An Experimental and Numerical Study on the

An Experimental and Numerical Study on the Fracturing Processes in Opalinus Shale by

Stephen Philip Morgan B.S., University of New Hampshire, 2009 S.M., Massachusetts Institute of Technology, 2011 Submitted to the Department of Civil and Environmental Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Civil Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2015 c ○2015 Massachusetts Institute of Technology. All rights reserved.

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Department of Civil and Environmental Engineering August 20, 2015 Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Herbert H Einstein Professor of Civil and Environmental Engineering Thesis Supervisor Accepted by. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heidi Nepf Donald and Martha Harleman Professor of Civil and Environmental Engineering Chairman, Department Committee on Graduate Theses

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An Experimental and Numerical Study on the Fracturing Processes in Opalinus Shale by Stephen Philip Morgan Submitted to the Department of Civil and Environmental Engineering on August 20, 2015, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Civil Engineering

Abstract The extraction of hydrocarbons from unconventional oil and gas reservoirs relies on a detailed understanding of the fracture processes in shale. Also, underground structures designed for nuclear waste repositories are typically constructed in shale due to its characteristic low permeability. To understand the behavior of shale it is important to know how cracks initiate, propagate and coalesce. Although there have been many studies on the cracking processes in rock, cracking in shale is not well understood mainly due to its anisotropy, which is caused by naturally formed bedding planes. Natural bedding planes are weak zones along which cracks can initiate and propagate. As a consequence, the effect of bedding planes on crack initiation and propagation has not been captured well in previous models. A series of unconfined compression tests were conducted on Opalinus shale extracted from the Mont Terri underground rock laboratory in Switzerland. These tests consisted of prismatic Opalinus shale specimens with two pre-existing flaws and various bedding plane orientations. High speed and high resolution imagery were used to capture crack initiation, -propagation and -coalescence between the flaw pairs. It was found that as the bedding plane angle increased, cracks initiating at the flaw tips tended to propagate more frequently along the bedding planes. FROCK, a model based on the Displacement Discontinuity Method (a type of Boundary Element Model) developed at MIT, was modified to incorporate the effect of bedding planes on the crack propagation patterns. A discontinuous critical strain criterion was implemented into the model, showing acceptable predictions of the crack initiation, -propagation pattern and -mode (tensile/shear) when compared to the experimental results. The results from this thesis can be used to further improve predictive crack propagation models in anisotropic rock.

Thesis Supervisor: Herbert H Einstein Title: Professor of Civil and Environmental Engineering 3

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Acknowledgments This thesis represents the result of a long and challenging journey at MIT. It would not have been possible without the support and help from some very important people. First, I would like the thank all of my colleagues that have lent their time and energy to me and my research over the years. This includes my fellow classmates Amer, Amy, Brendan and Jana, and my lab/office-mates Bing, Wei and Bruno. I must thank my committee members Dr. Antonio Bobet and Dr. Brian Evans for generously dedicating their time and minds to better my research. I want thank Dr. John Germaine for giving such crucial advice and hands-on assistance in the laboratory. Most importantly, thank you Dr. Herbert Einstein for being such an great influence on my research and spending a tremendous amount of personal time on me and my writing. I really appreciate the guidance and support you have given me over the years. My friends and family have been the foundation of my success at MIT. During my time I have had amazing luck, meeting some extremely awesome people. This includes my first year friends Geoff and Patty. My Sidney-Pacific Executive Council team; Stephanie Nam, Jenn Jarvis, Pierre Lepage, and George Chen. My housemates; Javier Sanchez-Yamagishi, Kelli Xu and Brain Spatocco. I’ve also brought some life long friends along with me from high school, so I must thank Brandyn Morton, Ryan Gunn, Quincy Stevens Gunn and Mike Bean. Finally, its important that I thank my family; Steven (dad), Kerry (mom), Jamie and Brandon. My family has always been there to support me, in such a positive and caring manner. I really appreciate the unconditional love and affection that you have given me throughout my time at MIT. Working hard isn’t hard work if it’s for the ones you love.

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Contents

1

2

Introduction 1.1

Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.2

Research Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Background 2.1

2.2

2.3 3

27

31

Shale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1.1

"Shale" Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.1.2

Origin and Diagenesis of Shale . . . . . . . . . . . . . . . . . . . . 33

2.1.3

Experimental Work on Shale . . . . . . . . . . . . . . . . . . . . . 42

2.1.4

Anisotropic Failure Theory . . . . . . . . . . . . . . . . . . . . . . 66

Rock Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.2.1

Fracture Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.2.2

Fracture Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 88

2.2.3

Modeling Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 102

2.2.4

Modeling Anisotropic Fracture . . . . . . . . . . . . . . . . . . . . 121

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Experimental Process 3.1

131

Material - Opalinus Shale . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 3.1.1

Sample Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

3.1.2

Mineralogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

3.1.3

Natural Water Content and Saturation Level . . . . . . . . . . . . . 136 7

3.1.4 3.2

3.3

4

Natural Fractures and Shells . . . . . . . . . . . . . . . . . . . . . 140

Specimen Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 3.2.1

Cutting Prismatic Specimens . . . . . . . . . . . . . . . . . . . . . 142

3.2.2

Cutting Flaws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

3.2.3

Preservation and Storage . . . . . . . . . . . . . . . . . . . . . . . 145

3.2.4

Cutting Pierre Shale . . . . . . . . . . . . . . . . . . . . . . . . . 146

Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 3.3.1

Specimens Tested . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

3.3.2

Testing Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

3.3.3

Analysis Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Experimental Results 4.1

4.2

4.3

161

Intact Opalinus Shale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.1.1

0o Bedding Planes . . . . . . . . . . . . . . . . . . . . . . . . . . 162

4.1.2

30o Bedding Planes . . . . . . . . . . . . . . . . . . . . . . . . . . 164

4.1.3

45o Bedding Planes . . . . . . . . . . . . . . . . . . . . . . . . . . 165

4.1.4

60o Bedding Planes . . . . . . . . . . . . . . . . . . . . . . . . . . 168

4.1.5

90o Bedding Planes . . . . . . . . . . . . . . . . . . . . . . . . . . 170

4.1.6

Summary of Intact Tests . . . . . . . . . . . . . . . . . . . . . . . 172

Flaw Pairs - Horizontal Bedding Planes (ψ = 0o ) . . . . . . . . . . . . . . 178 4.2.1

Coplanar Flaws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

4.2.2

Stepped Flaws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

4.2.3

Cracking Characteristics . . . . . . . . . . . . . . . . . . . . . . . 189

4.2.4

Coalescence Summary . . . . . . . . . . . . . . . . . . . . . . . . 198

4.2.5

Crack Initiation and Coalescence Stress Summary . . . . . . . . . 201

4.2.6

Comparison to Previously Tested Rocks . . . . . . . . . . . . . . . 203

Flaw Pairs - Inclined Bedding Planes . . . . . . . . . . . . . . . . . . . . . 210 4.3.1

Cracking Characteristics . . . . . . . . . . . . . . . . . . . . . . . 213

4.3.2

Coalescence Summary . . . . . . . . . . . . . . . . . . . . . . . . 215

4.3.3

Crack Initiation and Coalescence Stress . . . . . . . . . . . . . . . 216 8

4.3.4 4.4

Comparison of Horizontal/Inclined Bedding Flaw Pair Tests . . . . . . . . 218

4.5

Tested Specimen Saturation and Resaturation . . . . . . . . . . . . . . . . 219

4.6 5

4.5.1

Saturation Level . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

4.5.2

Resaturation of Opalinus Shale . . . . . . . . . . . . . . . . . . . . 219

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

Numerical Model 5.1

5.2

5.3

5.4 6

Comparison to Previously Tested Rocks . . . . . . . . . . . . . . . 216

227

Isotropic Strain-Based FROCK Model . . . . . . . . . . . . . . . . . . . . 228 5.1.1

FROCK Modifications and Improvements . . . . . . . . . . . . . . 230

5.1.2

Calibration and Input Parameters . . . . . . . . . . . . . . . . . . . 235

5.1.3

Isotropic FROCK Model Results . . . . . . . . . . . . . . . . . . . 244

5.1.4

Bedding Plane Limitation . . . . . . . . . . . . . . . . . . . . . . 251

Anisotropic Strain-based FROCK Model . . . . . . . . . . . . . . . . . . . 251 5.2.1

Proposed Anisotropic Cracking Criteria . . . . . . . . . . . . . . . 252

5.2.2

FROCK Modifications . . . . . . . . . . . . . . . . . . . . . . . . 257

5.2.3

Calibrating Parameters . . . . . . . . . . . . . . . . . . . . . . . . 260

5.2.4

FROCK Results Using The Anisotropic Criteria . . . . . . . . . . . 261

5.2.5

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

Future Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 5.3.1

Numerical Instabilities and Code Crashes . . . . . . . . . . . . . . 281

5.3.2

Anisotropic Strain Field Analysis . . . . . . . . . . . . . . . . . . 282

5.3.3

Calibration Method . . . . . . . . . . . . . . . . . . . . . . . . . . 282

5.3.4

Anisotropic Failure Criterion . . . . . . . . . . . . . . . . . . . . . 283

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

Digital Image Processing 6.1

6.2

287

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 6.1.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

6.1.2

DIC Method Fundamentals . . . . . . . . . . . . . . . . . . . . . . 290

Proof of Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 9

6.3

6.4 7

6.2.1

Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

6.2.2

Strain Verification . . . . . . . . . . . . . . . . . . . . . . . . . . 295

6.2.3

Crack Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

6.2.4

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

Opalinus Shale DIC Results . . . . . . . . . . . . . . . . . . . . . . . . . 299 6.3.1

Axial Strain Comparison . . . . . . . . . . . . . . . . . . . . . . . 299

6.3.2

Strain At The Flaw Tip . . . . . . . . . . . . . . . . . . . . . . . . 301

6.3.3

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

Summary and Further Considerations . . . . . . . . . . . . . . . . . . . . 304

Conclusions

307

7.1

Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

7.2

Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

7.3

Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . 312 7.3.1

Microscopic Cracking Behavior . . . . . . . . . . . . . . . . . . . 312

7.3.2

Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

7.3.3

Shale Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

7.3.4

FROCK Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

A Boring Logs

329

B Pierre Shale

333

B.1 Origin and Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 B.1.1

Bentonite Seams and Natural Shells/Fossils . . . . . . . . . . . . . 335

B.2 Specific Gravity, Natural Water Content, Mineralogy . . . . . . . . . . . . 336 B.3 Drying and Sealing Techniques . . . . . . . . . . . . . . . . . . . . . . . . 338 B.4 Cutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 B.5 Pierre Shale - Unconfined Compression Tests . . . . . . . . . . . . . . . . 341 C Scaling Experiments on Prismatic Specimens

345

C.1 Intact Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 C.2 30o Single Flaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 10

C.3 30o Co-Planar Flaws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 D Induction Proximity Sensor For Lateral Displacement Measurements: Gypsum Tests

351

D.1 Background: Defining Poisson’s Ratio . . . . . . . . . . . . . . . . . . . . 351 D.2 Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 D.3 Steel Brush End Platens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 D.4 Steel Block End Platens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 D.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 D.6 Limitations and Concerns . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 E Synchronizing High Speed Data Acquisition E.0.1

359

Brief History of Data Acquisition at MIT . . . . . . . . . . . . . . 359

E.1 Automatic Triggering Switch . . . . . . . . . . . . . . . . . . . . . . . . . 361 E.1.1

Relay Switch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

E.1.2

Triggering System . . . . . . . . . . . . . . . . . . . . . . . . . . 363

F Running FROCK on Windows Machines - Cygwin

365

F.1

Running FROCK on Windows PC . . . . . . . . . . . . . . . . . . . . . . 365

F.2

Installing Cygwin/X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

F.3

Running Cygwin and Startxwin . . . . . . . . . . . . . . . . . . . . . . . . 366

F.4

Running and Compiling FROCK . . . . . . . . . . . . . . . . . . . . . . . 366

F.5

Running and Compiling Post . . . . . . . . . . . . . . . . . . . . . . . . . 369

G Intact Test Analysis

371

G.1 Intact Opalinus Shale - 0o Bedding Planes . . . . . . . . . . . . . . . . . . 371 G.2 Intact Opalinus Shale - 30o Bedding Planes . . . . . . . . . . . . . . . . . 372 G.3 Intact Opalinus Shale - 45o Bedding Planes . . . . . . . . . . . . . . . . . 372 G.4 Intact Opalinus Shale - 60o Bedding Planes . . . . . . . . . . . . . . . . . 373 G.5 Intact Opalinus Shale - 90o Bedding Planes . . . . . . . . . . . . . . . . . 373 H Flaw Pair Test Analysis

375

11

12

List of Figures 2-1 Pressure and temperature plot of shale diagenesis . . . . . . . . . . . . . . 34 2-2 Physical Shale Formation Mechanisms . . . . . . . . . . . . . . . . . . . . 36 2-3 Pressure Solution of Grain Boundaries . . . . . . . . . . . . . . . . . . . . 37 2-4 Illite/Smectite Change Over Depth (Temperature and Pressure) . . . . . . . 39 2-5 Chlorite Change Over Depth (Temperature and Pressure) . . . . . . . . . . 40 2-6 Kaolinite Change Over Depth (Temperature and Pressure) . . . . . . . . . 41 2-7 Quartz Change Over Depth (Temperature and Pressure) . . . . . . . . . . . 41 2-8 Definition of load orientation angle and bedding angle . . . . . . . . . . . . 43 2-9 Green River Shale Tested by McLamore and Gray(1967) . . . . . . . . . . 44 2-10 Failure Modes of Anisotropic Shale Tested by McLamore and Gray (1967)

45

2-11 Definition of the loading orientation in triaxial samples tested by Niandou et al. (1997) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2-12 Effect of loading and bedding orientation on failure strength observed by Niandou et al. (1997) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2-13 Sketch of fracture patterns observed in triaxial tests conducted by Niandou et al. (1997) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2-14 Failure modes in shale observed by Aristorenas(1992) . . . . . . . . . . . . 49 2-15 Excess pore pressure observed in undrained tests by Aristorenas(1992) . . . 50 2-16 Fracture in extension tests proposed by Aristorenas(1992) . . . . . . . . . . 51 2-17 Fracture in compression tests proposed by Aristorenas(1992) . . . . . . . . 51 2-18 Typical response of an unconfined compression test on Opalinus shale) . . . 52 2-19 Cyclic unconfined compression tests conducted on Opalinus shale . . . . . 53 13

2-20 Stress-strain behavior of triaxial tests conducted on Opalinus shale by Corkum and Martin (2007) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2-21 Triaxial tests conducted by Naumann et al. (2007) on Opalinus shale at various bedding angles and confining pressures . . . . . . . . . . . . . . . 54 2-22 Opalinus shale specimen configuration tested by Amann et al.(2011) . . . . 55 2-23 Stress-strain-AE results on unconfined compression test of Opalinus shale (Amann et al., 2011) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2-24 Conceptual model of crack initiation and growth in a layered material (Amann et al., 2011) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2-25 Relationship between water content and suction for Opalinus shale . . . . . 58 2-26 Relationship between saturation and suction for Opalinus shale . . . . . . . 58 2-27 Unconfined compressive strength of Opalinus shale depending on suction level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2-28 Brazilian tensile strength of Opalinus shale depending on suction level . . . 60 2-29 Relation between water content and unconfined compressive strength for Opalinus shale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2-30 Specimen coring and test setup used by Tien et al.(2006) . . . . . . . . . . 65 2-31 Failure stresses of artificial anisotropic rock at different bedding angles and confining stresses(Tien et al., 2006) . . . . . . . . . . . . . . . . . . . . . 65 2-32 Initial cracking observed by Tien et al. (2006) for ψ=0o bedding angle test . 66 2-33 Four failure modes observed by Tien et al. (2006) . . . . . . . . . . . . . . 67 2-34 Definition of shear and normal stresses acting on on a discontinuity plane . 70 2-35 Plane of weakness theory u-curve . . . . . . . . . . . . . . . . . . . . . . 70 2-36 The variation of φ -c with loading angle for Angers schist tested by Duveau et al. (1998) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2-37 An example of the McLamore and Gray criterion fitted to tests on Angers schist by Duveau et al. (1998) . . . . . . . . . . . . . . . . . . . . . . . . 73 2-38 An example of the Ramamurthy et al. (1993) criterion fitted to tests on Angers schist by Duveau et al. (1998) . . . . . . . . . . . . . . . . . . . . 74 14

2-39 Orientation of bedding planes, loading direction and STN system by (Pei, 2008) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2-40 AMN criterion compared to Martinsburg slate tested by Donath (1964) . . . 76 2-41 Tensile strength variation with loading angle proposed by Buczek and Herakovich (1985) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2-42 Approximation of Tangential Stress at a Flaw Tip. . . . . . . . . . . . . . . 79 2-43 Modes of Cracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2-44 Sliding Crack Model (Tasdemir et al., 1990) . . . . . . . . . . . . . . . . . 82 2-45 Fracture-process zone ahead of a crack in concrete. (Anderson, 2005) . . . 84 2-46 White patching (fracture process zone) observed in Barre granite (Morgan et al., 2013) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 2-47 Micro-cracking observed in marble and gypsum process zones by Wong and Einstein (2009c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 2-48 Super-position of three crack scenarios proposed by Horii and Nemat-Nasser (1985) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 2-49 A body containing vertical cracks proposed by Costin (1985) . . . . . . . . 87 2-50 "Column" crack interaction proposed by Ashby & Hallam (1986) . . . . . . 88 2-51 Shear crack zone shapes in previous research . . . . . . . . . . . . . . . . 90 2-52 Seven crack types proposed by Wong and Einstein (2009b) . . . . . . . . . 91 2-53 Definition of Flaw Pairs Geometries . . . . . . . . . . . . . . . . . . . . . 92 2-54 Coalescence patterns observed by Shen et al.(1995) . . . . . . . . . . . . . 94 2-55 Coalescence patterns observed by Shen et al.(1995) . . . . . . . . . . . . . 95 2-56 Coalescence types proposed by Bobet and Einstein (1998a) . . . . . . . . . 96 2-57 Coalescence results for friction µ = 0.6 (Wong and Chau, 1998) . . . . . . 97 2-58 Coalescence results for friction µ = 0.7 (Wong and Chau, 1998) . . . . . . 98 2-59 Coalescence results for friction µ = 0.9 (Wong and Chau, 1998) . . . . . . 98 2-60 Nine coalescence patterns proposed by Wong and Einstein (2009a) . . . . . 100 2-61 Coalescence behavior of Barre granite (Morgan et al., 2013) . . . . . . . . 101 2-62 White patching observed in Barre granite between flaw pairs (Morgan et al., 2013) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 15

2-63 Extended Finite Element Model of Notch Test. . . . . . . . . . . . . . . . 103 2-64 FROCK stress field around crack tip. . . . . . . . . . . . . . . . . . . . . . 105 2-65 FROCK crack elements at the flaw tip. . . . . . . . . . . . . . . . . . . . . 106 2-66 A comparison of Stress-based and Strain-based FROCK results (Gonçalves da Silva and Einstein, 2013) . . . . . . . . . . . . . . . . . . . . . . . . . 107 2-67 FROCK strain field around crack tip. . . . . . . . . . . . . . . . . . . . . . 108 2-68 Diagram of Bonded Particle Model Interactions. . . . . . . . . . . . . . . . 109 2-69 Bonded Particle Model Cracking results from Zhang and Wong (2013). . . 110 2-70 Discrete block elements of the Universal Distinct Element Code (UDEC) . 111 2-71 Universal Distinct Element Code (UDEC) element interactions. . . . . . . . 111 2-72 UDEC fracturing results for unconfined compression and brazilian tests in plaster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 2-73 Element interface yielding and debonding (cracking) behavior in Y-Geo . . 114 2-74 Cohesive crack propagation model in the FDEM Y-Geo model . . . . . . . 114 2-75 Crack propagation in XFEM and DFEM . . . . . . . . . . . . . . . . . . . 115 2-76 Image processing mineral map of Brazilian test on granite . . . . . . . . . 116 2-77 Experimental and Y-Geo simulation results for a Brazilian test on granite. . 117 2-78 Kernel function definition for SPH . . . . . . . . . . . . . . . . . . . . . . 118 2-79 Implementation of the SPH model to a solid mechanics problem . . . . . . 119 2-80 Stress-strain results of SPH model on granite fracture . . . . . . . . . . . . 119 2-81 Fracture patterns observed in granite using SPH model . . . . . . . . . . . 120 2-82 Uniaxial compression simulation in Y-Geo using the discrete approach . . . 122 2-83 Variation of strength anisotropy in the Y-Geo smeared approach . . . . . . 123 2-84 Smeared approach results for various loading plane angles in the Y-Geo model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 2-85 Effect of loading angle on simulated compressive strength in Y-Geo . . . . 124 2-86 The effect of elastic anisotropy ratio on the FROCK crack-tip stress field . . 125 2-87 The effect of bedding angle (ψ) on the FROCK crack-tip stress field . . . . 126 2-88 Variation of anisotropic strength parameter in FRACOD . . . . . . . . . . 127 2-89 Fracture results for rock specimen with 65o bedding angles in FRACOD . . 128 16

2-90 Maximum axial stress for different bedding plane angles in FRACOD . . . 128 3-1 The Mont Terri Underground Laboratory . . . . . . . . . . . . . . . . . . . 133 3-2 The geology of the Mont Terri Underground Laboratory . . . . . . . . . . . 134 3-3 Map of the Mont Terri Underground Laboratory . . . . . . . . . . . . . . . 135 3-4 Grinding Opalinus shale . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 3-5 Specific gravity test volumetric flask . . . . . . . . . . . . . . . . . . . . . 138 3-6 Water content samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 3-7 Natural features found in Opalinus shale . . . . . . . . . . . . . . . . . . . 141 3-8 Opalinus shale borings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 3-9 Cutting prismatic specimens from a cylindrical core boring . . . . . . . . . 144 3-10 Cutting flaw pairs in prismatic Opalinus shale specimens . . . . . . . . . . 145 3-11 Opalinus shale sealing and storage . . . . . . . . . . . . . . . . . . . . . . 146 3-12 A shale specimen tested in this study . . . . . . . . . . . . . . . . . . . . . 147 3-13 Definition of flaw pair geometries . . . . . . . . . . . . . . . . . . . . . . 148 3-14 The co-planar flaw pair geometries tested . . . . . . . . . . . . . . . . . . 149 3-15 The stepped flaw pair geometries tested . . . . . . . . . . . . . . . . . . . 149 3-16 Flaw pair geometries tested at various bedding angles . . . . . . . . . . . . 150 3-17 Photograph of the intact Opalinus shale test setup in the current study . . . 152 3-18 Intact Opalinus shale test setup diagram . . . . . . . . . . . . . . . . . . . 152 3-19 Photograph of the test setup used for specimens with flaw pairs in the current study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 3-20 Flaw pair test setup diagram . . . . . . . . . . . . . . . . . . . . . . . . . 154 3-21 PhotronTM SA-5 high speed camera . . . . . . . . . . . . . . . . . . . . . 155 3-22 NikonTM D90 high resolution camera . . . . . . . . . . . . . . . . . . . . 156 3-23 Dolan-Jenner FiberliteTM MI-150 fiber optic lighting . . . . . . . . . . . . 156 3-24 Example analysis for Opalinus shale specimen with flaw pairs . . . . . . . 159 3-25 Example stress-strain graph for Opalinus shale specimen with flaw pairs . . 160 4-1 An unconfined compression test on intact Opalinus shale . . . . . . . . . . 162 4-2 Crack progression of intact Opalinus shale with 0o bedding planes . . . . . 163 17

4-3 Stress-strain data for unconfined compression tests loaded with bedding planes inclined at 0 degrees (ψ=0o ) . . . . . . . . . . . . . . . . . . . . . 164 4-4 Crack progression of intact Opalinus shale with 30o bedding planes . . . . 165 4-5 Stress-strain data for unconfined compression tests loaded 30 degrees with bedding planes inclined at 30 degrees (ψ=30o ) . . . . . . . . . . . . . . . . 166 4-6 Crack progression of intact Opalinus shale with 45o bedding planes . . . . 167 4-7 Stress-strain data for unconfined compression tests loaded 45 degrees with bedding planes inclined at 45 degrees (ψ=45o ) . . . . . . . . . . . . . . . . 167 4-8 Crack progression of intact Opalinus shale with 60o bedding planes . . . . 168 4-9 Stress-strain data for unconfined compression tests loaded 30 degrees with bedding planes inclined at 60 degrees (ψ=60o ) . . . . . . . . . . . . . . . . 169 4-10 Crack progression of intact Opalinus shale with 90o bedding planes . . . . 170 4-11 Stress-strain data for unconfined compression tests loaded 90 degrees with bedding planes inclined at 90 degrees (ψ=90o ) . . . . . . . . . . . . . . . . 171 4-12 Summary of Opalinus shale stiffness (E) and compressive strength (σc at different bedding plane angles determined from unconfined compression tests172 4-13 Single plane of weakness theory (Jaeger, 1960) applied to intact Opalinus shale results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4-14 Variable cohesion theory (Jaeger, 1960) applied to intact Opalinus shale results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 4-15 Variable strength theory (Ramamurthy et al., 1993) applied to intact Opalinus shale results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 4-16 Crack progression of Opalinus shale with a 2-0-0 flaw pair . . . . . . . . . 180 4-17 Crack progression of Opalinus shale with a 2-30-0 flaw pair . . . . . . . . 181 4-18 Crack progression of Opalinus shale with a 2-45-0 flaw pair . . . . . . . . 182 4-19 Crack progression of Opalinus shale with a 2-60-0 flaw pair . . . . . . . . 183 4-20 Crack progression of Opalinus shale with a 2-75-0 flaw pair . . . . . . . . 184 4-21 Crack progression of Opalinus shale with a 2-30-30 flaw pair . . . . . . . . 186 4-22 Crack progression of Opalinus shale with a 2-30-60 flaw pair . . . . . . . . 187 4-23 Crack progression of Opalinus shale with a 2-30-90 flaw pair . . . . . . . . 188 18

4-24 En-echelon cracking observed in Opalinus shale . . . . . . . . . . . . . . . 190 4-25 Natural en-echelon crack formation observed by Beach (1975) . . . . . . . 191 4-26 En-echelon cracking observed in Carrara marble tested by Wong (2008) . . 192 4-27 En-echelon cracking observed in Polymethylmethacrylate (PMMA) by Petit & Barquins (1988) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 4-28 Comparison of en-echelon cracking observed in Opalinus shale, Carrara marble and PMMA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 4-29 New type of tensile crack observed in Opalinus shale . . . . . . . . . . . . 194 4-30 White Patching Observed in Opalinus Shale. The above image shows the progression(a-c) of surface brightening (white patching) in Opalinus shale. The bottom images highlight the area of white patching and cracking. . . . 196 4-31 Optical USB VehoTM microscope . . . . . . . . . . . . . . . . . . . . . . 196 4-32 Microscopic images of cracking in Opalinus shale . . . . . . . . . . . . . . 197 4-33 Summary of the coalescence behavior observed in conducted on Opalinus shale with flaw pairs (bedding planes horizontal). . . . . . . . . . . . . . . 200 4-34 Coalescence behavior observed in Opalinus shale with respect to flaw angle (β ) and flaw pair bridging angle (α) . . . . . . . . . . . . . . . . . . . . . 201 4-35 Crack initiation stress of Coplanar Flaw Pairs . . . . . . . . . . . . . . . . 202 4-36 Crack initiation stress of Stepped Flaw Pairs. . . . . . . . . . . . . . . . . 203 4-37 Coalescence stress of Coplanar Flaw Pairs . . . . . . . . . . . . . . . . . . 204 4-38 Coalescence stress of Stepped Flaw Pairs . . . . . . . . . . . . . . . . . . 204 4-39 A comparison of the cracking processes observed in Opalinus shale, Barre granite, Hydrocal-B11 gypsum, and Carrara marble . . . . . . . . . . . . . 206 4-40 Coalescence behavior observed in Opalinus shale, gypsum, marble and granite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 4-41 Crack progression of Opalinus shale with a 2-30-30(30) flaw pair . . . . . . 211 4-42 Crack progression of Opalinus shale with a 2-30-30(60) flaw pair . . . . . . 212 4-43 Crack progression of Opalinus shale with a 2-30-30(90) flaw pair . . . . . . 213 4-44 Coalescence results of the bedding plane tests . . . . . . . . . . . . . . . . 216 4-45 Crack initiation stress of bedding plane tests with flaw pairs . . . . . . . . . 217 19

4-46 Coalescence stress of bedding plane tests with flaw pairs . . . . . . . . . . 217 4-47 Shale resaturation technique . . . . . . . . . . . . . . . . . . . . . . . . . 221 4-48 Change in water content of resaturated Opalinus shale . . . . . . . . . . . . 221 4-49 Resaturated versus sealed Opalinus coalescence stress behavior . . . . . . . 223 4-50 Resaturated versus sealed Opalinus cracking behavior . . . . . . . . . . . . 224 5-1 FROCK strain field around crack tip. . . . . . . . . . . . . . . . . . . . . . 229 5-2 New post-processing visualization code for FROCK . . . . . . . . . . . . . 231 5-3 Previous tensile-shear determination in FROCK developed by Gonçalves da Silva (2009) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 5-4 The process for determining crack mode (tensile/shear) in the updated FROCK model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 5-5 Crack tip designation used for automatic coalescence determination . . . . 234 5-6 A schematic of the automatic coalescence determination technique . . . . . 234 5-7 Calibration steps to determine the input parameters for Opalinus shale used in the isotropic strain-based FROCK developed by Gonçalves da Silva and Einstein (2013) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 5-8 The crack initiation stresses in FROCK determined for an array of εθcrit and γθcrit values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 5-9 Defining the crack initiation stress planes determined from FROCK . . . . 240 5-10 Total square error optimization of crack initiation stress based on εθcrit and γθcrit for simulations on all flaw pair geometries . . . . . . . . . . . . . . . 241 5-11 Calibration of εθcrit and γθcrit based on crack pattern comparison . . . . . . . 242 5-12 Calibration of µ based on crack pattern comparison . . . . . . . . . . . . . 243 5-13 Isotropic FROCK model results for Opalinus shale with coplanar flaw pairs 245 5-14 Isotropic FROCK model results for Opalinus shale with stepped flaw pairs . 246 5-15 Comparison of the crack initiation (CI) stress of coplanar flaws in the Isotropic FROCK model and the experimental results . . . . . . . . . . . . 248 5-16 Comparison of the crack initiation (CI) stress of stepped flaws in the Isotropic FROCK model and the experimental results . . . . . . . . . . . . . . . . . 248 20

5-17 Comparison of the coalescence stress of coplanar flaws in the Isotropic FROCK model and the experimental results . . . . . . . . . . . . . . . . . 250 5-18 Comparison of the coalescence stress of stepped flaws in the Isotropic FROCK model and the experimental results . . . . . . . . . . . . . . . . . 250 5-19 The limitations of the Isotropic FROCK model to predict bedding plane crack patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 5-20 Parameters and angles defined in the newly modified FROCK model . . . . 254 5-21 Definition of the discontinuous criterion implemented into FROCK . . . . . 255 5-22 Definition of the continuous criterion implemented into FROCK . . . . . . 257 5-23 An example of an input file (".inp") header with the new parameters added to FROCK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 5-24 Modified output visualizer for FROCK with bedding planes . . . . . . . . . 259 5-25 Modified critical propagation direction search method for FROCK . . . . . 260 5-26 An example of a FROCK numerical instability . . . . . . . . . . . . . . . . 262 5-27 Calibration steps used to determine the input parameters for Opalinus shale used in the anisotropic strain-based FROCK developed in this study . . . . 263 5-28 Discontinuous FROCK model bedding plane results for Opalinus shale . . . 266 5-29 Discontinuous FROCK model results for Opalinus with coplanar flaw pairs and horizontal bedding planes . . . . . . . . . . . . . . . . . . . . . . . . 267 5-30 Discontinuous FROCK model results for Opalinus with stepped flaw pairs and horizontal bedding planes . . . . . . . . . . . . . . . . . . . . . . . . 268 5-31 The crack initiation stress predicted by the Discontinuous FROCK model for various bedding planes compared to the experimental results . . . . . . 269 5-32 The crack coalescence stress predicted by the Discontinuous FROCK model for various bedding planes compared to the experimental results . . . . . . 270 5-33 Continuous FROCK model bedding plane results for Opalinus shale . . . . 273 5-34 Continuous FROCK model results for Opalinus shale with coplanar flaw pairs and horizontal bedding planes . . . . . . . . . . . . . . . . . . . . . 274 5-35 Continuous FROCK model results for Opalinus with stepped flaw pairs and horizontal bedding planes . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 21

5-36 The crack initiation stress predicted in the Continuous FROCK model for various bedding planes compared to the experimental results . . . . . . . . 277 5-37 The crack coalescence stress predicted in the Continuous FROCK model for various bedding planes compared to the experimental results . . . . . . 277 5-38 Definition of the proposed combined continuous/discontinuous criterion . . 283 6-1 DIC analysis conducted on Berea sandstone . . . . . . . . . . . . . . . . . 289 6-2 DIC analysis conducted on Neapolitan Fine Grained Tuff . . . . . . . . . . 289 6-3 The principle of DIC analysis . . . . . . . . . . . . . . . . . . . . . . . . . 290 6-4 Sub-pixel movement calculation in the DIC method . . . . . . . . . . . . . 291 6-5 Strain calculation in the NCORR DIC software . . . . . . . . . . . . . . . 292 6-6 DIC test setup for a gypsum specimen with a flaw . . . . . . . . . . . . . . 294 6-7 DIC test on gypsum with a painted speckle pattern . . . . . . . . . . . . . 294 6-8 Comparison of average axial stain from DIC and machine (load frame) cross-head measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 6-9 The strain transformations conducted on the DIC results . . . . . . . . . . 297 6-10 The minimum principal strains determined using DIC at the tip of a flaw in gypsum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 6-11 Comparison of average axial stain from DIC and machine measurements observed in a 2a-30-90 Opalinus shale specimen . . . . . . . . . . . . . . . 300 6-12 Opalinus shale specimen 2a-60-0C used to determine the strain at the inner tip of the left flaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 6-13 The minimum principal strain and max shear strains determined using DIC at the tip of a flaw in Opalinus shale . . . . . . . . . . . . . . . . . . . . . 302 7-1 The effect of bedding planes on cracking of Opalinus shale with flaw pairs . 310 A-1 Boring Log for Core #25 . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 A-2 Boring Log for Core #26 . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 A-3 Boring Log for Core #28 . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 B-1 Location of Pierre shale sample extraction in South Dakota . . . . . . . . . 334 22

B-2 Pierre shale outcrop dig site. . . . . . . . . . . . . . . . . . . . . . . . . . 334 B-3 Pierre shale samples wrapped and sealed in plastic after extraction . . . . . 334 B-4 Bentonite seams observed in Pierre shale outcrops. . . . . . . . . . . . . . 335 B-5 Fossilized Baculite shell fragment in Pierre shale. . . . . . . . . . . . . . . 335 B-6 Moisture barrier techniques and their effect on drying of Pierre shale . . . . 339 B-7 Pierre shale being cut in the bandsaw . . . . . . . . . . . . . . . . . . . . . 340 B-8 Wax dipping technique used to seal freshly cut surfaces of Pierre shale . . . 340 B-9 Chip wheel and pressurized air used to clean bandsaw blade when cutting Pierre shale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 B-10 Pierre shale specimen sdp-A tested in unconfined compression . . . . . . . 342 B-11 Pierre shale specimen sdp-B tested in unconfined compression . . . . . . . 342 C-1 Small scale gypsum specimens vs. full scale specimens . . . . . . . . . . . 346 C-2 Failure of intact small scale gypsum specimens . . . . . . . . . . . . . . . 347 C-3 Failure of small scale gypsum specimens with 30o single drilled and sawed flaws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 C-4 Failure of small scale gypsum specimens with co-planar flaws with flaw angles of 30o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 D-1 Transverse (l) and axial (L) displacements . . . . . . . . . . . . . . . . . . 352 D-2 Prismatic gypsum specimen tested using induction proximity sensors. . . . 353 D-3 Poisson testing equipment diagram . . . . . . . . . . . . . . . . . . . . . . 353 D-4 Brush end platen test setup . . . . . . . . . . . . . . . . . . . . . . . . . . 354 D-5 Steel block end platen test setup . . . . . . . . . . . . . . . . . . . . . . . 355 D-6 Plot of lateral strain vs. axial strain measured for all specimens . . . . . . . 355 D-7 Plot of stress [MPa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 E-1 Schematic of the previous synchronization process . . . . . . . . . . . . . 360 E-2 Schematic of the current synchronization process . . . . . . . . . . . . . . 362 E-3 angecraftTM W1117SIP-1 relay switch . . . . . . . . . . . . . . . . . . . . 362 E-4 Triggering system flow diagram . . . . . . . . . . . . . . . . . . . . . . . 364 23

F-1 The installation process for Cygwin used to run FROCK on windows . . . . 367 F-2 The icon placed on the desktop for Cygwin . . . . . . . . . . . . . . . . . 368 F-3 The terminal used for Cygwin . . . . . . . . . . . . . . . . . . . . . . . . 368 F-4 The X-win terminal used to run FROCK . . . . . . . . . . . . . . . . . . . 369 G-1 Lateral-axial strain results for unconfined compression tests loaded perpendicular to the bedding planes with bedding planes inclined at 0 degrees (ψ=0o ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 G-2 Lateral-axial strain results for unconfined compression tests loaded with bedding planes inclined at 30 degrees (ψ=30o ) . . . . . . . . . . . . . . . . 372 G-3 Lateral-axial strain results for unconfined compression tests loaded with bedding planes inclined at 45 degrees (ψ=45o ) . . . . . . . . . . . . . . . . 372 G-4 Lateral-axial strain results for unconfined compression tests loaded with bedding planes inclined at 60 degrees (ψ=60o ) . . . . . . . . . . . . . . . . 373 G-5 Lateral-axial strain results for unconfined compression tests loaded with bedding planes inclined at 90 degrees (ψ=90o ) . . . . . . . . . . . . . . . . 373

24

List of Tables 2.1

Summary of research conducted on Opalinus shale . . . . . . . . . . . . . 62

2.2

The four test series conducted by Wong and Einstein (2009a). Divided by ligament length and then coplanar (bridging angle of 0 degrees) and stepped flaws. All stepped flaws were oriented at an inclination of 30 degrees. 99

3.1

Material Properties of Mont Terri Opalinus Shale From The Literature . . . 132

3.2

Mineralogy of Mont Terri Opalinus Shale . . . . . . . . . . . . . . . . . . 136

3.3

Opalinus Shale Specific Gravity Tests . . . . . . . . . . . . . . . . . . . . 138

3.4

Water Content of Opalinus Shale Samples . . . . . . . . . . . . . . . . . . 140

3.5

List of Tests Conducted . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

3.6

Summary Of Techniques Used Based on Flaw Pair Geometry . . . . . . . . 157

4.1

Intact Opalinus Shale Tests - 0o Bedding Planes . . . . . . . . . . . . . . . 163

4.2

Intact Opalinus Shale Tests - 30o Bedding Planes . . . . . . . . . . . . . . 165

4.3

Intact Opalinus Shale Tests - 45o Bedding Planes . . . . . . . . . . . . . . 166

4.4

Intact Opalinus Shale Tests - 60o Bedding Planes . . . . . . . . . . . . . . 169

4.5

Intact Opalinus Shale Tests - 90o Bedding Planes . . . . . . . . . . . . . . 171

4.6

Intact Opalinus Shale Tests - Average Value Summary . . . . . . . . . . . . 172

4.7

Anisotropic Strength Theory Parameters . . . . . . . . . . . . . . . . . . . 176

4.8

Tests Observing En-echelon Cracking . . . . . . . . . . . . . . . . . . . . 190

4.9

Tests Observing New Crack: Type 4 Tensile . . . . . . . . . . . . . . . . . 194

4.10 Initial Crack Type Observed for Each Test . . . . . . . . . . . . . . . . . . 199 4.11 Initial Crack Type Observed for Bedding Tests . . . . . . . . . . . . . . . . 215 25

4.12 Saturation of Opalinus Shale Specimens . . . . . . . . . . . . . . . . . . . 220 4.13 Unconfined Compression Tests - Saturation Levels . . . . . . . . . . . . . 222 5.1

Crack Initiation Stress (σip ) Based on εcrit and γcrit . . . . . . . . . . . . . 239

5.2

Isotropic FROCK Model Input Parameters Used for Opalinus Shale . . . . 244

5.3

Evaluation System Used for FROCK Model Patterns . . . . . . . . . . . . 246

5.4

Evaluation of Isotropic FROCK Model . . . . . . . . . . . . . . . . . . . . 247

5.5

Summary of Isotropic FROCK Model and Experimental Test Results . . . . 251

5.6

Discontinuous FROCK Model Input Parameters Used for Opalinus Shale . 264

5.7

Evaluation of Discontinuous FROCK Model . . . . . . . . . . . . . . . . . 265

5.8

Summary of Discontinuous FROCK Model and Experimental Test Results . 271

5.9

Continuous FROCK Model Input Parameters Used for Opalinus Shale . . . 272

5.10 Evaluation of Continuous FROCK Model . . . . . . . . . . . . . . . . . . 275 5.11 Summary of Continuous FROCK Model and Experimental Test Results . . 278 5.12 Evaluation of FROCK Models (Total No. Xor XX) . . . . . . . . . . . . . 279 6.1

Flaw Tip Strains - 2a-60-0C Specimen in Opalinus Shale . . . . . . . . . . 303

7.1

Summary of FROCK Model Results . . . . . . . . . . . . . . . . . . . . . 312

B.1 Pierre Shale Specific Gravity Tests . . . . . . . . . . . . . . . . . . . . . . 336 B.2 Water Content of Pierre Shale Samples . . . . . . . . . . . . . . . . . . . . 336 B.3 Mineralogy of South Dakota Pierre Shale . . . . . . . . . . . . . . . . . . 337 B.4 Water Content of Pierre Shale Samples . . . . . . . . . . . . . . . . . . . . 343 C.1 Unconfined Compressive Strength of Intact Small Gypsum Specimens Compared to Normal Size Specimen by Wong (2008) and Janeiro (2009) . . . . 347 C.2 Peak Stress of 30o Single Flaw Scaled Gypsum Specimens . . . . . . . . . 348 C.3 Peak Stress of 30o Co-planar Flaw Pair Scaled Gypsum Specimens . . . . . 350 D.1 Summarized results for all specimens. The expected response from Hydrocal B-11 gypsum is also included. . . . . . . . . . . . . . . . . . . . . . . 357 H.1 Flaw Pair Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 26

Chapter 1

Introduction

Over the past few decades shale has begun to play a prominent role in the field of energy resource extraction and energy related waste storage/disposal. Specifically, the extraction of hydrocarbons from unconventional oil and gas reservoirs relies on a detailed understanding of the cracking processes in shale (Gale et al., 2007; Alexander et al., 2011; Howarth et al., 2011). Additionally, underground structures designed for nuclear waste repositories are typically constructed in shale due to its characteristic low permeability (Nagra, 2002). Specifically, how cracks initiate, -propagate and -coalesce in shale is important for the integrity, accuracy and safety of these procedures.

Although there have been many studies on the fracturing processes in rock, cracking in shale is not well understood mainly due to its anisotropy, which is caused by naturally formed bedding planes. Natural bedding planes are weak zones along which cracks can initiate and propagate. The effect of bedding planes on crack initiation and propagation has not been captured well in previous models. To understand the behavior of shale it is important to know how cracks initiate, propagate and coalesce. 27

1.1

Objective

This research consists of an experimental component, in which uniaxial compression tests are performed on shale, and a theoretical (analytical) component, in which the cracking behavior observed in the experiments is modeled.

Testing, the main part of this research, includes experiments on intact specimens (no flaws) and specimens with flaws of different geometries. Intact specimens are tested to determine the unconfined compressive strength, Poisson’s ratio and failure mechanism at various bedding plane inclinations. The flaw pair experiments are conducted following the methodologies used previously on other rocks. This consists of producing pre-existing cracks or flaws in specimens and then observing the crack initiation, - propagation and - coalescence process under applied uniaxial stress. High speed- and high resolution video is recorded during the tests to capture the cracking processes. The results of the experimental study will be systematically categorized and the observed trends compared to previous research conducted on brittle rocks.

The objective of the theoretical (analytical) part of this work is the application and additional development of the FROCK (Fractured ROCK) model to represent the cracking processes observed in the experiments. The strain-based FROCK model developed by Gonçalves da Silva (2009) will be applied. However, this model does not incorporate material anisotropy (elastic or strength). This research intends to determine any shortcomings of the model in predicting the cracking behavior in shale and then modify the model to improve its predictive capability.

In short, the objectives of this research are conduct comprehensive experiments on shale and advance analytical/models, such that this can be used to predict the fracture of shale. This has not been done before either at MIT or elsewhere and is, therefore, the major contribution of this research. 28

1.2

Research Structure

This thesis is structured as follows: chapter 2 presents the background and literature review on the state of experimental and modeling research being conducted on shale and other brittle rocks. In chapter 3, the material, i.e. Opalinus shale is discussed in detail as is the specimen preparation. The experimental setup is documented in chapter 3. The core of the thesis consists of chapters 4 and 5 with experimental results presented in chapter 4 and the analytical/numerical work in chapter 5. Chapter 6 consist of a brief study using the Digital Image Correlation (DIC) technique in Opalinus shale tests. The conclusions are given in chapter 7.

29

30

Chapter 2 Background This background chapter includes a detailed discussion on the definition and origin of shale as well as an extensive literature review of previous experimental and theoretical research conducted on both isotropic and anisotropic rocks including shale. A classification and definition of shale will be presented, and then the origin and diagensis mechanisms associated with the formation of shale will be discussed. A review of previous experimental work conducted on shale will be considered and several anisotropic failure criteria will be presented. This is followed by a brief review of fracture mechanics, fracture interaction theory and experimental work conducted on fracture interaction. Finally, this chapter will finish with a review of the modeling techniques currently available to predict fracture initiation, -propagation, and -interaction for both isotropic and anisotropic rocks such as shale.

2.1

Shale

The following section presents the definition and classification of shales as defined by classical geology and discusses how it differs from the definition used in petroleum engineering. Then the origin and diagenesis of shale will be presented. Finally, the history and current state of experimental work on shales will be discussed and several anisotropic fail31

ure criteria will be considered.

2.1.1

"Shale" Definition

Shale is a complex sedimentary rock with many different definitions, meanings and representations. The purpose of this section is to clarify the differences between how the term "shale" as defined by petroleum engineers compared to that of geologists. In a classical geology sense, "shale" is defined as a diagentic sedimentary rock comprised mostly of clay fraction particles ( Illite + Si4+

(2.1.2)

This change from smectite to illite can be seen as temperature and pressure increases (figure 2-4). Although some assumptions are necessary when comparing mineral composition at different depths, such as similar original composition and stress history, the change of smectite to illite is commonly used as an indicator of the maturity of the shale (Nadeau and Reynolds, 1981). Additional Clay Mineral Transformations In addition to smectite to illite transformations, there are many other minerological changes which can occur during shale formation. Not all mineral transformations can be discussed in detail in this study; however, it is critical to understand that other transformations may be more important depending on the specific mineral composition of the shale formation. The processes that will be discussed are the formation of chlorite, the loss of kaolinite and the increase in quartz with depth. Chlorite formation is believed to be a by-product of the smectite to illite reaction and the following reaction has been proposed by Boles and Frank (1979): 38

Figure 2-4: The change in the illite portion of the illite/smectite mixed layer observed by Hower et al. (1976) in Gulf of Mexico well borings. The clay fractions Illite +Chlorite + quartz + H +

(2.1.3)

Note that in this reaction there is no Al3+ , however, it is believed that the Al3+ necessary for the smectite to illite reaction is obtained by dissolving some of the semectite layers (Weaver, 1989). Therefore, an increase in chlorite has been observed with increasing depth in some shale formations (figure 2-5).

Figure 2-5: The change in the chlorite portion of the clay fraction observed by Hower et al. (1976) in shale. The clay fractions >2.0µm and 0.1-2.0 µm are shown. There is an increase in the proportion of chlorite has depth (temperature and pressure) increases. (Hower et al., 1976) Kaolinite is also a clay mineral which can be affected by temperature and pressure in shale formations. In certain conditions, when the pore water chemistry becomes more alkaline, kaolinite can transform to chlorite or illite (Weaver, 1989). This leads to a reduction in kaolinite content of the clay fraction with depth (figure 2-6). Finally, since there are some mineral transformations which include quartz as a component there may be some changes in the quartz content during shale formation. There has been some indication that quartz content will increase with depth (Hower et al, 1976) (figure 27). Although the relative change in quartz is not as dramatic as the transformation of other 40

Figure 2-6: The change in the kaolinite portion of the clay fraction observed by Hower et al. (1976) in shale. The clay fractions >0.1µm and 0.1-0.5 µm are shown. There is a decrease in the proportion of kaolinite as depth (temperature and pressure) increases. (Hower et al., 1976)

clay minerals, the formation of quartz can cause cementation of particles during diagenesis (Harwood et al., 2013).

Figure 2-7: The change in the quartz fraction observed by Hower et al. (1976) in shale. Various size fractions are shown. There is a slight increase in the proportion of quartz as depth (temperature and pressure) increases. (Hower et al., 1976)

41

Summary of Shale Diagensis

As previously discussed, shale diagensis is a very complex process due the variability of mineralogical composition in sedimentary deposits and the wide range of underlying mineralogy transformations which can occur under different temperature and pressure conditions. However, there are several key aspects which should be considered when selecting and testing a shale: ∙ Mineralogy composition: Clay, quartz and carbonate content ∙ Pressure and Temperature History ∙ Maturity: Formation age and deposition pathway ∙ Organic Content

2.1.3

Experimental Work on Shale

The following section gives a brief overview of the experimental research that has been conducted on shale. Since shale has been studied, in some form, for many years, only the studies most related to this thesis will be discussed in some detail. This will include the early experimentation on shale as well as the most important and relevant studies conducted on shale as it relates to this research. For both experimental and theoretical work conducted on anisotropic rocks, including shale, there are major inconsistencies regarding the definition of angles of load orientation and bedding planes. Therefore, this thesis will define the load orientation angle (θ ) as the angle between the bedding planes and the maximum principal stress direction (the vertical) (σ1 ) and the bedding plane angle (ψ) as the angle between the bedding planes and the minimum principal stress (the horizontal) (figure 2-8). It should be noted that although some studies prefer to use loading orientation while others use bedding plane angle, for the sake of consistency, the proposed terminology and notation (Refer to figure 2-8) will always be used when discussing previous studies conducted on materials with bedding planes. 42

Figure 2-8: Definition of load orientation angle and bedding angle. The load orientation angle (θ ) is the angle between the bedding planes and the maximum principal stress (σ1 ) direction (the vertical). The bedding plane angle (ψ) is the angle between the bedding plane and the minimum principal stress (σ3 ) direction (the horizontal). Initial experiments on Green River Shale will be discussed. Then experiments conducted on other shales such as Tournemire shale from France, Opalinus shale from Switzerland, Pierre shale from South Dakota and even artificially simulated shale will be presented. Emphasis will placed on studies which were conducted on Opalinus shale since this is the primary rock tested in the current study.

Green River Shale - McLamore and Gray (1967)

The study of shale failure and fracture has been studied for a long time. Although hydraulic fracturing and horizontal drilling techniques were not used as commonly as today, researchers at the time understood early on that increasing shale permeability by inducing fractures was very important for increasing oil extraction yield (Melton and Cross, 1968). The Green River oil shales were one of the first shales being studied experimentally for fracture potential and mechanical characteristics (McLamore and Gray, 1967; Melton and Cross, 1968; Chong et al., 1976). The Green River Shale, or sometimes referred to as the Colorado Oil Shale, has a high organic content and yields oil when heated to high temperatures (>371o C) (McLamore and Gray, 1967). 43

McLamore and Gray (1967) initially studied the anisotropic behavior of Green River shale. This study included triaxial tests of Green River Shale at different loading orientations and confining pressures. A reduction in fracture strength (peak deviatoric stress, σ1 − σ3 ) was observed at inclined loading angles (figure 2-9). McLamore and Gray fit theoretical predictions from Jaeger (1960) [single plane of weakness] and Walsh and Brace (1964), along with a proposed variable τ0 -tanφ theory (explained later in the thesis), to model the reduction in strength at various loading angles (Refer to figure 2-9). McLamore and Gray noted different failure mechanisms and summarized them into different categories (figure 2-10). As confining pressure increased, the failure mode of the shale changed from shearing to plastic flow or kinking (Refer to figure 2-10).

Figure 2-9: Green River Shale Tested by Mclamore and Gray(1967). The deviatoric failure strength (Fracture Strength) is plotted versus the loading orientation angle (the angle between the natural bedding angles and the minimum principal stress). Predictive models by Jaeger (1960) [single plane of weakness], Walsh and Brace (1964) and McLamore and Gray(1967) [variable τ0 -tanφ ] are shown. Modified from McLamore and Gray (1967)

44

Figure 2-10: Failure Modes of Anisotropic Shale Tested by McLamore and Gray (1967). Shear failure was observed at lower confining pressures and plastic and kink flow were observed at higher confining pressure. Modified from McLamore and Gray (1967)

Tournemire Shale - Niandou et al. (1997)

Niandou et al. (1997) conducted uniform compression and triaxial tests on Tournemire shale recovered from central France. In this study triaxial compression tests were performed on cylindrical samples (37mm x 75mm) at various loading orientations (angle θ between principal loading and bedding planes) and confining pressures (figure 2-11). Tournemire shale consists of about 15% carbonate, 19% quartz, and 55% clay (Niandou et al., 1997). From these tests the effect of loading direction and confining stress on failure stress were analyzed (figure 2-12). A decreased peak stress was observed for inclined loading direction with a minimum at a loading direction of about 30o (i.e. 60o bedding angle to horizontal) (Refer to figure 2-12). It should be noted that increasing confining pressures had little effect on the strength anisotropy (the difference between minimum and maximum failure stress with loading orientation, sometimes referred to as the depth ratio). In addition, Niandou et al. (1997) performed fracture observations of the failed specimens after the tests were complete (figure 2-13). These fractography observations were categorized by loading directions (θ = 0o -15o , 15o -60o , 60o -90o ) and confining pressure (low, 45

Figure 2-11: Definition of the loading orientation in triaxial samples tested by Niandou et al. (1997). Tournemire shale samples were tested in triaxial compression. The loading orientation angle θ is the angle between the maximum principal loading (σ1 ) direction and the bedding planes. Modified from Niandou et al. (1997)

Figure 2-12: Effect of loading and bedding orientation on failure strength observed by Niandou et al. (1997). The failure stress (σ1 − σ 3) vs. loading orientation is plotted for different confining stresses. Minimum failure stress is observed about 30o loading orientation for all confining stresses. Modified from Niandou et al. (1997)

46

high) (Refer to figure 2-13). The effect of confining pressure on the fracture pattern is only observed at low loading orientations (θ = 0o -15o ) where the specimen fractured along bedding planes at low confining pressures but sheared across the bedding planes at high confining pressures. This axial splitting could be observed at lower failure stresses at low confining pressures and loading angles (Refer to figure 2-13). Also, the consistency of fracture along bedding planes along moderately inclined loading directions (θ =15o -60o ) suggests that even at higher confining pressures bedding plane sliding still controls the failure strength of the material. This is consistent with the fact that confining pressure had little effect on the strength anisotropy observed (Refer to figure 2-12).

Opalinus Shale

Opalinus shale is the primary shale tested in the current study and therefore emphasis will be placed on previous experimental and theoretical work conducted on this material. Opalinus shale is located in the Jura mountain range, which is situated in the Northwestern part of Switzerland and extends into eastern France and southern Germany. Opalinus shale is, using traditional geologic terminology, a clayshale because it consists primarily of clay fraction particles. However, for this thesis it will be referred to as "Opalinus shale" as defined by the petroleum terminology discussed previously. A more extensive geological and mineralogical description will be presented in Chapter 3 of this report. Due to the challenge of tunneling through clayshale formations, as well as the possible benefit of using it as a nuclear waste host site, Opalinus shale has been an extensively tested material, both insitu and in the laboratory. Bellwald (1990) and Aristorenas (1992) Some of the early work conducted on Opalinus shale focused primarily on swelling and creep, as these are significant problems when tunneling in argillaceous (clay based) shales (Bellwald, 1990). Triaxial experiments were conducted by Bellwald (1990) and Aristorenas (1992) on many different clay based shales including Opalinus shale, Oxfordian shale and Lias Alpha shale, all from the same Jura mountain range as Opalinus shale. 47

Figure 2-13: Sketch of fracture patterns observed in triaxial tests conducted by Niandou et al. (1997). Post test fracture patterns were observe for various loading orientation (θ = 0o 15o , 15o -60o , 60o -90o ) and confining pressures (low, high). For both high and low confining pressures, fracture occurred along bedding planes for moderate loading orientations θ =15o 60o and across bedding planes for high loading orientations θ =60o -90o . At shallow load orientations θ = 0o -15o fracture along bedding planes occurred at low confining pressure and fracture across bedding planes occurred at high confining pressure. (Niandou et al., 1997)

48

Aristorenas (1992) preformed triaxial undrained compression (UC) and undrained extension (UE) tests on Opalinus and Lias shale. Two different failure modes were observed by Aristorenas (figure 2-14). Undrained compression tests typically showed multiple conjugate shear fractures at higher angles (ξ ≈50-70o ), while undrained extension tests observed parallel fractures at low angles (ξ ≈0o ) (Refer to figure 2-14). During these undrained tests, excess pore pressures were also measured (figure 2-15). Negative pore pressures, associated with dilatant behavior, were observed for both extension and compression tests.

Figure 2-14: Failure modes in shale observed by Aristorenas(1992). Type A failure mode was observed in undrained compression and Type B failure mode was observed in undrained extension. (Aristorenas, 1992) Although fractures could not be observed during their propagation stages in the test, the change in excess pore pressure (dilatant behavior) is explained by anisotropy and fracture. Aristorenas hypothesized, using post-test fractography methods, that dilatant behavior in the extension tests was due to inherent microcracks shearing and then propagating as tensile cracks along natural bedding planes (figure 2-16). Additionally, it was proposed that the dilatant behavior in compression could be explained by tensile cracks initiating from inherent microcracks, propagating and eventually coalescing through secondary cracks (figure 2-17). It should be noted that dilatant behavior (negative pore pressure) occurs at much 49

Figure 2-15: Excess pore pressure observed in undrained tests by Aristorenas(1992). Both undrained compression (positive q-values) and undrained extension (negative q-values) are shown. (Aristorenas, 1992)

lower absolute deviatoric stresses ("q") in the extensional direction (Refer to figure 2-15), which can be explained by the lower tensile strength along natarual bedding planes (Refer to figure 2-16). Corkum and Martin (2007) Corkum and Martin (2007) conducted a series of uniaxial and triaxial tests on Opalinus shale extracted from Mont Terri. The purpose of the study was to characterize the stressstrain response of Opalinus shale under different confining stresses. Corkum and Martin (2007) subdivided the typical stress-strain behavior of Opalinus shale tested in unconfined compression into five regions: ∙ Region I - Micro-crack closure and asperity crushing ∙ Region II - Linear elastic ∙ Region III - Onset of crack initiation ∙ Region IV - Unstable crack growth ∙ Region V - Post-peak 50

Figure 2-16: Fracture in extension tests proposed by Aristorenas(1992). Shear cracks develop through the specimen and then propagate as tensile cracks along bedding planes. (Aristorenas, 1992)

Figure 2-17: Fracture in compression tests proposed by Aristorenas(1992). Tensile cracks (wc) initiate and propagate from naturally existing microcracks and then secondary cracks (sc) initiate to coalesce the cracks. (Aristorenas, 1992)

51

These regions are shown in figure 2-18. It should be noted that the unloading stiffness observed in figure 2-18 is much higher than the stiffness in the linear-elastic region. Corkum and Martin (2007) attribute this increase in stiffness to the "work done against friction between the crack surfaces during the loading and unloading".

Figure 2-18: Typical response of an unconfined compression test on Opalinus shale. This example test was loaded parallel to the bedding planes. Five distinct regions of behavior were defined by Corkum and Martin (2007); micro-crack closure and asperity crushing, linear elastic, onset of crack initiation, unstable crack growth and post-peak. Modifed from Corkum and Martin (2007) The unconfined compression tests by Corkum and Martin (2007) were conducted on cylindrical Opalinus shale specimens with the bedding plane oriented parallel to the loading. Some of these tests were conducted using cyclic loading (loading and unloading) (figure 219). Substantial non-linearity was observed at low stresses during loading and unloading, even when the specimen was unloaded to zero load (Refer to figure 2-19). This meant that the non-linear behavior observed in Opalinus shale (Region I) was recoverable and likely due to a reversible mechanism such as micro-crack closure. Finally, triaxial tests were also conducted by Corkum and Martin (2007) on Opalinus shale with different confining pressures (figure 2-20). Non-linear and brittle behavior was consistently observed in Opalinus shale tests, even after increasing the confining pressure (Refer to figure 2-20). Non-linear behavior was also observed prior to the unstable crack growth, at about 80% to 90% peak stress. Additionally, stiffness increase was observed at higher 52

Figure 2-19: Cyclic unconfined compression tests conducted on Opalinus shale. Non-linear (crack closure) stress-strain behavior was observed even after unloading and reloading. (Corkum and Martin, 2007) confining pressures (Refer to figure 2-20).

Figure 2-20: Stress-strain behavior of triaxial tests conducted on Opalinus shale by Corkum and Martin (2007). Non-linear and brittle behavior was observed even at higher confining pressures (σ3 ). At higher confining pressures there was typically an increase in stiffness observed. (Corkum and Martin, 2007) Naumann et al.(2007) Naumann et al. (2007) conducted a series of cylindrical triaxial compression tests on Opalinus shale from the Mont Terri Rock Laboratory. Cylindrical specimens (100mm x 250mm) were tested in a conventional triaxial cell with various confining pressures and bedding 53

plane angles. The bedding plane angles were varied randomly to test a wide variety of bedding plane orientations. From the conventional triaxial test results Naumann et al. (2007) were able to determine the effect of bedding plane angle and confining pressure on the failure strength of Opalinus shale (Refer to figure 2-21).

Figure 2-21: Triaxial tests conducted by Naumann et al. (2007) on Opalinus shale at various bedding angles (ψ) and confining pressures. Naumann et al. (2007) refer to conventional triaxial as Karman tests. Confining pressures (sigma3 ) are color coded (White = 1 MPa, Light Gray = 3 MPa, Dark Grey = 6 MPa, Black = 10 MPa). Modified from Naumann et al. (2007) Additionally, Naumann et al. (2007) fit the Jaeger (1960) single plane of weakness theory to the results; however, it should be noted that there was a lack of tests conducted with bedding angles between 45o -80o (Refer to figure 2-21). Due to the lack of bedding angles tested the exact shape of this failure envelope is not very clear. As expected there is a clear effect of anisotropy on the failure strength of Opalinus shale. There appears to be a minimum failure strength (based on single plane of weakness model) at a bedding dip of about 50o , although there are no tests conducted at this angle. It should be noted that as confining pressure increases there is little to no change in the strength anisotropy (the difference between strength at 0o and 50o bedding) (Refer to figure 2-21). Therefore, over the range of confining pressures tested by Naumann et al. (2007), the 54

strength anisotropy exists also at higher confining pressures. Amann et al.(2011) Amann et al. (2011) conducted a series of cylindrical unconfined compression tests on Mont Terri Opalinus shale. In addition to axial displacements, radial displacements were measured with a circumferential strain gauge to determine radial and volumetric strains of the specimens (figure 2-22). Specimens were loaded perpendicularly to the bedding planes. Acoustic emissions were also measured with four piezoelectric sensors attached on the sides of the sensors (Refer to figure 2-22). Physical properties were determined before testing such as water content (Wc =7.5%), saturation (S = 1.00),and p-wave velocity perpendicular (V p⊥ = 2140 m/s) and parallel (V p‖ = 3020 m/s) to the bedding (Amann et al., 2011).

Figure 2-22: Opalinus shale specimen configuration tested by Amann et al.(2011). Circumferential and axial strain were measured along with axial load and acoustic emissions (Amann et al., 2011) Crack detection was done using both volumetric strains and acoustic emission data (figure 2-23). The crack initiation stress (σCI ) was determined by the departure of the volumetric strain from elastic behavior (Refer to figure 2-23). Unstable crack propagation (σCD ) was determined by the inflection point of the volumetric strain to dilative behavior. The crack initiation was observed at about 30% of peak stress, and it typically coincided with the beginning of acoustic emissions activity. This observation shows promising re55

sults for using acoustic emissions to detect crack initiation in Opalinus shale. It also makes this study unique because most studies on shale, especially Opalinus shale, do not include acoustic emissions data. However, acoustic event localization was not performed in this study.

Figure 2-23: Stress-strain-AE results on unconfined compression test of Opalinus shale. Crack initiation (σCI ) was determined by the deviation of volumetric strain from elastic behavior and unstable crack growth (σCD ) was determined by the inflection point of the volumetric stain to dilation behavior. Modified from Amann et al. (2011)

In order to explain the crack initiation and strain behavior of the Opalinus shale experiments Amann et al. (2011) developed a conceptual model of Opalinus shale as a layered system of stiff and soft layers (figure 2-24). This conceptual model assumes that as the specimen is loaded, cracks initiate in the the stiffer layers and then propagate and arrest at the interface between layers (Refer to figure 2-24). This crack propagation could be explained by the lower axial stiffness value observed after crack initiation (σCI ) occurs (Refer to figure 223). Afterwards, secondary cracks occur in the softer layer, potentially as shear cracks coalescing the tensile cracks in the stiffer layer, causing unstable crack propagation (σCD ) and leading to failure. It should be noted that this model is based on stress-strain results alone and not on any post-test fractography observations. 56

Figure 2-24: Conceptual model of crack initiation and growth in a layered material. (a) Initial configuration of a layered system with stiffer and softer layers. (b) Crack initiation and propagation in the stiffer layer arresting at the layer boundary. Secondary cracks then form in the soft layer, connecting previous tensile cracks in the stiff layer. (Amann et al., 2011) Wild et al. (2014) Wild et al. (2014) performed a detailed study on the effect of partial saturation in Opalinus shale. This included an investigation of the relationship between the suction and water content (figure 2-25) and suction and saturation level (figure 2-26). Suction values were obtained by exposing Opalinus shale to different relative humidity environments (Refer to figures 2-25 and 2-26). The relative humidity was controlled by using various supersaturated salt solutions. By using this technique to control the partial saturation (suction) value, unconfined and Brazilian tests were performed on Opalinus shale at different suction (saturation) levels (figures 2-27 and 2-28). The unconfined compression tests were performed with bedding planes perpendicular to the loading direction. The Brazilian tests were performed with bedding planes both perpendicular and parallel to the loading direction. From the unconfined compression tests, a linear increase in the unconfined compressive strength (UCS) of Opalinus shale was observed up to a suction of 86 MPa (Refer to fig57

Figure 2-25: Relationship between water content and suction for Opalinus shale. Suction values were obtained by exposing Opalinus shale to different relative humidity environments.

Figure 2-26: Relationship between saturation and suction for Opalinus shale. The fitted curve was determined using the Van Genuchten equation. (Wild et al., 2014)

58

ure 2-27). Beyond a suction 86 MPa there was no significant change in the UCS. For the Brazilian tests, an almost linear increase in the Brazilian tensile strength was observed up to a suction of 56.6 MPa (Refer to figure 2-28). Beyond a suction of 56.6 MPa a relatively constant Brazilian tensile strength was observed. Therefore, for both unconfined compressive strength (parallel to the bedding) and Brazilian tensile strength (parallel and perpendicular to bedding) there was a region of suction (saturation) values in which strength increased and then a region in which strength remained relatively constant with increasing suction.

Figure 2-27: Unconfined compressive strength (UCS) of Opalinus shale depending on suction level. A linear increase in the UCS of Opalinus shale was observed up to a suction of 86 MPa. Beyond a suction 86 MPa there was no significant change in the UCS. (Wild et al., 2014)

One of the major contributions of this work was that it showed that the unconfined compressive strength change due to partial saturation was different depending on the direction of bedding (figure 2-29). The change in UCS due to a drop in water content was larger for specimens loaded normal to the bedding compared to those loaded perpendicular to the bedding (Refer to figure 2-29). This meant that ratio of the unconfined compressive strength perpendicular and parallel to the bedding planes, sometimes referred to as the "strength anisotropy", changed with partial saturation level. However, it should be noted that ratio of Brazilian tensile strengths (parallel and perpendicular) remained relatively constant (Refer to figure 2-28). These anisotropic strength ratios are important if cracking occurs in multiple directions (across and along bedding planes) in partially saturated Opalinus shale. 59

Figure 2-28: Brazilian tensile strength of Opalinus shale depending on suction level. The tensile strength for specimens with bedding planes normal to the loading direction (σt,n ) and parallel to the loading direction (σ p,n ) are shown. An almost linear increase in the Brazilian tensile strength was observed up to a suction of 56.6 MPa. Beyond a suction of 56.6 MPa a relatively constant strength was observed. (Wild et al., 2014)

Figure 2-29: Relation between water content and unconfined compressive strength (UCS) for Opalinus shale. The UCS is plotted for specimens loaded parallel to the bedding (black) and normal to the bedding (gray). (Wild et al., 2014)

60

Bock (2009) Additionally, since the Mont Terri Rock Laboratory is located inside the Mont Terri Opalinus shale formation, there have been many technical reports describing the mechanical behavior Opalinus shale in the laboratory and in-situ. These technical reports and technical notes have been summarized by Bock (2009) The results of these summarized technical reports can give some general insight into mechanical properties not yet published in papers. The reports include data on the material properties, such as saturation, water content, grain density, and Atterberg limits, as well as mechanical properties, such as elastic modulus, Poisson’s ratio, compressive strength, tensile strength, cohesion, and friction angle of Opalinus shale. These technical report data will be used in this thesis for mechanical properties that have not be published in journal articles. Mineralogy It should also be noted that there have also been studies which investigated the mineralogy and composition of Opalinus shale (Van Loon et al., 2004; Courdouan et al., 2007; Wenk et al., 2008; Josh et al, 2012). Mineralogy and origin of Opalinus shale will be discussed in detail in Chapter 3. Opalinus Shale Literature Summary As previously discussed, Opalinus shale has been widely studied, with research conducted on its mechanical properties, geology, mineralogy and tunneling potential. It is not feasible to discuss all of these studies in detail. Therefore, a summary of the studies conducted on Opalinus shale have been summarized in table 2.1.

Pierre Shale

Pierre shale is a shallow (60% clay content) from the Niobrara and Denver Basin formations extending across Colorado, Wyoming, Nebraska and South Dakota. An Upper Cretaceous formation, Pierre shale is an organic formation known for 61

Table 2.1: Summary of research conducted on Opalinus shale Laboratory Testing and Mechanical Behavior Madsen (1976) Aristorenas (1992) Cripps and Taylor (1981) Einstein et al. (1995) IGB-Bericht (1981) Bock (2001) Madsen et al. (1985) Corkum and Martin (2007) ISRM (1989) Naumann et al. (2007) Madsen and Mitchell (1989) Bock (2009) Nüesch (1989) Soe et al. (2009) Bellwald (1990) Amann et al. (2011) N-16 Transjurane (1990) Valente et al. (2012) Nüesch et al. (1990) Jobmann et al. (2010) Nüesch (1991) Amann et al. (2015) Geology, Mineralogy and Fabric Grubenman and Letsch (1907) Wenk et al. (2008) Hartmann (1950) Josh et al. (2012) Peters (1962) Keller et al. (2011) Van Loon et al. (2004) Keller et al. (2013) Porewater Chemistry and Organic Content Meier (1997) Courdouan et al. (2007) Tunneling, In-situ Testing and Extraction Buxtorf (1916) IJA-BG (1992) Motor-Columbus (1981) N-16 Transjurane (1989) Etterlin (1987) Einstein (2000) Nagra (1988) Martin and Lanyon (2003) Steiner and Metzger (1988) Pei (2003) IG Wisenberg-Tunnel (1990) Bossart et al. (2012) IJA-BG (1991) Martin (2015) Modeling Lisjak et al. (2013) Salager et al. (2013) Lisjak et al. (2014)

Lisjak et al. (2015) Wild et al. (2015)

62

oil extraction, high fossil content and large bentonite seams. Although Pierre shale is part of oil and gas formations, only a limited amount of experimental testing has been done on its mechanical properties and fracture behavior. Since Pierre shale is an important source rock for hydrocarbons, it has been studied frequently by the United State Geological Survey (USGS). The USGS has extensively described the geology (Schultz, 1962), mineralogy (Tourtelot, 1964) and geochemistry (Schultz et al., 1980) of Pierre shale. Many experiments have been conducted on the consolidation and permeability behavior of Pierre shale due to its high smectite content (Olgaard et al., 1995). Finally, acoustic emissions techniques have been used to determine the attenuation (McDonal et al., 1958) and anisotropy (White et al., 1983) behavior of Pierre shale. However, there have been few studies describing laboratory tests on the mechanical properties of Pierre shale. Pierre shale is a clay shale that is highly sensitive to drying, and it is likely that conducting strength and fracture experiments on undisturbed Pierre shale is very difficult. One such study on the construction of the Oahe dam on the Missouri river near Pierre, South Dakota conducted unconfined compression and triaxial tests on Pierre shale bedrock (Johns et al., 1963). The unconfined compressive strength for Pierre shale bedrock was reported to be between 0.5 MPa to 17.4 MPa. In addition, elastic modulus was reported to be between 140 MPa to 970 MPa. It is very clear from the laboratory tests conducted by Johns et al. (1963) that the mechanical properties of Pierre shale are highly variable. Recently, there have been additional studies conducted to study the effect of drying on Pierre shale (Schaefer and Birchmier, 2013). They conducted a series of XRD and Bromhead Ring Shear tests (modified from ASTM D6467) on Pierre shale with different numbers of weathering cycles (wet and drying). Specimens from both shallow and deep borings ′

showed a decrease in residual friction angle (φr ) and an increase in plastic limit (PL) and liquid limit (LL) with weathering cycles. Disturbance by wetting and drying cycles of the clay-based Pierre shale is a major concern for correctly determining proper strength parameters. 63

Artificially Simulated Anisotropic Rock

In an effort to reproduce a standard and systematic anisotropic material, Tien et al. (2006) developed a procedure to make a layered rock-like material from cement and kaolinite. The technique involved molding two different mixes of cement and kaolinite. One mix was higher in cement (4:1:1.2 cement:kaolinite:water ratio), which had higher stiffness (E = 21,700 MPa) and strength (σc = 104.2 MPa). The other mix was lower in cement content (1:1:0.6 cement:kaolinite:water ratio) and was a softer (E = 11,900 MPa) and weaker (σc = 43.3 MPa) material. These two mixes were molded and consolidated into individual blocks and then sliced into layers 5mm thick. Alternating layers of each material were stacked and then the composite material was consolidated in a mold (Ko consolidated) up to a stress of about 2 MPa. The final layered rock specimen was then cored (100mm x 50mm cylinders) at different angles (ψ=0o ,15o ,30o ,45o ,60o ,75o ,90o ) (figure 2-30a).

These cores were loaded in a triaxial apparatus in unaxial compression and with several confining pressures (σ3 = 6, 14, 35 MPa). As the test progressed a photographic scanner was rotated around the specimen to scan the surface and capture any cracking that occurred (1 scan every 0.02%-0.04% strain to failure or about 20 - 45 images total)(figure 2-30b). These scanned images could be unrolled flat, or 3-D images of the cracking processes could be reconstructed(figure 2-30c). It was not clear if the tests conducted with confinement also included observations with the scanner during the test or if the specimens were scanned after a jacket was removed. The failure stresses at various bedding angles and confining stresses followed the previously observed U-shaped anisotropic behavior (figure 2-31).

Using scanned images the cracking processes could be observed throughout the uniaxial tests. Tien et al. (2006) observed that in tests with low bedding plane angles (ψ ≤ 30o ) cracking initiated in the softer, weaker layer. This observation is contrary to the model proposed by Amann et al. (2011), which hypothesized that cracks initiate in stiffer layers and then propagate to the boundary of softer layers and coalesced by shear cracks (Refer to figure 2-24). 64

Figure 2-30: Specimen coring and test setup used by Tien et al.(2006). (a) Layered artificial rocks were sliced and layered into a composite rock and then cored at different angles (example at 30o ). (b) Test setup used for uniaxial and triaxial compression tests on specimens. A rotating charge-coupled device (CCD) imager was used to capture cracking processes on a cylindrical specimen during the test. (c) Scanned images could be rolled back up to render 3-D specimens with cracking. Modified from Tien et al. (2006)

Figure 2-31: Failure stresses (σ1 ) of artificial anisotropic rock at different bedding angles (ψ) and confining stresses (σ3 ). Modified from Tien et al. (2006)

65

Figure 2-32: Initial cracking observed by Tien et al. (2006) for ψ=0o bedding angle test. Cracking was observed to initiate in the softer, weaker material layer. Modified from (Tien et al., 2006) From the cracking observations in all of the tests conducted, Tien et al. (2006) were able to subdivide the failure mechanisms observed into four failure modes: sliding failure along discontinuities, non-sliding (tensile) failure along discontinuities, tensile failure across discontinuities and sliding failure across discontinuities (figure 2-33). In addition, for each failure mode Tien et al. (2006) were able to define zones of bedding angle (ψ) and confining stress (σ3 ) over which each failure mode was observed (Refer to figure 2-33). Included in the sliding failure across discontinuities mode was single sliding fractures or multiple conjugate sliding fractures. This observation was also seen by Aristorenas in Opalinus shale (Refer to figure 2-14). These experiments by Tien et al. (2006) are unique in that they not only try to match fracture behavior with bedding plane orientation and confining stress, but they are also able to reproduce results very similar to other anisotropic shales with an artificially made rock.

2.1.4

Anisotropic Failure Theory

As described previously, shale is a highly anisotropic material with natural bedding planes that develop during formation. Typically shale is formed with bedding planes during deposition and diagensis which leads to a transversely isotropic elastic and failure behavior. For the purpose of this study all shales discussed will be assumed to be transversely isotropic. Unlike purely isotropic materials, failure of transversely isotropic rocks is dependent on the 66

Figure 2-33: Four failure modes observed by Tien et al. (2006). Failure modes were observed in tests of artificially produced anisotropic rock with different bedding angles and confining stresses. (Tien et al., 2006)

stress state and the orientation of the bedding planes. Although there have been many different theories developed to predict the failure strength of shale with loading and bedding orientation, only the most relevant anisotropic failure theories will be discussed.

Intact Shear Failure

In order to fully understand anisotropic failure, it is necessary to define the failure of isotropic intact material first. One of the most common shear failure criteria used is the Coulomb theory. The Coulomb failure criterion can be expressed as:

τ = σ tan φ + c

(2.1.4)

where τ is the shear stress, σ is the normal stress, φ is the internal friction angle of the material and c is the cohesion. Failure will occur when a stress state is on the failure envelope. The stress state can be written as:

1 τ = (σ1 − σ3 )sin2θ 2 67

(2.1.5)

1 1 σ = (σ1 + σ3 ) − (σ1 − σ3 )cos2θ 2 2

(2.1.6)

Where σ1 is the major principal stress, σ3 is the minor principal stress and θ is the angle between the stress state plane and the plane of the principal stress. By substituting equations (2.1.6) and (2.1.5) into (2.1.4), failure along the Coulomb plane can be expressed as:

1 1 1 (σ1 − σ3 )sin2θ = [ (σ1 + σ 3) − (σ1 − σ3 )cos2θ ]φw + c 2 2 2

(2.1.7)

The critical direction of failure (θc ) can be defined as:

θc = 45o −

φ 2

(2.1.8)

By substituting the critical plane of failure equation (2.1.8) into equation (2.1.7), the major principal stress at failure (σ1 ) can be obtained as:

σ1 = σ3

cosφ 1 + sinφ + 2c 1 − sinφ 1 − sinφ

(2.1.9)

Equation (2.1.9) is a key expression for predicting the failure strength of intact material. It can be used to represent anisotropy by coupling it with the failure of a discontinuity or by having variable cohesion and friction terms. Several of these failure theories for anisotropic rocks will now be discussed.

Discontinuous - Single Plane of Weakness

Jaeger, 1960 One such theory which incorporated the failure of intact rock with the failure of a single discontinuity plane was introduced by Jaeger (1960). He called a Single Plane of Weakness 68

theory. This theory assumes that both the rock and a joint plane follow Coulomb theory, each having unique cohesion (c) and internal friction angles (φ ). By visualizing the stresses along a discontinuity induced by principal stresses (σ1 , σ3 ) (figure 2-34), the stresses along the discontinuity can be expressed as:

1 1 σn = (σ1 + σ 3) − (σ1 − σ3 )cos2θ 2 2

(2.1.10)

1 τn = (σ1 − σ3 )sin2θ 2

(2.1.11)

where σn and τn are the normal and shear stresses along the plane and θ is the angle between the maximum principal loading direction (σ1 ) and the discontinuity plane (bedding or joint plane) (Refer to figure 2-34). By substituting equations (2.1.10) and (2.1.11) into equation (2.1.4) the maximum principal stress causing failure along the discontinuity can be obtained as:

σ1c (θ ) = σ3 +

2(cw + σ3tanφw ) (1 − tanφw tan θ )sin2θ

(2.1.12)

where cw and φw are the cohesion and friction angle of the joint respectively. The cohesion and friction angle of the joint are usually different from those of the intact rock material. Since failure can occur by either failure of the intact material or failure along the discontinuity, the final failure stress is the smaller of equations (2.1.9) and (2.1.12):

σ1c = min[σ1c (θ ), σ1c−intact ]

(2.1.13)

Therefore the material will fail either through the intact material or along the plane of weakness, creating a u-shaped strength curve (figure 2-35). The regions of intact failure are typically referred to as the "shoulders" of the curve (Refer to figure 2-35). These shoulders may be at different stress levels. Since it was observed that failure strength of 69

anisotropic material with horizontal and vertical bedding planes (non-discontinuity sliding) were different, especially for unconfined compression tests, this theory was extended to have different intact failure strengths for θ =0o and θ =90o (Refer to figure 2-35).

Figure 2-34: Definition of shear and normal stresses acting on on a discontinuity plane.

Figure 2-35: Plane of weakness theory u-curve. Plane of weakness u-curve shows both intact failure and failure along the discontinuity plane. The region of intact failure is typically referred to as the "shoulders" of the curve. These shoulders may be at different stress levels. Modified from Duveau et al. (1998)

70

Empirical Approaches

Empirical methods based on experimental results have also been used to predict failure strength of anisotropic material. These strength theories typically incorporate material or strength constants which are calibrated based on tests at different loading angles and confining pressures. Jaeger Variable Cohesive Theory Jaeger (1960) proposed an empirical strength theory which introduced a variable cohesion term with a constant friction coefficient. The variable cohesion was expressed as:

c = A − B[cos2(θm − θ )]n

(2.1.14)

where θm is the loading angle at which c is a minimum (figure 2-36) and A and B are empirically determined constants. This criterion should not be confused with the single plane of weakness also proposed by Jaeger (1960) as it does not incorporate discontinuities directly. The variable cohesion term in (2.1.14) can be substituted into the failure strength of intact material from (2.1.9) and be expressed as:

σ1 = σ3

1 + sinφ cosφ + 2{A − B[cos2(θm − θ )]n } 1 − sinφ 1 − sinφ

(2.1.15)

McLamore and Gray, 1967 Extending on the Variable Cohesive theory proposed by Jaeger (1960), McLamore and Gray (1967) incorporated both a variable cohesion and friction angle based on the loading direction which can expressed as:

c = A1 − B1 [cos2(θm − θ )]n

for θ between 0o < θ < θm

c = A2 − B2 [cos2(θm − θ )]n

for θ between θm < θ < 90o 71

(2.1.16)

Figure 2-36: The variation of φ -c with loading angle for Angers schist tested by Duveau et al. (1998). The φ -c were determined from the experimental tests (exp) and then the variable tanφ -c relations proposed by McLamore and Gray (1967) were numerically fitted to the results (num). The loading angles (θ ) for the minimum φ and c were determined to be at about 45o for both φ and c. Modified from Duveau et al. (1998)

′

for θ between 0o < θ < θm

′

′

for θ between θm < θ < 90o

tan(φ ) = C1 − D1 [cos2(θ − θm )]n tan(φ ) = C2 − D2 [cos2(θ − θm )]n

′

(2.1.17)

where A1,2 B1,2 are coefficients for the variable cohesion term, C1,2 D1,2 are coefficients ′

for the variable friction coefficient and θm and θm are the angles at which minimum cohesion and friction are observed respectively (Refer to figure 2-36). The use of θm and the piece-wise function allows for the strength curve to have a different shape a low and high loading angles (figure 2-37). This accounts for the difference in strength observed between specimens loaded at 0o and 90o (Refer to figure 2-37).

Ramamurthy et al., 1993

Ramamurthy et al. (1993) developed an theory, which directly determines failure strength by using empirical constants. It is based on a non-linear failure envelope in the Mohr plane (τ-σ ) and can be expressed as: 72

Figure 2-37: An example of the McLamore and Gray criterion fitted to tests on Angers schist by Duveau et al. (1998). Using the experimental results (exp) the variable tanφ -c method proposed by McLamore and Gray (1967) was used to numerically determine the strength (num). Triaxial tests with confining pressures of 5 and 40 MPa were conducted. Modified from Duveau et al. (1998)

σ1 − σ3 = Bj σ3



σc j σ3

α j (2.1.18)

where σc j is the uniaxial compression strength and α j and B j are material parameters at angle j(o ). The parameters α j and B j can be determined by:

αj = α90



σc j σc90

1−α90

Bj : = B90



α90 αj

0.5 (2.1.19)

where the subscript 90 corresponds to the parameters at θ =90o . The unixiaxal compression strength can be described as a function of θ as:

σc j = A1 − D1 [cos2(30o − θ )] o

σc j = A2 − D2 [cos2(30 − θ )]

for θ between 0o and 30o (2.1.20) o

for θ between 30 and 90

o

where A1,2 and D1,2 are material constants which can be determined from uniaxial com73

pression tests. Similar to McLamore and Gray (1967), this empirical theory can incorporate variations in vertical and horizontal strength(σc0 ̸= σc90 ); however, it has a fixed minimum strength at a 30o loading direction (figure 2-38).

Figure 2-38: An example of the Ramamurthy et al. (1993) criterion fitted to tests on Angers schist by Duveau et al. (1998). Triaxial tests with confining pressures of 5 and 40 MPa were conducted. Modified from Duveau et al. (1998)

Continuous Criteria

The theories previously discussed included discontinuity weakness plane and empirical methods. There are also methods, which use continuous functions to incorporate anisotropic strength criteria. Anisotropic Matsuoka-Nakai (AMN) Criteria (Pei, 2008) Pei (2008) proposed a modified Matsuoka-Nakai failure criterion with anisotropy. This model was defined using a failure function ( f ). When f is ≥ 0 failure occurs. The failure function can be expressed as: 74

′

′

′

′

′

′

f (r , θ , σst , σtm , σns ) = −(1 − θ )r 3 cos3θ + (1 − θs + θs2 )r 2 − θs2 ′

′

′

′

′

′

′

′

′

+2(1 − θs + θs2 )(σst2 + σtn2 + σns2 ) + 6(1 − θs )(σst2 rn + σtn2 rs + σns2 rt ) √ ′ ′ ′ ′ ′ ′ ′ −6 6(1 − θs )σst σtn σns − A(Z)θs2 (rn0 − rn ) + K(Z)(σtn2 + σns2 )

(2.1.21)

′

where R(Z), A(Z), K(Z) are functions and θs , rn0 are constants determined from experimental tests. rn , rt and rs are the stress deviators and can be expressed as:

1 rn = √ (2σn − σs − σt ) 6 1 rt = √ (2σt − σn − σs ) 6 1 rs = √ (2σs − σt − σn ) 6

(2.1.22)

σs , σt , σn , σst , σtn , and σtn are the stresses acting on the plane of anisotropy (figure 2-39) and be expressed in conventional triaxial space using the following expressions:

σs = σ3 σt = σ3 sin2 θ + σ1 cos2 θ

(2.1.23)

σn = σ3 cos2 θ + σ1 sin2 θ

σst = 0 σ1 − σ3 σtn = sin2θ 2

(2.1.24)

σns = 0

The modified AMN criterion expresses the effect of matrix anisotropy on strength, i.e. the fact that failure planes crossing bedding planes will affect strength. If failure occurs through sliding along bedding planes then the single plane of weakness comes into play. This is shown in figure 2-40. 75

Figure 2-39: Orientation of bedding planes (ψ), loading direction(θ ) and STN system by (Pei, 2008). The STN system (left) relates to the stresses that exist on the plane of anisotropy. Modified from Pei (2008)

Figure 2-40: AMN criterion compared to Martinsburg slate tested by Donath (1964). Martinsburg slate was tested at various loading angles and confining pressures by Donath (1964). The AMN criterion can be combined with the single plane of weakness (SPOW) theory by Jaeger (1960). When the SPOW predicts a lower strength than the AMN criterion then the latter it is shown with a dotted line. Modified from Pei (2008)

76

Buczek and Herakovich, 1985

The previously discussed theories incorporated the (Mohr-)Coulomb shear strength theory, however, they have typically neglected the tensile failure of the material. A continuous theory proposed by Buczek and Herakovich (1985) incorporated a relation for the change in tensile strength as:

σt = σt‖ sin2 (θ ) + σt⊥ cos2 (θ )

(2.1.25)

where θ is the angle between the principal tensile stress and the bedding plane direction and σt , σt‖ , and σt⊥ are the tensile strengths parallel to bedding and perpendicular to bedding respectively (figure 2-41). This theory is unique because it incorporates the variation in tensile strength (the negative portion of Mohr’s stress space) in the failure of an anisotropic material.

Figure 2-41: (a) Schematic of tensile loading orientation and (b) variation in tensile strength (σt ) with loading angle (θ ) proposed by Buczek and Herakovich (1985).

77

2.2

Rock Fracture

In addition to anisotropic failure criteria, it is necessary to understand some basic concepts of fracture mechanics such as the fracture initiation, -propagation and -coalescence mechanisms associated with shales. The following section describes the origin of fracture initiation and propagation theory, crack interaction theory, previous work conducted on crack interaction and coalescence in rocks, modeling fracture in rocks and finally, modeling fracture in anisotropic rocks such as shale.

2.2.1

Fracture Theory

The basis of stress concentrations around pre-existing cracks (flaws) was first approximated mathematically by Inglis in 1913 (figure 2-42). A flaw was represented by an elliptical opening within an infinite plate having a major axis (2a) much larger than its minor axis (2b). From this model, Inglis developed the tangential stress concentration at the flaw tip (σtip ) of a sharp elliptical hole (a»b) subjected to a far field tensile stress (σv ):

r σθ θ = σtip = 2σv

a ρ

(2.2.1)

where a is the flaw half length and ρ(= b2 /a) is the radius of curvature of the flaw tip. From this derivation, the stresses at the flaw tip increase as either the flaw half length (a) increases or the radius of curvature decreases. A "sharp" flaw can be represented by having a smaller minor axis (b) in comparison to the major axis the flaw half length (a) (a»b). This relationship can be intuitively understood since a circular hole should have lower stress concentrations than an extremely thin flaw. A flaw can be either "open" or "closed". A "closed" flaw refers to a thin flaw where the walls of the flaw are in contact and thus there is an interfacial sliding resistance. An "open" flaw refers to a flaw where the walls are not in contact. The terms "flaw" used to describe a pre-existing defect and the word "crack" is 78

used to describe a newly created fracture surface.

Inglis postulated that once the local stress exceeds the strength of the material, this flaw expands (crack initiation). This is the basis of the stress-based criterion for crack initiation.

Figure 2-42: Approximation of Tangential Stress at a Flaw Tip.

However, the crack initiation was observed in experiments were much lower stress than the actual strength of the atomic bonds in materials, therefore it was determined that other mechanisms are controlling the initiation of cracks (Griffith, 1920). Griffith (1920) expanded the theory of stress concentrations developed by Inglis (1913) and applied laws of thermodynamics and energy conservation to determine the stress required to initiate a crack at the flaw tip. This is the basis of the energy-based criterion for crack initiation. Griffith expressed the the change in strain work and potential energy of creating a surface crack in a plate (Refer to figure 2-42) as

d(W − Π ) =

d(Welastic) d(W sur f ace) − =0 da da 79

(2.2.2)

Where,

W = Welastic = Stored elastic strain energy =

πa2 σv E

Π = Wsur f ace = Potential Surface energy created by a crack = 4γa

(2.2.3)

(2.2.4)

where a is the crack half length, E is the Young’s Modulus of the plate and γ is the specific surface energy of the material. By substituting equations (2.2.3) and (2.2.4) into equation (2.2.2), the increment of stored elastic strain energy required to create a crack (surface energy) at the flaw tip and can be expressed as

d πa2 σv d ( ) = (4γa) da E da

(2.2.5)

Therefore, by integrating and solving for the stress (σv ), the necessary stress to produce crack initiation can be defined as followed:

r σv,crack =

2Eγ πa

(2.2.6)

Along with in plane tensile cracking (Mode I), there are two other modes of cracking which can occur; in-plane shear cracking (Mode II) and out-of-plane shear cracking (Mode III) (figure 2-43). Extending on the energy-based criterion developed by Griffith (1920), Irwin (1957) proposed an energy release rate concept which expresses the displacement and stresses around a flaw tip by a stress concentration factor, K.

√ K = −σv Q πa 80

(2.2.7)

Figure 2-43: Modes of Cracking (Anderson, 2005)

where Q is a geometric factor (Q =

2 π

for a penny shaped crack, Q = 1 for an infinitely thin

crack) and σv is the far field stress at which the crack initiates. There are three different stress concentration factors depending on the mode of cracking in a material (KI , KII , KIII ) (Refer to figure 2-43).

An extensively used model describing the cracking of an inclined flaw is called the sliding wing crack model (e.g. Brace and Bombolakis, 1963; Gramberg, 1965; Moss & Gupta, 1982; Germanovich & Dyskin, 2000). This model describes a kink crack which initiates at the tips of an inclined and propagates toward the direction of applied compressive loading (figure 2-44). As these kink cracks initiate and propagate, the top and bottom surfaces of the flaw must slide relative to each other; thus it is named the sliding wing crack model.

Tasdemir et al. (1990) propose a sliding crack scenario with closed-flaw under uniaxial compression. This model assumes that there is a closed inclined flaw in a compressive loading field. Therefore, it is assumed that the flaw will experience a frictional resistance along the sliding surface and that it cannot propagate in tension (figure 2-44). By applying equilibrium equations at the interface of the closed flaw, the resultant normal (σn ) and shear (σs ) stresses on the surface of the sliding crack can be determined as

σn = −σ sin2 θ 81

(2.2.8)

Figure 2-44: Sliding Crack Model. Modified from Tasdemir et al. (1990)

σs = −σ sinθ cosθ

(2.2.9)

Where θ is the angle of the flaw with respect to the direction of compressive loading and q is the far field compressive stress. Mellville (1973) applied the stress intensity factor theory to the sliding crack model to determine the stress intensity factors at the tip of the closed flaw as

f law

KII

√ = −σ πa(sin θ sinθ − µsin2 θ )

f law

KI

=0

(2.2.10)

(2.2.11)

Where µ is the coefficient of friction and a is the half flaw length. The mode I stress intenf law

sity factor (KI

= 0) is assumed to be zero because it is an inclined flaw in a compressive 82

loading field (Refer to figure 2-44). Since the criterion of sliding (mode II) is based on the components of normal and shear stress, there is a critical loading angle for which the sliding force is greater than the shear stress (| σs |> µ | σn |). This critical angle can be given as θc = arccot(µ). Therefore, if the loading angle is greater than the critical angle f law

(θ > θc ) the flaw cannot propagate in mode II (KII

= 0), however, if the loading angle

is less than the critical angle (θ < θc ) then the flaw can propagate in mode II and equation 2.2.10 applies.

As the load is increased a kink crack will initiate at the flaw tip (labeled A−A’ in figure 244). This kink crack can occur as either a shear or tensile crack. There are several criteria to calculate the initiation angle of this kink crack (Erdogan and Sih, 1963; Sih, 1974; Hussain et al., 1974). One common assumption is that the kink crack propagates in the direction of the maximum tensile stress intensity factor. Cottrell and Rice (1980) defined the stress intensity factors for a kink crack (A-A’ in figure 2-44) of infinitesimal length (l ≈ 0) initiating at the flaw tip (A or B in figure 2-44) by either mode I or mode II as:

f law

Where KII

κ 3κ 1 f law KIkink = KII [cos + 3cos ] 4 2 2

(2.2.12)

3 f law κ 3κ KIIkink = − KII [sin + sin ] 4 2 2

(2.2.13)

is the stress intensity factor calculated in equation (2.2.10) and κ is the angle

between the kink crack initiating at the flaw tip and the direction of compressive loading. It should be noted that the maximum value of KIkink (δ KIkink (κ)/δ κ=0) exists at approximately 71o . The sliding crack model is the basis for many theoretical and experimental studies on inclined flaws and it will shown later that it has been used in some current numerical models. 83

Fracture Process Zone Although early research assumed that linear elastic fracture mechanics (LEFM) was applicable in describing the brittle behavior of rock, more recent research has discovered that the behavior of cracking in rock is actually quasi-brittle (Irwin, 1961; Dugdale, 1960; Barenblatt, 1962). It has been shown that there is a plastic area (process zone) ahead of the propagating crack tip which is characterized by tractional bridging and microcracking (figure 2-45). This plastic zone can affect the stress concentrations predicted at the crack tip. As a result of these findings, there has been much research conducted to determine the existence of these process zones ahead of crack tips in brittle material (Friedman et al., 1972; Segall & Pollard, 1983; Maji and Wang, 1992; Anderson, 2005).

Figure 2-45: Fracture-process zone ahead of a crack in concrete. The process zone consists of the bridging and microcracking areas at the crack tip (Anderson, 2005). In Barre granite and Carrara marble, visible areas of grain lightening prior to cracking have been observed on the surface of the specimen (Morgan et al., 2013; Wong, 2008) (figure 2-46). These areas of material brightening, referred to as "white patching", have been observed previously in brittle rock materials and were determined to be networks of micro-cracks which develop prior to cracking (Wong and Einstein 2009c; Brooks 2010; Brooks et al. 2010) (figure 2-47). The micro-cracking areas, which are seen macroscopically as white patching, are believed to be fracture process zones of material weakening (Morgan et al., 2013). Fracture process zones have also been experimentally studied in granite using acoustic sensing (Labuz et al. 1987; Zietlow and Labuz 1998; Zang et al. 2000) as well as with optical examinations (Moore and Lockner 1995; Nasseri et al. 2006). However, fracture process zones (white patching) were not observed macroscopically in 84

uniaxial compression tests conducted on gypsum (Wong, 2008). By using ESEM techniques, micro-cracks could be observed near the tips of the fractures in gypsum (Wong, 2008) (Refer to figure 2-47).

Figure 2-46: White patching (fracture process zone) observed in Barre granite (Morgan et al., 2013). Areas of grain brightening referred to as "white patching" occur at the tips of flaws and fractures during uniaxial compression tests on Barre granite. (Morgan et al., 2013)

Figure 2-47: Micro-cracking observed in (a) marble and (b) gypsum process zones by Wong and Einstein (2009c). Marble images near flaws were captured using SEM techniques, while gypsum images were captured with ESEM. Modified from Wong and Einstein (2009c).

85

Crack Interaction Theory

Although predicting crack initiation and propagation in a brittle material can be used to estimate the strength of a body containing a flaw, the interaction and interconnection (referred to as coalescence) between cracks also plays an important role in the failure of a material with multiple flaws. There have been several analytical studies conducted to predict the interaction of cracks in a brittle material. One of the initial studies which theoretically predicted crack interaction was conducted by Horii and Nemat-Nasser (1985). They super-imposed a series of sub-scenarios consisting of homogenous material as well as multiple scenarios with a single crack in the matrix in order to model the interaction of multiple cracks in a homogenous material (figure 248). Unknown quantities from interacting cracks, referred to as "pseudo-tractions", are determined by solving the superimposed scenarios simultaneously (Refer to figure 2-48). This study served as the basis for several other analytical models of crack interaction.

Figure 2-48: Super-position of three crack scenarios proposed by Horii and Nemat-Nasser (1985). The illustration shows the crack interaction model using super-position of three sub-scenarios (From left to right: homogenous, a single horizontal crack, and a single slanted crack). The stresses on the cracks include both far field and "pseudo-tractions" from the interaction of adjacent cracks (Horii and Nemat-Nasser, 1985) Costin (1985) proposed a model in which a body contains a series of pre-existing cracks, 86

oriented along the direction of loading (figure 2-49). As these cracks propagate along the direction of loading the distance between the cracks decreases and interaction occurs based on the local tensile stress fields at the crack tips. Similar to the study conducted by Horii and Nemat-Nasser (1985), this model also used sub-scenarios with unknown pseudo-tractions representing the interaction of other cracks in the system. These sub-scenarios are used to simultaneously solve the system of cracks numerically and determine the stress intensity factor KI .

Figure 2-49: A body containing vertical cracks proposed by Costin (1985). The illustration shoes of a series of vertical cracks (length 2a) in a material with a distance between crack midpoints d1 after Costin (1985).

A series of studies by Ashby & Hallam (1986) and Hallam & Ashby (1990) proposed an alternative crack interaction model in which a series of inclined pre-existing cracks propagate at the crack tips in the direction of compressive loading. As these cracks propagate the area between the cracks creates a "column" of material which will be subject to a buckling effect under the compressive load (figure 2-50). This buckling effect changes the stress intensity factor KI in the cracks, thus representing the crack interaction. Finally, a model proposed by Kemeny & Cook (1987) expanded on the previous model from Ashby & Hallam (1986) by incorporating a curvilinear shape to the kink cracks that initiate from the pre-existing flaws as described in the sliding wing crack model. Similar to 87

the previous model by Ashby & Hallam (1986), the crack interaction is related to effective "columns" which form between cracks (Refer to figure 2-50). The crack interaction can be expressed by the failure of these "columns" in either axial compression (one column) or shear (multiple columns).

Figure 2-50: "Column" crack interaction proposed by Ashby & Hallam (1986). The illustration shows of a series of sliding wing cracks in a material which form "columns" that can be modeled to determine crack interaction after Ashby & Hallam, 1986, and Hallam & Ashby, 1990.

2.2.2

Fracture Experiments

Some of the first experimental tests on the fracturing in brittle materials were conducted by Brace & Bombolakis (1963). Since then there have been many experimental studies regarding the cracking processes in various natural and composite brittle materials. The following list of experimental research done on brittle rock-like and natural rock materials can be found in Wong (2008): Rock-like materials ∙ Columbia Resin 39 - (Brace & Bombolakis, 1963; Nemat-Nasser & Horii, 1982; Horii & Nemat-Nasser, 1985) 88

∙ Glass - (Hoek & Bieniawski, 1965; Bieniawski, 1967) ∙ Plaster of Paris - (Lajtai, 1970; Nesetova & Lajtai, 1973) ∙ Polymethylmethacrilate, or PMMA - (Petit & Barquins, 1988; Chaker & Barquins, 1996) ∙ Molded Gypsum - (Reyes, 1991; Reyes & Einstein, 1991; Shen et al., 1995; Bobet, 1997; Bobet & Einstein, 1998; Sagong, 2001; Sagong & Bobet, 2002; Wong & Einstein, 2009a) ∙ Sandstone-like Molded Barite - (Wong, 1997; Wong & Chau, 1997, 1998; Wong et al., 2001) ∙ Sandstone-like Concrete Mix - (Mughieda & Alzo’ubi, 2004) ∙ Ice - (Wang & Shrive, 1995) Natural rocks ∙ Sandstone - (Petit & Barquins, 1988) ∙ Granodiorite - (Ingraffea & Heuze, 1980) ∙ Limestone - (Ingraffea & Heuze, 1980) ∙ Granite - (Martinez, 1999; Miller, 2008) ∙ Marble - (Huang et al., 1990; Chen et al., 1995; Martinez, 1999; Li et al., 2005; Wong, 2008) It should be noted that these experimental tests were conducted on specimens of different sizes with various flaw lengths, orientations and apertures. The specimen parameters should be taken into account when comparing the observed cracking mechanisms from these different studies.

Single Flaw Experiments

One of the major inconsistencies in many of the previous studies conducted on single flaw specimens is the terminology associated with primary and secondary crack types. Primary cracking refers to the first crack to initiate from the flaw and is usually a tensile wing 89

crack. Secondary cracking includes any cracks which occur after primary cracking and in many cases are shear in nature. Due to this trend in chronology and cracking mode, many studies use the terms shear and secondary cracking interchangeably. This makes it difficult to interpret different research studies which use different terminology for the same types of cracking. However, in contrast to most of the previous studies which use the terms secondary and shear cracking interchangeably, this thesis refers to secondary cracks by its order and then specifically states as to whether it is tensile or shear in mode. There have been many studies regarding the brittle cracking processes associated with a single flaw. Most of the early research in this field focused on the cracking order (Primary, Secondary) and mode (Tensile, Shear) associated with different flaw sizes and inclinations. Although similar cracking order occurred in all studies (Primary tensile wing cracking and secondary tensile or shear cracking at the tips), some discrepancies existed regarding the areas and shape of shear crack zones or banding, which occurred near the tips of the flaw and extended to the edges of the specimen (figure 2-51).

Figure 2-51: Shear crack zone shapes in previous research. (a) Lajtai, 1974 (b) Huang et al., 1990 (c) Chen et al., 1995. Shear zones are shaded or cross-hatched. Tensile cracks are show as darker lines extending from the flaw tips toward the direction loading (top/bottom).

Wong and Einstein (2009a) conducted a series unaxial compression tests on molded gypsum and Carrara marble with flaws using a high speed camera to track the cracking processes. From these tests a set of crack types were proposed (figure 2-52). 90

Figure 2-52: Seven crack types proposed by Wong and Einstein (2009b). T= Tensile Cracks, S = Shear Cracks.

Multi-Flaw Experiments: Crack Interaction

As previously stated, the interaction between cracks plays a key role in the strength and behavior of a brittle material. In order to represent the interaction of multiple cracks in a material an additional flaw can be introduced and these flaws are referred to as a flaw pair. Flaw pairs can be geometrically varied by changing the ligament length between the flaws (L), the inclination of the flaw (β ) and the bridging angle between the two flaws (α) (figure 2-53a). An important distinction between different studies testing flaw pairs is the 91

terminology and definition of the flaw pair geometries. There are currently two different geometric definitions used in the research; ligament length and bridging angle or spacing and continuity (figure 2-53).

Figure 2-53: Definition of Flaw Pairs Geometries by (a) flaw inclination angle [β ], bridging angle [α] and ligament length [ L], or (b) flaw inclination angle [β ], continuity [c] and spacing [s] (Wong, 2009). The ligament length (L) and bridging angle (α) can be easily determined from a flaw pair defined by spacing (s) and continuity (c) by using the simple geometric relations

L=

p

s2 + c2

α = arctan(s/c)

(2.2.14)

(2.2.15)

Conversely, the spacing (s) and continuity (c) can be determined from the ligament length (L) and flaw inclination by

s = L sin(α)

(2.2.16)

c = L cos(α)

(2.2.17)

92

The behavior of flaw pairs has been tested experimentally as early as 1963 by Brace and Bombolakis. Since then many other experiments have been conducted regarding the coalescence patterns of different flaw length, flaw inclinations and flaw bridging angles (e.g. Horii and Nemat-Nasser, 1985; Reyes and Einstein, 1991; Chen et al., 1995). For the purpose of this literature review, only the most recent and relevant studies will be discussed. In 1995 Shen et al. conducted a series of uniaxial compression tests on molded gypsum with various geometric flaw (open and closed) configurations (bridging angle, flaw angle, ligament length). From these tests the cracking initiation process and the cracking mode of coalescence cracks were recorded (Figures 2-54 and 2-55) Shen et al. divided the coalescence into three different classifications based on the mode of the coalescence cracking; shearing, mixed shearing and tensile, and tensile. One of the most important conclusions from this study was the trend observed when varying bridging angles (α) between flaws (with a constant ligament length, L): ∙ Small positive bridging angles and small negative bridging angles: Coalescence generally occurred as a shear crack between the inner flaw tips ∙ Intermediate bridging angles: Coalescence generally occurred by both shear and tensile cracks. ∙ Large bridging angles: Coalescence generally occurred by a tensile crack In similar tests conducted by Bobet and Einstein in 1998, gypsum specimens with flaw pairs were tested under uniaxial and biaxial compression. In this test series, which used the spacing and continuity geometric definitions, the effect of geometric orientation as well as ligament length was investigated. A series of general coalescence types were proposed based on both cracking type and shape (figure 2-56). Bobet and Einstein (1998) concluded that the coalescence trends in regard to bridging angle are similar to those observed by Shen et al. (1995). In 1998 Wong and Chau performed a series of uniaxial compression tests on a synthetic sandstone type specimen with double flaw pairs. The observed coalescence patterns were 93

Figure 2-54: Coalescence patterns observed by Shen et al.(1995). Patterns were recorded for unaxial compression tests on molded gypsum with two flaws. Both open (nonfrictional) and closed (frictional) flaws were used. Specimens were defined using the flaw pair geometry configuration shown in figure 2-53.

94

Figure 2-55: Coalescence patterns observed by Shen et al.(1995). Patterns were recorded for unaxial compression tests on molded gypsum with two flaws. (continued from figure 254)

95

Figure 2-56: Coalescence types proposed by Bobet and Einstein (1998a). Tests were conducted in molded gypsum specimen. T = Tensile cracking, S = Shear cracking

96

divided into shear, mixed (shear and tensile), and wing tensile. All of the flaws were closed and the coefficient of friction (µ) was varied. From these tests Wong and Chau were able to plot each geometric configuration as a test point and define zones for each coalescence type as bridging angle and flaw angle vary (Figures 2-57 to 2-59).

Figure 2-57: Coalescence zones based on bridging angle and flaw inclination angle for a coefficient of friction µ = 0.6. S-regime represents geometric configurations in which shear coalescence occurs. M-regime represents geometric configurations in which mixed (shear and tensile) coalescence occurs. W-regime represents geometric configurations in which tensile wing coalescence occurs. Points labeled "?" showed no coalescence. (Wong and Chau, 1998) The coalescence regimes defined by Wong and Chau (1998) showed a trend in coalescence with regard to increasing bridging angle that was similar to previous research by Shen et al. (1995) (Refer to Figures 2-57 to 2-59). These coalescence regimes also showed that as the friction coefficient in the closed flaws decreases, shear coalescence occurred at higher bridging angles (Refer to Figures 2-57 to 2-59). It should be noted though that Wong and Chau (1998) used the terms secondary cracks and shear cracks interchangeable. Therefore, it is unknown whether a proper distinction was made between coalescence occurring from secondary cracking and shear coalescence. Additionally, experimental coalescence research was conducted by Wong and Einstein (2009a) on both natural Carrara marble and molded gypsum. This research was one of 97

Figure 2-58: Coalescence zones based on bridging angle and flaw inclination angle for a coefficient of friction µ = 0.7. All other captions are the same as those for figure 2-57 (Wong and Chau, 1998)

Figure 2-59: Coalescence zones based on bridging angle and flaw inclination angle for a coefficient of friction µ = 0.9. All other captions are the same as those for figure 2-57 (Wong and Chau, 1998)

98

the first studies regarding cracking and coalescence to incorporate the use of a high-speed camera to properly determine shear cracking and follow crack propagation. Four different test series were conducted on each of the two materials (Table 2.2). From the tests conducted on these flaw geometries Wong and Einstein (2009a) proposed a set of nine different coalescence patterns (Figure 2-60). The following conclusions were drawn: ∙ Bridging Angle (α): The effect of bridging angle was determined based on the stepped flaws test series with a constant flaw inclination angle of 30 degrees. It was concluded that at small bridging angles shear coalescence occurred. As the bridging angle increased mixed shear and tensile coalescence was observed. At higher bridging angles tensile coalescence was observed. This matched the previous conclusions made by Shen et al. (1995) and Bobet and Einstein (1998). ∙ Inclination Angle (β ): The effect of inclination angle was determined from the coplanar test series (constant 0o bridging angle). As the inclination increased, a trend from indirect or no coalescence, to shear coalescence, to tensile coalescence was reported. ∙ Ligament Length (L): Increasing the ligament length reduced the amount of coalescence (e.g. a higher number of non-coalescence), especially in the coplanar flaw geometries. Table 2.2: The four test series conducted by Wong and Einstein (2009a). Divided by ligament length and then coplanar (bridging angle of 0 degrees) and stepped flaws. All stepped flaws were oriented at an inclination of 30 degrees. Description coplanar stepped coplanar stepped

Ligament Length, L 2a 2a 4a 4a

Bridging Angle, α 0 -60, -30, 0, 30, 60, 90, 120 0 -60, -30, 0, 30, 60, 90, 120

Inclination Angle, β 0, 30, 45, 60, 75 30 0, 30, 45, 60, 75 30

Recently, additional flaw pair coalescence tests were conducted on Barre granite with flaw pairs (Morgan et al., 2013). These uniaxial compression tests on Barre granite were conducted with high speed video and then combined with the results from Martinez (1999) and Miller (2008) to develop a more complete and systematic representation of the coalescence behavior of granite (figure 2-61). The coalescence results observed in granite were 99

Figure 2-60: Nine coalescence patterns proposed by Wong and Einstein (2009a). Coalescence patterns were determined from uniaxial compression tests on Carrara marble and molded gypsum. T= Tensile Cracks, S = Shear Cracks. Crack types 1, 2, 3 refer to nomenclature introduced by Wong and Einstein (2009b) shown in figure 2-52. Coalescence categories 1 and 2 show no coalescence or indirect coalescence. Categories 3 and 4 show direct shear coalescence. Category 5 shows direct combined tensile-shear coalescence. Categories 6 through 9 show direct tensile coalescence. 100

similar in nature to those in previous research and could be categorized appropriately using the coalescence categories proposed by Wong and Einstein (2009a). However, there was typically more indirect coalescence behavior observed in granite compared to marble and gypsum. It was hypothesized that this could be due to the larger, stronger grains in granite causing more tensile fracturing along grain boundaries (Morgan et al., 2013). In addition to coalescence result, the progression and occurrence of white patching was systematically described for Barre granite as well (figure 2-62).

Figure 2-61: Coalescence behavior of Barre granite based on work by (a) Morgan et al. (2013), (b) Miller (2008) and (c) Martinez (1999). For each geometry, the Wong and Einstein (2009a) coalescence category is shown as well as whether coalescence pattern is indirect or direct. T = Tensile crack, S = Shear crack. (Morgan et al., 2013)

101

Figure 2-62: White patching observed in Barre granite between flaw pairs. The following progression of white patching is for flaw pairs with high combined flaw and bridging angles (α + β ≥ 60o ). (Morgan et al., 2013)

2.2.3

Modeling Techniques

Since crack propagation and interaction is a complex problem with no simple close-formed solutions, numerical modeling methods are often used. There are several different types of numerical models used today to predict fracture initiation, propagation and coalescence of fractures in rock. They can be sub-divided into three categories; continuum based models, discrete models, and hybrid continuum/discontinuum models.

Continuum Based Models

Continuum models define the problem as a continuous material body and include models such as eXtended Finite Element Method (X-FEM) and Boundary Element Method (BEM). Continuum models are able to capture constitutive relationships well; however, they have difficulty localizing failures and initiating fractures in uniform bodies. Therefore, continuum based fracture models typically require geometries with stress localizations or initial pre-existing fractures. The eXtended Finite Element Method (X-FEM) and FROCK, a 102

Boundary Element Method (BEM), will be discussed. Extended Finite Element Model Finite element methods use a mesh to determine stress-strain fields in a body based on constitutive laws. However, using finite element methods alone to model fracture is a troublesome process because after a crack propagates re-meshing is often required at the tip of the crack. Re-meshing is not efficient since it takes a considerable amount of computational power and time. In 1999, the extended Finite Element Method (X-FEM) was introduced by Moes et al. (1999) allowing finite element methods to be used to predict fracture propagation without re-meshing (figure 2-63). One drawback of using extended finite element modeling is that a finer mesh is typically used in an area where the crack might propagate. Therefore, extended finite element models typically require simple problems where the propagation direction of the crack is known beforehand. Since rock fracture is complex, the crack patterns are not simple and this issue is potentially one of the reasons that X-FEM is still not used commonly to predict fracture in rock material.

Figure 2-63: Extended Finite Element Model of Notch Test (Areias and Belytschko, 2005). The results of an extended finite element model of a notched beam bending test. A fine mesh is used in the area of potential propagation.

Boundary Element Method (BEM) Unlike other continuum methods, in which the entire body must be discretized, the Bound103

ary Element Method (BEM) only requires that the boundaries be discretized. In addition, BEM does not require external boundaries to be defined if far-field stresses are being applied. There are two types of BEMS, the Stress Discontinuity (SD) method and Displacement Discontinuity (DD) method. In the SD method elements transfer traction stresses across the element boundaries. In the DD method element transfer displacement discontinuities across the element boundaries. One model which implements both the SD and DD methods and has been used extensively to model fracture in rock is called FROCK.

Stress-based FROCK Model Numerical modeling of cracks has been done by the MIT rock mechanics group starting in 1986 by Chan. Chan (1986) developed the code ’FROCK’ (which stands for fractured rock). This is based on the Displacement Discontinuity Method (DDM) which incorporates Linear Elastic Fracture Mechanics principles and stress intensity factors to model crack initiation and initial crack propagation. The FROCK model consists of a crack initiation and propagation criterion implemented with a Displacement Discontinuity Method (a type of Boundary Element Model) and was originally developed by Chan (1986). FROCK was later improved by Bobet (1997), who incorporated a stress based criterion to better model crack initiation, propagation and coalescence. Additionally, Bobet (1997) compared the numerical model results to experimental test results observed in gypsum rock with flaw pairs. The FROCK model used by Bobet (1997) used a stress-based failure criterion. The FROCK model assumes a linear-elastic isotropic material given the elastic modulus (E) and Poisson’s ratio (ν). The stress-based model has four major input parameters which can be varied to calibrate the results: ∙ Critical tangential stress - σθcrit ∙ Critical shear stress - τθcrit ∙ Plastic radius - ro ∙ Coefficient of friction for existing flaws - µ The FROCK model applies a far field stress, calculates the stress-field and then searches 104

around all crack tip elements to determine if cracking should initiate (See figure 2-64). Due to the fact that stresses tend to infinity near a crack tip (r = 0), linear elastic theory does not apply in this region. Therefore, a zone of plastic influence (ro ) is assumed, and the model searches the stress-field around the crack tip, at a distance ro , to determine if a crack will initiate (Refer to figure 2-64). The criteria for which a tensile cracks will initiate or propagate are given as: ∙ At the distance ro from the tip of an existing crack δ σθ δθ

∙ In a direction θ in which σθ is max (σθ max ),

=0

δ 2σ θ δθ2

>0

∙ When σθ max = σθcrit , with σθcrit being the critical tangential stress and the criteria for which a shear crack will initiate or propagate are given as: ∙ At the distance ro from the tip of an existing crack ∙ In a direction θ in which τθ is max (τθ max ),

δ τθ δθ

=0

δ 2 τθ δθ2

>0

∙ When τθ max = τθcrit , with τθcrit being the critical shear stress

Figure 2-64: FROCK stress field around crack tip. Showing the cylndrical stresses of an element radial to the flaw tip (Bobet and Einstein, 1998).

Once a crack propagates by an increment (an element in the DDM) the process of applying the above mentioned criteria is repeated (See figure 2-65). Hence the entire crack initiation and propagation process can be modeled. 105

Figure 2-65: FROCK crack elements at the flaw tip. Once a crack propagates by an increment (an element in the DDM) the process of applying the criteria is repeated.

Strain-Based FROCK Model Although the Stress-based FROCK code was able to model many flaw pair geometries well, there were some limitations in the results obtained from this model and it had difficulty capturing behavior observed with certain flaw pair geometries, such as coplanar flaw pairs with high flaw angles (figure 2-66). In order to better capture the behavior, the original FROCK model was improved by implementing a strain-based failure criterion by Gonçalves da Silva and Einstein (2013) (Refer to figure 2-66). The strain-based FROCK model showed better results for predicting cracking behavior in gypsum, compared to flaw pair geometries that could not be completely represented with the original stress-based FROCK model. The strain-based FROCK model also assumes a linear-elastic isotropic material given the elastic modulus (E) and Poisson’s ratio (ν). The strain-based model has four major input parameters which can be varied to calibrate the results: ∙ Critical tangential strain - εθcrit ∙ Critical shear strain - γθcrit ∙ Plastic radius - ro ∙ Coefficient of friction for existing flaws - µ The strain-based FROCK model searches around all crack tip elements to determine if cracking should initiate (See figure 2-67). The criteria for which a tensile crack will initiate 106

Figure 2-66: A comparison of Stress-based and Strain-based FROCK results (Gonçalves da Silva and Einstein, 2013). Experimental test results were conducted by Bobet (1997) and Wong (2008). T = Tensile crack, W = Tensile Wing Crack, S = Shear Crack. (Gonçalves da Silva and Einstein, 2013)

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or propagate are given as: ∙ At the distance ro from the tip of an existing crack ∙ In a direction θ in which εθ is max (εθ max ),

δ εθ δθ

=0

δ 2 εθ δθ2

>0

∙ When εθ max = εθcrit , with εθcrit being the critical tangential strain and the criteria for which a shear crack will initiate or propagate are given as: ∙ At the distance ro from the tip of an existing crack ∙ In a direction θ in which γθ is max (γθ max ),

δ γθ δθ

=0

δ 2 γθ δθ2

>0

∙ When γθ max = γθcrit , with γθcrit being the critical shear strain

Figure 2-67: FROCK strain field around crack tip. Showing the cylndrical strains of an element radial to the flaw tip (Gonçalves da Silva and Einstein, 2013).

Discrete Element Method The Discrete Element Method (DEM) discretizes the entire body into particles or elements which have interaction forces and bond strengths. The benefit of models based on the DEM is that discontinuities in the material can be modeled on the grain scale and fracture initiation and propagation can scale up to the macroscopic behavior. This means that DEMs must determine the micro-scale particle parameters (bond strength, particle stiffness, etc.) by calibrating the model results to experimental macro-scale material parameters (Young’s modulus, compressive strength, strain to failure, etc.). Since there are typically many different micro-scale parameters governing the particle-to-particle interactions, this calibration 108

process can be very complex and sometimes difficult to relate to the macro-scale behavior. In addition, there is no need to mesh or re-mesh because the body is modeled as an array of discrete elements. There are several such models which use the DEM to predict fracture propagation and material behavior such as the Particle Flow Code (PFC) and the Universal Distinct Element Code (UDEC). Particle Flow Code (PFC) - Bonded Particle Model (BPM) One of the most common forms of discrete element models is the Particle Flow Code (PFC) (ITASCA, 2008a). One such PFC used to model fracture in granular rock material is called the Bonded Particle Model (BPM). In a Bonded Particle Model each particle is defined, along with its interactions (bonds and stress transmissions) with other particles (figure 268). When the bonds between two particles break, a crack forms. One of the limiting factors in Bonded Particle models is the size of particles that can be modeled. Since simulating large arrays of particles and all of their interactions can require a significant amount of computational power, there is a limit to how many particles can be modeled.

Figure 2-68: Diagram of Bonded Particle Model Interactions(Zhang and Wong, 2013). The Bonded Particle Model simulates discrete particles which interact to transfer stress and can be bonded together. When bonds are broken a cracks form.

Bonded Particle models have been used to simulate fracture initiation and propagation from a single flaw in rock material (Zhang and Wong, 2012), as well as flaw pairs (Zhang and Wong, 2013) (figure 2-69). The coalescence patterns observed in the BPM results agree well with previously observed coalescence patterns defined by Wong and Einstein (2009a) 109

(Refer to figure 2-60). Specifically, the indirect coalescence observed in the simulation results of tests conducted on a flaw pair geometry of 2a-15-0 (Refer to figure 2-69) are exceptionally close to the coalescence patterns observed in gypsum (Wong and Einstein, 2009a). This geometry has typically been very difficult to replicate in other models that attempt simulate crack coalescence such as FROCK (Bobet, 1998).

Figure 2-69: Bonded Particle Model Cracking results from Zhang and Wong (2013). Two flaw pair geometries loaded uniaxially are simulated with tensile bond breakage (tensile cracks) shown in white and shear bond breakage (shear cracks) shown in red. See Figure 2-53 for flaw pair geometry definitions. Universal Distinct Element Code (UDEC) The Universal Distinct Element Code (UDEC) is another discrete element method which has had some success predicting fracture in rock (ITASCA, 2008b). Similar to BPM, UDEC uses discretized elements with interaction forces and bonds between them. However, the discrete elements in UDEC are typically block elements without voids between them (figure 2-70). These block elements have bonds and frictional strength which can be modeled by their interaction forces (figure 2-71). The bond stiffness between these element can be linear, as shown in figure 2-71 (Kazerani and Zhao, 2010), or they can be non-linear (Kazerani et al., 2012). Kazerani et al.(2012)used UDEC to predict material stress-strain behavior and fracture patterns in plaster specimens. Both unconfined compression tests on cylindrical specimens and Brazilian tests were conducted experimentally and simulated with the UDEC. Micro110

Figure 2-70: Discrete block elements of the Universal Distinct Element Code (UDEC). Block elements do not have voids between them. (Kazerani and Zhao, 2010)

Figure 2-71: Universal Distinct Element Code (UDEC) element interactions. (a) A schematic sketch of a block element (Lisjak and Grasselli, 2014) and (b) forcedisplacement interactions between element contacts (Kazerani and Zhao, 2010). More complex models can incorporate non-linear stiffness curves and lower residual frictional strengths.

111

material (element) properties (ks , kn , Fs , Fn , etc.) were calibrated using macro-material behavior such as the tensile and compressive strength, Young’s modulus and Poisson’s ratio. Using these back calculated micro-material parameters, the fracture propagation leading to failure could be simulated (figure 2-72). The fracture pattern results were not compared with experimental results in this study. However, the model shows reasonable fracture patterns for both the unconfined compression fracture as well as the brazilian test (Refer to figure 2-72).

Figure 2-72: UDEC fracturing results for (a) unconfined compression and (b) Brazilian tests in plaster. Micro-parameters for elements and element interactions were calibrated using responses based on macro material properties such as stiffness and compressive strength. (Kazerani et al., 2012)

Hybrid Continuum/Discontinuum

Hybrid continuum/discontinuum models use continuum mechanics methods which can solve problems with discontinuities. These models are neither purely continuous or purely 112

discontinuous. Two such model types will be discussed, a combined Finite-Discrete Element Method (FDEM) called Y-GEO and the Smoothed Particle Hydrodynamic (SPH) model. Combined Finite-Discrete Element Method (FDEM) - Y-Geo The combined Finite-Discrete Element Method (FDEM) uses the finite element mesh technique to model the continuum properties of the material and additionally models each mesh element as a discrete element which interacts with adjacent elements. At each time step, a nodal point that exists in the model is governed by the equation:

M

δ 2x δx +C + Fint (x) − Fext (x) − Fc (x) = 0 2 δt δt

(2.2.18)

Where M is the mass diagonal matrix and C is the damping diagonal matrix. Fint , Fext and Fc are the vectors of internal resisting force, external loads and contact forces respectively. A four-node interface element exists between these three-node elements and can yield and break apart, becoming a broken crack element when it fully yields (figure 2-73 and figure 2-74). Elements can debond in either the tensile mode or the shear shear mode. Once the displacement of the interface bond reaches a critical level (Or , Sr ) a crack forms (Refer to figure 2-73). After bonds break there is no residual tensile strength; however, Y-Geo includes residual sliding friction after shear bonds are broken. The amount of energy associated with the crack is defined by the non-residual area under the force-displacement curve (GIc , GIIc ) (Refer to figure 2-73). It should be noted that in this model cracks can initiate as tensile, shear or mixed-mode, which is a combination of both tensile and shear yielding, depending on interface displacements (Refer to figure 2-73c). The mode of cracking can be defined by the equation:

m = 1+

s − sp sr − s p

113

(2.2.19)

Figure 2-73: Element interface yielding and debonding (cracking) behavior in Y-Geo. Elements can debond by either (a) tensile cracking or (b) shear cracking once the displacement reaches a critical level (Or , Sr ). The amount of strength associated with the crack is defined by the area under the force-displacement curve (GIc , GIIc ). (c) Additionally, cracks can initiate as tensile, shear or a mixed-mode depending on interface displacements. (Lisjak et al., 2013)

Figure 2-74: Cohesive crack propagation model in the FDEM Y-Geo model. The cohesive crack propagation model concentpualized by (a) Labuz et al. (1985) (b) Hillerborg et al. (1976) and modeled by (c) Y-Geo. As interfaces (crack element) between three-node elements yield, the four-node interface element becomes a yielding element and then becomes a broken crack element. (Lisjak et al., 2013)

114

Since these interface elements are able to yield, the Y-Geo FDEM is able to replicate, in some capacity, a cohesive crack model such as those proposed by Labuz et al.(1985) and Hillerborg et al.(1976) (Refer to figure 2-74). The length of this fracture process zone is defined by the number of elements that are allowed to yield. Also, it should be noted that cracks in Y-Geo can only occur at the interface between two body elements where the crack element exists. Therefore, one of the major differences between the extended finite element method and combined Finite-Discrete Element Method used by Y-Geo is that cracks can occur across elements in XFEM while FDEM requires that cracks occur at the interfaces of elements (figure 2-75).

Figure 2-75: Crack propagation in XFEM and DFEM. Crack propagation in (a) XFEM can occur across an element, however, in (b) DFEM such as Y-GEO shown above, cracks must occur at element interfaces. Modified from Moes et al. (1999) and Lisjak et al. (2014)

Y-Geo has been extensively used to model fracture in rock materials (Mahabadi et al., 2012; Lisjak et al., 2013; Lisjak et al., 2014). By using image processing techniques on granite to determine mineral distribution (figure 2-76) and assigning material parameters for each mineral type (feldspar, quartz, and biotite), Y-Geo has been able to model fracture in hetergenous material (figure 2-77). Y-Geo showed exceptional results for the predicted fracture pattern in a Brazilian test on granite (Refer to figure 2-77). The use of both image processing and mineral composition in the model is novel and unique. In addition, YGeo has been used to model fracture of rock with discrete fracture networks and bedding anistropy (Lisjak et al., 2013; ). Modeling anistropic fracture will be discussed in later sections. 115

Figure 2-76: Image processing mineral map of Brazilian test on granite. Mineral locations were identified in a Brazilian test conducted on granite and then modeled in Y-Geo. (Mahabadi et al., 2012)

116

Figure 2-77: Experimental and Y-Geo simulation results for a Brazilian test on granite. The (a) experimental results were compared to (b) Y-Geo fracture simulations for a Brazilian test on granite. The mineral map is shown in the Y-Geo results (Refer to figure 2-76 for mineral types). (Mahabadi et al., 2012)

117

Smoothed Particle Hydrodynamics (SPH) Models The Smoothed Particle Hydrodynamics (SPH) model is a mesh-free method originally developed to model fluid dynamics in the field of astrophysics to model non-spherical star formations (Gringold and Monaghan, 1977). Particles in SPH are modeled by Navier-Stokes equations (Potapov et al., 2001). SPH inherently uses a statistical smoothing method which averages a function of interest, such as density, over a finite area ("kernel") for each discrete particle (figure 2-78). By using this smoothing function, SPH is able to combine continuum modeling with particle interactions which can also capture discontinuities.

Figure 2-78: Kernel function definition for SPH. The kernel function (W) is defined over an area designated by radius kh around a particle of interest (i). This function will be influenced by adjacent particles (j) defined inside the area of interest. (Ma et al., 2011) Originally SPH was primarily used for modeling problems with fluid flow. However, Libersky and Petschek (1990) implemented the SPH model on solid mechanics problems. Libersky and Petschek (1990) modeled the defomed shape of a dynamic impact test on iron and compared it to a finite-element model called EPIC-2 (figure 2-79). Although the results from Libersky and Petschek (1990) did not match perfectly with the previously established FEM continuum model EPIC-2, the results were promising and established the possibility of SPH to model solid mechanics problems in the future. More recently the SPH model has been used to model fracture in rock-like materials (Ma et al., 2011). Ma et al. (2011) modeled the fracture patterns of unconfined compression tests 118

Figure 2-79: Implementation of the SPH model to a solid mechanics problem. The SPH model (points) was used to a simulate a plane-strain dynamic impact test on iron. The initial position is shown (left) as well as the position after 50 µs (right). The SPH results were then compared to the results of an FEM model EPIC-2 (line). A total of 1320 particles were used in the SPH simulation. (Libersky and Petschek, 1990) in a heterogeneous granite rock with SPH. The confining stresses were modified to study their effect on the fracturing of intact material under quasi-static loading conditions. Both stress-strain (figure 2-80) and fracture patterns (figure 2-81) could be observed using the SPH model proposed by Ma et al.(2011). Although the stress-strain results observed with the SPH model by Ma et al.(2011) did not predict the unconfined compression strength properly (desired σc =157.0 MPa), with more calibration better results were obtained and reasonable fracturing results were observed as well (Refer to figure 2-81).

Figure 2-80: Stress-strain results of SPH model on granite fracture. The SPH model was able to capture stress results (right ordinate) as well as incremental particle damage (left ordinate). The incremental particle damage is evidently shown as a histogram. A higher number of damaged particles can observed at failure. (Ma et al., 2011) 119

Figure 2-81: Fracture patterns observed in granite using SPH model. The locations of fracture initiation and propagation in a SPH simulation of a unaxial compression tests on granite can be observed. The stress-strain information corresponding to each frame (a-f) corresponds to the labels in figure 2-80. (Ma et al., 2011)

120

In summary, the SPH model is still relatively new to the field of fracture in rocks. Like discrete element methods, such as the particle flow code, SPH requires a lot of computational power which is directly related to the total number of particles that can be discretized. The amount of particles affects the minimum particle size and ultimately the resolution of the body being modeled. Therefore, it is likely that as computational power increases a model such as SPH will become more readily used to predict fracture in rocks.

2.2.4

Modeling Anisotropic Fracture

As previously discussed, fracture along planes of anisotropy is an important factor when modeling cracking in an anisotropic material such as shale. Therefore, models previously discussed, which assume isotropic strength conditions in all directions, will not properly predict cracking propagation and coalescence. In order to capture anisotropic behavior, researchers have been incorporating the anisotropy of both the elastic properties (stressfield) as well as the failure criteria into previously used modeling techniques. The following section will discuss current fracture propagation models which incorporate elastic and/or failure anisotropy.

Y-Geo (Lisjak et al., 2013; Lisjak et al., 2014)

Y-Geo, previously shown, is a hybrid discrete finite element method which is able to predict fracture initiation and propagation in heterogeneous rock (Lisjak et al., 2013). In addition, Y-Geo has been implemented to predict fracture in rocks with preferred strength anisotropy direction and anisotropic fracture networks (Lisjak et al., 2014). Both anisotropic elastic behavior and anisotropic strength criteria are incorporated into the Y-Geo model. Anisotropic elastic behavior is captured by the deformability of the individual elements. Anisotropic strength behavior can be captured by Y-Geo in two different ways. One approach is to introduce randomly distributed micro-fractures with preferential orientation along the direction of the bedding planes, referred to as the discrete approach. The alter121

native approach is to use reduced element bond strength values (e.g. cracking criteria) at different angles, referred to as the smeared approach. In the discrete approach proposed by Lisjak et al. (2013) use an isotropic homogenous medium with natural fractures that are aligned with the bedding plane (figure 2-82). These randomly distributed fractures are cohesionless. The concept behind the effective anisotropy caused by these oriented flaws is based on the sliding crack model which was previously discussed. In other words the crack initiation, propagation and coalescence between these flaws will be affected by the effective "bedding plane" angle. The major issue with the discrete approach is that it requires some previous knowledge of the fracture spacing and length. Therefore, scaling the discrete method from the lab scale to the field scale can be difficult due to different length scales.

Figure 2-82: Uniaxial compression simulation in Y-Geo using the discrete approach. The discrete approach models the material strength anisotropy by introducing randomly distributed natural flaws aligned with the bedding plane direction. (Lisjak et al., 2013) The other anisotropic strength method used by Lisjak et al.(2013) is called the smeared approach. For the smeared approach the strength parameters for crack formation (i.e. the energy required to create a crack G, etc.) were varied linearly with respect to angle (figure 2-83). This angle is the angle between the crack direction and the bedding plane direction, with 0o occurring when a crack is propagating along a bedding plane (lowest value) (Refer to figure 2-83). Using the smeared approach, the fracture patterns were simulated in 122

unconfined compression tests at various bedding plane angles (figure 2-84). From these results it could be observed that most cracking predicted was shear and that sliding occurred on specimens with inclined bedding planes (load angle, θ =15o , 30o , 45o , 60o ) (figure 2-85). The trend between predicted the uniaxial compressive strength and load angle could also be determined. A minimum unconfined compressive strength was observed at a 45o loading angle (θ = 45o ). As expected, the strength when the load is perpendicular to the bedding planes (θ = 90o ) is higher than when loaded parallel to the bedding planes (θ = 0o ) (Refer to figure 2-85).

Figure 2-83: Variation of strength anisotropy in the Y-Geo smeared approach. The smeared approach in Y-Geo assumes a linear change in the element bond strength properties ( ft , GIc , GIIc ) with respect to the angle between the crack element-to-element interface and the bedding plane. (Lisjak et al., 2013)

Figure 2-84: Smeared approach results for various loading plane angles in the Y-Geo model. The angle θ is the loading direction, therefore, the bedding plane angle with respect to the horizontal is ψ = 90o − θ . Shear cracks shown in red, tensile cracks are shown in blue. (Lisjak et al., 2013)

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Figure 2-85: Effect of loading angle on simulated compressive strength (UCS) in Y-Geo. The effect of the loading angle, θ (i.e. bedding plane angle, ψ = 90o − θ ) on the UCS is shown. Error bars associated with experimental UCS tests are shown by the dashed bars. (Lisjak et al., 2013)

FROCK with Anisotropic Stress-Field (Bobet and Martin, 2014)

As discussed previously, FROCK is a Boundary Element Method which predicts crack initiation, propagation and initiation in isotropic rocks. Although FROCK was able to accurately describe stress and strain fields in an isotropic material such as gypsum (Bobet, 1998; Gonçalves da Silva and Einstein, 2013), it was not able to accurately model stress and strain fields for anisotropic rock. The stress-based FROCK code developed by Bobet(1998) was updated to include anisotropic elastic stress fields. The effect of anistropy (Eyy to Exx ) was studied (figure 2-86) as well as the effect of bedding direction (figure 2-87). The results of FROCK were also compared to the numerical FEM model ABAQUSTM (Refer to figure 2-86). It can be seen from figures 2-86 and 2-87 that the anisotropy ratio and bedding plane direction affect the magnitude of both tangential and shear stresses near the crack tip. However, it appears as though the elastic anisotropy has little effect on the direction of maximum/minimum (critical) stress. Since the critical stress determines the direction of crack 124

Figure 2-86: The effect of elastic anisotropy ratio on the FROCK crack-tip stress field. The polar stress field around a crack tip in the FROCK code for (a) isotropic conditions (Eyy = Exx ) and (b) anisotropic conditions (Eyy = 2Exx ). σθ is the tangential stress (tensile crack initiation) and τ is the shear stress (shear crack initiation). The bedding plane angle (ψ) is 45o . Exx = 2400MPa, Gxy = 1043.4MPa, νxz = νyx = 0.15 and far-field stress σyy = 1MPa. Additionally, FEM predictions from ABAQUS are compared to the isotropic stress field (dashed line). Modified from Bobet and Martin (2014)

125

Figure 2-87: The effect of bedding angle (ψ) on the FROCK crack-tip stress field. The bedding angle (ψ) is defined from the horizontal (i.e. (ψ = 0o ). Exx = 7800MPa, Eyy = 2400MPa, Gxy = 830MPa, νxz = 0.22 νyx = 0.07 and far-field stress σyy = 1MPa. Refer to figure 2-86 for more explanation. Modified from Bobet and Martin (2014) propagation, it appears that including elastic anisotropy into FROCK may have little effect on the crack initiation direction and possibly crack propagation. Also, due the fact that the FROCK code with the anistropic stress field modification has been developed so recently it has not been studied for crack propagation or integrated with an anisotropic failure criterion.

FRACOD (Shen et al., 2015)

In addition to FROCK, there are other Boundary Element methods which have been developed to predict fracture. Once such BEM code called FRACOD was developed by Shen et al. in 2013. This code was recently modified to incorporate bedding plane anisotropy (Shen et al., 2015). This model does not include elastic anistropy, however, it does have a set of failure criteria which are based on the bedding direction (figure 2-88). The effect of linear or sinousiodal variations in the strength parameters were studied (Refer to figure 288). The strength parameters varied were the friction angle [φ (β )], cohesion [c(β )], and 126

tensile strength [σt (β )]. It was determined that the linear variation showed best results and it was used for simulations.

Figure 2-88: Variation of anisotropic strength parameter in FRACOD. Both sinusoidal and linearly varied strength parameters are shown. The strength parameters varied were the friction angle [φ (β )], cohesion [c(β )], and tensile strength [σt (β )](Shen et al., 2015)

From these variations in strength parameters, failure of a 2-D anisotropic rock mass could be modeled (1m x 2m) (figure 2-89). Fracture initiation and propagation can be observed to occur along the direction of pre-described bedding planes. A u-curve reduction in axial strength was observed in the FRACOD results, with a minimum at about 55o and a maximum at 10o (figure 2-90). As expected from the single plane of weakness theory, the minimum strength was observed at about 55o (Shen et al., 2015).

Interestingly the authors did not describe the initial material discontinuity state, such as initial fracture location and their spacing (if any). Clearly in the FRACOD results there is a finite and uniform spacing between each fracture (Refer to figure 2-89). However, one of the major drawbacks of BEM (or any continuum based model) for simulation of fracturing is that fractures typically need either a pre-described initial fracture (flaw) or a stress concentration (such as a notch or hole). Also, although the methodology and assumptions for FRACOD may be valid, the methodology leading to the results are not completely clear. 127

Figure 2-89: Fracture results for rock specimen with 65o bedding angles in FRACOD. Fracture results of the rock mass under biaxial compression are shown in two stages. (a) Stage 1 where some fractures have initiated and (b) Stage 2 where additional fractures have initiated and previous fractures have propagated. (Shen et al., 2015)

Figure 2-90: Maximum axial stress for different bedding plane angles (with respect to the horizontal) in FRACOD. The minimum strength is observed at about 55o .(Shen et al., 2015)

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2.3

Summary

The definition and origin of shale, experimental studies conducted on shale, aniostropic failure theories, theoretical and experimental work on fracture mechanics and fracture modeling techniques have been summarized in this chapter. As shown, there has been an extensive amount of work conducted on Opalinus shale. Additionally, crack initiation, -propagation and -interaction in many different rock-like materials has been well studied. However, there has not be a detailed and extensive experimental study conducted on the crack interaction in shale. Shale is anisotropic, and anisotropy plays a key role in the failure mechanisms of the material. It is important to understand, in a systematic and thorough way, how anisotropic bedding planes affect crack interaction and failure in shale. The following study will present an extensive series of experiments observing fracture/crack initiation, -propagation and -coalescence in Opalinus shale and then propose a new anisotropic fracture model to predict the fracture processes observed in the experimental results.

129

130

Chapter 3 Experimental Process This study consists of a series of uniaxial compression tests on prismatic Opalinus shale specimens with and without pre-cut flaw pairs at various bedding plane angles. This chapter will describe the extraction location and -method, mineralogy, specific gravity, natural water content and saturation of the Opalinus shale cores obtained for testing. The process of cutting and storing Opalinus shale will then be presented. Finally, the testing equipment and analysis procedures used for the tests conducted on Opalinus shale will be described.

3.1

Material - Opalinus Shale

Opalinus shale is a clay based shale from the Dogger formation (Middle Jurassic) found in the Jura mountains, which is located in Northwestern part of Switzerland and other areas in France and Southern Germany. Its mineral composition and material properties can vary from location to location. Therefore, Opalinus shale from the Mont Terri site may be different from other Opalinus shale samples, such as those from Benken or Schlattingen. One of the advantages of using Opalinus shale is that it is a very well documented material and has been extensively tested (both in-situ and in the laboratory). There have been many tests on the physical and mineralogical properties of Opalinus shale (summarized in table 131

3.1). The mechanical properties of Mont Terri Opalinus shale have been characterized in a number of studies and presented in technical reports (Bellwald, 1990; Aristorenas, 1992; Amann et al., 2001; Bock, 2001; Bock, 2009; Valente, 2012). There are also studies which investigated the mineralogy and composition of Opalinus shale (Van Loon et al., 2004; Courdouan et al., 2007; Wenk et al., 2008; Josh et al, 2012). For a detailed reference list of studies conducted on Opalinus shale refer to table 2.1. Since getting access to shale samples can be very difficult, determining every material property of Opalinus shale can be nearly impossible. However, the fact that Opalinus shale is well studied means that the reported literature values can be used with reasonable confidence. Table 3.1: Material Properties of Mont Terri Opalinus Shale From The Literature

Quartz Content 1 Clay Content 1 Carbonate Content 1 Porosity2 , n Water Content2 , wc Unconfined Compressive Strength2⊥ , σc⊥ Unconfined Compressive Strength2‖ , σc‖ Young’s Modulus2⊥ , E50 ⊥ Young’s Modulus2‖ , E50 ‖ Poisson’s Ratio2 , ν12 = ν13 Poisson’s Ratio2 , ν23 Fracture Toughness2 , KIC⊥ Fracture Toughness2 , KIC‖

18 [%] 60-68 [%] 10-19 [%] 13.7 [%] 6.4 [%] 14.9 [MPa] 11.6 [MPa] 1500 [MPa] 3800 [MPa] 0.25 0.35 0.53 [MN/m1.5 ] 0.12 [MN/m1.5 ]

1: Wenk et al., (2008) 2: Bock (2009)

3.1.1

Sample Origin

The Swiss government organization, Swisstopo, which is responsible for the Mt. Terri underground laboratory for radioactive waste disposal made available 3 meters of 11 cm diameter core. The cores were very carefully extracted through a triple tube overcoring techniques (Bossart et al., 2012). They arrived at MIT at the end of July 2012. The Mont 132

Terri Underground Laboratory is located in an extension of an access tunnel1 for a motorway tunnel through Mont Terri (figure 3-1). This underground laboratory was developed to conduct in-situ tests and extract test specimens of Opalinus shale. The location and geology of the boring samples can be seen in figures 3-2 and 3-3.

Figure 3-1: The Mont Terri Underground Laboratory. Shale samples were extracted from the underground laboratory that was built off of an access (safety/security) tunnel of the Mont Terri motorway. (Modified from www.mont-terri.ch)

1 This

was initially a pilot tunnel but it is now a safety/security tunnel.

133

Figure 3-2: The geology of the Mont Terri Underground Laboratory. (Modified from www.mont-terri.ch)

134

135

Figure 3-3: Map of the Mont Terri Underground Laboratory. The location (FE-A niche) and orientation (Azimuth = 242.10o ) of core boring extracted from the Mont Terri Underground Laboratory (Modified from Bossart et al., 2012)

3.1.2

Mineralogy

As stated before, shale mineralogy can vary greatly from location to location. Therefore, samples from the Opalinus shale cores used in this study were sent to Macaulay Scientific Consulting Limited for mineralogical XRD analysis. The results are presented in table 3.2. The proportion of quartz, carbonate and clay content is very similar to that reported by Wenk et al. (2008) (Refer to table 3.1). Table 3.2: Mineralogy of Mont Terri Opalinus Shale Opalinus Shale [%] 13.5 0.5 2.2 1.1 12.8 1.3 0.7 0 0 0.4 0.9 0.9 3.4 44.2 18.1

Quartz K-Feldspar Muscovite Plagioclase Calcite Dolomite Siderite Gypsum Halite Anatase Apatite Pyrite Chlorite (Tri) I+I/S-ML Kaolinite Total Quartz, Feldspar, Mica Total Carbonate Total Clay

3.1.3

17.3 14.8 65.7

Natural Water Content and Saturation Level

The strength of a material with a high clay content, such as Opalinus shale, can be highly dependent on water content and saturation level. To determine in-situ saturation, the specific gravity and water content were determined from some initial core cuttings. 136

Specific Gravity In order to estimate the saturation level from the water content of the core material, specific gravity tests were conducted on the Opalinus shale cores. The specific gravity analysis was conducted following the ASTM Standard D854-58 using calibrated volumetric flasks and ground material. First, cuttings from the cores were mechanically ground by hand into a powder (figure 3-4). The Opalinus shale powder was blended into a slurry and poured into a volumetric flask with distilled water (figure 3-5). The Opalinus shale slurry in the volumetric flask was then deaired. After room temperature equilibrium was reached, the mass of the flask and Opalinus shale was measured. Three measurements were made over a 36 hour period and then the contents of the flask were oven dried (60o C) to determine the exact dry mass of the Opalinus shale added to the flask. Using the mass (MB ) and volume(VB ) of the flask, the mass of the volumetric flask with water and shale(MB+W +S ), the dry mass (MS ) of shale powder and the unit weight of water(γW T ), the specific gravity (GS ) of the Opalinus shale powder can be determined (equation 3.1.1). This procedure was repeated three times. The results of the specific gravity test conducted on Opalinus shale are summarized in table 3.3.

Figure 3-4: Grinding Opalinus shale. Opalinus shale was mechanically ground into a powder for the specific gravity analysis.

137

Figure 3-5: Specific gravity test volumetric flask. Opalinus shale was mixed with distilled water in a volumetric and then deaired to determine the specific gravity of the solid material.

GS =

MS (MB +VB ρW T + MS ) − MB+W +S

Where:

GS : is the specific gravity of the solid material MS : is the dry mass of the solid material in the flask (g) MB : is the mass of the volumetric flask (g) VB : is the volume of the volumetric flask (cm3 ) ρW T : the density of the water (g/cm3 ) MB+W +S : mass of the volumetric flask, water and solid material (g)

Table 3.3: Opalinus Shale Specific Gravity Tests Sample 1 - M4 2 - M6 3 - 250-2 Average

Specific Gravity, GS 2.706 2.709 2.712 2.709

138

(3.1.1)

In-situ Water Content

From the core cuttings, five water content samples were taken (table 3.4). These prismatic water content samples were cut from a slice in the core (figure 3-6). The detailed cutting techniques used on Opalinus shale will be discussed later. These samples were massed, oven dried (60o C) and then re-massed. The difference in mass was assumed to be attributed to water loss alone. The saturation can be estimated (table 3.4) by assuming a void ratio (e) reported in the literature (Pearson et. al, 2003) and using the water content (WC ) and specific gravity (GS ) previously determined (equation 3.1.2). From these measurements it can be assumed that the Opalinus shale specimens were delivered and tested at a high level of saturation (approximately 85%) (Refer to table 3.4).

S[%] =

GSWC × 100 e

Where:

S: is the saturation level (%), S = VVWV GS : is the specific gravity WC : is the water content, WC =

MW MS

e: is the void ratio, e = VVVT and MW : is the mass of water (g) MS : is the mass of solids (g) VW : is the volume of water (cm3 ) VV : is the volume of voids (cm3 ) VT : is the total volume (cm3 ) 139

(3.1.2)

Table 3.4: Water Content of Opalinus Shale Samples Sample S1 S2 S3 M1 M2 Avg *:

Water Content,WC [%] 5.9 6.2 6.0 6.1 5.8 6.0

Saturation, S[%]* 83.8 87.8 85.7 86.6 83.4 85.5

Calculated from specific gravity (2.709) and Opalinus shale void ratio reported by Pearson et al. (2003) of e = 0.19

Figure 3-6: Water content samples. Prismatic samples used to determine water content of Opalinus shale.

3.1.4

Natural Fractures and Shells

Opalinus shale is a natural material with many naturals flaws and heterogeneities that requires very careful specimen preparation (figure 3-7). "Natural fractures" refer to discontinuities, which occur naturally in the rock (Refer to figure 3-7). These natural fractures form weak zones that can propagate and break the specimen very easily with vibration and handling. This can make it very difficult to keep the Opalinus shale specimens intact while cutting. Additionally, there can be fossilized shells and concretions which may impeded cutting and can affect the strength and fracturing behavior of the specimen (Refer to 140

figure 3-7). Fossils were rare and typically specimens with fossils were not used for testing.

Figure 3-7: Natural features found in Opalinus shale. Opalinus shale is a natural material with many natural fractures (left) and heterogeneities such as shell fragments (right).

3.2

Specimen Preparation

In order to prepare the specimens for testing, Opalinus shale cores needed to be cut into prismatic specimens and then flaws needed to be introduced. This section describes a dry cutting technique to cut prismatic specimens with flaws in Opalinus shale, as well as how to seal and preserve these specimens so they do not dry out. One of the challenges of working with shale, specifically shale with a high clay content such as Opalinus shale, is that it can be very sensitive to water. When shale with significant clay content comes in contact with water, bonding between clay particles will break causing the material to lose strength and fall apart. This eliminated the possibility of cutting the Opalinus shale specimens with traditional wet cutting techniques such as wet jetting or lubricated wet saws. Since most rock cutting techniques use water this makes cutting Opalinus shale a challenge. A dry cutting process was developed for Opalinus shale and is detailed below. 141

3.2.1

Cutting Prismatic Specimens

In order to cut Opalinus shale without destroying the internal clay bonds, all cutting techniques needed to be dry. The edges of the specimens were cut with a traditional tabletop band-saw with specialized carbide-tipped abrasive saw blades. These carbide-tipped saw blades are specially made for very hard materials and cut using an abrasive action. Since the blade makes an abrasive cut, over-heating can be a concern when lubrication is not used. However, Opalinus shale is soft enough to cut without using a lubricant and over-heating was not a problem during cutting. The Opalinus shale specimens were prepared from the horizontal bore cores in the Mont Terri Rock Laboratory in Switzerland (Refer to figures 3-1 and 3-2). The bedding planes were approximately 30 degrees to the major axis of the boring (figure 3-8). The specimens were cut to fit within the cross-section of the boring in order to maximize the total number of specimens that could be tested. Also cutting the specimens within the cross-section of the boring allows one to rotate the bedding plane orientation in the specimen. Although the majority of tests presented in this report were loaded perpendicular to the bedding plane (Shown in figure 3-8), several test series were also conducted with alternative bedding orientations.

Figure 3-8: Opalinus shale borings. Horizontal boring samples obtained from the Mont Terri Underground Laboratory. Bedding planes are approximately 30 degrees from the major axis of the boring. Specimens were cut to fit within the cross section of the boring to maximize the number of specimens. 142

In order to cut prismatic specimens from cylindrical core borings, the specimens were first cut into slices (figure 3-9). A stencil was used to define the size and edges of the specimen, as well as align the specimen with the bedding planes, which were visible on the core slice. The first edges of the specimen were then cut by using a 9 inch disc sander with a fine grit sand paper. The other edges of the specimen were cut using the carbide-grit band-saw blade. The edges were aligned using the previously sanded edges and the guide rails of the band-saw. The specimen edges were smoothed out using the disc sander. Finally, the front surface of the specimen was hand polished using very fine grit sand paper (200+ grit).

3.2.2

Cutting Flaws

Dry cutting flaw pairs into prismatic specimens had never been done before. Previously tested materials with flaw pairs have either been molded with removable shims or cut with a water jet. It was determined that the flaws could be cut by drilling a small hole and inserting a thin removable scroll saw blade (figure 3-10). Due to the small size of the specimens, and corresponding flaw sizes, very thin drill bits needed to be used. Drill bits of 0.660 mm (0.026 in - # 71) and 0.787 mm (0.031 in - # 68) were used. These drill bits are typically not made to drill deep enough to go through a specimen that is 25 mm (1 in) thick. Specialized drill bits with extended length were used. Extended length drill bits made it possible to drill through the specimen thickness, however, drill bit stability and breakage was still a concern. Hence, rectangular scroll saw blades with a thickness of 0.381 mm (0.015 in) and a width of 0.813 mm (0.032 in) were inserted into the drilled holes and then used to cut the flaws (Refer to figure 3-10).

143

144

Figure 3-9: The process of cutting prismatic specimens from a cylindrical core boring. (a) Intact core borings were used to cut (b & c) discs of the appropriate thickness were first cut from the cylinder. (d) The bedding plane was identified and a stencil was traced on the specimen. (f) The top edge was sanded down (e) and the side edge was set (g) to align the cutting in the band-saw. (g & h) The edges were cut off and then (i) sanded down to the appropriate size. (j) Finally the front surface of the specimen was sanded and polished with a fine grit sand paper

Figure 3-10: Cutting flaw pairs in prismatic Opalinus shale specimens. (a) Flaws were drawn on the specimen using a laser-cut stencil. (b) At one tip of the flaw a hole was drilled and then (c) a scroll saw blade was inserted into the hole to cut the flaw.

3.2.3

Preservation and Storage

Since the degree of saturation can affect the strength, stiffness and other properties of Opalinus shale, the in-situ water content was preserved for all samples and specimens. Extra care was taken to preserve and store these specimens to reduce saturation loss. All cores and specimens were either vacuum sealed or stored in double zip-lock bags and then placed in a secondary containment cooler (figure 3-11). During and after testing, specimens were sealed in zip-lock bags to prevent further water loss. 145

Figure 3-11: Opalinus shale sealing and storage. Opalinus shale was stored in plastic vacuum seals to preserve insitu water content (left). These sealed samples were placed in a secondary cooler container for more protection (right).

3.2.4

Cutting Pierre Shale

In addition to the procedure developed for Opalinus, a similar cutting and sealing process was developed for Pierre shale (Appendix B). Pierre shale is a high porosity, high clay content shale, which is very sensitive to drying. Therefore, some of the cutting techniques developed for Opalinus shale did not work and additional procedures were developed. Refer to Appendix B for more information on the preparation techniques developed for Pierre shale. Additionally, preliminary tests were conducted on Pierre shale and are reported in Appendix B.

3.3

Experimental Procedure

As previously stated, this study consists of unconfined compression tests conducted on prismatic Opalinus shale specimens with or without pre-cut flaw pairs. The Opalinus shale specimens were cut to fit within the cross-section of the core provided by the Mont Terri 146

Rock Laboratory. Since the core diameter is 11 cm (4.33 in), the largest prismatic specimen size possible (while maintaining the 2:1 h:w ratio) is 10 cm x 5 cm (4 in x 2 in) (figure 3-12). The boring logs which describe where specimens were cut from can be seen in Appendix A. Prismatic specimens previously tested by Bobet (1998), Miller (2008), and Wong (2009) were traditionally 15 cm by 7.5 cm (6 in x 3 in). In order to validate that there is no size effect in the cracking processes, very small gypsum specimens (50mm x 25mm x 10mm) were tested and compared to the traditionally sized gypsum counterparts (Appendix C). The flaw pairs are defined by flaw inclination angles (β ), bridging angles (α), ligament lengths (L) and bedding plane angle (ψ) (Figures 3-12 and 3-13). Flaw pair geometries are always labeled and referred to in the format L-β -α(ψ). For convenience, flaw pair geometries with horizontal bedding plane orientation (ψ=0o ) omit the bedding plane from the label (L-β -α).

Figure 3-12: A shale specimen tested in this study. Flaw Length (2a) = 0.33"

3.3.1

Specimens Tested

The tests conducted on Opalinus shale can be divided into three separate groups; tests on intact specimens, specimens with flaw pairs with bedding planes perpendicular to loading and tests conducted on flaw pairs with varying bedding plane orientations (table 3.5). Tests were conducted on intact Opalinus shale, loaded at different bedding plane angles (ψ) to determine bulk material characteristics (E, σc , ν) with different bedding plane directions(ψ 147

Figure 3-13: Definition of flaw pair geometries. Flaw pair geometries were defined by (a) flaw inclination angle [β ], bridging angle [α]. Bedding planes are defined by the bedding plane angle [ψ]. They are always referred to in the format L-β -α(ψ). For convenience, flaw pair geometries with horizontal bedding plane orientation (ψ=0o ) omit the bedding plane from the label (L-β -α).

148

= 0o ,30o ,45o ,60o ,90o ). Tests conducted on specimens with flaw pairs loaded perpendicular to the bedding plane were divided into coplanar flaw pairs and stepped flaw pairs. Coplanar flaw pairs consisted of flaw pairs with 0 degree bridging angle and flaw angles of 0, 30, 45, 60 and 75 degrees (figure 3-14). Stepped flaw pairs consisted of flaw pairs with a constant flaw angle of 30 degrees and bridging angles of 0, 30, 60 and 90 degrees (figure 3-15). Finally a series of tests were conducted on flaw pairs with a 30 degrees flaw angle and 30 degrees bridging angle at 30, 60 and 90 degrees bedding plane angles (figure 3-16). For each test configuration, at least three repetitions were typically performed to assure consistency between results (Refer to Table 3.5). However, during the preparation of specimens with various bedding plane angles, many of the specimens were weak and broke easily. Due to this fact, along with material constraints, only two tests were conducted on the intact specimens with 30 degrees bedding plane angle (Refer to table 3.5).

Figure 3-14: The co-planar flaw pair geometries tested. Co-planar flaw pairs have a constant bridging angle of 0 degrees with varying flaw angles. The flaw angles tested were 0, 30, 45, 60 and 75 degrees. These tests were conducted on specimens with bedding planes perpendicular to the loading direction (ψ=0o ).

Figure 3-15: The stepped flaw pair geometries tested. The stepped flaw pairs have a constant flaw angle of 30 degrees with varying bridging angles. The bridging angles tested were 30, 60 and 90 degrees. These tests were conducted on specimens with bedding planes perpendicular to the loading direction (ψ=0o ).

149

Table 3.5: List of Tests Conducted Geometry/Test

# of Tests

1. Intact Tests Intact - ψ = 0o Intact - ψ = 30o Intact - ψ = 45o Intact - ψ = 60o Intact - ψ = 90o

4 2 3 3 3

2. Flaw Pairs - Horizontal Bedding Planes (ψ = 0o ) Co-Planar Flaw Pairs (0o Bridging Angle [α]) 2a-0-0 2a-30-0 2a-45-0 2a-60-0 2a-75-0 Stepped Flaw Pairs (30o Flaw Angle [β ]) 2a-30-30 2a-30-60 2a-30-90

3 4 3 3 3 3 3 3

3. Flaw Pairs - Varying Bedding Plane (ψ = 30o , 60o , 90o ) 2a-30-30(30) 2a-30-30(60) 2a-30-30(90)

3 3 3

Total

49

Figure 3-16: The flaw pair geometries tested at various bedding angles. The specimens tested at various bedding angles have a constant flaw pair geometry with a 30 degree flaw angle and a 30 degree bridging angle. The bedding planes tested were 30, 60 and 90 degrees to the horizontal.

150

3.3.2

Testing Setup

The purpose of the experimental test setup in this study is to determine stiffness, strength and observe cracking in Opalinus shale specimens under uniaxial compression and then associate these cracking events with stress-strain-time data. Two different loading frames were used depending on the experiment. A stiffer loading frame was used for intact tests and a loading frame with better data acquisition and trigger synchronization was used for flaw pair tests.

Intact Specimen Test Setup

Specimens of intact Opalinus shale were uniaxially loaded in a 100-KIP ( 440 kN) MTSTM loading frame. Due to the higher stiffness of the MTSTM compared to other loading frames, tests conducted with this load frame yielded more accurate values of Young’s Modulus, therefore, it was preferred for intact specimens because these tests focused primarily on specimen stiffness and strength. A photograph of the intact test setup is shown in figure 3-17 and a schematic of the test setup is shown in Figure 3-18.

In tests conducted on intact specimens, proximity sensors and aluminum targets were used to measure the lateral displacement (Refer to figure 3-18) (See appendix D for a detailed study developed for this research on the use of induction sensors to measure the lateral displacement of rock). Data were acquired at 2 data points per second, and the tests were load controlled at a rate of 1,200 lb/min (5.34 kN/min). Although this data collection rate is too slow to relate cracking processes with stress-strain data, it is suitable for determining strength and stiffness. For all tests, steel brush end platens were used on the specimen boundaries to reduce end effects [See Bobet (1997) for more information on the brush platens]. Dimensions were measured prior to testing to calculate the stress and strain. 151

Figure 3-17: Photograph of the intact Opalinus shale test setup in the current study.

Figure 3-18: Intact Opalinus shale test setup diagram. Data acquisition can be divided into high speed data readings (left) and traditional data readings (right). These data are synchronized by a triggering system which detects when a break in the specimen occurs

152

Flaw Pair Specimen Test Setup Specimens with flaw pairs were uniaxially loaded in a 60-KIP ( 270 kN) BaldwinTM hydraulic loading frame. The setup with the BaldwinTM loading frame can automatically trigger high-speed video, which is necessary to detect cracking processes such as those observed in specimens with flaw pairs. A photograph of the flaw pair test setup is shown in Figure 3-19 and a schematic of the test setup is shown in Figure 3-20. All flaw pair tests were load controlled at a constant load rate of 1,200 lb/min ( 5.3 kN/min). Load, axial displacement and time data were recorded at a rate of 50 data points per second from the transducers by ADMET’s MTESTW software (Refer to figures 3-19 and 3-20). Dimensions were measured prior to testing to calculate the stress and strain.

Figure 3-19: Photograph of the test setup used for specimens with flaw pairs in the current study.

Testing Imagery Imagery of the cracking mechanisms was recorded during the test, using both high speed video and periodic still imagery. High speed video was captured using a PhotronTM SA-5 153

Figure 3-20: Flaw pair test setup diagram. Data acquisition can be divided into high speed data readings (left) and traditional data readings (right). These data are synchronized by a triggering system which detects when a break in the specimen occurs

high speed camera with a TamronTM 90mm lens (figure 3-21). The high speed camera was controlled by a separate laptop computer. The high speed camera can vary the frame rate, record duration and resolution. There is a finite amount of memory, so there is a trade off between frame rate, record duration and resolution. For these tests, the high speed camera captured between 5,000 and 10,000 frames per second, for 4 to 8 seconds, at 512x512 pixel resolution. The test was also videotaped in real time with a Sony HandycamTM (DCRHC65) Camcorder. In addition to high speed imagery and real time video, still images were taken periodically during the test to capture key events (crack initiation, spalling, breaking etc.), which occurred before high speed footage capturing coalescence occurred. Still images were taken with either the high speed camera or a NikonTM D90 high resolution camera with a 105mm lens (figure 3-22). Two Dolan-Jenner FiberliteTM MI-150 fiber optic lighting units were used to evenly light the surface of the specimen during testing (figure 3-23). During the course of this research the technique of taking still imagery was developed and 154

improved from no still imagery initially to using the snapshot ability of the high speed camera to using snapshots from a high resolution camera. Since some tests used different techniques which may have different sensitivity, such as determining crack initiation from the lower resolution still images of the high speed camera versus the high resolution still images from the NikonTM D90, it is important to differentiate these procedures used for different specimens (table 3.6).

Figure 3-21: PhotronTM SA-5 high speed camera. This camera was used to capture high speed cracking events. It captured between 5,000 and 10,000 frames per second, for 4 to 8 seconds, at 512x512 pixel resolution.

Data Synchronization

The high speed camera was electronically connected to a triggering system which initiated when a break was detected (e.g. a significant drop in load). This triggering system was also used to synchronize the load-displacement-time data from the loading frame and proximity sensors to the high speed imagery. In some tests where there was not a significant drop in load observed after coalescence, the test was manually stopped in order to trigger and capture the high speed video of coalescence occurring. For a more detailed explanation of the triggering synchronization techniques used in this study see Appendix E. The still images could be synchronized with the load-displacement-time data by matching still images 155

Figure 3-22: NikonTM D90 high resolution camera. A NikonTM D90 high resolution camera with a 105mm lens was used to take periodic still images throughout testing.

Figure 3-23: Dolan-Jenner FiberliteTM MI-150 fiber optic lighting. Two fiber optic lightening units were used, each with dual branch fibers to redirect the light onto the specimen surface.

156

Table 3.6: Summary Of Techniques Used Based on Flaw Pair Geometry Geometry 2a-0-0 2a-30-0 2a-45-0 2a-60-0 2a-75-0 2a-30-30 2a-30-60 2a-30-90 2a-30-30(30) 2a-30-30(60) 2a-30-30(90)

High Speed Video x x x x x x x x x x

Low Res Stills x x

High Res Stills

x x x x x x x x x

Res = Resolution (taken 1-2 per second) to a frame observed with the high speed imagery (typically over 2 to 3 seconds of total duration), which was previously synchronized with the stress-strain data.

3.3.3

Analysis Procedure

The analysis procedure for determining and describing the cracking in Opalinus shale is a multi-stage process, which combines visual imagery results with load-displacement-time data (figure 3-24). First, the load-displacement data recorded by the data acquisition system were used along with the specimen dimensions to calculate the stress-strain results (figure 3-25). Then the specific images which show key events (crack initiation, coalescence, etc.) were determined for each test. The cracking was traced in each image to create a sketch using an image editing program called Paint.NetTM (Refer to figure 3-24). After all of the cracking events were traced for the entire test, for each crack the exact image (still or high speed) at which initiation occurred was determined. It should be noted that the exact frame of a crack’s initiation often differed slightly from the traced frame due to the fact that computer aided zooming is necessary to determine the first frame at which a crack initiated. 157

High speed video was synchronized using the trigger point. In the case of intact tests, the high speed video was manually triggered and the trigger point was recorded by the data acquisition system. In the flaw pair tests, the high speed video was automatically triggered when a large drop in load associated with a break was detected (a drop below 70% peak load) or the test was stopped (i.e. the last recorded data point). The high resolution imagery was synchronized by using the best match between the high resolution image closest to failure and its corresponding frame in the high speed video. If a high resolution image was not taken during high speed video recording, then the end of high speed video was matched to the closest high resolution image. By synchronizing the stress-strain data with the images, the stress-stain of crack initiation could be determined (Refer to figure 3-25). The order of cracking was determined and labeled alphabetically (A, B, etc.) by using the time at which each crack occurred. Then the mode (shear or tensile) and type [defined by Wong and Einstein (2009a)] of cracking was determined and explained verbally in each analysis (Refer to figure 3-24). The analysis process can be summarized as follows: 1) Calculate the stress-strain-time from the load-displacement-time data 2) Determine the key image frames (crack initiation, -propagation, -coalescence, etc.) 3) Trace the cracking processes in each frame 4) Determine the exact frame when each crack initiated 5) Determine the high-speed and high resolution imagery synchronization points with data 6) Order the cracks (A, B, etc.) based on image number and associated time 7) Determine the crack mode (tensile shear) and type (Refer to figure 2-52) 8) Collate and present the cracking processes with stress-strain data All completed test analyses are shown in Appendices G and H.

158

159

Figure 3-24: Example analysis for Opalinus shale specimen with flaw pairs. This example analysis shows one photo and sketch frame for a test conducted on the flaw pair geometry 2a-75-0. Crack and coalescence types based on Wong and Einstein (2009)

Figure 3-25: Example stress-strain graph for Opalinus shale specimen with flaw pairs. This example graph is for a test conducted on the flaw pair geometry 2a-75-0. It shows the crack initiation, coalescence and peak stress. It also contains a point for each new crack initiation and each sketch frame in the analysis. Digital Image Correlation - DIC Additionally, some preliminary image analysis using computational Digital Image Correlation (DIC) was used. DIC uses the images captured during the test to recognize deformations on the surface of the specimen and determine localized strain fields throughout the test. This technique is still relatively new to the field of rock mechanics and it could be a valuable bridge between localized behavior and model parameters and predictions. A detailed explanation of this process, as well as its effectiveness on tests conducted with rock, will be presented in Chapter 6.

160

Chapter 4 Experimental Results A total of forty-nine tests were conducted on Opalinus shale (Refer to table 3.5). These Opalinus shale tests were subdivided into tests conducted on intact specimens, double flaw pair specimens (horizontal bedding planes, ψ=0o ) and 2a-30-30(ψ) flaw pair specimens tested at various bedding plane orientations. Additionally, the saturation level of the test specimens will be presented along with a study conducted on the effect of resaturating Opalinus shale.

4.1

Intact Opalinus Shale

Unconfined compression tests on intact Opalinus shale were conducted at five different bedding plane directions; 0o , 30o , 45o , 60o and 90o (Refer to figure 3-16 and table 3.5). The cracking behavior was captured using high speed and high resolution imagery. For each test, load and displacement were measured to determine the elastic modulus (E) (at approximately 50% peak stress) and unconfined compressive strength (σc ). Additionally, the lateral displacements were measured using proximity sensors to determine the Poisson’s ratio (ν) (figure 4-1). The following sections will discuss the mechanical and cracking behavior of intact Opalinus shale for each bedding plane orientation tested. The lateral 161

strain plots for all of the tests conducted on intact Opalinus shale are presented in Appendix G.

Figure 4-1: An unconfined compression test on intact Opalinus shale. The specimens were loaded at various bedding plane orientations. Induction proximity sensors and aluminum targets were used to measure lateral displacements.

4.1.1

0o Bedding Planes

The cracking processes observed in the specimen can be seen in figure 4-2. Failure of intact Opalinus with 0o bedding planes consisted of tensile cracking which propagated across the bedding planes (Refer to figure 4-2). Therefore, it can be assumed that there was little to no effect of bedding planes on the crack propagation which led to failure. 162

Figure 4-2: Crack progression of intact Opalinus shale with 0o bedding planes. Failure of intact Opalinus with 0o bedding planes consisted of tensile cracking which propagated across the bedding planes. The stress-stain behavior for the tests conducted on intact Opalinus shale with 0o bedding planes is shown in figure 4-3. A summary of the mechanical properties (E, σc , ν) is presented in table 4.1. A hardening effect (increase in stiffness) was observed due to crack closure during the beginning of the test (Refer to figure 4-3). Then a region of linear elastic response was observed. In some tests, softening or weakening occurred before failure. At failure, the stress-strain response showed a steep drop off, resulting in a brittle failure response. A brittle failure response has been observed in unconfined compression tests on shale before, such as the tests conducted on Opalinus shale by Amann et al. (2011) (Refer to figure 2-23). Table 4.1: Intact Opalinus Shale Tests - 0o Bedding Planes Om-I(0) -A Peak Stress, σc(0) [MPa] 22.4 Modulus, E50(0) [MPa] 1090 Poisson’s Ratio, ν12(0) 0.279

Om-I(0) -B Om-I(0) -C 20.8 17.7 1095 1070 0.256 0.299

163

Om-I(0) -D 16.8 1050 0.255

Average 19.4 1080 0.272

Figure 4-3: Stress-strain data for unconfined compression tests loaded with bedding planes inclined at 0 degrees (ψ=0o ).

4.1.2

30o Bedding Planes

The cracking processes observed in the 30o intact specimens can be seen in figure 4-4. Failure consisted of a combination of tensile cracking which propagated across the bedding planes as well as shear cracking which propagated along bedding planes simultaneously (i.e. order could not be distinguished) (Refer to figure 4-4). This was unique because many anisotropic failure theories, such as the single plane of weakness theory by Jaeger (1960), assume that failure occurs either through the material (across bedding planes) or along a sliding surface (shear along bedding). However, both of these failure mechanisms occurred in the intact Opalinus shale with 30o bedding planes. It should be noted that one of the specimens [OM-I(30)-B] broke before testing could occur and due to the lack of additional material only two tests were conducted at 30o . Since the behavior of the two tests conducted was relatively similar, it appeared that two test repetitions were sufficient. The stress-stain behavior for the tests conducted on intact Opalinus shale with 30o bedding planes is shown in figure 4-5. A summary of the mechanical properties (E, σc , ν) is presented in table 4.2. A single value of the Poisson’s ratio could not be calculated due to the non-linear lateral strain observed at low stresses (Refer to appendix G). Tests on intact 164

Figure 4-4: Crack progression of intact Opalinus shale with 30o bedding planes. Failure of intact Opalinus with 30o bedding planes consisted of tensile cracking, which propagated across the bedding planes, as well as shear cracking along the bedding planes. Opalinus shale with 30o bedding planes showed a much smaller range of crack closure (strain hardening) compared to the tests conducted on 0o bedding planes, which lead to a linear elastic response (Refer to figure 4-5). There was also much less softening before the brittle failure occurred compared to the specimens with 0o bedding planes. Table 4.2: Intact Opalinus Shale Tests - 30o Bedding Planes Om-I(30) -A Peak Stress, σc(30) [MPa] 8.00 Modulus, E50(30) [MPa] 1132 Poisson’s Ratio, ν12(30) N/A

4.1.3

Om-I(30) -C 9.88 1140 N/A

Average 8.94 1136 N/A

45o Bedding Planes

The cracking processes observed in the specimen can be seen in figure 4-6. Failure of intact Opalinus shale with 45o bedding planes consisted mostly of shear cracking which propagated along bedding planes (Refer to figure 4-6). The significance of this finding 165

Figure 4-5: Stress-strain data for unconfined compression tests loaded with bedding planes inclined at 30 degrees (ψ=30o ) meant that at 45o bedding planes, Opalinus shale failed along a sliding surface and not through the bulk material. Failure along a weak sliding plane is a fundamental assumption in the single plane of weakness theory (Jaeger, 1960), which appeared to occur in these specimens. The stress-stain behavior for the tests conducted on intact Opalinus shale with 45o bedding planes is shown in figure 4-7. A summary of the mechanical properties (E, σc , ν) is presented in table 4.3. The Poisson’s ratio was again not calculated due the non-linear lateral strain observed at low stresses. Crack closure followed by a linear elastic response was observed in the 45o tests as well (Refer to figure 4-7). However, softening before failure was not observed. Since the failure mechanism consisted of sliding along bedding planes, a large displacement was observed when failure occurred. Table 4.3: Intact Opalinus Shale Tests - 45o Bedding Planes Om-I(45) -A Peak Stress, σc(45) [MPa] 2.43 Modulus, E50(45) [MPa] 1200 Poisson’s Ratio, ν12(45) N/A

166

Om-I(45) -B Om-I(45) -C 4.27 2.83 1110 820 N/A N/A

Average 3.18 1043 N/A

Figure 4-6: Crack progression of intact Opalinus shale with 45o bedding planes. Failure of intact Opalinus with 45o bedding planes consisted of shear cracking along the bedding planes. Secondary tensile cracking occurred after initial failure (bottom left).

Figure 4-7: Stress-strain data for unconfined compression tests loaded with bedding planes inclined at 45 degrees (ψ=45o )

167

4.1.4

60o Bedding Planes

The cracking processes observed in the specimen can be seen in figure 4-8. Similar to the fracturing processes of the 45o specimens, the failure of intact Opalinus with 60o bedding planes consisted of shear cracking which propagated along bedding planes (Refer to figure 4-8).

Figure 4-8: Crack progression of intact Opalinus shale with 60o bedding planes. Failure of intact Opalinus with 60o bedding planes consisted of shear cracking along the bedding planes. The stress-stain behavior for the tests conducted on intact Opalinus shale with 60o bedding planes is shown in figure 4-9. A summary of the mechanical properties (E, σc , ν) is presented in table 4.4. For similar reasons as those in the tests conducted at 30o and 45o , the Poisson’s ratio was not calculated. Crack closure followed by linear elasticity was observed in intact Opalinus shale with 60o bedding planes (Refer to figure 4-9). Large displacements were observed at failure due to the sliding failure mechanism, which occurred along the bedding planes. It should be noted that the test conducted on OM-I-(60)-B had an incorrectly low data acquisition rate making the stress-strain curve unusable. Therefore, only the peak strength was recorded for this test. 168

Figure 4-9: Stress-strain data for unconfined compression tests loaded with bedding planes inclined at 60 degrees (ψ=60o )

Table 4.4: Intact Opalinus Shale Tests - 60o Bedding Planes Om-I(60) -A Peak Stress, σc(60) [MPa] 1.61 Modulus, E50(60) [MPa] 1180 Poisson’s Ratio, ν12(60) N/A

169

Om-I(60) -B Om-I(60) -C 0.81 2.34 N/A 1120 N/A N/A

Average 1.59 1150 N/A

4.1.5

90o Bedding Planes

The cracking processes observed in the specimen can be seen in figure 4-10. The failure of intact Opalinus with 90o bedding planes consisted of tensile cracking which propagation along bedding planes (Refer to figure 4-10). This was the only test series which showed tensile opening along bedding planes. As the test progressed, these tensile cracking fully propagated and then coalesced together.

Figure 4-10: Crack progression of intact Opalinus shale with 90o bedding planes. Failure of intact Opalinus with 90o bedding planes consisted of shear cracking along the bedding planes. The stress-stain behavior for the tests conducted on intact Opalinus shale with 90o bedding planes is shown in figure 4-11. A summary of the mechanical properties (E, σc , ν) is presented in table 4.5. Crack closure followed by linear elastic behavior was observed in tests conducted at 90o as well (Refer to figure 4-11). The failure processes of the 90o specimens was progressive, including tensile opening of the bedding planes and then coalescence of these cracks. This can be observed in figure 4-11 by the slight drops in load associated with tensile cracking along the bedding and then a sharp drop in load associated coalescence of tensile cracks. 170

Figure 4-11: Stress-strain data for unconfined compression tests loaded with bedding planes inclined at 90 degrees (ψ=90o )

Table 4.5: Intact Opalinus Shale Tests - 90o Bedding Planes Om-I(90) -A Peak Stress, σc(90) [MPa] 13.8 Modulus, E50(90) [MPa] 4194 Poisson’s Ratio, ν12(90) 0.211

171

Om-I(90) -B Om-I(90) -C 15.2 14.2 4270 5147 0.271 0.294

Average 14.4 4537 0.255

4.1.6

Summary of Intact Tests

The average values of the Young’s modulus and unconfined compressive strength for each bedding plane inclination presented earlier are summarized in table 4.6 and figure 4-12. There is a decrease in the unconfined compressive strength at inclined bedding angles (ψ=30o ,45o ,60o ). This u-shape trend in unconfined compressive strength vs. bedding plane angle, with a minimum unconfined compressive strength at 60o bedding plane direction, has been shown for other anisotropic materials (Refer to figure 4-12). Additionally, the difference in strength at 0o and 90o highlights the intrinsic material anisotropy that many anisotropic failure models incorporate (Jaeger, 1960; McLamore and Grey, 1967; Ramamurthy et al., 1993; Pei, 2008). Table 4.6: Intact Opalinus Shale Tests - Average Value Summary Bedding Angle, ψ [o ] 0 30 45 60 90

Young’s Modulus, E [MPa] 1080 1136 1043 1150 4537

Compressive Strength, σc [MPa] 19.4 8.94 3.18 1.59 14.4

Figure 4-12: Summary of Opalinus shale stiffness (E) and compressive strength (σc at different bedding plane angles determined from unconfined compression tests. 172

The results of the unconfined compressive strength determined in this study (σc0 =19.9 MPa, σc90 =14.4 MPa) are slightly higher than the values reported by Bock (2009) (σc0 =14.9 MPa, σc90 =11.6 MPa) (Refer to Table 3.1). The results of the elastic modulus (E0 =1080 MPa, E90 =4537 MPa) are comparable to the previously reported values (E0 =1500 MPa, E90 =3800 MPa) (Bock, 2009). Since Opalinus shale is a layered system, consisting of soft (weak) layers and stiff (strong) layers, it is expected that the response at low bedding plane inclinations (0o ) should be softer yet stronger. This is because the deformation is dominated by soft layers but failure of the strong layers dominates the strength. In contrast, at high bedding plane inclinations (90o ) the stiff layers dominate the deformation and failure along the weak layers can occur at lower stresses. This trend was observed in the 0o and 90o specimens. However, the Young’s modulus did not show much variation at inclined bedding planes (30o , 45o , 60o ). Other research observed, or predicted, an increase in modulus with bedding plane angle, such as the behavior predicted in the Y-Geo model by Lisjak et al. (2013) (Refer to figure 285).

Stress-Strain Behavior

As shown before, the stress-strain behavior of the tests conducted on Opalinus shale differed depending on the bedding plane orientation. All bedding plane orientations showed some kind of crack closure (stiffness increase) at the beginning of the test and then some linear behavior (Refer to figures 4-3, 4-5, 4-7, 4-9, 4-11). However, the failure response varied depending on the failure mechanism observed for that bedding plane orientation. This can be summarized as follows: Tests conducted on 0o bedding planes showed more softening near failure, which occurred as a sharp drop in load associated with cracking across bedding planes and through the intact material. Similarly, specimens at 30o showed similar softening and a jump in axial strain associated with cracking through intact material and sliding along bedding planes. The tests on 45o and 60o bedding plane angles showed relatively no softening or pre-failure 173

drop in load and then a large axial strain jump associated with failure as sliding along bedding planes occurred. Finally, specimens with 90o bedding plane showed a progressive failure, which included tensile cracking along bedding planes and then coalescence of these tensile cracks.

Comparison Of Intact Results With Anisotropic Strength Theories

Several different anisotropic failure criteria discussed in section 2.1.4 can be applied to the results of the unconfined compressive strength of intact Opalinus shale. Three methods will be applied and then discussed. Due to the limited amount of data collected, such as the limited number of bedding plane angles and confining pressures tested, only simple anisotropic strength models will be applied. The first theory used is the single plane of weakness model by Jaeger (1960). Then the variable cohesive model proposed by Jaeger (1960) is applied. Finally, the empirical strength model developed by Ramamurthy et al. (1993) is discussed. These theories were explained in detail, including their formulas, in section 2.1.4. Single Plane of Weakness - Jaeger, 1960 The single plane of weakness theory (Jaeger, 1960) was applied to results from intact tests on Opalinus shale. This theory is a discontinuous strength theory with material properties (φ and c) for intact failure and for sliding along a predefined bedding plane (φw and cw ) . Additionally, different material properties for σ1−90 (φ90 , c90 ) and σ1−0 (φ0 , c0 ) were used. Due to the fact that only one confining stress (no confinement, σ3 =0) was tested, either a friction angle (φ ) or cohesion (c) needed to be assumed to determine the intact material parameters measured at the 0o and 90o bedding plane angles. A friction angle for the intact material, for both 0o and 90o , was assumed to be 22o based on the tests reported by Bock (2009). The cohesion terms for the intact material failure were then determined (c0 and c90 ). The φw and cw of the sliding plane were determined by using the strengths results at inclined bedding plane angles (ψ=30o ,45o ,60o ) with curve fitting functions with MATLABTM . The results of the single plane of weakness fitting are presented in figure 4174

13. A summary of the material parameters used are shown in table 4.7. The minimum strength was predicted at a bedding angle of about 57.8o (Refer to figure 413). The zone of discontinuity sliding (u-curve portion) extends from the bedding plane angles of approximately 27.6o to 87.2o . This theory is thus capable of predicting the strength observed for most bedding plane angles. The model does slightly underpredicts the strength at 45o and slightly over predict the strength at 60o .

Figure 4-13: Single plane of weakness theory (Jaeger, 1960) applied to intact Opalinus shale results. This theory assumes the rock fails either through intact material (constant strength) or sliding on a predefined weak plane (u-shaped curve). Different strength values were used at 0o and 90o . Variable Cohesive Theory - Jaeger, 1960 The variable cohesive theory proposed by Jaeger (1960) was also applied to the intact strength data. This strength theory assumes that the friction angle of the material is constant and the cohesion is a function of the bedding angle. Again, a friction angle of 22o was assumed. The exponential term (n) was assumed to be 1. The parameters for the cohesion term were determined from curve fitting with MATLABTM . The results are presented in figure 4-14. A summary of the variable cohesion term parameters used are shown in table 4.7. The minimum strength was predicted at a bedding angle of about 52.2o (Refer to table 4.7). This is slightly lower than the experimentally observed minimum around 60o . The model 175

Table 4.7: Anisotropic Strength Theory Parameters Single Plane of Weakness (Jaeger, 1960) φw = 25.5o cw = 0.68 MPa φ0 = 22.0o * c0 = 6.55 MPa φ90 = 22.0o * c90 = 4.85 MPa Variable Cohesion (Jaeger, 1960) A= 5.71 MPa B= 4.72 MPa ψmin = 52.2o φ= 22.0o * n= 1.0* Variable Strength (Ramamurthy et al., 1993) A1 = 13.8 MPa D1 = 11.8 MPa A2 = 26.8 MPa D2 = 24.8 MPa n= 1.0* *:Assumed value predicts the compressive strengths at 0o , 45o and 90o ; however, the model underpredicts the strength at 30o and overpredicts the strength at 60o . It appears that this model is not capable of closely predicting the strengths at all of the bedding plane orientations tested. Empirical Strength Variation - Ramamurthy et al., 1993 Finally, the anisotropic strength theory proposed by Ramamurthy et al. (1993) was fit to the intact Opalinus shale test results. Again the exponential constant (n) was assumed to be 1 and the four material parameters (A1 , A2 , D1 , D2 ) were determined from curve fitting with MATLABTM . This theory typically assumes a minimum strength at 60o , which applies well for this data set. The results are presented in figure 4-15 and the parameters are shown in table 4.7. The theory slightly underpredicts the strength at 30o and slightly overpredicts the strength at 45o . 176

Figure 4-14: Variable cohesion theory (Jaeger, 1960) applied to intact Opalinus shale results. This theory assumes that the cohesion term varies with bedding plane orientation, while the friction angle is constant.

Figure 4-15: Variable strength theory (Ramamurthy et al., 1993) applied to intact Opalinus shale results. This theory assumes a purely empirical relation between the strength and bedding plane orientation.

177

Summary From the applications of these three strength theories it appears that the Single Plane of Weakness theory and variable strength theory proposed by Ramamurthy et al. (1993) are capable of accurately predicting the strengths observed in intact test on Opalinus shale. Ideally, more tests conducted at a larger number of bedding plane angles and different confining pressures (σ3 ) would produce more accurate predictions and interpretations. Especially, the shape of the strength-bedding plane behavior could be better compared to the model predictions if additional tests were conducted at more bedding plane angles. For example, is the actual intact strength more accurately predicted by the sliding plane theory with a u-shape curvature (Refer to figure 4-13) or a more gradual decrease in strength such as the theory proposed by Ramamurthy et al. (1993)? Other Anisotropic Strength Models Although only three strength theories were applied, it should be noted that there are many other anisotropic strength theories that may predict the strength behavior observed in intact Opalinus shale even better. However, many of these theories, such as McLamore and Gray (1967) and Pei (2008), require many different variables be fit to the data. Since there was a limited number of intact experimental tests conducted on Opalinus shale, too many assumptions would need to be made to fit a unique solution of these models to the data. For this reason, these models were not fit to the data presented in this study.

4.2

Flaw Pairs - Horizontal Bedding Planes (ψ = 0o)

As previously stated, 25 tests were conducted on Opalinus shale with flaw pairs loaded perpendicular to the bedding planes (ψ=0o ). The goal of these tests was to systematically understand the crack initiation, -propagation and -coalescence behavior by using high speed and high resolution imagery techniques. From these tests, the progression of cracking events could be described. The following sections will present a summary of the typical 178

cracking behavior observed for each geometry tested, i.e. photos and sketches of the typical crack process are shown (Full details of the cracking process are shown in Appendix H). Distinct cracking characteristics such as the initial crack type, en-echelon cracks, and potential white patching observed in Opalinus shale will then be presented. The coalescence crack type as well as the crack initiation and coalescence stress for all flaw pair geometries with horizontal bedding planes will be shown. For the sake of simplicity, all flaw pair geometries with horizontal bedding planes have the bedding plane label (ψ) in the flaw pair label [L-β -α(ψ)] omitted. Tests on flaw pair geometries with inclined bedding planes (ψ>0o ) will be shown in section 4.3. It should be noted that in this study tensile cracks are characterized by an opening of the crack faces and shear cracks are characterized by a relative sliding along the crack faces. Combined tensile-shear cracking refers to cracks which initiate as one crack type, tensile or shear, and then propagate as the other crack type.

4.2.1

Coplanar Flaws

This section includes results from tests conducted on Opalinus shale with co-planar flaw pairs (0 degree bridging angle) and horizontal bedding planes. The crack and coalescence types defined by Wong and Einstein (2009a) and Wong and Einstein (2009b), respectively, will be used extensively in this section (Refer to figures 2-60 and 2-52).

2a-0-0 An example test of the 2a-0-0 flaw pair geometry is shown in figure 4-16. Cracking typically initiated from the flaw tips, as well as the top and bottom faces of the flaws (See Figure 4-16). Cracking also initiated from the top and bottom edges of the specimen and propagated down toward the flaw pairs. As cracks propagated, cracking at the flaw tips connected with cracks extending from the specimen boundary (indirect coalescence). Indirect coalescence was observed in two out of the three tests with this geometry. The third 179

test showed similar initial cracking behavior; however, there was no coalescence between flaws.

Figure 4-16: Crack progression of Opalinus with a 2-0-0 flaw pair. Shown is the typical crack progression of an unconfined compression test on Opalinus shale with a flaw pair geometry of 2a-0-0. Cracks are labeled with letters (chronologically whenever possible), the type of crack (T = tensile or S = shear) and the crack type as defined by Wong and Einstein (2009b) (Refer to figure 2-52).

Significant closure of flaws was observed in tests conducted on 2a-0-0 (Refer to figure 416). This flaw closure observed in specimens with the 2a-0-0 flaw pair geometry only occurred significantly in the 2a-0-0 and 2a-30-0 geometries. Although initial cracks typically initiated from the flaws, the final crack leading to failure typically did not propagate through a flaw [cracks G(T) and H(T) in figure 4-16]. This final crack, which typically was observed through the center of the specimen at failure, appears very similar to the failure crack observed in intact specimens with horizontal bedding plane (Refer to figure 4-2). Although this might suggest that failure of specimens with the 2a-0-0 geometry was dominated by the material failure, the coalescence stress (coinciding with failure) of 2a-0-0 (10.1 MPa) was much lower than the material compressive strength at 0o bedding angles (19.4 180

MPa). Crack initiation and coalescence stresses will be discussed later in section 4.2.5.

2a-30-0 An example test of the 2a-30-0 flaw pair geometry is shown in figure 4-17. Initial cracking included both wing (type I) and anti-wing (type III) tensile cracks at the flaw tips. As these crack propagated, shear cracks initiated along the interior tensile anti-wing cracks and connected to the other anti-wing crack (indirect coalescence). Indirect coalescence was observed in all three tests with this geometry.

Figure 4-17: Crack progression of Opalinus shale with a 2-30-0 flaw pair. Shown is the typical crack progression of an unconfined compression test on Opalinus shale with a flaw pair geometry of 2a-30-0. Cracks are labeled with letters (chronologically whenever possible), the type of crack (T = tensile or S = shear) and the crack type as defined by Wong and Einstein (2009b) (Refer to figure 2-52). Crack closure was also observed in tests conducted on 2a-30-0 (Refer to figure 4-17). An unclassified crack type, which initiated at the center of a flaw surface and then propagated back around to the same flaw surfaced, was observed in some flaw pair geometries [See cracks E(T) and G(T) in figure 4-17]. This crack type was not observed in gypsum or 181

marble and; therefore, was not a crack type classified by Wong and Einstein (2009a).

2a-45-0 An example test of the 2a-45-0 flaw pair geometry is shown in figure 4-18. Cracks typically initiated from the flaw tips as tensile wing cracks (type I). As cracks propagated further, cracking initiated from the inner flaw tips as either a tensile or shear crack, and connected between the flaws (direct combined tensile-shear coalescence). Direct combined tensileshear coalescence was observed in two of the three tests with this geometry. The third test showed a single shear crack between the two flaws.

Figure 4-18: Crack progression of Opalinus shale with a 2-45-0 flaw pair. Shown is the typical crack progression of an unconfined compression test on Opalinus shale with a flaw pair geometry of 2a-45-0. Cracks are labeled with letters (chronologically whenever possible), the type of crack (T = tensile or S = shear) and the crack type as defined by Wong and Einstein (2009b) (Refer to figure 2-52).

2a-60-0 An example test of the 2a-60-0 flaw pair geometry is shown in figure 4-19. Cracking typ182

ically initiated as tensile wing cracks (type I) at the outer flaw tips. Initial cracking also included horsetail tensile cracks (type II) or type I shear cracks at the inner flaw tips. As cracking propagated further, the cracks at the inner flaw tips connected together (direct combined tensile-shear coalescence). Direct combined tensile-shear coalescence was observed in all three test with this geometry.

Figure 4-19: Crack progression of Opalinus shale with a 2-60-0 flaw pair. Shown is the typical crack progression of an unconfined compression test on Opalinus shale with a flaw pair geometry of 2a-60-0. Cracks are labeled with letters (chronologically whenever possible), the type of crack (T = tensile or S = shear) and the crack type as defined by Wong and Einstein (2009b) (Refer to figure 2-52).

2a-75-0 An example test of the 2a-75-0 flaw pair geometry is shown in figure 4-20. Cracking typically initiated at the flaw tips as either tensile wing crack (type I) or tensile horsetail cracks (type II). As cracking progressed, tensile cracks which initiated at the inner tips of the flaw propagated and connected between the flaws (direct tensile coalescence). Direct tensile coalescence was observed in all three tests with this geometry. 183

Figure 4-20: Crack progression of Opalinus shale with a 2-75-0 flaw pair. Shown is the typical crack progression of an unconfined compression test on Opalinus shale with a flaw pair geometry of 2a-75-0. Cracks are labeled with letters (chronologically whenever possible), the type of crack (T = tensile or S = shear) and the crack type as defined by Wong and Einstein (2009b) (Refer to figure 2-52).

184

Coplanar Flaw Pair Cracking Summary

Cracking initiated from the flaw tips in all coplanar flaw pair geometries. Initial cracking was typically tensile wing cracking. Shear cracking was observed in 2a-30-0, 2a-45-0 and 2a-60-0 flaw pair geometries. As the flaw angle increased, coalescence trended from indirect to direct combined shear-tensile to direct tensile coalescence.

4.2.2

Stepped Flaws

The following section includes results from tests conducted on Opalinus shale with stepped flaw pairs, fixed 30 degree flaw angle at various bridging angles, and horizontal bedding planes.

2a-30-30 An example test of the 2a-30-30 flaw pair geometry is shown in figure 4-21. Cracking typically initiated at the flaw tips as distinct tensile wing cracks (type I). As cracking progressed, shear cracks initiated at inner flaw tips and then extended as a single tensile crack to connect the inner flaws (direct combined tensile-shear coalescence). Direct combined tensile-shear coalescence was observed in all three tests with this geometry.

2a-30-60 An example test of the 2a-30-60 flaw pair geometry is shown in figure 4-22. Cracking typically initiated at the flaw tips as tensile wing cracks (type I). As cracking progressed, the tensile wing cracks at the inner flaw tips propagated and connected between the flaws (direct tensile coalescence). Significant tensile en-echelon cracking can be observed in the coalescence region between the cracks (Refer to figure 4-22). Direct tensile coalescence was observed in all three tests with this geometry. 185

Figure 4-21: Crack progression of Opalinus shale with a 2-30-30 flaw pair. Shown is the typical crack progression of an unconfined compression test on Opalinus shale with a flaw pair geometry of 2a-30-30. Cracks are labeled with letters (chronologically whenever possible), the type of crack (T = tensile or S = shear) and the crack type as defined by Wong and Einstein (2009b) (Refer to figure 2-52).

186

Figure 4-22: Crack progression of Opalinus shale with a 2-30-60 flaw pair. Shown is the typical crack progression of an unconfined compression test on Opalinus shale with a flaw pair geometry of 2a-30-60. Cracks are labeled with letters (chronologically whenever possible), the type of crack (T = tensile or S = shear) and the crack type as defined by Wong and Einstein (2009b) (Refer to figure 2-52).

187

2a-30-90 An example test of the 2a-30-90 flaw pair geometry is shown in figure 4-23. Cracking typically initiated at the flaw tips as tensile wing cracks (type I). Secondary cracking then initiated as type II horsetail tensile cracks or type III tensile anti-wing cracks. As cracking progressed further, tensile cracks propagated from one flaw tip to the other flaw tip (direct tensile coalescence). Direct tensile coalescence was observed in all three tests with this geometry.

Figure 4-23: Crack progression of Opalinus shale with a 2-30-90 flaw pair. Shown is the typical crack progression of an unconfined compression test on Opalinus shale with a flaw pair geometry of 2a-30-90. Cracks are labeled with letters (chronologically whenever possible), the type of crack (T = tensile or S = shear) and the crack type as defined by Wong and Einstein (2009b) (Refer to figure 2-52).

Stepped Flaw Pair Cracking Summary

Cracking initiated from the flaw tips in all stepped flaw pair geometries. Initial cracking was typically tensile wing cracking. Shear cracking was observed in the 2a-30-30 flaw 188

pair geometry. As the bridging angle increased, coalescence trended from direct combined shear-tensile to direct tensile coalescence.

4.2.3

Cracking Characteristics

The following section will discuss some unique cracking characteristics observed in Opalinus shale tested with horizontal bedding planes, specifically en-echelon cracking and a new crack type that has not been previously categorized. Brightening prior to cracking observed in zoomed high resolution and optical microscopy imagery will also be discussed. Finally, the initial cracking type for all flaw pair tests with horizontal bedding planes will then be discussed.

En-Echelon Cracking

En-echelon cracking in Opalinus shale was observed in several flaw pair geometries tested. Flaw pair geometries in which en-echelon cracking was observed are summarized in table 4.8. En-echelon cracking was observed both between the flaws, as part of coalescence cracking, as well as part of secondary cracking on the outer tips of the flaws (figure 4-24). En-echelon cracking was common in co-planar geometries 2a-60-0 and 2a-75-0, as well as, stepped flaw pair geometries 2a-30-30 and 2a-30-60 (Refer to table 4.8). This range of tests which showed en-echelon cracking can be related to a combined flaw/bridging angle (β +α): en-echelon cracking was observed when the combined flaw/bridging angle (β +α) was between 60o and 90o . En-echelon cracking can be observed in nature as an array of tensile cracks induced by shear deformations (Lutton, 1970; Beach, 1975; Pollard et al., 1982) (figure 4-25). It appears that these en-echelon cracking formations in Opalinus are due to predominance of shear zones in tests with inclined flaw pairs. The outer tips of the flaws have been shown in previous research to be areas of high shear stresses (Refer to figure 2-51). This explains the occurrence of en-echelon cracking at the outer and inner tips of the flaws in certain flaw 189

Table 4.8: Tests Observing En-echelon Cracking Geometry 2a-0-0 2a-30-0 2a-45-0 2a-60-0 2a-75-0 2a-30-30 2a-30-60 2a-30-90 Total

Number of Tests 0/3 1/3 0/3 2/3 3/3 3/4 3/3 0/3

Test Repetition Label (B) (A,C) (A, B, C) (C, D, E) (A, B, C)

48 %

Figure 4-24: En-echelon cracking observed in Opalinus shale. En-echelon cracking was observed in about 50% of test geometries in Opalinus shale near the flaw tips as either coalescence echelon cracking at the inner tips or secondary cracking at the outer tips. The example is from a test conducted on the 2a-30-60 geometry. pair geometries. The flaw pair geometries which showed en-echelon cracking typically had more inclined flaw pairs (60o ≤ β +α ≤ 90o ) with shear cracks occurring, especially during coalescence. Flaw pair geometries with less inclined flaws (β +α < 60o ), or overlapping flaws (β +α > 90o ) in the case of 2a-30-90, showed much more tensile cracking and less shear and enechelon cracking. The occurrence of en-echelon cracking in experimental testing on flaw pairs is not exclusive to Opalinus shale. En-echelon cracking was also observed in Carrara marble by Wong 190

Figure 4-25: Natural en-echelon crack formation observed by Beach (1975). (a) Natural en-echelon formations observed by Beach (1975) in the Morcles Nappe (Swiss Alps) and (b) schematic representation of shear induced en-echelon crack formation and subsequent propagation. Modified from Beach (1975). (2008) (figure 4-26). It was observed in both macro-scale cracking processes as well as in microscopic studies of white patching zones at the tip of flaws (Refer to figure 4-26). Additionally, it was observed in tests on Polymethylmethacrylate (PMMA), also known as acrylic, with flaws by Petit & Barquins (1988) (figure 4-27). In all cases, en-echelon cracking was typically observed in the shear zone developed at the tip of the flaws (figure 428).

New Crack Type

As shown before, the cracks observed in tests on Opalinus shale were classified using the crack types defined by Wong and Einstein (2009a). Most cracks fit these crack type categories well but some cracks did not fit into any of these categories. Some cracks in Opalinus shale initiated in the matrix or from the edge of a specimen. These cracks are not consistent and can be difficult to classify. However, there was one type of crack consistently observed in Opalinus shale, which was not previously defined by Wong and Einstein (2009a) This new crack type, defined in this study as a tensile type 4 (IV) crack or a tensile "loop 191

Figure 4-26: En-echelon cracking observed in Carrara marble tested by Wong (2008). Enechelon cracking was observed at both the macro-scale (left) and the micro-scale (right). Modified from (Wong, 2008).

Figure 4-27: En-echelon cracking observed in Polymethylmethacrylate (PMMA) by Petit & Barquins (1988). PMMA, also known as acrylic, showed distinct en-echelon cracking at the tips of the flaw. Modified from (Petit & Barquins, 1988).

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Figure 4-28: Comparison of en-echelon cracking observed in Opalinus shale, Carrara marble and PMMA. En-echelon cracking was observed in all three materials near the tips of the flaws, in the "shear zone" area. Carrara marble was tested by Wong (2008). PMMA was tested by Petit & Barquins (1988).

crack", typically initiated at the face of the flaw and then propagated back around to the corresponding tip closest to its initiation point (forming a loop) (figure 4-29). Experiments showing this type of cracking are summarized in table 4.9. Type 4 tensile cracking was commonly observed in tests with slightly inclined flaw angles (β = 30o or 45o ) (Refer to table 4.9). Type 4 tensile cracking was not observed in tests higher angle flaws (β = 60o or 75o ) or tests with 0o flaw angle. It should be noted that this crack was not observed in previously tested rock materials such as gypsum, marble or granite (Bobet, 1997; Martinez, 1999; Miller, 2008; Wong, 2009). Therefore, there is likely something unique to Opalinus shale that causes this type of crack to occur. This could be the horizontal bedding planes or the drilling used at the flaw tips. One explanation for this crack type and its associated trend with flaw pair geometries is that higher tensile stress exist on the faces of the flaw when the flaw is at a low inclination. In tests with lower flaw inclination (β ≤ 45o ), the face of the flaw is oriented more perpendicularly to the loading direction causing a higher tensile stress on the flaw face. In addition, the bedding plane orientation is horizontal. It is possible that these cracks are tensile cracks, opening along the bedding planes near the tensile zone of the flaw tip. 193

Figure 4-29: New type of tensile crack observed in Opalinus shale. This tensile crack typically initiated from the face (near the center of the flaw) and then propagated to the flaw tip. This crack type was not observed in gypsum, marble or granite tests conducted in previous studies. Therefore, it is classified as a new type; Type 4 tensile crack.

Table 4.9: Tests Observing New Crack: Type 4 Tensile Geometry 2a-0-0 2a-30-0 2a-45-0 2a-60-0 2a-75-0 2a-30-30 2a-30-60 2a-30-90 Total

Number of Tests 0/3 3/3 3/3 0/3 0/3 1/4 1/3 2/3 40 %

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Test Repetition Label (A, B, C) (A, B, C)

(E) (C) (A, B)

Brightening Prior to Cracking

Micro-cracking and micro-crack development has been well studied for the previously tested rocks discussed earlier (e.g. gypsum, marble and granite). With the addition of the high resolution camera, areas of surface brightening (referred to as white patching in marble and granite) could also be observed in magnified images of Opalinus shale prior to cracking (See figure 4-30). It should be noted that these patches were not a common occurrence in Opalinus shale and it is not clear if they were a surface effect or another unknown phenomenon, which occurs only in shale. However, it is possible that these are micro-cracking zones (white patching), which occurred prior to cracking in Opalinus shale, are similar to the behavior seen in marble and granite (Refer to [Wong and Einstein, 2009c] and [Morgan et al., 2013]). In order to fully understand the differences in cracking and coalescence behavior between shale and these rocks, a full microscopic study should be conducted. This may include (but should not be limited to) a microscopic study using SEM or ESEM to determine how the cracking process zone develops.

Preliminary Microscopic Analysis

Although electron microscopy was not conducted on Opalinus shale in this study, a powerful optical microscope was used on post-test specimens. A USB controlled VEHOTM optical microscope was used to capture 20x and 400x imagery of cracks which occurred in the 2a-30-60B specimen (figure 4-31). This specimen was chosen because tensile wing cracks arrested and did not fully propagate even after coalescence occurred. Microscopy was conducted on the location of crack initiation at the flaw tip, the arrested tip of a wing crack, and a crack which sliding occurred along (figure 4-32). Brightening was also observed in optical microscopy images at the tip of an arrested crack (Refer to figure 4-32). 195

Figure 4-30: White Patching Observed in Opalinus Shale. The above image shows the progression(a-c) of surface brightening (white patching) in Opalinus shale. The bottom images highlight the area of white patching and cracking.

Figure 4-31: Optical USB VehoTM microscope. This USB microscope was used to image post-test cracking processes in Opalinus shale.

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Figure 4-32: Microscopic images of cracking in Opalinus shale. The images were taken with an optical microscope on a post-test 2a-30-60B specimen.

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Initial Crack Type

The initial crack mode (tensile or shear) and type [defined by Wong and Einstein (2009a)] can be determined for each test conducted with flaw pairs and horizontal bedding planes (table 4.10). For all flaw pair geometries tensile cracking was observed first. The most common initial crack type observed was tensile wing cracking (type I). Tensile wing cracking occurred in approximately 80% of all flaw pair geometries (Refer to table 4.10). The only geometries to not have exclusively tensile wing cracking (type I) as the initial cracking were the 2a-30-0 and 2a-45-0 geometries. These flaw pair geometries also observed type II tensile cracking and type IV tensile cracking first, the new crack observed only in Opalinus shale and is defined in section 4.2.3. It should be noted that crack initiation always occurred before high speed video recording. Therefore, crack initiation was determined using snapshots captured periodically (every 12 sec) throughout the test. Some of the first specimens tested in the study, before the use of high resolution cameras (2a-0-0, 2a-30-0, 2a-30-30, 2a-30-60), used snapshots from the high speed camera, which lacks the precision of high resolution imagery when detecting crack initiation (Refer to table 3.6). These specimens may have lower observed initial crack stresses than those that used higher resolution imagery, which aided in detecting small initial cracks. Additionally, since initial cracks initiated and propagated very slowly, it was nearly impossible to determine if the crack initiated in the shear mode. In other words, all initial cracks appeared to open slowly over time in the periodic snapshot images; however, it is possible, although unlikely, that some of these cracks initiated as shear and then opened in a tensile mode.

4.2.4

Coalescence Summary

The coalescence type for all the results presented above are summarized in figure 4-33. For each flaw pair geometry the most common coalescence behavior is shown and the number of tests showing this behavior is stated. Each coalescence behavior observed was 198

Table 4.10: Initial Crack Type Observed for Each Test Geometry 2a-0-0 2a-30-0 2a-45-0 2a-60-0 2a-75-0 2a-30-30 2a-30-60 2a-30-90 Total

Test Repetition A B C D (T)I (T)I (T)I (T)II (T)IV (T)I (T)II* (T)IV (T)I* (T)I (T)I (T)I (T)I (T)I (T)I (T)I (T)I (T)II (T)I (T)I (T)I (T)I (T)I (T)I (T)I

Type I: Wing Crack 3/3 1/3 1/3 3/3 3/3 3/4 3/3 3/3 20/25 (80%)

T = Tensile, S = Shear. Roman numerals represent the cracking type defined by Wong and Einstein (2009a). Type IV tensile cracking is defined in section 4.2.3. *: The first crack initiated in the matrix away from the flaw. The first crack initiating from a flaw is shown instead.

assigned the Wong and Einstein (2009a) coalescence category number and labeled with either "Indirect" for coalescence of two or more cracks which coalesce at a point outside the immediate bridging zone or "Direct" for coalescence involving cracks which directly coalesce between the flaws. From the coalescence results shown, there is clearly a trend that as flaw- or bridging angles increase, there is a trend from "indirect" to "direct shear or combined tensile-shear" to "direct tensile" coalescence.

Another way to present the coalescence results is by subdividing coalescence behavior into "indirect", "direct shear or combined tensile shear" and "direct tensile" coalescence groups. Then by plotting where these coalescence behaviors observed in the test are located with respect to flaw and bridging angle, coalescence zones can be developed (See figure 4-34). Presenting coalescence behavior by using zones not only makes comparing coalescence behaviors and trends of previously tested material much simpler but also makes it potentially easier to predict coalescence behavior of flaw geometries not yet tested. 199

Figure 4-33: Summary of the coalescence behavior observed in conducted on Opalinus shale with flaw pairs (bedding planes horizontal). L is the ligament length between the two inner flaw tips (a is half the flaw length = 0.165 in. [4.2 mm]), β is the flaw angle with respect to the horizontal, and α is the bridging angle between the flaws. Each coalescence behavior observed was assigned the Wong and Einstein (2009a) coalescence category number (Refer to figure 2-60) and labeled with either "None" for no coalescence, "Indirect" for coalescence of two or more cracks which coalesce at a point outside the immediate bridging zone or "Direct" for coalescence involving cracks which directly coalesce between the flaws. The number of tests showing the particular behavior is shown in parenthesis. 200

Figure 4-34: Coalescence behavior observed in Opalinus shale with respect to flaw angle (β ) and flaw pair bridging angle (α). For all tests flaw spacing was L = 2a. The Wong and Einstein (2009a) coalescence number (See figure 2-60) is shown below each geometry tested.

4.2.5

Crack Initiation and Coalescence Stress Summary

As shown in the previous section, the initial cracking was always tensile. The stress level at which cracking initiated (CI) was determined for all coplanar (See figure 4-35) and stepped (See figure 4-36) flaw pairs. Although there is some variability between tests, there is a reasonably clear difference between the maximum and minimum crack stress at initiation. For co-planar flaws the minimum crack initiation stress occurred for the geometry 2a-45-0 and the maximum for 2a-75-0. For stepped flaws the maximum tensile initiation stress occurred for 2a-30-30 and the minimum occurred for 2a-30-90. It should be noted that in some tests, crack initiation was detected using still images from the high speed camera. These lower resolution images, when compared to the high resolution camera images, have a lower precision regarding crack detection and could thus lead to a delay in crack detection, which can over- estimate the crack initiation stress. These tests are labeled with an asterisk (*) (Refer to figures 4-35 and 4-36). In addition to crack initiation stresses, the stress level at which coalescence between the flaws occurred was also determined (See figures 4-37 and 4-38). Coalescence stress levels for coplanar flaw pairs were very similar for all geometries except 2a-75-0, which had the 201

Figure 4-35: Crack initiation of Coplanar Flaw Pairs. The crack initiation stress (CI) for tests conducted on Opalinus shale with co-planar flaw pairs (0 bridging angle). The crack initiation stress [MPa] is plotted with respect to varying flaw angles. *: crack initiation was detected using still images from the high speed camera. These lower resolution images, when compared to the high resolution camera images, have a lower precision of crack detection and could thus lead to a delay in crack detection and an overestimation of crack initiation stresses.

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Figure 4-36: Crack initiation stress of Stepped Flaw Pairs. The crack initiation stress (CI) for tests conducted on Opalinus shale with stepped flaw pairs (30 flaw angle). The crack initiation stress [MPa] is plotted with respect to varying bridging angles. *: crack initiation detected using lower resolution still images.

highest coalescence stress of coplanar flaw pairs. This correlates with the high crack initiation stress for 2a-75-0. For stepped flaw pairs most tests had similar coalescence stresses except 2a-30-60, which had the lowest coalescence of the stepped flaw pair tests. This test 2a-30-60, also had a lower crack initiation stress than most of the other stepped flaw pair geometries. There appears to be a relation between the magnitude of crack initiation stress and coalescence stress. However, given the relatively small number of tests no direct conclusions can be drawn at this point.

4.2.6

Comparison to Previously Tested Rocks

The results on cracking processes in Opalinus shale with flaw pairs can be compared to results conducted on previous rocks, such as Hydrocal B-11 gypsum (Wong and Einstein, 2009a), marble [Vermont White and Carrara] (Wong and Einstein, 2009a; Martinez, 1999) and Barre granite (Martinez, 1999; Miller, 2008; Morgan et al., 2013). 203

Figure 4-37: Coalescence stress of Coplanar Flaw Pairs. The coalescence stress for tests conducted on Opalinus shale with coplanar flaw pairs (0 bridging angle). The coalescence stress [MPa] is plotted with respect to varying flaw angles.

Figure 4-38: Coalescence stress of Stepped Flaw Pairs. The coalescence stress for tests conducted on Opalinus shale with stepped flaw pairs (30 flaw angle). The coalescence stress [MPa] is plotted with respect to varying bridging angles.

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Crack Initiation Similar to previously tested materials, initial cracks occurred at the flaw tips, typically in the form of tensile wing cracking (figure 4-39). The shape of initial wing cracking (initiation point and curvature) observed in Opalinus shale matches best with gypsum and marble (Refer to figure 4-39). That is, tensile wing cracking in Opalinus shale, gypsum and marble initiated perpendicular to the flaw and then propagated toward the principal loading direction. Tensile cracking in granite typically had much less curvature. This could be due to the large grain size observed in granite causing cracking to follow grain boundaries [average grain size (Morgan et al., 2013): gypsum =50µm, marble =125µm, granite = 1,705µm,Opalinus shale = 0

∙ When εθ max = εθcrit , with εθcrit being the critical tangential strain and the criteria for which a shear crack will initiate or propagate are given as: ∙ At the distance ro from the tip of an existing crack 228

∙ In a direction θ in which γθ is max (γθ max ),

δ γθ δθ

=0

δ 2 γθ δθ2

>0

∙ When γθ max = γθcrit , with γθcrit being the critical shear strain

Figure 5-1: FROCK strain field around crack tip. Showing the cylindrical strains of an element radial to the flaw tip (Gonçalves da Silva and Einstein, 2013).

The following section describes the input parameters that needed to be calibrated and then presents a newly developed method to calibrate the input parameters more accurately and quantitatively. The results of this model will then be presented and compared to the experimental test results with horizontal bedding planes.

Throughout this chapter, many different terms will be used to describe the cracking processes in FROCK. It is important to define the terminology used in this chapter before proceeding. The term crack "mode" will be used to refer to the mechanism in which the crack propagates. Cracking mode is defined in figure 2-43. Mode I, or opening mode, refers to tensile cracking. Mode II, or sliding mode, refers to shear cracking. The term crack mode "location" is used to describe the location of tensile or shear cracks. The location of tensile or shear cracking is important when determining the accuracy of crack mode predictions in the model results compared to the experimental results. Lastly, crack "pattern" is used to refer to the cracking location (initiation and propagation) and shape (curvature, length, etc.). Crack "mode", "location", and "pattern" will be used extensively when comparing qualitative cracking results in the model to experimental results. 229

5.1.1

FROCK Modifications and Improvements

With the recent advances in graphical software, some visual post-processing modifications were added to the original strain-based FROCK to improve the observable results. One of the drawbacks of the original FROCK code is that is was heavily reliant on older postprocessing functions, specifically X-windows developed for UNIX based systems. These functions in X-11 work well for their original purpose; however, modifying and accessing their information can be rather difficult for new users who are not familiar with UNIX R based C-code. In order to improve this, a code was written in MATLAB○ to visualize the

cracking processes as well as to allow the user to modify and determine cracking parameters much more easy (figure 5-2). This code has the ability to display any parameter that the user desires, such as the applied stress level (σv ) or the input parameters (critical strain values) of the simulation being conducted. In this chapter, the term "simulation" refers to the results of a FROCK analysis conducted for a single flaw pair geometry and set of input parameters. The code also has the capability to automatically process many different simulations in a row and save visual screen captures of the cracking behavior at any set point during the R allows for further improvements simulation. Finally, its implementation with MATLAB○

such as tensile-shear crack visualization and automatic coalescence identification.

Shear and Tensile Crack Determination and Visualization

One aspect that the original FROCK code (Chan, 1986; Bobet, 1997) lacked was the ability of the user to determine if a crack was shear or tensile. This is quite important because one of the key components of the experimental results is the amount of shear and tensile cracking observed. The crack propagation criterion for FROCK is either shear or tensile; however, without knowing if a crack is propagating in shear or tensile in the model the user can only guess by the shape or direction of the crack. Determining the mode (shear or tensile) of the crack was originally implemented into FROCK by Gonçalves da Silva (2009). In many cases these crack types appear to be cor230

Figure 5-2: New post-processing visualization code for FROCK. This post-processing R function developed for MATLAB○ is capable of visually plotting the crack elements determined at each step in the FROCK model, including the applied stress and whether or not the crack is tensile or shear. The code is also capable of determining where coalescence occurs. It uses the ’.post’ output file from FROCK.

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rect; however, sometimes the flaws would plot in different colors (figure 5-3). The reason for this inconsistency is potentially how the crack type was plotted. After each new element was created, it was sent to a plotting function, which plotted all of the calculated elements. All elements were re-plotted every time a new element was formed. It is likely that under certain circumstances, which are still unknown, this plotting function in X-11 has some issues with color changes and re-plotting (Refer to figure 5-3). Incorrectly plotting flaw elements is not a problem; however, these inconsistencies could potentially be occurring in the propagated crack elements as well. An improved approached was developed to determine crack mode, output it with the crack element information and potentially prevent these types of plotting issues from occurring.

Figure 5-3: Previous tensile-shear determination in FROCK developed by Gonçalves da Silva (2009). This modified version of FROCK was capable of plotting tensile cracks as black and shear cracks as blue. Most results showed acceptable shear and tensile location such as seen in (a); some simulations appeared to have inconsistencies and odd results, such as flaws turning blue (shear) as seen in (b). The version of FROCK proposed in this thesis appears to correctly predict shear and tensile cracking (Refer to figure 5-2).

In the new approach, the cracking mode was determined after the critical crack direction was determined (figure 5-4). Once the critical direction was determined the strains in that direction were recalculated and then the critical cracking mode was then determined and stored (Refer to figure 5-4). The cracking mode for each element was recorded in a newly added column in the ".post" file; "0" for a flaw element, "1" for a tensile crack, and "2" for a 232

R shear crack. The MATLAB○ script was implemented with a function capable of interpret-

ing this output column in the ".post" file and then plotting the element in the appropriate color (Refer to figure 5-2). This method showed more consistent crack mode results. The crack mode results will be shown later in this chapter.

Figure 5-4: The process for determining crack mode (tensile/shear) in the updated FROCK model. (a) The strains are calculated around the tip of the element and then (b) the critical crack direction are determined. From this critical crack direction, (c) the crack mode (tensile/shear) are determined and then stored.

Determining Coalescence of Flaws Automatically R , the crack initiation stress, cracking mode With the post-processing script in MATLAB○

and cracking pattern for many different simulations could be determined very quickly; however, determining coalescence automatically can be complicated. An additional function was developed to determine when a crack from one flaw connected to a crack from another flaw. When cracks from each flaw connect the code stores the applied stress level and location of intersection. Since FROCK limits crack initiation to the tips of the flaws, the cracks are associated with the tip which they initiate from. By using this assumption, each new crack element at a tip is assigned a number (1-4) with (1,3) being from the left flaw and (2,4) being from the right flaw (figure 5-5). If an element propagates from an element previously assigned one of these crack numbers, the new crack element inherits the same crack number as the previous element (it was an extension of the same crack) ((figure 5-6). If an extension of a crack element from one flaw (ex. 1,3) intersects another crack 233

element (shares the same nodal location) from the opposite flaw (ex. 2,4) then coalescence occurs at that nodal location (Refer tofigure 5-6). The applied stress at which coalescence occurs is then stored.

Figure 5-5: Crack tip designation used for automatic coalescence determination. Each tip is assigned a number, 1, 3 for the left and right tips of the left flaw respectively and 2,4 for the left and right tip of the right flaw respectively.

Figure 5-6: A schematic of the automatic coalescence determination technique. Each crack is assigned a number; 1, 3 for cracks initiating from the left and right tips of the left flaw respectively and 2,4 for a cracks initiating from the left and right tip of the right flaw respectively. If a crack from the left flaw (1,3) intersects a crack from the right flaw (2,4) coalescence occurs (Refer to figure 5-2).

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5.1.2

Calibration and Input Parameters

As stated before, the tests with horizontal bedding planes were used for calibrating the Isotropic FROCK model. This model assumes that the material is linear-elastic and isotropic. Therefore, the Young’s modulus (1,080 MPa) and Poisson’s ratio (0.27) determined from the intact tests on Opalinus shale with horizontal bedding planes (0o ) were used. There are four material input parameters in the strain-based Isotropic FROCK model (εθcrit , γθcrit , ro , µ) that needed to be calibrated based on the experimental test results. However, determining four input parameters simultaneously by comparing cracking patterns can be very time consuming and nearly impossible with current calibration techniques. Therefore, the standard procedure to calibrate these parameters in the past was essentially trial and error. By varying each parameter individually and determining the best set of results that compared most closely to the experimental observations (based on stress levels and crack patterns), optimal ranges for these parameters were determined. From the optimal ranges identified for each geometry, a single range of acceptable values for all test series was determined. This typically meant that there was not one single set of values for the material parameters which worked for all flaw pair geometries. Additionally, there was no way to precisely balance, in a quantitative way, the effect of the parameters on quantifiable measurements, such as crack initiation stress. To improve the calibration process, this thesis proposes a multi-stage process, in which the optimal input parameters are determined and a single set of input parameters is established for all test geometries.

Semi-Automated Simulations One of the improvements to the FROCK model parameter calibration was the introduction of a semi-automated method used to run FROCK simulations. FROCK automation made it possible to run a higher number simulations, thus determining the effect of each parameter on the output. It consists of generating input files and then running multiple simulations. R A MATLAB○ code was written to produce multiple input files (100+) which varied a pa-

235

rameter(s) in FROCK. This code is capable of generating multiple input files with selected input parameters modified. Therefore, multiple FROCK simulations can be conducted over desired range of values for any selected input parameter. These input files are named in a sequential order (’IT0001.inp’, ’IT0002.inp’, etc.). Typically, these input files are generated with two parameters varied at once in ten increments for each parameter, resulting in 100 input files (100 simulations). In order to run multiple simulations at once, a computer assisted software called Jitbit Macro Reader was used to input the file name and run the FROCK code for each generated input file. Using the post-processing script described in section 5.1.1, the ’.post’ output files for these simulations could be processed in a large batches and the parametric effect of selected input parameter(s) on the output results (crack initiation stress, crack shape, crack mode, etc.) could be determined.

Calibration Process

Even with the incorporation semi-automated simulations, optimizing all four input parameters (εθcrit , γθcrit , ro , µ) simultaneously is still not possible. It is especially difficult because even with numerical comparisons such as crack initiation stress, there is still a component of the FROCK calibration which relies on visual crack patterns. Without computer aided techniques, such as computational image analysis, cracking patterns need to be checked visually by the user. Therefore, it was decided that the best way to calibrate the material parameters was to calibrate them "semi-independently". This means: ro and µ were calibrated independently; however, since the responses of the critical tangential (εθcrit ) and shear (γθcrit ) strain are coupled (crack initiation, pattern, etc.), they were calibrated at the same time. The process describing how each parameter was determined will now be presented and the steps are shown in figure 5-7. 1) ro - Plastic Radius The plastic radius (ro ) is essentially the search distance at the tip of a crack element. It can 236

Figure 5-7: Calibration steps to determine the input parameters for Opalinus shale used in the isotropic strain-based FROCK developed by Gonçalves da Silva and Einstein (2013). The calibration process consisted of assuming a value for ro and then using the crack initiation stresses and crack patterns to determine εθcrit and γθcrit . In the final step, the crack patterns were used to calibrate µ. be one of the most difficult parameters to calibrate because the crack element length has an effect on it. The length of a crack element in FROCK is determined by the length of the flaws and number of elements in it. Each flaw in the Opalinus shale tests was 0.84 cm long. Previous studies on FROCK showing the best results used fifteen total crack elements per flaw (Bobet, 1997; Gonçalves da Silva, 2009). Therefore, fifteen elements per flaw were used in this study. Thus, each crack element was 0.056 cm long. Gonçalves da Silva (2009) did a detailed study analyzing the effect of different "ro to crack element length" ratios. From this study, it was determined that a ro equal to 0.4 times the crack element length showed the best results, i.e. ro is equal to 0.022 cm. The same ro was used for all simulations. 2) εθcrit , γθcrit - Tangential and Shear Strains εθcrit and γθcrit were calibrated in several steps using the automated technique described previously. For these simulations a starting value of µ needs to be assumed. Based on µ 237

used for previous studies conducted on FROCK (Bobet, 1997; Gonçalves da Silva, 2009), a value of µ=0.66 is assumed for the the calibration of εθcrit and γθcrit . In the first calibration step, the values of εθcrit and γθcrit are determined based on the crack initiation stress solely. The crack initiation stress is the far-field stress required for initial cracking to occur. For each flaw pair geometry, the εθcrit is varied from -0.0135 to -0.0110 and γθcrit is varied from 0.035 to 0.039. Clearly, these limits require some pre-existing knowledge of the level of acceptable crack initiation stress; therefore, some trial and error simulations were run before to determine the approximate range of acceptable results. From the simulations and the batch post-processing, the effect of both εθcrit and γθcrit on the crack initiation stress can be determined (figure 5-8).

Figure 5-8: The crack initiation stresses σip ) in FROCK determined for an array of εθcrit and γθcrit values. The crack initiation stress surface can be divided into two planes. A plane where crack initiation is shear (change in εθcrit has no effect) and a plane where crack initiation is tensile (change in γθcrit has no effect). From these two planes, two equations for crack initiation stress can be determined based on εθcrit and γθcrit . This figure shows an example for the crack initiation surface for the 2a-30-0 geometry. This process is conducted for all geometries to determine the effect of εθcrit and γθcrit on the crack initiation stress in each geometry. Since the direction of maximum tensile and shear strain at the flaw tips remains the same 238

(independent of different critical strain values), the change in crack initiation stress is linearly related to the critical strains. Therefore, the two planes can be defined depending on whether the initial crack was tensile (σip−tensile ) or shear (σip−shear ) (figure 5-9). A linear equation describing the crack initiation stress based on the εθcrit or γθcrit is determined for each plane (table 5.1). The actual crack initiation stress (σip ) for each geometry (i), which is shown in figure 5-8, is the minimum of these two planes (Refer to figure 5-9). This can be described by:

σip = min(σip−tensile , σip−shear )

(5.1.1)

This process was conducted for all flaw pair geometries and the planes defining their crack initiation stress were determined (Refer to table 5.1). Table 5.1: Crack Initiation Stress (σip ) Based on εcrit and γcrit Geometry

σip−tensile

σip−shear

2a-0-0 2a-30-0 2a-45-0 2a-60-0 2a-75-0 2a-30-30 2a-30-60 2a-30-90

-1040εθcrit -690εθcrit -750εθcrit -1060εθcrit -2130εθcrit -650εθcrit -700εθcrit -750εθcrit

250γθcrit 220γθcrit 230γθcrit 280γθcrit 420γθcrit 210γθcrit 230γθcrit 240γθcrit

It should be noted that this not the only way to determine these equations predicting crack initiation based on the critical strain values. Another approach would be to determine the "applied stress-local strain" concentration factors in the direction of critical tensile strain and critical shear strain for each flaw pair geometry. Using these concentration factors in the critical directions, the strains at the tip could be calculated from the far-field stress. From this, the crack initiation stress based on critical strains could be determined. From the crack initiation equations (Refer to 5.1), a set of optimal critical strains can be determined. This is accomplished by minimizing the total squared error (TSE) between 239

Figure 5-9: Defining the crack initiation stress planes determined from FROCK. The crack initiation stress surface can be divided into two planes; (a) a plane where crack initiation is tensile (change in γθcrit has no effect) and (b) a plane where crack initiation is shear (change in εθcrit has no effect). (c-d) From these two planes, the critical crack initiation stress can be determined based on εθcrit and γθcrit .

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the predicted initiation stress (σip ) and the experimentally observed initiation stress (σie ), which can be expressed as:

T SE = Σ (σip − σie )2

(5.1.2)

where the subscript (i) represents the flaw pair geometry. The TSE results are visually represented in figure 5-10. By minimizing the TSE, the values of εθcrit =-0.0112 and γθcrit =0.0383 were determined (Refer to figure 5-10). This calibration process optimizes the crack initiation stresses; however, it is necessary that the predicted crack patterns and mode (tensile/shear) match the experimental results as well, which will now be discussed.

Figure 5-10: Total square error optimization of crack initiation stress based on εθcrit and γθcrit for simulations on all flaw pair geometries. By minimizing the total square error the optimal εθcrit of -0.0112 and γθcrit of 0.0383, based on crack initiation stress, is determined. The final step to calibrate the εθcrit and γθcrit input parameters consists of a visual analysis to fine-tune the critical strain values based on the most acceptable cracking patterns. For each simulation, the predicted crack pattern is compared to the crack pattern observed in the experiments and then classified as a "Good match", "Average match" or "Poor 241

match"(figure 5-11). The set of critical strain values which is related to the highest amount of "Good" and "Average" matches is determined. From the visual comparison, one can conclude that the previously determined critical values, εθcrit =-0.0112 and γθcrit =0.0383, are acceptable enough to proceed to the calibration of µ.

Figure 5-11: Calibration of εθcrit and γθcrit based on crack pattern comparison. The crack patterns observed from the Isotropic FROCK simulations (right) are compared with experimental results (left). They are divided into three categories; "Good Match", "Average Match" and "Poor Match". The pattern comparison shown was conducted for the 2a-30-30 flaw pair geometry. 3) µ - Friction of a Newly Formed Crack Surface It is important to understand that µ, the friction of a newly formed crack, does not affect the crack initiation stress because µ requires a new crack element (non-flaw element) to be formed first. However, it does affect the crack pattern and crack coalescence stress. Coalescence stress is difficult to use as a calibration metric because not all geometries showed 242

coalescence. Therefore, only cracking pattern matching is used to calibrate µ. Similar to the visual cracking pattern assessment conducted for εθcrit and γθcrit , the acceptable cracking patterns based on µ are determined for each geometry (figure 5-12). This results in a final value of µ=0.74 for all flaw pair geometries.

Figure 5-12: Calibration of µ based on crack pattern comparison. The crack patterns observed from the Isotropic FROCK simulations (right) are compared with experimental results (left). They are divided into three categories; "Good Match", "Average Match" and "Poor Match". The pattern comparison shown was conducted for the 2a-30-30 flaw pair geometry.

Final Input Parameters Used Based on the crack initiation stress and final crack patterns observed in Opalinus shale tests, a single set of input parameters (ro , εθcrit , γθcrit , µ) was determined (table 5.2). The following section will present the results of this model. These results will include the model’s ability 243

to simulate the crack initiation, -propagation, -coalescence and -mode. Additionally, the crack initiation and coalescence stress will be presented as well. It is important to note that these parameters are specific to Opalinus shale with horizontal bedding planes. Table 5.2: Isotropic FROCK Model Input Parameters Used for Opalinus Shale εcrit -0.0112

5.1.3

γcrit 0.0383

ro 0.022

µ 0.74

Isotropic FROCK Model Results

Using the input parameters discussed above, the cracking processes were predicted with FROCK and then compared with the experimental results for both coplanar (See figure 513) and stepped (See figure 5-14) flaw pairs with horizontal bedding planes. The results show the cracking progression in two stages (Refer to figures 5-13 and 5-14). In order to systematically compare the crack patterns predicted with FROCK to the experimental results, a qualitative classification system was developed (table 5.3). This evaluation categorizes the crack initiation, coalescence and final pattern, and cracking mode results predicted by FROCK into three categories; "Poor Match", "Average Match", "Good Match" (Refer to table 5.3). Most importantly, this categorization system provides a measure of the qualitative results, which can be used to compare future model results. The evaluation of the crack pattern results is summarized in table 5.4. Overall, the progression of cracking predicted by the FROCK model was very similar to that of the experimental test results. The model was capable of capturing the observed initial cracking behavior in six out of the eight flaw pair geometries tested (Refer to table 5.4). The model also accurately predicted coalescence for five out of the eight flaw pair geometries (2a-45-0, 2a-60-0, 2a-75-0, 2a-30-30, and 2a-30-60). Cracking mode was also generally acceptable showing an average match or better in six out of the eight flaw pair geometries. The coalescence observed in tests conducted on flaw pair geometries 2a-0-0 and 2a-30-0 still cannot be captured. Similar results for these flaw pair geometries (2a-0-0 and 2a-30-0) 244

Figure 5-13: Isotropic FROCK model results for Opalinus shale with coplanar flaw pairs. A comparison can be made between the experimental test results and the FROCK model results. Experimental test results are shown on the left side and the FROCK model results are shown on the right. "Snapshots" are shown for each geometry (crack initiation = 1st stage and coalescence = 2nd stage).

245

Figure 5-14: Isotropic FROCK model results for Opalinus shale with stepped flaw pairs. See figure 5-13 for a detailed explanation.

Table 5.3: Evaluation System Used for FROCK Model Patterns Evaluation Poor Match=x

Initial Crack None of the initial cracking direction and shape match

Average Match=X

Some of the initial cracking direction and shape may match All or most of the initial cracking direction and shape match

Good Match=XX

Final Pattern None of the coalescence or final crack shape patterns match Some of the coalescence or final crack shape patterns may match All or most of the coalescence and final crack shape patterns match

246

Crack Mode None of the shear/tensile cracking match Some of the shear/tensile cracking may match All or most of the shear/tensile cracking match

were observed by Bobet (1998) and Gonçalves da Silva (2009) in FROCK simulations for gypsum. One reason why FROCK cannot capture indirect coalescence is because cracks can only initiate and propagate from previously existing crack elements (i.e. not in the intact matrix). FROCK is also not capable of predicting crack branching. Indirect coalescence typically occurred in experiments when two cracks, emanating from the flaws, were connected by a third crack, such as the behavior observed for 2-30-0 (Refer to figure 5-13). For this reason, FROCK still has difficulty predicting this coalescence type. Table 5.4: Evaluation of Isotropic FROCK Model Geometry 2a-0-0 2a-30-0 Coplanar 2a-45-0 2a-60-0 2a-75-0 2a-30-30 Stepped 2a-30-60 2a-30-90 Total X/XX x = Poor Match

Initial Crack X X x X X XX X X 7/8

Final Pattern x x XX X XX X X x 5/8

X= Average Match

Crack Mode x X XX X X X X X 7/8

XX= Good Match

Crack Initiation and Coalescence Stress

In addition to the cracking patterns and behavior, the crack initiation and coalescence stresses were determined with the Isotropic FROCK model. The following section will discuss crack initiation stress first and then the coalescence stress results. Crack Initiation Stress The crack initiation stress was determined for coplanar (figure 5-15) and stepped (figure 516) flaw pairs. From these plots, a reasonably close agreement can be seen between the model predictions and the experimental results for crack initiation stress (Refer to figures 5-15 and 5-16). However, it is no surprise that crack initiation predictions match the experimental values so well since the critical strain input parameters were calibrated based on 247

crack initiation stresses. Crack initiation stress was slightly under-predicted for coplanar flaws at 0o and 30o flaw angles and stepped flaws at 0o and 30o bridging angles. The model over-predicted stepped flaw pairs at a 90o bridging angle.

Figure 5-15: Comparing the crack initiation (CI) stress of coplanar flaws (bridging angle, α=0o ) in the Isotropic FROCK model and the experimental results. Simulations were conducted at one degree angle increments. *: crack initiation detected using lower resolution still images.

Figure 5-16: Comparing crack initiation (CI) stress of stepped flaws (flaw angle, β =30o ) in the Isotropic FROCK model and the experimental results. Simulations were conducted at one degree angle increments. *: crack initiation detected using lower resolution still images. 248

Crack Coalescence Stress Additionally, using the automated coalescence detection function developed for this study, the coalescence stresses for coplanar (figure 5-17) and stepped (figure 5-18) were determined as well. Simulations were conducted in one degree increments and only the flaw pair geometries producing coalescence were plotted (Refer to figures 5-17 and 5-18). For both coplanar and stepped flaw pairs, a large variability in the coalescence stress obtained with FROCK can be observed. This is due to the fact that coalescence is extremely sensitive to small variations in the FROCK model, such as the initial crack location and the coalescence type. Additionally, coalescence, more than crack initiation stress, is most affected by numerical instabilities which may occur at specific flaw and bridging angles. For example, there was a clear deviation from the general trend in coalescence stress observed for stepped flaw pairs between 59o and 69o bridging angles (Refer to figure 5-18). These coalescence results highlight some of the limitations of FROCK in predicting the coalescence stress. However, coalescence results have not been analyzed in such detail in FROCK before (one degree increments). It could be that the model still has some difficulties consistently predicting coalescence stresses at certain flaw and bridging angles. However, it should be noted that, excluding the angles where the coalescence stresses are abnormally high or low, the general trend of the coalescence stresses predicted is similar to the experimental tests (Refer to figures 5-17 and 5-18). For example, the coalescence stress for the coplanar flaw pairs (bridging angle, α=0o ) increases with flaw angle and for stepped flaw pairs (flaw angle, β =30o ) decreases slightly at higher bridging angles. This is an encouraging result since the coalescence stresses were not included as part of the calibration process. Summary of Crack Initiation and Coalescence Stresses The crack initiation and coalescence stresses for the flaw pair geometries experimentally tested can compared to the FROCK results and are summarized in table 5.5. The FROCK simulations compared well to both the crack initiation and coalescence stress for all but 249

Figure 5-17: Comparison of the coalescence stress of coplanar flaws in the Isotropic FROCK model and the experimental results. Simulations were conducted at one degree angle increments. Only the geometries producing coalescence are shown.

Figure 5-18: Comparison of the coalescence stress of stepped flaws in the Isotropic FROCK model and the experimental results. Simulations were conducted at one degree angle increments. Only the geometries producing coalescence are shown. one geometry. The FROCK prediction for the coalescence stress of 2a-30-60 was much higher (20.3 MPa) than the experimentally obtained value (9.2 MPa). As stated before, the coalescence stresses of stepped flaw pairs predicted at bridging angles between 59o and 69o were oddly high. This range included the 2a-30-60 flaw pair geometry. Overall, the Isotropic FROCK model was capable of predicting generally acceptable cracking patterns, 250

as well as crack initiation and coalescence stresses, for the flaw pairs tested in Opalinus shale with horizontal bedding planes. Table 5.5: Summary of Isotropic FROCK Model and Experimental Test Results Geometry

Coplanar

Stepped

2a-0-0 2a-30-0 2a-45-0 2a-60-0 2a-75-0 2a-30-30 2a-30-60 2a-30-90

FROCK Output [MPa] CI Coalescence 9.7 (No Coal.) 7.8 (No Coal.) 8.4 12.0 10.8 12.4 16.1 17.0 7.2 13.7 7.8 20.3 8.4 (No Coal.)

Experimental Test Avg. [MPa] CI Coalescence 10.1* 12.1 10.1* 13.9 8.5 14.1 10.8 14.0 15.7 19.1 11.7* 12.5 7.3* 9.2 6.0 12.8

CI = Crack Initiation Stress *: crack initiation was detected using still images from the high speed camera. These lower resolution images, when compared to the high resolution camera images, have a lower precision of crack detection and could thus lead to a delay in crack detection and an over-estimation of crack initiation stresses.

5.1.4

Bedding Plane Limitation

Although the strain-based FROCK model showed relatively acceptable results for the tests conducted with flaw pairs in Opalinus shale with horizontal bedding planes (e.g. the effect of flaw and bridging angle), it did not capture cracking which occurred due to bedding planes (figure 5-19). This is due to the fact that the strain-based FROCK model cannot incorporate bedding plane or directional weakness since it assumes an isotropic cracking criterion. Therefore, directionally dependent critical strains were incorporated into the original strain-based FROCK model.

5.2

Anisotropic Strain-based FROCK Model

As just stated, the strain-based Isotropic FROCK model is not capable of capturing the effect of bedding planes on cracking. Therefore, it was necessary to modify the exist251

Figure 5-19: The limitations of the Isotropic FROCK model to predict bedding plane crack patterns. The results predicted for the Isotropic FROCK model on 2a-30-30 flaw pair geometry (top) matched the calibrated 2a-30-30 test with horizontal bedding planes; however, it was not capable of varying the crack pattern based on bedding planes. The experimental results for 2a-30-30 with 0o , 30o , 60o and 90o bedding planes are shown at the bottom. ing FROCK model and incorporate a cracking criterion which captures both the effect of bridging and flaw angles (horizontal bedding plane) as well as the effect of inclined bedding planes. The following section will discuss two different anisotropic cracking criteria; including their theoretical basis, calibration method and results. In addition, other modifications incorporated into to the original strain-based FROCK code, such as new input parameters and a modified critical angle search function, will be discussed. Finally, a single anisotropic cracking criterion in FROCK will be recommended for predicting cracking in Opalinus shale.

5.2.1

Proposed Anisotropic Cracking Criteria

In order to capture the effect of bedding plane weakness in FROCK, it is necessary to incorporate directionally dependent critical strains. The modified FROCK model integrates 252

an anisotropic failure criterion; however, it still assumes linear elasticity. It also assumes that the critical strains along the bedding plane are always less than those in the intact material. By assuming that the critical strains of the bedding plane cracking are a reduced proportion of the intact material critical strains, the critical tangential strain (ε) and shear strain (γ) along the bedding can be expressed as:

ε(θ ) = Rε (θ )εintact

(5.2.1)

γ(θ ) = Rγ (θ )γintact

(5.2.2)

where εintact and γintact are the critical strains of the intact material, ε(θ ) and γ(θ ) are the critical strains at angle θ , and Rε (θ ) and Rγ (θ ) are reduction factors as a function of the search angle, θ , ranging from 0 to 1. The definition of the search angle (θ ) is shown in figure 5-20. It is important to note that this assumes that bedding plane weakness can occur at any location and therefore does not have a predefined location. Using a predefined bedding plane thickness would have made the implementation of bedding planes very complex because the bedding plane width is likely scale dependent and can be very difficult to determine in clay based shale such as Opalinus shale. The variation of the critical strain with angle θ is based on previously proposed anisotropic strength criteria (Jaeger, 1960; Buczek and Herakovich, 1985). Since there have been many other anisotropic failure criteria used to predict the strength of rocks, such as those presented in sections 2.1.4, this variation in strength is based on similar principles as these previous theories. Models such as the single plane of weakness theory (Jaeger, 1960), variable strength criterion by Ramamurthy et al. (1993) and the variable tensile strength theory by Buczek and Herakovich (1985) are used as a basis to define directionally dependent critical strains. Similar approaches were taken by Shen et al. (2015), who proposed both linear and sinusoidal strength criteria for the anisotropic FRACOD model, which were discussed in section 2.2.4. 253

Based on these previously established anisotropic failure criteria two different directional dependent critical strain cracking criteria are proposed and implemented into FROCK; the discontinuous criterion and the continuous criterion. The FROCK model implementing the discontinuous criterion will be referred to as the "Discontinuous FROCK model". The FROCK model implementing the continuous criterion will be referred to as the "Continuous FROCK model". The model previously used in section 5.1, which does not incorporate bedding planes, will still be referred to as the "Isotropic FROCK model". All three models use the strain-based cracking criteria. The following section will discuss the fundamental differences between the discontinuous and continuous criteria, which were implemented in FROCK. The terminology and angle definitions in FROCK will be used again. For reference figure 5-20 presents the key parameters which will be used during this discussion. Most importantly is the angle of interest (θ ) which defines the angle (from the horizontal) at which FROCK calculates the strains and compares them to the critical strains input by the user (Refer to figure 5-20). This should not be confused with the loading direction θ defined in chapter 2. For this reason, the bedding plane angle (ψ) will always be used to define the direction of the bedding planes in the results and not the loading angle. The bedding plane angle (ψ) is always defined as the angle between the planes and the horizontal (Refer to figure 5-20).

Figure 5-20: Parameters and angles defined in the newly modified FROCK model. θ defines the angle (from the horizontal) from an element tip to the location of a potential crack and it is where FROCK calculates the strains (εθ ,εr ,γrε ). The bedding plane angle (ψ) is the angle of the bedding plane with respect to the horizontal. 254

Discontinuous Criterion

The discontinuous criterion assumes that the material in the bedding plane direction is discretely weaker than that intact material and it therefore has two sets of critical strains associated with cracking. One set of critical strains represents the intact material cracking behavior (εintact ,γintact ). The other set of critical strains is used for cracking along bedding planes (εbedding ,γbedding ) (figure 5-21). This concept of a single set of material parameters representing a weak bedding direction is based on the single plane of weakness theory developed by Jaeger (1960).

Figure 5-21: Definition of the discontinuous criterion implemented into FROCK. The discontinuous criterion assumes a discrete reduction in the critical strains when the angle of interest (θ ) is in the bedding plane direction (ψ). A visual example (left) is shown with a minimum critical strain of 0.030 along the bedding plane and 0.040 across the bedding planes (intact). A graphical representation of the discontinuous criterion is shown as well (right).

The cracking criteria of the Discontinuous FROCK model can be represented as a piecewise function based on the angle of interest:

εθ θ =

  εθ θ −bedding , θ = ψ or θ = ψ + 180o  εθ θ −intact ,

θ ̸= ψ or θ 255

̸ ψ + 180o =

(5.2.3)

γ=

  γbedding , θ = ψ or θ = ψ + 180o  γintact ,

θ ̸= ψ or θ

(5.2.4)

̸ ψ + 180o =

where εθ θ −bedding and γbedding are the critical tangential and shear strains of the bedding plane respectively, εθ θ −intact and γintact are the critical tangential and shear strains of the intact material respectively, θ is the angle of interest being calculated by FROCK and ψ the bedding plane angle with respect to the horizontal. As stated before, the critical strains in the bedding plane direction can be represented as a proportion of the intact critical strain using reduction factors (R):

εθ θ −bedding = Rε εθ θ −intact

(5.2.5)

γbedding = Rγ γintact

These reduction factors (Rγ ,Rε ) are input into FROCK by the users along with the intact material critical strains and bedding plane angle (ψ).

Continuous Criterion

The continuous criterion models the critical strains (ε,γ) as a continuous function of the angle of interest (figure 5-22). This is based on a trigonometric variation of critical strains such as those proposed by the McLamore and Gray (1967), Buczek and Herakovich (1985) and Ramamurthy et al. (1993). The continuous criterion implemented into FROCK uses the same form as Buczek and Herakovich (1985) for both tensile and shear critical strain variation (Refer to figure 5-22) and can be expressed as:

εθ θ = εθ θ −intact sin2 (θ − ψ) + εθ θ −bedding cos2 (θ − ψ) 256

(5.2.6)

γ = γintact sin2 (θ − ψ) + γbedding cos2 (θ − ψ)

(5.2.7)

Figure 5-22: Definition of the continuous criterion implemented into FROCK. The continuous criterion assumes a continuous trigonometric reduction in critical strain with respect to the angle of interest (θ ). A visual example (left) is shown with a minimum critical strain of 0.030 along the bedding plane and 0.040 across the bedding planes (intact). A graphical representation of the continuous criterion is shown as well (right). The reduction factors (Rγ ,Rε ) for the continuous criterion define the minimum critical strain values, i.e. the critical strains in the direction of the bedding planes (γbedding and εθ θ −bedding ) (Refer to figure 5-22). Only this minimum critical strain value can be modified, which means this theory does not incorporate any changes to the shape of the continuous function. In other words, the drop or "sharpness" in reduction cannot be modified in this functional form. This was the only continuous function used; however, other forms of continuous reduction functions could have been implemented with different shapes such as the saw-tooth function (linear decrease) used by Shen et al. (2015) in the FRACOD model and Lisjak et al. (2013) used in the Y-Geo model.

5.2.2

FROCK Modifications

By adding a directional cracking criterion to FROCK due to bedding planes, additional modifications to the code and output visualizer were necessary. These changes included a 257

modified input file, bedding plane visualization and a more rigorous critical angle searching function.

Input File Modification Three additional input parameters were added to the input file format. In the modified FROCK code, the input file requires that the bedding plane angle, minimum tangential reduction factor and minimum shear reduction factor are input (figure 5-23). These input parameters were implemented by adding additional columns in the input file which were read by the code (Refer to figure 5-23).

Figure 5-23: An example of an input file (".inp") header with the new parameters added to FROCK. The bedding plane angle (ψ), tangential reduction factor (Rε ) and shear reduction factor (Rγ ) are added to the material parameter row in the input file and then read by the new FROCK code.

Bedding Plane Visualization With the addition of bedding planes, the newly developed post-processing visualizer in R MATLAB○ was modified to display the bedding plane angles (figure 5-24). As stated

before, both anisotropic criteria assume that the bedding plane weakness occurs at any location. Therefore, FROCK does not model actual bedding planes with a discrete spacing or location. This means that displaying the bedding planes in the output is only used to show the user the angle of the bedding planes. This method helps the user to visualize the bedding plane angle and to easily compare it to the crack propagation directions. 258

R Figure 5-24: Modified output visualizer for FROCK with bedding planes. The MATLAB○ viewer for the FROCK results is capable of displaying the bedding planes along with the bedding plane angle. These angles are for the user to visually understand the angle of the bedding planes with respect to the flaw pair geometry and do not represent discrete bedding planes.

Critical Propagation Angle Search Function

Lastly, incorporating anisotropic cracking criteria requires that the previous function used to determine the critical propagation direction be modified. Previously, the FROCK code used an optimized function to determine the critical propagation direction (figure 5-25a). First it used a coarse search to determine the strain at ten degree increments (Refer to figure 5-25a). By using the slope between the calculated strains, it determined if there is a peak between two previously determined points. Then it calculated the strain between these two points by halving the increment and repeating the process until the search parameter was reduced to a one degree increment, at which point it stored the peak strain (Refer to figure 5-25a). This worked well in the original FROCK code because the critical strains were constant at all angles; however, in the newly proposed anisotropic cracking FROCK the critical strains 259

change with the search angle. This can be significant in the case of the discontinuous criterion where the critical strain changes abruptly at a specific search angle (Refer to figure 5-25b). Therefore, a new (modified) search function was implemented into FROCK, which calculates the strain at every angle (at one degree increment). Clearly this takes much more computational time than the optimized function used before but it assures that all angles are checked.

Figure 5-25: Modified critical propagation direction search method for FROCK. (a) The search function used in previous versions of FROCK, which uses a coarse search, the slope between these points and then halves the increments to determine the peak efficiently. (b) The new search method which determines the strain in all directions and compares it do the directional dependent critical strain.

5.2.3

Calibrating Parameters

By incorporating bedding planes into FROCK, two newly introduced input parameters, Rε−min and Rγ−min , needed to be calibrated. As shown before, material parameters can be calibrated in FROCK by comparing crack initiation stresses from FROCK to experimental results and by comparing the cracking mode and shape from FROCK to the cracking observed in experiments. Calibration of the isotropic strain-based FROCK in section 5.1.2 was done by combining both crack initiation stress and crack shape comparisons. However, there were some issues when trying to calibrate the minimum reduction values for the bedding plane models based on crack initiation. Typically, FROCK predicted incorrect or unrealistic crack patterns (figure 5-26). Numerical instabilities at certain material parameters were a common issue in FROCK and were observed in previous work on FROCK 260

by Bobet (1998) and Gonçalves da Silva (2009). For this reason, the minimum reduction factors for both the discontinuous and continuous criteria were determined using only the crack pattern (figure 5-27). The calibration using crack patterns was shown previously (Refer to figure 5-11). Calibration of the minimum reduction factors, (Rε−min and Rγ−min ), was accomplished by using the previous input parameter values of ro , εθcrit , γθcrit and µ determined from the Isotropic FROCK model calibration (Refer to section 5.1.2). The minimum reduction factors Rε−min and Rγ−min were then varied and the cracking and coalescence pattern of tests were used to determine the most accurate matches. In the final step, the crack patterns of all geometries (even those with horizontal bedding planes) were checked and minor changes were made to εθcrit , γθcrit and µ. As shown in Section 4.1.6, Opalinus shale is anisotropic and stiffness can vary with bedding plane direction. However, the FROCK models assume that the material is linear-elastic and isotropic. Although the proposed FROCK models incorporate anisotropic cracking criteria, they do not incorporate anisotropic elastic stress-fields. Therefore, a single value of Young’s modulus and Poisson’s ratio needs to be used. Since the Young’s modulus for intact tests on Opalinus shale with bedding plane angles of 0o , 30o , 45o , 60o (shown in Chapter 4) is relatively constant, ranging between 1,043 and 1,150 MPa, the Young’s modulus (1,080 MPa) and Poisson’s ratio (0.27) determined from the 0o intact tests were used. However, the Young’s modulus at 90o was 4,537 MPa. This means that the stressstrain behavior at 90o inclination will likely not captured by the model. This is currently a limitation of the models.

5.2.4

FROCK Results Using The Anisotropic Criteria

The following section presents the results of the FROCK simulations conducted with both the discontinuous and continuous criteria implemented in FROCK. For each criterion, the calibrated input parameters and cracking patterns associated with the 2a-30-30(ψ) geometry at various bedding planes will be presented. Then the cracking patterns predicted for the 261

Figure 5-26: An example of a FROCK numerical instability. Numerical instabilities occurring at certain ranges of input parameters can cause FROCK to predict unusual and incorrect crack patterns. This issue occurred in both the current and previous versions of FROCK used by Bobet (1997) and Gonçalves da Silva (2009). This example shows a simulation conducted on the 2a-30-30(20) flaw pair geometry. flaw pair tested with horizontal bedding planes will shown. The ability of these FROCK models to predict cracking of flaw pairs with horizontal bedding planes will be used to compare their results to the original FROCK model. The crack initiation and coalescence stresses predicted will then be discussed. Finally, a summary will be presented along with a recommendation based on which criterion most accurately predicts cracking behavior in Opalinus shale.

Discontinuous FROCK results

The following section presents the results of the Discontinuous FROCK model. The calibrated input parameters will be presented. Then the crack pattern results for inclined bedding planes as well as horizontal bedding planes will be shown and compared to the experimental results. Finally, the crack initiation and coalescence stresses predicted by the model will be discussed. 262

Figure 5-27: Calibration steps used to determine the input parameters for Opalinus shale used in the anisotropic strain-based FROCK developed in this study. The calibration process consisted of using values of ro , εθcrit , γθcrit and µ from the isotropic calibration and then varying the minimum reduction factors Rε−min and Rγ−min . The cracking and coalescence shape of tests with 2a-30-30 flaw pairs and various bedding planes are used to determine the most accurate reduction factors. In the final step, the crack patterns of all geometries are used and minor changes in εθcrit , γθcrit and µ are made.

263

Calibrated Input Parameters For the Discontinuous FROCK model, the reduction factors (Rε , Rγ ) needed to be calibrated. The method used for calibrating the reduction factor input parameters was presented in section 5.2.3. As previously discussed, this process used the intact material parameters determined for the original Isotropic FROCK model shown in section 5.1 as a starting point. In the final step, the intact material parameters were adjusted slightly as part of the iterative calibration process. The material properties used in the Discontinuous FROCK model, as well as those used in the isotropic FROCK model, are presented in table 5.6. The original input parameters are essentially the same as the ones used in the Isotropic FROCK model (Refer to 5.6). A slightly higher γcrit of 0.0384 was used because it showed better results. Reduction factors of 0.70 for Rε and 0.73 for Rγ were also determined from the inclined bedding calibration process. Table 5.6: Discontinuous FROCK Model Input Parameters Used for Opalinus Shale Model Type Isotropic Discontinuous

εcrit -0.0112 -0.0112

γcrit 0.0383 0.0384

ro 0.022 0.022

µ 0.74 0.74

Rε 0.70

Rγ 0.73

Cracking Behavior As stated before, FROCK (both the original and the version with bedding planes) is very sensitive to material parameters and at certain input values it can become unstable, predicting unusual or incorrect cracking patterns (Refer to figure 5-26). The Discontinuous FROCK model had a similar instability issue when the crack initiating at a tip was in the direction of the bedding planes. For this reason, the initial crack at the flaw tip includes only the critical strains of the intact material (εintact ,γintact ), used in the Isotropic FROCK model. Once initial cracking occurs at the flaw tips, the discontinuous criterion then applies. The crack patterns predicted for inclined bedding planes by the Discontinuous FROCK model are presented in figure 5-28. The predictions for flaw pairs with horizontal bedding 264

planes are presented in figures 5-29 and 5-30. Using the evaluation method established previously, the behavior was assessed systematically and is summarized in table 5.7. The Discontinuous FROCK model showed acceptable results for almost all aspects of cracking in the flaw pairs with inclined bedding planes (Refer to figure 5-28 and table 5.7). Only the indirect coalescence and cracking mode of the flaw pair geometry with 90o could not be captured. The Discontinuous FROCK model predicted only shear cracks for 2a-30-30(90), in contrast to the exclusively tensile behavior observed in the experiments. As stated before, FROCK is still not capable of capturing indirect coalescence so this was expected. The results for the flaw pair with horizontal bedding planes were essentially the same as those observed by the Isotropic FROCK model. It is very important that the Discontinuous FROCK model is capable of predicting cracking along bedding planes as well as retaining the appropriate crack patterns for flaw pairs with horizontal bedding planes. Table 5.7: Evaluation of Discontinuous FROCK Model Geometry 2a-0-0 2a-30-0 2a-45-0 2a-60-0 ψ=0o 2a-75-0 2a-30-30 2a-30-60 2a-30-90 2a-30-30(30) ψ ̸= 0o 2a-30-30(60) 2a-30-30(90) Total X/X+ x = Poor Match

Initial Crack X X x X X XX X X XX X XX 8/11

X= Average Match

Final Pattern x x XX X XX X X x XX XX x 7/11

Crack Mode x X XX X X X X X X X x 9/11

XX= Good Match

Crack Initiation and Crack Coalescence Stresses As stated before, the Discontinuous FROCK model restricted the initial cracking at the flaw tips from occurring along the bedding plane direction. This showed more accurate cracking 265

Figure 5-28: Discontinuous FROCK model bedding plane results for Opalinus shale. Experimental test results are shown on the left side and the Discontinuous FROCK model results are shown on the right. "Snapshots" are shown for each geometry (crack initiation = 1st stage and coalescence = 2nd stage).

pattern results; however, the predicted crack initiation stress was therefore not dependent on bedding plane angle. For the purpose of determining the effect of bedding plane angle on the crack initiation stress, a separate version of the code allowing crack initiation to occur along the bedding planes was used to determine the effect of bedding plane angle on the crack initiation stress (figure 5-31). Currently, the Discontinuous FROCK model can predict either the cracking patterns or crack initiation stress based on bedding plane angle, but not both simultaneously. 266

Figure 5-29: Discontinuous FROCK model results for Opalinus with coplanar flaw pairs and horizontal bedding planes. See figure 5-28 for a detailed explanation.

267

Figure 5-30: Discontinuous FROCK model results for Opalinus with stepped flaw pairs and horizontal bedding planes. See figure 5-28 for a detailed explanation. The crack initiation stress for the Discontinuous FROCK model showed a slight decrease at inclined bedding plane angles (ψ=36o -85o ) with a minimum at approximately 60/61o . However, the minimum crack initiation stress at inclined bedding planes (60o ) predicted by the Discontinuous FROCK was not nearly as low as the minimum crack initiation stress observed in the experimental tests. It is likely that higher reduction factors would be necessary to reduce the minimum crack initiation stress. However, these parameters were calibrated based on crack patterns, and crack patterns were typically not acceptable at lower reduction factors. The coalescence stresses of the Discontinuous model are shown in figure 5-32. Similar to the process conducted for the Isotropic FROCK model, the coalescence was determined at one degree bedding angle increments and only the tests which produced coalescence were plotted (Refer to figure 5-32). The coalescence stress results showed a relatively constant 268

Figure 5-31: The crack initiation stress predicted by the Discontinuous FROCK model for various bedding planes compared to the experimental results. Simulations were conducted at one degree angle increments. coalescence stress (14.2 MPa) until a bedding angle of about 27o . Between bedding angles of 28o and 38o there are is a large scatter in the coalescence stress. It is unknown why this occurs, it could be due the bedding plane angle being close to the flaw angle (30o ) and interacting with coalescence. After 38o the coalescence stress has a smooth reduction as bedding angle increases. At higher bedding angles, likely where indirect coalescence would be observed in the experimental tests, coalescence does not occur. Since FROCK cannot predict indirect coalescence, this result is expected for these angles. The crack initiation and coalescence results of the Discontinuous FROCK are be summarized in table 5.8. The Discontinuous FROCK model showed reasonable crack initiation and coalescence stress predictions, with the exception of the crack initiation stress for the inclined bedding plane geometries and the coalescence stresses for 2a-30-60, 2a-30-30(30) and 2a-30-30(60). The coalescence stresses for 2a-30-60 and 2a-30-30(30) were shown to be at bridging and bedding angles which showed oddly variable or high coalescence stresses in FROCK. As stated before, coalescence stresses predicted in FROCK are highly sensitive to small variations and can be difficult to accurately predict. For the flaw pairs with inclined bedding planes, the Discontinuous FROCK model was not 269

Figure 5-32: The crack coalescence stress predicted by the Discontinuous FROCK model for various bedding planes compared to the experimental results. Simulations were conducted at one degree angle increments. Only geometries producing coalescence are shown. capable of showing large enough drops in crack initiation and coalescence stress. Therefore, the crack initiation and coalescence stresses for 2a-30-30(30), 2a-30-30(60) and 2a30-30(90) were over-predicted. This is likely an indication that the reduction factors used were too high; however, as stated before, lower reduction factors showed incorrect crack patterns. Summary In summary, the Discontinuous FROCK model shows good results for predicting the crack pattern for both the flaw pairs with inclined bedding planes and horizontal bedding planes. More importantly, the results on flaw pair with horizontal bedding planes from the Isotropic FROCK model remain basically the same. The Discontinuous FROCK model also showed generally acceptable results for crack initiation and coalescence stress for flaw pairs with horizontal bedding. However, it overpredicted the crack initiation- and coalescence stresses for the flaw pairs with inclined bedding planes. Although the minimum crack initiation and coalescence stress was observed at inclined bedding planes (38o - 85o ), the absolute minimum crack initiation and coalescence stresses were not as low as those observed in the experimental test results. 270

Table 5.8: Summary of Discontinuous FROCK Model and Experimental Test Results Geometry

Coplanar

Stepped

Inclined ψ

2a-0-0 2a-30-0 2a-45-0 2a-60-0 2a-75-0 2a-30-30 2a-30-60 2a-30-90 2a-30-30(30) 2a-30-30(60) 2a-30-30(90)

FROCK Output [MPa] CI Coalescence 9.7 (No Coal.) 7.8 (No Coal.) 8.3 12.0 10.9 10.9 16.2 19.8 7.2 14.2 7.3 21.0 7.9 (No Coal.) 7.2 20.3 6.0 * * 7.2 7.2 (No Coal.)

Experimental Test Avg. [MPa] CI Coalescence 10.1* 12.1 10.1* 13.9 8.5 14.1 10.8 14.0 15.7 19.1 11.7* 12.5 7.3* 9.2 6.0 12.8 4.6 6.6 2.0 2.3 4.6 7.7

CI = Crack Initiation Stress *: crack initiation was detected using still images from the high speed camera. These lower resolution images, when compared to the high resolution camera images, have a lower precision of crack detection and could thus lead to a delay in crack detection and an over-estimation of crack initiation stresses. **: Crack initiation determined using a separate version of the Discontinuous FROCK code, which allowed for initial cracking along bedding planes. Continuous FROCK results

The following section presents the results of the Continuous FROCK model. As before, the calibrated input parameters will be presented as well as the results for inclined and horizontal bedding planes. These results will be compared to the experimental results, and then the crack initiation and coalescence stresses predicted by the model will be discussed. Calibrated Input Parameters As stated before, the reduction factors needed to be calibrated for the Continuous FROCK model. This process started with the parameters determined by the Isotropic FROCK model and was iterative, re-evaluating the intact critical strains (εcrit ,γcrit ) and a new crack friction coefficient (µ) as well. The results of the calibration conducted for the Continuous FROCK model are presented in table 5.9. Due to poor results for flaw pair with horizontal bedding planes, substantial changes were necessary for the εcrit and µ parameters (Refer to table 271

5.9). Table 5.9: Continuous FROCK Model Input Parameters Used for Opalinus Shale Model Type Isotropic Continuous

εcrit -0.0112 -0.0105

γcrit 0.0383 0.0384

ro 0.022 0.022

µ 0.74 0.70

Rε 0.70

Rγ 0.73

Cracking Behavior The crack patterns predicted by the Continuous FROCK model are presented in figure 533. The predictions conducted on flaw pairs with horizontal bedding planes are presented in figures 5-34 and 5-35. Using the evaluation method established previously, the behavior were assessed systematically and are summarized in table 5.10. The Continuous FROCK model showed very good crack pattern results for the flaw pairs with inclined bedding planes (Refer to figure 5-33). Crack propagation direction (pattern), -coalescence and -mode (tensile/shear) were acceptable for almost every inclined bedding plane geometry. Only the indirect coalescence observed in 2a-30-30(90) was not captured, a coalescence type FROCK is not capable of predicting. The coalescence behavior of 2a30-30(60) was predicted particularly well, with zones of tensile/shear cracking and then shear cracking along the bedding planes. However, the Continuous FROCK crack pattern results for horizontal bedding planes were not as good (Refer to figures 5-34 and 5-35). Typical coalescence patterns, which were not captured before, such as those for 2a-0-0, 2a-30-0 and 2a-30-90, still cannot be predicted. However, other coalescence patterns, which were predicted by the Isotropic and Discontinuous FROCK models, could not be predicted by the Continuous FROCK model, such as 2a-75-0 and 2a-30-60. In fact only 3 out of the 8 horizontal bedding plane geometries observed coalescence in the Continuous FROCK model. Additionally, much more shear cracking was predicted in the Continuous FROCK model compared to the experimental results, which mainly observed tensile cracking (Refer to figures 5-34 and 5-35). In summary, the Continuous FROCK model was capable of predicting cracking of flaw pairs with inclined bedding planes well but could not accurately predict the behavior of 272

flaw pairs with horizontal bedding (Refer to table 5.10).

Figure 5-33: Continuous FROCK model bedding plane results for Opalinus shale. Experimental test results are shown on the left side and the Continuous FROCK model results are shown on the right. "Snapshots" are shown for each geometry (crack initiation = 1st stage and coalescence = 2nd stage). Crack Initiation and Crack Coalescence Stresses For the Continuous FROCK model the crack initiation and coalescence stresses were determined (figures 5-36 and 5-37). The crack initiation stresses for different bedding planes predicted in the Continuous FROCK model did not accurately match the experimental results (Refer to figure 5-36). Specifically, the Continuous FROCK model actually showed a higher crack initiation stress between 20o -63o bedding plane angles and a lower crack ini273

Figure 5-34: Continuous FROCK model FROCK model results for Opalinus shale with coplanar flaw pairs and horizontal bedding planes. See figure 5-33 for a detailed explanation.

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Figure 5-35: Continuous FROCK model results for Opalinus with stepped flaw pairs and horizontal bedding planes. See figure 5-33 for a detailed explanation. Table 5.10: Evaluation of Continuous FROCK Model Geometry 2a-0-0 2a-30-0 2a-45-0 2a-60-0 ψ=0o 2a-75-0 2a-30-30 2a-30-60 2a-30-90 2a-30-30(30) o ψ ̸= 0 2a-30-30(60) 2a-30-30(90) Total X/X+ x = Poor Match

Initial Crack X X x X X XX X X X X X 10/11

X= Average Match

275

Final Pattern x x XX X x XX x x X XX x 5/11

Crack Mode x X X x x XX X X X XX X 8/11

XX= Good Match

tiation stress at bedding plane angles less than 20o and greater than 63o . This clearly does not follow the decrease in crack initiation stress at inclined bedding plane angles, with a minimum at 60o , that was observed in the experimental tests. One possible explanation for the reduction in crack initiation stress observed at higher and lower bedding plane angles is possibly the low tensile reduction factor (Rε ) compared to the shear reduction factor (Rγ ). Since the Continuous FROCK model assumes some reduction in the material property in all directions (except perpendicular to the bedding direction), as the bedding plane orients towards the direction of maximum tensile or shear strain there is a reduction in crack initiation stress. It was determined in the model that the initial cracking was shear between 20o -63o bedding plane angles. Outside of these angles (0o -19o and 64o -90o ) the initial cracking was tensile (Refer to figure 5-36). Therefore, the lower crack initiation stresses observed at high and low bedding plane angles means that the tensile crack reduction factor (Rε ) used was too low compared to the shear crack reduction factor. Additionally, the initial shear cracking occurring at bedding plane angles between 20o -63o mean that a lower shear crack reduction factor (Rγ ) was needed to produce the lower minimum crack initiation stress necessary at these angles. However, the crack patterns observed for the Continuous FROCK model were not satisfactory at lower reduction factors. The coalescence stresses were predicted for the Continuous FROCK model as well (figure 5-37). These coalescence results typically showed a large variation in coalescence stress, especially at low bedding angles (0o -20o ) (Refer to figure 5-37). However, the average trend of these coalescence stresses appeared to follow a trend similar to the experimental results. The coalescence stress at 0o is the highest and it decreases to a minimum at about 60o . Finally, a summary of the crack initiation and coalescence stresses predicted by the Continuous FROCK model for each flaw pair geometry tested is presented in table 5.11. As shown before, most tests did not show coalescence. The tests which did show coalescence appear to match the experimental values. As also shown before, the crack initiation and coalescence stresses were over-predicted for the 2a-30-30(60) geometry. 276

Figure 5-36: The crack initiation stress predicted in the Continuous FROCK model for various bedding planes compared to the experimental results. Simulations were conducted at one degree angle increments. The bedding planes angles showing initial shear and tensile cracking in the Continuous FROCK model are labeled.

Figure 5-37: The crack coalescence stress predicted in the Continuous FROCK model for various bedding planes compared to the experimental results. Simulations were conducted at one degree angle increments. Only geometries producing coalescence are shown. Summary In summary, the Continuous FROCK model predicted the crack patterns of flaw pairs with inclined bedding planes very well. However, the crack patterns observed in the model for flaw pairs with horizontal bedding planes were below average. Crack initiation and 277

Table 5.11: Summary of Continuous FROCK Model and Experimental Test Results Geometry

Coplanar

Stepped

Inclined ψ

2a-0-0 2a-30-0 2a-45-0 2a-60-0 2a-75-0 2a-30-30 2a-30-60 2a-30-90 2a-30-30(30) 2a-30-30(60) 2a-30-30(90)

FROCK Output [MPa] CI Coalescence 8.9 (No Coal.) 6.7 (No Coal.) 7.4 14.4 9.9 9.9 14.5 (No Coal.) 7.2 13.7 6.7 (No Coal.) 7.1 (No Coal.) 6.4 7.7 6.0 6.1 5.1 (No Coal.)

Experimental Test Avg. [MPa] CI Coalescence 10.1* 12.1 10.1* 13.9 8.5 14.1 10.8 14.0 15.7 19.1 11.7* 12.5 7.3* 9.2 6.0 12.8 4.6 6.6 2.0 2.3 4.6 7.7

CI = Crack Initiation Stress *: crack initiation was detected using still images from the high speed camera. These lower resolution images, when compared to the high resolution camera images, have a lower precision of crack detection and could thus lead to a delay in crack detection and an over estimation of crack initiation stresses. coalescence stress predictions were adequate for most flaw pair geometries excepted the flaw pairs oriented at 60o bedding planes [2a-30-30(60)].

5.2.5

Discussion

This discussion section presents a comparison of the Discontinuous and Continuous FROCK models and is subdivided into two parts; a comparison of the crack patterns and a comparison of the crack initiation/coalescence stresses.

Crack Pattern Comparison

A comparison between the crack pattern results of the Discontinuous and Continuous FROCK models is summarized in table 5.12. In general, both the Discontinuous and Continuous FROCK models showed similarly acceptable initial cracking, final crack pattern and cracking modes for the flaw pair geometries with inclined planes (Refer to table 5.12). 278

The Continuous FROCK model had some issues predicting the crack patterns for flaw pairs with horizontal bedding planes and was only capable of correctly predicting three out of the eight geometries. In contrast, the Discontinuous FROCK model showed much better results for the flaw pairs with horizontal bedding planes. In fact, the Discontinuous FROCK model showed the same amount of acceptable cracking behavior for the horizontal bedding plane geometries as the Isotropic FROCK model (Refer to table 5.12). The ability of the Discontinuous model to predict cracking for flaw pair geometries with horizontal and inclined bedding plane is important since the flaw pair geometries with inclined bedding planes are only a subset of the flaw pair geometries tested in Opalinus shale. Table 5.12: Evaluation of FROCK Models (Total No. Xor XX) Model Horizontal Bedding Inclined Bedding

Isotropic Discontinuous Continuous Discontinuous Continuous

Initial Crack 7/8 7/8 7/8 3/3 3/3

Final Pattern 5/8 5/8 3/8 2/3 2/3

Crack Mode 7/8 7/8 5/8 2/3 3/3

The difference in the two models (Continuous and Discontinuous) results is likely due to the calibration method used. Both models used the input parameters calibrated for the Isotropic FROCK model. The Isotropic FROCK model calibrated the input parameters based on the tests with horizontal bedding planes. When the bedding planes are horizontal for the 2a-3030 geometry, the tensile and shear strain in the direction of the bedding plane (horizontal) is likely small or negligible. Therefore, the Isotropic FROCK model could be calibrated with the assumption that the material is isotropic and cracks did not preferentially propagate in the horizontal direction. This was confirmed in the experiments as crack propagation along bedding planes was rarely observed in the experimental tests with horizontal bedding planes. This assumption that the original four input parameters (εθcrit , γθcrit , ro , µ) are independent of bedding planes affects the Continuous and Discontinuous FROCK models differently. Since the Discontinuous FROCK models cracking either as cracks in the intact material or 279

cracks along a bedding plane, using the parameters calibrated using the Isotropic FROCK model was a good baseline for the horizontal bedding plane case, where cracks didn’t propagate along this bedding plane. However, the Continuous model assumes a continuous change in strength based on the direction. This means that even in the horizontal bedding plane case there is a reduction in strength at angles slightly offset to the horizontal (θ ̸=0). The solution to this issue is not a simple one. The calibration process for the Continuous FROCK model likely needs to have all five calibrated input parameters modified and calibrated at the same time (εcrit ,γcrit , µ, Rε−min and Rγ−min ). Ideally, calibrating all five input parameters at the same time would improve both models; however, it is likely that this is more important for the Continuous FROCK model which had trouble properly modeling flaw pairs with horizontal bedding planes. Unfortunately, the current process of calibrating would not be sufficient to optimize all of these parameters simultaneously. Using a bruteforce search technique, similar to the procedure previously shown for (εcrit ,γcrit ), would take a significant amount of time and effort. For example, conducting ten increments for each variable, for each of the five variables, would result in 100,000 simulations (105 ). Visually analyzing the cracking patterns of 100,000 simulations is not feasible.

Crack Initiation and Coalescence Stresses

For all of the horizontal bedding plane geometries both models showed generally acceptable crack initiation and coalescence stresses (for geometries in which coalescence was predicted). However, both models had difficulty predicting the minimum crack initiation stress at inclined bedding planes. Additionally, neither model was capable of accurately predicting the difference between crack initiation stresses at 0o and 90o bedding plane angles. Both models showed decreasing coalescence stresses at inclined bedding planes (30o -80o ). Also both models showed very high coalescence stresses for some geometries [2a-30-60, 2a-30-30(60)]. It should be noted that large deviations between coalescence stress observed in the experiment value and those predicted by the FROCK model were also observed by 280

Bobet (1997). In some cases, the coalescence stress predicted in the stress-based FROCK model by Bobet (1997) were 60% higher than the experimental values. In this study, some of the predicted coalescence stresses were over 100% larger than the expected experimental values. This could be due to a slightly higher µ value used in the Isotropic and Discontinuous FROCK model (µ=0.74) compared to that used by Bobet (1997) (µ=0.70), since the friction coefficient of newly formed cracks typically increases the stress required to propagate new cracks and can cause higher coalescence stresses.

5.3

Future Considerations

The following section will address the future considerations that need to be taken in FROCK to improve the model results. This includes the issue of numerical instabilities, the anisotropic strain-field analysis and the calibration method used.

5.3.1

Numerical Instabilities and Code Crashes

One concern with the Discontinuous FROCK model was the inability of the model to predict crack initiation along a bedding plane in the Discontinuous FROCK model due to numerical instabilities. Numerical instabilities and code crashes were also observed in some cases in the Isotropic and Continuous FROCK models as well. Additionally, these odd crack patterns were shown in previous FROCK studies (Gonçalves da Silva, 2009). Therefore, it is likely a fundamental issue in the FROCK code. Not only is this an issue for predicting crack initiation along bedding planes in the Discontinuous FROCK model but it also could be preventing more accurate calibration of input parameters due to incorrect crack patterns being shown at certain input parameters. A systematic study needs to be conducted on the instabilities observed in FROCK. 281

5.3.2

Anisotropic Strain Field Analysis

The isotropic strain field used for this model is a major assumption considering anisotropic rock, especially Opalinus shale has anisotropic elasticity. A modified version of FROCK using anisotropic stress-fields was developed by Bobet and Martin (2014). That version of FROCK incorporated anisotropic elastic properties to determine the effect of anisotropy on the stress concentrations at the flaw tips. However, that model did not incorporate cracking processes. Due to the difficulty of combining this stress based model into the strain-based Discontinuous and Continuous FROCK models, the anisotropic strain-field was not used in this study. In the future, a model which includes both an anisotropic cracking criterion as well as an anisotropic strain-field should be developed. This would likely improve the accuracy of crack patterns as well as the crack initiation and coalescence stresses for geometries with inclined bedding planes. It is likely that the parameters would be affected by an anisotropic strain-field and they would need to be re-calibrated.

5.3.3

Calibration Method

A more rigorous calibration method, based on the semi-automated one proposed in this thesis, should be developed to incorporate more computational and optimization techniques in the calibration of the FROCK model parameters. The crack pattern analysis could be combined with an automatic coalescence detection shown previously to determine how well the model predicts the cracking compared to experimental tests. This could include where cracks occur (spatial location), what mode they propagate in (tensile/shear), where and at what stress coalescence occurs, and at what angles do the cracks propagate. Using numerical methods, these metrics would be easily implemented into an automatic calibration code. Finally, future calibration methods should include some type of feedback optimization. Currently, the calibration methods require that many simulations be done (brute-force search), which include many incorrect crack patterns. However, with more sophisticated 282

techniques, the calibration process could modify a parameter automatically, determine its effect on the output observed and then adjust the next parameter modification accordingly. This feedback controlled process would greatly improve the speed and hence the accuracy of the parameter calibration. This type of method would require that the issue of numerical instabilities be solved first because these instabilities could occur in specific parameter ranges causing the automated calibration process to not work properly.

5.3.4

Anisotropic Failure Criterion

Previously developed anisotropic failure models, such as the modified AMN criterion developed by Pei (2008), have shown that there are two forms of anisotropy; anisotropy in the failure of the intact material and anisotropy from a plane of weakness failure. These two forms of anisotropy are analogous to the continuous and discontinuous criteria, respectively. The FROCK model could be better improved by incorporating both the continuous and discontinuous criteria into the same model (figure 5-38). A combined discontinuous/continuous FROCK model would have a continuous reduction in strength towards the bedding plane angle, as well as a discretely reduced strength along the bedding planes (Refer to figure 5-38).

Figure 5-38: Definition of the proposed combined continuous/discontinuous criterion. This combined criterion would assume a continuous trigonometric reduction in critical strain with respect to the angle of interest (θ ), as well as a discrete reduction in the critical strains when the angle of interest (θ ) is in the bedding plane direction (ψ). 283

5.4

Summary

In this chapter three variations of the strain-based FROCK model were used. The original strain-based Isotropic FROCK model developed and used by Gonçalves da Silva (2009) was applied to Opalinus shale with flaw pairs and horizontal bedding planes. This Isotropic FROCK model was modified to include two different anisotropic failure criteria based on the direction of bedding planes. These models were called the Discontinuous and Continuous FROCK models. The Discontinuous FROCK model assumed two sets of critical strains; one set for the intact material and one set in the direction of bedding planes. The Continuous FROCK model assumed a continuous variation of critical strains with respect to bedding plane angle, with a minimum critical strain in a direction of the bedding planes and a maximum perpendicular to the bedding plane direction. From the results on all three models the following conclusions for each model can be drawn: Isotropic FROCK Model ∙ The Isotropic FROCK model was capable of predicting most crack patterns correctly for Opalinus shale with horizontal bedding planes. ∙ Most crack initiation and coalescence stresses predicted by the model were comparable to the experimental values. ∙ Some coalescence stresses predicted by the model were abnormally high compared to the others. This was observed in the 2a-30-60 specimen (Model: 20.29 MPa, Experiment: 9.17 MPa). ∙ The Isotropic FROCK model was not capable of predicting cracking along bedding planes and, thus, could not predict the crack patterns in Opalinus shale with flaws and inclined bedding planes. Discontinuous FROCK Model ∙ The Discontinuous FROCK model was capable of predicting the crack patterns of Opal284

inus shale with flaw pairs and inclined bedding planes. ∙ The model was also capable of predicting the crack patterns for Opalinus shale with flaw pairs and horizontal bedding planes. ∙ The crack initiation and coalescence stresses predicted by the Discontinuous FROCK model were comparable to the experiments with horizontal bedding planes but were over-predicted for the inclined bedding plane geometries. Continuous FROCK Model ∙ The Continuous FROCK model was capable of predicting the crack patterns of Opalinus shale with flaw pairs and inclined bedding planes very well. ∙ The model showed below average predictions of the crack patterns for flaw pairs with horizontal bedding planes, predicting the correct coalescence pattern in only 3 out of the 8 geometries. ∙ The crack initiation stresses for horizontal bedding planes were well predicted; however, crack initiation stresses for inclined bedding plane geometries were not well predicted. ∙ The crack coalescence stresses predicted by the Continuous FROCK model were comparable to all experiments, except 2a-30-30(60). Recommendation From the summary of all of the results (crack pattern, crack initiation stress, coalescence stress), the Discontinuous FROCK model is the only model showing satisfactory results for both horizontal and inclined bedding plane geometries. The Isotropic or Continuous FROCK models were not able to model both horizontal and inclined bedding plane geometries individually. Therefore, the Discontinuous model is the recommended model to predict crack patterns in Opalinus shale.

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286

Chapter 6 Digital Image Processing This chapter discusses a image analysis technique called Digital Image Correlation (DIC). A brief background on the DIC method and some studies using DIC in rock testing will be presented. A uniaxial test applying the DIC method to gypsum will be shown, and then it will be applied two tests on Opalinus shale previously presented (Refer to chapter 4). The results of this analysis will be discussed and then compared to the associated input parameters in the FROCK model.

6.1

Introduction

One of the biggest problems with numerical models is accurately calibrating and determining the input parameters. In many models these parameters may not have a direct relation to physical behavior and therefore, are difficult to extract from experimental testing. Many of the input parameters in FROCK are analogous to the micro-scale mechanical behavior, such as the critical tangential and shear strains at the tip of a crack. However, measuring and determining these material parameters, which are localized at such small scales, can be extremely difficult. Previously used experimental techniques such as, crack mouth opening devices (CMOD) and surface mounted strain gauges, require pre-existing knowledge of 287

the exact crack propagation direction and can potentially influence the cracking behavior. Recent developments in computational image processing have made it possible to measure surface displacements and calculate localized strain fields from imagery captured throughout the test. There are two popular types of displacement based image processing methods used in experimental testing. One is called Particle Image Velocimetry (PIV) and is used mostly for experiments with fluid flow. PIV focuses on particle velocities and flow field contouring. One of the most common and easily accessible PIV codes is called PIVLAB. The other technique is called Digital Image Correlation (DIC) and is typically used to determine strain fields observed in solid materials. Although both can be applied to tests conducted on solid bodies, such as rock, DIC software typically incorporates more functions to determine the strain field, as opposed to pixel velocity; therefore, DIC will be the primary image processing technique used in this study.

6.1.1

Background

This section provides a brief review of the Digital Image Correlation (DIC) technique and discusses some previous studies, which incorporated DIC with tests on rock. One of the first studies to propose a digital image technique to determine displacements was conducted by Peters and Ranson (1982), which utilized laser speckle interferometry to determine subpixel displacements. Based on this digital image technique, the Digital Image Correlation (DIC) method was proposed by Buck et al. (1989). Many of the early experiments conducted using DIC were done on homogeneous elasto-plastic materials such as metal (Peters and Ranson, 1982; Toussaint et al, 2008). Only recently has DIC been implemented on brittle rock materials. DIC has been used in rock testing in several recent studies (Lin et al, 2010; Nguyen et al., 2011; Zhao et al., 2015). The main purpose of many of these studies was to determine the existence of a fracture process zone before crack propagation (figure 6-1) or to detect cracking at the sub-pixel scale (figure 6-2). These studies have shown some promising 288

results for DIC analysis applied to rock testing. The next section will discuss the basis of the DIC method and some considerations that should be addressed before applying it to a material which is fracturing.

Figure 6-1: DIC analysis conducted on Berea sandstone, specifically near the notch of a three-point bending test. (a) The pre-test image and (b) the horizontal displacement at 98% peak load from the DIC analysis are shown. The fracture process zone (FPZ) is shown at the tip of the flaw in the DIC analysis. Modified from (Lin et al., 2010)

Figure 6-2: DIC analysis conducted on Neapolitan Fine Grained Tuff. This analysis determines the crack location at sub-pixel scale. (a) The original image which shows a crack propagating from the bottom. (b) The displacement contour surface for this image, showing the extension of the crack well into the image. (c) The locations of high displacement are determined on the image and then (d) the crack is traced with the DIC analysis. Modified from (Nguyen et al., 2011)

289

6.1.2

DIC Method Fundamentals

DIC fundamentally associates the movement of pixels within a region of interest (subset) to another location based on grayscale pixel intensity changes (figure 6-3). From these computations, displacements at each pixel, or region of interest, can be determined for each image. Typically DIC software uses the first image in a series as the reference image and then the displacements for each consecutive image in the sequence are calculated based on that first reference frame.

Figure 6-3: The principle of DIC analysis. DIC analysis uses a subset area of a reference image (left) and then determines the displacement and deformation of that subset in the subsequent images (right). (Pan, 2009) Theoretically, the identification and determination of the deformation in the image is based on a correlation coefficient. There are several different correlation coefficients used in a DIC analysis; however, the most common one used is called the zero-mean normalized cross correlation (ZNCC) coefficient. The ZNCC coefficient can be described by the following expression:

¯ ∑ ( f (Xi ) − f¯) × (g(xi ) − g) ZNCC = 1 − r

i∈D

¯ ∑ ( f (Xi ) − f¯) × ∑ (g(xi ) − g)

i∈D

(6.1.1)

i∈D

Where D is the subset region of the image, f (Xi ) and g(xi ) are the grayscale intensities 290

of the reference and deformed images respectively, at pixel locations Xi and xi and f¯ and g¯ are the mean pixel intensity values for the reference and deformed images respectively. Incorporating the mean pixel intensity value accounts for variations in lighting conditions that may cause the entire image to increase or decrease the average pixel intensity. The location at which a subset location has the highest ZNCC coefficient is where it has likely deformed to. It should be noted that these pixel locations are not always integers because the deformation can occur at the subpixel level. Since DIC uses the pixel intensity changes to determine the object deformation, the technique is capable of determining sub-pixel resolution movement (Refer to figure 6-4). A simplified example shown in figure 6-4 uses a bi-linear interpolation; however, there are more sophisticated DIC techniques, which use other interpolation functions such as a bicubic spline interpolation.

Figure 6-4: Sub-pixel movement calculation in the DIC method. This simplified schematic shows three pixel movement scenarios of a single black dot in a white matrix; a full pixel shift, a half pixel shift in one direction and a half diagonal pixel shift. This example shows the gray-scale intensity of the pixel (from 0-256) with black being 256 and white being 0. This is an example of a bilinear interpolation; however, DIC can use much more complex shifting algorithms to determine the movement of pixel shapes between consecutive images.

The strain-field can be determined using the differential of the determined displacement 291

field (figure 6-5). The size of the area over which strain is calculated determines how ’smooth’ the strain field will be. The window of this calculation is also important when determining rigid body movement. Rigid body movement is identified when all of the displacement vectors move in the same direction and magnitude, thus having no differential. However, if the strain calculation window is too large, the rigid body movement of two bodies moving in different directions may be mistaken as stain. This is fundamentally an issue when determining cracking where to two rigid bodies (crack surfaces) are moving apart. The DIC method is based on continuum deformation, and this is an important concept when using DIC on rock, or fracturing in general. Discontinuous fracture surfaces may predict strains that represent crack opening and not actual material strains. It can be very difficult, and almost impossible at this point, to differentiate when a strain is associated which crack initiation at a micro-scale. For this reason, the possible effect of crack initiation on the observed strain localization should be made clear and will be addressed in the following results. The issue of crack surface displacement and process zones will be discussed later when addressing localized strains associated with crack initiation.

Figure 6-5: Strain calculation in the NCORR DIC software. The strains are calculated in the NCORR DIC software by determining the change in pixel displacements. The strain interpolation size can be increased, referred to as the "strain radius", and thus determines how smooth the strain will be. The image (a) shows the x-direction displacement field for a test conducted on gypsum with a circular hole. Part (b) shows the strain calculation (εxx ) for a point marked in the displacement field (a). Displacements are measured in pixel lengths. There are several reasons to use DIC on images taking during rock testing. One goal is to determine if localized strain fields around the flaw or crack tip can be observed and if these 292

strain fields are consistent with a process zone. Additionally, it is desirable to determine if these strain localizations are associated with crack initiation and how this compares to visual crack detection techniques. It is especially interesting to know if these DIC results are capable of determining or estimating numerical material parameters such as critical cracking strain.

6.2

Proof of Concept

In order to determine the viability of using DIC on Opalinus shale, the method was applied to a uniaxial compression test conducted on a Hydrocal-B11 gypsum specimen with a preexisting flaw. There were two goals in this test. The first, and most important goal, was to verify that the strains being calculated with the DIC software were consistent with the strains observed in the test using conventional methods. Since gypsum is a well studied material, the elastic properties of the material are well known and consistent. The second goal of this study was to determine if cracking or pre-cracking strains could be determined before cracking was observed as is possible using traditional visual analysis on the images. Once the capabilities of the process were confirmed, the process could then be used on Opalinus shale with confidence.

6.2.1

Test Setup

This test on gypsum was loaded uniaxially using the same BaldwinTM loading frame described in section 3.3.2. The same constant loading rate of 1,200 lb/min was used in this test as well. High resolution imagery of the flaw was taken periodically during the test as the specimen was loaded(figure 6-6). The surface of the gypsum specimen was originally a light gray; however, to improve the performance of the DIC software, black spray paint was misted onto the surface creating a speckled pattern (figure 6-7). This speckle pattern helped the DIC better distinguish small displacements due to the heterogeneous nature of the speckle pattern. In contrast, one could imagine a surface with a uniform color 293

intensity, which would have the same grayscale intensity at all locations in the imagery. Localized displacements would show no change in grayscale intensity and therefore, the surface would appear to remain the same.

Figure 6-6: DIC test setup for a gypsum specimen with a flaw. Imagery was taken very close to the flaw tip to get the best resolution possible.

Figure 6-7: DIC test on gypsum with a painted speckle pattern. Black spray paint was used to mist the surface of the originally gray specimen (left) to a create a black speckle patter on the surface (right) to help improve the DIC analysis results.

294

6.2.2

Strain Verification

When using a new technique it is important to verify its accuracy and understand its limitations before applying it to newly tested material. One of the first and simplest checks for the DIC method was to determine if the bulk material strain was comparable to the strain obtained from the conventional stress-strain-time results recorded by the loading frame. The NCORR DIC software used in this study was capable of outputting x and y displacement fields, as well as calculating the strain in the y, x and xy (shear) directions. In order to determine the bulk axial strain response for the test on gypsum with a single pre-existing flaw, a location far from the flaw (outside of the influence of the stress-field induced by the flaw) was identified and then strain in the axial direction (εyy ) over this area was averaged (figure 6-8). This axial strain from the DIC was then compared to the strain determined using the cross-head displacement measured during the test (Refer to figure 6-8). The DIC appears to measure a slightly lower axial strain than the strain determined from the cross-head displacement (Refer to figure 6-8). This agrees with the expectation that the strain determined from the cross-head displacement includes the machine deformation and platen deformation; therefore, is typically slightly greater than the actual axial strain in the material. Since gypsum is much softer than the stiffness of the machine this difference is not much; however, the difference does increase as the load increases and the machine deforms more. Additionally, the strain induced by the flaw was not included in the DIC calculation. The flaw would likely cause the axial strain determined from the machine displacement, which incorporates the strain of the entire specimen, to be slightly more than the strain calculated away from the flaw using DIC. In the future, it may be better to conduct the DIC analysis on the whole specimen, averaging the strains in the flaw region as well. Although the DIC strain doesn’t match the "machine strain" perfectly, these results still verify that the DIC method is capable of capturing the general bulk strain response in gyp295

sum. Overall, the response between the expected (machine) strain and the DIC is generally the same, with similar slopes, especially early in the test when the loads are lower. If the machine stiffness is the reason for the difference between DIC and the expected strain response, the DIC method might be more accurate in determining the actual axial specimen strain compared to using cross-head displacement.

Figure 6-8: Comparison of average axial stain from DIC and machine (load frame) crosshead measurements. The DIC axial strain was determined by taking the average axial direction (yy) strain over time at a location away from the flaw. The experimental axial strain was determined from the cross-head displacement of the machine. The loading rate was held constant at 1,200 lb/min.

6.2.3

Crack Detection

With the DIC method showing promising results in regards to axial strain measurements, the next step was to determine if the DIC could help identify cracks initiating from the flaw tips. The strain fields (εyy ,εxx ,εxy ) output from NCORR were transformed in order to appropriately determine the minimum principal strains (εmin ) in the material at any point 296

(figure 6-9). Locations where the minimum principal strain are negative should be associated with tensile opening. The reason for transforming the strain fields to principal directions is because tensile cracking may not be oriented in the y or x directions.

Figure 6-9: The strain transformations conducted on the DIC results. The NCORR DIC software outputs εyy ,εxx , and εxy strains. These strains were transformed into the minimum principal (εmin ), maximum principal (εmax ) and maximum shear strains (εxymax ).

Using this transformed strain field, the minimum principal strains could be determined (figure 6-10). The minimum principal strain was then averaged in an area near the tip of the flaw (Refer to figure 6-10). Additionally, the axial stress-strain response from the machine and the point at which visual cracking was observed on the images could be compared to the localized strain at the tip. There was clear drop in the minimum principal strain at the tip around 0.15 % axial strain. However, visual cracking was not observed with the high resolution imagery until about 0.5 % axial strain. This potentially shows the ability of the DIC analysis to identify crack initiation much earlier than it can be observed visually. Interestingly, the minimum principal strain at the tip starts to decrease linearly at about 0.15% axial strain (Refer to figure 6-10). This constant change could be associated with the opening of the crack face. It is difficult to determine if this behavior was due to a fracture process zone at the flaw tip or crack opening without a microscopic analysis. 297

Figure 6-10: The minimum principal strains determined using DIC at the tip of a flaw in gypsum. The minimum principal strains at the tip of a flaw were averaged over a small ’area of interest’ (left) and then compared to the stress-strain behavior determined from the machine cross-head displacement and load transducers (right). Crack initiation was visually determined much later than when the strain localization was observed at the flaw tip.

6.2.4

Summary

The Digital Image Correlation (DIC) technique was implemented using the NCORR software in an unconfined compression test on gypsum with a pre-existing flaw. The bulk axial strain results from the DIC method matched relatively well with the axial strain measurements determined from the loading machine cross-head displacement. However, the machine axial strain was slightly higher than the DIC strain, especially later in the test at high loads. This was expected since the machine strain typically includes the deformability of the platens and machine, causing it to be higher than the true specimen strain.

DIC was also used to detect the initiation of a crack in gypsum. It appears that localized strains begin to develop well before the crack was observed visually at about 25% of the visual crack detection load. It is not clear if this increase in the localized strain is due to crack initiation or a fracture process zone. Regardless, DIC appears to recognize the location of crack initiation well before visual detection methods are capable of determining crack initiation. 298

6.3

Opalinus Shale DIC Results

DIC techniques were also applied to the high resolution imagery captured during two tests on Opalinus shale. The two specimen geometries used in the DIC analysis were 2a-3090A and 2a-60-0C. One was used for an axial strain comparison(2a-30-90A) and the other was used to determine the crack initiation strains at the tip of a flaw (2a-60-0C). These tests were used because of the high quality of their high resolution imagery. It should be noted that although high speed imagery was recorded during these tests, due to the lower resolution of high speed imagery, DIC results conducted on the high speed imagery were not as consistent or clear as the results using high resolution imagery. The main goal of conducting DIC analysis on Opalinus shale was to determine the applicability of the DIC method when using Opalinus shale. These tests on Opalinus shale were not speckle painted and the ability of the software to detect strain was unknown.

6.3.1

Axial Strain Comparison

The imagery captured during the test on 2a-30-90A was used to determine the average axial strain with the DIC method (figure 6-11). The unique aspect of this DIC analysis was that it was conducted on the image of the entire specimen; therefore, the strain in the axial direction (yy) was averaged over most of the specimen surface (Refer to figure 6-11). This area included the flaw pairs. Including the flaw pairs should produce more accurate axial strain measurements since this produces the same average strain response as observed with the machine displacement transducers. Similar to the DIC tests conducted on gypsum, the strain determined by DIC on Opalinus shale was slightly lower than the strain determined by the machine cross-head displacement. This test was also loaded at a constant rate of 1,200 lb/min. It is likely that the over-estimation of the strain by the cross-head displacement is due to the fact that machine displacement incorporates machine and platen deformation. 299

Figure 6-11: Comparison of average axial stain from DIC and machine measurements observed in a 2a-30-90 Opalinus shale specimen. The DIC axial strain was determined by taking the average axial direction (yy) strain over time in an area of interest including the flaw pair. The axial strain was also determined from the cross-head displacement of the machine. The loading rate was held constant at 1,200 lb/min. The snapshot of the DIC analysis (left) corresponds to a point on the axial-time curve (right). Bands of higher strain axial strain were observed in the direction of the bedding planes.

Also, bands of higher strain axial strain were observed in the direction of the bedding planes (Refer to figure 6-11). The most obvious explanation for these areas of higher strain is that Opalinus shale is a layered system with soft layers and stiff layers. As the specimen was loaded, the local strains were higher in the soft layers than in the stiffer layers. It should be noted that the axial strain field around the flaws from the DIC method appears to match the theoretically expected strain-field. The axial compressive strains are higher at tips of the flaws (in the shear zone) and lower between the flaws (Refer to figure 6-11). This area of lower strain is expected as the flaws ’shadow’ this area and redistribute the stress. Additionally, areas of negative (extensional) axial strain are observed at the locations where wing cracks initiate. These negative strains in the axial direction are likely yy-components of extensional strains at the flaw tips, or the presence of tensile wing cracking (Refer to figure 6-11). Since the tensile strains at the flaw tip, and crack initiation, are not oriented with the principal loading direction, a component of the tensile strains at the tip is observed in the yy-direction. In summary, the bulk axial strain calculated by the DIC method in Opalinus shale is comparable to that of the axial strain determined by using machine cross-head displacement. 300

The slightly lower strain calculated in the DIC method is likely due to machine and platen strain; however, it is encouraging that the DIC method was capable of calculating reasonable results for Opalinus shale. It appears that the heterogeneous appearance of Opalinus shale is enough for the DIC software to determine strains without using a speckle paint pattern.

6.3.2

Strain At The Flaw Tip

Lastly, the strains at the tip of a flaw of the 2a-60-0C in Opalinus shale were determined (figure 6-12). In order to get the best DIC results, the region at the tip was cropped from the test images (Refer to figure 6-12). This helped to improve (reduce) the size of the subset regions and increases the precision of the results. Computationally it can take a long time, and the program can become unstable if small subset regions are used on large (high resolution) images. Since determining the strain at the tip was the purpose of this analysis, a small subset region was necessary to better capture the strains. Eighteen images over the entire test were used in the analysis.

Figure 6-12: Opalinus shale specimen 2a-60-0C used to determine the strain at the inner tip of the left flaw. Consecutive images leading up to crack initiation were used to determine the local strains at the tip of the flaw. The minimum principal strain and shear strain at the inner tip of the left flaw were averaged 301

over the duration of the experiment and compared to the machine stress-strain response and the point when visual cracking was observed in the images (figure 6-13). Both the shear and minimum principal local strains showed an increase (localization) well before visual cracking was observed (Refer to figure 6-13). A large jump in the local strain was observed when crack initiation was visually seen in the images. For each strain (minimum principal ′

′

and shear), the point at which initial local strain occurred (εxymax , εmin ) and the point at ′′

′′

which a large jump in the local strain (εxymax , εmin ) occurred, were identified (Refer to ′

′

figure 6-13). The initial local strain (εxymax , εmin ) was defined by the point when the local strain began to deviate from bulk strain in the material. As stated before, it is not known if the behavior of local strains at the tip is associated with a fracture process zone or crack propagation. However, it could be that these observed strains may be associated with micromechanical material properties of Opalinus shale. More DIC analysis is necessary to fully understand if these strains are associated with the material or just the fracture geometry.

Figure 6-13: The minimum principal strain and max shear strains determined using DIC at the tip of a flaw in Opalinus shale. The strains at the tip of a flaw were averaged over a small ’area of interest’ (left) and then compared to the stress-strain behavior determined from the machine cross-head displacement and load transducers (right). Crack initiation was visually determined around the large jump in the local strain at the flaw tip. The points ′ ′ ′′ ′′ where the initial local strain (εxymax , εmin ) and a large jump in local strain (εxymax , εmin ) occurred are also marked.

302

Comparison to Model Parameters

One of the goals of this study was to determine the applicability of DIC to determine, or estimate, FROCK material parameters from microscopic and local strain fields at the crack tip. The local strains at the flaw tip in the 2a-60-0C specimen and the Discontinuous ′

FROCK model critical strains are summarized in table 6.1. The initial local strains (εxymax , ′

εmin ) are significantly lower than the critical strains used in the FROCK model; however, the ′′

′′

local strains at visual crack initiation (εxymax , εmin ) were somewhat similar to the critical strains calibrated for the Discontinuous FROCK model (Refer to table 6.1). The local strains determined using the DIC method were about 50-60% greater than the critical strain calibrated in FROCK. It should be noted that FROCK was calibrated using the applied stress at which visual crack initiation occurred, previously referred to as the crack initiation stress. Therefore, the local strain at visual crack initiation may be most applicable to the critical strains used in FROCK.

Table 6.1: Flaw Tip Strains - 2a-60-0C Specimen in Opalinus Shale Strain Type Tangential, εmin Shear, εxymax

Initial Visual Cracking -0.14% -1.82% 0.28% 2.93%

Discontinuous FROCK Model -1.12% 1.92% (γ=2εxymax =3.84%)

Due to the fact that the FROCK model is highly sensitive to small variations in the input parameters it is likely that this kind of precision would not be satisfactory to use directly as FROCK input parameters. However, DIC analysis might be a useful tool to get a quick and easy initial estimate of the material parameters to begin calibration. It is also possible that this type of analysis could be conducted on a large number flaw tips and test specimens and that the average local strains at crack initiation may be closer to the approximate FROCK model. 303

6.3.3

Summary

In summary, two sets of DIC analysis were performed on Opalinus shale. One analysis, conducted on the 2a-30-90A specimen in Opalinus shale, calculated the average axial strain of the specimen during the test. This axial strain calculated with DIC was compared to the axial strain determined from the machine cross-head displacement. The strain calculated was typically lower than the strain determined by the machine; however, this was also observed in the DIC test conducted on gypsum. The DIC analysis conducted on the 2a-600C specimen in Opalinus shale calculated the local strains at one of the flaw tips. Local strains at the tip increased significantly well before visual cracking was observed. At visual cracking the local strains at the tip showed an large jump, likely due to crack propagation and opening. These local strain averages at crack initiation were compared to the critical strains calibrated for the Discontinuous FROCK model. The local strains calculated by the DIC method were about 50-60% higher than the critical strains calibrated for FROCK. It appears that although the DIC method calculates strains at the flaw tip on the same order of magnitude as the critical strains used in FROCK, they cannot be used directly in the FROCK model.

6.4

Summary and Further Considerations

This chapter discussed the use of DIC analysis on a series of tests in gypsum and then Opalinus shale. In both materials, the axial strain can be calculated and is typically lower than the strain determined using machine cross-head displacement. This difference between the DIC and machine axial strains increased as the test proceeded. The likely reason for this is that the DIC analysis is capable of determining the actual strain in the specimen better than the machine cross-head displacement, which includes the machine and end platen deformability. Additionally, local strains at the flaw tip were determined. Both materials observed strain 304

localization well before visual cracking was observed using conventional manual image analysis. In Opalinus shale, a large jump in strain localization was observed at the point in which visual cracking was observed. These strains were compared to the critical strains used in the FROCK model; however, the strains calculated from the DIC method were typically higher than the critical strains in FROCK. It is likely that the DIC method needs more refinement or additional analysis to determine if the critical strains at the tip of a flaw can be estimated for the FROCK input parameters.

305

306

Chapter 7 Conclusions This thesis has shown that the cracking processing in rock can be very complex. When studying an anisotropic rock, such as shale, these cracking processes may become even more complicated as they are affected by naturally occurring bedding planes. Many previous studies have researched the effect of bedding plane orientation on the strength of anisotropic rocks. However, there is very little research that fully characterizes the cracking processes (crack initiation, -propagation, and -coalescence) in anisotropic rocks. More specifically, it is important to understand how bedding planes can affect these cracking processes. This study presented a series experimental tests conducted on Opalinus shale. They included tests on intact specimens and tests on specimens with flaw pairs. Intact specimens were tested to determine the unconfined compressive strength, elastic modulus, Poisson’s ratio and failure mechanism of Opalinus shale at various bedding plane inclinations, 0o , 30o , 45o , 60o and 90o (to the horizontal). Specimens with flaw pairs were tested to determine the crack initiation, -propagation and -coalescence behavior with different flaw, bridging and bedding plane angles. Flaw pair specimens were subdivided into two groups; tests which varied bridging and flaw angle with horizontal bedding planes and tests which varied bedding plane angle (constant flaw and bridging angle). The results of the experimental study were systematically categorized and the observed trends were compared to 307

previous research conducted on brittle rocks. Using the experimental results, a model called FROCK (Fractured Rock) was used to predict the cracking behaviors observed. The strain-based FROCK model developed by Gonçalves da Silva (2009) was first applied to Opalinus shale with flaw pairs and horizontal bedding planes. However, geometries with inclined bedding planes that usually involve significant cracking along bedding planes could not be predicted by the previously developed strain-based FROCK model. Therefore, two directionally dependent strain criteria were developed and implemented into the model; a discontinuous cracking criteria and a continuous cracking criteria. The following sections will discuss, in detail, the conclusions from the experimental and numerical model results. Recommendations for future work will then be discuss.

7.1

Experimental

A series of experiments were conducted on intact Opalinus shale and Opalinus shale with pre-existing flaw pairs. The stress-strain and failure mechanisms of intact tests with various bedding plane orientations were determined. At all bedding plane orientations, intact Opalinus shale showed crack closure (inferred from strain hardening at low stresses) and then linear elastic behavior leading to brittle failure. The failure mechanism, and stress-strain response at failure differed with the bedding plane orientation. Tensile cracking across bedding planes was observed for intact tests with 0o bedding planes. Tensile cracks across bedding planes, as well as shear cracks along bedding planes, were observed for intact tests conducted with 30o bedding planes. For intact tests with 45o and 60o bedding planes, shear cracking along bedding planes was observed. Finally, tensile cracks along bedding planes were observed for tests conducted on intact Opalinus shale with 90o bedding planes. The unconfined compressive strength varied with bedding angle, showing a minimum at 60o and a maximum at 0o . The Young’s modulus was relatively constant for 0o , 30o , 45o , 60o bedding plane angles and approximately 4 times higher for 90o than the other bedding 308

plane angles. The unconfined compressive strength could be predicted well using the single plane of weakness theory by Jaeger (1960) and the variable strength theory by Ramamurthy et al. (1993). For tests on flaw pair specimens, observations were made on the crack initiation, -propagation, and -coalescence processes. Most tests on Opalinus shale with flaw pairs and horizontal bedding planes (80%) showed tensile wing cracks (type I) as the first cracks to initiate at the flaw tips. A new crack type (type 4 tensile) not observed in previously tested material could also be observed (40% of horizontal bedding plane tests). Additionally, several flaw pair geometries (48% of horizontal bedding plane tests) showed en-echelon cracking occurring in the shear zones located near the inner and outer the tips of the flaws. Preliminary optical microscopy showed brightening near the crack tip. The crack coalescence trended from indirect to direct combined shear-tensile to direct tensile with increasing bridging or flaw angle. This crack coalescence observed in Opalinus shale matched most closely with that observed in Carrara marble tested by Wong (2008). For tests conducted on Opalinus shale with inclined bedding planes and a constant 2a-30-30 flaw pair geometry, tensile wing cracks (type I) was the first cracks to initiate in most tests (78%). This initial cracking behavior was similar to that observed for tests conducted on flaw pairs with horizontal bedding planes. However, the bedding plane orientation played a significant role in the crack propagation and coalescence behavior. The crack initiation and coalescence stresses varied with bedding angle, showing a minimum at 60o and a maximum at 0o . The crack coalescence trended from direct combined shear-tensile across bedding planes to direct combined shear-tensile with shearing along bedding planes to indirect tensile along bedding planes with increasing bedding angle (figure 7-1). Finally, the tested specimens were over-dried and their water-content/saturation was calculated. It was determined that all the specimens were tested at similar water content levels (5.2% by mass +/-0.5%). Then a resaturation technique was applied. This resaturation method was used to test resaturated specimens with flaw pairs. The resaturated specimens showed a decrease in strength compared to specimens that were simply sealed after cut309

Figure 7-1: The effect of bedding planes on cracking of Opalinus shale with flaw pairs. As the bedding plane angle (the angle between the minimum principal stress direction and the bedding planes) increased, the crack coalescence trended from direct combined shear-tensile across bedding planes to direct combined shear-tensile with shearing along bedding planes to indirect tensile along bedding planes. ting; from a crack coalescence stress of 12.06 MPa for sealed specimens to 7.38 MPa for the resaturated specimens. However, similar cracking processes were observed for both resaturated specimens and those that were sealed. It was determined that cracking processes were not affected by the saturation level.

7.2

Numerical Model

Three alternate strain-based FROCK models were presented and then used to predict the fracturing behavior in Opalinus shale. The original strain-based Isotropic FROCK model developed and used by Gonçalves da Silva (2009) was applied to Opalinus shale with flaw pairs and horizontal bedding planes. This "Isotropic" FROCK model was then modified to include two different anisotropic failure criteria based on the direction of bedding planes. 310

These models were called the "Discontinuous" and "Continuous" FROCK models. The Discontinuous FROCK model assumed two sets of critical strains; one set for the intact material and one set in the direction of bedding planes. The Continuous FROCK model assumed a continuous variation of critical strains with respect to bedding plane angle, with a minimum critical strain in the direction of bedding planes and a maximum perpendicular to the bedding plane direction. The original Isotropic FROCK model was capable of predicting most crack patterns correctly for Opalinus shale with horizontal bedding planes (table 7.1). The crack initiation and coalescence stresses predicted by the Isotropic FROCK model were also comparable to the experimental values. Some of the coalescence stresses predicted by the model were abnormally high compared to the others. However, the Isotropic FROCK model was not capable of predicting cracking along bedding planes and, thus, could not predict the crack patterns in Opalinus shale with flaws and inclined bedding planes. The Discontinuous FROCK model was capable of predicting the crack patterns of Opalinus shale with flaw pairs and inclined bedding planes (Refer to table 7.1). The model was capable of predicting the crack patterns for Opalinus shale with flaw pairs and horizontal bedding planes as well. The crack initiation and coalescence stresses predicted by the Discontinuous FROCK model were comparable to the experiments with horizontal bedding planes but were slightly over-predicted for most of the inclined bedding plane geometries. The Continuous FROCK model was capable of predicting very well the crack patterns of Opalinus shale with flaw pairs and inclined bedding planes (Refer to table 7.1). However, the model showed less than satisfactory predictions of the crack patterns for flaw pairs with horizontal bedding planes. The crack initiation and coalescence stresses for horizontal bedding planes were well predicted; however, crack initiation stresses of inclined bedding plane geometries were not as well predicted. In summary, the Discontinuous FROCK model showed the best results, capable of modeling the crack patterns of flaw pair geometries with horizontal and inclined bedding planes. Therefore, this is the recommended model for predicting cracking in anisotropic rock, such 311

Table 7.1: Summary of FROCK Model Results FROCK Model Isotropic* Discontinuous Continuous

Crack Pattern Crack Initiation Stress Crack Pattern Crack Initiation Stress Crack Pattern Crack Initiation Stress

Horizontal Bedding

Inclined Bedding

Matches well Matches well Matches well Matches well Does not match well Matches well

Does not predict at all Does not predict at all Matches well Does not match well Matches well Does not match well

*: Strain-based FROCK model developed by Gonçalves da Silva (2009). as Opalinus shale.

7.3

Recommendations for Future Work

The following section will discuss some future recommendations for this research. This includes a microscopic crack study, additional use of image processing techniques, obtaining additional shale material and modifications to the FROCK model.

7.3.1

Microscopic Cracking Behavior

A brief study of the microscopic cracking behavior was also presented in this thesis. This used an optical microscope capable of 400x magnification. Since many other studies on the cracking behavior in brittle rocks include high magnification observations using techniques such as Scanning Electron Microscopy (SEM) and Environmental Scanning Electron Microscopy (ESEM), a similar study should be conducted on Opalinus shale. It would be beneficial to partially load a specimen to the crack initiation stress and then observe the microscopic behavior at the tip of the flaw and initiation crack. This could provide some insight as to whether or not there is an observable fracture process zone occurring in Opalinus shale. 312

7.3.2

Image Processing

There is still a lot of potential for using image processing techniques on the Opalinus shale tests, such as the Digital Image Correlation (DIC) technique. These pixel-based strain calculation techniques are still relatively new in the field of rock fracture mechanics and the full capabilities of these methods are still unknown. A full study on cracking and DIC should be conducted and the effects of discontinuous edges on the strain field should be analyzed. This could verify whether or not strain fields observed at the tips of cracks in Opalinus shale are fracture process zones or strains due to crack opening. There were many tests conducted on Opalinus shale in this study that were not analyzed with the DIC technique. High speed and high resolution imagery from these tests may contain much more information if image processing techniques, such as DIC, are applied rigorously and systematically to all of the tests conducted. Additionally, these image processing techniques may be able to automatically detect cracking. This could include automatic detection of crack initiation, -propagation and -coalescence. Automatic crack detection from test imagery would greatly improve the speed of the analysis procedure, which is currently conducted visually by the researcher.

7.3.3

Shale Material

Opalinus shale was the primary shale tested in this study. Shale can vary in clay-, carbonateand quartz content, which can affect the material and cracking behavior. Additionally, the pre-consolidation pressure (depth of extraction) and bedding structure can vary between different shale. These properties can also affect the cracking behavior, specifically the cracking along bedding planes. Ideally, it would be desirable to conduct similar tests on a shale that is a reservoir rock related to hydraulic fracturing. The behavior of a rock currently used in hydraulic fracturing practices would be very applicable to the oil and gas industry. 313

7.3.4

FROCK Model

Finally, there are some improvements that could be made to the FROCK model. The FROCK model had some issues with certain parameters, which caused it to predict unrealistic cracking patterns. This type of behavior was observed in the FROCK model used by both Bobet (1997) and Gonçalves da Silva (2009). Therefore, it is likely that there is an assumption or part of the code (rounding error, variable allocation, etc.), which is causing an instability to occur at certain material parameters. This issue could potentially be preventing more acceptable results from being observed, since certain ranges of material parameters show unacceptable results that may be due to instabilities in the code. This was especially an issue when using low bedding plane reduction factors and when crack initiation occurred along the bedding plane direction in the Discontinuous FROCK model.

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Bibliography Abdulhadi, N.O. (2009) An experimental investigation into the stress-dependent mechanical behavior of cohesive soil with application to wellbore instability. PhD Thesis, Massachusetts Institute of Technology. Akono, A.T. (2013) Assessment of Fracture Properties and Rate effects on Fracture of Materials by Micro Scratching: Application to Gas Shale. Phd Dissertation, Massachusetts Institute of Technology. pp 227. Alexander, T., Baihly, J., Boyer, C., Clark, B., Waters, G., Jochen, V., Calvez, J.L., Lewis, R., Miller, C.K., Thaeler, J., Toelle, B.E. (2011) Shale Gas Revolution. Oilfield Review Autumn 2011. 23, no. 3:40-55. Amann, F., Button, E.A., Evans, K. F., Gischig, V.S., and Blumel M. (2011) Experimental Study of the Brittle Behavior of Clay Shale in Rapid Unconfined Compression. Rock Mech Rock Eng. 44: 415-430. Amann, F., Wild, K.M., Martin, C.D. (2015) The role of capillary suction and dilatancy on the interpretation of the confined strength of clay shales. Proceedings of the 13th International Congress of Rock Mechanics, Montreal, Canada. 617 Anderson, T.L. (2005) Fracture Mechanics: Fundamentals and Applications, Edition 3, CRC Press, Boca Raton, FL. Areias, P.M.A., Belytschko, T. (2005) Analysis of three-deminsional crack initiation and propagation using the extended finite element method. International Journal for Numerical Method in Engineering. 63:760-788. Aristorenas (1992) Time-Dependent Behavior of Tunnels Excavated in Shale. Phd Dissertation, Massachusetts Institute of Technology. pp 553. Ashby, M.F. and Hallam, S.D. (1986) the failure of brittle solids containing small cracks under compressive stress states. Acta Metall., 34, pp. 497-510. Ashby, M.F. and Sammis. (1990) The damage mechanics of brittle solids in compression, Pure Appl. Geophys., 133, pp 489-521. 315

Baecher, G.B. and Einstein, H. H. (1981) Size Effect in Rock Testing. Geophysical Research Letters. 8(7):671-674. Barenblatt, G.I. (1962) The mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechanics, Vol. VII, Academic Press, NY, pp. 55-109. Beach, A. (1975) The geometry of en-echelon vein arrays. Tectonophysics. 28(4):245-263. Bellwald (1990) A Contribution to the Design of Tunnels in Argillaceous Rock. Phd Dissertation, Massachusetts Institute of Technology. pp 427. Bienawski. (1967) Mechanisms of brittle fracture of rock, part II - experimental studies, Int. J. Rock. Mech. Min. Sci. 4:407-423. Bobet, A. (1997) Fracture Coalescence in Rock Materials: Experimental Observations and Numerical Predictions, Sc.D. Thesis, Massachusetts Institute of Technology. Bobet, A. and Einstein, H.H. (1998) Fracture coalescence in rock-type materials under uniaxial and biaxial compression. Int. J Rock Mech Min Sci, 35(7), pp. 863-88. Bock, H. (2001) Mont Terri Project. RA experiment: Rock mechanics analyses and synthesis; data report on rock mechanics. Mont Terri Consortium, Technical Report TR 2000-02. Bock, H. (2009) Mont Terri Project. RA experiment: Updated Review of the Rock Mechanics Properties of the Opalinus Clay of the Mont Terri URL based on Laboratory and Field Testing. Q + S Consult, Technical Report TR 2008-04. Bossart, P., Burrus, F., and Nussbaum C. (2012) Techniques de forages et d’excavations dans les argiles à Opalinus, laboratoire souterrian du Mont Terri. Swiss Bull. Angew. Geol. 17(1), pp. 3-28. Brace, W.F. and Bombolakis, E.G. (1963) A note on brittle crack growth in compression. J. Geophys Res, 68(12), pp. 3709-3713. Britt, L.K., Schoeffler, J. (2009) The geomechanics of a shale play: what makes a shale prospective! Society of Petroleum Engineers. Paper 125525 presented at the 2009 SPE eastern regional meeting held in Charlston, WV. Brooks, Z. (2010) A Nanomechanical Investigation of the Crack Tip Process Zone of Marble. MSc Thesis, Massachusetts Institute of Technology. Brooks, Z., Einstein, H.E.E, and Ulm, F. (2010) A Nanomechanical Investigation of the Crack Tip Process Zone. 44th U.S. Rock Mechanics Symposium and 5th U.S.-Canada Rock Mechanics Symposium, June 27-30, 2010 , Salt Lake City, Utah. Brooks, Z. (2013) Fracture process zone : microstructure and nanomechanics in quasibrittle materials. PhD Thesis, Massachusetts Institute of Technology. 316

Bruck, H.A., McNeill, S.R., Sutton, M.A., Peters, W.H. (1989) Digital Image Correlation Using Newton-Raphson Method of Partial Differential Correction. Experimental Mechanics. 29(3):261-267 Bucher, K., Grapes, R. (2011) Petrogenesis of Metamorphic Rocks. New York: SpringerVerlag. Buczek, M., Herakovich, C. (1985) A normal stress criterion for crack extension direction in orthotropic materials. J Compos Mater 19:44-553. Buxtorf, A. (1916) Prognosen und Befunde beim Hauensteinbasis- und Grenchenbergtunnel un die Bedeuteung der letzteren für die Geologie des Jurabirges. Verh. Natf. Ges. Basel 27: 184-254 Casey, B. A. (2011) The significance of specimen end restraint in high pressure triaxial testing of cohesive soil. MSc Thesis, Massachusetts Institute of Technology. Chan, M. Automatic Two-Dimensional Muti-Fracture Propagation Modeling of Brittle Solids with Particular Application to Rock. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge. U.S.A. 1986. Chaker and Barquins. (1996) Sliding effect on branch cracks, Phys. Chem. Earth. 21(4):319323. Corkum, A.G., Martin, C.D. (2007) The Mechanical Behavior of weak mudstone (Opalinus Clay) at low stresses. International Journal of Rock Mechanics and Mining Sciences 44: 196-209. Cottrell, B., Rice, J.R. (1980) Slightly curved or kinked cracks, International Journal of Fracture, 16(2):155-169. Costin, L.S. (1985) Damage mechanics in the post-failure region. Mech. Mat. 4:149-160. Courdouan, A., Christl, I., Meylan, S., Wersin, P., and Kretzschmar, R. (2007) Characterization of dissolved organic matter in anoxic rock extracts and in situ pore water of the Opalinus Clay. Applied Geochemistry. 22: 2926-2939. Cripps, J.C., Taylor, R.K. (1981) The engineering properties of mudrocks. Q.J. eng. Geol. London 14:325-346 Dugdale, D.S. (1960) Yielding in steel sheets containing slits, Journal of the Mechanics and Physics of Solids, 8:100-104. Duveau, G, Shao, J.F., and Henry J.P. (1998) Assessment of some failure criteria for strongly anisotropic geomaterials. Mech. Cohes.-Frict. Mater. 3:1-26 Dyni, J.R. (2006) Geology and resources of some world oil-shale deposits. U.S. Geological Survey. Reston, Va. v. 42 317

Einstein, H.H., Bobet, A., Aristorenas, G. (1995) Feasibility study of Opalinuston. Nagra, Q +S Consult, Germany. Einstein, H.H. (2000) Tunnels in Opalinus Clayshale - a review of case histories and new developments. Tunnelling and Underground Space Technology 15(1): 13-29. Engel, M.H., Macko, S.A. (1993) Organic Geochemistry: Principles and Applications. Springer Science & Business Media. New York. Erdogan, F. and Sih, G.C. (1963) On the crack extension in plates under plane loading and transverse shear, Journal of Basic Engineering, 85:305-321. Eslinger, E., Pevear, D. (1988) Clay minerals for petroleum geologists and engineers. Vol Notes No. 22: Society of Economic Paleontologists and Mineralogists. Etterlin, A. (1987) Rekonstruktion Hauenstein-Basistunnel. Hasler & Holtz. Friedman, M., Handin, J. and Alani, G. (1972) Fracture energy of roc ks. Int. J. Rock. Mech. Min. Sci. 9:757-766. AAPG Bulletin. 91(4):603-622. Gale, J.F.W., Reed R.M., Holder, J. (2007) Natural fractures in the Barnett Shale and their importance for hydraulic fracture treatments. Germanovich, L.N. and Dyskin, A.V. (2000) Fracture mechanisms and instability of openings in compression, Int. J. Rock. Mech. Min. Sci. 37:263-284. Gingold, R.A., Monaghan J.J. (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon Not R Astron Soc. 181:375-89. Google Maps (2015) South Dakota. Retrieved from https://www.google.com/maps/place/ South+Dakota/@44.1488568,-100.1986896,902452m/. Gonçalves da Silva, B. (2009) Modeling of crack initiation, propagation and coalescence in rocks. MSc Thesis, Massachusetts Institute of Technology. pp. 356. Gonçalves da Silva, B., Einstein, H.H. (2013) Modeling of crack initiation, propagation and coalescence in rocks. Int J Fract. 182:167-186. Gramberg, J. (1965) Axial cleavage fracturing, a significant process in mining and geology, Engineering Geology. 1(1):31-72. Griffith, A.A. (1920) The phenomenon of rupture and flow in solids. Philosophical Transactions, Series A. 221:163-198. Grubenmann, U., Letsch, E. (1907) Die schweizerischen Tonlager. Beitr. Geol. Schweiz, Geotechn. Serie, IV Lieferung, Herausg: Geol. Komm. Schweiz. Naturf. Ges. Hartmann, A. (1950) Der untere braune Jura des Aargaus als wichtiger Tonlieferant der Schweiz. Mitt. aarg. natf. Ges., Heft 23:-3-20 318

Harwood, J., Aplin, A.C., Fialips, C.I., Kozdon, R., Ushikubo, T., Valley, J.W. (2013) Quartz cementation history of sandstones revealed by high-resolution sims oxygen isotope analysis. Journal of Sedimentary Research. 83:522-533. Hoek and Bieniawski. (1965) Brittle fracture propagation in rock under compression, Int. J. of Fracture Mechanics. 1:137-155. Howarth, R.W., Ingraffea, A., Engelder, T. (2011) Natural gas: Should fracking stop? Nature. 477:271-275 Hower, J., Eslinger, E. V., Hower, M. E., and Perry, E. A. (1976) Mechanism of burial metamorphism of argillaceous sediment: mineralogical and chemical evidence. Geol Society American Bulletin. 87:725-237. Horii, H. & Nemat-Nasser, S. (1985) Compression-induced microcrack growth in brittle solids: Axial splitting and shear failure. Journal of Geophysical Research. 90(B4):31053125. Huang J.F., Chen G.L., Zhao Y.H. and Wang R. (1990) An experimental study of the strain field development prior to failure of a marble plate under compression. Tectonophysics, Vol. 175, pp. 283-290. Hussain, M.A., Pu, S.L., Underwood, J. (1974) Strain energy release rate for a crack under combined mode I and mode II. Fract Anal ASTM STP 560:2-28. IG Wisenberg-Tunnel (1990) Bahn 2000-Wisenberg Tunnel. Schlitzpressenmessungen in der Auskleidung des Hauenstein-Basistunnels, Bericht. IGB-Bericht (1981) Nationalstrasse 3, Bözbergtunnel - Untersuchungen über das Quellverhalten von Opalinuston. IGB-Beritch 3899/1 IJA-BG Consortium d’études du Tunnel du Mont Terri (1991) Galéria de reconnaissance, Rapport d’essais et de mesures dans l’Aalénien. No 2593-266 IJA-BG Consortium d’études du Tunnel du Mont Terri (1992) Tunnel du Mont Terri, Interpretation des mesures de convergence dans le tunnel routier, Rapport Intermédiaire. No 2593-346 Inglis, C.E. (1913) Stresses in a plate due to the presence of cracks and sharp corners. Inst. Naval Architecture, London. 55:219-230. Ingraffea, A. and Wawrzynek, P.A. (2001) Encyclopedia of Materials: Science and Technology, pp. 1745-1750. Irwin, G.R. (1957) Analysis of stresses and strains near the end of a crack traversing a plate. Journal of Applied Mechanics. 24:361-364. ISRM (1989) Suggested Methods for Laboratory Testing of Argillaceous Swelling Rock. 319

ITASCA Consulting Group, Inc. (2008a) PFC2D and PFC3D manuals. ITASCA Consulting Group, Inc. (2008b) UDEC and 3DEC manual. Janeiro, R. P. (2009) The Effect of Inclusions in Brittle Material. MSc Thesis, Massachusetts Institute of Technology. Jobmann, M., Wilsnack, Th., Voigt, H.D. (2010) Investigation of damage-induced permeability of Opalinus clay. International Journal of Rock Mechanics and Mining Sciences 47(2): 279-285. Johns, E.A., Burnett, R.G., Craig, C.L. (1963) Oahe dam: influence of shale on Oahe power structures design. Journal of Soil Mechanics and Foundation Division. 89:95-114. Josh, M., Esteban, L., Delle Piane, C., Sarout, J., Dewhurst, D.N., Clennel, M.B. (2012) Laboratory characterisation of shale properties. Journal of Petroleum Science and Engineering 88-89: 107-124. Kazerani, T., Zhao, J. (2010) Micromechanical parameters in bonded particle method for modelling of brittle material failure. International Journal for Numerical and Analytical Methods in Geomechanics. 34(18):1877-1895. Kazerani, T., Yang, Z.Y., Zhao, J. (2012) A discrete element model for predicting shear strength and degradation of rock joint by using compressive and tensile test data. Rock Mech Rock Eng. 45(5):695-709. Keller, L.M., Holzer, L., Wepf, R., Gasser, P. (2011) 3D geometry and topology of pore pathways in Opalinus clay: Implications for mass transport. Applied Clay Science 52(1): 85-95. Keller, L.M., Schuetz P., Erni, R., Rossell, M.D., Lucas, F., Gasser, P., Holzer, L. (2013) Characterization of multi-scale microstructural features in Opalinus Clay. Microporous and mesoporous materials 170: 83-94. Kemeny, J.M. and Cook, N.G.W. (1987) Crack models for the failure of rocks in compression. In Proc. 2nd Int. Conf. on Constitutive Laws for Engineering Materials, Tucson, Arizona. Labuz, J.H., Shah, S.P. and Dowding, C.H. (1987) The Fracture Process Zone in Granite: Evidence and Effect. Int J. Rock Mech. Min Sci. and Geomech. Abstr., Vol. 24, No. 4, pp 235-246. Lajtai, E.Z. 1970. A theoretical and experimental evaluation of the Griffith theory of brittle fracture, Tectonophysics. 11:129-156. Lankford, J. (1976) Dynamic Strength of Oil Shale. Society of Petroleum Engineers Journal. SPE 5327:17-22. 320

Li YP, Chen LZ, Wang YH (2005) Experimental research on pre-cracked marble under compression. Int J of Solids Struct 42:2505-2516. Lin, Q., Labuz, J.F. (2011) Digital Image Correlation and the Fracture Process in Rock. 44th U.S. Rock Mechanics Symposium and 5th U.S.-Canada Rock Mechanics Symposium, June 27-30, 2010 , Salt Lake City, Utah. Lisjak, A., Grasselli, G., Vietor, T. (2013) Continuum-discontinuum analysis of failure mechanisms around unsupported circular excavations in anisotropic clay shales. Int J of Rock Mech and Min Sci. 65:96-115. Lisjak, A. and Grasseli, G. (2014) A review of discrete modeling techniques for fracturing processes in discontinuous rock masses. Journal of Rock Mechanics and Geotechnical Engineering. 6:301-314. Lisjak, A., Tatone, B.S.A., Grasselli, G., Vietor, T. (2014) Numerical Modelling of the Anisotropic Mechanical Behaviour of Opalinus Clay at the Laboratory-Scale Using FEM/DEM. Rock Mech Rock Eng. 47:187-206. Lisjak, A., Mahabadi, O.K., Grasselli, G., Marschall, P., Muller, H.R., Garitte, B., Vietor, T. (2015) Proceedings of the 13th International Congress of Rock Mechanics, Montreal, Canada. 377. Libersky, L.D., Petschek, A.G. (1990) Smoothed particle hydrodynamics with strength of materials. Advances in the Free Lagrange Method. Lecture Notes in Physics, vol 395:248257. Lutton, R.J. (1970) Tensile fracture mechanics from fracture surface morphology. 12th U.S. Symposium on Rock Mechanics. 561-571. Madsen, F.T. (1976) Quelldruckmessung an Tongesteinen und Berechnung des Quelldruckes nach der DLVO-Theorie. Mitt. IGB ETH Zürich. 108:1-65. Madsen, F.T., Mitchell, J.K. (1989) Chemical effects on clay hydraulic conductivity and their determination. Mitt. IGB ETH Zürich. 135:1-67. Madsen, F.T., Müller-Vonmoos, M. (1985) Swelling pressure calculated from mineralogical propertires of a Jurassic opalinum shale, Switzerland. Clays Clay Minerals. 33:501-509 Mahabadi, O.K., Lisjak, A., Munjiza, A., Grasselli, G. (2012) Y-Geo: New Combined Finite-Discrete Element Numerical Code for Geomechanical Applications. ASCE International Journal of Geomechanics. 12:676-688. Maji, A.K. and Wang, J. (1992) Fracture mechanics of a tension-shear macrocrack in rocks.Experimental Mechanics. 32(2):190-196. Martin, C.D., and Lanyon, G.W. (2003) Measure of In-situ Stress in weak rocks at Mont Terri Rock Laboratory, Switzerland. International Journal of Rock Mechanics and Mining 321

Sciences 40: 1077-1088. Martin, C.D. (2015) Proceedings of the 13th International Congress of Rock Mechanics, Montreal, Canada. 891. Martinez, A.R. (1999) Fracture coalescence in natural rock. MSc Thesis, Massachusetts Institute of Technology. McDonal, F.J., Angona, F.A., Mills, R.L., Sengbush, R.L., Van Nostrand, R.G., White, J.E. (1958) Attenuation of shear and compressional waves in Pierre Shale. Geophysical Prospecting. 4:404-407. Meier, L.P., and Menegatti, A.P. (1997) A new, efficient, one-step method for the removal of organic matter from claycontaining sediments. Clay Minerals. 32: 557-563 Melville, P.H. (1977) Fracture mechanics of brittle materials in compression, Int. J. Frac. 13:532-534. Miller, J.T. (2008) Crack coalescence in granite, MSc Thesis, Massachusetts Institute of Technology, 474 p. Moes, N., Dolbow, J., Belytschko, T. (1999) A finite element method for growth without remeshing. Internation Journal for Numerical Methods in Engineering. 46:131-150. Moore D.E. and Lockner, D.A. (1995) The role of microcracking in shear-fracture propagation in granite. Journal of Structural Geology, Vol. 17, No. 1, pp 95-114. Mont Terri Project. October, 17 (2011) http://www.mont-terri.ch/ May, 13 (2013). Morgan, S.M., Johnson, C.A., and Einstein, H.H. (2013) Cracking Processes in Barre Granite: Fracture Process Zones and Crack Coalescence. International Journal of Fracture, 180:177-204. Moss, W.C. and Gupta, Y.M. (1982) A constitutive model describing dilatancy and cracking in brittle rocks. Journal of Geophysical Research. 87(B4):2985-2998. Motor-Columbus (1981) Bericht über die mögliche Hohlraumgrösse bei Endlagerstätten im Opalinuston in Frunktion der Tiefenlage. Bericht an Nagra. Mughieda, O. and Alzo’ubi, A.K. (2004) Fracture mechanics of offset rock joints - a laboratory investigation, Geotechnical and Geological Engineering. 22:545-562. N-16 Transjurane, Galérie de reconnaissance - Mont Terri (1990) Essais de laboratorie, Forages TGR1-TGR2 et niches TM474-TM888, Etude demandée par le Serfice des Ponts et Chaussées de la République et Canton du Jura, R169. Ecole Polytechnique Fédérale de Lausanne. 322

N-16 Transjurane, Galérie de reconnaissance - Mont Terri (1989) Essais de charge Niche TM 888, Etude demandée par le Service des Ponts et Chausséess de la République et Canton du Jura, R69, Rapport. Ecole Polytechnique Fédérale de Lausanne. Nadeau, P.H., and Reynolds, R.C. (1981) Burial and contact metamorphism in the Mancos shales. Clay and Clay Minerals. 29(4):249-259. Nagra (1988) Sedimentstudie Endlager für hochaktive Abfälle, Bau and Betrieb. Inter Bericht. 88-26 Nagra (2002) Project Opalinus Clay: Safety Report. Demonstration of disposal feasibility (Entsorgungsnachweis) for spent fuel, vitrified high-level waste and long-lived intermediatelevel waste. Nagra Technical Report NTB 02-05, Nagra, Wettingen, Switzerland. Nasseri, M.H.B., Mohanty, B., and Young, R.P. (2006) Fracture Toughness Measurements and Acoustic Emission Activity in Brittle Rock. Pure appl. geophys., 163. 917-945. Naumann, M., Hunsche, U., Schulze, O. (2007) Experimental investigations on anisotropy in dilatancy, failure and creep of Opalinus Clay. Physics and Chemistry of the Earth. 32:889-895. Nesetova and Lajtai. (1973) Fracture from compressive stress concentrations around elastic flaws, Int J. Rock Mech. Min. Sci. and Geomech. Abstr. 10:265-284. Nguyen, T.L., Hall, S.A., Vacher, P., Viggiani, G. (2011) Fracture mechanisms in soft rock: Identification and quantification of evolving displacement discontinuities by extended digital image correlation. Tectonophysics. 503:117-128. Niandou, H., Shao, J.F., Henry, J.P., Fourmaintraux, D. (1997) Laboratory investigation of the mechanical behaviour of Tournemire Shale. Int. J. Rock Mech. Min. Sci. 34(1):3-16 Nüesch, R. (1989) Felsmechanische Resultae aus Untersuchungen an Opalinuston. Nagra Inter Bericht. NB 89-17 Nüesch, R., Hass, R., Madsen, F.T. (1990) Eignung von Ton als Deponiebarriere. Abfallspektrum Dez. 6:11-15 Nüesch, R. (1991) Das mechanische Verhalten von Opalinuston. Thesis, Zürich. pp 243. Pan, B. (2009) Reliability-guided digital image correlation for image deformation measurement. Applied Optics. 48(8):1535-1542. Passchier, C.W., Trouw, R.A.J. (2005) Microtectonics (2nd edition). New York: SpringerVerlag. Pei, J. (2004) Strength of transversely isotropic rocks. PhD Thesis, Massachusetts Institute of Technology. 323

Peters, T.J. (1962) Tonmineralogische Untersuchungen an Opalinuston und einem Oxfordienprofil im Schweizer Jura. SMPM 42:359-380. Peters, W.H., Ranson, W.F. (1982) Digital imaging techniques in experimental stress analysis. Optical Engineering. 21(3):427-431. Pollard, D.D., Segali, P., Delaney, P.T. (1982) Formation and interpretation of dilatant echelon cracks. Geological Society of America Bulletin. 93:1291-1303. Potapov, A.V., Hunt, M.L., Campbell, C.S. (2001) Liquid-solid flows using smoothed particle hydrodynamics and the discrete element method. Powder Technology. Vol. 116:204213. Petit, J. and Barquins, M. (1988) Can natural faults propagate under Mode II conditions, Tectonics. 7(6):1246-1265. Ramamurthy, T., Venkatappa Rao, G., Singh, J. (1993) Engineering behavior of phyllites. Engineering Geology. 33:209-225. Reyes, 0. (1991) Experimental study and analytic modeling of compressive fracture in brittle materials, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA. Reyes, O. and Einstein, H.H. (1991) Failure mechanism of fractured rock - a fracture coalescence model, Proceedings of the 7th International Congress of Rock Mechanics, Aachen, Germany. 1:333-340. Sagong, M. (2001) The study of the fracture of multiple flaw specimens, Ph.D. Thesis, Purdue University, Indiana. Sagong, M. and Bobet, A. (2002) Coalescence of multiple flaws in a rock-model material in uniaxial compression, Int. J. Rock. Mech. Min. Sci. 39:229-241. Schaefer, V.R., Birchmier, M.A. (2013) Mechanisms of Strength Loss during Wetting and Drying of Pierre Shale. Proceedings of the 18th International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013. Schultz, L.G. (1964) Analytical Methods in Geochemical investigations of the Pierre Shale. USGS Geological Survey Professional Paper 391-C. Schultz, L.G., Tourtelot, H.A., Gill, J.R., Boerngen, J.G. (1980) Composition and Properties of the Pierre Shale and Equivalent Rocks, Northern Great Plains Region. USGS Geological Survey Professional Paper 1064-B. Shen, B., Stephansson, O., Einstein, H.H. and Ghahreman, B. (1995) Coalescence of fractures under shear stress experiments, J. Geophys Res, 100(6), pp. 5975-90. Shen, B., Stephansson, O., Rinne, M. (2013) Modelling rock fracturing processes: a fracture mechanics approach using FRACOD. Springer, Berlin. 324

Shen, B., Siren, T., Rinne, M. (2015) Modelling Fracture Propagation in Anisotropic Rock Mass. Rock Mech Rock Eng. 48:1067-1081. Sih, G.C. (1974) Strain-energy-density factor applied to mixed mode crack problems, International Journal ofFracture, 10(3):305-321. Steiner, W., Metzger, R. (1988) IG Wisenberg-Tunnel, Bahn 2000-Wisenbergtunnel. Erfahrungen aus Tunneln in quellendem Gestein. Tasdemir, M.A., Maji, A.K. and Shah, S.P. (1990) Crack propagation in concrete under compression. Journal of Engineering Mechanics. 116(5):1058-1076. Tien, Y.M., Kuo, M.C., Juang, C.H. (2006) An experimental investigation of the failure mechanisms of simulated transversely isotropic rocks. Innt. J. Rock Mech. Min. Sci. 43:1163-1181 Tourtelot, H.A. (1962) Preliminary investigation of the geological setting and chemical composition of the Pierre Shale great plain region. USGS Geological Survey Professional Paper 390. Toussaint, F., Tabourot, L., Vacher, P. (2008) Experimental study with a Digital Image Correlation (DIC) method and numerical simulation of an anisotropic elastic-plastic commercially pure titanium. Archives of Civil and Mechanical Engineering. 8(3):131-143. Turner, F.J. (1981) Metamorphic Petrology. New York: McGraw-Hill. Valente, S., Fidelibus, C., Loew, S., Cravero, M., Iabichino, G., Barpi, F. (2011) Analysis of Fracture Mechanics Tests on Opalinus Clay. Van Dyke, K. (1997) Fundamentals of Petroleum. Petroleum extension service and the University of Texas at Austin publishing. Tulsa, Oklahoma. Van Loon, L. R., Soler, J.M., Muller, W. and Bradbury M. H. (2004) Anisotropic Diffusion in Layered Argillaceous Rocks: A Case Study with Opalinus Clay. Environ. Sci. Technol. 38: 5721-5728. Vargas, A.G.B (2013) The mechanical behavior of heavily overconsolidated resedimented Boston Blue Clay. MSc Thesis, Massachusetts Institute of Technology. Wang, E.Z. and Shrive, N.G. (1995) Brittle fracture in compression: mechanics, models and criteria, Engineering Fracture Mechanics. 52(6):1107 - 1126. Weaver, C.E. (1989). Clays, muds, and shales. Vol. 44: Elsevier. Wei, X., Jun, L., Dong, G.G., Han, S.Y., Kok, J., Zhao, J., Xin, Z. (2014) Global cross application of shale gas development work flows revealed. The 20th formation evaluation symposium of Japan, October. 325

Wenk, H. R., Voltolini, M., Mazurek, M., Van Loon, L.R., and Vinsot, A. (2008) Preferred Orientations and Anisotropy in Shales: Callovo-Oxfordian Shale (France) and Opalinus Clay (Switzerland). The Clay Minerals Society, Vol. 56, No. 3, pp-285-306. Weyl, P.K. (1959) Pressure solution and the force of crystallization: a phenomenological theory. Journal of Geophysical Research. Vol 64, No. 11, pp. 2001-2025. White, J.E., Martineau, L., Monash, C. (1983) Measured anisotropy in Pierre Shale. Geophysical Prospecting. 31:709-725. Wild, K.M., Amann, F., Martin, C.D. (2015) Proceedings of the 13th International Congress of Rock Mechanics, Montreal, Canada. 505. Wong RHC, Chau KT (1998) Crack coalescence in a rock-like material containing two cracks. Int J Rock Mech Min Sci 35(2):147-164. Wong RHC, Chau KT, Tang CA, Lin P (2001) Analysis of crack coalescence in rock-like ˘ Tpart ˇ materials containing three flawsâA I: experimental approach. Int J Rock Mech Min Sci 38:909-924. Wong, L.N.Y., Einstein, H.H. (2009a) Crack coalescence in molded gypsum and Carrara marble: part 1 - Macroscopic observations and interpretation. Rock Mech and Rock Eng 42(3):475-511. Wong, L.N.Y., Einstein, H.H. (2009b) Systematic Evaluation of Cracking Behavior in Specimens Containing Single Flaws under Uniaxial Compression. Int. J of Rock Mech and Min Sci 46(2) 239-249. Wong, L.N.Y., Einstein, H.H. (2009c) . Crack Coalescence in Molded Gypsum and Carrara Marble: Part 2 - Microscopic Observations and Interpretation. Rock Mech and Rock Eng 42(3): 513-545. Wong, N.Y. (2008) Crack coalescence in molded gypsum and Carrara Marble. Dissertation, Massachusetts Institute of Technology. Zang, A., Wagner, F.C., Stanchits, S., Janssen, C., and Dresen, G. (2000) Fracture process zone in granite. Journal of Geophysical Research, Vol. 105, No. B10, pp 23,651-23,661. Zhang, X.P., Wong, L.N.Y (2012) Cracking Processes in Rock-Like Material Containing a Single Flaw Under Uniaxial Compression: A Numerical Study Based on Parallel BondedParticle Model Approach. Rock Mech Rock Eng. 45:711-737. Zhang, X.P., Wong, L.N.Y (2013) Crack Initiation, Crack Initiation, Propagation and Coalescence in Rock-Like Material Containing Two Flaws: a Numerical Study Based on Bonded-Particle Model Approach. Rock Mech Rock Eng. 46:1001-1021. Zhao, C., Bao, C., Zhao, C., Tian, J. (2015) Experimental study on the damage and failure properties of rock-like material with pre-existing double flaws under uniaxial compression. 326

Proceedings of the 13th International Congress of Rock Mechanics, Montreal, Canada. Zietlow W.K. and Labuz J.F. (1998) Measurement of the intrinsic process zone in rock using acoustic emission. International Journal of Rock Mechanics, Mining Science and Geomechanic Abstracts, Vol. 35, No. 3, pp 291-299.

327

328

Appendix A Boring Logs

329

Natural Fracture

Water Content Samples

O1L

O1R

O2L

O2R

OM1 OM2 OM3 OM4 OM5 OM6 OM7 OM8 OM9 OM10 OM11 OM12 OM13

Natl. Frac.

25.230

Wc

25.275

O1L O1R

25.300 25.475 8/14/12 25.300 25.475 8/14/12

O2L O2R

25.475 25.630 8/15/12 25.475 25.630 8/15/12

OM1 OM2 OM3 OM4 OM5 OM6 OM7 OM8 OM9 OM10 OM11 OM12 OM13

25.630 25.657 25.684 25.711 25.738 25.765 25.792 25.819 25.846 25.873 25.900 25.927 25.954

Notes

Test Date

Test

Date Cut

End Cut

Start Cut

Sample #

~200-250 m

25.300 8/14/12 Wc

25.659 25.689 25.718 25.747 25.765 25.792 25.819 25.845 25.873 25.900 25.927 25.954 25.981

8/17/12 2a-30-30A 9/18/12 8/17/12 2a-30-30B 9/18/12 8/17/12 2a-30-30C 9/18/12 8/17/12 2a-30-30D 9/18/12 11/2/12 11/2/12 x x 11/2/12 x x 11/2/12 11/2/12 x x 11/2/12 x x 1/21/13 1/21/13 1/21/13 x x

Broke Broke Broke Broke

Broke

Figure A-1: Boring Log for Core #25. The visual representation of where cuts were taken is on the left and the detail of which test and exact cut location is shown on the right side. Note that samples were numbered as they were cut (OM#) and then assigned a geometry afterwards.

330

26

OM14 OM15 OM16 OM17 OM18 OM19 OM20 OM21 OM22 OM23 OM24 OM25 OM26 OM27 OM28 OM29 OM30 Natural Fracture OM31 OM32 OM33 OM34 OM35 OM36 OM37 OM38 OM39 OM40 OM41 OM42 OM43 OM44 OM45 OM46 End Cut

26 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9

OM14 OM15 OM16 OM17 OM18 OM19 OM20 OM21 OM22 OM23 OM24 OM25 OM26 OM27 OM28 OM29 OM30

26.005 26.035 26.064 26.092 26.131 26.159 26.187 26.216 26.244 26.272 26.300 26.326 26.354 26.381 26.409 26.438 26.464

26.035 26.064 26.092 26.131 26.159 26.187 26.216 26.244 26.272 26.300 26.326 26.354 26.381 26.409 26.438 26.464 26.495

Natl Frac

26.500 26.545

OM31 OM32 OM33 OM34 OM35 OM36 OM37 OM38 OM39 OM40 OM41 OM42 OM43 OM44 OM45 OM46 End

26.545 26.575 26.605 26.625 26.655 26.685 26.715 26.745 26.775 26.805 26.830 26.855 26.885 26.915 26.940 26.965 26.985

26.575 26.6.05 26.6.25 26.6.55 26.6.85 26.715 26.745 26.775 26.805 26.830 26.855 26.885 26.915 26.940 26.965 26.985 27.000

3/4/13 3/4/13 3/4/13 3/4/13 3/4/13 3/4/13 3/4/13 3/4/13 3/4/13 3/4/13 3/4/13 3/4/13 3/4/13 3/4/13 3/4/13 3/4/13 3/4/13

8/19/13 8/19/13 8/19/13 8/19/13 8/19/13 8/19/13 8/19/13 8/19/13 8/19/13 8/19/13 8/19/13 8/19/13 8/19/13 8/19/13 8/19/13 8/19/13 8/19/13

Notes

Test Date

Test

Date Cut

End Cut

Start Cut

Sample #

~200-250 m

~2a~ 30-30E 3/26/13 30-30F 3/26/13 Broke? @Saw 0-0A 0-0B 0-0C 30-0A 30-0B 30-0C 30-60A 30-60B 30-60C Resat Resat 60-0 A

3/26/13 3/26/13 3/26/13 5/22/13 5/22/13 5/22/13 5/22/13 5/22/13 3/26/13

Broke @ saw Broke @ saw

30(90)A 9/18/13 Edged Edged Split

30(30)A 9/18/13 60-0 B 9/17/13 60-0 C 9/17/13 75-0 A 9/17/13 75-0 B 9/17/13 75-0 C 9/17/13 30-90A 9/17/13 30-90B 9/18/13 30-90C 9/18/13

27

Figure A-2: Boring Log for Core #26. The visual representation of where cuts were taken is on the left and the detail of which test and exact cut location is shown on the right side. Note that samples were numbered as they were cut (OM#) and then assigned a geometry afterwards.

331

28

OM47 OM48 OM49 OM50 OM51 OM52 OM53 OM54 OM55 OM56 OM57 ` OM57 OM58 OM59 OM60 OM61 `

28 OM47 OM48 OM49 OM50 OM51 OM52 OM53 cut cut OM54 OM55 OM56 OM57 OM58 OM59 OM60 cut cut OM61

28.1 28.2 28.3 28.4 28.5

28.010 28.041 28.069 28.100 28.129 28.160 28.189 28.218 28.242 28.242 28.273 28.306 28.336 28.360 28.402 28.428 28.460 28.473 28.473

28.041 28.069 28.100 28.129 28.160 28.189 28.218 28.242 28.273 28.306 28.336 28.360 28.402 28.428 28.460 28.473

~2a~ 9/13/13 9/13/13 9/13/13 45-0 A 9/18/13 9/13/13 30(90)B 9/18/13 9/13/13 30(90)C 9/18/13 9/13/13 9/13/13 -

Notes

Test Date

Test

Date Cut

End Cut

OM46

Start Cut

Sample #

~200-250 m

broke broke

@ saw @ drill

broke broke

@ drill @ drill

9/13/13 30(60)A 9/18/13 9/13/13 30(60)B 9/18/13 9/13/13 30(60)C 9/18/13 9/13/13 intact 1/9/14 Parallel 9/13/13 intact 1/9/14 Parallel 9/13/13 intact 1/9/14 Parallel 9/13/13 45-0 B 9/18/13

28.500 9/13/13 45-0 C 9/18/13

28.6 Natural Fracture

28.7 Natl Frac

28.8 28.9

End

29

Figure A-3: Boring Log for Core #28. The visual representation of where cuts were taken is on the left and the detail of which test and exact cut location is shown on the right side. Note that samples were numbered as they were cut (OM#) and then assigned a geometry afterwards.

332

Appendix B Pierre Shale In addition to Opalinus shale, cutting and preservation techniques for Pierre shale were also developed. Pierre shale is a high porosity, high clay content shale which is very sensitive to drying. Therefore, some of the cutting techniques developed for Opalinus shale did not work and additional procedures were developed. The origin, specific gravity, water content and mineralogy for Pierre shale are presented. Then the cutting and preservation techniques developed for Pierre shale will be discussed. Finally, a few unconfined compression tests were conducted on Pierre shale and the results will be presented.

B.1

Origin and Extraction

Pierre shale is a clay based shale from the Niobrara formation which extends across Colorado, Wyoming, Nebraska and South Dakota. The samples obtained were extracted from freshly uncovered outcrops just outside of Rapid City, South Dakota (figure B-1). The depth of extraction was approximately 1 to 2 meters (3-6 feet) (figure B-2). Pierre shale is highly sensitive to drying and cracks due to drying could be observed on the surface within minutes. Therefore, all of the samples were wrapped and sealed in plastic as they were extracted (figure B-3). 333

Figure B-1: Location of Pierre shale sample extraction in South Dakota. Samples were extracted from outcrops just east of Rapid City, South Dakota. (Google Maps, 2015)

Figure B-2: Pierre shale outcrop dig site. Pierre shale samples were extracted from freshly uncovered outcrops about 1-2 meters (3-6 feet) deep.

Figure B-3: Pierre shale samples wrapped and sealed in plastic after extraction. Pierre shale is very sensitive to drying, therefore, sealing the samples quickly after extraction was important.

334

B.1.1

Bentonite Seams and Natural Shells/Fossils

It should be noted that Pierre shale has many heterogeneities which can compromise the integrity of intact samples. On a large scale, bentonite seams were observed on the outcrop (figure B-4). These bentonite seams were very weak and they were avoided when extracting samples. Additionally, there were many fossils and shells found in the Pierre shale formation (figure B-5). These outcrops are very well known by fossil hunters in South Dakota for ammonite and bacculite shells. Care was taken to avoid large shells when extracting samples, however, many shells persisted within the samples extracted.

Figure B-4: Bentonite seams observed in Pierre shale outcrops. These bentonite seams were very weak and were avoided when extracting samples.

Figure B-5: Fossilized shells fragment in Pierre shale. Bacculite and ammonite shell fragments, as well as concretions, were very common in Pierre shale outcrop samples. 335

B.2

Specific Gravity, Natural Water Content, Mineralogy

The specific gravity, in-situ water content and mineralogy for the Pierre shale samples were determined. The specific gravity was determined using the same procedure described in section 3.1.3. The specific gravity is summarized in table B.1. The specific gravity of Piere shale (2.781) is very similar to that measured for Opalinus shale (2.709) (Refer to table 3.3). Table B.1: Pierre Shale Specific Gravity Tests Sample 1 - M4 2 - M6 3 - 250-2 Average

Specific Gravity, GS 2.779 2.777 2.785 2.781

The natural water content determined from small cuttings is presented in table B.2. The porosity of Pierre shale is not well studied in the literature and those studies which document the porosity have large variations. Therefore the saturation was not determined. However, by using the determined water content and assuming 100% saturation, the porosity of the Pierre shale samples would be 37.2%. This is much higher in comparison to the porosity of Opalinus shale, which is only about 13.7% (Bock, 2009). Table B.2: Water Content of Pierre Shale Samples Sample S9 S73 Z-29 Avg

Water Content,WC [%] 22.9 21.6 21.3 21.9

XRD methods were used by Macaulay Scientific Consulting Limited to determine the mineralogy of Pierre shale (table B.3). In comparison to the mineralogy of Opalinus shale (refer to table 3.2), Pierre shale contains similar amounts of clay, however, it has more quartz and less carbonate compared to Opalinus shale. This likely means that there is less 336

carbonate cementation in Pierre shale than in Opalinus shale. It should be noted that this mineralogy analysis does not differentiate between smectite and illite in the Illite/SmectiteMixed Layer (I/S-ML). Due to the swelling and drying sensitive in Pierre shale it is likely that there is more smectite in Pierre shale than Opalinus shale. Also, since smectite is an indirect measure of shale maturity (depth and temperature), it is logical that there would be more smectite in the samples of Pierre shale that were close to the surface than in the deeply excavated samples of Mont Terri Opalinus shale. Table B.3: Mineralogy of South Dakota Pierre Shale Opalinus Shale [%] 20.4 3.8 1 7 0.9 1.1 0.3 0.5 0.5 0.1 0.7 1.3 3.7 50.5 8.3

Quartz K-Feldspar Muscovite Plagioclase Calcite Dolomite Siderite Gypsum Halite Anatase Apatite Pyrite Chlorite (Tri) I+I/S-ML Kaolinite Total Quartz, Feldspar, Mica Total Carbonate Total Clay

32.2 2.3 62.5

From the specific gravity, water content and mineralogy of Pierre shale, it can be seen that although both Opalinus and Pierre shale have high clay content, they differ in many ways. Pierre shale is very sensitive to drying in atmospheric conditions due to higher water content, porosity, and likely higher smectite content (a swelling clay mineral). This presents many difficulties when trying to store, cut and test a material like Pierre shale. The following sections will describe some techniques that were developed to handle a drying sensitive shale, such as Pierre shale. 337

B.3

Drying and Sealing Techniques

As previously stated, Pierre shale is highly sensitive to drying, and the cutting techniques used for Opalinus shale would take too long to cut Pierre shale under atmospheric conditions and lead to drying out. Additional moisture barriers, or sealing techniques had to be developed to cut Pierre shale. A study was conducted to determine the effect of different moisture barriers on the water content loss (drying) of Pierre shale (figure B-6). These techniques included; dipping and coating the Pierre shale in silicon oil, coating the surface with silicon grease, and coating the surface with wax (Refer to figure B-6). The techniques were compared to a baseline case of doing nothing. Silicon oil had little effect on the drying processes. Using silicon grease on the surface of the sample showed improved reduced drying. Finally, dipping the specimen in silicon oil and using a grease coating or using wax coating showed a satisfactory reduction in drying. However, silicon grease can be messy and difficult to work with compared to wax. Therefore, it was determined from this study that coating the specimens with wax was the best way to preserve the water content of the specimens.

B.4

Cutting

Pierre shale was cut with a tabletop bandsaw (figure B-7). When cutting wax coated Pierre shale, the freshly cut surface is exposed to atmospheric drying. Therefore, the specimens of Pierre shale were dipped in wax after each cut to prevent cut surfaces from being dried out during cutting (figure B-8). After the specimens were fully cut and sealed with wax, they were sealed with vacuum sealed plastic as well. Additionally, wax covered Pierre shale is very soft and can become stuck in the teeth of the bandsaw blade. Therefore, a bi-metal 14-18 variable tooth bandsaw was used with pressurized air on the blade teeth and a passive spinning chip wheel to clean the blade (figure B-9). 338

Figure B-6: Moisture barrier techniques and their affect on drying of Pierre shale. (top) Photos are shown for the different moisture barrier techniques [A-E] proposed for Pierre shale and (bottom) the water content loss (drying) over time for each of these techniques. Wax was determined to be the best moisture barrier technique for sealing Pierre shale. 339

Figure B-7: Pierre shale being cut in the bandsaw. Pierre shale was cut with a 14-18 variable tooth bandsaw blade.

Figure B-8: Wax dipping technique used to seal freshly cut surfaces of Pierre shale. After the specimen was cut, a fresh surface was exposed to air. This (a) freshly cut surface was (b) dipped into wax, (c) removed to let it dry and therefore (d) sealed from losing any additional water.

340

Figure B-9: (a) Chip wheel and (b) pressurized air used to clean bandsaw blade when cutting Pierre shale.

B.5

Pierre Shale - Unconfined Compression Tests

After developing cutting and sealing procedures for Pierre shale, two unconfined compression tests were conducted on sealed specimens of Pierre shale ( 1 in x 2 in x 2 in). From these tests the compressive strength and elastic modulus were determined (figures B-10 and B-11). Although there were no pronounced bedding planes in Pierre shale, such as those seen in Opalinus shale, the specimens were loaded in the vertical direction (as they were extracted), for consistency. Prior to testing, the wax surface of one of the specimen (sdp-A) was shaved down thinly to remove some wax to better observe the cracking processes. Since these Pierre shale specimens failed at very low stresses, the data collection rate was far too slow (2 data points per second) for the applied loading rate (1,200 lb/min). Therefore the full shape of the stress-strain curve is not very well defined (Refer to figures B-10 and B-11). The average results of these unconfined compression tests are summarized in table B.4. Both the strength and stiffness of Pierre shale is very low (Refer to table B.4) compared to that of Opalinus shale (Refer to tables 3.1 and 4.6). The Young’s modulus for these tests on 341

Figure B-10: Pierre shale specimen sdp-A tested in unconfined compression. Pierre shale specimens were sealed in wax to prevent drying. The wax surface of the specimen sdp-A was shaved down thinly to remove some wax to better observe the cracking processes.

Figure B-11: Pierre shale specimen sdp-B tested in unconfined compression. Pierre shale specimens were sealed in wax to prevent drying. The wax surface of the specimen sdp-B was not shaved down. The textured pattern on the wax comes from the impression left by the vacuum sealing plastic material.

342

Pierre shale was determined using cross-head displacement, therefore, it includes the wax coating on top and bottom specimens of the surface. It is possible that the soft wax causes the stiffness to appear lower than it actually is. Table B.4: Water Content of Pierre Shale Samples Characterization Parameter Pierre Shale Young’s Modulus, E [MPa] 48 Uniaxial Compressive Strength, σc [MPa] 0.47

343

344

Appendix C Scaling Experiments on Prismatic Specimens This study investigates the effect of two differences when testing shale specimens compared to previously tested materials. Shale specimens are of smaller size (controlled by core size, see section 2.3.1) and the flaws will need to be cut out using a drill and scroll saw technique. The availability of certain materials, such as shale rock or highly over-consolidated clay, is very limited and testing specimens of the size tested previously by the MIT rock mechanics group may not be possible. It is therefore necessary to do tests with smaller sized specimens. This, however, requires that there is no significant size effect. For this reason tests were conducted to verify the strength and cracking mechanisms of different size specimens. Gypsum specimens of approximately one third the size (50mm x 25mm x 10mm) of the original sized specimens (150mm x 75mm x 25mm) were created by cutting small specimens out of previously molded full sized gypsum blocks (See figure C1). In addition to size effects, this study also investigates the effects of using a drill and scrollsaw technique to cut the flaws out of specimens on the cracking mechanisms in gyspum. Shale rocks with high clay content are sensitive to water and traditional flaw creation tech345

niques such as molding specimens or wet-jet cutting flaws cannot be used (See section 2.1). Therefore, the flaws were created by drilling a hole through the gypsum specimen and then cutting the flaw out with a scroll-saw. Cutting flaws out will introduce some defects and disturbance around the flaw which typically does not occur when using molded specimens in gypsum.

Figure C-1: Small scale gypsum specimens vs. full scale specimens. Comparison of small scale gypsum specimens to full scale gypsum specimens previously test by the MIT rock mechanics group. Small scale gypsum specimens (left) were approximately one third the size (50mm x 25mm x 10mm) of the traditionally sized specimens (150mm x 75mm x 25mm) (right) were created by cutting small specimens out of previously molded full sized gypsum blocks.

C.1

Intact Specimens

Three intact specimens were tested to determine any size effects on the strength of smaller specimen sizes (See figure C-2). All three experienced much lower strengths than previously reported by Wong (2008) and Janeiro (2009) (Table C.1). Usually we expect that smaller sized specimens should have higher strengths than larger ones (Baecher and Einstein, 1981). However, these tests experience out-of-plane splitting or crushing at the end 346

which may have caused failures to occur a lower than normal strengths (Refer to figure C2). It is believed that due to the much smaller cross sectional area of these specimens, the slightest misalignment with the machine can cause a large eccentricity on the specimen. Since previous specimens were much larger, even if the specimen was slightly off center, axial splitting or reduced strengths were typically not observed. It should also be noted that the specimens were cut using a saw and not molded, which introduces defects at the edges. Extra care was used when testing specimens with flaws but splitting and crushing was still observed in some tests which will be described in the following sections.

Figure C-2: Failure of intact small scale gypsum specimens. Specimens Gs-I-A (a) and Gs-I-B (b) split out of plane and specimen Gs-I-C (c) failed by crushing on one end.

Table C.1: Unconfined Compressive Strength of Intact Small Gypsum Specimens Compared to Normal Size Specimen by Wong (2008) and Janeiro (2009)

σc [MPa]

Gs-I-A

Gs-I-B

Gs-I-C

Average

Wong

Janeiro

24.04*

22.06*

19.85+

21.98

33.85

37.20

* Split

out of plane

+ Crushed

at the ends

347

C.2

30o Single Flaw

In addition to intact specimens, smaller size specimens with drilled and sawed flaws at 30o angles in gypsum were also tested. A series of three tests were conducted; two out of the three experienced premature failure due to splitting (See figures C-3b,c). One test showed very similar results to those of Wong (2008) (See figure C-3a). In this test, tensile wing cracks first developed at the flaw tips and then a secondary shear crack initiated at the flaw tip. Also, the failure stress of this test was very close to the average failure stress of specimens with single 30o flaws in full scaled gypsum tested by Wong (2008) (Table C.2).

Figure C-3: Failure of small scale gypsum specimens with 30o single drilled and sawed flaws. Specimen Gs-30-A (a) showed typical wing crack behavior and specimens Gs-30-B (b) and Gs-30-C (c) split out of plane. T = Tensile Crack - S = Shear Crack.

Table C.2: Peak Stress of 30o Single Flaw Scaled Gypsum Specimens Gs-30-A Peak Stress [MPa] 28.14

Gs-30-B Gs-30-C 19.61* 24.32*

* Split

1 Determined

Average 24.03

Wong1 28.57

out of plane from Tensile Crack Initiation Stress and Tensile Crack Initiation:Peak Stress Ratio

348

C.3

30o Co-Planar Flaws

Drilled and sawed flaw pairs were also tested in the scaled down gypsum specimens (See figure C-4). Two out of the three tests showed coalescence behavior, the third test failed prematurely due to crushing at one end. The first test (Refer to figure C-4a) showed direct shear coalescence, which is comparable to that observed by Wong (2008) for coplanar flaws oriented at 30o . The second test (figure figure C-4b) experienced indirect coalescence with a shear crack. Although Wong (2008) always observed direct shear cracking for this flaw pair geometry, indirect coalescence was typically observed for coplanar flaw pairs of slightly lower flaw angle (