TIME-LAPSE ULTRASONIC IMAGING OF ELASTIC ANISOTROPY IN SATURATED SANDSTONE UNDER POLYAXIAL STRESS STATE

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1 TIME-LAPSE ULTRASONIC IMAGING OF ELASTIC ANISOTROPY IN SATURATED SANDSTONE UNDER POLYAXIAL STRESS STATE by Mehdi (Sherveen) Ghofrani Tabari A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto Copyright 2015 by Mehdi Ghofrani Tabari

2 Time-Lapse Ultrasonic Imaging of Elastic Anisotropy in Saturated Sandstone under Polyaxial Stress State Abstract Mehdi (Sherveen) Ghofrani Tabari Doctor of Philosophy Physics Department University of Toronto 2015 Although true-triaxial test (TTT) of rocks is now more extensive worldwide, stress-induced heterogeneity is not accounted for and usually simplified anisotropic models are used. Data from a TTT on a cubic sample of Fontainebleau sandstone is used in this study to evaluate our velocity imaging methodology. An anisotropic P wave velocity tomography method was developed using a geometrical approach based on an ellipsoidal P wavefront surface. During the two non-damaging phases of the experiment, saturation of the rock sample with water resulted in inaccurate tomographic images; however, during the final elasto-plastic phase of the experiment comprising major AE activities, tomographic images demonstrated reasonable anomalies. Thus, the P-S1-S2 velocity survey was utilized to obtain an accurate and reliable velocity image of the sample during the two non-damaging phases. This was accomplished using a numerical investigation by FLAC3D on the non-uniform distribution of stress over the sample to II

3 estimate the compaction pseudo-boundary surfaces within the rock. Thus, the problem of breakdown in the expected symmetry of shear wave velocities was resolved. It was discovered that a homogeneous anisotropic core in the center of the sample is formed under the standard polyaxial setup where elastic parameters could be computed. Offdiagonal elastic tensor parameters were obtained by a combination of various velocity survey data and justified the ellipsoidal model as being the most appropriate and facilitated the calculation of Thomsen parameters. The ellipsoidal heterogeneous velocity model was also verified by AE event location of transducer shots through the cubic rock specimen especially at the final phase of the experiment consisting lower-velocity zones bearing partially saturated fractures. AE of the rock during the whole experiment recorded by the surrounding transducers were investigated by location methods developed for anisotropic heterogeneous medium. AE events occurred in the vicinity of the dilation pseudo-boundaries where, a relatively large velocity gradient was formed and along parallel fractures in the σ 1 /σ 2 plane. This research facilitated the computation of anisotropic parameters for rock during polyaxial tests contributing to enhanced AE interpretation of fracture growth processes in the rock under laboratory true-triaxial stress conditions. III

4 Acknowledgement I would like to express my deep appreciation to Prof. R. Paul Young for his wise and constructive supervision and support during the course of my PhD program and for providing and offering extremely helpful resources such as the RFDF laboratory with its unique experimental setup. I would like to offer my special thanks to Laszlo Lombos at the ErgoTech Company for designing the True-Triaxial Geophysical Imaging Cell and to Dr. Farzin Nasseri and Sebastian Goodfellow who played a major role in the operation and data analysis of the polyaxial experiments. I am also thankful for the supportive advice I received from Prof. Richard C. Bailey and Prof. Bernd Milkereit. Finally, my gratefulness is extended to the staff of the Physics department, Ms. Krystyna Biel and Ms. Teresa Baptista for their generous assistance during my graduate studies at the University of Toronto. IV

5 Research Contribution The Rock Fracture Dynamics Facility (RFDF) laboratory was constructed under the leadership of Prof. R. Paul Young starting from January The facility was mainly funded by the Canadian Foundation for Innovation (CFI). It contained a custom-made MTS loading frame as well as a True Triaxial Geophysical Imaging Cell (TTGIC) built by Laszlo Lombos. RFDF also used a specifically designed Acoustic Emission (AE) streaming equipment for continuous recording of the AE waveforms. A set of four polyaxial experiments were conducted on Fontainebleau sandstone samples during the years of 2011 and 2012 where the last two were successful tests with acceptable results. The data from the last True Triaxial Test (TTT) which was also called FTB4 is used in this thesis. All the sample preparations were done by the author of this thesis, Sherveen G. Tabari. Also, the piping task connecting the TTGIC system with the permeability measurement equipment was accomplished by Sherveen under the supervision of Laszlo Lombos before the tests. The TTTs was designed and led by Dr. Farzine Nasseri, the RFDF lab manager, and the tests were accomplished by Sherveen, Sebastian Goodfellow and Hamed Owlade-Ghaffari under the supervision of Prof. Young. A few discussions were made after the experiment within the group where Hamed also participated effectively. The main journal paper as a comprehensive report of this unique FTB4 test was published in 2014 (Nasseri et al., 2014) containing the preliminary data analysis on permeability, apparent velocity imaging and AE source locations. The FLAC3D simulations were developed in a workshop by Itasca and V

6 performed in collaboration with Sebastian. The advanced analysis of the velocity survey data was all accomplished by Sherveen illustrated in this thesis which includes the development of anisotropic tomography, AE location, and other inversion codes in MATLAB. VI

7 Table of Contents Abstract... II Acknowledgement... IV Research Contribution... V Table of Contents... VII List of Tables... XII List of Figures... XIII 1 Introduction Nature of Our Research Objectives of Research Thesis Outline Literature Review Crack-Induced Anisotropy in Rocks Effect of Stress on Crack-Induced Anisotropy Effect of Fluid Saturation on Elastic Properties of the Rock Laboratory Studies on Velocity Anisotropy in Rocks Physics and Mechanics of Rocks in Polyaxial Experiments Elastic Wave Propagation in Ellipsoidal Media Anisotropic Tomography and Laboratory Studies AE Event Location Theory of Elastic Anisotropy Wave Propagation in Anisotropic Media Christoffel s Equation VII

8 3.3 Phase and Group Velocity Stiffness Tensor Orthorhombic Anisotropy Ellipsoidal Anisotropy Rasolofosaon s Theory Daley s Theory Thomsen Parameters Anisotropic Velocity Model Parameters Geophysical Imaging Cell and Laboratory Setup True-Triaxial Test Procedure on Fontainebleau Sandstone Fontainebleau Sandstone Sample Preparation True-Triaxial Test Procedure Ultrasonic Velocity Measurements in Three Principal Directions (P-S1-S2 Velocity Surveys) Evolution of P, S1, and S2 Velocities Transducer-to-Transducer Velocity Survey Stereonet Representation of the P wave Velocity Error Analysis for Velocity Measurements Anisotropic Travel-Time Tomography Ray-based Tomography in Ellipsoidal Media Anisotropic Tomography Images Simulation of Stress Distribution Velocity Anisotropy Analysis Compaction Pseudo-Boundary Surfaces Shear Wave Velocities in the Minimum and Intermediate Principal Stress Directions87 VIII

9 4.7.3 Compressional Wave Velocities Velocity Domains of the Cubic Rock and Diagonal Stiffness Tensor Parameters Off-Diagonal Stiffness Tensor Parameters Rudzki s Ellipticity Condition Cheadle s Method for Orthorhombic Anisotropy Daley s Method for Inversion of Anellipsoidal Parameters Strength of Anisotropy with Thomsen Parameters Discussion and Conclusions Effect of Velocity Structure on AE Source Locations AE and Microseismic Event Location Source Location Theory in an Isotropic Homogeneous Medium Collapsing Grid Search Location Results in a Synthetic Isotropic Homogeneous Medium Real Transducer Survey Shot Locations Source Location in an Anisotropic Homogeneous Medium Real Transducer Survey Shot Locations Source location in an Anisotropic Heterogeneous Medium Location Results in a Synthetic Anisotropic Heterogeneous Model Real Transducer Survey Shot Locations AE Source Locations during the Polyaxial Experiment Discussion and Conclusions Conclusions Local Heterogeneities and Anisotropic Tomography Velocity Structure Induced in the Polyaxial Experiments AE Source Locations in Polyaxial Experiments IX

10 6.4 Summary of Scientific Contributions Recommendations for Future Studies References Appendix Appendix A. Ultrasonic Travel-Time Tomography A.1 Introduction A.2 Transducer Setup for Acoustic Imaging A.3 Ray-Path Segment Matrix A.3.1 An Alternative Arrangement of Sensor Configuration Design A.4 Travel-Time Inversion A.4.1 Singular Value Decomposition (SVD) A.4.2 Picard Cut-Off Number A.5 Isotropic Synthetic Modelling A.5.1 Inversion of the Synthetic Results Built upon Hypothetical Isotropic Models A.5.2 Checker-Board Model A.6 Anisotropic Travel-Time Tomography A.7 Anisotropic Synthetic Modeling A.7.1 Precision of Measurement in Arrival-Times A.7.2 Evaluating the Anisotropic Tomography Method for Cubic Block-Sized Heterogeneities A.7.3 Anomaly Pattern Recognition Test Using the Inaccuracy Factor A.7.4 Detectability of Heterogeneous Cells A.8 Tomography Results form the Polyaxial Experiment A.8.1 Ray Coverage A.8.2 Cross-Correlation for Real-Time Monitoring X

11 A.8.3 Consistency with Micro CT Images and AE Locations A.9 Interpretation and Discussion Appendix B. Flowchart of Data Acquisition and Processing XI

12 List of Tables Table 3.1: The relationship between elastic constants to the generic Thomsen parameters. All the relations are valid for any strength of anisotropy Table 4.1: Transformation of Ergotech notation for wave velocity propagation directions and polarizations to Cartesian coordinate system notation Table 5.1: Isotropic homogeneous velocity model Table 5.2: Anisotropic homogeneous velocity model Table 5.3: Synthetic anisotropic heterogeneous velocity model Table 5.4: Anisotropic heterogeneous velocity model XII

13 List of Figures Figure 2.1: Schematic illustration of seismic birefringence or shear wave splitting in distributions of stressaligned fluid-saturated parallel vertical micro-cracks aligned normal to the direction of minimum horizontal stress,. For nearly vertical propagation the polarisation of the faster split shear wave is parallel to the strike of the cracks, parallel to the direction of maximum horizontal stress, (Crampin et al., 2005) Figure 2.2: Diagrammatic view of Imperial College s true-triaxial loading frame (King, 1997) Figure 3.1: The group velocity (ray) vector pointing from the source to the receiver (angle ψ) in a homogeneous anisotropic medium. The corresponding phase velocity (wave) vector is orthogonal to the wavefront (angle θ) (Tsvankin, 2001) Figure 4.1: Polyaxial servo-controlled MTS system. Each of the six stainless steel platens with three AE sensors are installed on the True-triaxial Geophysical Imaging Cell (TTGIC) designed and manufactured by ErgoTech Figure 4.2: Evolution of the main principal stress in the Z direction (blue) and the cumulative number of located AE events representing the AE activity (green). The red arrow shows the onset of high AE activity of the specimen indicating the failure of the sample Figure 4.3: Evolution of the main principal stress in the Z direction (blue) and the evolution of strain in the X-axis (green). The red arrow shows the onset of dilation of the rock in the X direction Figure 4.4: Schematic view of different stages of the experiment on a stress strain curve in the main principal stress direction Figure 4.5: Ergotech Diagram of the P-S1-S2 velocity measurements Figure 4.6: Cross-section of TTGIC in the YoZ plane containing the sample in the middle and two opposite platens. The chamfered edges and P-S1-S2 acoustic stacks are displayed (Lombos et al., 2012). 48 Figure 4.7: P-S1-S2 wave velocity measurements during the FTB4 experiment Figure 4.8: Real laboratory sensor arrangement. Left: transducer locations. There are 3 transducers embedded on each platen. Locations of the transducers on each pair of the opposite platens are the same. Right: Ray-paths between the 18 transducers covering the cubic sample Figure 4.9: A waveform window and the first-arrival picking in the InSite seismic processing software Figure 4.10: Evolution of the rock in FTB4 experiment represented by directional apparent P wave velocities demonstrated by stereonets at different stress states with constant minimum and intermediate principal stress ( ) and increasing maximum principal stress ( ) Figure 4.11: Frequency spectrum of an ultrasonic waveform arrived from a source of piezoelectric transducer pulse. The red window shows the frequency filter applied to the data for data analysis XIII

14 Figure 4.12: Ellipsoidal anisotropic velocity model Figure 4.13: Stereonet representation of the apparent velocity data at ( ) where, the average velocity of the 2 rays in the X direction is 4.9 km/s, the average velocity of the 3 rays in the Y direction is 5.2 km/s, and the average velocity of the 2 rays in the Z direction is 5.6 km/s Figure 4.14: Anisotropic P wave travel-time Tomography ( ) from the transducer survey at ( ). The minimum inverted velocity is 5.1 km/s and the maximum inverted velocity is 6.2 km/s Figure 4.15: Stereonet representation of the apparent velocity evolution at ( ) from its previous velocity survey that was performed at ( ) Figure 4.16: Evolution of the velocity structure of the rock in the Z direction presented by a tomography image between the two velocity surveys at ( ) and ( ) Figure 4.17: Stereonet representation of the apparent velocity data at ( ), where the average velocity of the 2 rays in the X direction is 4.4 km/s, the average velocity of the 3 rays in the Y direction is 5.0 km/s, and the average velocity of the 2 rays in the Z direction is 5.5 km/s Figure 4.18: Anisotropic P wave travel-time Tomography from the transducer survey at ( ). The minimum inverted velocity is 4.4 km/s and the maximum inverted velocity is 6.6 km/s Figure 4.19: Stereonet representation of the apparent velocity evolution at ( ) from its previous velocity survey that was performed at ( ) Figure 4.20: Evolution of the velocity structure of the rock in the Z direction presented by a tomography image between the two velocity surveys at ( ) and ( ) Figure 4.21: Cross-sectional view of the TTT system including the three platens and the cubic sample in FLAC3D Figure 4.22: Schematic view of the cubic sample with chamfered edges and the cross-sectional slab in green that is simulated in FLAC3D for the YoZ and XoZ planes Figure 4.23: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample along with part of the platen touching the sample at stress state Figure 4.24: Apparent velocity survey demonstrating 106 ray-paths (Right) and stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) under the stress state. The average velocity of the 2 rays in the X direction is m/s. The average velocity of the 3 rays in the Y direction is m/s. The average velocity of the 2 rays in the Z direction is m/s. The P wave velocity measurements that are available from the ultrasonic P, S1 & XIV

15 S2 acoustic stack are, km/s in the X direction, km/s in the Y direction and km/s in the Z direction Figure 4.25: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample along with part of the platen touching the sample at stress state. The dashed line shows the contour interface between the maximum stress and the next lower stress zone Figure 4.26: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocity of the 2 rays in the X direction is m/s. The average velocity of the 3 rays in the Y direction is m/s. The average velocity of the 2 rays in the Z direction is m/s Figure 4.27: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample along with part of the platen touching the sample at stress state Figure 4.28: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocity of the 2 rays in the X direction is m/s. The average velocity of the 3 rays in the Y direction is m/s. The average velocity of the 2 rays in the Z direction is m/s Figure 4.29: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample along with part of the platen touching the sample at stress state Figure 4.30: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocity of the 2 rays in the X direction is m/s. The average velocity of the 2 rays in the X direction is m/s. The average velocity of the 3 rays in the Y direction is m/s. The average velocity of the 2 rays in the Z direction is m/s Figure 4.31: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample (Above) at stress state and, the corresponding graphs presenting the main principal stress value at the center of the rock along the X and Y directions Figure 4.32: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocities in the X, Y and Z directions are m/s, m/s, and m/s respectively XV

16 Figure 4.33: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample (Above) at stress state and, the corresponding graphs presenting the main principal stress value at the center of the rock along the X and Y directions Figure 4.34: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocities in the X, Y and Z directions are m/s, m/s, and m/s respectively Figure 4.35: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample (Above) at stress state and, the corresponding graphs presenting the main principal stress value at the center of the rock along the X and Y directions Figure 4.36: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocities in the X, Y and Z directions are m/s, m/s, and m/s respectively Figure 4.37: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample (Above) at stress state and, the corresponding graphs presenting the main principal stress value at the center of the rock along the X and Y directions Figure 4.38: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocities in the X, Y and Z directions are m/s, m/s, and m/s respectively Figure 4.39: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample along with part of the platen touching the sample (Above) at stress state and, the corresponding graphs presenting the main principal stress value at the center of the rock along the X and Y directions Figure 4.40: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample (Above) at stress state and, the corresponding graphs presenting the main principal stress value at the center of the rock along the X and Y directions Figure 4.41: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocities in the X, Y and Z directions are m/s, m/s, and m/s respectively XVI

17 Figure 4.42: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample (Above) at stress state and, the corresponding graphs presenting the main principal stress value at the center of the rock along the X and Y directions Figure 4.43: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocities in the X, Y and Z directions are m/s, m/s, and m/s respectively Figure 4.44: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample (Above) at stress state and, the corresponding graphs presenting the main principal stress value at the center of the rock along the X and Y directions Figure 4.45: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocities in the X, Y and Z directions are m/s, m/s, and m/s respectively Figure 4.46: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample (Above) at stress state and, the corresponding graphs presenting the main principal stress value at the center of the rock along the X and Y directions Figure 4.47: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocities in the X, Y and Z directions are m/s, m/s, and m/s respectively Figure 4.48: Direction of propagation and polarization of the nine components of velocity with respect to the principal stress axes, and planes of aligned cracks produced (King et al., 1997) Figure 4.49: S wave velocities as a function of stress during the cracking cycle with (King et al., 1997) Figure 4.50: Compaction pseudo-boundary criterion in minor and intermediate principal stress directions Figure 4.51: Green dashed line: LS zone effective thickness in the Y direction. Blue line: Major stress. Red line: Cumulative number of AE events. Green dots: AE events plotted by distance from the Y surface.. 86 Figure 4.52: Green dashed line: LS zone effective thickness in the X direction. Blue line: Major stress. Red line: Cumulative number of AE events. Green dots: AE events plotted by distance from the X surface.. 87 Figure 4.53: Measured shear wave propagating in the X direction and polarized in the Z direction along with the magnitudes in the compact central domain, and in the near-facet or lowerstressed domain in the specimen XVII

18 Figure 4.54: Measured shear wave propagating in the Y direction and polarized in the Z direction along with the magnitudes in the compact central domain, and in the near-facet or lowerstressed domain in the specimen Figure 4.55: Velocity contrast at the minimum principal stress direction compaction pseudo-boundary (red) and at the intermediate principal stress direction compaction pseudo-boundary (dark blue) Figure 4.56: Shear wave velocities in the central compacted zone and each pair of the near X- and Y- facets propagating in the XoY plane Figure 4.57: Average of the measured and shear wave velocities compared to the average of their calculated values in the central compacted domain Figure 4.58: Measured P wave velocities along the three principal stress directions from P-S1-S2 surveys Figure 4.59: Average P wave velocities propagating in each of the three principal stress directions from transducer-to-transducer surveys Figure 4.60: Measured and calculated P wave velocities propagating in the minimum principal stress direction in different domains Figure 4.61: Measured and calculated P wave velocities propagating in the intermediate principal stress direction in different domains Figure 4.62: P wave velocity contrasts at the compaction pseudo-boundaries in the X and Y directions. 99 Figure 4.63: A schematic view of the rock with resolved shear and compressional wave velocities Figure 4.64: Diagonal stiffness tensor coefficients for the central compacted domain of the rock specimen Figure 4.65: Off-diagonal stiffness coefficients calculated from Rudzki s ellipticity conditions Figure 4.66: Transmission measurements between opposing edges of the phenolic cube. The propagation directions were at 45 degrees to two of the principal axes and perpendicular to the third (Cheadle et al., 1991) Figure 4.67: Ray-paths chosen for calculation of off-diagonal stiffness tensor coefficients by Cheadle s method Figure 4.68: Velocity evolution of the four ray-paths that are used in calculation of off-diagonal stiffness tensor coefficients by Cheadle s method. r28 lies on the XoY plane with lowest velocity, r48 lies on the XoZ plane, r105 is the first ray mentioned in the text that lies on the YoZ plane and r100 is the second ray on the YoZ plane Figure 4.69: Cheadle s resolved stiffness tensor coefficient along with those coefficients obtained from Rudzki s equations XVIII

19 Figure 4.70: Thirty two ray-paths with lengths larger than 80 mm that were used for the inversion of the anellipsoidal deviation vector Figure 4.71: Non-diagonal stiffness matrix components resolved by Daley s method Figure 4.72: Non-diagonal stiffness matrix components resolved by Daley s method using the uncorrected measured P wave velocities Figure 4.73: Tsvankin s version of Thomsen parameters for orthorhombic media in the YoZ plane ( ) Figure 4.74: Tsvankin s version of Thomsen parameters for orthorhombic media in the XoZ plane ( ) Figure 4.75: Tsvankin s version of Thomsen parameters for orthorhombic media in the XoY plane ( ) Figure 5.1: Location results of the synthetic travel-time data in an isotropic homogeneous velocity model. The stars show the location results Figure 5.2: Mean time residual (R) of the real data and synthetic travel-times calculated by all the located transducers using a single value of 5500 (m/s) for the isotropic homogeneous velocity model for separate transducer surveys at 20 different stress states Figure 5.3: Mean location misfit of the transducer location results from the fixed transducer positions in 20 different transducer surveys in various stress states using a single value of 5500 (m/s) for the isotropic homogeneous velocity model Figure 5.4: Mean time residual (R) of the real data and synthetic travel-times calculated by all the located transducers using the varying isotropic homogeneous velocity model for separate transducer surveys at 20 different stress states Figure 5.5: Mean location misfit of the transducer location results from the fixed transducer positions in 20 different transducer surveys in various stress states using the varying isotropic homogeneous velocity model Figure 5.6: Mean time residual (R) of the real data and synthetic travel-times calculated by all the located transducers using the transversely isotropic homogeneous velocity model for separate transducer surveys at 20 different stress states Figure 5.7: Mean location misfit of the transducer location results from the fixed transducer positions in 20 different transducer surveys in various stress states using the transversely isotropic homogeneous velocity model Figure 5.8: Mean time residual (R) of the real data and synthetic travel-times calculated by all the located transducers using the ellipsoidal homogeneous velocity model for separate transducer surveys at 20 different stress states XIX

20 Figure 5.9: Mean location misfit of the transducer location results from the fixed transducer positions in 20 different transducer surveys in various stress states using the ellipsoidal homogeneous velocity model Figure 5.10: The anisotropic heterogeneous velocity structure of the rock obtained in chapter 4 by the P- S1-S2 velocity survey analysis Figure 5.11: Mean time residual (R) of the real data and synthetic travel-times calculated by all the located transducers using the ellipsoidal heterogeneous velocity model for separate transducer surveys at 20 different stress states Figure 5.12: Mean location misfit of the transducer location results from the fixed transducer positions in 20 different transducer surveys in various stress states using the ellipsoidal heterogeneous velocity model Figure 5.13: First phase of the AE source locations in a 3D and three orthogonal cross sectional images Figure 5.14: Second phase of the AE source locations in a 3D and three orthogonal cross sectional images Figure 5.15: Third phase of the AE source locations in a 3D and three orthogonal cross sectional images Figure 5.16: Fourth phase of the AE source locations in a 3D and three orthogonal cross sectional images Figure 5.17: Fifth phase of the AE source locations in a 3D and three orthogonal cross sectional images Figure 5.18: Sixth phase of the AE source locations in a 3D and three orthogonal cross sectional images Figure 5.19: Seventh phase of the AE source locations in 3D and three orthogonal cross-sectional images Figure 5.20: Eighth phase of the AE source locations in a 3D and three orthogonal cross sectional images Figure 5.21: Ninth phase of the AE source locations in a 3D and three orthogonal cross sectional images Figure 5.22: Tenth phase of the AE source locations in a 3D and three orthogonal cross sectional images Figure 5.23: Eleventh phase of the AE source locations in a 3D and three orthogonal cross sectional images XX

21 Figure 5.24: Twelfth phase of the AE source locations in a 3D and three orthogonal cross sectional images Figure 5.25: Thirteenth phase of the AE source locations in a 3D and three orthogonal cross sectional images Figure 5.26: Fourteenth phase of the AE source locations in a 3D and three orthogonal cross sectional images Figure 5.27: Fifteenth phase of the AE source locations in a 3D and three orthogonal cross sectional images Figure 5.28: AE source location patterns versus CT scan image of the XoZ cross section of the rock at the middle of the Y axis Figure A.1: Real laboratory sensor arrangement. Left: transducer locations. There are 3 transducers embedded on each platen. Locations of the transducers on each pair of the opposite platens are the same. Right: Ray-paths between the 18 transducers covering the cubic sample Figure A.2: Ray travel scenarios through the square and cubic cells in 2D and 3D conditions respectively. In the 2D case, there are only 3 different scenarios, while in the 3D case there are 11 different scenarios for rays to travel between each opposite and parallel interfaces of the cells. Numbers in red indicate each interface through which the rays pass. The left-right arrow also represents the intersection point intersecting with the boundary between the four interfaces Figure A.3: A ray-path and the probed cells along its path between two sensors Figure A.4: Representation of the total ray length in each block for a sample with 18 transducers and 5 blocks along each edge. n = 135 data points. m = 125 blocks. Number of blank blocks = Figure A.5: Representation of the total ray length in each block for a sample with 18 transducers, and 7 blocks along each edge. n = 135 data points. m = 343 blocks. Number of blank blocks = Figure A.6: Hypothetical sensor arrangement. Left: transducer locations. There are 5 transducers embedded on each platen. Locations of the transducers on each pair of the opposing platens are the same. Right: Ray-paths between the 30 transducers covering the cubic sample Figure A.7: G matrix representation of the 30 transducers array for a sample with 5 blocks along each edge. n = 375 data points. m = 125 blocks. Number of blank blocks = Figure A.8: Schematic representation of Picard Plot (Aster et al., 2004) Figure A.9: Left: Synthetic isotropic model with one heterogeneous zone of Δv = - 1% ; Right: Recovered velocity model with Pout= Figure A.10: Hypothetical isotropic checker-board model with 5% heterogeneity Figure A.11: Inversion of the hypothetical isotropic checker-board model with 18 sensors Figure A.12: Inversion of the hypothetical isotropic checker-board model with 30 sensors XXI

22 Figure A.13: Schematic representation of oriented penny-shaped cracks under true-triaxial stress state ( ) Figure A.14: Waveform visualizer in the InSite software Figure A.15: The cubic heterogeneous block with the same size of the cells will move onto each cell in every corresponding synthetic model Figure A.16: Left: a) Stereonet representation of the real velocities in a stage of the experiment where,. b) the calculated default homogeneous ellipsoidal anisotropic velocity model that is used as the background velocity structure of the synthetic model. It is obtained from averaging the real laboratory travel-time data: Vx= 4729 (m/s), Vy= 5211 (m/s), Vz= 5633 (m/s). Difference between heterogeneous model travel-times and homogeneous one is incorporated in the inversion calculations. Right: The flowchart for anomaly recognition test Figure A.17: Left: Mean absolute error between the overall synthetic and inverted model. Right: Inaccuracy factor vs. Picard cut-off number Figure A.18: Detectability of Heterogeneity. Inaccuracy factor of the 125 different synthetic models with same Picard Cut-off Number (30) as well as the same background anisotropic velocity structure versus different heterogeneity percentages Figure A.19: Ray coverage of the cubic sample during the polyaxial experiment Figure A.20: Coverage of the cubic sample during the polyaxial experiment Figure A.21: Coverage of the cubic sample during the polyaxial experiment Figure A.22: The cross-correlation window in InSite software Figure A.23: Left: picture of the rock surface after the FTB3 experiment on the same kind of sandstone. Right: CT scan image of the FTB4 experiment rock from the XoZ plane side (Nasseri et al., 2014) Figure A.24: CT scan image of the rock superimposed with located AE events that occurred in stage D of the experiment. The red dots indicate the AE events that happened between the two velocity surveys at ( ) and ( ) above the blue dots that indicate those AE events that already happened before XXII

23 1 Introduction 1.1 Nature of Our Research It is now 185 years since the first time the idea of direction-dependency in velocity of the elastic waves, called anisotropy, got significant attention (Helbig et al., 2005). In the broader sense, a material is anisotropic if its properties, when measured at the same location, change with direction. A material is inhomogeneous if its properties, when measured in the same direction, change with location. As Winterstein [1990] stated, all anisotropy originates in inhomogeneity, and every inhomogeneous material is also, at some scale, anisotropic in this broad sense. For most of the 19th century, anisotropic wave propagation was studied mainly by mathematical physicists, and the only applications were in crystal optics and crystal elasticity. Seismic anisotropy began in 1895 with Maurice Rudzki, the first official professor of geophysics, who was certain of the anisotropy of rocks. At the turn of the 20th century, Rudzki stressed the significance of seismic anisotropy from a theoretical perspective. However, it was in the last two decades of the 20th century that anisotropy changed from a nuisance to an opportunity, for which special multicomponent surveys were undertaken. Aligned discontinuities are one of the reasons rocks indicate anisotropic characteristics in which the elastic behaviours such as elastic wave propagation velocity vary with direction in different scales. For example, in the scale of earthquake seismology which is a leading geophysical field, shear wave splitting as the most significant fingerprint of anisotropy was recognised in seismograms in igneous and metamorphic rocks above small earthquakes. Shear wave splitting was also observed in seismic reflection surveys and vertical seismic profiles in engineering scales of the sedimentary hydrocarbon reservoirs (Crampin et al., 2005). Due to the urgent need for safe engineering construction at great depths underground, laboratory studies of the mechanisms of rock deformation and failure, where the in-situ stress and hydraulic pressure could be simulated, became of great importance. Specifically, monitoring of induced seismicity and research on behaviour of rocks with systems of aligned discontinuities gained importance in a broad range of industrial operations from hydrocarbon reservoirs to underground mines and geothermal 1

24 projects. The passive microseismic monitoring mainly aims at identifying fractures, which is of special interest for safety and productivity reasons. For example, in civil engineering, mining engineering and underground disposal, it is very important to be able to quantify the extension of the rock mass zone disturbed by underground excavations. Moreover, seismic imaging using micro-earthquakes induced by hydraulic fracturing that produces a three-dimensional (3D) velocity model of the fractured zone improves the calculated locations of the micro-earthquakes, and will lead to better estimates of fracture plane orientations, fracture density, and water or oil flow paths. Oil companies have recently been investing billions of dollars in seismic time-lapse surveys of producing reservoirs, where (usually) three-component reflection surveys over extensive areas are repeated with similar source-to-receiver geometries in order to monitor the movement of fluids in producing reservoirs. The rock mass discontinuity sizes differ over several orders of magnitude from large scale faults and fractures to micro-cracks. If fractures are embedded in an otherwise isotropic host rock, the symmetry of the crack-induced anisotropy is close to orthorhombic for any fracture orientations, shapes, and types of infill (Tsvankin et al., 2010). As well, the regional stress field as a symmetric second-rank tensor has at least three mutually perpendicular planes of mirror symmetry, which is the defining property of an orthorhombic elastic solid. Therefore, orthorhombic symmetry is common in most regions of the crust. This stress field generally consists of two unequal horizontal (tectonic) principal stresses and a vertical principal stress caused by the overburden weight. Among the anisotropic symmetries, ellipsoidal and transversely isotropic (TI) models are also very well known in subsurface geomechanics. Ellipsoidal symmetry is a degenerate case of orthorhombic symmetry but with 6 independent elastic parameters instead of 9. TI symmetry represents anisotropy as an ellipsoid having a vertical rotation axis of symmetry with 5 independent elastic parameters. Mathematical demonstration of the abovementioned anisotropic symmetries is given in chapter 3. Laboratory rock mechanics resemble in-situ stress patterns but in the small scale which can be studied under some controlled processes. A significant privilege of a laboratory with fracture mechanics and microseismic facilities is the real-time monitoring of the active seismicity or passive seismic properties of the rock specimen, either by screening its Acoustic Emission (AE) events, or through non-destructive methods such as 2

25 ultrasonic tomography. Consequently, laboratory studies help us obtain a better understanding of fracturing processes and patterns as well as the internal damage of the deforming rock, which ultimately assist in the recognition of true-triaxial stress effects in natural seismogenic zones where, ( ). Vertical fracturing in reservoirs, and in the caprock overlying the reservoir, significantly affects the flow characteristics and permeability of the reservoir. Therefore, an important question is the extent and orientation of fracturing in a shale caprock or sandstone reservoir. The answer to this question has a strong impact on reservoir characterization and delineation 1, which in turn is vitally important for any horizontal drilling program. With the continuing increase in horizontal drilling to drain thin reservoirs efficiently, it becomes more important than ever to be able to quantify anisotropy in-situ. Density and orientation of fracture sets are determined by investigations in seismic anisotropy. As Schoenberg et al. [1997] stated, With the advent of new drilling technology and the capability to drill wells following more complicated 3D trajectories comes the potential to carry out ever more sophisticated seismic experiments. The laboratory scale analyses are very useful in gaining an understanding of the fundamental processes of the mechanics of geomaterials and how deformation alters elastic properties, or conversely, how elastic properties might allow characterisation of deformation effects. Such analyses might also help us understand observations of spatial and temporal changes in elastic properties of the subsurface detected in seismic data. Ultrasonic measurements for the study of changes in elastic properties with loading have a relatively long history in laboratory soil and rock mechanics. Time-lapse or fourdimensional (4D) seismic monitoring refers to repeated imaging of producing subsurface reservoirs using non-destructive and remote methods (Hall, 2009). That is also where geophysics and geomechanics meet in laboratory experiments for investigating the elastic properties due to deformation of rock associated with loading. The conventional compression test (CCT), in which a solid cylindrical specimen is loaded axially, is one of the most common experimental practices in rock mechanics. However, of the three principal stresses, two are taken to be equal to one another in 1 Delineation drilling is carried out to gain a better understanding of the structure and extent of a deposit in order to decide whether to mine it or not. It is used in tandem with exploration drilling. 3

26 most cases, which mean the stress paths are only confined to a certain plane in the space ( ). On the other hand, a true-triaxial test (TTT) for the deformation of rock, also known as a polyaxial test, is an experimental setup that aims to simulate the in-situ stress state of rocks in the laboratory. In the polyaxial experiments, loads are applied on three opposite pairs of faces of cubic samples ( ). Therefore, the most effective experimental method for revealing the deformation and failure mechanism of rocks is the TTT which can simulate high stress, high deviatoric stress and the complex stress path at a great depth. However, due to the existence of several loading boundary effects, especially friction between the surfaces of platens and rock, stress is not distributed uniformly in the sample. This results in a rather complicated velocity structure within the specimen that combines heterogeneity and rock anisotropy that needs to be deconvolved in order to determine the true anisotropic properties of the rock. This study therefore focuses on using ultrasonic surveys to develop a method to account for these effects and then use the anisotropic velocity structure to improve acoustic emission (AE) analysis for an enhanced interpretation of induced fracturing. In 2005, technical specifications were prepared by the Rock Fracture Dynamics Facility (RFDF), Lassonde Institute, University of Toronto, Canada, for ErgoTech Ltd., to construct a polyaxial or true-triaxial geophysical imaging cell (TTGIC) that would enable simultaneous application of the three principal stresses, acoustic monitoring, directional permeability and electrical resistivity measurements, all under elevated pore pressures and temperatures (Lombos et al., 2012). Beside the TTGIC, the RFDF laboratory contains a purpose-designed servo-controlled loading frame from MTS for polyaxial loading. This was the first of its type that MTS had built. The TTGIC and the custom-made MTS along with specifically designed AE streaming equipment formed a unique facility for true-triaxial testing. The outstanding advantage of this system is the capability of real-time monitoring of the rock versus the usual, less expensive, postmortem analysis of test specimens such as CT scanning of the rock sample, in which a lot of information about deformation of the rock and evolution of fractures are already missed. A number of tests were carried out on Fontainebleau sandstone cubic specimens in this polyaxial imaging cell to elevate our understanding about the role of the 4

27 intermediate principal stress on deformation, fracturing, fluid permeability and acoustic activity of the rock under different sets of stress states while saturated with water under drained conditions during the course of the experiments. The measurements and results of these experiments are reported in Young et al. [2012] and Nasseri et al. [2014]. We used the data from one of the experiments called FTB4 on a cubic sample of Fontainebleau sandstone in this study to evaluate our velocity imaging methodology. 1.2 Objectives of Research The primary objective of this study was to investigate the elastic properties and anisotropic velocity models of rock, specifically ellipsoidal anisotropy, under different sets of laboratory true-triaxial stress states and examine their applicability in velocity imaging and AE monitoring of the rock. This objective was fulfilled through analyzing the ultrasonic velocity survey data using the theory of elastic wave propagation in anisotropic media along with tomography and AE location techniques. The effect of an applied non-hydrostatic stress is to close cracks in some directions and leave cracks open in other directions. This simple explanation by Nur and Simmons [1969], which was observably inherited from many former researchers, is the basis for many experimental studies on the effect of stress on anisotropy in elastic properties of rocks. As Crampin [1987] explained, Distribution of stress-aligned fluid-filled microcracks creates extensive-dilatancy anisotropy which alters the elastic wave velocity in different directions. Dilatancy would also be associated with acoustic emission, increase of volumetric strain and permeability, and decrease of ultrasonic wave velocities. However, in the true-triaxial experiment, the rock sample is clamped by three mutual platen surfaces and dilations do not occur as easily as in the uniaxial and the conventional triaxial tests with cylindrical specimens. Haimson et al. [2000] observed in the polyaxial experiment, that a widening of the quasi-linear elastic range occurs as the intermediate principal stress is increased for the same least stress. This was evidenced by a rise in the onset of dilatancy relative to the peak major principal stress. One of the most important requirements for a true-triaxial testing machine is that external loads can be applied uniformly to the sides of the specimen. However, due to the existence of several loading boundary effects such as loading eccentricity, stress concentrations due to the existence of edges and corners in the sample, loading a blank corner, and the non- 5

28 uniform deformation of platen and end friction, it cannot be expected that the actual distribution of stresses in the specimen loaded will be uniform (Shi et al., 2012). This will cause some compaction and dilatancy boundary surfaces within the specimen. Compaction boundary surface as a barrier property of the rock happens by non-uniform distribution of closed cracks under compressional stress during the quasi-linear stressstrain stage of the experiment. Dilatancy boundary surface happens by exceeding a certain differential stress which causes the transition from non-dilatant to dilatant behaviour. Before the dilatancy happens, the scenario is all about compaction of the rock by closing of the pre-existing micro-cracks. Therefore, the question is how much the rock is compacted under a non-uniform distribution of stress due to edge effects, friction, and other laboratory constraints in the polyaxial experiments. This comes down to the problem of detecting different domains of the rock sample with different elastic wave velocities separated by compaction pseudo-boundary interfaces. Using these explanations, we aim at studying the anisotropic models and velocity patterns by imaging a Fontainebleau sandstone cubic specimen under true-triaxial stress state. In this work, we will estimate the compaction pseudo-boundary surfaces caused by imaging the stress induced non-uniform distribution of voids and micro-cracks through investigating the velocity anisotropy using ultrasonic P and S wave velocity surveys. In a study by King et al. [2012], transversely isotropic velocity model was considered and used in the laboratory AE locations in sandstone under true-triaxial stress state. However, it was proven in theory that isotropic elastic media, when triaxially stressed, constitute a special subset of orthorhombic media, called "ellipsoidal media" (Rasolofosoan, 1998). In ellipsoidal media, the P wave slowness vectors in different directions as well as the P wavefront directional velocities form ellipsoidal distributions in space. Once the elastic parameters are obtained, the next step would be the implementation of the deduced elastic moduli of fractured rock with ellipsoidal anisotropy in ray-based anisotropic velocity tomography algorithm to enhance the capabilities for detection of the heterogeneous zones in the passive anisotropic tomography methods and to improve the AE event locations. In our study, a Fontainebleau Sandstone sample was tested in Prof. R Paul Young s RFDF Laboratory with a Geophysical Imaging Cell manufactured in ErgoTech Company mainly by Laszlo Lombos (Lombos et al., 2012; Young et al., 2012; Nasseri et 6

29 al., 2014). At different stages of the experiment the True-Triaxial Geophysical Imaging Cell (TTGIC), armed with an ultrasonic and AE monitoring system, performed several velocity surveys to image velocity structure of the sample. Going beyond a hydrostatic stress state (poro-elastic phase), the rock sample went through a non-dilatational elastic phase, a dilatational but non-damaging elasto-plastic phase containing initial AE activity and finally a dilatational and damaging elasto-plastic phase up to the failure point. The experiment was divided into these phases based on the information obtained from strain, velocity and AE streaming data. In order to estimate the anisotropic parameters in the form of elastic coefficients and study their variation as a function of time, we utilized conventional measurements of transmission velocities of P wave and two perpendicularly polarized S waves along each of the three principal axes of the cubic sample, as well as transducer-to-transducer P wave velocity surveys. Transducer-to-transducer velocity surveys employed by multiple transducer shots, which were surrounding the sample, enabled imaging of the velocity structure by means of an active ray-based tomography. P wave travel-time tomography as a non-destructive method outfitted a map of wave velocity in the sample in order to detect the localised heterogeneities and provide the spatial variation and temporal evolution of induced damages in rocks at various stages of loading. In the next step, compaction pseudo-boundary surfaces and the velocity contrasts were estimated along the minimum and intermediate principal stress axes by simulating the stress distribution in the cubic rock with FLAC3D (Itasca, 2012) and analyzing the nine orthogonal P and S wave ultrasonic surveys (P-S1-S2 velocity surveys). The location of AE events was also improved by incorporating the heterogeneous anisotropic velocity structure for the sake of more accurate detection of fractures and their evolution. This operation was performed under different velocity structures obtained from different anisotropic models considered as the background velocity model. 1.3 Thesis Outline The second chapter of the thesis reviews the literature on crack-induced anisotropy in rocks and the effects of stress and water saturation on elastic properties of the rock. Also, general descriptions about elastic parameters and anisotropic symmetries are given along with an overview on the corresponding laboratory studies. Physics and mechanics 7

30 of rocks in polyaxial experiments are explained in detail followed by a controversial discussion about elastic wave propagation in ellipsoidal media. Furthermore, a fair amount of literature on anisotropic velocity tomography and the corresponding laboratory experiments are explained, and lastly, the advantages of AE event locations are discussed. Chapter three describes the theory of elastic anisotropy. By focusing on Christoffel s equations for wave propagation in anisotropic media, the relationship between P and S wave velocities with stiffness tensor is represented. Shear wave splitting is introduced and phase and group velocities are discussed. Elastic tensors for orthorhombic and ellipsoidal anisotropies are demonstrated along with their theoretical aspects through Rasolofosaon s and Daley s theories. Finally, Thomsen s Parameters for orthorhombic media are depicted. Time-lapse ultrasonic velocity imaging of the rock using the transducer-totransducer P wave velocity survey data as well as the P-S1-S2 velocity survey data is illustrated in chapter four along with an anisotropic symmetry analysis of the rock by calculating the stiffness tensor parameters. First, the polyaxial experiment procedure on Fontainebleau sandstone cubic specimen is described and information about the Geophysical Imaging Cell and laboratory setup are provided. The experiment is divided into four different stages based on the shift in behavior of physical parameters representing the micro-crack properties of the rock demonstrated in the stress-strain curve and the P-S1-S2 velocity survey data. Also, stereonet representations of directional velocity evolution are obtained by mapping the P wave apparent velocities on stereograms. The apparent velocities in the main three orthogonal cubic directions are used as raw data for building a mean spatial distribution model of anisotropy ratios. Then, time-lapse images of anisotropic velocity model of the rock is provided by inverting the apparent P wave velocities. The initial velocity images of the rock are obtained mainly for stage D of the experiment by use of an anisotropic tomography method further described in Appendix A. Then, physical parameters are acquired through a combination of a stress distribution model with different measurements including P, S1, and S2 wave velocities along the three principal axes, and transducerto-transducer velocity surveys. Numerical simulations of time-lapse stress distribution in the cubic sample are accomplished mainly for stages B and C of the experiment by 8

31 using FLAC3D software. Compaction pseudo-boundary surfaces are estimated by stress distribution analysis along the minimum and intermediate principal axes to calculate the time-lapse evolution of shear and P wave velocities in the central compacted part and the outer lower-stressed domains of the rock. Different velocity domains of the cubic rock are displayed in a schematic view with computed P and S velocities. Thereafter, diagonal and off-diagonal stiffness tensor parameters are derived and compared based on different theories for ellipsoidal and orthorhombic medium. Finally, Thomsen s parameters are calculated to give a sense of the anisotropy strength in the rock. In chapter five, theories and algorithms for AE event locations in anisotropic heterogeneous media are developed. Ellipsoidal anisotropic symmetry for the P wave wavefront surface is used to calculate the travel-times and apparent velocities of the rays. Location results for transducer shots with different isotropic, anisotropic and heterogeneous mediums are obtained and compared with each other. The results show an improvement in the accuracy of the transducer shot locations by using an anisotropic heterogeneous model for the velocity structure of the sample. Chapter six presents conclusions and contributions of the research. Suggestions for further study and other possible applications of this modelling technique are recommended. Appendix A is about the development of an anisotropic tomography method based on ellipsoidal P wave velocity surface. The rock sample is partitioned into cubic grid cells as model space. A ray-based tomography method measuring body wave traveltimes along ray-paths between pairs of emitting and receiving transducers is used to calculate isotropic ray-path segment matrix elements ( ) which contain segment lengths of the ith ray in the jth cell in three dimensions. Synthetic P wave travel-times are computed as data space between pairs of transducers in a hypothetical isotropic heterogeneous cubic rock sample. A 3D strain of the squeezed rock and the consequent geometrical deformation is also included in computations for further accuracy. The Singular Value Decomposition (SVD) method is used for the inversion from data space to model space. A Synthetic checker-board model is used to examine the workability of our isotropic tomography method. In the next step, the anisotropic ray-path segment matrix and the corresponding data space are computed for hypothetical anisotropic heterogeneous samples based on the ellipsoidal anisotropic model of velocity. The 9

32 method is examined for several different synthetic heterogeneous models. An Inaccuracy factor is utilized to inquire the accuracy of inversion results for anisotropic tomography to obtain an optimal number of singular values for inversion as well as a minimum heterogeneity percentage criterion that can be recovered. 10

33 2 Literature Review 2.1 Crack-Induced Anisotropy in Rocks Fractures and cracks generally increase rock compliance (reduce stiffness), and thus reduce seismic wave speeds. If the distribution or orientation of fractures is not uniform and isotropic, then significant heterogeneity or anisotropy can be observed in seismic velocity. Tectonic stress fields or temperature gradients induce cracks with preferred orientations in the rocks in different microscopic or macroscopic scales. Therefore, rock masses can be expected to be anisotropic even if the matrix is isotropic. Due to the importance of crack- and fracture-induced anisotropy for seismic wave propagation, several theoretical studies have been reported in the literature on this topic (Nur, 1971; Anderson et al., 1974; Crampin, 1987; Schoenberg et al., 1995; Berryman et al., 1999; Sarkar et al., 2003; Crampin et al., 2005; Berryman, 2007; King, 2009 and many more!). As Winterstein [1990] has put it, Whether a change of properties with direction distinguishes a solid as anisotropic or inhomogeneous is a function of scale. The relevant scale for geophysicists varies but is related to the wavelengths of the seismic waves propagating through them. For seismic waves, the scale may be several tens of meters for earthquakes; decimeters for well logs; and, a few millimeters for laboratory measurements. Hence, for clarity and effect, anisotropy as a concept of geophysics should not be applied to every heterogeneous material, but only to those that may be treated as homogeneous on the scale of the wavelengths used to probe them. The main question is how the presence of the fracture systems can affect the elastic moduli of the fractured rock. 2.2 Effect of Stress on Crack-Induced Anisotropy Nonhydrostatic stress is a main source of seismic anisotropy forming the shape and density of fractures and cracks. The cracks tend to be aligned perpendicular to the direction of minimum compressional stress, just like hydraulic fractures induced by the injection of high-pressure fluid in the oil and gas or geothermal industry. Thus, the rock mass in the subsurface behaves as if it contains systems of parallel fractures decreasing the stiffness in some directions. As mentioned above, these stiffness reductions 11

34 contribute to the velocity anisotropy of seismic waves with wavelengths much larger than the spacing of the fractures. Amos Nur was one of the first scientists who studied the effects of stress on velocity anisotropy in rocks with cracks. He mentioned in his 1971 paper that anisotropic crack distribution causes elastic anisotropy with associated acoustic birefringence 1. Shear wave splitting (birefringence) is the most reliable indicative of seismic anisotropy for geophysicists. A schematic demonstration of shear wave splitting is given in figure (2.1). Figure 2.1: Schematic illustration of seismic birefringence or shear wave splitting in distributions of stressaligned fluid-saturated parallel vertical micro-cracks aligned normal to the direction of minimum horizontal stress,. For nearly vertical propagation the polarisation of the faster split shear wave is parallel to the strike of the cracks, parallel to the direction of maximum horizontal stress, (Crampin et al., 2005). Crampin [1987] introduced the concept of extensive-dilatancy anisotropy (EDA). Dilatancy is the increase in the volumetric strain which is caused by the increase in volume as micro-cracks open before failure in rock specimens subjected to laboratory 1 Seismic birefringence also called double refraction or shear wave splitting is the phenomenon that occurs when a polarized shear wave enters an anisotropic medium and splits into two parallel shear wave rays polarized perpendicularly. 12

35 stresses greater than half the eventual fracture strength of the intact sample. Extensivedilatancy anisotropy is the hypothesized distribution of stress-aligned fluid-filled microcracks pervading most rocks in the Earth's crust. Crampin [1987] suggested that, the geometry of these cracks and the aligning stress-field can be monitored by analysing the waveforms of shear waves propagating through the rock mass. 2.3 Effect of Fluid Saturation on Elastic Properties of the Rock Every sedimentary rock retains water in its interstices after deposition, and the long term regional stress-fields will tend to align the pore space normal to the minimum compressional stress. Therefore, distributions of stress-aligned fluid-filled micro-cracks must be expected in the crust. In isotropic medium, porosity does not directly affect elastic properties of the grains, but since it affects the velocities, it affects the ratio (Gassmann, 1951). Gassmann s well-known results show that when isotropic porous elastic media are saturated with any fluid, the fluid has no mechanical effect on the shear modulus µ, but can have a significant effect on the bulk modulus, and therefore on λ. Two main predictions are made by the Biot Gassmann theory: first, the S wave velocity in liquid-saturated porous rocks will always be less than in dry rocks (the shear modulus is assumed in the theory to remain the same on saturation but increases the density). Second, the P wave velocity in saturated rocks will generally be higher than in dry rocks, except for rocks having very high bulk moduli. However, ultrasonic S wave velocities for liquid saturated rocks could sometimes be higher than the dry rock S wave velocities in some cases. The reason for the anomalous S wave velocity behaviour noted above is due to "squirt" flow. One of the mechanisms that affect seismic velocities in saturated rocks is called local flow or squirt which is related to grain-scale microscopic flow field. When a low frequency wave passes through a fully liquidsaturated porous rock, there is sufficient time for any pore pressure gradients generated to dissipate. When, however, a high-frequency wave passes through the same rock, pressures built up in cracks and pores of small aspect ratio have insufficient time to dissipate fully or "squirt" into well-rounded pores. The porous rock in its fully saturated state, therefore, has a higher shear modulus than in its dry state, thus invalidating one of the assumptions required for Biot Gassmann theory (Mukerji et al., 1994; King, 13

36 2009). Therefore, the degree of compressional wave anisotropy may either increase or decrease with saturation depending on the crack distribution, the effective pressure, and the frequency at which the measurements are made. Consequently, velocity anisotropy for measurements at low frequencies, typical of in-situ observations, will generally be different from those at high frequencies, typical of the laboratory. Thin liquid-filled cracks have very little effect on P wave propagation and the isotropic P wave velocity models. Hence, shear wave splitting is likely to be the most reliable indicator of anisotropy in the underground rocks. Crampin [1987] stated that, Shear wavetrains contain three or four times more information about the structure along the ray-path and the source than the equivalent P wavetrain. Therefore, Crampin suggested, since modifying EDA cracks is the most direct effect of changes of stress before earthquakes, and since analyzing EDA cracks is the only way that the essential anisotropic nature of all stress-induced effects can be explored, monitoring shear wave splitting with shear wave VSP 1 s could be a very promising technique for earthquake prediction research. Another important aspect of laboratory rock mechanics tests is whether the rock is under drained or undrained conditions. When permeability is finite, a rock can be drained or undrained - where these expressions are statements about the boundary conditions of some portion of the rock mass as a medium in nature or a rock core in a laboratory. Undrained conditions imply the system is actually wrapped in some impenetrable jacketing material, or that a disturbance is applied so rapidly that the fluid pressure inside a certain small region can approximately equilibrate locally, but has insufficient time to equilibrate on some larger length scale. Drained means that the rock sample has no obstructions at its boundaries and plenty of time to equilibrate with the pressure of an outside reservoir of fluid. Drained conditions also usually imply that the presence of the fluid has no mechanical effect on the behaviour of the rock and so the actual fluid can be replaced by air or vacuum in a laboratory experiment, since drained conditions also imply long time scales. The result of drained conditions in nature is that after an increase in confining stress on the solid frame of the rock (i.e., to 1 Vertical Seismic Profile (VSP) is a measurement made in vertical wellbore using geophones inside the wellbore and a source at the surface near the well. It used for correlation with surface seismic data, for obtaining images of higher resolution than surface seismic images and for looking ahead of the drill bit. 14

37 simulate actual overburden stresses corresponding to in-situ conditions), the pore fluid pressure is equalized throughout the sample to a constant value, which may be either to atmospheric pressure or to a set value to approximate the pore pressure state of a rock at some depth in the earth. If a porous system has connected pores and, therefore, finite permeability, then, after a mechanical disturbance of the system (compression or extension, for example, or during the propagation of a seismic wave), it is possible for the entire pore-fluid mass inside the rock to relax to a state in which the pore pressure of the fluid is the same everywhere (Hornby, 1998; Berryman, 2007). 2.4 Laboratory Studies on Velocity Anisotropy in Rocks Laboratory experiments, in general, have many benefits compared to the in-situ experiments. First of all, many factors such as stress state, pore pressure, temperature, fluid flow, etc. can be controlled and repeated in a relatively short time. The repeatability factor strengthens and improves the data quality and helps in validating the results. Mukerji & Mavko [1994] raised a general criticism about laboratory experiments on velocity measurements for crack-induced anisotropic rocks. They shared a couple of features that can limit the usefulness of laboratory velocity measurements, particularly when comparing laboratory and field data. They first mentioned that laboratory velocity measurements are based on idealized (and unrealistic) crack geometries such as ellipsoidal cracks. Therefore, they argued that they are intrinsically limited to low crack densities. Also, they complained that the anisotropic mineral matrix is often ignored, and their treatment of frequency-dependent saturation effects is incomplete. However, at the laboratory scale, ultrasonic wave measurements have been used for a relatively long time as a non-destructive method in laboratory soil and rock mechanics to study the changes in elastic properties due to deformation associated with loading. Additionally, recordings of AEs by transducers, which constitute an extension of earthquake monitoring carried out routinely at the real-world scale, has since (around the 1960s) become a popular means to track the evolution of damage in deforming rocks (Hall, 2009). The most basic AE sensor incorporates a simple disc, or other simple shape, composed of a piezoelectric ceramic, which gets directly attached to the rock specimen in the laboratory to detect vibrations. The piezoelectric element generates a voltage in 15

38 response to stresses caused by the acoustic energy impinging upon it and vice versa. Therefore, piezoelectric sensors can both detect and emit acoustic emission waves. Ultrasonic wave-speed measurements via emitting and receiving piezoelectric elastic pulse excitations constitute the most widely used method for the determination of the elastic constants of anisotropic solids (Every & Sachse, 1992). Also, there has been extensive work on the characterisation of crack-induced anisotropy and its loaddependence through experimental observations. Research on intrinsic anisotropy of rocks was initiated by simple studies. For example, Thill et al. [1973] measured compressional wave velocity omni-directionally in a rock sphere by the ultrasonic pulse method. They plotted and contoured the results on a Schmidt equal area net (stereonet) to disclose the amount and symmetry of velocity variation in the Barre granite in both dry and water saturated conditions. They observed strong orthorhombic symmetry patterns in the velocity under all moisture conditions. As for stress- or crack-induced anisotropy in rocks, a Model by Nur and Simmons [1969] proposed that an isotropic rock matrix has a uniform distribution of randomly oriented cracks that, in the absence of stress, keeps the rock isotropic. Axial stresses close cracks preferentially perpendicular to stress directions, making the rock anisotropic (Winterstein, 1990). Richard O Connell and Bernard Budiansky [1974] studied the velocity measurements from partially saturated rocks in the laboratory and compared it to the velocity data before the San Fernando earthquake. By modeling their experiments with crack density theories, they observed that seismic wave velocities always decrease with increasing crack density. They also realized that the velocity ratio decreases for dry cracks and increases for saturated cracks and this can be used as a precursory indicator for seismic events at many scales such as earthquakes. To obtain elastic constants, velocities and anisotropies in different rocks such as shale were traditionally measured on multiple adjacent core plugs with different orientations. For example, for a transversely isotropic rock, three plugs must be measured separately (one parallel, one perpendicular, and one 45 to the symmetry axis) to derive the five independent elastic constants (Vernik et al., 1992). Instead of the common three-plug method, Wang [2002] used a single-plug method for measuring 16

39 seismic velocities and anisotropic parameters of transversely isotropic shale rocks in laboratory by fitting the piezoelectric transducers to the top, bottom, and sides of the cylindrical sample. Therefore, he reduced the sample preparation and velocity measurement time by more than two-thirds. He used horizontal core plugs which had much higher permeability than vertical core plugs in his method. Using another methodology, Cheadle et al. [1991] did not core any plugs out of the specimen. Instead, they measured shear- and compressional-wave velocities in the three orthogonal principal axes related to the fast, medium, and slow directions through a cube of an industrial laminated Phenolic. They also measured those wave velocities in directions between opposing edges of the cubic sample to support the determination of the orientations of the planes of symmetry. Their purpose was to derive the expressions for stiffness matrix components for the case of orthorhombic symmetry as a function of the measured velocities. Hornby [1998] discussed methods for the measurement of ultrasonic P and S wave phase velocities on fluid saturated shale samples under drained conditions as a function of confining pressure. In order to reconstruct the elastic parameters using P and S wave velocity measurements, he argued that an effort must be made to use as much data as possible on core samples. In particular, he suggested that the critical elastic parameter (or Thomsen's δ) needs at least two off-axis measurements to be reconstructed. In a most recent work, Mahmoudian et al. [2014] obtained the group velocities in various directions through a phenolic medium with orthorhombic anisotropy to estimate its orthorhombic stiffness coefficients. Phenolic materials, because of their micro-layered texture, can be used to simulate finely layered rocks, such as some sandstones and shale, or fractured limestone. They estimated the off-diagonal stiffness coefficients using an approximate explicit expression for the P wave group velocity suggested by Daley et al. [2006] in terms of the nine orthorhombic parameters. 2.5 Physics and Mechanics of Rocks in Polyaxial Experiments One of the major roles of rock mechanics testing in a laboratory is to characterize deformation and strength behaviors under in-situ stress states. The Kármán-type triaxial test proposed by Kármán [1911] has been widely used in experimental rock mechanics because of the simplicity of the equipment and the convenient preparation of 17

40 specimens. In a Kármán-type triaxial test, a cylindrical specimen is loaded axially under confinement and the intermediate and minimum principal stresses making the confining pressure are equal ( ). It is also called Conventional Triaxial Test (CTT) as being easily and frequently used with no intermediate principal stress existing to contribute to the deformation and failure of the rock. However, due to the simplicity of this test, in rock mechanics references the characteristics of rock strengths, deformation and failure have mainly been obtained from CTT. True-triaxial tests which can simulate the in-situ stress state ( ) were mainly developed in order to study the effects of the intermediate principal stress on deformation of the rock. As Mogi [1977] stated, the most important result of rock deformation under general triaxial stress states is the high anisotropic dilatancy under high values of the intermediate principal stress. This anisotropic dilatancy is explained by the opening of cracks perpendicular to the minimum principal stress and should cause the anisotropy of the anomalous change in the seismic velocity. Mogi [1977] noted that this anisotropic dilatancy which is observed in the laboratory may be expected before earthquakes and can therefore be assumed as a precursor. There have been approximately one hundred TTT machines reported in the studies on soil, concrete and rock (Shi et al., 2012). TTT machines are divided into three types based on the loading methods and the boundary conditions. In type-i, a rigid load is applied in all three directions. In type-ii, flexible loading in all three directions is applied; and in type-iii, a mixed loading is utilized which is also called Mogi type (Mogi, 1967). Due to the limitations of the loading capacity, the type-ii TTT machine is rarely used in the field of rock mechanics. Our polyaxial experiment was a type-iii test. In the early 1990s at Imperial College London, a true-triaxial (polyaxial) stress loading system with the capability of varying each of the principal stresses independently was developed (Figure 2.2) for determining the ultrasonic velocities and attenuation, fluid permeability and elastic properties of cubic rock specimens of 51 mmside as they were loaded to failure (King et al., 1995; King, 2002). The TTGIC in Prof. Young s RFDF laboratory which is enhanced by AE streaming system is an enhanced version of the true-triaxial stress loading system at Imperial College London, having more measurement capabilities in investigation of the inter-relationships between 18

41 mechanical, thermal and hydraulic stress, crack mechanisms, anisotropy, fluid flow and rock integrity. King et al. [1997] studied cubic samples of sandstone in their polyaxial stressloading system. They measured and studied all nine components of P and polarized S wave velocities and attenuation in the principal directions, and fluid permeability in the major principal stress direction. Later in 1998, Shakeel & King used Nishizawa's [1982] theory to calculate the elastic constants and Thomsen s anisotropic parameters from the P and S wave velocity measurements of the same test. They found that in dry sandstones the anisotropy parameter delta is the most sensitive to the crack density. Later in 2002, Colin Sayers used the same data and studied the effect of the orientation distribution and normal and shear compliances of the discontinuities in the rock on its elastic wave velocity anisotropy. He inverted elastic wave velocity measurements using discontinuity formulas written in terms of a second-rank and a fourth-rank tensor. His results were used to determine the ratio of the normal to shear compliance of the discontinuities. Figure 2.2: Diagrammatic view of Imperial College s true-triaxial loading frame (King, 1997). Sayers et al. [1990] had already studied P and S wave velocities in three orthogonal directions in a 50 mm cubic sample of Berea sandstone in a true-triaxial 19

42 experiment. By inverting the velocity data to the micro-crack density and orientation distribution, they confirmed that failure is preceded by the growth of micro-cracks parallel to the direction of the maximum principal compressive stress. Consequently, a relationship between P and S wave velocities was achieved by Sayers & Van Munster [1991] using the same dry experiment data. More recently, Popp & Salzer [2007] used a multi-anvil apparatus to study the stress dependent onset of dilatant deformation of cubic rock specimens under truetriaxial stress state. The aim of their spatial velocity measurements in a multi-anvil apparatus was not only to quantify the seismic anisotropy of the Opalinus clay but also to identify the onset of dilatancy and to monitor its evolution at various states of stresses. By investigating the anisotropy and differences in the sensitivity of and they found that the crack-sensitive P and S wave velocities are powerful tools for the determination of the so-called dilatancy boundary. Dilatancy boundary criterion is a barrier property of a solid rock (Cristescu et al., 1998; Hunsche et al., 2003). It happens by exceeding a certain differential stress which causes the transition from non-dilatant to dilatant behaviour. Popp & Salzer [2007] argued that progressive onset of dilatancy would be associated with AE, increase of volumetric strain and permeability, and decrease of ultrasonic wave velocities. They showed that the onset of dilatancy in Opalinus Clay, as indicated by the reversal of V S with respect to V P, begins at significantly lower stress levels than the failure boundary. In polyaxial experiments, friction has a major effect on stress distribution in the rock sample. Mogi [1971] mentioned that in the true-triaxial loading, if stresses are applied through steel end platens, friction between platens and specimens and stress concentration at the ends of the specimens could introduce marked errors. Although, a few researchers had tried to eliminate the friction effects by using rubber sheets with grease. However, the effect of lubrication caused some errors, because intrusion of the lubricant into the ends of the specimen is likely to set up a radial tensile stress. More explanations are given with this regard in the discussion part of chapter 4. Also, Shi et al. [2012] concluded that due to the existence of several loading boundary effects such as loading eccentricity, loading a blank corner, the non-uniform deformation of platen and end friction, it cannot be expected that the actual distribution of stresses in the specimen loaded will be uniform. Instead, a numerical analysis needs to be done in order 20

43 to simulate the non-uniform stress distribution in the rock. Shi et al. [2012] used FLAC3D to simulate the contact friction between specimen and platen in the loading process of Mogi-type true-triaxial test to study the loading boundary effects. They found that the corner effect causes a negative effect on the uniform distribution of intermediate principal stress. 2.6 Elastic Wave Propagation in Ellipsoidal Media In anisotropic media, energy will not necessarily propagate with the phase velocity in the direction of the wave normal as it does for isotropic media. In the scale of rock experiments, one measures group velocity. However, there are certain experimental configurations which allow direct calculation of the magnitude of the phase velocity from time delay measurements, eliminating the need for any type of modification for group velocity measurements. Some information must be known about the orientation of the sample relative to its material crystallographic symmetry in order to make use of the measurement of the magnitude of the phase velocity (Sahay et al., 1992). In our case with the configuration of polyaxial experiments, where opening and closure of parallel micro-cracks due to true-triaxial stress state are responsible for anisotropy, the phase velocities are being measured in the three main principal directions. It is also important to bear in mind that, if the anisotropy is weak, the azimuthal dependences of group and phase velocities are the same (Backus, 1965). Aside from the fact that elastic wave velocity and slowness surfaces vary in materials with different anisotropic symmetries, the shape of the velocity surface is also very controversial in the geophysics literature. Helbig et al. [2005] believed that ellipsoidal geometry is not a good approximation for the elastic wavefront propagating in materials with moderate intrinsic anisotropy. In 1911, Rudzki had already found that in transversely isotropic media, the P wavefront is elliptical (and SV wavefront spherical) under specific conditions. However, Thomson [1986] maintained that elliptical anisotropy is an inadequate approximation most of the time. Schoenberg and Helbig [1997] who are among the critics of ellipsoidal anisotropy stated that for transversely isotropic media, only the SH wavefront is ellipsoidal; the SV wavefront is never an ellipsoid, and the P wavefront is an oblate ellipsoid under certain circumstances (Mahmoudian et al., 2010). 21

44 All the above assumptions about deficiency of ellipsoidal geometry of the wavefront surface in orthorhombic materials are reasonable. However, for the specific case of laboratory polyaxial experiment setup where the rock sample is under 3D stress state, Rasolofosaon [1998] theoretically analyzed the problem and proved that isotropic elastic media, when triaxially stressed, constitute a special sub-set of orthorhombic media, called "ellipsoidal media". Ellipsoidal anisotropy is the natural generalization of elliptical anisotropy. Ellipsoidal anisotropy is to orthorhombic symmetry what elliptical anisotropy is to transversely isotropic (TI) symmetry. Elliptical anisotropy is a special case of ellipsoidal anisotropy restricted to TI media. In other words, ellipsoidal anisotropy degenerates in elliptical anisotropy in TI media. In ellipsoidal media the P wave slowness surface is always an ellipsoid (Rasolofosaon, 1998). Only six independent elastic constants (instead of nine in conventional orthorhombic media) define an ellipsoidal medium. These constants can be chosen as the diagonal elements of the elasticity matrix. In orthorhombic media, and in particular in ellipsoidal media, these elastic constants correspond to the moduli P and S waves propagating and polarized along the coordinate axes. Furthermore, Ettrich et al. [2001] defined ellipsoidal media as a media with ellipsoidal shape of the slowness and group velocity surface. They derived linear relationships for the coefficients of the ellipsoid based on the elastic coefficients of the anisotropic medium and suggested that they are widely appropriate for analytical calculations of travel-times. The experimental results obtained by Sarkar et al. [2003] on Berea Sandstone confirmed the theory given by Rasolofosaon, who first proved that isotropic materials subjected to an arbitrary triaxial stress field become, effectively, an orthorhombic medium with elliptical anisotropy in each symmetry plane. The symmetry planes observed in this orthorhombic medium are aligned with the orientations of the principal stresses, and anisotropic parameters can reveal information about the stress magnitudes. Thus, Sarkar et al. [2003] suggested that time-lapse monitoring of changes in anisotropy can potentially provide information on temporal variations in the stress field too. 22

45 2.7 Anisotropic Tomography and Laboratory Studies Seismic tomography is an inversion technique that exploits information contained in seismic records to constrain 1D, 2D or 3D models of velocity, attenuation or other rock properties of the Earth s interior. It generally requires the solution of a large inverse problem to obtain a heterogeneous seismic model that is consistent with observations. Seismic tomography has been a vital tool in probing the Earth's internal structure and enhancing our knowledge of dynamical processes in the Earth's crust and mantle. Also, seismic tomography and microseismic monitoring are techniques that have been used to characterize rock masses in the oil, gas and geothermal fields. On the other hand, ultrasonic tomography provides the possibility to assess the spatial, and eventually temporal, variations in ultrasonic velocity (and attenuation) within geomechanics test specimens in the laboratory which can help in rock mass characterization. Existence of aligned cracks and non-hydrostatic stress conditions cause seismic anisotropy. The tomography method assuming anisotropy of media is called anisotropic tomography. The anisotropic stress field ( ) usually plays the major role in anisotropy in the rock. Many researchers use the transverse isotropy approximation ( ), which represents anisotropy as an ellipsoid having a vertical rotation axis of symmetry. In the nature, this approximation is suitable when the anisotropy is caused by the effect of sedimentation and thin beddings in the near-horizontal layer which is targeted by petroleum explorations. However, in the vicinity of wellbore walls, excavation tunnels or in the field of rock engineering, the assumption of transverse isotropy of media is not always valid. For example, the orientation of the anisotropy axis may not coincide with the analysis axis. In many cases, the effect of anisotropy is heterogeneous in the object area and the heterogeneity caused by anisotropy is very important information for understanding rock mass structure (Watanabe et al., 1996). Watanabe et al. [1996] used a sinusoidal approximation with respect to the angle from the anisotropic axis for anisotropy of slowness in a field-size medium. They represented anisotropy by the minimum slowness, the maximum slowness and its orientation for every cell. Their numerical simulations included a ray-tracing method based on Huygens Principle and both iterative least-squares technique and singular 23

46 value decomposition inversion methods. Their results showed that the anisotropic tomography successfully reconstructed the velocity structure having 1-10% velocity anisotropy in case of noise free data. On the other hand, the ordinary isotropic tomography failed to reconstruct the anisotropic structure. Abraham et al. [1998] examined the rock mass by microseismic logging and seismic tomography around a test gallery that was excavated in shale at a depth of 500 m to study the impact of excavation on the behavior of an anisotropic rock mass. They used two different approaches for introducing weak transverse isotropy in a simultaneous iterative reconstruction technique (SIRT) tomographic inversion algorithm. They argued that the ellipticity hypothesis of anisotropy was not derived from the physical properties of the materials; rather, it was a mathematical simplification. However, both of their proposed anisotropy models gave almost similar inversion results for their experimental data and the elliptical anisotropy method was found to result in a very modest increase in calculation time. Wu and Lees [1999] in a superb work developed a new method for inverting P and S wave travel-times based on a tensorial velocity representation for seismic anisotropy on a near-surface local scale. They found that random noise is less important than ray directional coverage in anisotropic inversions. Another anisotropic tomography work in the field scale belongs to Meglis et al. [2005]. They also tried to characterize the damage induced during excavation of a test tunnel in granite by a 2D ultrasonic wave velocity tomography between 16 one-meter deep boreholes using anisotropic simultaneous iterative reconstruction technique for an elliptical anisotropy approximation. They argued that the obtained velocity anisotropy indicates that the dominant orientation of micro-crack planes is everywhere tangential to the tunnel wall, even though the minimum principal stress is tangential and tensile in the sidewall. In the laboratory scale, one of the major research areas in rock fracture mechanics is to simulate and study induced earthquakes that can be triggered by water flow or change of underground water pressure in nature. Induced earthquakes are also studied to characterize the artificial reservoir in a hot dry rock geothermal area. Seismic and ultrasonic velocity tomography is highly used as monitoring tools for detection of induced seismicity. Masuda et al. [1990] monitored the fluid flow in a cylindrical Inada granite sample under transversely isotropic stress by means of AE hypocenter locations 24

47 and P wave velocity tomography using Lead zirconium titanate (PZT) piezoelectric transducers in the laboratory. Their velocity structure, reconstructed by seismic tomography, showed that after the water injection, P wave velocity first increased rapidly to its peak value and then decreased gradually. They argued that the decrease of P wave velocity is due to under-saturation as the rate of cracking exceeds the rate of fluid flow. Falls et al. [1992] recorded ultrasonic tomography and AE data during laboratory hydraulic fracturing tests through injecting water into the hole that was bored down the central axis of an unconfined cylindrical sample of granite to investigate the its micromechanical response to the stress field generated by hydraulic fracturing. Tao and King [1990] carried out a systematic laboratory study of ultrasonic velocity and attenuation of compressional (P) and orthogonally polarized shear (S1 and S2) waves in anisotropic dry rock samples under constant axial stress. Their main purpose was attenuation measurement studies but they observed that anisotropy in attenuation was generally accompanied by anisotropy in velocity. Other than non-hydrostatic loading, thermally induced fracturing would also generate acoustic emissions and result in acoustic velocity changes in the rock. Debski and Young [2002] accomplished an excellent work on tomographic imaging of thermally induced fractures in granite using Bayesian inversion. Jansen et al. [1993] claimed that a combination of tomographic methods and AE source location would create an ideal tool to study any possible macroscopic fracture patterns induced by thermal gradients. Using an elliptical anisotropy model, a simultaneous iterative reconstruction and a cross-correlation technique, they studied thermally induced micro-fracturing in an unconfined 15 cm side cube of Lac du Bonnet granite. Similar techniques were used by Chow et al. [1995] who employed 3D ray-path coverage of transmitted ultrasonic waves to assess the progressive development of micro-crack damage in a cylinder of Lac du Bonnet grey granite subjected to uniaxial cyclic loading. Their results showed an increase in slowness and a change in orientation of the anisotropy of wave propagation with continued load cycling. Also, AE monitoring indicated that their sample exhibited both the Kaiser effect 1 (AE were detected only after the previous cycle's peak load was 1 The Kaiser effect was discovered by Joseph Kaiser who performed his experiments on small specimens of metals, wood and sandstone in the early 1950s. Kaiser effect is an absence of acoustic emission at loads not exceeding the 25

48 exceeded) and the Felicity effect 1 (during the final stages of the experiment, AE commenced at loads less than the previous peak load). 2.8 AE Event Location Acoustic Emission (AE) has been used fairly extensively as a laboratory tool to investigate the micro-mechanisms of deformation and fracture by which a material responds to stress during mechanical testing. It has also found increasing application as a non-destructive evaluation technique for detecting and locating flaws in structures subjected to stress. AE is the term used for the elastic waves generated by abrupt localized changes in stress in a solid. Opening, closure, or any deformation in cracks and fractures are the most usual cases for such localized AE sources in the rocks. The waves propagate away from the source and cause transient (nanosecond to millisecond time scale) surface displacements that may be measured with transducers (Wadley et al., 1984). There have been an enormous number of studies on the seismic, microseismic and AE event location. In the global scale, the determination of earthquake locations requires a good velocity model for the region of interest. Most earthquake location procedures commonly used today may be viewed in terms of an optimization problem. In all the location algorithms, four event parameters (three components of the hypocenter location in the Cartesian coordinate system and the origin time ) are to be inverted. Nowadays, the industry is moving increasingly towards more unconventional hydrocarbon resources such as tight gas, shale gas, and coal bed methane. Such reservoirs need fractures, either naturally occurring or induced by hydraulic fracture stimulation, in order to be produced economically. Therefore a growing need exists to develop techniques that can robustly identify fractures in-situ. The consequences of neglecting material anisotropy would be the introduction of systematic errors in source location results, hence, a smaller degree of confidence in them. Since event source previous maximum load level when material undergoes repetitive loading patterns. Discontinuities created in material during previous steps do not move or expand until former stress is exceeded resulting in Kaiser effect. 1 Felicity effect is an effect in acoustic emission that reduces Kaiser effect at high loads of material. Under Felicity effect the acoustic emission resumes before the previous maximum load was reached. 26

49 locations are the most informative pieces of information in the identification of geological features at the origin of seismicity, the inclusion of material anisotropy in the source location procedure would be a natural step in the improvement of source location results (King and Talebi, 2007). Among the first AE source locations in rock mechanic studies, Mogi [1972] located the source of AE events during failure of rock subjected to a bending moment, and thus was able to study the growth of tension fractures preceding catastrophic failure. Scholz [1970] studied the AE in granite under uniaxial compression, and by using six transducers attached to the surface of the sample, was able to locate some of the AE events prior to shear failure. Also, Byerlee [1975] developed a system to study AE in rock which was subjected to high confining pressure, differential stress, and pore pressure. Verdon and Kendall [2011] showed how to detect the induced anisotropy in the microseismic data while Wuestefeld et al. [2010] provided an automated workflow to analyze the degree of anisotropy in large microseismic data sets. Within the engineering scales, by solving for effective anisotropy simultaneously with locations of microseismic events, Grechka et al. [2011] found that while velocity heterogeneity does not need to be introduced to explain the data acquired at each stage of an engineering operation such as hydraulic fracturing, the obtained models are suggestive of possible time-lapse changes in the anisotropy parameters that characterize the stimulated reservoir volume. They concluded that accounting for seismic anisotropy improves description of the recorded travel-times and reduces uncertainties in the locations of microseismic events compared to those obtained in purely isotropic models. King et al. [2012] published AE location results of a test on Crosland Hill sandstone under polyaxial loading conditions leading to the formation of sets of aligned micro-cracks. This experiment which was accomplished in 1995 with addition of an enhanced AE streaming system formed the starting point for design of TTGIC in the RFDF laboratory. During the experiment, minor principal stress was maintained at a low value while the two other principal stresses were increasing until failure of the rock. The anisotropic P wave velocity structure that was used in the AE event location algorithm was transversely isotropic velocity model. However, due to ellipsoidal anisotropic nature of P wave velocity in the rock under TTT, considering an ellipsoidal 27

50 velocity model with three independent P wave velocities would be helpful in order to enhance the monitoring accuracy of the AE activity in laboratory experiments. This will also help us revisit the accuracy of the location results by quick and cost effective location methods using transversely isotropic velocity model. The literature review given above discussed around the multi-disciplinary field of true-triaxial tests involving the laboratory rock mechanics technical considerations, the effects of stress distribution and fluid saturation, and the acoustic emission and microseismic monitoring aspects such as tomography and AE event locations. In the next chapter, we will explore more into the theory of wave propagation in anisotropic media and we show the mathematical demonstrations of ellipsoidal symmetry and the corresponding Thomsen parameters which we used in our work. 28

51 3 Theory of Elastic Anisotropy 3.1 Wave Propagation in Anisotropic Media A medium (or a region of a continuum) is called anisotropic with respect to a certain parameter if this parameter changes with the direction of a measurement. If an elastic medium is anisotropic, seismic waves of a given type propagate in different directions with different velocities. This velocity anisotropy implies the existence of a certain structure (order) on the scale of seismic wavelength imposed by various physical phenomena which results from the summation of contributions of grains, discontinuities, intra-granular cracks, fractures and layers, over a range of length scales. In typical subsurface formations, velocity changes with both spatial position and propagation direction which make the medium heterogeneous and anisotropic. The notions of heterogeneity and anisotropy are scale-dependent, and the same medium may behave as heterogeneous for small wavelengths and as anisotropic for large wavelengths. The laboratory scale analyses are very useful in gaining an understanding of the fundamental processes of the mechanics of geomaterials and how deformation alters elastic properties, or conversely, how elastic properties might allow characterisation of deformation effects. Such analyses might also help with understanding observations of spatial and temporal changes in elastic properties of the subsurface detected in seismic data. The main purpose of revisiting anisotropic wave propagation in this work is to present several analytic results in the form most suitable for application in seismic inversion and processing, and to establish a convenient notation that enhances analysis of seismic data and seismic imaging practices. 3.2 Christoffel s Equation One of the main scientists who conducted a great deal of research about anisotropy in Geophysics is Ilya Tsvankin. He formulates Christoffel s governing anisotropy equations very well in his book, titled Seismic signatures and analysis of reflection coefficients in anisotropic media, published in The wave equation for general anisotropic heterogeneous media follows from the second Newton's law applied to a 29

52 volume ΔV within a continuum. Expressing the tractions (contact forces) acting across the surface of ΔV in terms of the stress tensor yields, (3.1) where ρ is the density, is the displacement vector, is the body (external) force per unit volume, t is time and are the Cartesian coordinates. Summation over j = 1, 2, 3 is implied; while, i = 1, 2, 3 is a free index. In the limit of small strain, which is sufficiently accurate for most applications in seismic wave propagation, the stress-strain relationship is linear and is described by the generalized Hooke's law: (3.2) Here is the fourth-order stiffness or elastic tensor responsible for the material properties (it is discussed in detail below), and is the strain tensor defined as (3.3) The elastic coefficients constitute the most condensed manner in which to characterize the elastic behavior of the rock. Substituting Hooke's law (3.2) and the definition (3.3) of the strain tensor into the general wave equation (3.1), and assuming that the stiffness coefficients are either constant or vary slowly in space (so that their spatial derivatives can be neglected), we find the equation outlined below. Also physically, the homogeneous wave equation describes a medium without sources of elastic energy. Therefore, by dropping the body force f, (3.4) Now, as a solution of equation (3.4), a harmonic (steady-state) plane wave is used that is represented by (3.5) Where are the components of the polarization vector U, ω is the angular frequency, V is the velocity of wave propagation (usually called phase velocity), and n is the unit vector orthogonal to the plane wavefront (the wavefront satisfies ). 30

53 Substituting the plane wave (3.5) into the wave equation (3.4) leads to the so called Christoffel equation for the phase velocity V and polarization vector U: (3.6) Here is the Christoffel matrix (or as mentioned in Cheadle et al. [1991] the socalled Kelvin-Christoffel stiffnesses), which depends on the medium properties (stiffnesses) and the direction of wave propagation: (3.7) Equation (3.6) can be rewritten in a more compact form, (3.8) The Christoffel equation (3.6) or (3.8) describes a standard 3 3 eigenvalue - eigenvector (U) problem for the symmetric matrix G. The eigenvalues are found from (3.9) which leads to a cubic equation for. For any given phase (slowness) direction n in anisotropic media, the Christoffel equation yields three possible values of the phase velocity V, which correspond to the P wave (the fastest mode) and two S waves. Solutions of equation (3.9) in terms of the elements can be presented as explicit expressions for three different phase velocities V in arbitrary anisotropic homogeneous media and can be found in Appendix 1A of Tsvankin s 2001 book. The plane-wave polarization vector in isotropic media is either parallel (for P waves) or orthogonal (for S waves) to the slowness vector. In the presence of anisotropy, however, polarization is governed not only by the orientation of the vector n, but also by the elastic constants of the medium which determine the form of the Christoffel matrix G. Since matrix G is real and symmetric, the polarization vectors of the three modes (i.e., the eigenvectors) are always mutually orthogonal, but none of them is necessarily parallel or perpendicular to n. Thus, except for specific propagation directions, there are no pure longitudinal and shear waves in anisotropic media. For that reason, in anisotropic wave theory the fast mode is often called the "quasi-p" wave and 31

54 the slow modes are called "quasi-sl" and "quasi-s2". However, for simplicity, we are referring to them as P and S waves here in this thesis. 3.3 Phase and Group Velocity The group velocity vector determines the direction and speed of energy propagation (i.e., it defines seismic rays) and, therefore, is of primary importance in seismic travel-time modeling and inversion methods. The difference between the group and phase velocity vectors may be caused by velocity variations with either frequency (velocity dispersion) or angle (anisotropy). As illustrated by the 2D sketch in Figure (3.1), the group velocity vector in a homogeneous medium is aligned with the source-receiver direction, while the phase velocity (or slowness) vector is orthogonal to the wavefront. Figure 3.1: The group velocity (ray) vector pointing from the source to the receiver (angle ψ) in a homogeneous anisotropic medium. The corresponding phase velocity (wave) vector is orthogonal to the wavefront (angle θ) (Tsvankin, 2001). Plotting the phase velocity of a given mode as the radius vector in all propagation directions defines the phase velocity surface. Likewise, plotting the inverse value in the same fashion results in the slowness surface, where topology is directly related to the shape of the corresponding wavefronts. The ray direction (i.e., the direction of the group velocity vector shown above) is orthogonal to the slowness surface. In homogeneous isotropic media, the phase velocity and slowness surfaces, along with the corresponding wavefronts are spherical, but it is later shown that the slowness surface is ellipsoidal in anisotropic media under true-triaxial stress. 32

55 In the transmission experiment, travel-time measurements yield the phase velocity if the transducers are relatively wide compared to their separation, or they yield the group velocity if the transducers are very small compared to their separation (Dellinger and Vernik, 1994; Vestrum, 1994; Auld, 1973). Therefore, it is believed that in transmission experiments on a model with large enough dimensions, the measured firstbreak determines the group velocity, which is not equal to the phase velocity except for the principal directions. Nevertheless, Dellinger and Vernik [1994] indicated that finding the phase velocity is problematic. Due to the small sample size, there are rather large errors associated with picking the first-break and consequently, it may result in phase velocity measurements with high error. Not being able to measure phase velocities substantially complicates the determination of elastic constants in pulse transmission measurements (Vestrum et al., 1999; Mah and Schmitt, 2001). The traditional way of estimating the acoustic velocity is to calculate the time difference between the first-break of the transmitted and received signals. The firstbreaks are defined as the points at which the wave energy is seen to arrive. The corresponding velocity is related to the high-frequency components of the signal (Rathore et al., 1993). This can be significantly different from the dominant lowerfrequency phase velocity for a dispersive and anisotropic medium. The alternative method, also being used here in this thesis, is to calculate the time interval between the first zero crossovers of the transmitted and received signals. To a first approximation, the zero crossover of the signal amplitude can be regarded as a common phase point for all frequency components in the signal (this being exact for the case of an antisymmetric signal around this point). The zero crossover point will move with the phase velocity in a non-dispersive medium. This may also provide a good estimate of the phase velocity of the dominant frequency, even when the medium is dispersive, as long as the dispersion is small for the dominant frequencies in the signal. Hence, Rathore et al. [1993] were convinced that the first zero crossovers are associated with the phase velocities. However, especially in our case with a rather noisy data, we also tried crosscorrelations between the successive velocity survey waveforms (Appendix A) in order to find the time difference between them and calculate the associated apparent P wave velocities shown in chapter 4. The results from both methods were in consistency with each other. 33

56 3.4 Stiffness Tensor The contribution of the medium symmetry to the wave equation (3.4) and the Christoffel equation (3.6) is controlled by the stiffness tensor, whose structure determines the Christoffel matrix (3.7) and, consequently, the velocity and polarization of plane waves for any propagation direction. While a general fourth-order tensor has components, possesses several symmetries that reduce the number of independent elements. The medium with the lowest possible symmetry is described by a total of 21 stiffness elements, and the tensor can be represented in the form of a 6 6 matrix. This operation is usually accomplished by replacing each pair of indices (ij and kl) by a single index according to the Voigt recipe : And, the resulting "stiffness matrix" is symmetric. For example, the most general anisotropic model is called triclinic which has 21 independent stiffness components and no material is ever identified with this symmetry model from seismic measurements. The simplest realistic symmetry for most rock formations is often orthorhombic with nine independent elastic tensor parameters (the symmetry of a brick) described in the next sub-section. Transversely isotropic (TI) symmetry, with five independent elastic tensor parameters, and ellipsoidal anisotropic symmetry, with six independent elastic tensor parameters, are degenerate cases of orthorhombic anisotropy with fewer numbers of independent elastic parameters. Therefore, geophysicists commonly use the latter two less complicated anisotropic models for their calculations as opposed to orthorhombic models, which are harder to deal with. 3.5 Orthorhombic Anisotropy Orthorhombic (or orthotropic) symmetry models are common in seismic anisotropy studies. Orthorhombic model is characterized by three mutually orthogonal planes of mirror symmetry and is usually caused by parallel vertical fractures embedded in a finely layered medium (Tsvankin, 2001). Also, an orthorhombic model is the simplest azimuthally anisotropic model used to describe realistic fractured reservoirs (e.g., 34

57 vertical cracks embedded in an anisotropic background media or orthogonal fracture systems in a purely isotropic matrix). In the coordinate system associated with the symmetry planes (i.e., each coordinate plane is a plane of symmetry), orthorhombic media have nine independent stiffness components and the symmetrical stiffness matrix is written as: (3.11) Where,, and. The above stiffness matrix would describe isotropic media in the special case when, and where, λ and µ are Lame parameters. The phase velocity V and the displacement (polarization) vector U of plane waves in orthorhombic media can be found from the Christoffel equation (3.6). The Christoffel matrix (3.7) for orthorhombic symmetry can be found in the following form: (3.12) As before, n is the unit vector in the phase (slowness) direction. In a specific case when the wave propagates in the three main principal directions of the Cartesian coordinate system in an orthorhombic material, the Christoffel matrix components will be more simplified in terms of the stiffness components. For example, if the slowness direction is confined to the X direction,, then the Christoffel matrix components reduce to: (3.13) And equation (3.6) becomes 35

58 (3.14) For this rather simple case, that of propagation along a principal direction, there are three obvious eigenvalues which will zero the determinant of the 3 3 matrix. For each of these, the associated eigenvector is the polarization (or unit-particledisplacement) vector: (3.15) where, i and j indices in represent the direction of propagation and polarization of the waves, respectively. By applying the same approach on the slowness directions confined to the other two main principal directions (Y and Z directions) in orthorhombic media, we find the expressions for six diagonal stiffness coefficients (Cheadle et al., 1991): (3.16) Still, three off-diagonal independent coefficients of the orthorhombic stiffness matrix are remained to be solved. Mahmoudian et al. [2014] have given a solution for off-diagonal stiffness components of an orthorhombic medium illustrated in section of this thesis. 3.6 Ellipsoidal Anisotropy Ellipsoidal anisotropy is a degenerate case of orthorhombic symmetry in which the velocity or slowness wavefront surface will be spatially ellipsoidal. The stiffness matrix in ellipsoidal symmetry looks exactly like the stiffness matrix in orthorhombic symmetry; however, it has six independent components despite the orthorhombic 36

59 symmetry that has nine independent components. There are two different approaches presented in the next sub-sections that illustrate ellipsoidal anisotropy in detail Rasolofosaon s Theory Schwartz et al. [1994] demonstrated that two very different rock models, namely a cracked model and a weakly consolidated granular model, always lead to elliptical anisotropy when uniaxially stressed. However, Rasolofosaon [1998] stated that isotropic elastic media, when true-triaxially (or polyaxially) stressed, constitute a special sub-set of orthorhombic media, here called "ellipsoidal media" where, the P wave slowness surface is always an ellipsoid. Johnson and Rasolofosoan [1996] proposed a relationship between the elastic property of the rock and the main principal stresses. They assumed that is the P wave modulus in the presence of a triaxial stress where M designates a wave modulus, that is the unstressed density multiplied by the "natural" velocity (i.e., the length of the "acoustical path" in the unstressed state divided by the wave travel-time in the stressed state) and derived the following equation for : (3.17) The equation below is a consequence of equation (3.17) and simply means that the P wave slowness surface is always an ellipsoid in isotropic media, which are triaxially stressed. This constitutes the main result of Rasolofosoan s work. (3.18) Where, a, b and c are the 3 principal P wave slownesses along the eigen-directions of stress demonstrated as follows: (3.19) And, X, Y and Z are the components of the P wave slowness vector (provided that the P wave modulus is density normalized) are represented as, (3.20) 37

60 Thus, an initially isotropic media, when triaxially stressed, forms an ellipsoidal media where the P wave slowness surface, and as a consequence, the P wave velocity surface (because these surfaces are reciprocal) are always strictly ellipsoidal Daley s Theory As Mahmoudian et al. [2014] also mentioned, ellipsoidal anisotropy refers to the assumption regarding approximating the non-spherical wavefront in anisotropic media to an ellipsoid. There are many arguments about ellipsoidal anisotropy assumption in the geophysics literature. Thomson [1986] maintained that elliptical anisotropy is an inadequate approximation most of the time. Schoenberg and Helbig [1997] stated that for transversely isotropic media, only the SH wavefront is ellipsoidal; the SV wavefront is never an ellipsoid, and the P wavefront is an oblate ellipsoid under specific conditions. However, Daley and Kebes [2006] showed that in the degenerate ellipsoidal case, the P wave eikonal becomes (3.21) Where, components of the slowness vector are defined in terms of the P phase (wavefront normal) vector components. Also, in Voigt notation, the density normalized elastic anisotropic parameters,, have the dimensions of velocity squared. To describe an orthorhombic medium, nine independent density normalized elastic constants are required, plus. The velocity measurements related to symmetry directions have simple expressions that allow particular elastic constants to be obtained. For an orthorhombic solid, taking the symmetry axes of the sample as principal axes, three P wave velocities along principal axes determine the ; three S wave velocities also along the principal axes determine. The off-diagonal elastic constants can not be determined from P wave velocity measurements independently; they can only be determined in combination with other elastic constants. Equation (3.21) is a simplified form of equation (3.22) under specific situations where the three defined symmetry plane anellipsoidal deviation parameters are zero. The equation below is a consequence of Daley s work from 38

61 determining the linearized phase velocity in an anisotropic medium of orthorhombic symmetry for a quasi-compressional (qp) wave using the method given in Backus [1965] which produces the subsequent first order linearized approximation in a weakly anisotropic medium: (3.22) with the quantities being the linearized forms of the anellipsoidal deviation terms in the plane or equivalently in the (slowness) plane, and defined as: (3.23) For a general (non-elliptical) orthorhombic medium, Daley and Kebes [2006] obtained a reasonably accurate expression for P wave group velocity by manipulating the eikonal equation, considering the phase velocity expression as in equation (3.22). They obtained a general expression for P wave group velocity V(N) in an orthorhombic medium as: (3.24) The equation above can be applied to invert for off-diagonal elastic constants, which is used by Mahmoudian et al. [2010]. The method is described in section of this thesis. 3.7 Thomsen Parameters Thomsen s paper (Thomsen, 1986) is the single most-cited article in the history of GEOPHYSICS (Peltoniemi, 2005). Thomsen [1986] derived a set of three dimensionless anisotropy parameters to describe weak to moderate transverse isotropy of a medium. These parameters were defined in terms of the five components of the stiffness tensor (,,,, ) relating stress and strain for the transversely isotropic (TI) medium. The dimensionless Thomsen parameters (Thomsen, 1986) which characterize the anisotropy of a medium are industry standard and are used in many commercial software products. 39

62 For P wave propagation in the earth near the vertical direction, the important anisotropy parameter is δ. For SV wave propagation near the vertical, the combination ( ) plays essentially the same role as δ does for P waves. For SH waves, the pertinent anisotropy parameter is γ. All three of the Thomsen parameters vanish for an isotropic medium. Thomsen parameters for an orthorhombic symmetry with nine independent stiffness tensor components are given and used by Mensch and Rasolofsoan [1997], Tsvankin [1997], Ruger [1997], Rasolofsoan [1998], Shakeel and King [1998], Tsvankin [2001], Rüger [2001], Sarkar et al. [2003], Mahmoudian et al. [2014] and many others. Once all the elastic constants of the material are determined, the Thomsen anisotropy coefficients δ λ can be calculated. Following Tsvankin [1997] and Rüger [2001], the relationship between elastic constants to generic Thomsen parameters (for the principal planes of XoZ, YoZ, and XoY) for a medium with ellipsoidal anisotropy, which is a degenerate form of orthorhombic symmetry, are listed in Table (3.1). Note that the calculated Thomsen parameters are exact values valid for any strength of anisotropy, which is different from week-anisotropy Thomsen parameters introduced by Thomsen [1986]. Thomsen Parameter y-z plane x-z plane x-y plane Table 3.1: The relationship between elastic constants to the generic Thomsen parameters. All the relations are valid for any strength of anisotropy. 40

63 The coefficients conveniently quantify the magnitude of velocity anisotropy in orthorhombic media, both within and outside the symmetry planes. By design, these parameters provide a simple way of describing seismic signatures in the symmetry planes of orthorhombic media using the known equations for VTI media expressed through Thomsen parameters. The parameters and are close to the fractional difference between vertical and horizontal P wave velocities in the YoZ and XoZ planes (respectively) and, therefore, yield an overall measure of the P wave anisotropy in these planes. Similarly, the coefficients and govern the magnitude of the velocity variation of the elliptical SH wave in the vertical symmetry planes. In the vertical symmetry planes, the needed parameters are the vertical P wave velocity (the scaling factor) and a pair of the anisotropic coefficients: and (XoZ plane) or and (YoZ plane). As in VTI media, the coefficients and are responsible for phase and group velocity in different ranges of phase angles, which is extremely convenient for purposes of seismic processing and inversion. Specifically, the coefficients (XoZ plane) and (YoZ plane) determine near-vertical P wave velocity variations as well as the anisotropic term in the expression for normal-moveout velocity from horizontal reflectors (Tsvankin, 2001). As outlined in this chapter, the anisotropic symmetry induced in the true-triaxial test is ellipsoidal anisotropy. Having the elastic wave propagation fundamental features and properties in ellipsoidal media set in mathematical equations, we will apply them in the next chapter to obtain a velocity image of the rock specimen as well as its elastic parameters. 41

64 4 Anisotropic Velocity Model Parameters 4.1 Geophysical Imaging Cell and Laboratory Setup A state of the art True-Triaxial Geophysical Imaging Cell (TTGIC) is used at the Rock Fracture Dynamic Facility (RFDF) laboratory at University of Toronto under supervision of Prof. R. Paul Young to study rock fracture physics under polyaxial compressive stress regime. Figure (4.1) shows the detailed components of TTGIC, designed and manufactured by Ergotech Company, which is placed in a custom made polyaxial servo-controlled Minneapolis Testing System (MTS) loading frame. The MTS loading frame can produce up to 3400 kn laterally (X and Y axes) and 6800 kn vertically (Z-axis). Therefore, the system is able to apply up to 1059 MPa as for the major principal (vertical) stress, and up to 530 MPa as for the intermediate and minor principal (horizontal) stresses on 80 mm-side cubic rock specimens, all capable of being applied independently. Also, a maximum pore pressure capability of 35 MPa and a maximum strain capability of 5-8% can be provided. Sealing technology is also a special characteristic of this system that enables it to operate directional permeability measurements. Figure (4.1) also shows the Polyaxial servo-controlled MTS system along with the TTGIC. Seismic operations in the TTGIC are carried out with Applied Seismology Consultants (ASC) InSite 1 seismic processing software. Each loading platen hosts P, S1 and S2 wave piezo-ceramic (800 khz resonant frequency) transducers embedded behind the platen surface. Thus, ultrasonic P and S wave velocities can be measured along the three principal stress directions at various stages of loading in this cell. Also, transducerto-transducer velocity surveys can be performed between 18 Lead Zirconium Titanate (PZT) transducers, where three transducers are embedded on the surface of each of the six loading platens in a triangular configuration, and a special mechanism maintains a constant contact pressure of approximately 3 MPa between the transducer and the 1 ASC's software, InSite, provides an integrated tool for seismic processing, data management and visualisation. ASC is part of IMaGE, Itasca Microseismic & Geomechanical Evaluation, combining microseismic excellence with state-ofthe-art geomechanical simulations, offering a new standard in engineering hydraulic fracture treatments for unconventional reservoirs. Itasca is a Minnesota-based engineering consulting and software firm, working primarily with the geomechanics, hydrogeological and microseismics communities. 42

65 sample. PZT transducers are sensitive to vertical motions and to the frequencies ranging from 100 khz to 1 MHz after the applied filters during the waveform recordings by preamplifiers. The waveforms from 18 AE sensors are continuously recorded with a Volts dynamic range. Data from 18 channels is acquired across nine computer units each streaming two channels. Each computer unit has a storage capacity of about one Terabytes, which allows for roughly seven hours of continuous data acquisition for a 10 MHz sampling rate and 12-bit resolution. Figure 4.1: Polyaxial servo-controlled MTS system. Each of the six stainless steel platens with three AE sensors are installed on the True-triaxial Geophysical Imaging Cell (TTGIC) designed and manufactured by ErgoTech. 4.2 True-Triaxial Test Procedure on Fontainebleau Sandstone Fontainebleau Sandstone Sample Preparation A Fontainebleau sandstone cubic sample of size was prepared using a Wasino CNC grinding machine with an accuracy of 10 micron flatness on each face after being cut in cubic shape with a slab saw. After the sample was machined, it was 43

66 oven-dried at 70 C for 24 hours and then saturated with distilled water under vacuum for 24 hours. Fontainebleau sandstone is pure sandstone of Oligocene age (Stampian) and is found in the Ile de France region around Paris. It is a very homogeneous rock containing 98% quartz. Fontainebleau sandstone has several porosities due to the varied cementations and is widely characterised as well-sorted sandstone with constant grainsize around 250 µm. The variation of porosity is enormous and goes from 2% to 30% (Bourbie et al., 1985; Charalampidou, 2011). Porosity of the Fontainebleau sandstone that was used in our experiment was 5%. Sandstone is a clastic sedimentary rock composed mainly of sand-sized minerals or rock grains. Clastic rocks are composed of fragments of pre-existing minerals and different rocks. Clastic sedimentary rocks are composed of silicate minerals and rock fragments that were transported by moving fluids and were deposited when these fluids came to rest. Most of the sandstones are composed of quartz and/or feldspar since these are the most common minerals in the Earth's crust. Sandstone is the ideal rock for ground water and will house substantial aquifers. Petroleum is also a fluid that flows through sandstone and sandstone is one of the best oil reservoirs (Fjar et al., 2008). In-situ velocity measurements are much easier than permeability measurements in geothermal or oil reservoirs. As a result, finding a relation between velocity and permeability by tying acoustic and hydraulic properties of sandstone in the laboratory is desirable. Fontainebleau sandstone is an appropriate type of rock for the purpose of laboratory AE and permeability measurements True-Triaxial Test Procedure Being placed within the True-Triaxial Geophysical Imaging Cell (TTGIC), the cubic Fontainebleau sandstone sample went through different loadings exerted by the MTS machine during the experiment. The Fontainebleau sandstone specimen (FTB4) was tested first at hydrostatic stress of 5 MPa. Then, the loading pattern moved on to a true-triaxial loading state with a stress ratio where, and were kept at a constant stress of 5 and 35 MPa respectively during the experiment. In the meanwhile, gradually increased up to the failure of the rock at 490 MPa (figures 4.2, 4.3 and 4.4). In this experiment, and were applied under load control mode on two paired horizontal actuators (along X and Y directions) while simultaneously maintaining a 44

67 constant load on two opposite sides of the cubic specimen. However, was raised under a constant displacement control rate of mm/s up to failure (Nasseri et al., 2014). Strain along each axis was measured using the average of the three linear variable differential transformers (LVDT), which were mounted close to the platen-rock interface. The figure below displays the evolution of the main principal stress in the Z direction with time by the blue line and the AE activity of the rock specimen by the green line represented by the cumulative number of located AE events. As it can be seen for stress pattern, there are many instances during the experiment that stress is kept constant to perform permeability measurements. At the end, the rock specimen drastically failed. Figure 4.2: Evolution of the main principal stress in the Z direction (blue) and the cumulative number of located AE events representing the AE activity (green). The red arrow shows the onset of high AE activity of the specimen indicating the failure of the sample. As Nasseri et al. [2014] also reported, Haimson and Chang [2000] designed and fabricated a true-triaxial testing system and determined that dilatancy is more pronounced at low intermediate stress magnitudes but diminishes at higher levels. Indeed, in figure (4.3), where axial strain in the minimum principal stress direction (X direction) along with the evolution of main principal stress versus time is illustrated, it is indicated that dilation happens right at the red arrow exactly at the same time as the onset of high AE activity shown in figure (4.2). This is where developing micro-cracks open up primarily in the minimum principal stress direction. 45

68 The experiment was divided into four main stages based on dilatancy of the rock. Stage A is about closure of cracks hydrostatically where the volume of the rock shrinks as those voids are closed. Stage B occurred from around the main principal stress of 35 MPa to 150 MPa and consists of no dilation. Stage C happened at maximum stresses larger than 150 MPa up to around 350 MPa with a little AE activity and a slight dilation but can still be approximated as no dilation due to the strain data. It was at stage D where the rock heads down to failure from maximum stress of 350 MPa starting with a high rate of AE activity up to around 490 MPa where an abrupt dilation and then failure ensued. Stage A Stage B Stage C Stage D Figure 4.3: Evolution of the main principal stress in the Z direction (blue) and the evolution of strain in the X-axis (green). The red arrow shows the onset of dilation of the rock in the X direction. Stage C Stage D Stage B Stage A Figure 4.4: Schematic view of different stages of the experiment on a stress strain curve in the main principal stress direction. 46

69 The combination of principal stress magnitudes was designed to simulate the in situ stress state in sedimentary rocks approaching failure when subjected to large differences between each of the principal effective stresses, as found adjacent to a deep borehole during drilling. Figure (4.4) shows a schematic view of different stages of the experiment on a stress strain curve in the main principal stress direction. During the experiment, two different Ultrasonic wave measurements were performed that is explained in the next sections in detail. The P-S1-S2 velocity measurements analyses revealed the relations between the stress distribution and heterogeneity in compacted domain of the rock in a true-triaxial experiment. 4.3 Ultrasonic Velocity Measurements in Three Principal Directions (P-S1- S2 Velocity Surveys) Using an ASC integrated pulser-amplifier system software, the ultrasonic wave velocity survey measured compressional ( ) and two shear wave velocities ( and ) along all three principal stress directions at various stages of loading. Figure 4.5: Ergotech Diagram of the P-S1-S2 velocity measurements. Conventionally, the color green is chosen by the manufacturer to represent the Z direction, blue for the Y direction and red for the X direction. We also follow this set of colors in our figures. 47

70 Table (4.1) demonstrates the transformation of Ergotech notation for velocity directions and polarizations (P, S1 and S2) to the notations in the Cartesian coordinate system where the first index is the direction of propagation and the second index is the polarization direction. Propagation Polarization P S1 S2 X Y Z Table 4.1: Transformation of Ergotech notation for wave velocity propagation directions and polarizations to Cartesian coordinate system notation. Figure below shows the ZoY plane cross-section of the TTGIC. Ultrasonic P and S wave measurements are accomplished with a stack of transducers called the ultrasonic P, S1 & S2 acoustic stack mounted in each of the six platens. Chamfered edge of the rock sample Z X Y P, S1 & S2 stack Figure 4.6: Cross-section of TTGIC in the YoZ plane containing the sample in the middle and two opposite platens. The chamfered edges and P-S1-S2 acoustic stacks are displayed (Lombos et al., 2012). 48

71 An overview of the results of a number of experiments using the polyaxial stress arrangement is reported in King [2002]. A set of parallel fractures and cracks was introduced in a number of cubic sandstone specimens presented by King [2002], increasing the major and intermediate principal stresses in unison to failure, while maintaining the minor principal stress at a low level. It was emphasized that the reciprocal equality in measured S wave velocities would indicate the uniform state of stress within the specimens at failure. We have seen, however, that this was not the case in our experiment as it was not the case for any of the true-triaxial experiments unless a soft metal with low stiffness was used on the surface of the platen in order to reduce the friction between the platen-sample interface, which would have sacrificed the test at high stresses Evolution of P, S1, and S2 Velocities The P, S1 and S2 velocity measurements along all three principal axes are reported for different stages of the TTT experiment in Nasseri et al. [2014]. Figure (4.7) also demonstrates the exact same data but with consideration of strain data in calculations. Due to the symmetric nature of the orthorhombic anisotropy, we expect that pairs of measured shear wave velocities should be equal to each other. Observation of their inequality, appeals us to go through an appropriate analysis in order to accurately interpret the anisotropic velocity characteristics of the rock during the experiment. The four aforementioned stages of the experiment are again observed by the velocity evolution of the rock in the three principal directions. Stage A is associated with compaction and closure of the voids. The elastic deformation does not occur in Stage B until some AE activities start near the minimum principal stressed facets of the cubic rock at the onset of Stage C where some velocity drop occurs. At Stage D, the velocities drastically drop and a huge number of AE events turn out and the specimen goes toward failure. We only focus on stages B and C until the rock maintains its elastic behaviours in the absence of any through-going fractures at its center, where all the compressional waves and shear waves ray-paths (P, S1, and S2) along the three principal directions meet. Our main goal is to draw out the elastic tensor parameters at the central part of the cubic rock, where assumingly, the applied stress is almost uniformly distributed. Figures 13 and 14 in Nasseri et al. [2014] also demonstrate the 49

72 AE activity zones at different stages of the experiment and verify that the deformations mainly start from the corners, edges and areas near facets of the cubic rock sample. Stage A Stage B Stage C Stage D Figure 4.7: P-S1-S2 wave velocity measurements during the FTB4 experiment. The expected symmetry conditions shown in equation (4.1) between the reciprocal S wave velocities are violated because of the existence of heterogeneities in the rock: 50

73 (4.1) In our experiment, the P wave velocities propagating in the directions of the three maximum, intermediate and minimum principal stresses, respectively, do satisfy our expectations as they follow the order ( ). However, no simple pattern is followed by the S waves in the data shown above and it is clear from the graph that there are pairs of almost equal velocities with similar evolving patterns existing. These velocity pairs are (4.2) Each of the three velocity pairs mentioned in equation (4.2) have similar patterns, which is the most important and reliable characteristic of this velocity data. This indicates dominance of the effect that the direction of propagation has on the shear wave velocities in this data rather than the effect of the direction of polarization. The shear waves propagating in the X and Y directions (,,, and ) are particularly sensitive to the growth and presence of the aligned fractures as they already start decreasing during stage C and drastically drop at the onset of stage D. Also, as can be observed in the data, the rock shows a small initial anisotropy where, ( ) but, the anisotropic pattern gets aligned with the TTT principal stress directions in higher stresses and that is what we are interested to study in this thesis. 4.4 Transducer-to-Transducer Velocity Survey Piezoelectric sensors typically exhibit some ringing when detecting wave arrivals. Later wave modes often arrive while the transducer is still ringing from an earlier arrival. Therefore, in practice, the first longitudinal wave (P wave) arrival is usually more distinct and accurately measurable than the other arrivals. Particularly, if the wave travel distance is short, the next modes of oscillations after the P wave first-break such as the S wave first-break will arrive too soon and get mixed with the P wave signal, making it impossible to determine the S wave arrival-time. 51

74 There are 18 transducers located on the six platens surrounding the cubic rock specimen that make seismic surveys during the experiment, called Milne survey named after the seismic acquiring system. What happens in this survey is that each of the 18 transducers emits P wave pulses into the rock, and the other 17 transducers receive and record it. During each velocity survey, waveforms were stacked 10 times to reduce the background noise. There would be a total number of paths between all the transducers. However, on the same platen, there are two adjacent transducers to the triggering one that should not be taken into account, as the ray-path between them does not pass through the rock sample but it travels along the surface of the rock-platen interface. Therefore, the total number of ray-paths will be reduced to. The figure below demonstrates the locations of the transducers as well as the 135 ray-paths between the transducers in our laboratory configuration. Figure 4.8: Real laboratory sensor arrangement. Left: transducer locations. There are 3 transducers embedded on each platen. Locations of the transducers on each pair of the opposite platens are the same. Right: Ray-paths between the 18 transducers covering the cubic sample. The velocities are the phase velocities of the dominant frequency propagating waves, estimated by timing the propagation of a recognizable phase point through the sample. As Rathore et al. [1994] outlined the phase velocity determination, we also used the method of travel-time calculation by measuring the time interval between the first zero crossovers of the transmitted and received signals. 52

75 The validity of the straight ray-path assumption depends crucially on the imaged structure. As Debski and Young [2002] described, if the velocity gradients and variations are small enough that the departure of an actual ray from a straight line at the receiver position is smaller than a few wavelengths, the straight-line approximation may be used. This is usually the case in tomography imaging of small objects like rock samples or pillars in mines. In such applications, modeling errors introduced by the straight-line ray-path approximation only decrease the contrast of the images Stereonet Representation of the P wave Velocity As mentioned earlier, apparent velocities given in these figures are phase velocities of the dominant low-frequency propagating waves, estimated by timing the propagation of a recognizable phase point through the sample. The figure below displays an InSite seismic processing software window in which the first arrival is picked. It also demonstrates the difficulty in picking the arrival-times in the presence of noise. Figure 4.9: A waveform window and the first-arrival picking in the InSite seismic processing software. In order to plot stereonet diagrams, the velocities computed for each ray-path hitting the lower hemisphere are projected and plotted on a horizontal circular area called stereonets at the point corresponding to the trend and plunge of that ray-path. The vertical direction of the stereonet (0 plunge) corresponds to the Z axis direction of the sample, while the bottom of the stereonet (90 or 270 trend/90 plunge) is chosen to correspond with the Y axis direction. Not all the ray-paths obtained from each velocity scan were used to construct the stereonets, because some waveforms corresponding to two of the transducers on Y and X platens were too noisy and removed from the calculations. This made the total number of the involved ray-paths equal to 106 between 16 transducers instead of 135 ray-paths between 18 transducers. Those ray- 53

76 paths with the same trend and plunge that passed through different parts of the sample (and therefore may have had different velocities) were averaged. Figure (4.10) displays the evolution of the apparent P wave velocity images of the rock as the main principal stress in the Z direction vary, while the other two intermediate and minimum principal stresses (Y and X directions, respectively) are kept constant during the FTB4 experiment. Figure 4.10: Evolution of the rock in FTB4 experiment represented by directional apparent P wave velocities demonstrated by stereonets at different stress states with constant minimum and intermediate principal stress ( ) and increasing maximum principal stress ( ). The blue line shows the maximum principal stress in the Z direction. The green line presents the cumulative number of located AE events and the apparent velocity stereonets are plotted at different stages from each ultrasonic velocity survey data. Between each of the two consequent surveys, any change in the P wave velocity along 54

77 each ray-path is obtained and plotted later in this chapter on stereonets to demonstrate the directional velocity evolution in the sample. In the next sub-section we will see that the slowness difference data corresponding to such directional velocity changes are used for constructing tomographic images and provide a detailed three-dimensional picture of the progressive development of damage in the rock specimen. The required accuracy of travel-time measurement in tomographic applications also demands at least a visual check on all automatically picked arrival-times in ASC s InSite software. In the initial stages of the experiment, the fast and slow velocity directions in the sample were not consistent with the applied loading pattern, however, clearly in main principal stresses bigger than 75 MPa, the fast and slow velocity directions were both approximately perpendicular to each other and parallel to the Z and X axes directions, respectively, while the intermediate velocity direction was parallel to the Y axis direction Error Analysis for Velocity Measurements As Hornby [1998] mentioned, the wave velocities that are computed using traveltimes of transmitted ultrasonic pulses, can be calculated by the following expression: (4.3) Where, L is sample length, is the measured travel-time through the sample, and is a reference travel-time from the head-to-head measurement, which is 1.3 µs in our experiment. Expanding in terms of partial derivatives gives us an expression that can be used for the error analysis, (4.4) Where, and are the errors in the travel-time picks for the measured and reference travel-times and is the magnitude of the error in the sample length measurement. A bound on the absolute error in the velocity estimation is evaluated as (4.5) The maximum absolute error in the measurement is (4.6) 55

78 Where,. To estimate errors, we consider a typical sample length of 80 mm and a precision of the Wasino CNC grinding machine sample length measurement of mm. For P wave propagation, a typical value for is 17 µs. Precision of time measurement in the recording system is half of the sampling rate (0.1 µs), which is 0.05 µs. However, the maximum arrival-time picking error due to the noisy waveforms is at least half of the typical waveform period at the onset of P wave arrival which is about 1 µs. Therefore, setting a precision of the travel-time picking equal to 0.5 µs, we estimate an absolute error of the velocity measurement of about ±280 m/s. This corresponds to a relative error in the velocity estimation of approximately 6% for an average P wave velocity of 4700 m/s. 4.5 Anisotropic Travel-Time Tomography The first attempt toward obtaining a velocity image of the rock under polyaxial stress state used the transducer-to-transducer P wave velocity survey data from our experiment. The approach was based on anisotropic tomography of the rock in which an anisotropic velocity model was considered as the background velocity model where heterogeneous zones were supposed to be recovered on top of it. A full description of the developed methodology for anisotropic ray-based tomography is given in Appendix A. It is the relevant scales between the wavelengths (equivalent to frequencies) and heterogeneities that determine the nature of the seismic wave-medium interaction. The recorded voltage signals and waveforms by transducers were band pass filtered from 100 khz to 1 MHz using preamplifiers manufactured by the Applied Seismology Consultants (ASC) Company. These ASC preamplifiers have a switching mechanism that allows for each transducer to act as a source as well as a receiver during transducer-to-transducer P wave velocity surveys. During the survey, each of the 18 piezoelectric transducers sends an AE pulse equivalent to 500 electric volts pulse into the rock which is heard by all the other transducers and this periodic process is performed for all 18 transducers in sequence. During velocity surveys, waveforms were stacked 10 times to reduce background noise. Marion et al. [1994] investigated the relationship between wave velocity measurements in layered media at different frequencies and scales encountered in laboratory measurements with frequency and wavelength at ( 56

79 ), logging ( ), and seismic ( ) measurements. They emphasized that both theoretical and experimental results from a periodic media composed of steel and plastic discs indicated that in stratified media, when the wavelength is much larger than the layer spacing instead of illustrating heterogeneous attributes, layered media behave as homogenous (transversely anisotropic) media. It was proven by them that velocities in stratified media are controlled by the ratio of wavelength to layer spacing. They compared velocities based on ray theory, which involves averaging slownesses and velocities based on effective medium theory, which involves averaging compliances (slowness squared), and they concluded that the long wavelength (effective medium) velocity is always slower than the short wavelength (ray theory) velocity. Marion et al. [1994] explained that when the ratio of seismic wavelength to layer spacing is small,, wave propagation may be described using ray theory with velocities that are faster than in the effective medium limit. The frequency content and wavelength of ultrasonic waveforms is a critical component to determine the resolution limitations. Based on the 500 khz to 1 MHz peak frequency, the Fresnel zone of the ray tube (from a banana-doughnut 1 shape perspective) passing through a 5 (km/s) rock medium will be the square root of multiplication of wavelength and the propagation distance (Williamson et al., 1993; Hung et al., 2001; Viggiani et al., 2012) which in our case, will be about. Thus, the detectable Fresnel zone size will depend on the distance traveled by the ray that is between about 40 to 80 millimetres, which makes it between and with an average of about 12 mm. Therefore, it is reasonable to divide each of the 8 cm long sides of the cubic sample to 5 portions making each portion 16 mm. Thus, we have blocks in each cubic sample. 1 Banana-doughnut is a technical term in global seismology referring to the tele-seismic raypaths that appear as banana if visualized in the plane of propagation, but as doughnut on a cross-section perpendicular to the ray. The traveltime is sensitive to the above-mentioned probed area. 57

80 Figure 4.11: Frequency spectrum of an ultrasonic waveform arrived from a source of piezoelectric transducer pulse. The red window shows the frequency filter applied to the data for data analysis. Consequently, for the laboratory scales dealing with ultrasonic waves with few milimeter wavelengths, assuming velocity anisotropy in the background of the medium, one can aim for finding local anomalies in the velocity structure caused by compaction, dilation or damages which we call heterogeneities. Therefore, in our polyaxial experiment on a cubic Fontainebleau sandstone specimen, we tried to use travel-time tomography methods to investigate whether we can obtain real-time velocity images of the rock using the P wave velocity measurements from transducer-to-transducer surveys Ray-based Tomography in Ellipsoidal Media Figure (4.12) demonstrates the 3D ellipsoidal anisotropic model of velocity that we applied for either the bulk of the cubic sample or in each grid cell of the sample. The relation between the components of the velocity vector within an ellipsoid in the three principal directions is,,, (4.7) Where, the primed parameters are components of the apparent velocity vector data ( ), and the non-primed vectors are semi-principal axes lengths that are line segments from the origin to the surface of the ellipsoid in three principal directions. Our goal is to extract the magnitude of these semi-principal axes as local velocity characteristics of the medium. 58

81 The velocity vector can be decomposed in the Cartesian coordinate system as below,,, (4.8) While, the magnitude of the velocity vector is equal to, (4.9) Z θ φ Y X Figure 4.12: Ellipsoidal anisotropic velocity model. Each ray-path (i) with a fixed orientation can have a different velocity magnitude ( ) in different cubic grid cells (j). Therefore, in each cell, components of the ray-path velocity in the X, Y and Z directions (,, ) depend on the three orthogonal ellipsoidal velocities in its principal anisotropic axes, which are characteristics of that particular cell. Combining equations (4.7) and (4.8) gives, (4.10) The equation above can also be simplified by using the α and β coefficients. 59

82 (4.11) Thus, components of the local velocity vector for any ray-path with a certain direction of propagation in each probing grid cell can separately be obtained through the following equation based upon the semi-principal axes lengths ratios, α and β, (4.12) The first arrival-time of a seismic waveform is corresponded to the first motion P wave that has passed through the shortest path between source and receiver, which is a straight line for a homogeneous model. The main idea in our anisotropic tomography equations is that the aligned micro-cracks which are located along the straight ray-path between the transmitter/receiver sensors during a transducer-to-transducer survey cause the earliest arrived P wave to pass through a diverted path which is longer than a straight line distance but shortest in travel-time. Such distance variations for each raypath along a certain orientation can then be projected on the velocity structure anisotropic variations through transformation from distance to velocity in the tomography equation, and vice versa. Therefore, in the anisotropic G Matrix, lengths of each ray in each block will differ from the isotropic G matrix based on the dip and strike of the ray. The equation can be calculated for either of the X, Y, or Z directions, resulting in the inverted velocity structure in the corresponding direction. Then, the velocity structure of the other two directions can be extracted using α and β coefficients. In the following equation, the grid cell velocity for a specific ray with a certain direction of propagation is written based upon grid cells z direction velocity, where α and β coefficients are assumed to be approximately constant all over the sample, (4.13) Where, the i subscript belongs to a specific ray and the j subscript belongs to a specific cell. Equation above is the basis for the forward modeling for velocities in 60

83 hypothetical heterogeneous models. Hence, the inversion equation for anisotropic tomography will be, (4.14) Where, the anisotropic G matrix in the equation above is, (4.15) Thus, the abovementioned ellipsoidal velocity model will be used as the reference or background velocity model in the tomography process and the tomographic results for heterogeneities would show local deviations from this model in each of the three principal directions. Those deviations can be assumed as specific heterogeneous domains that have two characteristics. The first of the two characteristics is that those heterogeneous domains will have a lower velocity than the background anisotropic velocity in the presence of micro-cracks; and the second, based on the different directional velocities in each cubic grid cell, a different local anisotropic model will be effective in the corresponding chunk of the rock specimen Anisotropic Tomography Images After all the transducer-to-transducer survey arrival-time data was produced and collected through the recorded waveforms in the InSite software, the data was exported into a text file. After being managed into a proper format in Excel, the arrival-time data was imported into Matlab. By knowing the first triggering transducer arrival-time data point, the travel-times of the waves traveling between the shot and receivers were calculated and sorted accordingly to be used as input in our anisotropic tomography inversion. On the other hand, in each of the three principal stress directions, the average P wave velocities of ultrasonic waves traveling between each pair of the transducers located in front of each other on opposite platen surfaces were calculated for separate transducer surveys. These average velocities were used to construct the ellipsoidal anisotropic velocity structure of the bulk of the sample as a background velocity model for our anisotropic tomography procedure. 61

84 Due to water saturation of the rock sample, detectability of compacted central part versus lower-stressed domains was not possible by P wave travel-time tomography during stages B and C of the experiment according to figure (4.7). Only at the final stage of the experiment (Stage D), the tomographic images showed reasonable anomalies near the X-facets, where velocity reduction occurred due to fracture development and local damages mainly because of dilation of the cubic rock specimen in the minimum principal stress direction. In the two figures below, the apparent velocity data from transducer-to-transducer ultrasonic surveys along with their resultant tomographic images are demonstrated. According to figure (4.10), the last two investigated transducer-to-transducer ultrasonic surveys were performed at the principal stress state of ( ) and ( ). According to figure (4.10), it is between these two instances of the experiment where, a storm of AE events occur as a result of the dilation of the rock by bearing larger differential stress values. Also, according to figure (4.7), ultrasonic wave velocities drop drastically in stage D of the experiment. Therefore, this has to be the phase of the experiment in which apparent velocities drop more or less as a function of their direction of propagation which could happen uniformly all over the bulk of the rock sample; however, velocity tomography could illustrate the local changes that mainly contributed to the variations happen in the apparent velocities. Y Z X Figure 4.13: Stereonet representation of the apparent velocity data at ( ) where, the average velocity of the 2 rays in the X direction is 4.9 km/s, the average velocity of the 3 rays in the Y direction is 5.2 km/s, and the average velocity of the 2 rays in the Z direction is 5.6 km/s. 62

85 Z Y X Figure 4.14: Anisotropic P wave travel-time Tomography ( ) from the transducer survey at ( ). The minimum inverted velocity is 5.1 km/s and the maximum inverted velocity is 6.2 km/s. The stereonet plot of the last survey clearly demonstrates the anisotropic signature of a medium under true-triaxial stress state. The fastest apparent velocity is in the Z axis direction where the maximum stress is applied. Intermediate and minimum apparent velocities are in the Y and X axes directions, where the intermediate and minimum principal stresses are applied, respectively. The cross-sectional images are plotted in the XoZ plane where the maximum and minimum principal stresses are applied. The decay trend can be seen in the tomographic images, where an apparent velocity drop occurs in the X axis direction, which is a reason for developing aligned cracks and dilation of the rock near the X1 and X2 facets of the cubic sample. 63

86 Y Z X Figure 4.15: Stereonet representation of the apparent velocity evolution at ( ) from its previous velocity survey that was performed at ( ) Z Y X Figure 4.16: Evolution of the velocity structure of the rock in the Z direction presented by a tomography image between the two velocity surveys at ( ) and ( ). 64

87 Y Z X Figure 4.17: Stereonet representation of the apparent velocity data at ( ), where the average velocity of the 2 rays in the X direction is 4.4 km/s, the average velocity of the 3 rays in the Y direction is 5.0 km/s, and the average velocity of the 2 rays in the Z direction is 5.5 km/s Z Y X Figure 4.18: Anisotropic P wave travel-time Tomography from the transducer survey at ( ). The minimum inverted velocity is 4.4 km/s and the maximum inverted velocity is 6.6 km/s. Again, we can see a clear velocity drop in the X axis direction and even a little velocity gain in the Z direction as a consequence of dilation of the rock in the X 65

88 direction. The tomographic images also confirm the velocity drop near the X facets along the X direction, which show the spatial velocity variations that correlate with our results from the last chapter. Y Z X Figure 4.19: Stereonet representation of the apparent velocity evolution at ( ) from its previous velocity survey that was performed at ( ) Z Y X Figure 4.20: Evolution of the velocity structure of the rock in the Z direction presented by a tomography image between the two velocity surveys at ( ) and ( ). 66

89 The anomalies obtained in velocity images of the rock at the last stages of the experiment show the opening of new cracks that are still unsaturated or partially saturated. The heterogeneity recovery percentage criteria which could be useful in the interpretation of the results, is given in section (A.7.4). Only heterogeneities larger than 5% can be detected by our anisotropic tomography method. Although, this criteria will increase with the 6% error in P wave velocity measurements given in section (4.4.2). However, the tomographic images are not reliably recovering clear borders of anomalous zones of the heterogeneous areas in the rock. Therefore, there still remains a necessity to solve for velocity structure of the rock that may also shed light on the rock structure during stages B and C of the experiment. That leads us to use the P-S1-S2 velocity survey data which involved shear waves that are sensitive to fluid-filled cracks. 4.6 Simulation of Stress Distribution During the experiment, coalescence of micro-cracks occurs on a pseudo-boundary around the domains with highest stress gradients as well as highest velocity gradients in the rock specimen. This pseudo-boundary is a consequence of two factors combined together. One is the geometrical and physical constraint of the rock specimen including edge chamfers and friction on the platen-rock surfaces. The second cause is the applied polyaxial stress pattern. Shi et al. [2012] also mentioned that in the TTT, in order to avoid contact between the axial and lateral platens, the length of the lateral end pieces is slightly shortened to be less than the length of the rock specimen. This slight difference in the size, results in lack of compressional load at the edges and especially at the corners of the cubic specimen, which is called the corner effect. Also, in our TTT Geophysical Imaging Cell, for the purpose of independent directional permeability measurements, a unique cubic skeleton rubber seal (CSRS) system is developed which needs the rock sample to be chamfered on its 12 edges. Shi et al. [2012] also clarified that due to the differences between the elastic parameters of the platen and the specimen as well as the existence of friction between them, a centripetal term of radial shear stress on the end of the specimen is generated, causing a clamping effect at the end. This brings about stress concentration at the edges and hampered fracture propagation at the end. Also, stress is disturbed by the so-called loading eccentricity, which is the loading center offset from the geometric center of 67

90 specimen, and can be induced by different reasons including machinery inaccuracy, a non-aligned assembly of the platens and the specimen as well as non-uniform loading throughout the test. Thus, modeling the stress distribution within the cubic rock specimen is necessary. FLAC3D is a numerical modeling code for advanced geotechnical analysis of soil, rock, and structural support in three dimensions developed by Itasca Consulting Group, Inc. A numerical model of our true-triaxial testing system was created using this three-dimensional explicit finite-difference method based software that solves a coupled equation of motion and the constitutive stress-strain relationship. Due to symmetrical nature of the problem, only one eighth of the cubic sample along with one fourth of each of the three perpendicular platens was simulated with mirror boundaries on their cut surfaces. In the stress distribution demonstrated in figure (4.21), the loads are applied at the end of each platen. While the steel platens transfer the stress to the surface of the specimen, the stress is damped. A quarter of the X, Y and Z Platens A quarter of the Cross-section of the specimen 4 cm 28 cm Figure 4.21: Cross-sectional view of the TTT system including the three platens and the cubic sample in FLAC3D. Figure (4.22) demonstrates the cubic sample that is chamfered at its 12 edges. In each of the FLAC3D simulations, the 2D image is a quarter of one of the two cross- 68

91 sectional surfaces in either of the XoZ or YoZ planes going through the center of the rock. The symmetry, with respect to the three coordinate planes, means that a single octant (one eighth of the cubic sample) contains all the significant information about the spatial distribution of stress. Z L/2 L Y L/2 L/2 X L L Figure 4.22: Schematic view of the cubic sample with chamfered edges and the cross-sectional slab in green that is simulated in FLAC3D for the YoZ and XoZ planes. In the first step, we approached the shear wave velocities in the XoZ and XoY planes. We used stress distributions simulated with FLAC3D over the rock specimen at every stage of the experiment for analysis of the aforementioned velocities. The following estimated physical parameters of the sandstone sample and the stainless steel platens are used to simulate the stress distribution in FLAC3D: Fontainebleau Sandstone Sample: Density =, Bulk modulus = Pa, Shear modulus = Pa. Stainless Steel Platens: Density =, Bulk modulus = Pa, Shear modulus = Pa. 69

92 Z platen Y=L/2 X=L/2 Z platen L/2 σz X platen Y platen σz L/2 σz (Pa) L/2 = 40 mm L/2 = 40 mm Figure 4.23: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample along with part of the platen touching the sample at stress state. There is a P wave velocity survey available at this stage of the experiment under the stress from 106 ray-path P wave velocities that is displayed in a stereonet representation shown below: Y Z X Figure 4.24: Apparent velocity survey demonstrating 106 ray-paths (Right) and stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) under the stress state. The average velocity of the 2 rays in the X direction is m/s. The average velocity of the 3 rays in the Y direction is m/s. The average velocity of the 2 rays in the Z direction is m/s. The P wave velocity measurements that are available from the ultrasonic P, S1 & S2 acoustic stack are, km/s in the X direction, km/s in the Y direction and km/s in the Z direction. 70

93 Z platen Z platen L/2 σz X platen Y platen σz L/2 ΔL x = 40 mm ΔL y = 40 mm L/2 = 40 mm L/2 = 40 mm Figure 4.25: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample along with part of the platen touching the sample at stress state. The dashed line shows the contour interface between the maximum stress and the next lower stress zone. The P wave velocity survey at the following stereonet representation: is also displayed in Y Z X Figure 4.26: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocity of the 2 rays in the X direction is m/s. The average velocity of the 3 rays in the Y direction is m/s. The average velocity of the 2 rays in the Z direction is m/s. The P wave velocity measurements that are available at stress state from the ultrasonic P, S1 & S2 acoustic stack are km/s in the X direction, km/s in the Y direction and 5 km/s in the Z direction. 71

94 Z Z platen L/2 σz X platen Y platen σz L/2 ΔL x = 40 mm ΔL y = 40 mm L/2 = 40 mm L/2 = 40 mm Figure 4.27: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample along with part of the platen touching the sample at stress state. The directional evolution of P wave velocities between the two different stages of and is also displayed in the following stereonet representation from the transducer-to-transducer velocity survey data: Y Z X Figure 4.28: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocity of the 2 rays in the X direction is m/s. The average velocity of the 3 rays in the Y direction is m/s. The average velocity of the 2 rays in the Z direction is m/s The P wave velocity measurements that are available at stress state from the ultrasonic P, S1 & S2 acoustic stack are km/s in the X direction, km/s in the Y direction and km/s in the Z direction. 72

95 Z Z L/2 σz X platen Y platen σz L/2 ΔL x = 40 mm ΔL y = 40 mm L/2 = 40 mm L/2 = 40 mm Figure 4.29: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample along with part of the platen touching the sample at stress state. The P wave velocity survey at is also displayed in the following stereonet representation. The directional evolution of P wave velocities from the previous stage is also displayed: Y Z X Figure 4.30: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocity of the 2 rays in the X direction is m/s. The average velocity of the 2 rays in the X direction is m/s. The average velocity of the 3 rays in the Y direction is m/s. The average velocity of the 2 rays in the Z direction is m/s. The P wave velocity measurements that are available at stress state from the ultrasonic P, S1 & S2 acoustic stack are km/s in the X direction, km/s in the Y direction and km/s in the Z direction. 73

96 Z platen Z platen L/2 σz X platen Y platen σz L/2 ΔL x = 22 mm Y, Z = L/2 = 40 mm ΔL y = 20 mm X, Z = L/2 = 40 mm Figure 4.31: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample (Above) at stress state and, the corresponding graphs presenting the main principal stress value at the center of the rock along the X and Y directions. Y Z X Figure 4.32: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocities in the X, Y and Z directions are m/s, m/s, and m/s respectively. 74

97 Z platen Z platen L/2 σz X platen Y platen σz L/2 ΔL x = 17 mm Y, Z = L/2 = 40 mm ΔL y = 15 mm X, Z = L/2 = 40 mm Figure 4.33: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample (Above) at stress state and, the corresponding graphs presenting the main principal stress value at the center of the rock along the X and Y directions. Y Z X Figure 4.34: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocities in the X, Y and Z directions are m/s, m/s, and m/s respectively. 75

98 Z platen Z platen L/2 σz X platen Y platen σz L/2 ΔL x = 15 mm ΔL y = 13 mm Figure 4.35: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample (Above) at stress state and, the corresponding graphs presenting the main principal stress value at the center of the rock along the X and Y directions. Y Z X Figure 4.36: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocities in the X, Y and Z directions are m/s, m/s, and m/s respectively. 76

99 Z platen Z L/2 σz X platen Y platen σz L/2 ΔL x = 15 mm ΔL y = 13 mm Figure 4.37: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample (Above) at stress state and, the corresponding graphs presenting the main principal stress value at the center of the rock along the X and Y directions. Y Z X Figure 4.38: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocities in the X, Y and Z directions are m/s, m/s, and m/s respectively. 77

100 Z platen Z platen L/2 σz X platen Y platen σz L/2 ΔL x = 15 mm ΔL y = 13 mm Figure 4.39: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample along with part of the platen touching the sample (Above) at stress state and, the corresponding graphs presenting the main principal stress value at the center of the rock along the X and Y directions. There was no transducer-to-transducer P wave velocity survey available from this stage of the experiment. Therefore, for attaining P wave velocities at this stage, we calculated the average of P wave velocities between the previous and next stage of transducer-to-transducer surveys. The P wave velocity measurements that are available from the ultrasonic P, S1 & S2 acoustic stack are km/s in the X direction, km/s in the Y direction and km/s in the Z direction. 78

101 Z platen Z platen L/2 σz X platen Y platen σz L/2 ΔL x = 15 mm ΔL y = 13 mm Figure 4.40: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample (Above) at stress state and, the corresponding graphs presenting the main principal stress value at the center of the rock along the X and Y directions. Y Z X Figure 4.41: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocities in the X, Y and Z directions are m/s, m/s, and m/s respectively. 79

102 Z platen Z platen L/2 σz X platen Y platen σz L/2 ΔL x = 15 mm ΔL y = 13 mm Figure 4.42: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample (Above) at stress state and, the corresponding graphs presenting the main principal stress value at the center of the rock along the X and Y directions. Y Z X Figure 4.43: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocities in the X, Y and Z directions are m/s, m/s, and m/s respectively. 80

103 Z platen Z platen L/2 σz X platen Y platen σz L/2 ΔL x = 15 mm ΔL y = 13 mm Figure 4.44: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample (Above) at stress state and, the corresponding graphs presenting the main principal stress value at the center of the rock along the X and Y directions. Y Z X Figure 4.45: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocities in the X, Y and Z directions are m/s, m/s, and m/s respectively. 81

104 Z Z platen L/2 σz X platen Y platen σz L/2 ΔL x = 15 mm ΔL y = 13 mm Figure 4.46: Flac3D simulation of stress distribution in a quarter of the cross-section of the rock sample (Above) at stress state and, the corresponding graphs presenting the main principal stress value at the center of the rock along the X and Y directions. Y Z X Figure 4.47: Stereonet representation of apparent directional velocities of the bulk of the rock sample (Left) and stereonet representation of apparent directional evolution after the previous survey (Right) under the stress state. The average velocities in the X, Y and Z directions are m/s, m/s, and m/s respectively. There was no transducer-to-transducer velocity survey available from the exact moment of this last stage of the experiment at, but there was data available from a later survey where the Z principal stress is about 375 MPa. 82

105 4.7 Velocity Anisotropy Analysis Compaction Pseudo-Boundary Surfaces Unlike P wave velocities that would not seriously get affected by passing through the lower-stressed (LS) domains that may get damaged at later stages - containing voids and micro-cracks in a water saturated rock, the shear waves get affected because shear wave velocity of fluids are almost zero. King et al. [1997] measured shear wave velocities in sandstone specimens under polyaxial stress state and studied the evolution of the velocities as they changed the stress state. They divided their experiment into three different periods or cycles. In the first cycle, the rock specimen was fresh and uncracked during the application of an increasing hydrostatic state of stress. In the second cycle, called cracking cycle, a system of aligned cracks was formed in the rock specimen. During the third cycle, called crack-closing cycle, a further increasing hydrostatic state of stress was applied to close the cracks that were formed. Figure 4.48: Direction of propagation and polarization of the nine components of velocity with respect to the principal stress axes, and planes of aligned cracks produced (King et al., 1997). Figure (4.48) shows the direction of propagation and polarization of P and S wave velocities with respect to the principal stress axes in their experiment setup. Their indices are different from the indices employed in this thesis since the standard indices from Cartesian coordinate system were used in this work, where, (1=X=minimum stress 83

106 direction, 2=Y=intermediate stress direction, 3=Z =maximum stress direction); but, they determined the indices based on the magnitude of principal stresses where, (1=Z=maximum stress direction, 2=X=intermediate stress direction, 3=Y=minimum stress direction). They noticed, as was increased above 100 MPa during the cracking cycle, the magnitudes of became higher than and diverged from those of. King et al. [1997] argued that the reason for this behaviour probably lied in the inhomogeneous nature of the state of stress in the rock specimen at these high and stress levels. The propagation paths for and lied in the plane at the centre of the specimen, where confinement caused by the loading platens leading to the prevention of crack formation was at a maximum. The propagation paths for and, on the other hand, were in the direction and must pass through all the aligned cracks. This behaviour suggested that the aligned crack density towards the extremities of the specimen in the direction was higher than in the centre for values of greater than 100 MPa. They could not provide a more accurate explanation to this phenomenon. Figure 4.49: S wave velocities as a function of stress during the cracking cycle with al., 1997). (King et Sayers [2002] studied the anisotropic symmetry of sandstone under polyaxial experiment that was experimented by Chaudhry [1995]. Sayers [2002] mentioned it is clear that most micro-cracks induced by loading have normals approximately parallel to 84

107 the minimum stress direction as expected. Those cracks also strongly affected the S wave velocities,, and since those corresponded to the propagation or polarization direction being approximately perpendicular to the crack faces. He also noticed that the symmetry, necessary in an orthotropic medium was violated at high stress. He argued that this probably originated from the constraints imposed on the sample by the loading plates for propagation along the maximum stress direction. For this reason, he only used and to obtain the and elastic parameters in order to minimize the effect of constraint by the loading plates. However, we did not follow the above-mentioned simplifications and used an accurate solution to this problem. We used the concept of compaction pseudo-boundary surface due to heterogeneous distribution of stress as a result of the geometrical effects of edge chamfers and friction on the platen-rock surfaces. While in the TTT, σ 3 and σ 2 play the clamping role and σ 1 plays the compaction role, the compaction pseudoboundary criterion can be defined as boundaries within the rock specimen in the σ 3 and σ 2 directions separating the central compacted domain from the domains where less stress is induced to compress the pre-existing micro-cracks and voids. We estimated the pseudo-boundaries by the dashed lines in figures (4.25) to (4.46) which show the stress distribution contour interfaces between the maximum stress zones connected to the Z platen and the adjacent lower stress zones within the rock specimen. Except at the starting phase of stage B at stress state, all the next pseudo-boundaries in the next stages of the experiment are chosen approximately at the 50% criteria between the maximum stress value at the center and the minimum stress value at the two edges of the rock sample in the minimum and intermediate principal stress directions. Z dly dlx Y X Figure 4.50: Compaction pseudo-boundary criterion in minor and intermediate principal stress directions. 85

108 Following the FLAC3D simulations, figure (4.50) demonstrates that the compaction pseudo-boundary surface extends from the interior of the specimen toward the surfaces. There are 48,500 AE events located by the InSite software during the experiment reported by Nasseri et al., [2014]. They are plotted with green dots as illustrated in figures (4.51) and (4.52) along with the compaction pseudo-boundary criterion plotted with the dashed green line. These two data sets are plotted at certain distances from one side of the specimen ranging from 0 to 40 mm demonstrated on the right vertical axis of the plot, assuming symmetry on both sides for the boundary. Although there was no symmetry in AE event locations, a summation of these AE events on both sides is plotted. The side length of the square cross-section of the rectangular cubic volume passing through the center of the rock along both Y and X axes in which the AE events are picked to be plotted in green dots is 40 mm. Stage B Stage C Stage D Lower-Stressed Zone Effective Thickness in the Y direction Figure 4.51: Green dashed line: LS zone effective thickness in the Y direction. Blue line: Major stress. Red line: Cumulative number of AE events. Green dots: AE events plotted by distance from the Y surface. There is not much AE events located above the LS domain effective thickness line in the Y and X directions as shown by the green dots representing the AE events distances from the edges of the cubic sample in figures (4.51) and (4.52). It indicates that no events occur in the central domain of the rock within the compaction pseudoboundary surfaces and this confirms that the central domain of the rock stays intact 86

109 during stages B and C of the experiment. By the AE locations and the CT scan images, we see that the compaction pseudo-boundary surfaces during stage D of the experiment are certainly transformed into dilatancy pseudo-boundary surfaces which lie between the central intact domain of the rock and the domain indicating a higher stage of damage near facets of the specimen. Stage B Stage C Stage D Lower-Stressed Zone Effective Thickness in the X direction Figure 4.52: Green dashed line: LS zone effective thickness in the X direction. Blue line: Major stress. Red line: Cumulative number of AE events. Green dots: AE events plotted by distance from the X surface Shear Wave Velocities in the Minimum and Intermediate Principal Stress Directions The equations that explain the relationship between the S1 and S2 velocities and the effective thicknesses of the LS zones in the X and Y directions are given below. On the XoZ plane, considering that in the TTT, the minor and intermediate principal stresses play the clamping role while the major principal stress plays the compaction role, would be consistent along the Z direction and therefore,. 87

110 (4.16) Where, is the shear wave velocity through the lower-stressed domains near the two opposite X-facets in the X direction and polarized in the Z direction (XoZ plane) and, is its slowness. is the shear wave velocity in the center of the rock, where the maximum principal stress in the Z direction is approximately uniform and, is the corresponding slowness. The above equation is written based on the credible assumption that within the central compacted domain of the rock sample, the and which is the measured shear wave velocity propagating in the Z direction and polarized in the X direction are equal due to symmetrical conditions. The results are shown in the figures below. 3D Strain data is also incorporated in the calculations but they are too small and their contribution to the results is negligible. The figure below shows the evolution of the shear wave velocity propagating in the minimum principal stress direction and polarized in the maximum principal stress direction under different loading stages during stages A, B and C of the experiment. Figure 4.53: Measured shear wave propagating in the X direction and polarized in the Z direction along with the magnitudes in the compact central domain, and in the near-facet or lowerstressed domain in the specimen. 88

111 As shown above, all results depend on the measured values of shear waves that propagate in the maximum principal stress direction. For the shear wave velocity propagating in the intermediate principal stress direction and polarized in the maximum principal stress direction in the YoZ plane, knowing that, (4.17) Where, is the shear wave velocity through the lower-stressed domains near the two opposite Y-facets in the Y direction and polarized in the Z direction (YoZ plane). And, is the shear wave velocity in the center of the rock, where the maximum principal stress in the Z direction is approximately uniform. The above equation is written based on the reliable assumption that within the central compacted domain of the rock sample, and are equal due to symmetrical conditions. Figure 4.54: Measured shear wave propagating in the Y direction and polarized in the Z direction along with the magnitudes in the compact central domain, and in the near-facet or lowerstressed domain in the specimen. Comparing the shear wave velocities shown above in the XoZ and YoZ planes, the contrast between the central compacted (semi-uniform-stressed) and lower-stressed domains for shear waves propagating in the minimum principal stress direction is larger than the waves propagating in the intermediate principal stress direction. 89

112 Figure 4.55: Velocity contrast at the minimum principal stress direction compaction pseudo-boundary (red) and at the intermediate principal stress direction compaction pseudo-boundary (dark blue). The difference between the shear wave velocities in the XoZ and YoZ planes and the velocity contrasts at the effective thicknesses ( and ) or compaction boundaries are plotted at each stage of the experiment in figure (4.55). However, unlike the XoZ and YoZ planes, in the XoY plane both of the S2 waves propagating in the direction of minimum and intermediate principal stresses are passing through lower-stressed domains and they contain no measurable velocity similar to shear wave velocity propagating in the Z direction among them to be used as a comparable reference. Therefore, in order to solve for near the two opposite X-facets, as well as near the two opposite Y-facets and also or in the center of the rock, we need to follow a different approach. Fortunately, at this point, we have an estimation of the values of and in different domains at various stages of the experiment. As mentioned before, looking at the measured velocities in figure (4.7), similar patterns of fluctuation in shear wave velocities with the same directions of propagation are recognizable for all the three orthogonal directions. Thus, on one side, effective thicknesses and on the other side, the fraction of shear wave velocities in the same direction of propagation between the lower-stressed domains and uniform compacted central domains in both X and Y directions are the two aspects that lead us to obtaining the shear wave velocities in the XoY plane. First, having the effective thicknesses ( and ) we can write: 90

113 (4.18) Considering the anisotropic symmetry rules in the compacted center of the rock where,, we can eliminate the central velocity from the system of equations above. (4.19) The second above-mentioned useful a priori information here is the value of the other two shear waves propagating in the X and Y directions but polarized in the Z direction in both lower-stressed and central compacted domains. Assuming that the lower-stressed domain inclusions are approximately disc-shaped micro-cracks indicates similar change in shear wave velocities propagating in either of the X or Y directions, once passing through the compaction pseudo-boundaries. Therefore, we presume similar fraction of shear wave velocities between the lower-stressed domain and the central compacted domain for shear waves propagating in the same direction as is formulated below. (4.20) Again, taking advantage of the reciprocal equality condition ( the system of equations: ) simplifies (4.21) Combining the above equations (4.19) and (4.21), we can calculate the following results for near the X-platens, near the Y-platens, considering the constraint or in the center of the rock as a homogeneous elastic medium: 91

114 (4.22) Hence, one of the three above-mentioned velocities, obtained as below. near the two Y-platens is (4.23) Then, we can easily calculate near the two X-platens. (4.24) And finally, in the central compacted domain can be acquired. (4.25) Nevertheless, implementing the above assumptions and formulas provide numerical results for near the X-platens, near the Y-platens, and in the center of the rock, which lack integrity and compatibility with the experiment conditions. The proper results are not obtained by the above equations, specifically as a consequence of the velocity symmetry rule constraint for the central compacted domain applied on the equation, which causes great errors in the results calculated from the velocity measurement data. Therefore, the above equations and 92

115 approaches don t offer a robust solution for shear wave velocities ( and ) in the XoY plane. However, once we formulate the equations for and velocities separately, based only on different central and lower-stressed domain thicknesses without applying the velocity symmetry rule as a constraint in the central compacted domain, we will get robust results. (4.26) Combining the equations (4.18) and (4.26), we can derive the following equation relationship between the measured shear wave velocities and their values in the central compacted domain. (4.27) Therefore, a formula is obtained to calculate shear wave velocities in the central compacted domain. (4.28) Once we get the values for and, we can calculate the values of near the X-platens, as well as near the Y-platens through equation (4.25). The figure below shows the shear wave velocities in the XoY plane. It demonstrates that although the velocity symmetry rule constraint for the central compacted domain is not applied in the calculations, they are however, almost equal in the results. 93

116 Figure 4.56: Shear wave velocities in the central compacted zone and each pair of the near X- and Y- facets propagating in the XoY plane. The new obtained velocities for central and velocities have a significant difference with the measured and velocities. Comparing their average in the figure (4.57) can see this difference. Figure 4.57: Average of the measured and shear wave velocities compared to the average of their calculated values in the central compacted domain Compressional Wave Velocities To this point, by dividing the experiment into four stages A, B, C and D, we investigated the velocity evolution in the first three stages. In stage A, where the poroelastic deformation is happening, we just assumed that all parts of the rock are approximately squeezing together homogeneously. In stages B and C, which are at the 94

117 center of our attention and where we claim our analysis to be more accurate, we applied the effective thickness of the lower-stressed domain near the X and Y facets of the cubic rock to calculate the shear wave velocities in lower-stressed near-facets as well as central compacted domains. On the other hand, we observed that P wave velocities evolve with stress too which is demonstrated in the following figure. As shown in the figure below, the strain of the rock is also considered in the demonstration of the measured P wave velocities although it has an ignorable effect. Figure 4.58: Measured P wave velocities along the three principal stress directions from P-S1-S2 surveys. Aside from errors in travel-time measurement, migration of water and healing are the two reasons that cause fluctuations in the measured values of the wave velocities propagating in the same direction along different stages of the experiment. Delay in migration of water through micro-cracks is a consequence of changing the stress and deforming the rock within a relatively short time. This effect will cause some local pressure rise at points where micro-cracks are closed while some other dilated microcracks will be left unsaturated for a period of time. Therefore, mechanical properties of the rock will locally change with time and affect the wave velocity. At some parts of the experiment, the loads are kept constant for the purpose of permeability measurements. Therefore, there is sometimes a pause on a specific stress state for a while during the experiment when more than one survey is performed at the same state of stress. During this pause, the voids and micro-cracks have some time to readjust and heal. This phenomenon results in more micro-crack closures and increases the stiffness of the rock. 95

118 The figure below demonstrates the average P wave velocities propagating in each of the three principal stress directions from transducer-to-transducer surveys. An example for healing points can be shown in the figure below, where two different velocity surveys are performed at a constant principal stress of 300 MPa. Healing points Figure 4.59: Average P wave velocities propagating in each of the three principal stress directions from transducer-to-transducer surveys. The compressional waves pass through water; however, P wave velocities still get affected by the saturated voids and micro-cracks in the rock. As the sound wave speed in water is about 1.5 km/s, which is different from the background rock matrix velocity, anisotropy is expected to be observed as reported in measured P wave velocities due to anisotropic orientation distribution of voids in the rock under polyaxial stress state. Therefore, we also try to formulate equations and solve for and velocity values separately based on different central and lower-stressed domains thicknesses and contrasts. The difference for shear waves between the lower stressed zone and the compacted zone propagating along the X axis can be written as such: (4.29) We know that velocity difference between the two different domains is due to existence of the voids and micro-cracks. However, these voids will affect the P wave and 96

119 S wave velocities differently because P waves can pass through the water content of the voids, although with smaller velocities than the matrix of the rock, but S waves don t transfer through water at all. Consequently, the effect of the voids on the P wave is relatively less than that of the S wave, and could be considered as a proportion of the effect of voids on the S wave velocity contrast. This proportionality is shown by the empirically derived proportional coefficients and in the X and Y directions, respectively. We can estimate this proportionality for the P wave propagating along the X axis with a linear relationship between the shear wave fraction and the compressional wave fraction as below: (4.30) The bigger the crack density, the bigger the difference under the effect of watersaturated voids, hence, the bigger the proportional coefficient or should be. In the case of no voids in the lower stressed zone, the proportional coefficient is equal to zero, and in the hypothetical case that the whole volume in the lower stressed zone is filled with water, the proportional coefficient goes toward. This proportion depends mainly on the crack density and their saturation level in the lower-stressed domain. This means, with no cracks existing in the lower-stressed domain, not only there will be no difference between the shear wave velocities in all domains along the wave propagation path, but also there will be no difference between the P wave velocities. However, in the case of the whole volume in the lower stressed zone being filled with water, the shear wave velocity in the lower stressed zone will be equal to zero while the corresponding P wave velocity will be equal to the speed of sound in water. Therefore, the proportional P wave velocities along the minimum and intermediate principal stress directions are, (4.31) write, In order to derive velocities in different domains from the measured values, we can 97

120 (4.32) Hence, the P wave velocities in the central compacted domain of the rock propagating in the X and Y directions shown as ( and ) can be calculated from the equation below. (4.33) In order to choose the right empirical values for and coefficients, an upper limit also needs to be considered, where the resulting P wave velocities in the central compacted domain of the rock in the three principal directions follows the condition,. As mentioned earlier, the bigger the crack density, the smaller the proportional coefficients and should be. The crack density in the lower-stressed domain near X- facets of the rock is bigger than that of the Y-facets. Also, it has to be considered that the voids in both lower-stressed domains are only pre-existing micro-cracks that are just not completely closed compared to the central compacted domain, thus, their crack density absolute values are very small. Therefore, the empirical average for proportional coefficients ( and ) which are close to zero due to small values of crack densities and satisfy the condition are acceptable. The calculated P wave velocities in the X and Y directions in the central part and near the facets are obtained and plotted as below. 98

121 Velocity (Km/s) Velocity (km/s) Velocity (km/s) 5.6 P wave velocity V11 in the X direction Measured Velocity V11 Central V11 V11 on X-facet σz = Z-direction Stress (MPa) Figure 4.60: Measured and calculated P wave velocities propagating in the minimum principal stress direction in different domains. As can be seen, the P wave velocities in the lower-stressed zones decay with increase in the effective stress. 5.6 P wave velocity V22 in the Y direction Measured Velocity V22 Central V22 V22 on Y-facet σz = Z-direction Stress (MPa) Figure 4.61: Measured and calculated P wave velocities propagating in the intermediate principal stress direction in different domains P wave velocity contrasts near X and Y facets V11 contrast near X-facet V22 contrast near Y facet σz = Z-direction Stress (MPa) Figure 4.62: P wave velocity contrasts at the compaction pseudo-boundaries in the X and Y directions. 99

122 The P wave velocity contrasts at the compaction pseudo-boundaries in the X and Y directions are calculated and presented in figure (4.62). The results show that the P wave velocity contrasts at the X-facet compaction pseudo-boundaries increase more rapidly than the contrasts near the Y-facets Velocity Domains of the Cubic Rock and Diagonal Stiffness Tensor Parameters Figure (4.63) shows a schematic view of the rock with its undisturbed central compacted domain and four lower-stressed domains near the X and Y facets along with resolved shear and compressional wave velocity values under polyaxial stress state. In such a standard setting of the polyaxial setup, an interesting result is that the central compacted domain of the rock gets almost equal values for those shear wave velocities that are either propagating or are polarized in the minimum principal stress direction. However, it is not yet similar to the transversely isotropic symmetry because in the TI anisotropy, the compressional wave velocities in the minimum and intermediate principal stress directions are equal, unlike the case in the central compacted domain of our experiment. As seen before in the Orthorhombic Anisotropy section, the diagonal stiffness tensor components can be obtained by solving the Cristoffel equation: ρ ρ ρ ρ ρ (4.34) ρ ρ ρ ρ Diagonal stiffness tensor coefficients for the central compacted domain of the rock with a rock density of 2.49 (gr/cc) is shown in figure (4.64). For calculating the coefficient, the average of S1 shear wave velocities in the XoY plane is counted in. 100

123 Velocity (Km/s) Velocity (Km/s) Velocity (Km/s) Measured P-wave velocities in 3 principal directions VPx=V Vpy=V VPz=V σz = Z-direction Stress (MPa) Figure 4.63: A schematic view of the rock with resolved shear and compressional wave velocities X Z Y P-wave velocity V22 in the lower-stressed zone near the Y-facet V σz = Z-direction Stress (MPa) P-wave velocity V11 in the lower-stressed zone near the X-facet V σz = Z-direction Stress (MPa) 101

124 Stiffness factors (GPa) 85 Diagonal Stiffness Tensor Coefficients C11 C22 C33 C44 C55 C σz = Z-direction Stress (MPa) Figure 4.64: Diagonal stiffness tensor coefficients for the central compacted domain of the rock specimen. 4.8 Off-Diagonal Stiffness Tensor Parameters There are still three off-diagonal coefficients of the orthorhombic stiffness matrix that remain to be solved. Cheadle et al. [1991] have solved them for orthorhombic media. However, there are two other approaches for calculation of the off-diagonal coefficients based on ellipsoidal anisotropy symmetry. These are to be tested due to our purpose of theoretical investigation for the symmetry model of a material under triaxial stress state, which suggests ellipsoidal anisotropy as an appropriate background symmetric model for tomography. The first approach known as Rudzki s ellipticity conditions for ellipsoidal anisotropy (Helbig, 1983 and, Rasolofosaon, 1998) is based upon the assumption that the magnitude of the components of the stress deviator is small compared to the wave moduli (eg. stiffness coefficient). Then, it elaborates on the strain tensor equation with second-order elastic (SOE) and third-order elastic (TOE) tensor. It derives a relationship between the stiffness coefficients and the three principal stresses. The second approach derived by Daley et al. [2006] constructed eikonal equations using the standard linearized approximation of the phase velocity for quasi- 102

125 compressional (qp) wave propagation in a weakly anisotropic orthorhombic medium. Both methods suggest a degenerate (ellipsoidal) case of qp wave propagation in an orthorhombic medium with an ellipsoidal slowness surface. They derive equations to obtain the three off-diagonal stiffness coefficients from the other six independent diagonal stiffness matrix components. Here, we examine all three methods and compare their results starting with Rudzki s condition for ellipsoidal anisotropy, which is the simplest method Rudzki s Ellipticity Condition For many researchers who studied the effect of stress on the elastic properties of solids, the magnitude of the stress was assumed small compared to the elastic moduli. Using a perturbation theory, Dahlen [1972] demonstrated that the stress-induced P wave anisotropy is only of the second order in the ratio of the stress magnitude to the elastic moduli. Nikitin and Chesnokov [1984] showed that this is true only if the unstressed medium is isotropic. Furthermore, these authors showed that an initially isotropic medium is necessarily of higher symmetry than orthorhombic when triaxially stressed. Rasolofosaon [1998] mentioned that since the P wave slowness surface is always an ellipsoid, then the section of this surface by the coordinate planes are ellipses. Thus, under the specific situation where the applied stress on an initially isotropic medium is small compared to the elastic moduli, Rudzki s 3 ellipticity conditions must be verified (Helbig, 1994) by the stressed elastic constants: (4.35) Using Rudzki s ellipticity conditions, the off-diagonal stiffness tensor components (,, ) are calculated below: (4.36) 103

126 Stiffness coefficients (GPa) The off-diagonal stiffness coefficients calculated from Rudzki s ellipticity conditions are plotted for different states of stress during our experiment illustrated in the figure below Nondiagonal Stiffness Tensor Coefficients σz = Z-direction Stress (MPa) C23=C32 C13=C31 C12=C Figure 4.65: Off-diagonal stiffness coefficients calculated from Rudzki s ellipticity conditions. Equation (3.18) implies that initially isotropic media, when triaxially stressed, form a well-defined sub-set of orthorhombic media, which are not characterized by nine independent elastic coefficients, as conventional orthorhombic media, but by only six independent coefficients. Yet, Rasolofosaon [1998] mentioned that these three relations have different forms by different authors and do not seem to have a simple physical interpretation Cheadle s Method for Orthorhombic Anisotropy In order to determine the three independent off-diagonal stiffness coefficients in an orthorhombic medium, one must measure velocities for ray-paths along many different directions. Hornby [1998] discussed methods for the measurement of ultrasonic P and S wave phase velocities on fluid saturated shale samples under drained conditions as a function of confining pressure. In order to reconstruct the elastic parameters using P and S wave velocity measurements, he argued that effort must be made to use as much data as possible on core samples. In particular, he suggested that the critical elastic parameter (or Thomsen's δ) deserves special attention. He stated that a minimum of two measurements off-axis must be used in the reconstruction of this parameter. The 104

127 use of a single measurement of P wave velocity off-axis in the reconstruction of elastic constants can lead to unacceptable and unquantifiable errors in the reconstruction of the critical parameter. Cheadle et al. [1991], who experimented on phenolic cubic samples to study their orthorhombic anisotropy, managed to derive all the elastic tensor coefficients. They mentioned that after the main principal stress directions, the next simplest directions to consider would seem to be those in principal planes at 45 degrees to each of the two principal directions. The expressions for three off-diagonal stiffness tensor coefficients are derived in the Appendix of the article by Cheadle et al. [1991] as shown below: (4.37) (4.38) (4.39) Where, is density of the rock; with (i=1,2,3) representing X, Y and Z directions, respectively, are directional cosines for each ray-path. And,, and are three P wave velocities for ray-paths at 45 degrees to the symmetry directions shown in the figure below. Figure 4.66: Transmission measurements between opposing edges of the phenolic cube. The propagation directions were at 45 degrees to two of the principal axes and perpendicular to the third (Cheadle et al., 1991). 105

128 Our experimental setup allows us to analyze P wave velocities with specific raypaths from the transducer-to-transducer surveys that are implemented separately around the same time and the same stress state as the P-S1-S2 ultrasonic surveys are done. Four appropriate ray-paths were chosen and used for this purpose with angles either equal to or close to 45 degrees from the principal axes of the planes they were lying in. Two of the rays were lying in the YoZ plane while, the other two rays were lying in the XoY and the XoZ planes, respectively. The figure below shows these four rays along with the specific transducers between which they are traveling. Figure 4.67: Ray-paths chosen for calculation of off-diagonal stiffness tensor coefficients by Cheadle s method. On the XoY plane, The two directional cosine values for the ray (r28) traveling between transducers 3 on the X2 platen and 6 on the X1 platen are and, respectively. That means, the two angles between the ray and the X and Y axes are and degrees, respectively. On the XoZ plane, both of the directional cosine values for the ray (r48) traveling between transducers 4 on the X2 platen and 16 on the Z2 platen are That means, both of the angles between the ray traveling through transducers 4 to 16 in the XoZ plane with X and Z axes are 45 degrees, respectively. On the YoZ plane, the two directional cosine values for the first ray (r105) 106

129 Velocity (m/s) traveling between transducers 14 on the Z1 platen and 17 on the Z2 platen are and , respectively. The two angles between the ray and the Y and Z axes are and degrees, respectively. As for the second ray (r100) on the YoZ plane, which is traveling between transducers 12 on the Y2 platen and 18 on the Z2 platen, both of the directional cosine values are That means both of the angles between the ray and the Y and Z axes are 45 degrees. The figure below shows the P wave velocities along each of those four ray-paths from transducer-to-transducer surveys during the experiment that are used in calculation of off-diagonal stiffness tensor coefficients by Cheadle s method Evolution of P-wave velocities for four rays r28 r r100 r σz = Z-direction Stress (MPa) Figure 4.68: Velocity evolution of the four ray-paths that are used in calculation of off-diagonal stiffness tensor coefficients by Cheadle s method. r28 lies on the XoY plane with lowest velocity, r48 lies on the XoZ plane, r105 is the first ray mentioned in the text that lies on the YoZ plane and r100 is the second ray on the YoZ plane. It has to be reminded that we are calculating the off-diagonal stiffness matrix coefficients for the central compacted domain of the rock. However, the fourabovementioned rays are also passing through the lower-stressed domains and get affected by the lower velocities within derived effective thicknesses along their path. 107

130 Stiffness coefficients (GPa) Therefore, we managed to only incorporate velocity of these rays in the central compacted domain as shown below to be used in Cheadle s equations. As an example, ray 48 kicks off from transducer 4 on the X2 platen surface, passes through the sample s x-facet effective thickness with an angle and then, passes through the central part of the specimen until it arrives to transducer 16 on the Z2 platen surface. Hence, the only domain that affects this particular ray s velocity is the lower-stressed domain near the x-facet with thickness equal to. Consequently, the length of the ray that passes through this domain is, (4.40) We have already solved the P wave velocity in the X direction near the x-facet. Assuming that the total travel-time is equal to the addition of two travel-times of the ray through the LS and central domain, respectively, we can write the following equation, (4.41) And then, we solve for the velocity of the ray in the central part of the rock in each different stage of the experiment: (4.42) 1-4 Nondiagonal Stiffness Tensor Coefficients σz = Z-direction Stress (MPa) C12=C21 C13=C31 C23=C32 Cheadle's C Figure 4.69: Cheadle s resolved stiffness tensor coefficient along with those coefficients obtained from Rudzki s equations. 108

131 Figure (4.69) demonstrates the calculation result for only one of the off-diagonal stiffness tensor coefficients as the other two could not be resolved by Cheadle s equations. Due to numerical constraints, only one off-diagonal stiffness tensor coefficient ( ) could be obtained by Cheadle s equation for orthorhombic anisotropy. Only the acoustic emission ray-path, which traveled in the XoZ plane by 45 degree angle with both of the X and Z axes was resolved. This means, although the resolved values are not too off from Rudzki s condition, however, the equations based on purely orthorhombic anisotropy are not suitable to deal with elastic properties of the rock under our experimental setup with polyaxial stress state Daley s Method for Inversion of Anellipsoidal Parameters We already know that group and phase velocities are equal in the principal directions. The diagonal elastic constants are already determined directly from the P and S waves measurements along the principal axes (equation 3.16). To determine the off-diagonal stiffness coefficients by Daley s method (Daley and Kebes, 2006), as discussed earlier in this report in section 3.6.2, the linear relation between the P wave group velocity and anellipsoidal deviation terms is elaborated here in the following equation (Mahmoudian et al., 2014). Knowing the values of, the redundant measurements of P wave velocity becomes the data vector in a least-squares inversion to estimate the values. The P wave measurements from the transmission experiments can be inverted for the three unknowns using linear equation (3.24) by defining the coefficients D, B, F, L as follows, (4.43) 109

132 Hence, equation (3.24) can be written in the new form below, (4.44) Incorporating n data points for P wave velocity measurements as the data ( ), the linear equation (4.43) can be used to express a linear system of equations with three unknowns : (4.45) If we define the BFL matrix as below, (4.46) The inversion solution to the anellipsoidal deviation vector is, (4.47) Thus, having the anellipsoidal deviation vector, the three non-diagonal matrix coefficients can be obtained through equation (4.43) as below, (4.48) Figure 4.70: Thirty two ray-paths with lengths larger than 80 mm that were used for the inversion of the anellipsoidal deviation vector. 110

133 Only ray-paths with lengths larger than 80 mm were selected as P wave velocity measurements for this inversion. The 32 ray-paths that met these criteria out of the total 106 available rays are shown in figure (4.70). The results have been tested with two different inversion methods including the simple matrix inversion as well as the Singular Value Decomposition method. Both of the methods show similar results. As lower-stressed domains near x-facets and y-facets affect the P wave velocity, we also corrected the ray-path velocities to obtain their velocities in the central compacted domain of the rock. Velocities of the ray-paths, which kicked off or arrived at either X or Y platens, were corrected considering the distance through which they passed to the lower-stressed domains as below, (4.49) (4.50) Velocity of rays in the central domain can be achieved by the equation below, where is different for near X or Y facets. (4.51) The results for the non-diagonal stiffness matrix components of the central compacted domain by incorporating the velocities from equation (4.50) are shown below: Figure 4.71: Non-diagonal stiffness matrix components resolved by Daley s method. 111

134 In comparison, we also solved for the non-diagonal stiffness components, not with the central P wave velocities, but with measured P wave velocities of those 32 rays. We see the results in the figure below that the values are mixed with each other and significantly differ from the solution with corrected velocities. Figure 4.72: Non-diagonal stiffness matrix components resolved by Daley s method using the uncorrected measured P wave velocities. 4.9 Strength of Anisotropy with Thomsen Parameters Since the strength of anisotropy is hidden in the elastic constants, calculation of the Thomsen anisotropy parameters from the elastic constants is helpful for understanding this strength. The dimensionless anisotropic coefficients conveniently characterize the magnitude of anisotropy and represent a natural tool for developing weak-anisotropy approximations outside the symmetry planes of orthorhombic media (Tsvankin, 1997) and hence, in ellipsoidal media. The parameters are all zero for an isotropic medium and their deviation from zero represents the degree of anisotropy. Sarkar et al. [2003] applied non-hydrostatic stress to an initially transversely isotropic solid with vertical symmetry axis, which resulted in an effective medium having almost orthorhombic symmetry. They emphasized that the stiffness tensor of an intrinsically orthorhombic medium differs from the effective stiffness tensor of a stressinduced orthorhombic medium by virtue of the asymmetry in the latter. They explained that some theoretical predictions suggest that this asymmetry should be negligible and can be ignored, but their observed data showed considerable asymmetry. They explained 112

135 that the asymmetry in their data, that for stressed media, shear velocities (for propagation along the -axis and polarization along the -axis) and (for propagation along the -axis and polarization along the -axis) are unequal, resulting in an asymmetric stiffness tensor ( ). Therefore, Sarkar et al. [2003] stated that the anisotropic parameters defined for a symmetric orthorhombic tensor by Tsvankin [1997] are not strictly applicable to their data from a stress-induced orthorhombic medium, and they adopted approximations to Tsvankin s parameters using the fourth order elastic tensor components. However, in our data, where we have managed to solve the asymmetry problem, we can still use the upgraded form of Thomsen parameters by Tsvankin [1997]. As shown by Thomsen [1986], the parameters are less than 0.2 in magnitude for weak to moderate anisotropy. The Thomsen parameters for the central compacted domain of our Fontainebleau sandstone sample are plotted in the three figures below in three orthogonal planes of the rock following the equations in table (3.1). The resolved values of in three different orthogonal planes of symmetry presented in figures (4.73), (4.74), and (4.75) meet the above criteria for weak to moderate anisotropy. Figure 4.73: Tsvankin s version of Thomsen parameters for orthorhombic media in the YoZ plane ( ). Shakeel and King [1998] calculated the Thomsen parameters of Penrith sandstone cubic samples under polyaxial stress state. Although they used Thomsen s equations for transversely isotropic media to calculate the three traditional Thomsen parameters 113

136 for their polyaxially stressed cubic rock, however, they gained useful insights about the interpretation of these parameters. Going through three stages of increase in hydrostatic compressional stress, cracking, and crack closing cycles during their experiment, Shakeel and King [1998] remarked that cracks close very quickly during the initial loading, resulting in a consequent rapid decrease in the value of the anisotropy parameters. When the stress is increased further, a major fraction of the crack surface area comes into close contact, which slows down the closure of cracks and the anisotropy parameters decrease much more slowly than before. They added that the higher rate of increase of Thomsen anisotropy parameters for stresses would be due to the nucleation and coalescence of the majority of the aligned cracks. Figure 4.74: Tsvankin s version of Thomsen parameters for orthorhombic media in the XoZ plane ( ). Figure 4.75: Tsvankin s version of Thomsen parameters for orthorhombic media in the XoY plane ( ). 114

137 At each stress level indicates that the anisotropy in S wave velocities is greater and more sensitive to the crack density than the anisotropy in P wave velocities. Elliptical anisotropy is only possible in the weak anisotropic region (anisotropy parameters < 0.2), which occurs only at low crack densities. In our experiment, there are instances especially in the YoZ plane that the anisotropy becomes elliptical ( = δ) for stresses at Discussion and Conclusions Computational codes were developed in Matlab as a ray-based anisotropic tomography tool to implement any set of transducer distribution and any 3D model space resolution of cubic samples in a true-triaxial test. In the 3D anisotropic inversion, there were supposed to be many more unknown parameters than in the isotropic case, and therefore many more data points would have been needed to obtain reliable results. However, the ellipsoidal anisotropic tomography applied on real data used in this work gave reasonable results as it reduced the number of unknown parameters. Singular Value Decomposition inversion method calculation time in Matlab was relatively fast (an order of tenth of a second for a single inversion procedure). Therefore, different resolutions and sensor arrays could be implemented in our tomography algorithm in Matlab to be examined. Tomographic images were generated by inverting the apparent P wave velocities that were displayed by stereonet images. As mentioned earlier, thin liquid-filled cracks have little effect on P wave propagation and the isotropic P wave velocity models. Stereonets showed velocity drop in the minimum principal stress direction at stage D due to the opening of new unsaturated fractures. Also, primary AE locations as well as CT scan images verified that fractures were spread with surface normal vectors perpendicular to the minimum principal stress direction and located adjacent to the two X facets of the rock. Monitoring the evolution of the rock during an experiment depends on rock porosity and whether the sample is dry or saturated with water. With saturated rocks, it was harder to detect the heterogeneous zones and therefore, only images obtained from stage D of the experiment were determined to be reliable. During the last stage of the experiment (Stage D), dilation expands and coalescence of micro-cracks happens on 115

138 the boundary between the highest stress and velocity gradients in the rock specimen. That boundary is a consequence of two reasons combined together. One is the geometry and specific conditions of the rock specimen including edge chamfers and friction on the platen-rock surfaces. The second reason is the true-triaxial stress pattern, which acts as a constraint to the fracture developments within the rock, and hence, delays the dilation of the rock to higher stresses. The anomalies obtained in velocity images of the rock at the last stages of the experiment show the opening of new cracks that are still unsaturated or partially saturated. The heterogeneity recovery percentage criteria in our method signified that heterogeneities larger than 5% can be detected by our tomography method in case of a noise free data. However, due to saturated pore spaces and lack of ray coverage the tomographic images reconstructed incomplete anomalies with uncertain values. Therefore, a solution was developed based on S wave velocity measurements to image the rock specimen. Prof. King s research group conducted a set of polyaxial experiments more than a decade ago (Chaudhry, 1995; King et al., 1997; Shakeel and King, 1998; King, 2002; Pettitt and King, 2004). In order to reduce frictional effects between the rock specimen and the platens, five milimeter thick magnesium metal plates, matching approximately the elastic properties of the sandstone specimen, were used between each of the six faces of the rock specimen and the loading platens. That resulted in obtaining almost equal shear wave velocities between each symmetric pair of mutually perpendicular direction of propagation and polarization. In another work, Popp & Salzer [2007] sprayed graphite on the end faces of the sample cubes before the experiments to minimize friction on the rock-platen interface. In another attempt within the experiments done by Prof. King and Prof Young s groups, ceramic was being used as the building material for platens but, beside the fact that it was not appropriate for resistivity measurements, it also cracked under high pressures. Therefore, these features were not used in our experimental setup as it would have affected the AE waveform recordings as well as the permeability measurement results and they may not have functioned well under high stress loads while the steel platen built in the RFDF polyaxial system functioned well enough under high stresses up to 500 MPa. Also, the main purpose of using the steel platens was that, steel is a good compliance match for crystalline rocks such as granite. 116

139 Hence, as well as many other standard polyaxial experiments practiced all over the world (Mogi-type, etc), there are indeed specific restrictions in this category of laboratory rock mechanics experiment that are also mentioned as loading and boundary conditions in Shi et al. [2012] and Nasseri et al. [2014]. The main problem that we managed to solve in this chapter was the asymmetry and unequal shear wave velocities between each pair of mutually perpendicular direction of propagation and polarization due to a complicated velocity structure induced into the rock. The laboratory conditions did not allow us to embed any stress or strain gauge or any other sensor to measure the crucial parameters inside the rock and we can only monitor the rock from outside and take remote measurements. Therefore, we estimated anisotropic properties of the central homogeneous part of the cubic rock specimen by a combination of stress distribution modeling and analysis of compressional and two perpendicularly polarized shear wave velocity measurements in each of the principal stress directions as well as P wave velocity measurements from the transducer-to-transducer surveys. One could initially deduce different interpretations about the scenario of velocity structure in the rock specimen under polyaxial stress state. The platens are made up of steel with Young s modulus of around 200 GPa while the sandstone has Young s modulus of around 1-20 GPa. Therefore, the platen is much stiffer than the rock. Control of the displacement of the platens affects the strain field along the axis of the corresponding direction (Z axis) and causes the rock to have different compressions or dilations along the associated principal directions depended on the applied stress. In case of the compression stress, the rock has to be blocked at its surface contacting with the fixed stiff platens and no displacement should be allowed, hence maximum stress should be locally applied whereas the stress should be at its minimum level at the center of the rock. As a result, we may simplistically conclude that the rock is relatively stiffer at its surfaces in direction of the applied compressive stress in each of the principal stress directions and more relaxed in the center. However, unlike a uniaxial stress pattern, in a true-triaxial stress pattern there are two other principal stresses applied simultaneously perpendicular to the main principal stress that cause compactions, clamps, frictions and torsions that affect the overall complicated stress distribution along the two axes of the corresponding directions. FLAC3D simulations demonstrated that the cubic specimen was not stiffer at its surfaces in direction of the applied 117

140 compressive stresses and the central part was compacted, and hence, stiffer due to the large maximum principal stress before the cracking stage. Therefore, under the specific true-triaxial or ployaxial state of stress, where the intermediate and minimum principal stress are greater than some minimum criteria which is yet unknown to us, the stresses are coupled with each other in the way their effect on the rock is different from three perpendicular uniaxial stress states, and makes the stress and strain field more complicated in the bulk of the rock which needs to be studied in more detail. Similar to the observations by Popp and Salzer [2007], the observed differences between and in our results suggest that S waves are more sensitive to opening or closure of flat micro-cracks due to localized stress concentrations, while in order for the P waves to be affected, progressive opening of fractures and damage is required. This observation corresponds to compaction as well as dilatancy concepts with which the interpretation of the coupled thermal, mechanical, and hydraulic processes in rocks undergoing various states of stresses get more straightforward. It is also observed that the compaction or dilatancy pseudo-boundary surfaces separate the non-dilatant compaction domain, where mechanical healing is possible from the dilatant domain where damage occurs at stresses significantly below the failure boundary. Depending on the state of stress, rock deformation was associated with an increase of permeability. Nasseri et al. [2014] showed the correlation between directional velocities and permeabilities within this experiment. Knowing the anisotropic heterogeneous velocity model of the rock, analysis of the anisotropic permeability data can also be improved to a more sophisticated level. 118

141 5 Effect of Velocity Structure on AE Source Locations 5.1 AE and Microseismic Event Location Identifying seismically active zones in rocks is of interest for a wide array of applications. Seismic activity is caused by rock failure, which, for example in mines, reduces the stability and thus the safety of excavations. In hydrocarbon reservoirs, cracks and fracturing can connect the pores of the reservoir rock, creating anisotropic permeability, which influences the extraction efficiency. In geothermal projects, fractures increase not only permeability but also the effective contact surface easing the heat transport from rock to the transfer fluid. Also, to prevent unwanted leakage, CO2 storage projects require that cracks do not propagate into the overburden. In all these cases, it is therefore desirable to have detailed knowledge of crack and fracture geometry, which in turn allows optimization of drilling strategies. Finally, on a more regional scale, these fractures and cracks can form (macro-scale) faults, which increase the hazard in volcanic and seismically active regions. Naturally, it is desirable to identify and study these active zones in-situ. Seismic methods arguably offer the best approach. Locating seismic events (i.e., ruptures) offers a rough guide on where these weak zones are (Wuestefeld, 2010). AE monitoring is a passive technique similar to earthquake monitoring but on a much smaller length scale. AE and microseismic monitoring techniques have been used in the field of material science as a non-destructive test for structural flaws and fatigue. If the source of the events can be located, the sites of potential weakness in the structure can be identified. Human activities in hydrocarbon reservoirs, mining areas, large water reservoirs or geothermal sites induce microseismicity. Long-term microseismic monitoring in hydrocarbon reservoirs has the potential to investigate the progress of fluid fronts during production (Maxwell and Urbancic, 2001). Particularly for the mining industry, long-term microseismic monitoring provides valuable input for safety and hazard assessments. The real time aspect of microseismic monitoring becomes predominantly important for application to hydraulic fracturing, which is associated with high microseismic activity. Real-time monitoring of hydraulic fracturing has become a more common practice since it provides immediate feedback of the progress of 119

142 fracturing and can help to image the extent and growth of fractures during this process (Oye and Roth, 2003). However, there are several obstacles to the practical use of the microseismic monitoring technique. First of all, the fracturing induced seismic events are generally weak and difficult to detect. Then, for the detected events, accurate picking of the first P or S wave arrival-times and reliably determining their arriving back-azimuths are sometimes difficult due to noise contamination. Finally, even with good readings, the location of microseismic events is still prone to error due to the lack of an accurate velocity model, for example, in gas/oil shale cases where strong VTI anisotropy (up to 30 percent) is commonly seen (Li et al., 2013). Therefore, obtaining the anisotropic heterogeneous velocity structure of the medium will improve the accuracy in AE or microseismic source locations. Here in this chapter, we first developed our AE source location for first arrival-times in homogeneous medium. Starting by synthetic models, we then applied a collapsing grid search location method on real travel-time data from transducer-to-transducer surveys at various stages of the experiment. Accuracy of the method is examined through calculating the standard deviation and average location uncertainties of the fixed shot transducers. Afterward, we move forward to developing the method to anisotropic homogeneous and anisotropic heterogeneous models and take the same steps to investigate the effect of velocity structure on AE source locations within the cubic rock sample in a polyaxial experiment. 5.2 Source Location Theory in an Isotropic Homogeneous Medium As mentioned above, AE source locations accuracy can be enhanced using a more realistic velocity structure of the medium. We use one of the simplest location methods using the first arrival-times of the P waves. The following location algorithms can also be tested by locating the transducers through the transducer-to-transducer survey results. We assume that an AE event occurs at an unknown time t, and at an unknown position of the source as. If the receiver transducers located at detect the AE events at real arrival-time data, this arrival-time as the observed or recorded data will depend on the origin time and the travel-time between the source and the station : 120

143 (5.1) By knowing the velocity structure, the synthetic arrival-times at the receivers (data) can be computed from an assumed model vector composed of the source location and origin time (5.2) For example, in an isotropic homogeneous velocity structure (v), the synthetic arrival-time data is (5.3) The purpose of an inverse problem is to find a model that the consequent synthetic arrival-times fit the real arrival-times better. In order to do this, we can begin with a starting model, which is an estimate of a model that we hope is close to the solution we seek. By changing the starting model, we try to find the fittest solution model: (5.4) However, if we don t have a starting model but only have the arrival-times at transducers, we need to eliminate the origin time from the calculations and we will approach the problem using a reference time which is the time that the wave takes to arrive to the closest transducer. Then, we incorporate as the time difference between the time of arrival to transducer i-th and the closest or reference one. (5.5) Considering transducers i and j, Expanding and simplifying, we get, (5.6) 121 (5.7)

144 In the above equation, are the four unknown parameters which need at least four set of independent equations to be solved. As the unknown variable itself includes the arrival-time information of a reference transducer, the reference transducer cannot be contributed as the i-th or j-th transducers involved in the equations. Therefore, at least six arrival-time data at different transducers, including the reference transducer, are needed to solve the inversion problem of the system of four equations. The formulation of the synthetic arrival-times using a starting model is not linear. In order to linearize the problem, the first order of Taylor series expansion is useful: (5.8) Consequently, the linearized inversion equation will be written as the difference between arrival-times of the best model and starting model where the synthetic arrivaltimes of the best model can be substituted by the real arrival-times: (5.9) The common form of an inversion equation is written as the transformation matrix (G) is a partial derivative matrix defined as below, where (5.10) The transformation matrix components for an isotropic homogeneous medium can be displayed as, (5.11) 122

145 As mentioned before, we need to have at least six arrival-time data at transducers to solve the inversion problem. However, using the grid search method, four arrival-time data suffice to calculate the four AE location parameters and therefore, it also enables us to investigate more AE events during the experiment Collapsing Grid Search By trying the regular matrix inversion methods, we did not get proper results. Since our goal was not to develop a fast and cost efficient event location, but to examine the effect of velocity structure on accuracy of the location results, we chose the simplest location method, which is grid search. The collapsing grid search method utilized in Matlab was not cost effective and it sometimes took several days to solve for a few thousand AE event locations, however, it provided accurate results. In a grid search, the test specimen is discretized or divided into a particular grid, and the travel-times from any point in the grid to each sensor are calculated. Since the number of grid points depends on the available computer memory, the grid can be made as fine as what is computationally possible. We started the grid with a range of cubic window with 5 mm space intervals in each of the three orthogonal directions as well as a range of time window with 1 (µs) time intervals. We then collapsed the grid size twice down to a final range of cubic window with 0.1 mm space intervals in each of the three orthogonal directions as well as a range of time window with 0.1 (µs) time intervals. Comparison of the real travel-time data with the synthetic travel-time data computed using hypocentral location and the origin time determines the point of best agreement between the observed and calculated travel-times. Several methods can be used to measure the point of best agreement. A common approach is the method of least squares (L2 norm) which leads to root mean squared residuals (where the residual is the difference between the observed and calculated travel-times). The point having the lowest misfit (L2 norm) is usually the best hypocenter with the corresponding source time. We computed the AE hypocenter locations by minimizing the time residual, given by, (5.12) 123

146 Where, is the computed travel-time and is the real travel-time data. We solved this problem by repeatedly altering the location and time zero variables to obtain the best solution for which gives a smaller R Location Results in a Synthetic Isotropic Homogeneous Medium A synthetic isotropic homogeneous model with a constant velocity of 5500 (m/s) was built. Travel-time data between each of the 18 shot transducers and the 15 receiving transducers on the other platens for P waves propagating in the synthetic isotropic homogeneous model with a constant velocity of 5500 (m/s) was generated. Six of the travel-time data were picked to be inverted, but before the inversion a noise was also added to the travel-time data. The range of travel-time data between transducers in the cubic rock sample varies between 3.5 and 20 µs. Therefore, adding a random noise with a maximum of 0.1 µs could make up to 2.8% noise to signal ratio in the travel-time data. Figure 5.1: Location results of the synthetic travel-time data in an isotropic homogeneous velocity model. The stars show the location results. The location results using the grid search method recovered the source time and source location by (µs) residual and almost 1 mm misfit in the spatial location. 124

147 5.2.3 Real Transducer Survey Shot Locations First, a single isotropic homogeneous velocity model with a constant velocity of 5500 (m/s) was used during the experiment for transducer survey shot locations at different stages of the experiment. 20 different transducer survey travel-time data were analyzed. The figure below shows the mean time residual (R) for the 16 shot locations as well as the mean location misfits between all the shots at each of the transducer surveys. Figure 5.2: Mean time residual (R) of the real data and synthetic travel-times calculated by all the located transducers using a single value of 5500 (m/s) for the isotropic homogeneous velocity model for separate transducer surveys at 20 different stress states. Figure 5.3: Mean location misfit of the transducer location results from the fixed transducer positions in 20 different transducer surveys in various stress states using a single value of 5500 (m/s) for the isotropic homogeneous velocity model. 125

148 In the next step, we used a different value for the isotropic homogeneous velocity model. The isotropic homogeneous velocity model used for transducer survey shot locations at different stages of the experiment is provided in the table below. # Main Principal stress (MPa) at different Transducers Surveys Main Principal stress (MPa) at different P-S1-S2 surveys P Wave Velocity (km/s) from P-S1-S2 Survey in all directions for the homogeneous model Table 5.1: Isotropic homogeneous velocity model In order to assess the accuracy of our location method, data from transducer-totransducer surveys at different stages of the experiment were used to locate the transducers. The velocity models applied in our location program for different stages of the experiment were the ones obtained from P-S1-S2 surveys which were operated around the same time and stress state of the test. In the table above, the first column shows the main principal stresses at which the transducer-to-transducer surveys were 126

149 operated and the second column shows the main principal stresses at which P-S1-S2 surveys were operated. Then, the cubic rock sample was modeled by three sets of velocity models representing P wave velocities in directions of the three orthogonal principal stresses each containing data points. As the length of each side of the cubic rock is 80 mm, the data points are hence distributed with 1 mm length intervals in each of the three directions. Each of these three models was also divided into central as well as lower-stressed domains near the X and Y facets with different velocities. In this case of isotropic velocities, all P wave velocity values in different directions and domains were equal. Figure 5.4: Mean time residual (R) of the real data and synthetic travel-times calculated by all the located transducers using the varying isotropic homogeneous velocity model for separate transducer surveys at 20 different stress states. Figure 5.5: Mean location misfit of the transducer location results from the fixed transducer positions in 20 different transducer surveys in various stress states using the varying isotropic homogeneous velocity model. 127

150 5.3 Source Location in an Anisotropic Homogeneous Medium For an anisotropic homogeneous velocity structure, there is a different velocity ( ) for each seismic ray that will depend on the orientation of ray vector demonstrated by dip and strike. The synthetic arrival-time will be (5.13) Where, can be computed using the following equation (5.14) In which, and are directional coefficients for the ray-path orientation. Also, and are fractional coefficients which are obtained from velocities in the minimum and intermediate principal directions with regard to, the velocity in the main principal stress direction., (5.15) Therefore, the transformation matrix components for an anisotropic homogeneous medium will differ from the transformation matrix in an isotropic homogeneous medium by the anisotropic velocity ( ) values for each ray-path. After generating the synthetic travel-times in an anisotropic homogeneous model and adding the abovementioned noise to it, we again used the grid search method for location of the transducers by the same synthetic anisotropic homogeneous model. The recovered source time and source locations resulted in 0.06 (µs) residual and almost 0.6 mm misfit in the spatial location. We also inverted the synthetic travel-times that were generated by an anisotropic homogeneous model to locate the transducers using isotropic homogeneous models. In this case, the recovered source time and source locations resulted in (µs) residual and almost 2.45 mm misfit in the spatial location. 128

151 5.3.1 Real Transducer Survey Shot Locations The anisotropic homogeneous P wave velocity model which is also demonstrated in figure (4.7) was used for transducer survey shot locations at different stages of the experiment and is provided in the table below. Main Principal Main Principal P Wave Velocity (km/s) from P-S1-S2 Survey in direction of: Stress (MPa) Stress (MPa) Main Intermediate Minimum # at Different at Different Principal Principal Principal Transducer P-S1-S2 Stress Stress Stress surveys surveys Table 5.2: Anisotropic homogeneous velocity model Before getting the results for location of AE hypocenters with an ellipsoidal homogeneous velocity model, we got the location results using a transversely isotropic homogeneous velocity model. We averaged the P wave velocity in the X and Y 129

152 directions and used this value for the horizontal component of the elliptical velocity model. Figure 5.6: Mean time residual (R) of the real data and synthetic travel-times calculated by all the located transducers using the transversely isotropic homogeneous velocity model for separate transducer surveys at 20 different stress states. Figure 5.7: Mean location misfit of the transducer location results from the fixed transducer positions in 20 different transducer surveys in various stress states using the transversely isotropic homogeneous velocity model. Represented below are the results for location of AE hypocenters with an ellipsoidal homogeneous velocity model. 130

153 Figure 5.8: Mean time residual (R) of the real data and synthetic travel-times calculated by all the located transducers using the ellipsoidal homogeneous velocity model for separate transducer surveys at 20 different stress states. Figure 5.9: Mean location misfit of the transducer location results from the fixed transducer positions in 20 different transducer surveys in various stress states using the ellipsoidal homogeneous velocity model. There was little difference between the recovered source location results between the two methods using the transversely isotropic homogeneous velocity model and the ellipsoidal anisotropic homogeneous velocity model. 5.4 Source location in an Anisotropic Heterogeneous Medium For an anisotropic heterogeneous velocity structure, each block also has different velocities for the same seismic ray. If is corresponded to velocity of ray i with arrivaltime that the ith sensor detects when passing through the qth block, then the synthetic arrival-time is 131

154 (5.16) Where is the distance that the ith ray travels through the qth block. is a portion of the total length of the ray vector between source and receiver and leads to defining a constant relative to the corresponding block velocity in order to write the following equation (5.17) Then, the synthetic arrival-time can be calculated as (5.18) Again, having a starting model, the G matrix components for an anisotropic heterogeneous medium can be displayed as: (5.19) Location Results in a Synthetic Anisotropic Heterogeneous Model The synthetic anisotropic heterogeneous model was built based on the results from chapter 4 of this thesis by the P-S1-S2 velocity survey analysis (figure 4.63) plus an estimated model for stage D. The four near facet domains in the minimum and intermediate principal stress direction as well as a compacted central domain of the rock was included. The velocities that were used for our synthetic model are given in the table below. After generating the synthetic travel-times in such an anisotropic heterogeneous model and adding the said noise to it, we again used the convenient grid search method for location of the transducers by the same synthetic anisotropic homogeneous model. The recovered source time and source locations resulted in 0.07 (µs) residual and almost 0.45 mm misfit in the spatial location. 132

155 Domain Center Y facets X facets Direction X Y Z X Y Z X Y Z Velocity (km/s) Table 5.3: Synthetic anisotropic heterogeneous velocity model Figure 5.10: The anisotropic heterogeneous velocity structure of the rock obtained in chapter 4 by the P- S1-S2 velocity survey analysis Real Transducer Survey Shot Locations The anisotropic heterogeneous velocity model used for transducer-to-transducer survey shot locations at different stages of the experiment is given in the table below. We applied this structure all over the test including all A to D stages of the experiment. Table (5.4) includes P wave velocities from the P-S1-S2 velocity survey data that were analyzed and obtained in chapter four. It presents P wave velocities in three different domains of the rock specimen consisting of the central, near X facets and near Y facets of the cubic sample in each of the three principal directions. Figure (5.11) illustrates the results for location of AE hypocenters with ellipsoidal heterogeneous velocity models in different stress states. 133

156 Stress at Stress at P Wave Velocity (km/s) from P-S1-S2 Survey Transducer P-S1-S2 Center Y facets X facets # Surveys Surveys X Y Z X Y Z X Y Z (MPa) (MPa) Table 5.4: Anisotropic heterogeneous velocity model Figures (5.11) and (5.12) signify that the AE source locations using the anisotropic heterogeneous model has not made a significant change in results compared to using the transversely isotropic or ellipsoidal homogeneous velocity model in our laboratory scale rock specimen. However, the results in the final velocity survey are remarkably different with almost 2 mm improvement in accuracy of the transducer locations. 134

157 Figure 5.11: Mean time residual (R) of the real data and synthetic travel-times calculated by all the located transducers using the ellipsoidal heterogeneous velocity model for separate transducer surveys at 20 different stress states. Figure 5.12: Mean location misfit of the transducer location results from the fixed transducer positions in 20 different transducer surveys in various stress states using the ellipsoidal heterogeneous velocity model. 5.5 AE Source Locations during the Polyaxial Experiment After the experiment, continuous AE waveforms were harvested to extract discrete events. An AE event, defined as four0 or more transducer channels, recorded a voltage of more than 100 mv within a window length of 512 sample points (51.2 μm). If this criterion was met, a waveform with a length of 2048 data points acquired those data points that were posited 25% of the total data points before the first arrival and 75% of the total data points after it. Then, all of these acquired channel data were written to a new file representing a data file for a specific AE event (Nasseri et al., 2014). In total, 135

158 about 58,000 AE events were source located. We enhanced the locations of the same AE events that were picked in Nasseri et al. [2014]. Figures (5.13) to (5.27) illustrate the 3D pictures as well as the XoY, XoZ and YoZ cross sectional source location images of the different phases of AE activities during the experiment. The figure below represents the onset of the AE activities within the cubic sample. As we can see, the AE activity starts at the state of the experiment, where the main principal stress is. We need to recall that once the main principal stress was lifted to values larger than 35 MPa, the minimum principal stress ( ) and the intermediate principal stress ( ) were kept at a constant stress value of 5 and 35 MPa, respectively, during the experiment. Figure (4.3) represented the evolution of the stress state as well as the hit counts of the AE activity. The first 200 AE events happened between and. This phase is within Stage C of the experiment according to figure (4.4). The AE events initiated from the corner of X2- Y1-Z1 which seems to be the weakest part of the rock at that phase. Figure 5.13: First phase of the AE source locations in a 3D and three orthogonal cross sectional images. The second phase of the AE activity includes 1100 AE events happening between and, after having a relatively long pause at according to figure (4.3). The events have migrated and spread from the corner of X2- Y1-Z1 toward the side of X2- Z1 facets cross-section. Figure 5.14: Second phase of the AE source locations in a 3D and three orthogonal cross sectional images. 136

159 The third phase of the AE activity includes 200 AE events happening between and, after having a very short pause at. Besides those events migrating from the side of X2-Z1 facets down the Z axis, some new events also initiated from the corner of X1-Y1-Z1 facets as well. Figure 5.15: Third phase of the AE source locations in a 3D and three orthogonal cross sectional images. The fourth phase of the AE activity includes 300 AE events happening between and. We see that after a short time from the previous phase, aside from those events migrating from the corner of X1-Y1-Z1 facets toward the middle of the X1-Z1 facets side, a new trend of AE activity starts from the corner of X1-Y1-Z2 facets. Figure 5.16: Fourth phase of the AE source locations in a 3D and three orthogonal cross sectional images. The fifth phase of the AE activity includes 1800 AE events happening between and, especially during a relatively long pause at according to figure (4.3). There are three active spots which are already activated and continue on the AE activity. The first spot is located on the upper half of the X2 platen along a zone extended between the Y1 and Y2 platens. The second and third ones are on the side of X1-Z1 and X1-Z2 facet cross-sections. During the fifth phase, a new trend of AE activity on the side of X2-Z2 facet cross-section takes off. 137

160 Figure 5.17: Fifth phase of the AE source locations in a 3D and three orthogonal cross sectional images. The sixth phase of the AE activity includes 2000 AE events happening between and. Since the previous phase, the past four active zones continue their AE activities more near the X2 facet and less near the X1 facet while the X2- Z1 facets cross-section zone has again been activated too. Figure 5.18: Sixth phase of the AE source locations in a 3D and three orthogonal cross sectional images. The seventh phase of the AE activity includes 4000 AE events happening between and. The AE activities continue almost all over the X2 facet while the X1 facet of the rock is quiet. This phase is the onset of Stage D of the experiment according to figure (4.4). This is a storm of AE events at this phase of the AE activities occurring within 70 seconds (roughly, 60 AE events per second) by only 5 MPa raise in the main principal stress value. Figure 5.19: Seventh phase of the AE source locations in 3D and three orthogonal cross-sectional images. 138

161 The eighth phase of the AE activity includes 7000 AE events happening between and. At this phase, migration of AEs toward the center starts for both X1 and X2 facets. Two upper and lower sides of the X1 facet have been activated again but this time, their active zones start approaching toward each other. Likewise, AE events on the three upper, middle and lower parts of the X2 facet start migrating slightly toward the middle of the rock while they are connected together. Figure 5.20: Eighth phase of the AE source locations in a 3D and three orthogonal cross sectional images. The ninth phase of the AE activity includes 5000 AE events happening between and. Here at this phase of the AE activities, another storm of the AE events occurred within 50 seconds (roughly, 100 AE events per second) by only 3 MPa raise in the main principal stress value. At this phase, no activity is detected near the X1 facet and the AE activities near the X2 facet are being migrated and concentrated more toward the middle of the X2 facet and toward the center of the rock specimen. Figure 5.21: Ninth phase of the AE source locations in a 3D and three orthogonal cross sectional images. The tenth phase of the AE activity includes 8000 AE events happening between the main principal stresses of and, after having a relatively long pause at according to figure (4.3). In this phase, the AE activities near the X2 facet have not gone further toward the center of the rock. In addition to 139

162 the resumption of activities near the X1 facet, a specific feature consisting three fractures can be observed near the X1 facet al.ong the Z direction. Although X1 facet had less AE activity than the X2 facet, this phase was the most active phase for the X1 facet. Figure 5.22: Tenth phase of the AE source locations in a 3D and three orthogonal cross sectional images. The eleventh phase of the AE activity includes 4400 AE events happening between and. At this phase, the AEs from the X2 facet have migrated and coalesced close to the center of the rock while there is not much AE activity left behind at the X2 facet sides and corners. The AE activities near the X1 facet are also carried on but here, more aligned along the Z direction. Figure 5.23: Eleventh phase of the AE source locations in a 3D and three orthogonal cross sectional images. The twelfth phase of the AE activity includes AE events happening between and. Here at this phase of the AE activities, another storm of the AE events occurred near the X2 facet within 110 seconds (again, roughly 100 AE events per second) by 7 MPa raise in the main principal stress value. A semi-curved pattern connecting the AE source locations can be seen in the image, which is a typical pre-failure fracture pattern in the polyaxial experiments. The main principal stress at is the highest value that the main principal stress reached during this 140

163 polyaxial experiment. After this point, the rock proceeded to the failure process bearing an AE avalanche. Figure 5.24: Twelfth phase of the AE source locations in a 3D and three orthogonal cross sectional images. The thirteenth phase of the AE activity includes 7000 AE events happening during a pause on the main principal stress while dropping to. This phase is the onset of a dynamic failure of the rock that was not caused by an increase in the main principal stress, but it accelerated by reaching the critical stress state after the rock specimen was weakened enough with previous static failure phases. The coalesced active zone near and between the X2 facet and center of the rock quickly migrated toward the bottom of the rock near the Z2 facet in the shape of a planar cluster with a slope of almost 20 angle with respect to the vertical direction. These AE events occurred within 45 seconds (roughly, 150 AE events per second). Figure 5.25: Thirteenth phase of the AE source locations in a 3D and three orthogonal cross sectional images. The fourteenth phase of the AE activity includes 5000 AE events and is the drastic failure stage of the rock while the main principal stress is dropping 108 MPa from to. The planar cluster, which had moved downward near the Z2 facet in the previous phase, now rose upward ending up exactly in the center of 141

164 the rock. On the other hand, another failure process started from an area near the X1- Z1 facet toward the center of the rock. However, the specimen had not completely failed because it still maintained 381 MPa main principal stress. Figure 5.26: Fourteenth phase of the AE source locations in a 3D and three orthogonal cross sectional images. The fifteenth and last phase of the AE activity includes 1000 AE events and is the last kick and harshest failure of the rock, while the main principal stress is dropping 202 MPa from to. As we can see, a coalescence of the AE events reaching each other from two different sides ensued in this phase. Therefore, the failure process got almost complete as the two semi-vertical fractures from the bottom and top of the rock along the main principal stress direction met each other. Figure 5.27: Fifteenth phase of the AE source locations in a 3D and three orthogonal cross sectional images. This final phase involved the last AE events that could be detected as the rock got too noisy after that and the waveforms got saturated with big noises and made it impossible to capture any more AE events. 142

165 5.6 Discussion and Conclusions In the transducer location results, the calculated source time zero also affects the resultant time residual (R). Therefore, sometimes the location may have even bigger misfit in location but with a relatively smaller time residual (R). The diameter of the transducer surface is about 2 mm and as a result, a minimum of 2 mm uncertainty in locations of the shot transducers is inevitable and reasonable. Although the location results in the anisotropic heterogeneous model did not make a significant change compared to the transversely isotropic or even to the ellipsoidal homogeneous velocity model, however, the transducer positions in the final velocity survey at stage D were recovered with almost 2 mm improvement. Overall, the shot location results were improved by implementation of different velocity models for the sample starting from isotropic and homogeneous models going toward anisotropic and heterogeneous models. The shot locations showed a major improvement after the velocity model corrections had been applied especially at the final phase of the experiment. This location improvement validated our velocity model at the final phase of the experiment consisting lower-velocity zones bearing partially saturated fractures. This indicates that the anisotropic heterogeneous velocity structure with dilatancy pseudo-boundaries is entirely effective after the weakening near the facets in the minimum principal stress direction by the deformational AE activities. Considering the structures with engineering scales of tens of meters or the field scales with at least hundreds of meters in different dimensions, the effect of anisotropy (ellipsoidal, elliptical, etc) on source locations, especially with much more localized location of the receivers with respect to the object scales, will be remarkable. As Byerlee and Lockner [1977] mentioned, when a rock sample approaches failure, the AE activity rises dramatically and will often saturate the system. They argued that in such cases, there will be even bogus events for which the recorded arrival-times will be meaningless. Therefore, the least-square solutions for such events will blow up and the computer program will reject the event. However, unlike our system, their system only started recording a waveform only in case a triggering criterion was met and therefore, it suffered from lack of abundant information from continuous AE streaming waveforms. 143

166 In total, we were able to recover a good deal of the fracturing process by the AE source locations especially with the aid of the continuous waveforms from the AE streaming system. Below is a CT scan image of the rock after failure in the XoZ plane cross-section in the middle of the rock along the Y axis. The features in yellow as well as the semi-curves in brown were almost observed by the AE source locations. Also, figures (4.20) and (4.63) confirm these weak zone features in the rock sample. However, the feature on top of the rock signified by a red ellipse could not be recovered due to saturation of the AE sensors by noise of the rock at final stages of the failure process that was mentioned above. Z Y X Figure 5.28: AE source location patterns versus CT scan image of the XoZ cross section of the rock at the middle of the Y axis. The semi-curved features have also been observed by King et al. (2012) in their polyaxial experiment on 50 mm-side cubic samples of Sandstone. Even Reches and Dieterich (1983) who performed several polyaxial experiments on 21 mm-side cubic Berea Sandstone specimens mentioned that their fault surfaces were semi-planar and curved, while a few curved surfaces were composed of two coalescent semi-planar segments. 144

167 At stage D of the experiment, the compaction pseudo-boundaries were replaced by dilatancy pseudo-boundaries with a similar corresponding anisotropic heterogeneous velocity structure. Dilatancy is the increase of volume relative to elastic changes caused by deformation. The onset of dilatancy is the point along the strain versus time plot (Figure 4.3) in the minimum principal stress direction, where the initially linear curve begins to deviate towards a negative strain, which is accompanied by a volumetric increase. This mildly starts in stage C but mainly occurs with the onset of stage D. Haimson et al. [2000] suggested that in polyaxial experiments, the onset moment of dilation of the rock similar to stage D of our experiment, entailing the rise in AE activity, depends on the loading path used in the test and specifically on the intermediate principal stress magnitude. They illustrated that there is a direct correlation between the magnitude of the ( ) difference and the extent of the rock quasi-linear elasticity which is the state of the rock before dilation. This implies that a larger ( ) difference delays the onset of failure initiation or micro-cracking. Maxwell and Young [1995] proposed that the high velocity zone corresponds to where the rock mass is very highly stressed and "clamped", so that the region is essentially acting as an asperity. Conversely, in the low velocity zone the rock mass is fractured and no longer capable of supporting stresses in order to be the site of seismic events. The seismicity is located in a region of moderate velocity, where the normal stresses are sufficient to enable the rock mass to store strain energy, but not high enough to "clamp" the rock mass. In our experiment, the region of moderate velocity was around the dilation pseudo-boundary surface at stage D. Thus, based on the velocity images that we obtained by the P-S1-S2 velocity survey analysis as well as the anisotropic tomography, the location of the AE events were also in consistence with them, as they were also concentrated around the compaction and dilation of those pseudo-boundary zones with maximum velocity gradients. 145

168 6 Conclusions The elastic state of the rock and the state of the cracks are interrelated in several ways. Cracks reduce the effective elastic moduli of the rock and permit a larger strain for a given stress, in comparison to a case with no cracks. Also, changes in stress can vary the crack density by opening closed cracks or by completely closing open cracks. Open cracks can be extended by a fracture; consequently, the moduli are affected. A water saturated Fontainebleau Sandstone cubic sample was tested in Prof. R Paul Young s RFDF Laboratory within a True Triaxial Geophysical Imaging Cell to study the fracturing process and the effects of polyaxial loading on physical and mechanical properties of the rock (Young et al., 2012; Nasseri et al., 2014). The rock went under a loading pattern of different stress states with an increasing main principal stress while the other two principal stresses were kept constant at relatively low pressures to simulate the in situ stress states in deep underground structures such as the vicinity of wellbore walls or excavation tunnels. Our polyaxial experiment on saturated Fontainebleau sandstone was divided into four stages (A, B, C, and D) based on the stress path, strain, AE activity and of course the velocity evolution of the rock in the three principal directions. Stage A was a poroelastic phase associated with the compaction and closure of the voids. Stage B was an elastic phase with no dilation or AE activities. At the onset of Stage C, some AE activities started near the facets of the cubic rock while a slight velocity drop also occurred but it was still considered as a non-damaging phase. At Stage D which was an elasto-plastic phase, the apparent velocities drastically dropped, a huge number of AE events took place and the specimen went toward failure. The fundamental objective of this thesis was the time-lapse ultrasonic imaging of elastic anisotropy to obtain an anisotropic heterogeneous velocity structure of the saturated Fontainebleau sandstone under true-triaxial stress state. In order to image the rock under true-triaxial stress state, an anisotropic velocity tomography method was developed to invert the transducer-to-transducer P wave velocity surveys. The tomographic images reconstructed correct features at final stages of dilation of the rock but with incomplete anomalies and uncertain values due to saturated pore spaces and lack of a good ray coverage. However, conventional measurements of transmission 146

169 velocities of P wave and two perpendicularly polarized S waves along each of the three principal axes of the cubic sample (P-S1-S2 velocity surveys) were successfully utilized to reconstruct an image of the rock before dilation. Elastic parameters of the central compacted domain of the rock under TTT were then computed. Also, we aimed at the acoustic emission monitoring of the failure of the cubic rock sample under laboratory true-triaxial stress state. This objective was approached by examining different velocity models for location of transducer shots and eventually, it was fulfilled by using the fittest anisotropic heterogeneous velocity model in the algorithm applied for AE event source location. The fracture process was mapped accurately and consistent with the CT scan images while confirming the anisotropic heterogeneous velocity models of the rock under various stress states as well. A flowchart of the data acquisition and data processing is provided in Appendix B. 6.1 Local Heterogeneities and Anisotropic Tomography As Hall [2009] described well, there is always a need to use some form of remote sensing technique, such as seismic imaging, to provide information on the in-situ properties of the rocks and their fracture sets in reservoirs. Especially, this need is more urgent in existence of many reservoir characterisation challenges to identify, quantify or characterise fracturing in the subsurface while borehole data is one-dimensional and limited. Therefore, developing fracture monitoring techniques by ultrasonic imaging especially in anisotropic medium under the in situ stress state conditions is necessary. Using the concept of ellipsoidal geometry of P wavefront surface, computational codes were developed in Matlab as a ray-based anisotropic tomography tool to implement any set of transducer distribution and any 3D model space resolution of cubic samples in our true-triaxial test. Singular value decomposition method was used as our inversion approach. The result of our numerical simulation showed that the anisotropic tomography of synthetic data successfully reconstructs the synthetic velocity structure, even having 5% velocity heterogeneities in case of noise free data. On the other hand, the ordinary isotropic tomography failed to reconstruct the anisotropic structure. Tomography images of the rock from transducer-to-transducer velocity surveys were generated by inverting the apparent P wave velocities displayed by stereonet 147

170 images. Stereonets showed velocity drop in the minimum principal stress direction at stage D due to the opening of new unsaturated fractures and dilation of the rock. Also, primary AE locations as well as CT scan images (Nasseri et al., 2014) verified that curvi-planar fractures were spread with surface normal vectors perpendicular to the minimum principal stress direction and located adjacent to the two X facets of the rock. The evolution of tomographic images was in good conformity with what we expected in terms of relative local variations in the velocity values. Knowing that thin liquid-filled cracks have very little effect on P wave propagation and the isotropic P wave velocity models, the anisotropic P wave velocity tomography could only detect the dilation of the sample edges in stage D. There were restrictions in the tomography process such as source-receiver arrangement limitations, the existence of noise, the absence of perfect point location for transducer positions, and most importantly, the influence of water saturation of the sample on the P wave data that affect the accuracy of the reconstructed images. In spite of these restrictions, our anisotropic tomography method could detect some heterogeneous zones of the anisotropic rock. Hence, this anisotropic tomography method which works under these circumstances has the potential to be of help in better understanding fluid saturated subsurface structures especially before the formation of new fractures. 6.2 Velocity Structure Induced in the Polyaxial Experiments The standard polyaxial experiments practiced all over the world (Mogi-type, etc), suffer from specific restrictions that are also mentioned as loading and boundary conditions in Shi et al. [2012] and Nasseri et al. [2014]. For example, Sayers [2002] also investigated the effects of cracks on the S wave velocities in his study of the anisotropic symmetry of sandstone under polyaxial experiment that was experimented by Chaudhry [1995]. He specifically noticed that the symmetry, necessary in an orthotropic homogeneous medium was violated in a laboratory polyaxial experiment. Considering the constraints imposed on the sample by the loading plates as a reason of this asymmetry, he maintained the assumption of homogeneity of the rock sample. Therefore, he only used and to obtain the and elastic parameters. Thus, the main problem that we managed to solve and justify in chapter 4 was the problem of asymmetry and unequal shear wave velocities between each pair of mutually 148

171 perpendicular direction of propagation and polarization due to a complicated stress as well as velocity structure induced into the rock that was not solved before (Chaudhry, 1995; King et al., 1997; Sayers, 2002; Sarkar et al., 2003; Popp and Salzer (2007); Nasseri et al., 2014). We estimated anisotropic properties of the central compacted domain of the cubic rock specimen by a combination of stress distribution modeling and an analysis of compressional and two perpendicularly polarized shear wave velocity measurements in each of the principal stress directions as well as P wave velocity measurements from the transducer-to-transducer surveys. However, we used the predilation concept of compaction pseudo-boundary surface due to anisotropic heterogeneous distribution of stress and therefore, anisotropic heterogeneous distribution of velocity was considered as a result of the geometrical effects of edge chamfers and friction on the platen-rock surfaces. The compaction pseudo-boundary criteria were defined as boundaries in the minimum and intermediate principal stress directions separating the compacted domain from the other domains under less compressional stress. A stress distribution simulation by FLAC3D helped in the estimation of compaction pseudo-boundary surfaces and the velocity contrasts along the minimum and intermediate principal axes. Then, anisotropic parameters could be recovered in the form of elastic coefficients and therefore, studying their variation under different stress states was made possible. Besides reconstructing velocity images of the rock, the hypothesis of ellipsoidal anisotropy under a true-triaxial stress state was examined and approved through appropriate methods for off-diagonal components of the stiffness tensor solution. Hence, we solved the anisotropic heterogeneous velocity structure of the rock using P-S1-S2 velocity survey data analysis in stages B and C while the rock maintained its elastic behaviors in the absence of any through-going fractures at its center. We obtained the elastic tensor parameters as well as Thomsen parameters at the central homogeneous part of the cubic rock, where the applied stress was almost uniformly distributed. 6.3 AE Source Locations in Polyaxial Experiments The delineation and characterisation of fracturing is important in the successful exploitation of many hydrocarbon reservoirs, as fractures can significantly affect 149

172 permeability and thus production, e.g. during horizontal drilling or deep geological disposal investigations (Hall, 2009). Polyaxial experiments which provide an environment for the rock sample similar to the in-situ stress states can help in the characterization of fractures using AE monitoring of the location of damages and deformation of the rock. The relationship between the velocity structure of the medium and its microseismicity was studied by many researchers such as Maxwell & Young [1992]. They suggested that the zones of decreased velocity correspond to an increase in the velocity gradient. This can cause the localized destressing and relaxation of the clamping forces which can result in microseismicity. Resolution enhancement in AE monitoring and velocity imaging of fractured rocks enables us to study the correlations between the location of seismic events, velocity structure and stress pattern more extensively. We started with AE source locations of transducer shots in synthetic velocity models. The synthetic velocity models included an isotropic homogeneous model, two anisotropic homogeneous models, and finally, an anisotropic heterogeneous model. After we built and examined the location program for synthetic data using the collapsing grid search strategy in Matlab, we applied this approach on real travel-time data from transducer-to-transducer surveys at various stages of the experiment. The accuracy of the method was examined through calculating the time residual (R) and average location misfit of the fixed shot transducers. Hence, through the comparison of transducer shot location results, we investigated the effect of velocity structure on AE source locations within the cubic rock sample in a polyaxial experiment. A minimum of 2 mm uncertainty in the locations of the shot transducers was inevitable and reasonable due to the fact that the diameter of the transducer surface is about 2 mm. Although the location results in the anisotropic heterogeneous model did not make a significant change to the transversely isotropic or ellipsoidal homogeneous velocity models, the transducer positions in the final velocity survey were recovered with an average of almost 2 mm improvement in location. This indicated that the anisotropic heterogeneous velocity structure with dilatancy pseudo-boundaries is totally effective after the weakening near the facets in the minimum principal stress direction by the deformational AE activities. Considering the structures with scales of tens of meters or the field scales with at least hundreds of meters in different dimensions, the effect of 150

173 anisotropy (ellipsoidal, elliptical, etc.) on source locations, especially with more localized locations of the receivers with respect to the object scales, will not be negligible. After locating the transducer shots in velocity surveys, the AE events sources were located using the same grid based location approach using the anisotropic heterogeneous velocity model. Investigating the AE activity of the rock under truetriaxial stress state showed interesting processes of fracturing. During the last stage of the experiment (Stage D), the coalescence of micro-cracks occurred on the boundary between the highest stress and velocity gradients in the rock specimen. That boundary was a consequence of two reasons combined. One was the geometry and specific conditions of the rock specimen including edge chamfers and friction on the platen-rock surfaces. The second reason was the true-triaxial stress pattern which acted as a constraint to the fracture developments within the rock, and hence, delayed the dilation onset in the rock. 6.4 Summary of Scientific Contributions Anisotropic tomography methods were developed based on ellipsoidal symmetry in P wavefront geometry in fractured rock under a true-triaxial stress state. An anisotropic ray-path segment matrix reduced the number of unknown parameters in the velocity model to be inverted. Heterogeneities larger than 5% could be detected by our anisotropic tomography method in case of a noise free data. The tomographic velocity images of the saturated rock after dilation were reconstructed. The reasonable anomalies obtained in the tomographic velocity images of the rock at the last stages of the experiment proved that even in a fully water-saturated rock, it is still possible to detect the opening of new cracks that are still unsaturated or partially saturated especially before failure of the rock. The velocity structure of saturated Fontainebleau sandstone was obtained using the concept of compaction pseudo-boundaries as a result of the specific experimental conditions. Stress distribution simulations in FLAC3D proved the formation of an intact homogeneous central part of the rock under laboratory true-triaxial stress state and this technique gave an accurate image of the rock in TTT for the first time. 151

174 Our methodology enabled us to compute the elastic parameters as well as Thomsen parameters for the rock specimen under laboratory true-triaxial stress state in its discovered central compacted zone. Ellipsoidal anisotropy under laboratory true-triaxial stress state was approved by calculating the off-diagonal stiffness tensor components. The effect of different velocity structures on AE source locations under truetriaxial stress state was investigated through transducer shot locations and improved the location results. The anisotropic heterogeneous velocity model proved to be the best model for location of AE events in the polyaxial experiment especially after dilation of the rock. The fracturing patterns and crack formations in polyaxial experiment under the effects of intermediate principal stress were studied by computing AE source locations during different stages of the experiment. The M-shaped fracture pattern observed through the CT scan images was also confirmed by putting all the AE locations together. Application of the anisotropic heterogeneous velocity structure enhanced the resolution of AE event locations. It resulted in further discovery of the evolution of AE activities in a polyaxial experiment starting from the corners, weakening the slabbing edges for almost all of the experiment until they migrated toward the center of the rock. Our approach could explicitly enhance current practice since for the first time in a unique experiment, the technology of real-time monitoring of a rock specimen was used to obtain and verify the time-lapse anisotropic heterogeneous velocity structure of a saturated porous rock under laboratory true-triaxial stress state. The stress distribution simulated by FLAC3D and subsequently, the velocity structure discovered by compressional and shear wave velocity data analysis was in good correlation with results obtained from the AE source locations as well as the CT scan images. This makes it a promising approach for time-lapse imaging of the seismic properties and AE activity of the rock under true-triaxial stress state. 152

175 6.5 Recommendations for Future Studies Similar to most of the other studies, this work could be an end and at the same time, a beginning for some scientific implications. For example, we realized that, in such laboratory scales of a few centimeters, some simplifications of the background velocity model such as using TI model are still workable in AE source locations. Therefore, the question of the effect of TI versus the ellipsoidal anisotropic model on the AE source locations is now insignificant in laboratory scales. However, we took a step forward in digging a couple of layers to revealing the velocity structure of the rock under laboratory true-triaxial stress state. Therefore, the next step forward would be excavating a deeper layer by transforming and interpreting the elastic parameters into crack distribution properties such as crack density tensor similar to the approach used by Sayers et al. [1990]. This study can also help in further investigations of pre-seismic phenomena such as fracture propagation due to a local increase in velocity gradients, crack density or stress concentration by seismic monitoring techniques in a water saturated rock under in-situ true-triaxial stress state. Reches [1983] explained that the reduction of the shear stresses on the surface of a cubic specimen is essential for the success of a polyaxial experiment. However, as explained in the discussion part of chapter four, it is almost impractical to eliminate friction from the polyaxial experiments. In the meantime, the true-triaxial tests aim to minimise or take into consideration the effects of friction. We also proved in chapter four that the true-triaxial conditions are generated away from the boundaries of the cubic sample. Therefore, the minimization of friction is still a big challenge in the polyaxial experiments and needs to be customized for different rocks. Further investigations can be done regarding the evolution of anisotropic symmetry resulting in an M-shaped failure geometrical pattern of the rock specimen under a true-triaxial stress state (figure 5.28). Discovering a relationship between the anisotropic elastic properties of the rock and the fracturing angle may lead us to a criterion for the prediction of the failure patterns of the rock. However, this may need several polyaxial experiments with a full monitoring system, which will be relatively 153

176 expensive due to the cost of implementation and maintenance of velocity and AE monitoring tools and techniques during the experiment. A new arrangement of transducer positions suggested in Appendix A provides more coverage of the rays through the bulk of the rock sample, and therefore, can definitely enhance the tomographic images of the rock if it is practically possible to be implemented. AE events waveforms P wave arrival-times can be added to our travel-time data to be used in velocity tomography. More travel-time data can enhance the ray coverage, and thus, the resolution of an inverted velocity structure. Therefore, once the accuracy of the AE source locations are verified, a simultaneous tomography can also be performed going back and forth between the two following steps until the problem reaches to the minimum misfit point. The first step should be locating AE events using the anisotropic heterogeneous velocity structure. The second step would be incorporating AE arrival-time data into velocity tomography. Overall, the possibility of controlling and repeating different factors such as stress state, pore pressure, temperature, fluid flow, etc. in laboratory experimental rock physics and mechanics is an advantage for improving data quality. In particular, using true-triaxial tests where the in-situ stress states are simulated produces conditions similar to rocks in nature. In conclusion, any results obtained from TTT will significantly enhance our understanding of the elastic and AE properties of rocks which are considerable for engineering applications. 154

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188 Appendix Appendix A. Ultrasonic Travel-Time Tomography A.1 Introduction Heterogeneities in the rocks are usually caused by variations in lithology, porosity, permeability, pore fluid properties, and conditions of pore pressure, temperature, and inhomogeneous distribution of stress. The main source of anisotropy may be aligned crystals, aligned grains, periodic thin layers, aligned cracks or orientation-dependent stress. Treating the change in properties of the rock with direction as anisotropic or heterogeneous is entirely scale dependent. Ray-based tomography using straight rays is an acceptable method in the lab scales and ray bending effects will not significantly alter a straight ray image. Young et al. [1989] (II) who applied geotomographic imaging and microseismic monitoring techniques validated by laboratory calibration trials to study mining induced seismicity also signified that their synthetic and experimental results had shown that when velocity contrasts exceeded 15-25%, the errors caused by assuming straight ray propagation became significant. A.2 Transducer Setup for Acoustic Imaging In many previous works on tomography of rock samples in the laboratory, the samples were usually removed from the loading frame following each cycle of loading or after the end of the experiment and were placed in a scanning frame of transducers, which allowed tomographic imaging (Hall, 2009). However, we performed a real time monitoring of the rock at various stages of loading during the experiment as a nondestructive method in order to accomplish a live active velocity tomography. Having the maximum number of 135 ray-paths between all these transducers and their corresponding travel-time data, our geophysical imaging cell is able to perform a 3D P wave travel-time tomography of the rock specimen at each specific stress state of the rock during the experiment. Therefore, the compressional anisotropic velocity 166

189 structure of the cubic rock sample as well as the fractured zones can be investigated. Figure (A.1) displays all the ray-paths from a transducer-to-transducer survey. Based on the definition of velocity and slowness, (A.1) Tomography aims to obtain a velocity structure of the medium through which the seismic wave passes. The medium is divided into a certain number of cells that the raypaths probe. (A.2) Figure A.1: Real laboratory sensor arrangement. Left: transducer locations. There are 3 transducers embedded on each platen. Locations of the transducers on each pair of the opposite platens are the same. Right: Ray-paths between the 18 transducers covering the cubic sample. Where x is distance, v is velocity, s is slowness (inverse of velocity), and t is travel-time. The index i belongs to the ray-path number, and the index j is the number of the block through which the ray is passing. is the distance that the ray i travels in the jth block. The elements make the Ray-path segment (G) matrix components. Arrival-times are the data, and velocity structure represented by velocity of the blocks is the model that we are looking at to determine its solution, (A.3) 167

190 The number of G matrix rows is equal to the number of ray-paths, and the number of G matrix columns is equal to the number of blocks in the cubic sample. Naming the slowness as our model and the travel-times as our data, it becomes a matrix equation, (A.4) Ray-path segment matrix elements ( ) are made up of lengths of the ith ray in the jth cell. The matrix representation of the problem is as below. (A.5) The goal of tomography methods is to solve for the model space by inversion techniques. In this process, the ray-path segment matrix (G matrix) plays the key role. Now, the problem with anisotropic tomography is that each of the 125 blocks of the cubic sample contains three different velocities in the three orthogonal directions. That makes number of the unknown parameters up to, but we only have 135 travel-time data to use in the system of equations. Hence, our main goal is to find a physically acceptable approach to minimize the number of unknown parameters in the anisotropic tomography problem. We aim that goal by using the relationship between the three different velocities through the ellipsoidal shape of the propagating wavefront velocity or slowness as proved in the previous chapter to reduce the number of unknowns. However, we first construct our tomography method for isotropic media, and then develop it for the anisotropic media. A.3 Ray-Path Segment Matrix We used straight rays in our calculations. As it is explained by Vigianni & Hall [2012], if the velocity variations are small, a reasonable approximation can still be achieved with straight rays (which means that no relinearisation is necessary), although resulting in some loss in spatial resolution. Generating the G matrix was the most laborious part of the tomography program. The rock was divided into equal sized cubic blocks in all three dimensions, e.g. the rock 168

191 sample would be divided into 125 blocks if there were 5 blocks in each direction. Then, a P wave velocity can be assigned to each of these blocks. The validity of the straight ray-path assumption depends crucially on the imaged structure. If the velocity gradients and variations are small enough so that the departure of an actual ray from a straight line at the receiver position is smaller than a few wavelengths, the straight line approximation may be used. This is usually the case in tomography imaging of small objects like rock samples or pillars in mines. In such applications, modeling errors introduced by the straight-line ray-path approximation only decrease the contrast of the images (Debski and Young, 2002). 3D 2D Figure A.2: Ray travel scenarios through the square and cubic cells in 2D and 3D conditions respectively. In the 2D case, there are only 3 different scenarios, while in the 3D case there are 11 different scenarios for rays to travel between each opposite and parallel interfaces of the cells. Numbers in red indicate each interface through which the rays pass. The left-right arrow also represents the intersection point intersecting with the boundary between the four interfaces. The figure above compares different possible situations of rays passing through 2D and 3D cells used for calculating the G matrix elements. In the 2D case, there are only three simple scenarios but in the 3D case, there are 11 different scenarios that should be considered in computations. Going through each ray-path, the 3D G matrix program first finds the direction with the longest distance of the ray length to simplify the 169

192 computations. Then, going from one interface of the cubic cells to the next one in the chosen direction along the ray, the program finds out the location of interface plane intersections, border intersections, or corners as nodes that the ray passes in between the two interfaces along the determined direction. At the same time, the length of each chunk of the ray-path is calculated between each of the two consequent nodes. This gives the length portion of each ray-path in each cell. These values will then be assigned to their corresponding rows and columns of the G matrix which represent the number of ray-paths and specific cells respectively. The figure below demonstrates the cubic cells that are probed by a ray-path. The highlighted blocks are equivalent to the non-zero columns of the G matrix in the specified G matrix row corresponding to the ray-path number. The color bar represents the range of the ray-path length in each block in millimeters. Figure A.3: A ray-path and the probed cells along its path between two sensors. In order to make sure that the G matrix is calculated accurately, first, the probed cells for each of the 135 rays between the 18 transducers were plotted to examine their path locations. Second, the length of the ray in each cell was checked to not be bigger than the cubic block space diagonal. Third, the sum of the lengths of the same ray-path 170

193 portions in the probed blocks was checked to be equal to the distance between the two sensors. The important Parameters implemented in the G Matrix calculation for flexibility of the program include the possibility to incorporate any set of transducer locations. Also, any set of noisy transducers can be dropped out from the calculations. Any number of blocks along each edge can be considered for obtaining different resolutions except the cases in which some transducer locations take place on the edge of a cell. Figure A.4: Representation of the total ray length in each block for a sample with 18 transducers and 5 blocks along each edge. n = 135 data points. m = 125 blocks. Number of blank blocks = 10. The effect of small heterogeneities and precision of measurements are comparable with that of the strain of the rock on arrival-times. Thus, their effects on the inverted velocities are also of the same order. Therefore, strain of the rock measured by three linear variable differential transformers (LVDT) along X, Y and Z axes can be considered and implemented in the G matrix. In the case of any existing strain, the cubic sample as well as all the cells will be squeezed relatively and transducer locations will change. Therefore, the distance between transducers as well as the length of each 171

194 ray through each block will change, meaning that we will have a different G matrix. All the above features are implemented in the G matrix program which is coded in Matlab. Figures (A.4) and (A.5) are representations of the G matrix in samples divided to cubic cells and cubic cells respectively for the current configurations of the 18 sensors. The color bar indicates the lengths summation of all the rays that have passed through each specific block. In the current configuration design of sensors where corners and edges of the sample are not covered, number of blocks that are not probed by any ray-path is 10 and 51 respectively for the aforementioned figures. Figure A.5: Representation of the total ray length in each block for a sample with 18 transducers, and 7 blocks along each edge. n = 135 data points. m = 343 blocks. Number of blank blocks = 51. The figures representing the ray-paths in the 18 transducers arrangement show that a big part of the rock is not probed by any rays and there is a lack of ray coverage on the edges and corners of the sample. As a result, for the purpose of attaining better coverage of the sample, although it is practically not a low cost operation to build 172

195 platens with larger embedded number of transducers in them, we still propose an alternative arrangement of sensor configuration design to overcome the problems associated with ray coverage restrictions of the current source-receiver arrangement. A.3.1 An Alternative Arrangement of Sensor Configuration Design Restriction of source-receiver arrangements always degrades the resolution and generates some artifacts in the results. The following configuration design gives a better coverage of the rock body with rays especially at the edges and corners of the cube. However, the current arrangement is a compromise for capacity of the AE streaming system and number of channels as well as manufacturing of platens. Therefore, it needs to be acknowledged that a new design requires many other considerations to be feasible. Figure A.6: Hypothetical sensor arrangement. Left: transducer locations. There are 5 transducers embedded on each platen. Locations of the transducers on each pair of the opposing platens are the same. Right: Ray-paths between the 30 transducers covering the cubic sample. In the suggested arrangement, there are five transducers embedded on each platen. Locations of the transducers on each pair of the opposing platens are exactly the same, but differ between the X, Y and Z platens for avoiding short distance ray-paths and attaining a better coverage. The transducer locations are designed in a way to cover all corners and edges of the cubic sample. The number of ray-paths grows from 135 for the 18 transducers to for the 30 transducers as each of the 30 sensors are 173

196 communicating with the 25 sensors on the other platens. Also, as a transducer works both as a transmitter and receiver and a ray-path is shared between two transducers, the multiplied number needs to be divided by two. The figure below shows the matrix representation of the alternative arrangement with 30 transducers array for a cubic sample divided by five blocks along each edge. It is seen that the rays probe all of the cells, and there is no blank block left. Figure A.7: G matrix representation of the 30 transducers array for a sample with 5 blocks along each edge. n = 375 data points. m = 125 blocks. Number of blank blocks = 0. Thus, in the G matrix representation of the alternative configuration design of 30 sensors shown in figure (A.7) for a grid of cubic cells there is no cell left with zero probing rays. Also, the maximum total length of rays for individual blocks is elevated from about 160 millimeters to around 475 millimeters, which will also cause in an increase in the directional ray coverage in each block, and therefore, will affect the results of the anisotropic tomography of the rock. 174

197 Velocity values for all the grid cells, especially the ones sitting near the edges and the corners of the cubic rock sample, cannot be well resolved by tomography due to the incomplete ray coverage. Wu and Lees [1999] suggested that one method to estimate the effects of ray coverage is to perform a pre-inversion using a synthetic data set including travel-times from a constant-velocity model with zero residuals using the real ray coverage. The pre-inversion result will reflect only the effects of ray coverage since there are no structural variations in the constant velocity model. Blocks with poor ray coverage should generally be discounted in subsequent interpretations of the real data inversions. A.4 Travel-Time Inversion An over-determined problem, where the number of travel-time observations is greater than the number of model blocks, can be solved by standard techniques. The least squares solution follows, (A.6) However, in tomography problems this formula can almost never be used since the matrix is invariably singular or extremely ill-conditioned. Some of the ray-paths may be nearly identical while some of the blocks may not be sampled by any of the raypaths. These difficulties can be reduced in the case of small matrices with linear algebra techniques such as singular value decomposition (SVD) technique. A.4.1 Singular Value Decomposition (SVD) SVD technique is used to obtain a solution for the velocity structure. SVD is a method of analyzing and solving least squares problems that is of particular interest in ill-conditioned and/or rank-deficient systems. Formally, singular value decomposition of a matrix (G) is a factorization of the form, where U is an m by m orthogonal matrix with columns that are unit basis vectors spanning the data space. V is an n by n orthogonal matrix with columns that are basis vectors spanning the model space. And S is an m by n diagonal matrix with nonnegative diagonal elements called singular values. 175

198 If p singular values are nonzero, we can partition S as and, we can simplify the SVD of G into its compact form (Aster et al., 2004). A.4.2 Picard Cut-Off Number The idea behind the Picard cut-off number is to limit the degree of freedom in the model and fit the data to an acceptable level. A necessary condition for obtaining good regularized solutions is that the Fourier coefficients of the right-hand side, when expressed in terms of the generalized SVD associated with the regularization problem, on average decay to zero faster than the generalized singular values. This is the discrete Picard condition. Therefore, the Picard cut-off number is the number of singular values that are used in the solution, while the rest of the singular values are cut off from the SVD solution for more accuracy in the inversion result. By changing this number, the solution which has the minimum travel-time residual can be obtained. (A.7) The figure below is a schematic representation of the Picard plot in (Aster et al., 2004). It shows that the values for reach a noise floor after (i=11). On the other hand, the singular values continue to decay. As a consequence, the ratios increase rapidly. It is clear from this plot that we cannot expect to obtain useful information from the singular values beyond p = 11. Figure A.8: Schematic representation of Picard Plot (Aster et al., 2004). 176

199 Therefore, the Picard condition indicates that when singular values larger than a specific number are used in the solution, the resolved model will become unstable, and thus, erroneous. It is necessary for us to find out the appropriate Picard cut-off number to obtain reasonable solutions to our inversion problem. A.5 Isotropic Synthetic Modelling A.5.1 Inversion of the Synthetic Results Built upon Hypothetical Isotropic Models The synthetic P wave travel-times between the 18 transducers or any set of active transducers as mutual shot/receiver pairs can be calculated separately in a sample. A variety of heterogeneous zones with rectangular cubic shapes are applied with different sizes, geometries and numbers for synthetic models. The figure below is a sample of a synthetic isotropic model with one small heterogeneous zone as well as its recovered velocity structure result based on an isotropic inversion method in a sample divided to cubic cells. Each cubic cell size is. The heterogeneity is 1% less in value from the hypothetical intact rock velocity (5500 m/s), and the Picard cut-off number used is 35, meaning that 35 smallest singular values are thrown out of the calculations in the inversion process. Clearly, the block containing the heterogeneous rectangular cube is showing a very distinct velocity anomaly with respect to its surrounding blocks. However, the magnitude of the velocity is not perfectly solved as the recovered intact rock velocity is a bit higher than the initial synthetic velocity. Moreover, the recovered heterogeneous zone velocity equal to 5475 m/s is also higher than its equivalent initial synthetic one of 5445 m/s. This indicates that our inversion method makes slight changes in range of the velocity values. This effect will get elevated with a larger size and number of heterogeneous zones. However, as it will be shown below, accuracy of the recognized anomalies will stay decently correct. 177

200 Figure A.9: Left: Synthetic isotropic model with one heterogeneous zone of Δv = - 1% ; Right: Recovered velocity model with Pout=35. It needs to be mentioned that the arrival-times that are used in the inversion process are the time difference between travel-times in the heterogeneous sample subtracted from travel-times in the homogeneous sample (with a constant background velocity) for each ray-path. It is empirically confirmed that the inversion method gives much more accurate results using these reduced travel-time values as data. Thus, the figure above demonstrates a solid tomography result for inversion of travel-times in a homogeneous sample with no apparent discrepancies that may have happened due to the abovementioned suggestion by Wu and Lees [1999]. A.5.2 Checker-Board Model The Checker-board model is a standard, yet very complicated model, to be recovered by a tomography method for accuracy evaluation purposes. Travel-times are calculated based on an isotropic checker-board model with the heterogeneity of 5% below the value of the hypothetical intact rock velocity (5500 m/s). There are 135 and 375 Synthetic arrival-times of the transducer-to-transducer surveys respectively between 18 and 30 sensors calculated in the isotropic model and are inverted using the SVD method. 178

201 Figure A.10: Hypothetical isotropic checker-board model with 5% heterogeneity. The following figures show the inversion results from the two different sensor configuration designs. It is clear that with the 135 ray-paths, some blocks on the edges are not showing any anomaly because of the lack of ray coverage, but this problem is resolved in the case of 375 ray-paths. While velocity ranges in the inverted models (~ m/s) are not too different from the initial synthetic model ( m/s), in the interpretations, we still cannot rely on the inverted velocity values for different blocks, especially since all the blocks are not covered with the same amount of ray-path lengths and show different inverted velocities from block to block. However, for those blocks that are well covered by ray-paths, we can count on corresponded variations from their neighboring zones. 179

202 Figure A.11: Inversion of the hypothetical isotropic checker-board model with 18 sensors. Figure A.12: Inversion of the hypothetical isotropic checker-board model with 30 sensors. 180

203 A.6 Anisotropic Travel-Time Tomography As stated before, the anisotropy caused by aligned cracks in the rocks would have two major effects on seismic wave propagation. First, the velocities of both P waves (with longitudinal particle motion) and shear waves (with transverse motion) vary with the direction of propagation. Second, on entering a region of anisotropy, shear waves split into the two (or more) phases with fixed velocities and fixed polarizations that propagate in that particular direction through the anisotropy (Crampin, 1987). In our experimental setup, S wave tomography is not feasible because of the transducers used are essentially polarised orthogonal to the rock surface and therefore, radiation patterns for S waves are limited. Future sensors may be multicomponent but this is not without engineering and physics implications. Hence, in this chapter, we work with P wave travel-time data and can therefore focus on the affected P wave velocities. As mentioned in the previous chapters, in an anisotropic medium with orthorhombic symmetry, three axes of symmetry for crystals are mutually perpendicular. For materials with orthorhombic crystals, elastic P waves would travel with different velocities in three directions, and hence the velocity surface would convert to a near spheroid or ellipsoid in the case of weak anisotropy (Hirahara and Ishikawa, 1984). Hirahara [1988] discussed that since general anisotropic media have 21 independent elastic parameters, which are too many to be resolved in each block, we need to reduce the number of unknown parameters by assuming some symmetry in anisotropy. In his work, to investigate 3D azimuthal variations of velocity, Hirahara considered the use of a spheroidal P wave velocity surface. This was considered to be a geometrical approximation of the P wave velocity for the weakly anisotropic medium with orthorhombic symmetry, though this is not derived directly from the elastic wave equation. The figure below is a schematic representation of a material with aligned pennyshape cracks leading to either of the orthorhombic, ellipsoidal, azimuthal or elliptical anisotropies in it. As we have illustrated before, it is proven that the elastic properties of a rock that initially has a random crack distribution can attain ellipsoidal symmetry under three different principal stresses. 181

204 Figure A.13: Schematic representation of oriented penny-shaped cracks under true-triaxial stress state ( ). Our main goal was to find a physically acceptable approach to minimize the number of unknown parameters in the anisotropic tomography problem. Using the ellipsoidal anisotropic model provided in chapter 4 and the relationship between the three orthogonal velocities, we aimed at reducing the number of unknown parameters from 375 to 125 to be inverted by tomography. A.7 Anisotropic Synthetic Modeling A.7.1 Precision of Measurement in Arrival-Times Travel-time tomography in the anisotropic medium will not be as accurate as in the isotropic case because the variation between the 3D anisotropic image and the overall average velocity of the sample might exceed smaller heterogeneity percentages and cause blurry effects. Furthermore, in simulating the real data, a precision of measurement has to be applied due to the recording equipment accuracy limitations. Figure A.14: Waveform visualizer in the InSite software. 182

205 The precision of measurement or digital error in timing of waveforms in InSite is half of the sampling interval (0.1 Microsecond). That means, when we read a number it should be expressed as: the value. This is the minimum uncertainty in arrivaltime picking aside from the bigger automatic or manual picking error of arrival phases that was discussed in the previous chapter. Due to the fact that in a transducer-totransducer survey there are two mutual arrival-times for each path based on the fact that both transducers on the two sides of the ray can be transmitters and receivers, we have two measurements for a ray-path during each survey. In case of several measurement trials, one should estimate the value of the mean of the trials long with its corresponding uncertainty from the precision of measurement, which is, (A.8) Where, is uncertainty in one measurement and n is the number of measurement. Thus, error of the mean for each pair of the transducer-to-transducer survey arrival-time data is, (A.9) In order to simulate the real data, the calculated error will be randomly added to the synthetic travel-times of different ray-paths. The smallest travel-time between the closest sensors is about 4 μs, and therefore, the smallest noise/signal ratio in the traveltimes data is about 1%. Thus, we can conclude that only because of the inevitable precision of measurement, any heterogeneous zone with any geometry or size along the ray-path in an isotropic medium that makes less than 1% change in travel-time cannot be detected by the tomography method. Adding the real laboratory conditions of the stressed rock for an anisotropic sample as well as time-picking errors increases the minimum detectable heterogeneity percentage. Wu and Lees [1999] also showed that after adding 10 percent noise to their data, the inversion results deteriorated slightly, and incorrect anisotropic directions appeared even in the central, well-sampled portions of their target. Hence, we need a method to evaluate the accuracy and detectability of our anisotropic tomography method under different circumstances. Therefore, an inaccuracy factor is defined for the purpose of anomaly recognition and comparison of 183

206 synthetic models with recovered results to help in interpretation of the real data tomography. A.7.2 Evaluating the Anisotropic Tomography Method for Cubic Block- Sized Heterogeneities While the isotropic tomography method fails to reconstruct the synthetic anisotropic structure, which is also confirmed by (Watanabe et al., 1996), we still need to evaluate the accurady of our anisotropic tomographic results. Also, after the tomography images are obtained, one needs to interpret them. Besides the fact that the Picard cut-off number used for Singular Value Decomposition method directly affects the tomographic results, one needs to determine the minimum anomaly that is systematically possible to recover with the specific tomography method used under the specific conditions of the experiment including the state of stress and the background anisotropic velocity model. In order to evaluate the tomographic images, we try to program the algorithm that our eyes follow to recognize the anomalies and heterogeneous zones in an image. The cubic sample is divided to cubic cells, while there is n number of blocks on each edge of the sample. Imagine the synthetic travel-times are calculated for a synthetic heterogeneous model in which the heterogeneous zone has the shape and size of a cubic grid cell located exactly in place of one of the cells. After obtaining a tomography result, there are two approaches that can recognize a more acceptable result with a more distinguished anomalous zone. An anomaly is a change of value with respect to the surrounding cells. In the inverted model, we first characterize the anomaly in place of the supposed heterogeneous block. This is done by counting the number of neighbouring cells, which have a remarkable difference in their velocity magnitude with respect to the supposed heterogeneous block. Then, we search through all the other blocks of the sample to see whether they have similar or bigger anomalies relative to their neighbouring blocks. If we find any other block demonstrating a remarkable anomaly, it is an indication of a poor tomography result and increases the inaccuracy factor. 184

207 In order to program this condition, first the average velocity of all blocks except the supposed heterogeneous block is calculated in our inverted tomographic images built up of 125 cubic grid cells. (A.10) Then the code looks for cases in which the velocity difference between the heterogeneous block and its neighbours are even smaller than the velocity difference between the heterogeneous block and the average velocity of the bulk of the rock. For each neighboring block that is found to fulfill this condition, the inaccuracy factor which is initially zero, adds to one. The disagreeable condition investigated in our inverted tomographic images is as below, (A.11) Aside from the neighbourhood anomaly, any cell across the bulk of the sample that has comparable value with the supposed heterogeneous cell is also an indication of poor result. Thus, the code searches for any block whose velocity difference with the average velocity is equal to or bigger than the velocity difference between the heterogeneous block and average velocity. For any single block along the bulk of the sample that is found to fulfill this condition, the inaccuracy factor adds to one. The empirical condition set applied on our inverted tomographic images is as below, (A.12) Finally, the inaccuracy factor is normalized being divided by half of the total number of grid cells (125/2 as an empirical normalizing factor) and is obtained as one data point for each inversion model. In addition, having the synthetic and inverted tomographic images, the mean absolute error between the equivalent synthetic model velocities ( ) and inverted model velocities ( ) was also calculated to examine an overall difference between the velocity values of the model and the result. (A.13) 185

208 A.7.3 Anomaly Pattern Recognition Test Using the Inaccuracy Factor Specific heterogeneous models are built up by moving one heterogeneous block with the same size of cells all over the cubic sample each time fitting on one cell. The whole cubic sample is based on the background anisotropic homogeneous velocity structure that is obtained from real data (figure 4.10). In order to evaluate the anisotropic tomography with the SVD inversion method, two operations are accomplished to investigate the consistency between the synthetic and inverted models to determine an optimal value for Picard cut-off number in the SVD inversion. Also, the minimum heterogeneity percentage that can be recovered is investigated. Figure A.15: The cubic heterogeneous block with the same size of the cells will move onto each cell in every corresponding synthetic model. For each Picard cut-off number, synthetic models and their inverted results are generated. Each model is built up with a background anisotropic velocity structure with one heterogeneous zone by shape of a cube with the same size of the cells. The background homogeneous ellipsoidal anisotropic velocity model that is obtained from 186

209 averaging the real laboratory experimental data is Vx= 4729 (m/s), Vy= 5211 (m/s), Vz= 5633 (m/s). This velocity model was used to generate travel-time data for our synthetic model to be inverted by our tomography method. Figure A.16: Left: a) Stereonet representation of the real velocities in a stage of the experiment where,. b) the calculated default homogeneous ellipsoidal anisotropic velocity model that is used as the background velocity structure of the synthetic model. It is obtained from averaging the real laboratory travel-time data: Vx= 4729 (m/s), Vy= 5211 (m/s), Vz= 5633 (m/s). Difference between heterogeneous model travel-times and homogeneous one is incorporated in the inversion calculations. Right: The flowchart for anomaly recognition test. In total, 115 different Picard cut-off numbers were used to perform the SVD inversion method. For each case of SVD inversion with a specific Picard cut-off number, 125 different models were created according to the number of cells through which the 187

210 heterogeneous block is supposed to place. Therefore, in total, models were generated and travel-times between the fixed transducers were calculated. Precision of measurements was also randomly applied to the travel-time data. The mean absolute error between the synthetic and inverted models is presented in a log-normal plot. As we can see in the figure below, the semblance between the synthetic and inverted velocity image increases as the mean absolute error decreases by the Picard cut-off number. However, this numerical factor representing the values of the models does not necessarily mean that the anomalies are the same. That is why we have used the inaccuracy factor to solve for tomographic results with least ambiguity. All of the 125 mean absolute errors and inaccuracy factor parameters obtained for each Picard cut-off number were separately averaged or normalised by dividing the summation of them to the number of models (125). Under the specific modelling conditions adopted from the real test, the best picard cut-off number was chosen to be 30 based on the minimized mean absolute error and inaccuracy factor (both together). It is apparent from the figures that while the mean absolute error decreases versus Picard Cut-off Number, the accuracy of the result does not necessarily increase. Figure A.17: Left: Mean absolute error between the overall synthetic and inverted model. Right: Inaccuracy factor vs. Picard cut-off number. 188

211 A.7.4 Detectability of Heterogeneous Cells A feasibility study is also performed to evaluate the heterogeneity detectability of the method. After the best Picard cut-off number for the specific anisotropic velocity model obtained from the test is determined, then the Inaccuracy factor of the 125 different synthetic models with the same Picard Cut-off Number (30) as well as the same background anisotropic velocity structure are calculated, but for different heterogeneity percentages. The heterogeneity percentages were changing between 0.5% to 30% with intervals of 0.5% in order to find the minimum heterogeneities that can be recovered with our tomographic inversion method. Again, all of the 125 inaccuracy factors obtained for each heterogeneity value were separately averaged or normalised by dividing the summation of them to the number of models (125). The images produced from our inversion technique have been shown to be valid for velocity contrasts greater than 5% as the inaccuracy factor increases for smaller heterogeneities. Young et al. [1989] (II) also found that the geotomographic images produced from both theoretical and experimental modelings were valid for velocity contrasts greater than 5%. Figure A.18: Detectability of Heterogeneity. Inaccuracy factor of the 125 different synthetic models with same Picard Cut-off Number (30) as well as the same background anisotropic velocity structure versus different heterogeneity percentages. 189

212 We can also perform such a feasibility study to find an upper limit of heterogeneity detectability for further investigation. Knowing the minimum heterogeneities that can be recovered with our tomographic inversion method can help in interpreting the tomographic images from real data. A.8 Tomography Results form the Polyaxial Experiment A.8.1 Ray Coverage The figure below shows the available ray-paths in each transducer-to-transducer velocity survey. One of the transducers on the X1 platen and one on the Z1 platen had either noisy recordings or sent inappropriate shots and were not performing well during the experiment. Therefore, they were not included in our travel-time data used in tomographic inversions. Consequently, missing two datasets from the two noisy transducers even reduced the number of P wave ray-paths traveling between the transducers and the ray coverage over the bulk of the rock. We knew that this will affect our tomographic results and make the inverted image suffer from a shortage of velocity values. Also, Wu and Lees [1999] explained that random noise is even less important than ray directional coverage in anisotropic inversions. However, with reducing the number of unknown parameters under the ellipsoidal anisotropic model, we were able to tackle the underdetermined problem. Figure A.19: Ray coverage of the cubic sample during the polyaxial experiment. 190

213 Figure A.20: Coverage of the cubic sample during the polyaxial experiment. Figure A.21: Coverage of the cubic sample during the polyaxial experiment. 191

214 Figure (A.20) shows the blocks that are not probed by the 106 available ray-paths in our experiment. Our results, therefore, are limited to the white blocks shown below. Figure (A.21) shows the ray coverage in different blocks obtained from G-matrix calculation for 106 rays. The number of non-probed cells increased from 10 blank blocks for a system of 18 transducers with 135 rays down to 15 blank blocks for a system of 16 transducers with 106 ray-paths. The ray coverage image can help in the interpretation of the tomographic images from real data. A.8.2 Cross-Correlation for Real-Time Monitoring Generally, it is desirable to pick the first arrival-time, which corresponds to the moment at which the first energy arrives at a receiver; however, this can often be difficult to determine consistently between receivers, even by using automatic picking procedures. Therefore, using cross-correlation will be useful for detecting small changes in the waveforms with the same ray-paths. The delays between common ray-paths collected during consecutive velocity surveys can be measured using a cross-correlation technique. Travel-time delays measured using cross-correlation (CCR) techniques result in the most accurate sequential images (Maxwell & Young 1992). Sequential images will be able to isolate and map the time dependent effects of the state of stress and fracturing of a rock mass. Sequential imaging may also prove to be a practical tool for mapping the temporal changes in the seismic potential of a rock mass, by using the established seismicityvelocity correlation (Maxwell and Young, 1992). Maxwell & Young [1992] found that computing images of velocity difference by inverting changes in measured travel-times was more accurate than simply computing the difference between successive velocity images. Furthermore, measuring travel-time changes with cross-correlation techniques was found to be more accurate than measuring the difference between successive manually picked travel-times and resulted in more accurate velocity difference images. The advantage of the technique is that by mathematically comparing the two arrivals, a much more precise and accurate time difference is measured than if both arrivals were manually picked. This is because each manual pick has a relatively large uncertainty due to where the operator can visibly pick the arrival-time from the noise level. By over-sampling the data, the CCR is able to measure a time shift that can be 192

215 smaller than a single real sample point. Such high precision is required so as to obtain sensitivity to small changes in apparent velocity. Figure A.22: The cross-correlation window in InSite software. Waveforms for rays traversing identical paths during successive cycles can be cross-correlated with those from the initial reference scan by windowing the first portion of the arrival. This procedure can yield the change in travel-time along that path with a resolution of 0.01 micro sec, from which the difference in the slowness (the inverse of velocity) is computed. A.8.3 Consistency with Micro CT Images and AE Locations The figure below shows the picture of the cracks and fractures in two different experiments. The first image on the left is a picture of the Y2 surface of the rock (in the XoZ plane) after a previous polyaxial experiment in which the loading pattern resembled our investigated polyaxial experiment; but unlike the last experiment, the main principal stress did not increase up to the criteria of the failure of the rock. Therefore, the rock was still intact when it was taken out of the geophysical imaging cell and we could take a picture of the fractures on it. The image on the right is a micro computed tomography (CT) image from the middle of the rock in the XoZ plane viewpoint. Our Specimen was sent to the University of Texas in Austin for Micro-CT imaging. It was scanned under a high energy system for a duration of 16 hours at the order of 0.2 mm slice spacing and about 0.12 mm in-plane pixel (Nasseri et al., 2014). 193

216 Figure A.23: Left: picture of the rock surface after the FTB3 experiment on the same kind of sandstone. Right: CT scan image of the FTB4 experiment rock from the XoZ plane side (Nasseri et al., 2014). In both cases, there are bending parallel fractures near the two X1 and X2 facets that are the characteristic type of fractures for the true-triaxial tests also seen in (King et al., 2011). These fractures occurred before the rock failure stage, which was the case for the previous experiment shown in the left image - the test was stopped before failure and the rock was unloaded. But in the latter experiment shown in the CT scan image on the right - the rock was loaded more up to the failure stage where big intersecting fractures were visibly crossing the middle of the rock. These figures confirm that the damages mainly happen near the facets of the rock specimen in the minimum principal stress direction, which correlate with our results from the tomography method shown in chapter 4. Nasseri et al. [2014] also reported the AE events in our polyaxial experiment that were located through the InSite software. P wave arrivals were automatically picked and event hypocenter locations were calculated by the downhill Simplex method using a time-varying transverse isotropic velocity model. Their transverse isotropic velocity model for hypocenter determination was used to average the X and Y-axis velocities. The below CT scan image is superimposed with those located microseismic events that happened during stage D between the two abovementioned ultrasonic transducer surveys from which tomography images were produced. The AE event locations that occurred all over the bulk of the rock were projected on the plane normal to the intermediate principal stress direction (XoZ plane). These images confirm that the tomographic results obtained from the velocity variation shown in chapter 4 correlates well with the AE locations regarding the location of the induced damages. In figure 194

217 (4.20), the left and right hand side of all the cross-sectional images of the cubic sample, which show the tomographic anomalies near the X platens, obviously show a reduction in velocity representing the developing unsaturated fractures and is consistent with the AE events locations corresponding to the parallel fractures on the sides Figure A.24: CT scan image of the rock superimposed with located AE events that occurred in stage D of the experiment. The red dots indicate the AE events that happened between the two velocity surveys at ( ) and ( ) above the blue dots that indicate those AE events that already happened before. These also indicate that X facets of the rock are dilated while the middle of the rock is compacted with increased, or at least, unchanged velocity which makes a velocity gradient resulting in more AE activities. Further interpretation of the results is explained in the discussions that follow. A.9 Interpretation and Discussion Stress changes cause pre-existing micro-cracks in the rock to open and close, thereby decreasing and increasing the velocity. Furthermore, Maxwell and Young [1995] reminded us that opening micro-cracks in the tensile region in the sidewalls of the rock also probably contributes to the velocity reductions in the mining or laboratory environment. Microseismic events in mines are also associated with a zone of velocity decrease. The mechanism of the association between the seismicity and velocity decrease may be due to unclamping of the fractures. Such an unclamping would facilitate the release of stored strain energy. Damage from the induced microseismic event may also contribute to the observed velocity decrease, and could be isolated by applying the 195

218 sequential imaging technique to the foreshock/aftershock sequences (Maxwell and Young, 1993). Increasing stress is likely to increase crack densities and aspect ratios by promoting crack growth rather than by opening new cracks, which occurs only at high stress. Changing the dimensions of cracks by crack growth or changing aspect-ratio is likely to alter the pore-fluid pressure as well (at least temporarily). Detection of the saturation boundary in rocks has been the object of study for many researchers in petroleum and engineering rock mechanics for a long time. As an example, Masuda et al. [1990] showed by P wave velocity tomography using piezoelectric (PZT) transducers in the laboratory, that the decrease of P wave velocity a while after water injection is due to undersaturation as the rate of cracking exceeds the rate of fluid flow. On the other hand, significant increases of aspect-ratio may lead to under-saturation and the introduction of small vapour-filled bubbles into the cracks. Variations of pore-fluid pressure and particularly the introduction of small bubbles have little direct effect of seismic velocities and are unlikely to be easily recognized in patterns of particle displacement. The largest effect of the introduction of bubbles will be to increase attenuation. Also, Crampin [1987] explained that the dilatancy-diffusion theory of earthquake precursors suggests that changes of stress could cause fluids to drain from dilated cracks modifying wave propagation through the cracked rock. The abovementioned physics of the rock under stress is what has been occured at stage D of our polyaxial experiment, where we could reconstruct meaningful tomography images shown in chapter 4. Image resolution is also an important facet of interpretation. It may not be possible to image individual anomalies significantly smaller than the dominant seismic wavelength recorded. Young et al. [1989] (I) mentioned that the dominant wavelength is largely a function of seismic ray-path length, increasing with ray-path length and thus physical size of the object being imaged. It is also a function of source/receiver characteristics and attenuation/rock mass quality characteristics of the imaged rock volume. Thus the size of the physically smallest detectable anomaly within an object will be governed by a complicated function of transducer characteristics and spacing, and the physical properties and size of the entire object being imaged. Accurate quantitative resolution limits need to be determined as part of the interpretation strategy. The use of ray theory in tomography is based on an assumption that the 196

219 wavelength is at least several times smaller than the spatial variations of the velocity field. For many of the cases of interest in laboratory geomechanics, this will not be a fully valid assumption. Therefore, whilst reasonable results can often be achieved using ray based inversion, improvements may be possible using approaches that are not limited by the high-frequency approximation of ray-based methods. Thus, an approximated limit can be set for the dimension of the smallest features that may be imaged to be of the order of, where L is the propagation distance and λ is the wavelength (Williamson et al., 1993; Hung et al., 2001; Viggiani et al., 2012). Therefore, based on the definition of Fresnel zone in microseismic imaging to be the square root of multiplication of wavelength and the propagation distance, we chose the size of our grid cells equal to cubes of 16 mm long sides. After all, Daley et al. [2006] also quoted the statement by Mensch and Farra [1999], "Examples obtained in a homogeneous orthorhombic medium show that a reference media with ellipsoidal anisotropy is a better choice to develop the perturbation approach than an isotropic reference medium," which gives further indication that this is a reasonable approach in which to proceed. In our anisotropic tomography method, ellipsoidal anisotropic model was adapted as a reference media to detect the heterogeneities as perturbation to it. S wave velocity tomography was almost impossible due to the geometrical constraints. According to the short length of ray-paths, which results in short P and S wave travel-time difference, the S wave arrival phase picking is not possible because it will be mixed with the P wave. As a result, we only used P wave velocities in our inversion technique for tomography due to the difficulty in picking S wave arrival-times. The result of our numerical simulation shows that the anisotropic tomography successfully reconstructs the velocity structure having even 1% velocity anisotropy in case of noise free data. On the other hand, the ordinary isotropic tomography fails to reconstruct the anisotropic structure. Although both the restriction of source-receiver arrangement and the existence of noise as well as absence of point location for transducers affect the accuracy of the reconstructed images, our anisotropic tomography method can detect some heterogeneous zones of the anisotropic rock, which might be of help for better understanding subsurface structures, especially before formation of new 197

220 fractures. This may also be useful as a precursor of the new growing fractures in a saturated rock formation. 198

221 Appendix B. Flowchart of Data Acquisition and Processing ASC Pulser-Amplifier System PZT Transducer Supporting Both P-S1-S2 and Transducer-to-Transducer Velocity Surveys Acquired by Cecchi and Milne Systems True-Triaxial Geophysical Imaging Cell ASC Richter System ASC Milne System ASC Cecchi System AE Events in InSite Transducer-to-transducer Velocity Survey in InSite P-S1-S2 Velocity Survey in InSite Z dly dlx Y X AE Locations P Wave Velocity Structure & Tomography 199 Anisotropic Velocity Image

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