17

63.50

11

12

31.75

10

9

z (mm) 0.00

6

4 -31.75

5

1

-63.50 25.40 12.70 0.00

y (mm)

(a)

25.40 12.70 0.00 -12.70 x (mm) -12.70 -25.40 -25.40

(b)

Figure 18: a) Sandstone sample for the acoustic emission experiment. b) Model of the acoustic emission experiment sample. The borehole has a diameter of 5.2 mm. The piezoelectric sensors are numbered (solid black dots). fractured model, uncertainty in the moment tensor components was large. Presence of both isotropic and deviatoric components in the moment tensor indicates a very complex source mechanism. We assume a Ricker wavelet source time function with a central frequency of 500 kHz. The sampling interval for the seismogram recordings is set to 4 ns. We compute full waveforms for both isotropic and anisotropic models.

Figure 19: Spectral-element mesh for the acoustic emission experiment sample. The mesh is partitioned into 8 domains. Figure 20 shows snapshots of P-wave and S-wave potentials. We observe reflected and converted wavefields from the borehole surface. Shear wave splitting is not clearly visible due to a weak anisotropy. We observe only delays in arrival times due to the lower velocity of the anisotropic model (Figure 21a). First-arrival times generally match with the observed data (Figure 21b). Since the piezoelectric sensors used in the experiment are only sensitive to radial 18

Figure 20: P-wave potential (left) and S-wave potential (right). High values are shown in red and low values are shown in blue. The side plane in each plot represents a slice through the borehole.

19

0

Receivers

Receivers

12 11 10 9 8 7 6 5 4 3 2 1 1

2

3

4

12 11 10 9 8 7 6 5 4 3 2 1 0

1

2

3

4

Time (s)

Receivers

5

2

3

5

4

5

Time (s)

−5

x 10

Receivers

Receivers

Time (s)

12 11 10 9 8 7 6 5 4 3 2 1 6 −5

x 10

12 11 10 9 8 7 6 5 4 3 2 1 2

3

4 Time (s)

−5

x 10

5

6 −5

x 10

12 11 10 9 8 7 6 5 4 3 2 1 0

1

2

3 Time (s)

4

5 −5

x 10

Figure 21: Synthetic waveforms (left column) computed for an isotropic (black) and an anisotropic (red) model of triaxial acoustic-emission experiment sample, and observed waveforms (right column) for the same sample. Top to bottom are x, y, and z components respectively. For the observed data only the x and y components were available. Superimposed are the P- (black) and S-wave (blue) first-arrival times computed analytically. Seismograms are normalized to their absolute maximum.

20

motion, S-wave motion is not fully captured. As a result, P-wave amplitudes appear stronger than the S-wave amplitudes in the observed waveforms. Sensors 8 and 10 point towards the x direction, and 7 and 9 towards the y direction; therefore those sensors show signals only on the respective components. We observe more complicated waveforms in the real data. The model used to generate synthetic data does not represent all features of the real fractured sample. In the real sample, the wavefield may interact with the borehole as well as induced fractures resulting in more complicated waveforms.

5

Discussion

We simulated wave propagation for microearthquakes in an underground ore mine and a rock slope, and acoustic emissions in a laboratory experiment using the spectral-element method. Although the hexahedral meshing may be a challenging task for complex velocity and structural models, the SEM is a stable and efficient tool. For the mine model, structural complexity and high velocity contrasts caused by the mined-out cavities pose the main challenges for the simulation. Since these structural heterogeneities are in the order of the wave length of the sources, a strong influence is observed in the complexity of the signals. The results of the 2D simulations show that unless we are interested on the wavefield inside the air regions, we can safely discard those regions during meshing reducing the computational costs significantly. For very complex models such as regularly gridded velocity models obtained from tomography, it is also possible to use the SEM using structured grid as the spectral-element mesh, but it may not always be feasible due to the unnecessarily finer sampling and the discrepancy in the velocity model due to interpolation. For the ˚ Aknes rock slope model, we have used a homogeneous velocity model. Therefore, the actual topography represents the only complexity for the simulation of wave propagation. Even with a simple explosion source mechanism, the computed wavefield is complicated due to its interaction with the free surface topography. For the cylindrical sample of the acoustic emission experiment, even a single borehole interacts with the wavefield and produces fairly complex signals. The effect of anisotropy is not clearly observed, in particular, in shear wave splitting, because the anisotropy is very weak. Since reliable velocity models and accurate source mechanisms are important for the accurate simulation of the wave propagation, it would be of future interest to simulate wave propagation more accurately in those models, once we have the more reliable velocity models and accurate source mechanisms. We plan to investigate the possibility of full waveform tomography in these models with the adjoint capabilities of the spectral-element method [50].

6

Acknowledgments

We thank Yang Luo and Christina Morency for their help on 2D meshing and modelling, Tina Kaschwich and Valérie Maupin for helpful discussions and suggestions, and Ricardo M. Garcia, Jr. and Emanuele Casarotti for their help on hexahedral meshing with CUBIT. We are grateful to Dimitri Komatitsch and Jeroen Tromp for their helpful remarks. We thank ˚ Aknes/Tafjord Beredskap IKS (www.aknes.no) for the ˚ Aknes digital elevation map. We thank Katja Sahala and ISS for access to the mine model and the in-mine data, and Eyvind Aker and Fabrice Cuisiat for access to the model and data of acoustic emission experiment. Parallel programs were run on the Titan cluster owned by the University of Oslo and the Norwegian metacenter 21

for High Performance Computing (NOTUR), and operated by the Research Computing Services group at USIT, the University of Oslo IT-department. The 3D data were visualized using the open-source parallel visualization software ParaView/VTK (www.paraview.org). This work was funded by the Norwegian Research Council, and supported by industry partners BP, Statoil and Total.

References [1] E. Aker, H. D. V. Khoa, F. Cuisiat, and M. Soldal. Acoustic emission experiments and microcrack modelling on porous rock. In EAGE 2nd Passive Seismic Workshop, 2009. [2] K. Aki. Scaling law of seismic spectrum. Journal of Geophysical Research, 72:1217–1231, 1967. [3] A. Bayliss, K. E. Jordan, B. J. Lemesurier, and E. Turkel. A fourth-order accurate finite-difference scheme for the computation of elastic waves. Bulletin of the Seismological Society of America, 76:1115–1132, 1986. [4] M. Bohnhoff, G. Dresen, W.L. Ellsworth, and H. Ito. New Frontiers in Integrated Solid Earth Sciences, (International Year of Planet Earth), chapter Passive Seismic Monitoring of Natural and Induced Earthquakes: Case Studies, Future Directions and Socio-Economic Relevance, pages 261–285. Springer, 2010. [5] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral methods in fluid dynamics. Springer, 1988. [6] C. Chevalier and F. Pellegrini. PT-SCOTCH: a tool for efficient parallel graph ordering. Parallel Computing, 34(6-8):318–331, 2008. [7] G. Cohen. Higher-order numerical methods for transient wave equations. Springer-Verlag, Berlin, Germany, 2002. [8] M. A. Dablain. The application of high-order differencing to the scalar wave equation. Geophysics, 51(1):54–66, 1986. [9] J. D. De Basabe and M. K. Sen. Stability of the high-order finite elements for acoustic or elastic wave propagation with high-order time stepping. Geophysical Journal International, 181(1):577–590, 2010. [10] M. O. Deville, P. F. Fischer, and E. H. Mund. High-Order Methods for Incompressible Fluid Flow. Cambridge University Press, Cambridge, United Kingdom, 2002. [11] M. Durufle, P. Grob, and P. Joly. Influence of Gauss and Gauss-Lobatto quadrature rules on the accuracy of a quadrilateral finite element method in the time domain. Numerical Methods for Partial Differential Equations, 25:526–551, 2009. [12] E. Faccioli, F. Maggio, R. Paolucci, and A. Quarteroni. 2D and 3D elastic wave propagation by a pseudo-spectral domain decomposition method. Journal of Seismology, 1:237–251, 1997. 10.1023/A:1009758820546. 22

[13] G. V. Ganerød, G. Grøneng, J. S. Rønning, E. Dalsegg, H. Elvebakk, J. F. Tønnesen, V. Kveldsvik, T. Eiken, L. H. Blikra, and A. Braathen. Geological model of the ˚ Aknes rockslide, western Norway. Engineering Geology, 102(1-2):1–18, 2008. [14] C. Geuzaine and J. F. Remacle. Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering, 79(11):1309–1331, 2009. [15] H. N. Gharti, V. Oye, and M. Roth. Travel times and waveforms of microseismic data in heterogeneous media. SEG Technical Program Expanded Abstracts, 27(1):1337–1341, 2008. [16] H. N. Gharti, V. Oye, M. Roth, and D. K¨ uhn. Automated microearthquake location using envelope stacking and robust global optimization. Geophysics, 75(4):MA27–MA46, 2010. [17] T. C. Hanks. Earthquake stress drops, ambient tectonic stresses and stresses that drive plate motions. Pure and Applied Geophysics, 115(1):441–458, 1977. [18] J. S. Hesthaven and C. H. Teng. Stable spectral methods on tetrahedral elements. SIAM Journal on Scientific Computing, 21(6):2352–2380, 1999. [19] R. S. Jih, K. L. McLaughlin, and Z. A. Der. Free-boundary conditions of arbitrary polygonal topography in a two-dimensional explicit elastic finite-difference scheme. Geophysics, 53:1045–1055, 1988. [20] D. Komatitsch, R. Martin, J. Tromp, M. A. Taylor, and B. A. Wingate. Wave propagation in 2-D elastic media using a spectral element method with triangles and quadrangles. Journal of Computational Acoustics, 9:703–718, 2001. [21] D. Komatitsch and J. Tromp. Introduction to the spectral element method for threedimensional seismic wave propagation. Geophysical Journal International, 139:806–822, 1999. [22] D. Komatitsch and J. Tromp. Spectral-element simulations of global seismic wave propagation – I. validation. Geophysical Journal International, 149:390–412, 2002. [23] D. Komatitsch and J. P. Vilotte. The spectral element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures. Bulletin of the Seismological Society of America, 88(2):368–392, 1998. [24] D. K¨ uhn, V. Oye, E. Aker, F. Cuisiat, and H. N. Gharti. Moment tensor analysis of acoustic emission data from a triaxial laboratory experiment. European Seismological Commission 32nd General Assembly, Montpellier, France, September 2010. [25] V. Kveldsvik, B. Nilsen, H. Einstein, and F. Nadim. Alternative approaches for analyses of a 100,000 m3 rock slide based on Barton-Bandis shear strength criterion. Landslides, 5(2):161–176, 2008. [26] S. Larsen and C. A. Schultz. ELAS3D: 2D/3D elastic finite difference wave propagation code: Technical Report No. UCRL-MA-121792. Technical report, 1995. 23

[27] W. H. K. Lee and S. W. Stewart. Principles and Applications of Microearthquake Networks. Academic Press, New York, 1981. [28] A. R. Levander. Fourth-order finite-difference P-SV seismograms. Geophysics, 53:1425– 1436, 1988. [29] G. Manthei. Characterization of acoustic emissions in a rock salt specimen under triaxial compression. Bulletin of the Seismological Society of America, 95(5):1674–1700, 2005. [30] E. D. Mercerat, J. P. Vilotte, and F. J. Sánchez-Sesma. Triangular spectral element simulation of two-dimensional elastic wave propagation using unstructured triangular grids. Geophysical Journal International, 166:79–69, 2006. [31] D. A. Nielasen and H. M. Blackburn. Gauss and Gauss-Lobatto element quadratures applied to the incompressible Navier-Stokes equations. In Proceedings of the 8th biannual conference: Computational Techniques and Applications (CTAC 97), 1997. [32] T. Ohminato and B. A. Chouet. A free-surface boundary condition for including 3D topography in the finite difference method. Bulletin of the Seismological Society of America, 87:494–515, 1997. [33] V. Oye, H. Bungum, and M. Roth. Source parameters and scaling relations for mining related seismicity within the Pyhaesalmi ore mine, Finland. Bulletin of the Seismological Society of America, 95:1011–1026, 2005. [34] V. Oye, H. N. Gharti, E. Aker, and D. K¨ uhn. Moment tensor analysis and comparison of acoustic emission data with synthetic data from spectral element method. SEG Technical Program Expanded Abstracts, 29(1):2105–2109, 2010. [35] V. Oye and M Roth. Automated seismic event location for hydrocarbon reservoirs. Computers & Geosciences, 29:851–863, 2003. [36] A. T. Patera. A spectral element method for fluid dynamics: laminar flow in a channel expansion. Journal of Computational Physics, 54:468–488, 1984. [37] P. Pebay, M. Stephenson, L. Fortier, S. Owen, and D. J. Melander. pCAMAL: An embarrassingly parallel hexahedral mesh generator. In Michael L. Brewer and David Marcum, editors, Proceedings of the 16th International Meshing Roundtable, pages 269– 284. Springer Berlin Heidelberg, 2008. [38] D. Peter, D. Komatitsch, Y. Luo, R. Martin, N. Le Goff, E. Casarotti, P. Le Loher, F. Magnoni, Q. Liu, C. Blitz, T. Nissen-Meyer, P. Basini, and J. Tromp. Forward and adjoint simulations of seismic wave propagation on fully unstructured hexahedral meshes. Geophysical Journal International, 2011. accepted. [39] P. Podvin and I. Lecomte. Finite difference computation of traveltimes in very contrasted velocity models: a massively parallel approach and its associated tools. Geophysical Journal International, 105:271–248, 1991. [40] H. Puustjärvi. Pyhaesalmi modeling project, section B. Geology. Technical report, Geological Survey of Finland, and Outokumpu Mining Oy., 1999. 24

[41] R. Rainsberger. TrueGrid User’s Manual. XYZ Scientific Applications, Inc., Livermore, CA, version 2.3.0 edition, 2006. Aknes, Norway. [42] M. Roth and L. H. Blikra. Seismic monitoring of the unstable rock slope at ˚ EGU General Assembly 2009, Vienna, Austria, abstract #EGU2009-3680, 11:3680, 2009. [43] E. H. Saenger and T. Bohlen. Finite-difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid. Geophysics, 69:583–591, 2004. [44] Sandia National Laboratories. CUBIT 13.0 User Documentation, 2011. [Online; accessed 27-May-2011]. [45] G. Seriani. 3-D large-scale wave propagation modeling by spectral element method on Cray T3E multiprocessor. Computer methods in applied mechanics and engineering, 164:235–247, 1994. [46] G. Seriani and S. P. Oliveira. Dispersion analysis of spectral-element methods for elastic wave propagation. Wave Motion, 45:729–744, 2008. [47] J. Shepherd and C. Johnson. Hexahedral mesh generation constraints. Engineering with Computers, 24:195–213, 2008. [48] M. A. Taylor and B. A. Wingate. A generalized diagonal mass matrix spectral element method for non-quadrilateral elements. Applied Numerical Mathematics, 33(1-4):259–265, 2000. [49] L. Thomsen. Weak elastic anisotropy. Geophysics, 51(10):1954–1966, 1986. [50] J. Tromp, D. Komatitsch, and Q. Liu. Spectral-element and adjoint methods in seismology. Communications in Computational Physics, 3(1):1–32, 2008. [51] J. Virieux. P-SV wave propagation in heterogeneous media: velocity-stress finitedifference method. Geophysics, 5:889–901, 1986.

25

Chapter 4 Elastoplastic failure

64

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2011; 00:1–34 Prepared using nmeauth.cls [Version: 2002/09/18 v2.02]

Application of an elastoplastic spectral-element method to 3D slope stability analysis Hom Nath Gharti1,2,∗ , Dimitri Komatitsch3,4 , Volker Oye1 , Roland Martin3 and Jeroen Tromp5 1

NORSAR, Gunnar Randers vei 15, N-2007 Kjeller, Norway 2 Department of Geosciences, Oslo University, Sem Sælands vei 1, N-0371 Oslo, Norway 3 G´ eosciences Environnement Toulouse CNRS UMR 5563, Observatoire Midi-Pyr´ en´ ees (OMP), ´ Universit´ e Paul Sabatier, 14 avenue Edouard Belin, 31400 Toulouse, France 4 Institut universitaire de France, 103 boulevard Saint-Michel, 75005 Paris, France 5 Department of Geosciences and Program in Applied & Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA

SUMMARY We implement a spectral-element method for 3D elastoplastic problems in geomechanics. As a first application, we apply the method to slope stability analysis from small to large scales. The implementation uses an element-by-element preconditioned conjugate gradient solver for efficient storage. The program can handle material heterogeneity and complex topography. Either simple or complex water table profiles may be used to assess effects of hydrostatic pressure. Both surface loading and pseudostatic seismic loading are implemented. A Mohr-Coulomb yield criterion is used in combination with an initial strain method —a visco-plastic algorithm— for nonlinear plastic iterations. The software is parallelized based on domain decomposition, using the Message Passing Interface (MPI) for large-scale problems. Strong-scaling measurements show that the parallelized software performs very efficiently for large-scale problems. We validate our spectral-element results against several other methods, and apply the technique to simulate failure of an earthen embankment and a c 2011 John Wiley & Sons, Ltd. mountain slope. Copyright key words: Finite-element method, Spectral-element method, Elastoplasticity, 3D Slope stability, Pseudostatic seismic loading, Parallel processing

1. INTRODUCTION The finite-element method (FEM) is a powerful tool for solving boundary value problems. In many areas of solid mechanics, it is probably the most widely used numerical method (e.g, [1, 2, 3]). The flexibility of unstructured meshing based on a variety of element types (e.g., triangles, tetrahedra, hexahedra, shells, etc.) and the adaptability of using an arbitrary order of integration via Gaussian quadrature make the FEM a highly versatile tool. However, in

∗ Correspondence

to: NORSAR, Gunnar Randers vei 15, N-2007 Kjeller, Norway

c 2011 John Wiley & Sons, Ltd. Copyright

2

GHARTI ET AL.

Gaussian quadrature interpolation nodes and integration points do not coincide. Consequently, the resulting mass matrix is seldom diagonal, complicating the time-marching scheme. There are some techniques for diagonalizing the mass matrix, e.g., mass lumping, but, unfortunately, these are approximate procedures requiring extra computation. Hence, using a higher-order FEM is often computationally expensive. The spectral-element method (SEM) —not to be confused with a frequency-domain finiteelement method, which is usually referred to as a spectral finite-element method and sometimes also as a spectral-element method— is a higher-order FEM in which integration over an element is based on nodal quadrature. In nodal quadrature —e.g., Gauss-Legendre-Lobatto quadrature, also known as Gauss-Lobatto or Lobatto quadrature— the interpolation nodes of an element and the numerical integration points are the same. The interpolation nodes are determined by the roots of a spectral polynomial, namely the derivative of a Legendre polynomial. The coincidence of integration and interpolation points has two main advantages: 1) the interpolating functions become orthogonal, resulting in a diagonal mass matrix, thereby greatly simplifying the time-marching algorithm because a fully explicit scheme may be used (e.g., [1]), and 2) interpolation is unnecessary to determine nodal quantities, thus simplifying computation of the stiffness matrix, strain, stress, etc. Therefore, higher-order elements are easier to implement for a high-degree of spatial accuracy. However, the SEM also has a number of drawbacks. Nodal quadrature was originally limited to specific types of elements, e.g., quadrilaterals in 2D and hexahedra in 3D. Hexahedral meshing is a challenging task and an area of active research (e.g., [4, 5]). Only a few hexahedral meshing tools are currently available, e.g., Cubit (cubit.sandia.gov), Gmsh [6] etc. Meshing is usually not fully automated, and careful mesh design is necessary. Fortunately, meshing generally is a one-time effort. Recently, there have been some successful implementations of the SEM on other types of elements (e.g., triangles in 2D, tetrahedra in 3D) using so-called Fekete points (e.g., [7, 8, 9, 10]). Another drawback is that since nodal quadrature includes the end points of the integration interval, the order of integration may not always be sufficiently high to reach an acceptable level of accuracy [11]. Although the tradeoffs between high spatial accuracy and low-order integration are not entirely clear, the influence of low-order integration on accuracy and convergence may not be significant, depending on the specifics of the problem [12, 13, 14]. Despite these considerations, the SEM is a versatile tool due to an efficient time-marching scheme and a high degree of spatial accuracy. The SEM was originally developed to address problems in computational fluid dynamics [15, 16]. Detailed reviews may be found in, e.g., Cohen et al. [17] and Deville et al. [18]. Recently, the method has also been widely used to simulate seismic wave propagation from local to global scales (e.g., [19, 20, 21, 22, 23, 24]). The nonlinear SEM has been used for 2D viscoplastic problems [25]. In this article we will deal with 3D elastoplastic problems in the context of slope stability analysis. Several problems in geomechanics, e.g., slope failure, mine or tunnel collapse, etc., may be described using elastoplastic deformation, which is inherently a nonlinear process. Such problems are usually solved using the nonlinear FEM (e.g., [1, 26, 27, 28, 3, 29]). Higher-order finite elements are often desirable for such nonlinear problems. Due to this nonlinearity and the use of higher-order elements, computation is frequently demanding, and therefore the SEM may be a suitable tool for these problems. Slope stability analysis is one of the important problems in geotechnical engineering, offering important guidelines for landslide hazard preparedness, and safe and economic design of c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls

Int. J. Numer. Meth. Engng 2011; 00:1–34

ELASTOPLASTIC SPECTRAL-ELEMENT METHOD

3

infrastructure, such as dams and roads (e.g., [30, 31]). Limit equilibrium analysis is probably the most widely used method for slope stability analysis (e.g., [30, 32]). This method is based on the concept of equilibrium of a rigid body above an assumed failure surface. Therefore, the choice of failure surface is critical in this method. Two forces are assumed to act on the rigid body: a driving force that tries to move the body, and a resisting force that prevents the slope from failing. Failure occurs if the total driving force acting on the rigid body exceeds the resisting force, thereby breaking equilibrium. The ratio between resisting and driving forces for the potential failure surface is often referred to as the factor of safety (FOS). Computation has to be repeated for several assumed failure surfaces to determine the potential failure surface. The actual forces are often estimated by dividing the rigid body into several pieces of a simpler geometry (e.g., vertical columns). There are several limit equilibrium analysis methods, e.g., the ordinary, modified Bishop, Morgenstern & Price, Spencer, and Jabu methods (details may be found in, e.g, Abranson et al. [30]), and there have been numerous improvements on the methods including 3D problems (e.g., [33, 34, 35, 36, 37, 38, 39]). The limit equilibrium method is simple and fast. Unfortunately, it is very difficult to define a realistic failure surface for a complex model with material heterogeneity, particularly in 3D. In addition to the failure surface, assumptions have to be made regarding the side forces on the columns (see e.g., [40]). Stress-deformation analysis is another method used for the slope stability analysis. In this approach, a numerical method is used to compute the displacement field to simulate stress-deformation behavior of the slope. Despite the fact that this approach has a large computational burden, it offers several distinct advantages: no assumptions need to be made regarding a potential failure surface or side forces, and the progressive nature of slope failure may be captured because the displacement field is computed at each stage (e.g., [41]). Due to the advent of modern computers, these methods are becoming increasingly popular. Methods based on this approach may be broadly classified into two main categories: 1) continuum modeling, and 2) discontinuum modeling. Continuum modeling describes the model as a continuous body. Therefore, this method is applicable to slopes whose behavior may be realistically reproduced under the continuum assumption, e.g., soils, massive sound rock, heavily jointed rock, etc. Commonly used continuum based methods are the FEM and the finite-difference method (FDM). While the FEM solves a weak (variational) form of the governing equation on an unstructured mesh, the FDM solves a strong form, often on a structured grid (e.g., [3]). Discretization and the solution procedure are generally relatively simple in the FDM. However, it is often difficult to model complex geometries with a structured grid. Moreover, in the strong formulation a separate equation for the traction free boundary condition must be solved. To the contrary, unstructured meshing is well suited for complex geometries. In addition, in the weak formulation traction free boundary conditions are automatically satisfied (e.g., [42]). Therefore, the FEM is capable of solving a wide range of problems in solid mechanics, and it generally gives more accurate results for complicated boundary geometries. The FEM has been applied to slope stability problems in 2D and 3D (e.g., [43, 44, 45, 46]), and it has been used to model the toppling failure of rock mass (e.g., [47]). There are also a few studies of slope stability based on the FDM (e.g., [48, 49]). Discontinuum modeling considers the model as an assemblage of discrete particles interacting through contact (e.g [50]). Commonly used methods are the distinct-element method, discontinuous deformation analysis (DDA), and particle flow codes (PFC). Collectively, these methods are referred to as discrete-element methods (DEMs). Originally, the DEM was c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


4

GHARTI ET AL.

proposed by Cundall [51, 52] for problems in rock mechanics. Since then, it has been used to address a number of problems, in particular for granular materials and blocky structures (e.g., [53]). It has also been used to study landslides, e.g., the Val Pola landslide [54], and seismic triggering of landslides in Kyrgyzstan [55]. The DEM has also been used to model catastrophic events, e.g, the 1991 Randa rock slide in Switzerland (e.g.,[56]), and mass flow avalanches due to the 1999 Chichi Earthquake in Taiwan (e.g., [57, 58]). Most of these studies were performed in 2D, although there have been a few 3D investigations (e.g., [59, 60, 61, 62]). Although the DEM is capable of simulating discontinuous models realistically, it requires a large number of particles. Hence, this method is often prohibitive for large-scale problems. In this regard, special types of FEM techniques, such as the extended finite-element method (XFEM) (e.g., [63, 64]) and the particle discretization scheme finite-element method (PDSFEM) [65, 66] or the combined finite-discrete element method [67] may be of interest. In the XFEM, so called enrichment functions are separately defined to simulate crack propagation. Recently, the XFEM has been applied to detect soil instabilities in elasto-plastic soils due to existing discontinuities [68]. In the PDS-FEM, displacement and stress fields are separately discretized over a conjugate pair of Voronoi blocks and Delaunay tessellations. This way the discontinuity is naturally introduced within an element along Voronoi boundaries. These methods may also be suitable for slopes, in which case discrete discontinuities or block structures are dominant. Nevertheless, continuum modeling is capable of addressing a wide range of slope stability problems. Simulations of flow after an avalanche have been performed based on granular theory (e.g., [69, 70, 71]), the DEM (e.g., [72, 73, 74]), or elastoplasticity using the Mohr-Coulomb criterion (e.g., [75, 76]). Such processes are beyond the scope of this article. We will deal only with slope stability analysis to determine potential failure surface and the factor of safety of the slope. Most of the existing tools for slope stability analysis are limited to either 2D or relatively simple and small 3D problems. In this article, we implement the SEM for 3D slope stability problems of various sizes. As described earlier, the SEM is a continuum-based approach. Our SEM software can handle material heterogeneity and a complex model with significant topography. Both simple and complex free surface profiles may be used based on a simple hydrostatic relation. We implement both surface loading and pseudostatic seismic loading. For efficient storage, we use an element-by-element preconditioned conjugate gradient method. The program is parallelized based on domain decomposition. We compare our results with classical FEM calculations and several limit-equilibrium-based methods, and apply the method to simulate failure of a reservoir embankment and slumping of a mountain slope.

2. FORMULATION 2.1. Discretization of the governing equation The governing equations for elastodynamics problems may be written in index notation as σij;j + fi = ρ u ï

,

(1)

where a dot over a symbol denotes partial differentiation with respect to time, and σij = Cijkℓ εkℓ represents the stress tensor, εkℓ = 12 (uk;ℓ + uℓ;k ) the strain tensor, Cijkℓ the fourthorder tensor relating stress and strain, fi the force term, ρ the mass density, and ui and u ï c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


5


particle displacement and acceleration, respectively. We use the summation convention for repeated indices, and a semicolon (;) denotes covariant differentiation. The weak form of the governing equations (1) is Z Z Z Z (2) wi fi dΩ + wi ti dΓ , wi;j σij dΩ = ρ wi u ï dΩ + Γ

Ω

Ω

Ω

where wi denotes a test function, Ω and Γ the volume and boundary of the domain, and ni the normal to the boundary. Finally, the traction on the boundary is denoted by ti = σij nj .

1

0

1

−1 −1 0

0 1 −1

(a)

(b)

Figure 1: (a) A typical finite element with 20 interpolation nodes (open circles). (b) The same finite element mapped to its natural coordinates. The Gauss points located inside the element (solid black circles) are used for numerical integration. Only the interpolation nodes on the three visible faces are shown for clarity.

1

0

1

−1 −1 0

0 1 −1

(a)

(b)

Figure 2: (a) A typical spectral element with five interpolation nodes in each dimension (open circles). (b) The same spectral element mapped to its natural coordinates. The GaussLegendre-Lobatto points (solid black circles) are used for numerical integration. Integration points and interpolation nodes coincide. Only the interpolation nodes on the three visible faces are shown for clarity. c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


6

GHARTI ET AL.

Spectral-element discretization and integration techniques are summarized in, e.g., [16, 20, 21, 17, 18, 23]. For completeness, we briefly summarize them here. In the spectral-element method (SEM), the displacement field is expressed as ui (ξ) =

N X

α uα i φ (ξ) ,

(3)

α=1 α where uα i and φ denote nodal displacements and interpolation functions, respectively. In each direction j = 1, 2 or 3 in the natural state with coordinates ξ = {ξj } , the Nj = n + 1 GaussLegendre-Lobatto (GLL) interpolation points are the roots of (1 − ξ 2 ) Pn′ (ξ) = 0 , where Pn denotes the Legendre polynomial of degree n . The total number of interpolation points is the product of the number of GLL points along each direction: N = N1 N2 N3 . The interpolation functions φα in natural coordinates are determined by the tensor product of one-dimensional Lagrange polynomials, 3 Y α (4) φj j (ξj ) , φα (ξ) = j=1

such that α

φj j (ξj ) =

Nj Y (ξj − ξjβ ) α

β=1 β6=αj

(ξj j − ξjβ )

,

(5)

where α is the index of a GLL point located at {α1 , α2 , α3 } , given for instance by α = N1 N2 (α3 − 1) + N1 (α2 − 1) + α1 using a 3D to 1D linear mapping. In a classical finite-element method (FEM), one uses a low degree polynomial —often of degree 1 or 2— as the interpolation function. Figure 1a shows a typical hexahedral mesh used in a FEM. In contrast, the SEM uses a high-degree polynomial. A typical spectral element is shown in Figure 2a. In a FEM, interpolation nodes are determined irrespective of the integration points, whereas in the SEM, interpolation nodes coincide with integration points, i.e., the GLL points. For numerical integration, a point x = {xi } in a deformed element is mapped to a point ξ = {ξj } in the natural element, as illustrated in Figures 1b and 2b, using the transformation x(ξ) =

Ng X

xα ψ α (ξ) .

(6)

α=1

Here ψ α denotes a shape function and Ng the number of geometrical nodes xα of an element. The Jacobian matrix of the transformation has elements given by Jij (ξ) = ∂xi (ξ)/∂ξj . Since the internal GLL points of a spectral element do not contribute to the interelement connectivity, these points can safely be excluded during the interpolation of the model geometry, i.e., N and Ng may be different. Therefore, the shape functions ψ α for the transformation (6) should be chosen carefully, because we cannot always determine these from tensor products similar to equation (4). Depending on element type and the numerical algorithm, fewer points may be sufficient to capture transformation (6). For example, for a planar faceted hexahedral element, only the 8 corner nodes are required to capture the geometry if the deformed mesh is not relevant for further simulations, e.g., straindisplacement matrix (so-called B matrix) computation. Therefore, we usually have N > Ng , c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls



7

and consequently the degree of the interpolating functions, φα , is greater than the degree of the shape functions, ψ α , leading to a subparametric formulation. The SEM is a continuous Galerkin method, in which the interpolation function φα is taken as the test function wi . Upon substituting wi = φα and ui , given by (3), in equation (2), we obtain a set of elemental linear equations that may be written conveniently in matrix-vector form: ¨ e + Ke Ue = Fe , (7) Me U where Me , Ke and Fe are known, respectively, as the mass matrix, stiffness matrix and force ¨ e and Ue are the acceleration and displacement vectors, vector of an element. Similarly, U respectively. Symbolically, we write Z ρ ΦT Φ dΩ , Me = Ωe Z BT C B dΩ , Ke = (8) Ωe Z Z ΦT t dΓ , ΦT f dΩ + Fe = Ωe

Γe

where a superscript T denotes the transpose and Ωe the volume of an element. The quantities Φ , B and C are known, respectively, as the interpolation function matrix, the straindisplacement matrix and the elasticity matrix. More specifically, the elemental mass matrix in equation (8) is determined by Z αβ ρ(x) φα (x) φβ (x) d3 x , (9) Me = Ωe

where α, β = 1, . . . , N . Using numerical integration based on GLL quadrature over the GLL points, we obtain N X Meαβ ≈ wγ ρ(ξ γ ) φα (ξ γ ) φβ (ξ γ ) J(ξ γ ) , (10) γ=1

γ

γ

where w and J(ξ ) are, respectively, the integration weights and the determinant of the Jacobian matrix evaluated at the γth integration point. Using the orthogonality of the interpolation functions, we obtain Meαβ ≈

N X

wγ ρ(ξγ ) δ αγ δ βγ J(ξ γ ) = δ αβ

γ=1

N X

wγ ρ(ξ γ ) J(ξ γ ) ,

(11)

γ=1

where δ αβ is the Kronecker delta. Therefore, the elemental mass matrix is diagonal, which is also true for the global mass matrix. This facilitates an efficient time-marching scheme, which is a significant advantage of the SEM. The diagonal mass matrix is a key difference between the SEM and the classical FEM, in which integration points do not coincide with interpolation points, as shown in Figure 1b. Therefore, in the FEM, interpolation is necessary to compute nodal quantities from their values at quadrature points, and vice versa. In a FEM the mass matrix is seldom diagonal, and hence an approximate diagonalization procedure is frequently necessary, e.g., proportional mass lumping, which requires extra computation. c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


8

GHARTI ET AL.

Upon assembling the elemental equations, we obtain a set of global equations: ¨ + KU = F . MU

(12)

In this article, we deal only with time-independent elastoplastic problems relevant to slope stability analysis, such that KU = F . (13) Thus the SEM advantage of a diagonal mass matrix is irrelevant for static slope stability problems, but we shall see that the SEM still offers benefits over the classical FEM. 2.2. External loading We consider three different slope loads: gravity loading, surface traction, and pseudostatic seismic loading. Gravity loading is caused by the self weight of the slope. It is computed using the volume integral Z ΦT f gr dΩ , (14) (Fgr )e = Ωe

gr

where f is the gravity force per unit volume. In our case, the gravity force always acts downward in the vertical direction, i.e., f gr = {0, 0, − γe } , where γe is the unit weight of slope material on the eth element. Similarly, the load due to surface traction is computed using the surface integral Z tr ΦT t dΓ . (15) (F )e = Γe

To numerically calculate this integral, it is often convenient to define interpolation and shape functions in 2D. These functions may be computed analogous to equations 4 and 6, but considering a quadrilateral element. In that case the 2D Jacobian is given by J(ξ1 , ξ2 ) = k∂ξ1 x × ∂ξ2 xk . Slope stability under earthquakes is commonly analyzed using a pseudostatic approach, in which the effects of an earthquake are represented by constant horizontal and/or vertical accelerations (e.g., [77]). These accelerations are used to determine the pseudostatic load applied to the slope. Therefore, the pseudostatic seismic load may be computed using the volume integral Z ΦT keq dΩ , (16) (Feq )e = γe Ωe

where keq denotes the pseudostatic earthquake coefficient, which may be expressed as the ratio of pseudostatic to gravitational accelerations. The vertical pseudostatic force has minimal influence on stability; therefore, only a horizontal pseudostatic coefficient (kh = k1 = k2 ) is considered. The direction of pseudostatic seismic loading is chosen such that it gives the worst condition for stability. Choosing an appropriate coefficient for seismic loading requires good judgment. The approach of expressing earthquake loading by pseudostatic forces was originally proposed by Terzaghi [78], who suggested a value of kh equal to 0.1 for severe earthquakes, 0.2 for violent earthquakes, and 0.5 for catastrophic earthquakes. Several authors have suggested alternative pseudostatic coefficients depending on the problem (e.g., [79, 80]). It should be noted that the pseudostatic approach is an approximate procedure, and it cannot reproduce actual behavior under cyclic earthquake loading. Sometimes, it can therefore give misleading c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


9


results (e.g., [81, 82]). Nevertheless, this approach offers several advantages. Namely, it is simple and straightforward, can be implemented easily in both limit equilibrium and stressdeformation approaches, and it gives a quantitative value that may be important for the qualitative analysis of earthquake hazard [77]. 2.3. Elastoplastic failure For elastoplastic slope deformation, we implement a Mohr-Coulomb yield criterion considering a non-hardening and non-associative material. Some other commonly used criteria, e.g., Tresca, Huber-von Mises, Drucker-Prager, etc., (e.g., [3]) have relatively simpler yield surfaces with fairly easy implementations. For complex models, such as the critical state family (e.g., [83]), some modifications may be necessary. The Mohr-Coulomb yield surface forms an irregular hexagon in principal stress space, as shown in Figure 3. Mathematically, we have

Figure 3: Mohr-Coulomb yield surface in effective principal stress space.

F = σm sin φ′ + σ ¯

1 cos θ − √ sin φ′ sin θ − c′ cos φ′ 3

,

(17)

where c′ and φ′ represent the effective cohesive strength and internal friction angle of the material, and σm , σ ¯ , and θ are the stress invariants known as the mean stress, deviatoric stress and Lode angle, respectively. The stress invariants are computed using the effective stress, which is obtained using the relation ′ σij = σij − p δij

,

(18)

where p denotes water pressure. A rigorous way of computing p would be to determine the flow net depending on permeability and appropriate seepage conditions (e.g., [3]), but for slope c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


10

GHARTI ET AL.

stability problems it is often considered sufficient and conservative (e.g., [40]) to approximate pressure with the simple hydrostatic relation p = γw h w

,

(19)

where γw denotes the unit weight of water and hw the depth of the water column. We solve the elastoplastic problem in an iterative manner using the initial strain method (also called the visco-plastic algorithm) [84, 85]. In this approach, the plastic strain is computed using the simple relation δεp = ∆t ε˙p

,

(20)

where ∆t and ε˙p denote the pseudo-time step and plastic strain rate (in our context it is also referred to as the visco-plastic strain rate). Cormeau [86] derived the pseudo-time step for unconditional stability of various yield criteria. For the Mohr-Coulomb yield criterion, it is given by 4(1 + ν)(1 − 2ν) , (21) ∆t = E (1 − 2ν + sin2 φ) where E , ν , and φ denote Young’s modulus, Poisson’s ratio, and the friction angle, respectively. For heterogeneous materials we take the smallest pseudo-time step. Similarly, the plastic strain rate ε˙p is given by ∂Q , (22) ε˙p = F ∂σ ′ where Q is a plastic potential, which is usually taken to be the same as the yield function F , ∂Q but with the friction angle φ replaced by the dilation angle ψ . The partial derivative ∂σ ′ may be conveniently expressed in terms of the stress invariants (e.g., [3]) as ¯ ∂Q ∂σm ∂Q ∂ σ ∂Q ∂θ ∂Q = + + ′ ′ ′ ∂σ ∂σm ∂σ ∂σ ¯ ∂σ ∂θ ∂σ ′

.

(23)

We solve the global equations using a constant stiffness approach. The method attempts to satisfy nonlinear behavior by iteratively correcting the loads and solving the linear system at each iteration (e.g., [87]): K Uk = F + (Fp )k−1

.

(24)

The force term, Fp at the kth iteration, depends on the incremental plastic strain, δεp , and is given by X Z (Fp )k = (Fp )k−1 + BT C (δεp )k dΩ . (25) elements Ωe

The force contributed by the incremental plastic strain is self-equilibrating so that the net loading remains the same. This load is accumulated in successive iterations until no integration point violates the yield criterion within a certain tolerance. In our numerical implementation, we use the maximum norm of the difference in displacements between two successive iterations as the convergence criterion. c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls



11

3. PARALLEL IMPLEMENTATION The program is parallelized based on a non-overlapping domain decomposition method, in which each domain contains a unique set of elements; no domains share elements, and only nodes on common domain interfaces are shared. The Message Passing Interface (MPI) is used as the parallel library (e.g., [88, 89]). For efficient parallel processing, the number of elements should be approximately divided equally among domains, and the number of nodes on domain interfaces should be minimal. Therefore, efficient mesh partitioning is required. Fortunately, there are some open-source tools for both sequential and parallel graph partitioning, e.g., Scotch [90, 91] and Metis [92]. In order to solve the system of linear equations, we use a parallel preconditioned conjugate gradient (PCG) method, which is an iterative solver widely used in the classical FEM. This approach has also been implemented in both overlapping and non-overlapping domain decomposition methods on distributed memory systems (e.g., [93, 94, 95]). In the domain decomposition method, each processor solves its own part of the system, occasionally communicating across processors to assemble entities along common interfaces. For more details, we present the parallel algorithm of the PCG in Algorithm 1. ˆ0 r0 := F − K U ˆ −1 r0 z0 := D p0 := z0 for k = 0 to MAX ITERATIONS do rT ˆ z αk := pˆTkK kpˆ k k ˆ k+1 := U ˆ k + αk p U ˆk ˆ k k∞ kp if kα ˆ k+1 k∞ ≤ TOLERANCE return kU rk+1 := rk − αk K p ˆk ˆ −1 rk+1 zk+1 := D rT

ˆ z

k+1 βk := k+1 rT zk k ˆ pk+1 := zk+1 + βk pk end for non-convergence

Algorithm 1: Parallel preconditioned conjugate gradient method to solve the system KU = F . The preconditioner is represented by D . For simplicity, we use a Jacobi preconditioner, i.e., D = diag(K) . All hatted quantities in Algorithm 1 represent the local part of the corresponding assembled (global) quantities. The matrix-vector multiplication K p ˆ may be performed on an element-by-element basis, never forming a global stiffness matrix as given by relation (26). This strategy prevents storage of large arrays. Specifically, X Kp ˆ= Ke p ê , (26) elements

where Ke and p ê are the elemental matrix and vector expanded to the same size as K and p ˆ , respectively, but having entries only in locations corresponding to the eth element. In c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


12

GHARTI ET AL.

practice, neither Ke nor p ê need be expanded. Only the elemental entries are scattered to their respective locations of the composite vector. The stiffness matrix never needs to be assembled along common interfaces; only certain vectors need to be assembled. We only need one assemblage of vector D outside the PCG loop (Algorithm 1), and only two vectors, z and p , need to be assembled inside the loop. The scalar ˆ k+1 k∞ ) have the same values across variables α , β , and L∞ -norms (i.e., kαk p ˆk k∞ and kU processors. Two L∞ -norms and four dot products involved in the numerator and denominator of α and β involve global operations. Each requires the communication of a scalar value across processors. For faster convergence, we may use the displacement field obtained in the previous ˆ 0 ) for the conjugate gradient solver. In the future, nonlinear iteration as the initial guess (U other algorithms which provide data locality, e.g., localized ILU preconditioning (e.g., [96]), will be of interest.

4. NUMERICAL RESULTS 4.1. Validation In order to validate the method, we first apply it to predict a factor of safety (FOS) and potential failure surface of a given model. To determine the factor of safety, we use the shear strength reduction technique (e.g., [97]), in which the strength parameters c′ and φ′ are reduced by a certain reduction factor as c′ tan φ′ ′ ′ cf = , (27) , φf = arctan SRF SRF where SRF is a strength reduction factor. The value of the SRF is increased repeatedly until the slope fails, thereby identifying the factor of safety of the slope. With this method, it would be possible to implement a different SRF for c′ and φ′ , but we choose to use the same SRF for both parameters. To judge the failure stage, we plot the maximum displacement against SRF. The SRF at which we observe a sudden increment in displacement gives the factor of safety [98]. As failure approaches, more Gauss points undergo plastic deformation, requiring a large number of iterations for convergence. Therefore, non-convergence within a predefined limit may also be used to judge failure (e.g., [3]). We should note that convergence is dependent on error tolerance, maximum number of iterations (iteration ceiling), and sometimes problem size. Therefore, some tests may be necessary to determine the iteration ceiling for non-convergence. There are several other procedures to judge failure, e.g., tests on slope bulging [99], limiting shear stress on the potential failure surface [100], the total equivalent plastic strain (TEPS) zone [101], the ratio of unbalanced force to applied loading [102], or formulation of an initial value problem to trace the critical slide line [103]. For validation purposes, we compute the FOS and potential failure surface for the model shown in Figure 4, which consists of a weak layer and water table, closely resembling a realistic field problem. The weak layer may be interpreted as a geosynthetic layer or a discontinuity. This model was originally used in 2D by Fredlund and Krahn [104], and later extended to 3D by Xing [35]. Since then, several authors have used it as a benchmark for validating their methods (e.g., [37, 34, 41, 36]). In order to test the robustness of our method, we consider four different cases motivated by the original problem, as follows c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


13

ELASTOPLASTIC SPECTRAL-ELEMENT METHOD 18.3 m

6.1 m 7.3 m 7.2 m

6.1 m

5.0 m Z Y 40.0 m

X 50.0 m

Figure 4: Slope model used for validation: dry region (light gray), wet region (dark gray), and weak layer with 0.5 m thickness (black). Top surfaces of the wet region mark the water table. The dry and wet regions have the same material properties of unit weight γ=18.8 kN/m3 , friction angle φ=200 , and cohesion c=29 kN/m2 . The weak layer has material properties such that γ=18.8 kN/m3 , friction angle φ=00 , and cohesion c=10 kN/m2 . The dilation angle is assumed to be 0 for both regions. Case Case Case Case

1. 2. 3. 4.

Homogeneous dry slope. Dry slope with a weak layer. Partially wet slope with a weak layer (original model). Partially wet slope with a weak layer and pseudostatic seismic loading.

We compute the displacement field for several strength reduction factors with three different programs: 1) a FEM adapted from Smith and Griffiths [85], 2) our serial SEM, and 3) our parallel SEM. In the FEM we use 20-node hexahedral elements with four Gauss points for numerical integration. This is a reduced integration, which is usually used to prevent the socalled “locking” phenomenon [85, 3]. The FEM package [85] currently we have, cannot handle the water table or pseudostatic seismic loading. Therefore, we use the FEM only for the first two cases. For both the serial and parallel SEM we use 3, 4 or 5 GLL points in each direction, i.e., the number of nodes per element is 27, 64 or 125. Since the model is symmetric about the xz-plane, we take advantage of this symmetry and solve only for the symmetric half of the model using appropriate boundary conditions on the symmetry plane. For the boundary conditions, the left and right faces are partially constrained along the x-axis. Similarly, the front and bottom faces are fully constrained, whereas the symmetry face is partially constrained only along the y-axis. To mesh the model, we use the mesh generation toolkit Cubit developed by Sandia National Laboratory [?]. For the FEM, 20-node hexahedral elements may be created directly within Cubit (Figure 5a). However, creating spectral elements with different GLL nodes is currently not possible within Cubit. Therefore, first we create 8-node hexahedral elements with Cubit, and subsequently, in the SEM program, spectral elements with the desired number of nodes are created based on the GLL points (Figure 5b). If necessary, spectral elements c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


14

GHARTI ET AL.

may also be created using any type of hexahedral element of higher degree. For the parallel SEM implementation, meshes are partitioned using the graph partitioning tool Scotch [90] (Figure 5c). We use a relative tolerance of 10−8 for conjugate gradient iterations and 5 10−4 for nonlinear iterations.

(a)

(b)

(c)

Figure 5: a) Finite element mesh of the model with 20-node hexahedral elements. b) Spectralelement mesh of the model with 3 GLL points in each spatial direction. c) Same as b), but partitioned into eight domains for parallel processing. The total number of elements in all cases is 1,670. Case 1: Homogeneous dry slope This is the simplest model, involving a homogeneous material. The gravity load caused by self weight is the only load acting on the slope. The meshes for the FEM, serial SEM, and parallel SEM are shown in Figures 5a-c. Each mesh consists of 1,670 elements with an average size of 2 m; in the SEM, the number of GLL points in each direction is 3. For the parallel SEM, the mesh is partitioned into eight subdomains. Figure 6 shows the maximum displacement computed at various strength reduction factors, and Figures 7a-b show the required total number of nonlinear plastic iterations and conjugate gradient iterations. Initially, we observe small displacements until the reduction factor reaches approximately 2.0. Up to this value, the model behaves largely as an elastic material, and the three different results accurately match. Only a few iterations are required to converge to the solution. When c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


15


large displacements occur, at a factor of approximately 2.18, the curve becomes almost vertical, suggesting a rapid increase in displacement corresponding to collapse of the slope. A large part of the model undergoes plastic deformation, and the number of iterations required to achieve convergence sharply increases. We define this value as the FOS of this slope. As expected, the results for the serial and parallel SEMs perfectly coincide in terms of both displacements and required iterations. After collapse initiation, although displacements computed by the FEM and SEM are slightly different, the discrepancy between the estimated FOSs is negligible. Regarding the computational cost, the FEM and SEM require roughly the same number of iterations before collapse begins, although the FEM always requires a slightly larger number of iterations. But after collapse initiation the number of iterations required for the FEM is noticeably higher than for the SEM, which may be significant for large scale problem. In Table I we compare the estimated FOS with results obtained by other authors. Our results are in excellent agreement. Despite using only 3 GLL points, we obtain fairly accurate results.

Maximum displacement (m)

0.065

0.055

FEM SEM serial SEM parallel

0.045

0.035

0.025 1.00

1.25

1.50 1.75 2.00 Strength reduction factor

2.25

Figure 6: Maximum displacement at various strength reduction factors for case 1. The resulting displacement vectors are plotted in Figures 8a-c, illustrating possible failure mechanisms. We observe similar patterns for the FEM, serial SEM and parallel SEM. As for the homogeneous slope model, we observe almost circular failure. In fact, we may scale-up the displacement field along these vectors and observe the actual failure pattern (Figures 9a-c), which appears quite realistic. Again the results of the SEM are very similar to those obtained with the FEM. Although we used the same element size in both the FEM and SEM, the SEM figures appear to have higher resolution, but this is a visualization issue. We are able to render 20-node hexahedra directly, but we need to decompose each 27-node hexahedral element into (N1 − 1)(N2 − 1)(N3 − 1) 8-node hexahedra to be able to visualize it with the graphics package that we use. Using the same model, we analyze convergence of the SEM with h-refinement (refining the mesh) and p-refinement (increasing the degree of interpolation) as compared to h-refinement c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


16

GHARTI ET AL.

4

400

10


Total conjugate gradient iterations

Number of plastic iterations

500

300

200

100

0 1.00

1.25


2.25

(a)

x 10

8


6

4

2

0 1.00

1.25


2.25

(b)

Figure 7: a) Total number of plastic iterations, and b) total number of conjugate gradient iterations required at various strength reduction factors for Case 1.

(a)

(b)

(c)

Figure 8: Displacement vectors at failure for Case 1. a) FEM. b) Serial SEM. c) Parallel SEM. Shaded surfaces represent interfaces of partitioned domains. c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


17


(a)

(b)

(c)

Figure 9: Deformed volume at failure for Case 1. a) FEM. b) Serial SEM. c) Parallel SEM. Actual deformation is increased along the displacement vector for clearer visualization. For FEM data visualization, 20-node hexahedral elements are used, whereas for the SEM each hexahedral element is decomposed into (N1 − 1)(N2 − 1)(N3 − 1) 8-node hexahedra.

Xing (1988) Bishop’s modified method (Lam & Fredlund, 1993) Janbu’s simplified method (Lam & Fredlund, 1993) CLARA (Lam & Fredlund, 1993) GLE (Lam & Fredlund, 1993) Chen et al (2001) Chen et al (2003) FEM (Griffiths & Marquez, 2007) FEM SEM

Case 1 2.122 – – – – 2.262 2.187 2.17 2.18 2.18

Case 2 1.553 1.607 1.558 1.62 1.603 1.717 1.603 1.58 1.57 1.57

Case 3 1.441 1.511 1.481 1.54 1.508 – – – – 1.49

Case 4 – – – – – – – – – 1.16

Table I: Comparison of FOS obtained by different methods. The serial and parallel SEM give identical results.

c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


18

GHARTI ET AL.

in the FEM. For the FEM, we perform three complementary computations with three different mesh sizes. For the SEM, we first perform three computations with three different mesh sizes, keeping the number of GLL points fixed at 3. Next, we perform three SEM computations in which we vary the number of GLL points from 3 to 4 to 5, keeping both the number of elements and the mesh size constant. The results of these calculations are summarized in Figures 10a-b. Before collapse begins, we observe almost the same maximum displacements in all cases. After collapse, we observe discrepancies between results obtained with different degrees of h- and p-refinement, but they all basically converge to the same FOS solution; the maximum discrepancy in the FOS is less than 1%. Griffiths and Marquez [41] performed a similar convergence test for their FEM implementation, and found that the FOS discrepancy never exceeded 2%. Mesh sizes of 1 m and 0.075 m result in, respectively, 8 and 22 times more elements than a 2 m mesh size. The results obtained using 4 and 5 GLL points with a 2 m mesh size (p-refinement) are even better than the corresponding h-refinement results. Therefore, the SEM is a very accurate method, which may be important for other elastoplastic problems in geomechanics. However, in the context of slope stability, if the FOS is more important than the actual displacements, 3 GLL points are sufficient, facilitating faster simulations.

0.07 0.06

SEM: Mesh size=2.00m SEM: Mesh size=1.00m SEM: Mesh size=0.75m FEM: Mesh size=2.00m FEM: Mesh size=1.00m FEM: Mesh size=0.75m

0.08 Maximum displacement (m)


0.08

0.05 0.04 0.03 0.02 1.00

1.25


(a)

2.25

0.07 0.06

SEM: GLL points=3 SEM: GLL points=4 SEM: GLL points=5 FEM: Mesh size=2.00m FEM: Mesh size=1.00m FEM: Mesh size=0.75m

0.05 0.04 0.03 0.02 1.00

1.25


2.25

(b)

Figure 10: Maximum displacements at various strength reduction factors for Case 1. a) FEM and SEM simulations with different mesh sizes. The total numbers of elements for mesh sizes of 2 m, 1 m, and 0.75 m (h-refinement) are 1,670, 13,240, and 37,233, respectively. The SEM simulations for mesh sizes of 1 m and 0.075 m were run on 16 and 24 processors, respectively. b) FEM simulations with different mesh sizes and SEM simulations with varying number of GLL points (p-refinement). For the SEM simulations the mesh size is 2 m. In h- and p-refinement, the model is more densely sampled, leading to more accurate results. However, it is interesting to see how p-refinement influences results for the same number of discretization points (spectral nodes). In practice, it is very difficult to retain the same number of spectral nodes with varying polynomial degree. Using the same model as above, we consider three different discretization schemes: 1) 3 GLL points with 1,670 elements, 2) 4 GLL points with 511 elements, and 3) 5 GLL points with 220 elements. These three schemes correspond c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


19


to 15,393, 15,862, and 16,233 spectral nodes, respectively, for a maximum variation of 5%. We summarize the results in Figures 11a-b. Although we observe significant discrepancies in displacements, particularly at failure, the influence on the FOS is negligible. As expected, higher order elements are capable of capturing nonlinear behavior more accurately than lower order elements. Obviously we obtain more accurate displacement fields using 220 elements and 5 GLL points than 1,670 elements and 3 GLL points, but due to the highly nonlinear behavior the former requires more plastic iterations than the latter, thereby increasing the computational burden.

0.065

1200

GLL points=3, Nodes=15393 GLL points=4, Nodes=15862 GLL points=5, Nodes=16233

Number of plastic iterations


0.075

0.055 0.045 0.035

1000

GLL points=3, Nodes=15393 GLL points=4, Nodes=15862 GLL points=5, Nodes=16233

800 600 400 200

0.025 1.00

1.25


(a)

2.25

0 1.00

1.25


2.25

(b)

Figure 11: a) Maximum displacement at various strength reduction factors, and b) number of plastic iterations for three different discretization schemes. The total number of spectral nodes is roughly the same in each case, the maximum variation being about 5%. Our convergence tests show that 3 GLL points may be sufficient for the prediction of the FOS and the potential failure surface of the slope. Therefore, we use 3 GLL points for the remaining three cases. Case 2: Dry slope with a weak layer In this test we add a weak layer to the model in Case 1. Again the only load acting on the slope is gravity. The FEM and SEM meshes are shown in Figures 12a-b. Each mesh consists of 1,890 elements with an average size of 2 m. However, the elements in the weak layer are smaller along the vertical axis. We obtain excellent agreement between the FEM, serial SEM, and parallel SEM (Figure 13). We estimate the value of the FOS to be about 1.57, which is in agreement with results obtained by other authors (Table I), including simulations based on CLARA [105] and the general limit equilibrium (GLE) method by Lam and Fredlund [36]. In the presence of a weak layer, the FOS decreases significantly by about 28%. In comparison to the previous case, we observe interesting failure patterns (Figures 14a-c), illustrating that the slope fails along the weak layer. Both the FEM and SEM simulations show a similar failure pattern. Case 3: Partially wet slope with a weak layer (original model) In this test we add a water table to the model in Case 2, i.e., we test the original model [104, 35], as shown in Figure 4. c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


20

GHARTI ET AL.

(a)

(b)

Figure 12: Mesh for a dry slope with a weak layer (Case 2). a) FEM. b) SEM. The weak layer is shown in black.


0.10 Case 4

0.08

Case 3 Case 2


Case 1

0.06

0.04

0.02 0.50

0.75

1.00 1.25 1.50 1.75 Strength reduction factor

2.00

2.25

(a)

Figure 13: Maximum displacement at various strength reduction factors for four different cases.

There is a gravity load acting on the slope. Additionally, water pressure acts on the wet region of the slope. The mesh for the SEM is shown in Figure 15a. Each mesh consists of 1,800 elements with an average size of 2 m, although it is denser in the weak layer and around the sharp corners of the model. We use homogeneous material properties above and below the water table, which is honored by the mesh. Honoring the water table makes it easier to identify submerged nodes during the calculation of hydrostatic pressure, but it may adversely affect mesh quality, e.g., around sharp corners (Figure 15a). We show an example of meshing without honoring the water table in Example 1. Figure 15b shows the pressure distribution c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls



(a)

21

(b)

(c)

Figure 14: Deformed volume at failure for Case 2. a) FEM. b) Serial SEM. c) Parallel SEM. Actual deformation is increased along the displacement vector for clearer visualization.

due to the water table. Figure 13 illustrates that we obtain a perfect match between the serial and parallel SEM (8 processors). As mentioned earlier, we cannot use our FEM package for this case. We estimate the value of the FOS to be approximately 1.49, which is a 5% reduction compared to Case 2 and a 32% reduction compared to Case 1. This suggests that the presence of a water table in some slopes will significantly reduce the FOS. Our FOS estimate is in close agreement with results obtained by other authors (Table I). The failure pattern (Figures 16a-b) again follows the weak layer, but the volume of failure is larger than in Case 2. Case 4: Partially wet slope with a weak layer and pseudostatic seismic loading We now add a pseudostatic seismic load with a coefficient kh = 0.1 to Case 3. Therefore, in addition to gravity and water pressure, an earthquake load acts on the slope. Because the model is constrained along the y-axis, the pseudostatic force will not have much significance along that direction. Similarly, the slope is oriented toward the positive x-axis, and therefore the worst effect is obtained if we apply the pseudostatic force along the positive x-axis. We use the same mesh as for Case 3. We estimate the value of the FOS to be about 1.16, which is 22% lower than in Case 3. The result seems to be reasonable for a value of kh = 0.1 [77]. As expected, the serial c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


22

GHARTI ET AL.

(a)

(b)

Figure 15: a) SEM mesh for a dry slope with a weak layer and a water table (Cases 3 and 4). The weak layer is shown in black, and the wet region is shown in dark gray. b) Hydrostatic pressure due to the water table.

(a)

(b)

Figure 16: Deformed volume at failure for Case 3. a) Serial SEM. c) Parallel SEM. Actual deformation is increased along the displacement vector for clearer visualization. and parallel SEM (8 processors) are in perfect agreement. Due to the fact that the pseudostatic seismic load extends failure more along the positive x-axis, the unstable volume is larger than in Case 3 (Figures 17a-b). 4.2. Example 1: Reservoir embankment Having validated both the serial and parallel SEM, let us consider the stability of an embankment. We construct a 3D model (Figure 18) from a 2D reservoir embankment model developed by Griffiths and Lane [40], which represents an actual reservoir with an earthen dam [106, 107]. There are two main differences between this problem and the previous four cases. First, the model consists of two slopes with different angles, and second, we need to include surface traction due to the reservoir (Figure 19). Therefore, we need to consider the effects of the gravity load, water pressure, and surface traction simultaneously. There is a constant c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


23


(a)

(b)

Figure 17: Deformed volume at failure for Case 4. a) Serial SEM. c) Parallel SEM. Actual deformation is increased along the displacement vector for clearer visualization. 7.3 m Free surface

17.1 m 7.3 m 180

Z Y 33.5 m X

200 m

28.6 m 230 124.4 m 33.5 m

Figure 18: Reservoir embankment model: dry region (light gray), and wet region (dark gray). The top surfaces of the wet region mark the free surface. Dry and wet regions have the same material properties of unit weight γ=18.2 kN/m3 , friction angle φ=370 , and cohesion c=13.8 kN/m2 . The dilation angle is assumed to be 0 for both regions.

traction on horizontal surfaces and a downwardly increasing traction on sloping submerged upstream surfaces (Figure 19). We again take advantage of the symmetry of the model and solve only for the symmetric half of the model with appropriate boundary conditions on the symmetry plane. The left and right faces are partially constrained along the x-axis, front and back faces are partially constrained along the y-axis, and the bottom face is fully constrained. Figure 20a shows the mesh for this model, which consists of 9,669 elements with an average size of 3 m. The mesh is partitioned into 16 domains for parallel processing. Contrary to Case 3, we do not honor the water table during meshing. This avoids a possible reduction in mesh quality, in particular around sharp corners generated by the water table. With this meshing scheme, however, we need to test all the nodes to determine their submerged condition. The c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


24

GHARTI ET AL.

computed hydrostatic pressure is shown in Figure 20b. Free surface 17.1 m

17.1

Figure 19: Cross-section of the embankment showing surface traction on upstream surfaces due to the reservoir. We compute the FOS and potential failure surface for two possible scenarios: 1) the reservoir is filled, as shown in Figure 18, and 2) the reservoir is empty, corresponding to the drawdown stage or the stage before the reservoir was filled. Obviously, surface traction and hydrostatic pressure need not be considered in the second case. We use a relative tolerance of 10−8 for conjugate gradient iterations and 5 10−4 for nonlinear iterations. From the maximum displacement curve (Figure 21) we estimate a FOS of about 2.54 for the empty reservoir and 1.91 for the filled reservoir. When the reservoir is filled, the FOS is reduced by about 25%, which is very important for safe design of the embankment. For the 2D model, Griffiths and Lane [40] used limit equilibrium to estimate a FOS of 2.42 and 1.90 for the two cases, and a FOS of 2.5 and 1.9 using a FEM. Our results are in good agreement with these results. It may seem surprising that the 2D and 3D FOS results are almost identical, because the FOS obtained in 3D is usually greater than in 2D (equivalent plain strain) [108, 33]. However, it is known that the 3D FOS approaches the 2D FOS as the ratio of slope width to slope height increases (e.g., [41, 39]). In our case, the ratio is relatively large (about 9.4, i.e., 200 m/21.3 m), which explains the similar values of the 3D and 2D FOS. The downstream slope is steeper than the upstream slope. In case of a filled reservoir, the horizontal component of traction on the sloping face acts along the orientation of the downstream slope. As a result, it is more vulnerable than the upstream slope in both cases. Therefore, the downstream slope fails as shown in Figures (22a-b). In the case of a filled reservoir, however, the extent of failure is larger than for an empty reservoir, as expected.

(a)

(b)

Figure 20: a) Mesh of the symmetric half of the reservoir embankment model. The mesh is partitioned into 16 domains for parallel processing. b) Hydrostatic pressure due to the reservoir.




25

With water


0.25 Without water

0.20

0.15

0.10

0.05 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 Strength reduction factor

Figure 21: Maximum displacement at various strength reduction factors for two examples of an earthen embankment: with and without water.

(a)

(b)

Figure 22: Deformed volumes at failure for an earthen embankment. a) Without water. b) With water. Actual deformation is increased along the displacement vector for clearer visualization.

4.3. Example 2: Large-scale mountain slope ˚knes, which is an unstable mountain Finally, we apply the method to a mountain slope at A slope located in western Norway (Figure 23). Its unstable flank is moving at a mean rate of about 6 cm/year and as fast as 14 cm/year [109]. The failure trace is visible on the slope surface. The sliding slope consists of mainly gneiss, densely jointed along the foliation, and its volume is approximately 35-40 million m3 [110, 111, 112]. Due to its massive volume and proximity to the fjord, it poses a potential tsunami risk. Several monitoring instruments are in place for early warning, including a seismic network (e.g.,[113]). Although we do not have detailed information on the failure surface, it is important to know approximately how it will fail. Therefore, rather than determining a factor of safety, we compute elastoplastic deformation for a rather large FOS (we select 2.0), and determine its possible failure pattern. We use a Digital Elevation Model map of the slope and build a 3D model as shown in Figure 24a. Kveldsvik et. al. c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


26

GHARTI ET AL.

˚knes slope (Table II). Therefore, we use [109] have estimated the material properties of the A a homogeneous model based on these material properties. We take an approximate value of cohesion of 200 kN/m2 for the densely jointed granitic rock.

Fjord

Åknes model

Figure 23: ˚ Aknes mountain slope in Norway. Source: www.norgei3d.no. Unit weight Young’s modulus Cohesion Friction angle Dilation angle

26.86 kN/m3 409 kN/m3 200 kN/m2 27.60 13.80 Table II: Material properties for the ˚ Aknes slope model.

The model is meshed with an average element size of 25 m, leading to a total number of 28,480 elements. The mesh is partitioned into 64 domains, as shown in Figure 24b. Due to the relatively coarse mesh, we use 4 GLL points in each direction, i.e., 64 GLL points per element. We set a conjugate gradient tolerance of 10−6 and a nonlinear tolerance of 5 10−4 . We compute displacement fields with two different boundary conditions. In both cases, the front, back and bottom surfaces are fully constrained, and the right surface is constrained only along the horizontal direction. In one case, the left surface is fully constrained, whereas in the second case the left surface is constrained only along the horizontal direction. If the rock is sound, the first case is more appropriate. On the other hand, the second case is more appropriate if the rock is weak. Figures 25a-b show the displacement fields for the two cases. As expected, we observe slightly different displacement fields depending on the case. Consequently, the collapse patterns are also slightly different (Figures 26a-b). In case of a partially constrained left face, larger displacements are observed at the left surface. The collapse patterns are more-or-less circular, due to the fact that we have taken a homogeneous model. c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls



27

949m

723m 1100m

Figure 24: a) ˚ Aknes mountain slope model. b) Spectral-element mesh of the model. The mesh is partitioned into 64 domains for parallel computing. The white lines represent the computational domain interfaces, i.e., the mesh slices for the parallel code.

(a)

(b)

Figure 25: Total displacement at a strength reduction factor of 2.0. a) Left face fully constrained. b) Left face partially constrained.

4.4. Parallel performance Using the ˚ Aknes model, we conduct a strong-scaling performance test, i.e., we measure the computation time for various numbers of processors while keeping the problem size fixed. We run the program on 16, 32, 48, 64 and 80 processors. The results shown in Figure 27 demonstrate that the code performs reasonably well for large-scale problems. In the future, it may be important to implement reverse Cuthill-McKee sorting of the degrees of freedom and bandwidth reduction of the element adjacency matrix [114], renumbering nodes to further enhance parallel performance (e.g., [115]). c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


28

GHARTI ET AL.

(a)

(b)

Figure 26: Displacement vectors at a strength reduction factor of 2.0. a) Left face fully constrained. b) Left face partially constrained. The vectors have the same scale in both cases. The shaded surfaces represent the interfaces of the partitioned domains used for parallel computing.

Total elapsed time (s)

40000

Actual Reference

20000

10000

5000 16

32 48 Number of processors

64

80

Figure 27: Total elapsed time for a fixed problem size run on 16, 32, 48, 64 and 80 processors, compared to a reference line computed using the total elapsed time for 48 processors, and assuming perfect linear scaling.

5. DISCUSSION AND CONCLUSIONS We have implemented a spectral-element method for 3D elastoplastic problems in geomechanics, and developed versatile software for slope stability analysis. The numerical method is parallelized based on domain decomposition. We successfully compared the results of our simulations with a classical finite-element method as well as other limit-equilibriumbased methods. We currently need to input possible strength reduction factors manually. In the future, it would be useful to implement an automatic procedure to find the possible FOS using simple root finding methods, such as the bisection method (e.g., [102]). Results obtained for an earthen embankment are in excellent agreement with 2D results from the literature. Finally, we simulated a mountain slope using a simplified model of an unstable rock slope at c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls



29

˚ Aknes, Norway. This simulation illustrates potential applications of the software to complex and large scale problems in geomechanics. Considering the fact that the rock mass at ˚ Aknes is heavily jointed, it would be of future interest to use the Hoek-Brown failure criterion [116]. We plan to include more realistic model parameters and boundary conditions to perform more relevant simulations. With such detailed models, it would be possible to predict the long-term creep behavior of the ˚ Aknes mountain slope. In our examples, we have used an almost uniform mesh size. In the future, it would be helpful to use a geometrically adaptive mesh for large scale problems. For other elastoplastic problems, using a more accurate and efficient integration algorithm for the elastoplastic constitutive relation may be important, e.g., a modified Euler method, in particular one with automatic error control on load stepping (e.g., [117, 118]) and a return mapping with tangent stiffness approach (e.g., [26, 119]). It would be interesting to see how the performance of the spectral-element method compares with hp-finite element methods (e.g., [120, 121]), in particular, for highly nonlinear problems. We envisage further developments for geodynamic problems related to glacial rebound and post-seismic relaxation with the inclusion of viscoelasticity, e.g., based on a generalized Maxwell rheology. Finally, thus far we have implemented only material nonlinearity. Geometrical nonlinearity could be implemented for the analysis of large displacements, e.g., based on adaptive mesh refinement. Our software is open-source, and until we make a formal release it is available upon request from the corresponding author.

ACKNOWLEDGEMENTS

We thank Michael Roth, Valérie Maupin and Daniela K¨ uhn for helpful discussions and suggestions. We thank ˚ Aknes/Tafjord Beredskap IKS (www.aknes.no) for the ˚ Aknes digital elevation map. The parallel programs were run at the Titan High Performance Computing facilities of the University of Oslo, Norway, and at the Princeton Institute for Computational Science and Engineering (PICSciE), USA. 3D data were visualized using the open-source parallel visualization software ParaView/VTK (www.paraview.org). The open-source graphics packages Inkscape and GIMP were used to create and edit some of the figures. This work was funded in part by the Norwegian Research Council, and supported by industry partners BP, Statoil and Total.

REFERENCES 1. Hughes TJR. The finite element method: linear static and dynamic finite element analysis. Prentice-Hall, 1987. 2. Bathe KJ. Finite Element Procedures. Prentice Hall, 1995. 3. Zienkiewicz OC, Taylor RL. The finite element method for solid and structural mechanics. Elsevier Butterworth-Heinemann, 2005. 4. Pebay P, Stephenson M, Fortier L, Owen S, Melander DJ. pCAMAL: An embarrassingly parallel hexahedral mesh generator. Proceedings of the 16th International Meshing Roundtable, Brewer ML, Marcum D (eds.). Springer Berlin Heidelberg, 2008; 269–284. 5. Shepherd J, Johnson C. Hexahedral mesh generation constraints. Engineering with Computers 2008; 24:195–213. 6. Geuzaine C, Remacle JF. Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering 2009; 79(11):1309–1331. c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


30

GHARTI ET AL.

7. Hesthaven JS, Teng CH. Stable spectral methods on tetrahedral elements. SIAM Journal on Scientific Computing 1999; 21(6):2352–2380. 8. Taylor MA, Wingate BA. A generalized diagonal mass matrix spectral element method for nonquadrilateral elements. Applied Numerical Mathematics 2000; 33(1-4):259–265. 9. Komatitsch D, Martin R, Tromp J, Taylor MA, Wingate BA. Wave propagation in 2-D elastic media using a spectral element method with triangles and quadrangles. Journal of Computational Acoustics 2001; 9:703–718. 10. Mercerat ED, Vilotte JP, S´ anchez-Sesma FJ. Triangular spectral element simulation of two-dimensional elastic wave propagation using unstructured triangular grids. Geophysical Journal International 2006; 166:679–698. 11. Durufle M, Grob P, Joly P. Influence of Gauss and Gauss-Lobatto quadrature rules on the accuracy of a quadrilateral finite element method in the time domain. Numerical Methods for Partial Differential Equations 2009; 25:526–551. 12. Nielasen DA, Blackburn HM. Gauss and Gauss-Lobatto element quadratures applied to the incompressible Navier-Stokes equations. Proceedings of the 8th biannual conference: Computational Techniques and Applications (CTAC 97), 1997. 13. Seriani G, Oliveira SP. Dispersion analysis of spectral-element methods for elastic wave propagation. Wave Motion 2008; 45:729–744, doi:10.1016/j.wavemoti.2007.11.007. 14. De Basabe JD, Sen MK. Stability of the high-order finite elements for acoustic or elastic wave propagation with high-order time stepping. Geophysical Journal International 2010; 181(1):577–590, doi:10.1111/j.1365-246X.2010.04536.x. 15. Patera AT. A spectral element method for fluid dynamics: laminar flow in a channel expansion. Journal of Computational Physics 1984; 54:468–488. 16. Canuto C, Hussaini MY, Quarteroni A, Zang TA. Spectral methods in fluid dynamics. Springer, 1988. 17. Cohen G. Higher-order numerical methods for transient wave equations. Springer-Verlag: Berlin, Germany, 2002. 18. Deville MO, Fischer PF, Mund EH. High-Order Methods for Incompressible Fluid Flow. Cambridge University Press: Cambridge, United Kingdom, 2002. 19. Seriani G. 3-D large-scale wave propagation modeling by spectral element method on Cray T3E multiprocessor. Computer methods in applied mechanics and engineering 1994; 164:235–247. 20. Komatitsch D, Vilotte JP. The spectral element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures. Bulletin of the Seismological Society of America 1998; 88(2):368–392. 21. Komatitsch D, Tromp J. Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophysical Journal International 1999; 139:806–822. 22. Komatitsch D, Tromp J. Spectral-element simulations of global seismic wave propagation – I. validation. Geophysical Journal International 2002; 149:390–412. 23. Tromp J, Komatitsch D, Liu Q. Spectral-element and adjoint methods in seismology. Communications in Computational Physics 2008; 3(1):1–32. 24. Oye V, Gharti HN, Aker E, K¨ uhn D. Moment tensor analysis and comparison of acoustic emission data with synthetic data from spectral element method. SEG Technical Program Expanded Abstracts 2010; 29(1):2105–2109, doi:10.1190/1.3513260. 25. di Prisco C, Stupazzini M, Zambelli C. Nonlinear SEM numerical analyses of dry dense sand specimens under rapid and dynamic loading. International Journal for Numerical and Analytical Methods in Geomechanics 2007; 31:757–788. 26. Simo JC, Hughes TJR. Computational inelasticity. Springer, 1998. 27. Belytschko T, Liu WK, Moran B. Nonlinear Finite Elements for Continua and Structures. Wiley, 2000. 28. Luccioni LX, Pestana JM, Taylor RL. Finite element implementation of non-linear elastoplastic constitutive laws using local and global explicit algorithms with automatic error control. International Journal for Numerical Methods in Engineering 2001; 50:1191–1212. 29. Dupros F, De Martin F, Foerster E, Komatitsch D, Roman J. High-performance finite-element simulations of seismic wave propagation in three-dimensional nonlinear inelastic geological media. Parallel Computing 2010; 36:308–325. 30. Abramson LW, Lee TS, Sharma S, Boyce GM. Slope Stability and Stabilization Methods. Wiley, 1995. 31. Cheng YM, Lau CK. Slope stability analysis and stabilization: new methods and insight. Spon Press, 2008. 32. Duncan JM, Wright SG. Soil strength and slope stability. John Wiley & Sons, 2005. 33. Hungr O. An extension of Bishop’s simplified method of slope stability analysis to three dimensions. G´ eotechnique 1987; 37:113–117. 34. Chen J, Yin JH, Lee CF. Upper bound limit analysis of slope stability using rigid finite elements and nonlinear programming. Canadian Geotechnical Journal 2003; 40:742–752. 35. Xing Z. Three-dimensional Stability Analysis of Concave Slopes in Plan View. Journal of Geotechnical c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls



31

Engineering 1988; 114(6):658–671. 36. Lam L, Fredlund DG. A general limit equilibrium model for three-dimensional slope stability analysis. Canadian Geotechnical Journal 1993; 30:905–919. 37. Chen Z, Wang X, Haberfield C, Yin JH, Wang Y. A three-dimensional slope stability analysis method using the upper bound theorem : Part I: theory and methods. International Journal of Rock Mechanics and Mining Sciences 2001; 38(3):369–378. 38. Zhu D, Lee CF, Jiang HD. Generalized framework of limit equilibrium methods for slope stability analysis. G´ eotechnique 2003; 53(4):377–395. 39. Michalowski RL. Limit analysis and stability charts for 3D slope failures. Journal of Geotechnical and Geoenvironmental Engineering 2010; 136(4):583–593. 40. Griffiths DV, Lane PA. Slope stability analysis by finite elements. G´ eotechnique 1999; 49:387–403. 41. Griffiths DV, Marquez RM. Three-dimensional slope stability analysis by elasto-plastic finite elements. G´ eotechnique 2007; 57:537–546. 42. Reddy JN. Introduction to the Finite Element Method. McGraw-Hill, 1993. 43. Smith IM, Hobbs R. Finite element analysis of centrifuged and built-up slopes. G´ eotechnique 1974; 24:531–559. 44. Zienkiewicz OC, Humpheson C, Lewis RW. Associated and non-associated viscoplasticity and plasticity in soil mechanics. G´ eotechnique 1975; 25:671–689. 45. Griffiths DV. Finite element analysis of walls, footings and slopes. Proceedings of the symposium on computer applications to geotechnical problems in highway engineering, Randolph MF (ed.), 1980; 122– 146. 46. Sainak AN. Application of three-dimensional finite element method in parametric and geometric studies of slope stability. Advances in geotechnical engineering (Skempton Conference), vol. 2, 2004; 933–942. 47. Kalkani EC, Piteau DR. Finite element analysis of toppling failure at Hell’s Gate Bluffs, British Columbia. Bulletin of the International Association of Engineering Geology 1976; 13(4):315–327. 48. Chung AK. On the boundary conditions in slope stability analysis. International Journal for Numerical and Analytical Methods in Geomechanics 2003; 27:905–926. 49. Kinakin D, Stead D. Analysis of the distributions of stress in natural ridge forms: implications for the deformation mechanisms of rock slopes and the formation of sackung. Geomorphology 2005; 65:85–100. 50. Jing L, Stephansson O. Fundamentals of Discrete Element Methods for Rock Engineering: Theory and Applications. Elsevier Science, 2007. 51. Cundall PA. A computer model for simulating progres-sive large scale movements in blocky rock systems. ISRM Symp., Proc. 2, Nancy, France, 1971; 129–136. 52. Cundall PA, Strack ODL. A discrete numerical model for granular assemblies. G´ eotechnique 1979; 29(1):47–65. 53. Itasca. Itasca software products-FLAC, FLAC3D, UDEC, 3DEC, PFC, PFC3D. Itasca Consulting Group Inc., Minneapolis 2004; . 54. Calvetti F, Crosta G, Tatarella M. Numerical simulation of dry granular flows: from the reproduction of small-scale expriments to the prediction of rock avalanches. Rivista Italiana Geotecnica 2000; XXXIV(2):21–38. 55. Havenith HB, Strom A, Calvetti F, Jongmans D. Seismic triggering of lanslides. Part B: Simulation of dynamic failure processes. Natural Hazards and Earth System Sciences 2003; 3:663–682. 56. Eberhardt E. Twenty-ninth Canadian geotechnical colloquium: the role of advanced numerical methods and geotechnical field measurements in understanding complex deep-seated roch slope failure mechanisms. Canadian Geotechnical Journal 2008; 45:484–510. 57. Kuo CY, Tai YC, F B, Mangeney A, Pelanti M, Chen RF, Chang KJ. Simulation of Tsaoling landslide, Taiwan, based on Saint Venant equations over general topography. Engineering Geology 2009; 104(14):181–189. 58. Tang CL, Hu JC, Lin ML, Angelier J, Lu CY, Chan YC, Chu HT. The Tsaoling landslide triggered by the Chi-Chi earthquake, Taiwan: Insights from a discrete element simulation. Engineering Geology 2009; 106:1–19. 59. Campbell CS, Cleary PW, Hopkins M. Large-scale landslide simulations: global deformation velocities and basal friction. Journal of Geophysical Research 1995; 100:8267–8283. 60. Zhu F, Stephansson O, Wang Y. Stability investigation and reinforcement for slope at Daye open pit mine, China. Proceedings of EUROCK 96: Prediction and Performance in Rock Mechanics and Rock Engineering, Turin, A.A. Balkema, Rotterdam, 1996; 621–625. 61. Valdivia C, Lorig L. Slope stability at escondida mine. Slope stability in Surface Mining, Hustrulid WA, McCarter MK, Van Zyl DJA (eds.), Society for Mining Metallurgy & Exploration, 2000; 153–162. 62. Corkum AG, Martin CD. Discrete element analysis of the effect of a toe berm on a large rockslide. Proceedings of the 55th Canadian Geotechnical Conference, Niagara Falls, Stolle et al. (Eds.): Canadian Geotechnical Society, Toronto, Canada, 2002; 633–640. c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


32

GHARTI ET AL.

63. Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering 1999; 45(5):601–620. 64. Sukumar N, Mo¨ es N, Moran B, Belytschko T. Extended finite element method for three-dimensional crack modeling. International Journal for Numerical Methods in Engineering 2000; 48(11):1549–1570. 65. Hori M, Oguni K, Sakaguchi H. Proposal of FEM implemented with particle discretization for analysis of failure phenomena. Journal of the Mechanics and Physics of Solids 2005; 53(3):681–703, doi: 10.1016/j.jmps.2004.08.005. 66. Oguni K, Wijerathne MLL, Okinaka T, Hori M. Crack propagation analysis using pds-fem and comparison with fracture experiment. Mechanics of Materials 2009; 41(11):1242–1252, doi: 10.1016/j.mechmat.2009.07.003. 67. Munjiza A. The Combined Finite-Discrete Element Method. John wiley and sons, 2004. 68. Sanborn SE, Pr´ evost JH. Frictional slip plane growth by localization detection and the extended finite element method (XFEM). International Journal of Numerical and Analytical Methods in Geomechanics 2010; doi:10.1002/nag.958. 69. Savage SB, Hutter K. The motion of a finite mass of granular material down a rough incline. Journal of Fluid Mechanics 1989; 199:177–215. 70. Hutter K, Svendsen B, Rickenmann D. Debris flow modeling: a review. Continuum Mechanics and Thermodynamics 1996; 8:1–35. 71. Dartevelle S. Numerical modelling of geophysical flows. 1. A comprehensive approach to granular rheologies and geophysical multiphase flows. Geochemistry Geophysics Geosystem 2004; 5:2003GC000 636. 72. Linares-Guerrero E, Goujon C, Zenit R. Increased mobility of bidisperse granular avalanches. Journal of Fluid Mechanics 2007; 593:475–504. 73. Goujon C, Dalloz-Dubrujeaud B, Thomas N. Bidisperse granular avalanches on inclined planes: a rich variety of behaviors. European Physical Journal E 2007; 23(2):199–215. 74. Staron L. Mobility of long-runout rock flows: a discrete numerical investigation. Geophyical Journal International 2008; 172(1):455–463. 75. Pouliquen O, Forterre Y. Friction law for dense granular flows: Application to the motion of a mass down an inclined plane. Journal of Fluid Mechanics 2002; 453:133–151. 76. Mangeney-Castelnau A, Vilotte J, Bristeau M, Perthame B, Bouchut F, Simeoni C, Yerneni S. Numerical modeling of avalanches based on Saint Venant equations using a kinetic scheme. Journal of Geophysical Research 2003; 108(B11), doi:10.1029/2002JB002024. 77. Kramer SL. Geotechnical earthquake engineering. Prentice Hall, 1996. 78. Terzaghi K. Engineering geology (Berkey) Volume, chap. Mechanisms of landslides. Geological Society of America, 1950. 79. Seed HB. Considerations in the earthquake-resistant design of earth and rockfill dams. G´ eotechnique 1979; 29:215–263. 80. Hynes-Griffin ME, Franklin AG. Rationalizing the seismic coefficient method. Technical Report, US Army Corps of Engineers waterways experiment station 1984. 81. Seed HB, Makdisi FI, Idriss IM, Lee KL. The slides in the San Fernando dams during the earthquake of February 9, 1971. Journal of the Geotechnical Engineering Division 1975; 101:651–688. 82. Marcuson WF, Ballard R, Ledbetter RH. Liquefaction failure of tailings dams resulting from the Near Izu Oshima earthquake, 14 and 15 January, 1978. 6th Pan American conference on soil mechanics and foundation engineering, Lima, Peru, 1979. 83. Schofield AN, Wroth CP. Critical state soil mechanics. McGraw-Hill, 1968. 84. Zienkiewicz O, Cormeau I. Visco-plasticity-plasticity and creep in elastic solids-a unified numerical solution approach. International Journal for Numerical Methods in Engineering 1974; 8(4):821–845. 85. Smith IM, Griffiths DV. Programming the finite element method. John Wiley & Sons, 2004. 86. Cormeau I. Numerical stability in quasi-static elasto/visco-plasticity. International Journal for Numerical Methods in Engineering 1975; 9:109–127. 87. Thomas JN. An improved accelerated initial stress procedure for elasto-plastic finite element analysis. International Journal for Numerical and Analytical Methods in Geomechanics 1984; 8:359–379. 88. Gropp W, Lusk E, Skjellum A. Using MPI, portable parallel programming with the Message-Passing Interface. MIT Press: Cambridge, USA, 1994. 89. Pacheco P. Parallel Programming with MPI. Morgan Kaufmann, 1997. 90. Pellegrini F, Roman J. SCOTCH: A software package for static mapping by dual recursive bipartitioning of process and architecture graphs. Lecture Notes in Computer Science 1996; 1067:493–498. 91. Chevalier C, Pellegrini F. PT-SCOTCH: a tool for efficient parallel graph ordering. Parallel Computing 2008; 34(6-8):318–331. 92. Karypis G, Kumar V. Multilevel k-way partitioning scheme for irregular graphs. Journal of Parallel and Distributed Computing 1998; 48(1):96–129. c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls



33

93. Sobh NA, Gustafson K. Preconditioned conjugate gradient and finite element methods for massively data-parallel architectures. Computer Physics Communications 1991; 65:253–267. 94. Khan AI, Topping BHV. Parallel finite element analysis using Jacobi-conditioned conjugate gradient algorithm. Advances in Engineering Software 1996; 25:309–319. 95. Liu Y, Zhou W, Yang Q. A distributed memory parallel element-by-element scheme based on Jacobiconditioned conjugate gradient for 3D finite element analysis. Finite Elements in Analysis and Design 2007; 43:494–503. 96. Nakajima K, Okuda H. Parallel iterative solvers with localizedILU preconditioning for unstructured grids on workstation clusters. International Journal of Computational Fluid Dynamics 1999; 12(3):315–322, doi:10.1080/10618569908940835. 97. Matsui T, San KC. Finite element slope stability analysis by strength reduction technique. Soils and foundations 1992; 32:59–70. 98. Manzari MT, Nour MA. Significance of soil dilatancy in slope stability analysis. Journal of Geotechnical and Geoenvironmental Engineerin 2000; 126(1):75–80. 99. Snitbhan N, Chen WF. Elastic-plastic large deformation analysis of soil slopes. Computers & structures 1976; 9:567–577. 100. Dunlop P, Duncan JM. Development of failure around excavated slopes. Journal of the Soil Mechanics and Foundations Division 1970; 96:471–493. 101. Chen JX, Ke PZ, Zhang G. Slope stability analysis by strength reduction elasto-plastic FEM. Key Engineering Materials 2007; 345-346:625–628. 102. Dawson EM, Roth WH, Drescher A. Slope stability analysis by strength reduction. G´ eotechnique 1999; 49:835–840. 103. Zheng H, Liu DF, Li CG. Slope stability analysis based on elasto-plastic finite element method. International Journal for Numerical Methods in Engineering 2005; 64:1871–1888. 104. Fredlund DG, Krah J. Comparison of slope stability methods of analysis. Canadian Geotechnical Journal 1977; 14:429–439. 105. Hungr O. CLARA 2.31: Slope stability in two or three dimensions for IBM compatible microcomputers. O. Hungr Geotechnical Research Inc., Vancouver 1988. 106. Paice GM. Finite element analysis of stochastic soils. PhD Thesis, University of Manchester, U.K. 1997. 107. Torres R, Coffman T. Liquefaction analysis for CAS: Willow Creek dam. Technical Report WL-8312-2, US Department of the interior, Bureau of Reclamation 1997. 108. Hutchinson JN, Sarma SK. Discussion on ’Three-dimensional limit equilibrium analysis of slope’. G´ eotechnique 1985; 35:215. 109. Kveldsvik V. Static and dynamic stability analyses of the 800m high ˚ aknes rock slope, western norway. PhD Thesis, Norwegian University of Science and Technology, Norway 2008. 110. Ganerød GV, Grøneng G, Rønning JS, Dalsegg E, Elvebakk H, Tønnesen JF, Kveldsvik V, Eiken T, Blikra LH, Braathen A. Geological model of the ˚ Aknes rockslide, western Norway. Engineering Geology 2008; 102(1-2):1–18. 111. Kveldsvik V, Nilsen B, Einstein HH, Nadim F. Alternative approaches for analyses of a 100,000 m3 rock slide based on Barton-Bandis shear strength criterion. Landslides 2008; 5(2):161–176. 112. Kveldsvik V, Einstein HH, Nilsen B, Blikra LH. Numerical analysis of the 650,000 m2 ˚ Aknes rock slope based on measured displacements and geotechnical data. Rock mechanics and rock engineering 2009; 42:689–728. 113. Roth M, Blikra LH. Seismic monitoring of the unstable rock slope at ˚ Aknes, Norway. EGU General Assembly 2009, Vienna, Austria, abstract #EGU2009-3680 2009; 11:3680. 114. Cuthill E, McKee J. Reducing the bandwidth of sparse symmetric matrices. Proceedings of the 24th National ACM Conference, ACM Press, New York, 1969; 157–172. 115. Komatitsch D, Labarta J, Mich´ ea D. A simulation of seismic wave propagation at high resolution in the inner core of the earth on 2166 processors of marenostrum. High Performance Computing for Computational Science - VECPAR 2008, Palma JM, Amestoy PR, Dayd´ e M, Mattoso M, Lopes JaC (eds.). Springer-Verlag: Berlin, Heidelberg, 2008; 364–377. 116. Hoek E, Carranza-Torres C, Corkum B. Hoek-Brown failure criterion – 2002 edition. Proceedings of the fifth North American rock mechanics symposium, vol. 1, 2002; 267–273. 117. Abbo AJ, Sloan SW. An automatic load stepping algorithm with error control. International Journal for Numerical Methods in Engineering 1996; 39:1737–1759. 118. Sloan SW, Abbo AJ, Sheng D. Refined explicit integration of elastoplastic models with automatic error control. Engineering Computations 2001; 18:121–194. 119. de Souza Neto EA, Peri´ c D, Owen DRJ. Computational methods for plasticity: theory and applications. Wiley, 2009. 120. Babuska I, Suri M. The p- and h-p versions of the finite element method, an overview. Computer Methods in Applied Mechanics and Engineering 1990; 80:5–26. c 2011 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


34

GHARTI ET AL.

121. Schwab C. p- and hp- Finite Element Methods: Theory and Applications to Solid and Fluid Mechanics. Oxford University Press, USA, 1999.



Chapter 5 Multistage excavation

99

Simulation of multistage excavation based on a 3D spectral-element method Hom Nath Ghartia,b,∗, Volker Oyea , Dimitri Komatitschc,d , Jeroen Trompe a

NORSAR, Gunnar Randers vei 15, N-2007 Kjeller, Norway Department of Geosciences, Oslo University, Sem Sælands vei 1, N-0371 Oslo, Norway c Géosciences Environnement Toulouse CNRS UMR 5563, Observatoire Midi-Pyrénées (OMP), ´ Université Paul Sabatier, 14 avenue Edouard Belin, 31400 Toulouse, France d Institut universitaire de France, 103 boulevard Saint-Michel, 75005 Paris, France e Department of Geosciences and Program in Applied & Computational Mathematics, Princeton University, Princeton, New Jersey, USA

b

Abstract We implement a 3D spectral-element method for multistage excavation problems. To simulate excavation in elastoplastic soils, we implement a Mohr-Coulomb yield criterion using an initial strain method. We parallelize the software based on non-overlapping domain decomposition using the MPI. We verify the uniqueness principle for multistage excavation in linear elastic materials. We validate our serial and parallel programs, and illustrate several examples of multistage excavation in elastoplastic materials. We finally apply our software to a model of the Pyhäsalmi ore mine in Finland. Strong-scaling performance tests involving multistage excavation show that the parallel program performs reasonably well for large-scale problems. Keywords: Spectral-element method, multistage excavation, mining, elastoplasticity, parallel processing

Corresponding author Email addresses: [email protected] (Hom Nath Gharti), [email protected] (Volker Oye), [email protected] (Dimitri Komatitsch), [email protected] (Jeroen Tromp) ∗

Preprint submitted to Computers & Structures

September 2, 2011

1. Introduction Excavation is a common process in many geotechnical engineering constructions, for example, underground caverns, tunneling, mining, road construction, etc. (e.g., [1]). During excavation, a certain portion of the model is removed in stages, thereby significantly changing the geometry. Accordingly, the domain, domain boundary, and boundary conditions of the problem change with each excavation stage (e.g., [2]). After excavation, newly generated surfaces become traction-free boundaries (e.g., [3]). Therefore, during the excavation process, not only the geometry of the excavation but also the resulting stress redistribution have important consequences for the stability of the excavated region. Simulation of multistage excavation provides progressive information during the excavation process, which may be important for economic design and efficient risk mitigation. The finite-element method (FEM) is commonly used to simulate the excavation process. The FEM is a powerful tool for solving boundary value problems, and is widely used in many applications of solid and fluid mechanics (e.g., [4–6]). The original idea behind FEM-based simulations of multistage excavation was to nullify contributions of nodes of an excavated element by considering an infinitesimal stiffness (e.g., [7–9]). In a linear elastic medium, multistage excavation satisfies the uniqueness principle first postulated by Ishihara (e.g., [10, 11]). The uniqueness principle is based on the principle of superposition; it states that the final solution is independent of the sequence of excavation. Based on the infinitesimal stiffness approach, it was not always possible to satisfy the uniqueness principle due to spurious contributions from excavated nodes. To overcome this problem, Desai and Sargand [12] proposed a hybrid method to compute nodal loads that satisfy the uniqueness principle. Similarly, Borja et al. [13, 14] developed a method employing infinitesimal stiffnesses and a variational formulation to satisfy the uniqueness principle. Comodromos et al. [15] suggested a method based on the so-called ‘variable domain vector’, which basically tracks intact and excavated entities so that contributions of excavated nodes may be removed by static condensation during the solution procedure (e.g., [5, 16]). Smith and Griffiths [2] used a similar technique, forming the global equations only for intact elements. There are several other studies that deal with excavation based on the FEM, e.g., 2D excavation models [17, 18], stability of vertical excavation in plane-strain and axisymmetric problems [19], and deep excavation [20]. There are also some studies involving 3D excavation, e.g., 3D modeling of tunneling [21], deep excavation in Shanghai [22], and 2

simulations based on parallel processing [23]. Other related research includes transient analysis of excavation [24] and excavation in poroelastic media [25]. The discrete-element method (DEM; e.g., [26, 27]) is also used to simulate excavation (e.g., [28]). The DEM is often suitable for discontinuous media to simulate brittle failure. For reliable simulations, a large number of discrete particles is often required, which involves large computational costs. Therefore, special types of FEMs, such as the extended finite-element method (XFEM; e.g., [29, 30]), the particle discretization scheme finite-element method (PDS-FEM; e.g., [31, 32]), or the combined finite-discrete element method (e.g., [33]) may be of interest. Most simulations of multistage excavation are limited to either 2D or simple 3D models. In this study, we develop a parallel 3D package based on a spectral-element method, which may be used to simulate excavation in complex 3D models of various scales. The spectral-element method (SEM) is a higher-order FEM in which integration over an element is based on nodal quadrature, e.g., GaussLegendre-Lobatto quadrature. In the nodal quadrature, interpolation and integration points coincide. These points are determined by the roots of a polynomial, namely the derivative of a Legendre polynomial. In view of nodal quadrature, the SEM has two main advantages: 1) interpolating functions become orthogonal on the quadrature points, resulting in a diagonal mass matrix, thereby greatly simplifying the time-marching algorithm because a fully explicit scheme may be used (e.g., [4]), and 2) interpolation is unnecessary to determine nodal or quadrature-point quantities, thus simplifying pre- and postprocessing. Therefore, higher-order elements are easier to implement. In 2D and 3D models, nodal quadrature was originally limited to specific types of elements, e.g., quadrilaterals and hexahedra. Hexahedral meshing is a challenging task and an area of active research (e.g., [34, 35]). Only a few hexahedral meshing tools are currently available, e.g., CUBIT [36], TrueGrid [37], and Gmsh [38]. Meshing is usually not fully automated, and careful mesh design is necessary. Recently, there have been successful implementations of the SEM with other types of elements, for instance triangles in 2D and tetrahedra in 3D, using so-called Fekete points (e.g., [39–41]). Since nodal quadrature includes boundary points, the order of numerical integration may not always be sufficiently high [42]. Due to the high spatial accuracy, however, the influence of low-order integration on accuracy and convergence may not be significant, depending on the problem (e.g., [43]). 3

The SEM was originally applied to problems in computational fluid dynamics [44, 45]. Detailed reviews may be found in, e.g., Cohen et al. [46] and Deville et al. [47]. Recently, it has been widely used to simulate seismic wave propagation from local to global scales (e.g., [48–53]). Similarly, the nonlinear SEM has been used for 2D viscoplastic problems [54]. Gharti et al. [55] have developed a software package for 3D slope stability analysis based on the elastoplastic spectral-element method. The software has been parallelized for large-scale problems. Several problems in geomechanics, e.g., excavation, slope failure, and mine or tunnel collapse, exhibit elastoplastic deformation, which is an inherently nonlinear process. Nonlinear problems are usually solved using a nonlinear finite-element method (e.g., [4, 6, 56–58]). Higher-order finite elements are often desirable to capture nonlinear behaviour efficiently. In this respect, general higher-order finite-element methods, such as hp-finite element methods (e.g., [59, 60]) may be more reliable. Due to nonlinearity and the use of higher-order elements, computation is often demanding for elastoplastic problems. Therefore, the SEM may be a suitable tool for these problems. In this article, we present a 3D spectral-element implementation for multistage excavation problems. This work is an extension to the software package of 3D slope stability analysis [55]. In the excavation algorithm, excavated elements are used solely to compute surface tractions on intact regions. Processing is performed only on the intact elements without ever computing stiffnesses for excavated elements. This approach leads to a robust and efficient algorithm. We use an element-by-element preconditioned conjugate-gradient solver for efficient storage (e.g., [61]). The software is parallelized based on non-overlapping domain decomposition using MPI (Message Passing Interface; see e.g., [62, 63]). We verify the uniqueness principle for multistage excavation in linear elastic materials. We validate both serial and parallel programs with an example of multistage excavation in an elastoplastic material, and apply the technique to an underground ore mine in Finland. Finally, we present strong-scaling performance tests of our parallel code.

4

2. Formulation 2.1. Discretization of the governing equations The governing equations for elastostatic problems may be written in index notation as σij;j + fi = 0 , (1) subject to the boundary conditions ui = uî ti = σij nj = tî

on Γu on Γt

,

(2)

where σij = Cijkℓ εkℓ represents the stress tensor, εkℓ = 21 (uk;ℓ + uℓ;k ) the strain tensor, Cijkℓ the fourth-order tensor relating stress and strain, and fi the force term. The normal to the boundary is denoted by ni , and uî and tî are the prescribed displacement and traction on boundaries Γu and Γt , respectively (Figure 1). We use the summation convention for repeated indices, and a semicolon (;) denotes covariant differentiation.

Figure 1: Schematic diagram of a multistage excavation. The excavation stages are numbered. The weak form of the governing equations (1) is Z Z Z wi ti dΓ , wi;j σij dΩ = wi fi dΩ + Ω

Ω

(3)

Γt

where wi denotes a test function, and Ω and Γ the volume and boundary of the domain, respectively. For multistage excavation problems, the domain, domain boundary, and boundary conditions change with each subsequent excavation stage. Therefore, equations (1), (2), and (3) are generalized to multistage excavation. 5

More specifically, the stress tensor at excavation stage s depends on the previous excavation stage s − 1 via σijs = σijs−1 + ∆σijs

,

(4)

where ∆σijs denotes the incremental stress due to the removal of certain portion at excavation stage s.

1

0

1

−1 −1 0

0 1 −1

(a)

(b)

Figure 2: (a) A typical spectral element with five interpolation nodes in each dimension (open circles). (b) The spectral element mapped to its natural coordinates. The Gauss-Legendre-Lobatto points (solid black circles) are used for numerical integration. Integration points and interpolation nodes coincide. Only nodes on the three visible faces are shown here for clarity. Spectral-element discretization and integration techniques are explained in detail in the literature (e.g., [45–47, 49, 52]). For completeness, we briefly summarize the procedure here. In the spectral-element method (SEM), the displacement field is discretized using interpolation functions defined over Gauss-Legendre-Lobatto (GLL) points (Figure 2) via ui (ξ) =

N X

uαi φα (ξ) ,

(5)

α=1

where uαi and φα denote nodal displacements and interpolation functions, respectively. Similarly, N denotes the total number of GLL points in an element; it is given byQthe product of the number of GLL points in each dimension, i.e., N = 3j=1 Nj . In the natural state with coordinates ξ = {ξj }, the Nj GLL points are determined by the roots of the polynomial (1 −ξ 2 ) Pn′ (ξ) = 0, where Pn denotes the Legendre polynomial of degree n = 6

Nj −1. The interpolation functions φα in natural coordinates are determined by the tensor product of one-dimensional Lagrange polynomials, α φj j (ξj )

=

Nj Y (ξj − ξjβ ) α

β=1 β6=αj

such that α

φ (ξ) =

3 Y

(ξj j − ξjβ )

α

φj j (ξj ) ,

,

(6)

(7)

j=1

where α is the index of a GLL point in a linear mapping of the 3D GLL points, corresponding to the location {α1 , α2 , α3 }. For numerical integration, a point x = {xi } in a deformed element is mapped to a point ξ = {ξj } in the natural element, as illustrated in Figures 2a-b, using the transformation x(ξ) =

Ng X

xα ψ α (ξ) .

(8)

α=1

Here ψ α denotes a shape function and Ng the number of geometrical nodes xα of an element. The Jacobian matrix of the transformation has elements determined by Jij (ξ) = ∂xi (ξ)/∂ξj . The same GLL points are used as quadrature points for numerical integration. Since the internal GLL points of a spectral element do not contribute to inter-element connectivity, these points can safely be excluded during interpolation of model geometry, i.e., N and Ng may be different. Depending on element type and numerical algorithm, fewer points may be sufficient to capture transformation (8). Therefore, we usually have N > Ng , and consequently the degree of the interpolating functions, φα , is greater than the degree of the shape functions, ψ α , leading to a subparametric formulation. The SEM is a continuous Galerkin method, in which the interpolation function φα is taken as the test function wi . Upon substituting wi = φα and ui , given by (5), in equation (3), we obtain a set of elemental linear equations that may be written conveniently in matrix-vector form: Ke Ue = Fe

,

(9)

where Ke and Fe are known, respectively, as the stiffness matrix and force vector of an element. Similarly, Ue is the displacement vector. Symbolically, 7

we write Z

BT C B dΩ , Z ZΩe T ΦT t dΓ , Φ f dΩ + Fe =

Ke =

(10)

Γe

Ωe

where a superscript T denotes the transpose and Ωe the volume of an element. The quantities Φ, B, and C are known, respectively, as the interpolation function matrix, the strain-displacement matrix, and the elasticity matrix. Upon assembling the elemental equations, we obtain a set of global equations: KU = F . (11) 2.2. Excavation load During excavation, a newly excavated surface becomes a traction-free surface. Therefore, a load equal and opposite to the load contributed by the self weight and stress state of the excavated region has to be applied on excavated surfaces. This excavation load is computed using only excavated elements, and it may be expressed as (e.g., [2]) Z Z T ex ΦT f gr dΩ , (12) B σ 0 dΩ − Fe = Ωe

Ωe

gr

where f represents the force due to gravity given by {0, 0, −γe }, where γe denotes a unit weight of material on the eth element. Similarly, σ 0 represents the initial stress in an element before an excavation stage. If the initial stress before excavation is unknown, it may be approximated using an overburden pressure determined by (e.g. [64]) Z z σv (z) = σv (0) + γ(z)dz , (13) 0

where γ(z) denotes a unit weight of material at depth z. Horizontal stress is often computed using the simple relation σh = K0 σv , where, K0 is the atrest lateral earth pressure coefficient. For a simple model, it is not difficult to compute the overburden pressure using relation (13). However, it is difficult to use this relation for complex and heterogeneous models. In such cases, we may use the spectral-element method itself to compute the stress field before excavation, taking into account the appropriate boundary conditions. This enables computation of initial stress fields in complex models. 8

2.3. Elastoplastic failure For elastoplastic materials, we implement a Mohr-Coulomb yield criterion with a non-associated flow rule. The Mohr-Coulomb yield criterion may be expressed as (e.g., [6]) 1 F = σm sin φ + σ ¯ cos θ − √ sin φ sin θ − c cos φ , (14) 3 where σm , σ ¯ , and θ are the stress invariants known as the mean stress, deviatoric stress, and Lode theta, respectively. The parameters c and φ represent the cohesive strength and internal friction angle of the material, which we assume to be elastic-perfectly plastic. We solve the elastoplastic problem in an iterative manner using an initial strain method (e.g., [65]). In this method, the material is allowed to sustain stresses outside the failure envelope for a finite period, and the plastic strain is computed using a concept of pseudo-viscosity. Hence the method is also referred to as the viscoplastic strain method (e.g., [2]). The method attempts to satisfy non-linear behavior by successively correcting loads and solving linear system (11) using constant stiffness: K Uk = F + (Fp )k

.

(15)

The force term, (Fp )k , at each iteration, k, depends on the force term at the previous iteration, k − 1, and incremental plastic strain, (δεp )k , and is given by X Z p k p k−1 BT C (δεp )k dΩ . (16) (F ) = (F ) + elements

Ωe

The force contributed by the incremental plastic strain is self-equilibrating so that net loading remains the same. This load is accumulated in successive iterations until convergence is achieved. Convergence is measured as kUk − Uk−1 k/kUk−1 k ≤ ǫ, where ǫ is a tolerance. 3. Numerical implementation and parallelization In our software, we implement an efficient and robust strategy for the simulation of multistage excavation. Surface traction due to excavated regions is computed solely based on excavated elements. The equations are formed only for active degrees of freedom corresponding to intact nodes, similar to the strategy of Comodromos et al. [15] and Smith and Griffiths 9

[2]. The stiffness matrix is computed and stored only for intact elements. In order to solve the linear equations, we use an element-by-element preconditioned conjugate-gradient method, which is an iterative solver widely used in the classical FEM (e.g., [5]). Both conjugate gradient and nonlinear iteration loops only involve intact elements. Since the numbers of excavated and intact elements change at each excavation stage, we may use dynamic-memory allocation to manage memory efficiently. For large-scale problems, the software is parallelized based on a nonoverlapping domain decomposition method. In this method, the mesh is divided into a number of subdomains. No subdomains share elements, and only nodes on common subdomain interfaces are shared. The Message Passing Interface (MPI) is used as the parallel language to facilitate communication across processors (e.g., [62, 63]). For efficient parallel processing, elements should be approximately equally distributed among subdomains, such that the number of nodes on subdomain interfaces is minimal. Consequently, efficient mesh partitioning is required. There are some open-source tools for both serial and parallel graph partitioning, e.g., SCOTCH [66] and Metis [67]; in this study we use SCOTCH. To simulate multistage excavation, we implement a fixed-partition strategy. In this strategy, the domain is partitioned once and for all before excavation begins. Since the excavated portion may involve any subdomains, the excavation load should be computed and distributed carefully. For example, Figure 3 illustrates an excavation in a fixed-partition model with three subdomains. The excavation region lies on partitions 1 and 3, and hence excavation loads are computed only on these partitions. Since processing involves only intact elements, some of the excavation loads are unused by partitions 1 and 3 although these are necessary for partition 2. Therefore, these unused loads are distributed to neighboring active partitions, partition 2 in this example. The fixed-partition strategy is relatively simple to implement. Once preprocessing is completed and communication topology for parallel processing is determined, modification of the preprocessing and the communication topology at subsequent excavation stages is straightforward. However, load-balancing may not always be perfect because the excavated region can involve any number of partitions, and the excavated portion in each partition may not be equivalent. Alternatively, we may repartition the intact domain at each excavation stage, but this requires preprocessing and determination of the communication topology at every stage. In this article, we use only the fixed-partition strategy for all examples dealing with parallel 10

1

2

3

Figure 3: Illustration of three partitions (numbered 1, 2, and 3) of an excavation model. Black lines mark boundaries of partitioned subdomains. The excavation region is indicated by the shaded rectangle and involves subdomains 1 and 3. The resulting load affects all three regions. processing. We use a parallel preconditioned conjugate-gradient solver to be consistent with the non-overlapping domain decomposition. This solver has previously been implemented in both overlapping and non-overlapping domain decomposition methods (e.g., [68–73]). In the domain decomposition method, each processor solves its own part of the system, occasionally communicating across processors to assemble entities along common interfaces. Depending on the boundary conditions, load balancing during the solution procedure may not be ideal, even if the mesh is evenly decomposed. One can also repartition the degrees of freedom so that close to perfect loadbalancing is achieved during the solution procedure (e.g., [74]). However, the communication topology for parallel processing may be complicated. In the future, other algorithms which provide data locality, e.g., localized ILU preconditioning (e.g., [75]), could be of interest. 4. Numerical results 4.1. Example 1: 1D excavation in a linear elastic medium In the first example, we take a simple four-stage excavation model with a total excavation height of 20 m (Figure 4a). The model consists of a linear elastic material with Young’s modulus E = 104 kN/m2 , unit weight γ = 1 kN/m3 , Poisson’s ratio ν = 0.2, and at-rest pressure coefficient K0 = 0.5. For the boundary conditions, nodes on the bottom surface are fixed in both directions. Nodes on side faces are fixed only along the normals to the corresponding surfaces. Therefore, we have ux = uy = uz = 0 on the plane z = zmin , ux = 0 on the planes x = xmin and x = xmax , and uy = 0 on 11

the planes y = ymin and y = ymax . Due to the boundary conditions and the excavation geometry of this particular model, displacement depends only on the z dimension. Therefore, this example is equivalent to a 1D problem. This particular model has been used by several authors to validate their excavation algorithms (e.g., [8, 9, 15]). The analytical solution for this problem gives an upward displacement of 0.036 m on the excavated surface at the final stage (e.g., [15]).

1 2 30

3 4 20

10

(a)

(b)

Figure 4: a) 1D excavation model with four excavation stages (numbered 1, 2, 3, and 4). b) Spectral-element mesh of the model with three GLL points in each dimension. The total number of elements is 256. Since the model has a simple geometry, meshing is straightforward. Here, we use the mesh generation tool kit CUBIT [36]. The generation of spectral elements with more than three GLL points in each dimension is currently not possible within CUBIT. Therefore, we first create an 8-node hexahedral mesh with CUBIT. The mesh is then converted to a spectralelement mesh with the desired number of GLL points within our main program. For this example, we use three GLL points in each dimension, resulting in a total of 27 nodes per element. We mesh the model with an element size of 5 m, resulting in a total of 256 elements (Figure 4b). Based on this mesh, we perform two simulations using our serial SEM program. In the first simulation, four layers are excavated sequentially in four stages. In the second simulation, four layers are excavated in one single stage. We set a relative tolerance of 10−8 for conjugate-gradient iterations. With both simulations, we obtain the same 0.036 m upward displacement on the excavated surface (Figures 5a-b). This displacement is in perfect agreement with the analytical solution. As expected, we obtain the 12

same displacement field and displacement vectors in the entire domain for single- and multistage excavations. Hence, the final solution is independent of the excavation sequence. That also verifies the uniqueness principle for linear elastic materials.

(a)

(b)

Figure 5: a) Displacement field at the final stage. b) Displacement vectors at the final stage. Results for single- and multistage simulations were identical.

4.2. Example 2: 2D excavation In this example, we consider a three-stage excavation model with a total excavation height of 6 m (Figure 6a). The material has unit weight 20 kN/m3 , Young’s modulus 105 kN/m3 , cohesion 31 kN/m2 , friction angle 0◦ , and dilation angle 0◦ . For initial stress, we consider only the overburden pressure, which we compute during the initial stage of the simulation. As boundary conditions, nodes on the bottom surface are fixed in both directions. Nodes on side faces are fixed only along the normal to the corresponding surfaces. Therefore, we have ux = uy = uz = 0 on the plane z = zmin , ux = 0 on the planes x = xmin and x = xmax , and uy = 0 on the planes y = ymin and y = ymax . Due to the boundary conditions and the excavation geometry of this particular model, the displacement depends only on the x and z dimensions. Therefore, this example is equivalent to a 2D plane strain problem. We mesh the model with an element size of 1 m, resulting in a total of 1000 elements. For the elastoplastic material we need to solve nonlinear constitutive equations. Therefore, we use a relatively finer mesh than in the previous example to capture nonlinear behavior of the model. Initially, we use three GLL points in each dimension. We set relative tolerances 13

1 2 3

(a)

(b)

Figure 6: a) 2D excavation model with three excavation stages (numbered 1, 2, and 3). b) Spectral-element mesh of the model with three GLL points in each dimension. The total number of elements is 1000. of 10−8 for conjugate-gradient iterations and 10−5 for nonlinear iterations. Similarly, we set the maximum number of nonlinear iterations to 5000. We perform four different simulations using the serial SEM program: 1) single-stage excavation in an elastic material, 2) multistage excavation in an elastic material, 3) single-stage excavation in an elastoplastic material, and 4) multistage excavation in an elastoplastic material. Figures 7a-b show displacement profiles along a vertical line in the middle of the vertical excavated surface. As expected, profiles for single- and multistage simulations computed in the elastic medium are identical. We observe small discrepancies between the profiles of single- and multistage simulations computed in the elastoplastic medium. Although the displacement field and displacement vectors appear similar for single- and multistage simulations in an elastoplastic medium, we observe small numerical discrepancies of order ∼ 1.4 × 10−4 m. One purpose of the simulation of the excavation process is to assess the stability of structures during excavation. We can estimate the limiting strength of the material at which the model collapses during the excavation process. In order to estimate this limiting strength (i.e., cohesion in our example), we compute displacement fields for a range of cohesion values. Figures 10a and 10b show the resulting maximum displacement and required number of nonlinear iterations, respectively. We observe a small displacement for excavation stages 1 and 2. Relatively few iterations are required for convergence at these stages, and the model exhibits mostly 14

−2

−2

−4

−4

z (m)

0

z (m)

0

−6

−6

−8

−10 −0.0030

−8 Single−stage Multistage −0.0010

0.0010

0.0030

Displacement (m)

−10 −0.0040

Single−stage Multistage −0.0020

0.0000

0.0020

0.0040

0.0060

Displacement (m)

(a)

(b)

Figure 7: a) Displacement profile along a vertical line at (x = 5 m, y = 5 m) for an elastic material. b) Same as a) but for an elastoplastic material. The displacement profiles for the elastic material are identical for singleand multistage simulations.

(a)

(b)

Figure 8: a) Displacement field at the final stage after four-stage excavation. b) Displacement field at the final stage after single-stage excavation. The displacement field in each figure is independently scaled to its displacement range.

15

(a)

(b)

Figure 9: a) Displacement vectors at the final stage after four-stage excavation. b) Displacement vectors at the final stage after single-stage excavation. The vector field in each figure is independently scaled to its displacement range. elastic behaviour for all strengths. For excavation stage 3, we observe small displacements until the strength reaches approximately 32 kN/m2 , and the corresponding nonlinear iterations are relatively few. At strength 31 kN/m2 the displacement begins to increase noticeably, requiring a larger number of nonlinear iterations. The displacement suddenly increases at 30 kN/m2 , and it does not converge within the given maximum number (i.e., 5000) of nonlinear iterations. This indicates a possible collapse of the model, and hence the limiting strength of the model is ∼ 30 kN/m2 . For the plane strain condition, the classical-limit plastic solution (e.g., [76, 77]) gives a value of limiting cohesion of approximately 31 kN/m2 for an excavation height of 6 m, which is in good agreement with the computed result. The maximum displacement and total number of nonlinear iterations appear similar for single- and multistage excavations with a maximum discrepancy of ∼ 5.8 × 10−5 m in displacement magnitude (Figures 10a-b). We observe a circular failure pattern after the final excavation stage, which is similar for single- and multistage excavations (Figures 8a-b). Although it is difficult to compute accurate displacements during collapse, we conduct multistage simulations for elastoplastic media considering different degrees of h- and p-refinements of the mesh, and see how the SEM results behave. We discretize the model with four different element sizes of approximately 2 m, 1 m, 0.75 m, and 0.5 m, resulting in a total number of elements of 150, 1000, 2548, and 8000, respectively. First, we perform four complimentary simulations of the multistage excavation based on these 16

0.0900

Total nonlinear iterations


0.0700 0.0600 0.0500 0.0400 0.0300 0.0200 0.0100 0.0000

5000

Multi−stage Single−stage

0.0800

Multi−stage Single−stage

4000 3000

1000

3 1

2

30

35 40 45 Cohesion (kN/m2)

50

0

12

30

3

35 40 45 Cohesion (kN/m2)

50

(b)

(a)

Figure 10: a) Maximum displacement at three excavation stages for various cohesion values. b) Total nonlinear iterations at three excavation stages for various cohesion values. The excavation stages are numbered 1, 2, and 3. four meshes, using three GLL points in each dimension. Second, we perform two complimentary simulations with the coarsest mesh (element size 2 m) using, respectively, four and five GLL points in each dimension. In each simulation, we compute the displacement field for a range of cohesion, successively decreasing its value until the solution does not converge within the maximum number of nonlinear iterations. Figure 11 summarizes the results of these simulations. With the coarsest mesh, using three GLL points, we observe a sudden increase in maximum displacement at a cohesion of 28 kN/m2 , and the solution does not converge thereafter. Accordingly, this gives an approximate limiting cohesion of 28 kN/m2 , which is ∼ 10% less than the correct limiting strength of 31 kN/m2 . With a finer mesh of element size 1 m, we obtain an approximate limiting cohesion of 30 kN/m2 , which is ∼ 3% less than the actual value. As we further refine the mesh, the limiting cohesion converges to the correct value. On the other hand, with the coarsest mesh and four GLL points, we obtain an approximate limiting cohesion of 31 kN/m2 , in agreement with the actual value. The results involving the coarsest mesh and four or five GLL points are similar to those obtained based on the finer meshes of element sizes 0.75 m and 0.5 m, respectively. 4.3. Example 3: 3D excavation In the third example, we simulate single- and multistage excavation in the model shown in Figure 12. We use similar material properties as in 17


0.7000

h = 2.00m, Ni = 3 h = 1.00m, Ni = 3

0.6000

h = 0.75m, Ni = 3 h = 0.50m, Ni = 3

Limit cohesion = 31 kN/m2

0.5000

h = 2.00m, Ni = 4 h = 2.00m, Ni = 5

0.4000

h = Element size Ni = Number of GLL points in each direction

0.3000 0.2000 0.1000 0.0000

30

35

40

45

50

2

Cohesion (kN/m )

Figure 11: Maximum displacement at the final stage for different degrees of h- and p-refinement computed for a range of cohesion values. The displacement for the lowest cohesion for each curve represents the non-converged solution. previous cases, with unit weight 20 kN/m3 , Young’s modulus 105 kN/m3 , cohesion 25 kN/m2 , friction angle 0◦ , and dilation angle 0◦ . We again consider only overburden pressure as initial stress. For boundary conditions, nodes on the bottom surface are fixed in both directions. Nodes on side faces are fixed only along the normal to the corresponding surfaces. Therefore, we have ux = uy = uz = 0 on the plane z = zmin , ux = 0 on the planes x = xmin and x = xmax , and uy = 0 on the planes y = ymin and y = ymax . Unlike in previous examples, the displacement in this model depends on both x, y, and z dimensions. The model is meshed using an average element size of 1 m, resulting in a total of 1700 elements (Figure 13a). In order to validate our parallel program, we also partition this mesh into eight subdomains using the graph partitioning tool SCOTCH [66] for parallel processing (Figure 13b). We perform four different simulations with each of the serial and parallel SEM programs: 1) single-stage excavation in an elastic material, 2) multistage excavation in an elastic material, 3) single-stage excavation in an elastoplastic material, and 4) multistage excavation in an elastoplastic material. We use three GLL points in each dimension; and set the 18

1

2 3

Figure 12: Excavation model with three excavation stages (numbered 1, 2, and 3).

(a)

(b)

Figure 13: a) Spectral-element mesh of a model with three GLL points in each dimension. b) Same as a), but partitioned into eight subdomains for parallel processing. The total number of elements is 1700.

19

same relative tolerances for conjugate gradient and nonlinear iterations, and the same maximum number of nonlinear iterations as in previous cases. For the elastoplastic case, we apply the excavation load in ten increments. Figures 14a-b show the displacement profiles along a vertical line on the front-left vertical excavated corner. As expected, the profiles for single- and multistage simulations computed in the elastic medium are identical. We observe small discrepancies between the profiles of single- and multistage simulations in the elastoplastic medium. Results computed by the serial and parallel programs are in excellent agreement.

−2

−2

−4

−4

z (m)

0

z (m)

0

−6

−6

−8

−10 −0.0015

Single−stage serial Single−stage parallel Multistage serial Multistage parallel −0.0005

0.0005

0.0015

Displacement (m)

(a)

0.0025

−8

−10 −0.0040

Single−stage serial Single−stage parallel Multistage serial Multistage parallel 0.0000

0.0040

0.0080

Displacement (m)

(b)

Figure 14: a) Displacement profile along a vertical line at (x = 10 m, y = 5 m) for an elastic material. b) Same as a) but for an elastoplastic material. The displacement profiles for the elastic material are identical for singleand multistage simulations. Figures 15a-f show the resulting displacement fields and displacement vectors obtained from the multistage simulation in an elastoplastic medium. We obtain a maximum numerical discrepancy on the order of ∼ 4.5×10−4 m between single- and multistage displacement fields. During the first excavation stage, the excavation height is small, therefore, upward displacement of the floor is dominant. As the excavation height increases, displacements on the side walls become dominant and hence failure may occur. Using the same model, we compute displacements for a range of cohesion values considering both single- and multistage excavations. We use both serial and parallel programs. Figures 16a and 16b show the resulting maximum displacements and corresponding numbers of nonlinear iterations, respectively, observed at different excavation stages. Both excavation stages 1 and 2 result in a small displacement requiring very few nonlin20

(a)

(b)

(c)

(d)

(e)

(f)

Figure 15: Displacement fields (left column) and displacement vectors (right column) computed at three excavation stages. Top to bottom rows show excavation stages 1, 2 and 3. The displacement and vector fields in each figure are independently scaled to their range. The results obtained by the serial and parallel programs were identical.

21

ear iterations. During these stages the model behaves mostly as an elastic material. On the other hand, during excavation stage 3, as cohesion decreases, the model undergoes plastic deformation, requiring larger numbers of nonlinear iterations to converge. Maximum displacements computed by the serial and parallel programs are in perfect agreement for all cohesion values (Figure 16a), and the numbers of nonlinear iterations are also similar (Figure 16b). Displacements computed for the single-stage excavation are similar to those for the multistage excavation, but some discrepancies are observed after the model undergoes plastic deformation. The required number of nonlinear iterations is different compared to the multistage excavation case. For this excavation model, the limiting value of cohesion is estimated to be ∼ 24 kN/m2 (Figures 16a-b). Multi−stage serial Multi−stage parallel Single−stage serial Single−stage parallel

0.0100

0.0050 3

2425

27

30 35 Cohesion (kN/m2)

(a)

40

Multi−stage serial Multi−stage parallel Single−stage serial Single−stage parallel

600 500 400 300 3

200

2 1 0.0000

700 Total nonlinear iterations


0.0150

0

2 1 24 25

27

30 35 Cohesion (kN/m2)

40

(b)

Figure 16: a) Maximum displacement at three excavation stages for various cohesion values. b) Total nonlinear iterations at three excavation stages for various cohesion values. The excavation stages are numbered 1, 2, and 3. 4.4. Example 4: Excavation in a mine In this example, we apply our parallel program to an underground ore mine, namely the Pyhäsalmi mine in central Finland. This mine consists of a volcanogenic massive sulphide (VMS) deposit, and produces mainly copper, zinc, and pyrite concentrates. The copper-zinc ore body in the mine extends down to a depth of ∼ 1.4 km (Figure 17a); and microearthquakes are frequently observed in this mine (e.g. [78]). The in-mine seismic network comprises eighteen geophones [79], which are used to record and locate microearthquakes induced by mining operations. Figure 18 illustrates the 3D wave-speed model used for the simulation of seismic wave propagation 22

(see e.g., [80]), which consists of only an ore body, host rock, and stopes (i.e., mined out voids) (Table 1). 3

Mass density (kg/m ) P-wave speed (m/s) S-wave speed (m/s) Cohesion (MPa) Friction angle (◦ ) Dilation angle (◦ )

Host rock 2000 6000 3460 14 39 0

Ore body 4400 6300 3700 14 39 0

Table 1: Material properties of the Pyhäsalmi mine model. To simulate multistage excavation, we estimate Young’s modulus and Poisson’s ratio from the seismic properties summarized in Table 1 (e.g., [81]). The estimated values of Young’s moduli for host rock and ore body are approximately 60 GPa and 149 GPa, respectively. The approximate Poisson’s ratios for ore body and host rock are 0.25 and 0.24, respectively. We consider only overburden pressure as initial stress, computed within the SEM program before the excavation loop begins. Due to the complex structure of, in particular, mined-out cavities, it is very difficult to generate a hexahedral mesh for the complete model (Figure 17b). Therefore, we simplify the original model, and consider only two major stopes (Figure 18). We assume four excavation stages, as shown in Figures 19a-d. Unlike previous examples, there are two different excavation regions within the model. Each region is excavated in two stages. We assume that the four excavation stages are performed sequentially, as shown in Figures 19a-d. Even with this simplified model, generation of a hexahedral mesh with the mesh generation tool kit CUBIT [36] is nontrivial. To be able to generate a high-quality mesh, we need to decompose the complex geometry into meshable volumes. We divide this particular model into 78 volumes that can be meshed with the functionalities available within CUBIT (Figure 20a). We mesh the model with an average element size of 9.5 m for rock and 10 m for the ore body, resulting in a total of 107,712 spectral elements (Figures 20a-b), which leads to a total of 7,161,572 nodes using three GLL points in each dimension. We partition the mesh into 32 domains for parallel processing (Figure 21). We use a fixed-partition strategy. Therefore, distribution of the 23

(a)

(b)

Figure 17: a) Pyhäsalmi mine with surrounding infrastructure: the copper/zinc ore body is shown in brown/pink, access tunnels are shown in yellow, the elevator shaft is shown in dark blue, and seismic stations are numbered. Passage for quarried ore is marked by KN1. b) 3D wave-speed model (see Table 1) of the Pyhäsalmi mine: stopes (i.e., mined-out cavities) are shown in blue and the ore body in brown. The remainder is host rock. Geophones are indicated by black dots and numbered.

Figure 18: Simplified 3D model of the Pyhäsalmi mine, including only the ore body (solid) and two major stopes (black). 24

(a)

(b)

(c)

(d)

Figure 19: Four excavation stages in the Pyhäsalmi mine model. a) Stage 1. b) Stage 2. c) Stage 3. d) Stage 4.

25

(a)

(b)

Figure 20: a) Spectral-element mesh for a 3D model of the Pyhäsalmi mine. b) Interior section of the mesh visualizing the ore body. Color represents the different volumes created for meshing. load among processors may vary and become less optimal as excavation progresses. For this example, we set relative tolerances of 10−8 for conjugate gradient iterations and 10−5 for nonlinear iterations, as in the previous examples. We apply the excavation load in 10 increments. Figure 22 shows maximum displacements computed at four excavation stages. Only one iteration per load increment is needed for convergence, suggesting fully elastic behavior, due to the fact that the rock and ore body are very sound, possessing large cohesion and friction. In fact, only a single increment suffices for this model. There was no plastic deformation during these excavation stages. Figures 23 and 24 illustrate displacement fields and displacement vectors, respectively, computed at four excavation stages. The upward displacements of roofs and downward displacements of floors are larger than displacements of side walls. In excavation stages 3 and 4, we observe regions of small displacements in the slab (i.e., the model portion between the two excavated cavities). The top surface of the slab experiences upward displacements and the bottom surface experiences downward displacements due to removal of material during respective excavations. However, small displacements do not necessarily imply stability of the slab. Such slabs may actually behave as a bending plate under certain circumstances (e.g., [82]). 26

Figure 21: Spectral-element mesh of a 3D model of the Pyhäsalmi mine partitioned into 32 subdomains. Thick black lines represent subdomain interfaces.


0.0045

0.0060

0.0055

0.0050

0.0045

1

2 3 Excavation stage

4

Figure 22: Maximum displacement at four excavation stages. The solutions at all stages converge in one iteration per load increment.

27

(a)

(b)

(c)

(d)

Figure 23: Displacement fields computed at four excavation stages, visualized in half the model. a) Stage 1. b) Stage 2. c) Stage 3. d) Stage 4. Displacement fields in each figure are independently scaled to their range.

28

(a)

(b)

(c)

(d)

Figure 24: Displacement vectors computed at four excavation stages, visualized in half the model. a) Stage 1. b) Stage 2. c) Stage 3. d) Stage 4. Shaded surfaces represent interfaces of partitioned subdomains. Vector fields in each figure are independently scaled to their range.

29

4.5. Parallel performance Finally, we conduct a strong-scaling performance test of our parallel program using the model of the Pyhäsalmi mine. We measure total computation times for different numbers of processors, keeping the problem size fixed. We run the parallel program on 16, 32, 48, 64, 80, and 96 processors. The results shown in Figure 25 illustrate that the code scales reasonably well for large problems. In the future, it may be important to implement efficient algorithms for node renumbering, e.g., the reverse Cuthill-McKee algorithm [83], to further improve parallel performance (e.g., [84]). Although we have used a fixed-partition strategy, parallel performance is reasonable. For the cases in which load-balancing is severely affected, partitioning of the intact region at every excavation stage may be a suitable option, despite requiring extra computation for preprocessing and determination of the communication topology.

Total elapsed time (s)

8.0e+04

Actual Reference

4.0e+04

2.0e+04

1.0e+04

5.0e+03

16

32 48 64 Number of processors

80 96

Figure 25: Total elapsed time for a fixed problem size run on 16, 32, 48, 64, 80, and 96 processors, compared to a reference line computed using the total elapsed time on 48 processors.

5. Discussion and conclusions We have successfully implemented a spectral-element method for 3D multistage excavation. The numerical method is parallelized based on domain decomposition using MPI. Our program satisfies the uniqueness principle for multistage excavation in linear elastic materials. We have validated both the serial and parallel programs, and demonstrated several simulations of multistage excavation in elastoplastic materials. We have simulated multistage excavation in a heterogeneous model of the Pyhäsalmi mine in Finland. This simulation illustrates a potential application of the software to 30

complex and large-scale excavations. Due to very sound rock in the mine, unstable zones are barely visible during the excavation stages. We plan future simulations with a more realistic model of the Pyhäsalmi mine, that better captures the in-situ stress state and actual excavation stages. Since mining operations are often initiated by blasting, it would be helpful to include blasting effects by considering brittle failure. We have used a mesh with uniform element size for purposes of demonstration. A mesh with local refinement, e.g., geometrically adaptive meshing, may be important for efficient simulations of large-scale problems. For load-controlled problems in elastoplastic media, the constant-stiffness approach overestimates the stiffness near collapse. As a result, a large number of iterations is required to achieve convergence (e.g., [2]). Hence, other more accurate and efficient integration algorithms for elastoplastic constitutive relationships may be important, e.g., modified Euler methods with drift correction and automatic error control (e.g., [85, 86]), or return mapping algorithms (e.g., [56, 87]). In some excavation problems, it may also be important to assess longterm creeping behaviour of structures such as tunnels, mines etc. This would require implementation of viscoelasticity or viscoplasticity. We have only implemented material nonlinearity, and in the future it will also be important to implement geometrical nonlinearity for the analysis of large displacements, e.g., based on adaptive mesh refinement. Our software is open-source under GNU GPL version 2.0, and until we make a final release it is available upon request from the corresponding author. 6. Acknowledgments We thank Daniela K¨ uhn, Michael Roth, and Valérie Maupin for helpful discussions and suggestions, Katja Sahala and ISS for access to the mine model and in-mine data, and Ricardo M. Garcia, Jr. for his help with meshing. Parallel programs were run on the Titan cluster owned by the University of Oslo and the Norwegian metacenter for High Performance Computing (NOTUR), and operated by the Research Computing Services group at USIT, the University of Oslo IT-department; and at the Princeton Institute for Computational Science and Engineering (PICSciE), USA. The 3D data were visualized using the open-source parallel visualization software package ParaView/VTK (www.paraview.org). This work was funded 31

in part by the Norwegian Research Council, and supported by industry partners BP, Statoil, and Total. References [1] E. Hoek, T. Brown, Underground Excavations in Rock, Institution of mining and metallurgy, 1990. [2] I. M. Smith, D. V. Griffiths, Programming the finite element method, John Wiley & Sons, 2004. [3] P. Dunlop, J. M. Duncan, Development of failure around excavated slopes, Journal of the Soil Mechanics and Foundations Division 96 (2) (1970) 471–493. [4] T. J. R. Hughes, The finite element method: linear static and dynamic finite element analysis, Prentice-Hall, 1987. [5] K. J. Bathe, Finite Element Procedures, Prentice Hall, 1995. [6] O. C. Zienkiewicz, R. L. Taylor, The finite element method for solid and structural mechanics, Elsevier Butterworth-Heinemann, 2005. [7] J. T. Christian, I. H. Wong, Errors in simulating excavation in elastic media by finite elements, Soils and Foundations 13 (1) (1973) 1–10. [8] G. W. Clough, A. I. Mana, Lessons learned in finite element analysis of temporary excavations, in: International conference on numerical method in geomechanics, ASCE, 1976. [9] A. I. Mana, Finite element analysis of deep excavation behavior in soft clay, Ph.D. thesis, Stanford University, Stanford, CA (1978). [10] K. Ishihara, Relations between process of cutting and uniqueness of solutions., Soils and Foundations 10 (1970) 50–65. [11] J. Ghaboussi, D. A. Pecknold, Incremental finite element analysis of geometrically altered structures, International Journal for Numerical Methods in Engineering 20 (11) (1984) 2051–2064. doi:10.1002/nme.1620201108. [12] C. S. Desai, S. Sargand, Hydrid FE procedure for soil-structure interaction, Journal of Geotechnical Engineering 110 (1984) 473–486. [13] R. I. Borja, S. R. Lee, B. R. Seed, Numerical simulation of excavation in elastoplastic soils, International Journal for Numerical and Analytical Methods in Geomechanics 13 (6) (1989) 231–249. doi:10.1002/nag.1610130302. [14] R. I. Borja, Analysis of incremental excavation based on critical state theory, Journal of Geotechnical Engineering 116 (6) (1990) 964–985. [15] E. Comodromos, T. Hatzigogos, K. Pitilakis, Multi-stage finite element algorithm for excavation in elastoplastic soils, Computers & Structures 46 (2) (1993) 289–298. doi:10.1016/0045-7949(93)90193-H. [16] E. L. Wilson, The static condensation algorithm, International Journal for Numerical Methods in Engineering 8 (1974) 198–203. doi:10.1002/nme.1620080115. [17] V. S. Chandrasekaran, G. J. W. King, Simulation of excavation using finite elements, Journal of Geotechnical Engineering Division 100 (9) (1974) 1086–1089. [18] P. T. Brown, J. R. Booker, Finite element analysis of excavation, Computers and Geotechnics 1 (3) (1985) 207–220. doi:10.1016/0266-352X(85)90024-2. [19] D. V. Griffiths, N. Koutsabeloulis, Finite element analysis of vertical excavations,

32

[20]

[21]

[22] [23] [24] [25]

[26] [27] [28]

[29]

[30]

[31]

[32]

[33] [34]

[35] [36]

Computers and Geotechnics 1 (3) (1985) 221–235. doi:10.1016/0266-352X(85)900254. C.-Y. Ou, D.-C. Chiou, T.-S. Wu, Three-dimensional finite element analysis of deep excavations, Journal of Geotechnical Engineering 122 (5) (1996) 337–345. doi:10.1061/(ASCE)0733-9410(1996)122:5(337). G. Galli, A. Grimaldi, A. Leonardi, Three-dimensional modelling of tunnel excavation and lining, Computers and Geotechnics 31 (3) (2004) 171–183. doi:10.1016/j.compgeo.2004.02.003. Y. Hou, J. Wang, L. Zhang, Finite-element modeling of a complex deep excavation in Shanghai, Acta Geotechnica 4 (2009) 7–16. doi:10.1007/s11440-008-0062-3. I. M. Smith, J. Leng, L. Margetts, Parallel three-dimensional finite element analysis of excavation, in: 13th ACME conference, University of Sheffield, UK, 2005. D. A. Holt, D. V. Griffiths, Transient analysis of excavations in soil, Computers and Geotechnics 13 (3) (1992) 159–174. doi:10.1016/0266-352X(92)90002-B. J. P. Hsi, J. C. Small, Simulation of excavation in a poro-elastic material, International Journal for Numerical and Analytical Methods in Geomechanics 16 (1992) 25–43. doi:10.1002/nag.1610160104. P. A. Cundall, O. D. L. Strack, A discrete numerical model for granular assemblies, Géotechnique 29 (1) (1979) 47–65. Itasca, Itasca software products-FLAC, FLAC3D, UDEC, 3DEC, PFC, PFC3D. Itasca Consulting Group Inc., Minneapolis. Z. Zhu, H. Li, Q. Liu, X. He, Numerical simulation for tunnel excavation in stratified rock mass by FLAC3D, in: Computer-Aided Industrial Design & Conceptual Design, IEEE 10th International Conference, 2009, pp. 2271–2274. doi:10.1109/CAIDCD.2009.5375152. T. Belytschko, T. Black, Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering 45 (5) (1999) 601–620. N. Sukumar, N. Moës, B. Moran, T. Belytschko, Extended finite element method for three-dimensional crack modeling, International Journal for Numerical Methods in Engineering 48 (11) (2000) 1549–1570. M. Hori, K. Oguni, H. Sakaguchi, Proposal of FEM implemented with particle discretization for analysis of failure phenomena, Journal of the Mechanics and Physics of Solids 53 (3) (2005) 681–703. doi:10.1016/j.jmps.2004.08.005. K. Oguni, M. L. L. Wijerathne, T. Okinaka, M. Hori, Crack propagation analysis using PDS-FEM and comparison with fracture experiment, Mechanics of Material 41 (11) (2009) 1242–1252. doi:10.1016/j.mechmat.2009.07.003. A. Munjiza, The Combined Finite-Discrete Element Method, John Wiley and Sons, 2004. P. Pebay, M. Stephenson, L. Fortier, S. Owen, D. J. Melander, pCAMAL: An embarrassingly parallel hexahedral mesh generator, in: M. L. Brewer, D. Marcum (Eds.), Proceedings of the 16th International Meshing Roundtable, Springer Berlin Heidelberg, 2008, pp. 269–284. J. Shepherd, C. Johnson, Hexahedral mesh generation constraints, Engineering with Computers 24 (2008) 195–213. Sandia National Laboratories, CUBIT 13.0 User Documentation, [Online; accessed

33

[37]

[38]

[39] [40]

[41]

[42]

[43]

[44] [45] [46] [47] [48]

[49]

[50]

[51] [52] [53]

27-May-2011] (2011). URL cubit.sandia.gov R. Rainsberger, TrueGrid User’s Manual, XYZ Scientific Applications, Inc., Livermore, CA, version 2.3.0 Edition (2006). URL www.truegrid.com C. Geuzaine, J. F. Remacle, Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, International Journal for Numerical Methods in Engineering 79 (11) (2009) 1309–1331. J. S. Hesthaven, C. H. Teng, Stable spectral methods on tetrahedral elements, SIAM Journal on Scientific Computing 21 (6) (1999) 2352–2380. M. A. Taylor, B. A. Wingate, A generalized diagonal mass matrix spectral element method for non-quadrilateral elements, Applied Numerical Mathematics 33 (1-4) (2000) 259–265. E. D. Mercerat, J. P. Vilotte, F. J. S´ anchez-Sesma, Triangular spectral element simulation of two-dimensional elastic wave propagation using unstructured triangular grids, Geophysical Journal International 166 (2006) 679–698. M. Durufle, P. Grob, P. Joly, Influence of Gauss and Gauss-Lobatto quadrature rules on the accuracy of a quadrilateral finite element method in the time domain., Numerical Methods for Partial Differential Equations 25 (2009) 526–551. D. A. Nielasen, H. M. Blackburn, Gauss and Gauss-Lobatto element quadratures applied to the incompressible Navier-Stokes equations, in: Proceedings of the 8th biannial conference: Computational Techniques and Applications (CTAC 97), 1997. A. T. Patera, A spectral element method for fluid dynamics: laminar flow in a channel expansion, Journal of Computational Physics 54 (1984) 468–488. C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral methods in fluid dynamics, Springer, 1988. G. Cohen, Higher-order numerical methods for transient wave equations, SpringerVerlag, Berlin, Germany, 2002. M. O. Deville, P. F. Fischer, E. H. Mund, High-Order Methods for Incompressible Fluid Flow, Cambridge University Press, Cambridge, United Kingdom, 2002. G. Seriani, 3-D large-scale wave propagation modeling by spectral element method on Cray T3E multiprocessor, Computer Methods in Applied Mechanics and Engineering 164 (1994) 235–247. D. Komatitsch, J. P. Vilotte, The spectral element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures, Bulletin of the Seismological Society of America 88 (2) (1998) 368–392. D. Komatitsch, J. Tromp, Introduction to the spectral element method for threedimensional seismic wave propagation, Geophysical Journal International 139 (1999) 806–822. D. Komatitsch, J. Tromp, Spectral-element simulations of global seismic wave propagation – I. validation, Geophysical Journal International 149 (2002) 390–412. J. Tromp, D. Komatitsch, Q. Liu, Spectral-element and adjoint methods in seismology, Communications in Computational Physics 3 (1) (2008) 1–32. V. Oye, H. N. Gharti, E. Aker, D. K¨ uhn, Moment tensor analysis and comparison of acoustic emission data with synthetic data from spectral element method, SEG Technical Program Expanded Abstracts 29 (1) (2010) 2105–2109. doi:10.1190/1.3513260.

34

[54] C. di Prisco, M. Stupazzini, C. Zambelli, Nonlinear SEM numerical analyses of dry dense sand specimens under rapid and dynamic loading, International Journal for Numerical and Analytical Methods in Geomechanics 31 (2007) 757–788. [55] H. N. Gharti, D. Komatitsch, V. Oye, R. Martin, J. Tromp, Application of an elastoplastic spectral-element method to 3D slope stability analysis, International Journal for Numerical Methods in Engineering submitted. [56] J. C. Simo, T. J. R. Hughes, Computational inelasticity, Springer, 1998. [57] L. X. Luccioni, J. M. Pestana, R. L. Taylor, Finite element implementation of nonlinear elastoplastic constitutive laws using local and global explicit algorithms with automatic error control, International Journal for Numerical Methods in Engineering 50 (2001) 1191–1212. [58] F. Dupros, F. De Martin, E. Foerster, D. Komatitsch, J. Roman, High-performance finite-element simulations of seismic wave propagation in three-dimensional nonlinear inelastic geological media, Parallel Computing 36 (2010) 308–325. [59] I. Babuska, M. Suri, The p- and h-p versions of the finite element method, an overview, Computer Methods in Applied Mechanics and Engineering 80 (1990) 5– 26. [60] C. Schwab, p- and hp- Finite Element Methods: Theory and Applications to Solid and Fluid Mechanics, Oxford University Press, USA, 1999. [61] T. J. R. Hughes, I. Levit, J. Winget, An element-by-element solution algorithm for problems of structural and solid mechanics, Computer Methods in Applied Mechanics and Engineering 36 (2) (1983) 241–254. doi:10.1016/0045-7825(83)90115-9. [62] W. Gropp, E. Lusk, A. Skjellum, Using MPI, portable parallel programming with the Message-Passing Interface, MIT Press, Cambridge, USA, 1994. [63] P. Pacheco, Parallel Programming with MPI, Morgan Kaufmann, 1997. [64] K. Terzaghi, Theoretical soil mechanics, J. Wiley, New York, 1943. [65] O. Zienkiewicz, I. Cormeau, Visco-plasticity-plasticity and creep in elastic solids-a unified numerical solution approach, International Journal for Numerical Methods in Engineering 8 (4) (1974) 821–845. [66] F. Pellegrini, J. Roman, SCOTCH: A software package for static mapping by dual recursive bipartitioning of process and architecture graphs, Lecture Notes in Computer Science 1067 (1996) 493–498. [67] G. Karypis, V. Kumar, Multilevel k-way partitioning scheme for irregular graphs, Journal of Parallel and Distributed Computing 48 (1) (1998) 96–129. [68] K. H. Law, A parallel finite element solution method, Computers & Structures 23 (6) (1986) 845–858. doi:10.1016/0045-7949(86)90254-3. [69] R. B. King, V. Sonnad, Implementation of an element-by-element solution algorithm for the finite element method on a coarse-grained parallel computer, Computer Methods in Applied Mechanics and Engineering 65 (1) (1987) 47–59. doi:10.1016/0045-7825(87)90182-4. [70] E. Barragy, G. F. Carey, A parallel element-by-element solution scheme, International Journal for Numerical Methods in Engineering 26 (1988) 2367–2382. [71] N. A. Sobh, K. Gustafson, Preconditioned conjugate gradient and finite element methods for massively data-parallel architectures, Computer Physics Communications 65 (1991) 253–267. [72] A. I. Khan, B. H. V. Topping, Parallel finite element analysis using Jacobi-

35

[73]

[74] [75]

[76] [77] [78]

[79] [80]

[81] [82] [83]

[84]

[85] [86] [87]

conditioned conjugate gradient algorithm, Advances in Engineering Software 25 (1996) 309–319. Y. Liu, W. Zhou, Q. Yang, A distributed memory parallel element-by-element scheme based on Jacobi-conditioned conjugate gradient for 3D finite element analysis, Finite Elements in Analysis and Design 43 (2007) 494–503. L. Margetts, Parallel finite element analysis, Ph.D. thesis, University of Manchester (2002). K. Nakajima, H. Okuda, Parallel iterative solvers with localized ILU preconditioning for unstructured grids on workstation clusters, International Journal of Computational Fluid Dynamics 12 (3) (1999) 315–322. D. W. Taylor, Stability of earth slopes, Journal of Boston Society of Civil Engineers 24 (1937) 197–246. J. Heyman, Simple plastic theory applied to soil mechanics, in: Symposium on the role of Plasticity in Soil Mechanics, 1973, pp. 151–172. V. Oye, H. Bungum, M. Roth, Source parameters and scaling relations for mining related seismicity within the Pyhaesalmi ore mine, Finland, Bulletin of the Seismological Society of America 95 (2005) 1011–1026. H. Puustjärvi, Pyhaesalmi modeling project, Section B. Geology, Tech. rep., Geological Survey of Finland, and Outokumpu Mining Oy. (1999). H. N. Gharti, V. Oye, M. Roth, Travel times and waveforms of microseismic data in heterogeneous media, SEG Technical Program Expanded Abstracts 27 (1) (2008) 1337–1341. doi:10.1190/1.3059162. S. Stein, M. Wysession, An Introduction to Seismology, Earthquakes and Earth Structure, Wiley-Blackwell, 2002. J. N. Reddy, Introduction to the Finite Element Method, McGraw-Hill, 1993. E. Cuthill, J. McKee, Reducing the bandwidth of sparse symmetric matrices, in: Proceedings of the 24th National ACM Conference, ACM Press, New York, 1969, pp. 157–172. D. Komatitsch, J. Labarta, D. Michéa, A simulation of seismic wave propagation at high resolution in the inner core of the earth on 2166 processors of marenostrum, in: J. M. Palma, P. R. Amestoy, M. Daydé, M. Mattoso, J. a. C. Lopes (Eds.), High Performance Computing for Computational Science - VECPAR 2008, SpringerVerlag, Berlin, Heidelberg, 2008, pp. 364–377. A. J. Abbo, S. W. Sloan, An automatic load stepping algorithm with error control, International Journal for Numerical Methods in Engineering 39 (1996) 1737–1759. S. W. Sloan, A. J. Abbo, D. Sheng, Refined explicit integration of elastoplastic models with automatic error control, Engineering Computations 18 (2001) 121–194. E. A. de Souza Neto, D. Perić, D. R. J. Owen, Computational methods for plasticity: theory and applications, Wiley, 2009.

36

Chapter 6 Concluding remarks 6.1

Conclusions

Source location and moment-tensor inversion I have developed a robust and versatile source-location method for microearthquake data with low SNR. Although the method requires higher computational cost than most traditional methods, it is fully automatic. In case that seismic phases can be picked, a semiautomatic scheme can be applied to increase the computational speed and the efficiency. Additionally, the timereversal method may be used for simultaneous source location and moment-tensor inversion for microearthquake data. The main advantage of this technique is the automatic correction for source mechanism during stacking of the signals, which increases the resolution of the microearthquake locations. Reliable velocity model and good coverage of the receiver network are important for accurate source location and moment-tensor inversion. Similarly, the computation of reliable synthetic data (i.e., arrival times and full waveforms) is equally important.

Arrival times and wave propagation I computed first arrival times and full wavefields for microearthquakes in an underground ore mine and a rock slope, and for acoustic emissions in a weakly anisotropic sample used in a laboratory experiment. Because of the high-frequency content, even small-scale structural and material heterogeneities strongly influence the wavefield resulting in complicated waveforms. 136

Due to the wavefront healing, however, the influence of such heterogeneities on the arrival times may not be significant if the receivers are situated relatively far from those heterogeneities. The discretization of model is simple in the finite-difference method (FDM), but the mapping of complicated boundaries and interfaces is very difficult. As a result, the FDM is less accurate in the presence of strong contrasts and complex free surfaces. On the other hand, the discretization of model is a challenge in the spectral-element method (SEM), in particular for complex 3D models, but the SEM is more accurate in mapping model boundaries and interfaces giving a high degree of precision even in the presence of strong contrasts and complex free surfaces.

Elastoplastic failure The accuracy and efficiency of the spectral-element method can be very useful to model failure processes in solid (geo)mechanics. I implemented a 3D spectral-element method to solve elastoplastic failure problems, and applied it to slope stability analysis. The resulting software package may be an important tool for landslide hazard preparedness. The parallelization of the code makes the software applicable to large scale landslides and other failure problems in solid (geo)mechanics.

Multistage excavation In several solid (geo)mechanics problems, in which the model geometry changes with time, it is also important to understand the stress redistribution that can influence the progressive failure. I extended the software package for elastoplastic failure for such problems. I applied the method to multistage excavation in a mine. With this new tool, it is possible to study stress redistribution and possible progressive failure during such kind of dynamic processes, for example, mining and tunneling.

137

6.2

Outlook

Failure and microseismicity With the fully automatic migration-based and time-reversal tools (see Chapter 2), it is possible to detect a much higher number of microearthquakes and to determine their mechanisms. Similarly, the tools based on the spectral-element method facilitate the simulation of realistic failure processes in complex media (see Chapter 4). The combined interpretation of these results can provide a clearer understanding of microearthquake mechanisms, occurrences, and potentially unstable zones.

Hydrofracturing Hydrofracturing is an important geomechanical work process in several fields, for example, rock grouting, geothermal energy, and oil and gas industry (e.g., Wong and Farmer, 1973; Aydin, 2000; Gudmundsson et al., 2002). I have developed a tool to simulate the nonlinear process of failure with a higher-order numerical method, namely the spectral-element method (see Chapter 4). The higher-order spectral elements are well-suited to solve solid-fluid coupled problems in poroelastic media. After the implementation of poroelasticity, the existing spectral-element method can be a powerful tool to simulate hydrofractures.

Geodynamic problems I have developed a high-performance tool to simulate geomechanical processes where the model geometry alters significantly with time (see Chapter 5). (De)glaciation of the Earth is one of such processes. With the implementation of viscoelasticity or viscoplasticity, it is possible to simulate such processes as well as long-term creeping behavior of rock slopes, mines, tunnels, etc. (e.g., Latychev et al., 2005; Shalabi, 2005). On a regional or global scale, such tools can also be applied to study the phenomenon of post-seismic relaxation (e.g., Pollitz, 1997).

138

References Aydin, A., 2000, Fractures, faults, and hydrocarbon entrapment, migration and flow: Marine and Petroleum Geology, 17, 797–814. Gudmundsson, A., I. Fjeldskaar, and S. L. Brenner, 2002, Propagation pathways and fluid transport of hydrofractures in jointed and layered rocks in geothermal fields: Journal of Volcanology and Geothermal Research, 116, 257–278. Latychev, K., J. X. Mitrovica, J. Tromp, M. E. Tamisiea, D. Komatitsch, and C. C. Christara, 2005, Glacial isostatic adjustment on 3-D Earth models: a finite-volume formulation: Geophysical Journal International, 161, 421–444. Pollitz, F. F., 1997, Gravitational viscoelastic postseismic relaxation on a layered spherical Earth: Journal of Geophysical Research, 102, 17,921–17,941. Shalabi, F. I., 2005, FE analysis of time-dependent behavior of tunneling in squeezing ground using two different creep models: Tunnelling and Underground Space Technology, 20, 271– 279. Wong, H. Y., and I. W. Farmer, 1973, Hydrofracture mechanisms in rock during pressure grouting: Rock Mechanics and Rock Engineering, 5, 21–41.

139

Appendix A Softwares packages

140

A.1

Major contribution

I designed the main algorithms and wrote most of the source codes of the following software packages.

A.1.1

MIGLOC

MIGLOC is a migration-based source imaging and location software package written in MATLAB (see Chapter 2). The software uses the synthetic first-arrival times computed by a 3D finite-difference eikonal solver (Podvin and Lecomte, 1991) for heterogeneous velocity models. A robust global optimization technique, namely differential evolution (e.g., Price et al., 2005; Buehren, 2008) is implemented for the location of sources. The software can utilize both single and multicomponent data, and it can be applied to complex and heterogeneous 3D velocity models.

A.1.2

SPECFEM3D_SLOPE

SPECFEM3D_SLOPE is a 3D slope stability analysis software package based on the spectralelement method (see Chapter 4). The software is written entirely in FORTRAN 90. It uses unstructured hexahedral mesh to model complex structures. An element-by-element preconditioned conjugate gradient solver is employed for the efficient storage (e.g., Hughes et al., 1983; Law, 1986; Barragy and Carey, 1988). The software is parallelized using domain decomposition and MPI (Message Passing Interface). An alternative version of the package is parallelized using FORTRAN coarray — a recent extension to the FORTRAN language for efficient parallelization. The domain decomposition is accomplished using the graph partitioning tool SCOTCH (Pellegrini and Roman, 1996). Gravity, surface, and pseudo-static earthquake loading can be applied. Both a simple as well as a complex groundwater tables can be used to implement the effect of hydrostatic pressure.

141

A.1.3

SPECFEM3D_EXCAVATION

SPECFEM3D_EXCAVATION is an extension of the software package SPECFEM3D_SLOPE to simulate multistage excavation in various geotechnical constructions, e.g., mining, tunneling, and underground caverns (see Chapter 5). The software is based on a fixed-partition strategy to decompose large-scale models. It can be applied to complex heterogeneous models ranging from small to large scales.

A.1.4

TREVERSAL

TREVERSAL is a software to determine source location and qualitative moment-tensor simultaneously (see Chapter 2). This program utilizes synthetic strain Green’s tensors for crosscorrelation and stacking to obtain the source image and the qualitative moment-tensor. It is parallelized based on the SPMD (Single Program Multiple Data) model, and it uses the MATLAB parallel toolkit.

A.2

Minor contribution

I also used following software packages extensively and added minor contributions.

A.2.1

SPECFEM3D_SESAME / SPECFEM2D

SPECFEM3D_SESAME is a software package for the simulation of wave propagation in 3D models of various scales. This software has originally been developed by Komatitsch et al. (e.g., Komatitsch et al., 2001; Tromp et al., 2008). The software is written in FORTRAN 90, and parallelized based on domain decomposition using MPI. The software uses unstructured hexahedral mesh, and can simulate wave propagation in anelastic and anisotropic media. I added routines for the weak anisotropy and parallel visualization. SPECFEM2D is a 2D version of SPECFEM3D_SESAME to simulate wave propagation in 2D models using unstructured quadrilateral mesh. This has also originally been developed by Komatitsch et al. (e.g., Komatitsch and Tromp, 1999; Tromp et al., 2008). I added routines for weak anisotropy and visualization. 142

A.2.2

E3D

E3D is a software package for the simulation of wave propagation in 2D and 3D models based on the viscoelastic finite-difference method. The software has been developed by Shawn Larsen (Larsen and Schultz, 1995). It is written in C, and parallelized based on domain decomposition using MPI. Both absorbing as well as free surface boundary conditions can be applied. Complex heterogeneous velocity models can be used. I added routines for the computation of strain Green’s tensors, and for visualization.

A.2.3

FDTIMES

FDTIMES is a 2D/3D finite-difference eikonal solver to compute first-arrival times. The solver is written in C, and it has been developed by Podvin and Lecomte (1991). Complex heterogeneous velocity models can be used. I added routines for structured input, computation of rays and fat rays, and visualization.

References Barragy, E., and G. F. Carey, 1988, A parallel element-by-element solution scheme: International Journal for Numerical Methods in Engineering, 26, 2367–2382. Buehren, M., 2008, Differential evolution: www.mathworks.com/matlabcentral/ fileexchange/18593. (Online; accessed 14-June-2011). Hughes, T. J. R., I. Levit, and J. Winget, 1983, An element-by-element solution algorithm for problems of structural and solid mechanics: Computer Methods in Applied Mechanics and Engineering, 36, 241–254. Komatitsch, D., R. Martin, J. Tromp, M. A. Taylor, and B. A. Wingate, 2001, Wave propagation in 2-D elastic media using a spectral element method with triangles and quadrangles: Journal of Computational Acoustics, 9, 703–718. Komatitsch, D., and J. Tromp, 1999, Introduction to the spectral element method for threedimensional seismic wave propagation: Geophysical Journal International, 139, 806–822.

143

Larsen, S., and C. A. Schultz, 1995, ELAS3D: 2D/3D elastic finite difference wave propagation code: Technical Report No. UCRL-MA-121792: Technical report. Law, K. H., 1986, A parallel finite element solution method: Computers & Structures, 23, 845– 858. Pellegrini, F., and J. Roman, 1996, SCOTCH: A software package for static mapping by dual recursive bipartitioning of process and architecture graphs: Lecture Notes in Computer Science, 1067, 493–498. Podvin, P., and I. Lecomte, 1991, Finite difference computation of traveltimes in very contrasted velocity models: a massively parallel approach and its associated tools: Geophysical Journal International, 105, 271–248. Price, K. V., R. M. Storn, and J. A. Lampinen, 2005, Differential evolution: a practical approach to global optimization: Springer. Tromp, J., D. Komatitsch, and Q. Liu, 2008, Spectral-element and adjoint methods in seismology: Communications in Computational Physics, 3, 1–32.

144

Modelling and inversion of microearthquakes

Modelling and inversion of microearthquakes

Suggest Documents

PATTERNS OF INDUCED MICROEARTHQUAKES ...

Advances in modelling and inversion of seismic wave propagation

Forward Modelling and Inversion of Multi-Source TEM Data

Advances in modelling and inversion of seismic wave propagation

for volcanic microearthquakes - Earth-prints

GIMLi â Geophysical Inversion and Modelling Library â programmers ...

Estimates of sub-national methane emissions from inversion modelling

Modelling the Bias of Abel Inversion Algorithm for ...

Source Parameters of Injection-Induced Microearthquakes at 9 km ...

Inversion, retrograde, and retrograde inversion - The University of ...

Inversion and Configuration of Faces

An inversion modelling method to obtain the acoustic ...

Dynamic triggering of microearthquakes in three ... - Wiley Online Library

The importance of microearthquakes in crustal ... - Wiley Online Library

The relocation of microearthquakes in the ... - Wiley Online Library

Moho depth determination from waveforms of microearthquakes in the ...

Tidal triggering of microearthquakes on the Juan de Fuca Ridge

The relocation of microearthquakes in the northern Mississippi

Source Parameters of Injection-Induced Microearthquakes at 9 km

Contrast between 2D inversion and 3D inversion ... - Science Direct

Measurement Of Inversion And Eversion Movements ...

inversion - Unicef

inversion - Unicef

partial derivatives and iterative inversion of seismic

Modelling and inversion of microearthquakes