Instabilities Modeling in Geomechanics

Instabilities Modeling in Geomechanics describes complex mechanisms which are frequently met in earthquake nucleation, geothermal energy production, nuclear waste disposal and CO2 sequestration. These mechanisms involve systems of non-linear differential equations that express the evolution of the geosystem (e.g. strain localization, temperature runaway, pore pressure build-up, etc.) at different length and time scales.

In order to study the evolution of a system and possible instabilities, it is essential to know the mathematical properties of the governing equations. Therefore, questions of the existence, uniqueness and stability of solutions naturally arise. This book particularly explores bifurcation theory and stability analysis, which are robust and rigorous mathematical tools that allow us to study the behavior of complex geosystems, without even explicitly solving the governing equations. The contents are organized into 10 chapters which illustrate the application of these methods in various fields of geomechanics.

Ioannis Stefanou is Associate Professor and Researcher at Laboratoire Navier, Ecole des Ponts Paris Tech, France.

Jean Sulem is Full Professor and Senior Researcher at Laboratoire Navier, Ecole des Ponts Paris Tech, France.

Introduction xi
Ioannis STEFANOU and Jean SULEM

Chapter 1. Multiphysics Role in Instabilities in Geomaterials: a Review 1
Tomasz HUECKEL

1.1. Introduction 1

1.2. General remarks 2

1.3. Solid phase material criteria 5

1.4. Material sample stability: experimental 10

1.5. Boundary value problems: uniqueness and stability at the field scale 19

1.5.1. Landslides 19

1.5.2. Thermal pressurization problem 24

1.5.3. Localization during drying of geomaterials 25

1.6. Conclusion 27

1.7. References 27

Chapter 2. Fundamentals of Bifurcation Theory and Stability Analysis 31
Ioannis STEFANOU and Sotiris ALEVIZOS

2.1. Introduction 31

2.2. Bifurcation and stability of dynamical systems 35

2.2.1. Definition of stability 36

2.2.2. Linear systems of ODEs 37

2.2.3. Nonlinear systems of ODEs 39

2.2.4. An example of LSA 41

2.3. Stability of two-dimensional linear dynamical systems 42

2.3.1. Classification of fixed points 43

2.3.2. Love mechanics: Romeo and Juliet 46

2.4. Common types of bifurcations 48

2.4.1. Saddle-node bifurcation 48

2.4.2. Transcritical bifurcation 50

2.4.3. Supercritical and subcritical pitchfork bifurcation 51

2.4.4. From one to two dimensions – limit cycles 53

2.4.5. Bifurcations in two dimensions – supercritical and subcritical Hopf bifurcation 54

2.4.6. Mathematical bifurcations in PDEs 59

2.5. From ODEs to PDEs 61

2.5.1. Deformation bands and the acoustic tensor 61

2.5.2. Deformation bands as an instability problem 65

2.6. Summary 68

2.7. Appendix 69

2.8. References 69

Chapter 3. Material Instability and Strain Localization Analysis 73
Jean SULEM

3.1. Introduction 73

3.2. Shear band model 75

3.2.1. Strain localization criterion 76

3.2.2. Strain localization, loss of ellipticity and vanishing speed of acceleration waves 79

3.3. Shear band formation in element tests on rocks 80

3.3.1. Drucker–Prager model 80

3.3.2. Non-coaxial plasticity 82

3.3.3. Cataclastic shear banding 82

3.3.4. Postlocalization behavior 83

3.4. Strain localization in fluid-saturated porous media 84

3.4.1. Strain localization criterion in fluid-saturated porous media 84

3.4.2. Stability analysis of undrained shear on a saturated layer 86

3.5. Conclusion 90

3.6. References 90

Chapter 4. Experimental Investigation of the Emergence of Strain Localization in Geomaterials 95
Pierre BÉSUELLE

4.1. Introduction 95

4.2. Methods 98

4.2.1. Digital image correlation 99

4.2.2. X-ray computed tomography 103

4.2.3. Experimental devices for in situ full-field measurements 104

4.3. Selected materials 110

4.3.1. Hostun sand 110

4.3.2. Caicos ooids sand 111

4.3.3. Vosges sandstone 111

4.3.4. Callovo–Oxfordian clayey rock 111

4.4. Strain localization in sands 112

4.4.1. Plane strain compression by FRS 112

4.4.2. Triaxial compression by X-ray CT and DIC 116

4.4.3. Triaxial compression by X-ray CT, the critical void ratio 122

4.5. Strain localization in porous rocks 124

4.5.1. Strain localization in Vosges sandstone 124

4.5.2. Strain localization in a clayey rock 130

4.6. Conclusion 135

4.7. References 136

Chapter 5. Numerical Modeling of Strain Localization 141
Panos PAPANASTASIOU and Antonis ZERVOS

5.1. Introduction 142

5.2. Cosserat continuum 145

5.2.1. Governing equations 145

5.2.2. Finite element formulation of Cosserat model 148

5.2.3. Material parameters 150

5.2.4. Failure in thick-walled cylinder test 151

5.2.5. Stability analysis of elliptical shape perforations 154

5.3. Gradient elastoplasticity 156

5.3.1. Governing equations 156

5.3.2. Finite element formulation 160

5.3.3. Material model 162

5.3.4. Modeling of the biaxial test 163

5.3.5. Modeling cavity expansion 167

5.4. Conclusion 169

5.5. Acknowledgments 170

5.6. References 170

Chapter 6. Numerical Modeling of Bifurcation: Applications to Borehole Stability, Multilayer Buckling and Rock Bursting 175
Euripides PAPAMICHOS

6.1. Introduction 175

6.2. Borehole stability 176

6.2.1. Primary loading path 177

6.2.2. Hole failure 180

6.2.3. Simulation of hollow cylinder experiments 183

6.3. Folding of elastic media as a bifurcation problem 187

6.3.1. Buckling of a layer under initial stress 188

6.3.2. Eigen-displacements and tractions at layer boundaries 190

6.3.3. Buckling of a layer system – the transfer matrix technique 191

6.3.4. Buckling of layered half-space 192

6.4. Axial splitting and spalling 194

6.4.1. Buckling of a half-space with surface parallel cracks 195

6.5. Conclusion 199

6.6. Acknowledgments 200

6.7. References 200

Chapter 7. Numerical Modeling of Multiphysics Couplings and Strain Localization 203
Frédéric COLLIN, Panagiotis KOTRONIS and Benoît PARDOEN

7.1. Introduction 203

7.2. Experimental evidences of strain localization 205

7.3. Regularization methods 205

7.3.1. Enrichment of the constitutive law 206

7.3.2. Enrichment of the kinematics 209

7.4. Coupled local second gradient model for microstructure saturated media 212

7.4.1. Balance equations for microstructure poromechanics 213

7.4.2. Coupled finite element formulation 219

7.4.3. Two-dimensional specimen under compression 224

7.5. Coupled local second gradient model for an unsaturated medium 229

7.5.1. Partial saturation conditions 229

7.5.2. Anisotropy of the intrinsic permeability 230

7.5.3. Compressibility of the solid grains 231

7.6. Modeling of a gallery excavation 233

7.6.1. Numerical model 233

7.6.2. Influence of stress and permeability anisotropies 237

7.6.3. Influence of second gradient boundary condition 239

7.6.4. Influence of Biot’s coefficient 239

7.6.5. Influence of gallery ventilation 240

7.7. Conclusion 246

7.8. References 246

Chapter 8. Multiphysics Couplings and Strain Localization in Geomaterials 253
Jean SULEM and Ioannis STEFANOU

8.1. Introduction 253

8.2. Thermo-chemo-chemical couplings and stability of shear zones 255

8.2.1. Problem statement 255

8.2.2. Stability of adiabatic undrained shear 257

8.2.3. Chemical weakening and earthquake nucleation 259

8.3. Dissolution weakening and compaction banding 264

8.3.1. Multiscale modeling of strong chemo-poro-mechanical coupling 264

8.3.2. Compaction banding in oedometric compression 268

8.4. Conclusion 273

8.5. References 274

Chapter 9. On the Thermo-poro-mechanics of Chemically Active Faults 279
Manolis VEVEAKIS

9.1. Introduction 280

9.2. Time-independent formation of shear zones from solid mechanics 282

9.2.1. Shear zone thickness at boundary temperature conditions 283

9.2.2. Shear zone thickness at elevated temperature 284

9.3. Time-dependent evolution of shear zones 285

9.3.1. Energy considerations 287

9.3.2. The Taylor–Quinney coefficient 288

9.3.3. Chemical reactions 289

9.4. Postfailure evolution of a shear zone 290

9.4.1. Analysis of the system’s response 293

9.4.2. Time scales of the system 295

9.5. Comparison to field observations 296

9.6. Application to ETS sequences 298

9.6.1. Regular sequences – Cascadia ETS sequence 299

9.7. Discussion 302

9.8. Appendix: poro-chemical model 305

9.9. References 306

Chapter 10. Analysis of Instabilities in Faults 313
Hadrien RATTEZ, Ioannis STEFANOU, Jean SULEM, Manolis VEVEAKIS and Thomas POULET

10.1. Introduction 314

10.2. Description of the model 316

10.2.1. Cosserat continuum theory 316

10.2.2. Constitutive equations for a Cosserat continuum 317

10.2.3. Mass balance equation 319

10.2.4. Energy balance equation 319

10.3. Bifurcation analysis 320

10.3.1. LSA for a Cosserat continuum with THM couplings 320

10.3.2. Localization conditions for a fault zone 322

10.3.3. Shear band thickness evolution in a fault zone 324

10.4. Numerical analysis 326

10.4.1. Regularization of the mesh dependency 326

10.4.2. Response and shear band thickness of a fault gouge 329

10.5. Conclusion 334

10.6. Bibliography 334

List of Authors 337

Index 339

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