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[资源] Poromechanics

Contents
Preface xi
Acknowledgements xiii
1 Deformation and Kinematics. Mass Balance 1
1.1 The Porous Medium and the Continuum Approach . . . . . . . . . . . . . 1
1.1.1 Connected and Occluded Porosity. The Matrix . . . . . . . . . . . 1
1.1.2 Skeleton and Fluid Particles. Continuity Hypothesis . . . . . . . . 2
1.2 The Skeleton Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Deformation Gradient and Transport Formulae . . . . . . . . . . . 2
1.2.2 Eulerian and Lagrangian Porosities. Void Ratio . . . . . . . . . . 5
1.2.3 Strain Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.4 Infinitesimal Transformation and the Linearized Strain Tensor . . 7
1.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Particle Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Strain Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Mass Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.1 Equation of Continuity . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.2 The Relative Flow Vector of a Fluid Mass. Filtration Vector. Fluid
Mass Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Advanced Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5.1 Particle Derivative with a Surface of Discontinuity . . . . . . . . 14
1.5.2 Mass Balance with a Surface of Discontinuity. The Rankine–
Hugoniot Jump Condition . . . . . . . . . . . . . . . . . . . . . . 15
1.5.3 Mass Balance and the Double Porosity Network . . . . . . . . . . 17
2 Momentum Balance. Stress Tensor 19
2.1 Momentum Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 The Hypothesis of Local Forces . . . . . . . . . . . . . . . . . . . 19
2.1.2 TheMomentum Balance . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.3 The Dynamic Theorem. . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 The Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1 Action–Reaction Law . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.2 The Tetrahedron Lemma and the Cauchy Stress Tensor . . . . . . 22
vi CONTENTS
2.3 Equation ofMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.1 The Local Dynamic Resultant Theorem. . . . . . . . . . . . . . . 24
2.3.2 The Dynamic Moment Theorem and the Symmetry
of the Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.3 Partial Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Kinetic Energy Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.1 StrainWork Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.2 Piola–Kirchhoff Stress Tensor . . . . . . . . . . . . . . . . . . . . 29
2.4.3 Kinetic Energy Theorem . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Advanced Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.1 The Stress Partition Theorem . . . . . . . . . . . . . . . . . . . . 30
2.5.2 Momentum Balance and the Double Porosity Network . . . . . . 32
2.5.3 The Tortuosity Effect . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Thermodynamics 37
3.1 Thermostatics of Homogeneous Fluids . . . . . . . . . . . . . . . . . . . 37
3.1.1 Energy Conservation and Entropy Balance . . . . . . . . . . . . . 37
3.1.2 Fluid State Equations. Gibbs Potential . . . . . . . . . . . . . . . 39
3.2 Thermodynamics of Porous Continua . . . . . . . . . . . . . . . . . . . . 40
3.2.1 Postulate of Local State . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.2 The First Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.3 The Second Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Conduction Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.1 Darcy’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.2 Fourier’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4 Constitutive Equations of the Skeleton . . . . . . . . . . . . . . . . . . . 51
3.4.1 State Equations of the Skeleton . . . . . . . . . . . . . . . . . . . 51
3.4.2 Complementary Evolution Laws . . . . . . . . . . . . . . . . . . . 54
3.5 Recapitulating the Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.6 Advanced Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.6.1 Fluid Particle Head. Bernoulli Theorem. . . . . . . . . . . . . . . 59
3.6.2 Thermodynamics and the Double Porosity Network . . . . . . . . 60
3.6.3 Chemically Active Porous Continua . . . . . . . . . . . . . . . . . 61
4 Thermoporoelasticity 71
4.1 Non-linear Thermoporoelastic Skeleton . . . . . . . . . . . . . . . . . . . 71
4.1.1 Infinitesimal Transformation and State Equations . . . . . . . . . 71
4.1.2 Tangent Thermoporoelastic Properties . . . . . . . . . . . . . . . . 73
4.1.3 The Incompressible Matrix and the Effective Stress . . . . . . . . 74
4.2 Linear Thermoporoelastic Skeleton . . . . . . . . . . . . . . . . . . . . . 75
4.2.1 Linear Thermoporoelasticity . . . . . . . . . . . . . . . . . . . . . 75
4.2.2 Isotropic Linear Thermoporoelasticity . . . . . . . . . . . . . . . . 75
4.2.3 Relations Between Skeleton and Matrix Properties . . . . . . . . . 78
4.2.4 Anisotropic Poroelasticity . . . . . . . . . . . . . . . . . . . . . . 81
4.3 Thermoporoelastic Porous Material . . . . . . . . . . . . . . . . . . . . . 84
4.3.1 Constitutive Equations of the Saturating Fluid . . . . . . . . . . . 84
4.3.2 Constitutive Equations of the Porous Material . . . . . . . . . . . 84
CONTENTS vii
4.4 Advanced Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.4.1 Non-linear Isotropic Poroelasticity . . . . . . . . . . . . . . . . . 87
4.4.2 Brittle Fracture of Fluid-infiltrated Materials . . . . . . . . . . . . 93
4.4.3 From Poroelasticity to the Swelling of Colloidal Mixtures . . . . . 98
4.4.4 From Poroelasticity to Chemoelasticity and Ageing Materials . . . 108
5 Problems of Poroelasticity 113
5.1 Linearized Poroelasticity Problems . . . . . . . . . . . . . . . . . . . . . 113
5.1.1 The Hypothesis of Small Perturbations . . . . . . . . . . . . . . . 113
5.1.2 Field Equations and Boundary Conditions . . . . . . . . . . . . . 115
5.1.3 The Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . 116
5.2 Solved Problems of Poroelasticity . . . . . . . . . . . . . . . . . . . . . . 117
5.2.1 Injection of a Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2.2 Consolidation of a Soil Layer . . . . . . . . . . . . . . . . . . . . 125
5.2.3 Drilling of a Borehole . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3 Thermoporoelasticity Problems . . . . . . . . . . . . . . . . . . . . . . . 133
5.3.1 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.3.2 Half-space Subjected to a Change in Temperature . . . . . . . . . 135
5.4 Advanced Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.4.1 Uniqueness of Solution . . . . . . . . . . . . . . . . . . . . . . . 137
5.4.2 The Beltrami–Michell Equations . . . . . . . . . . . . . . . . . . 140
5.4.3 Mandel’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.4.4 Non-linear Sedimentation . . . . . . . . . . . . . . . . . . . . . . 145
6 Unsaturated Thermoporoelasticity 151
6.1 Mass andMomentum Balance . . . . . . . . . . . . . . . . . . . . . . . . 151
6.1.1 Partial Porosities and Degree of Saturation . . . . . . . . . . . . . 151
6.1.2 Mass andMomentum Balance . . . . . . . . . . . . . . . . . . . . 152
6.1.3 Mass and Momentum Balance with Phase Change . . . . . . . . . 152
6.2 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.2.1 Energy and Entropy Balance for the Porous Material . . . . . . . 153
6.2.2 Skeleton State Equations. Averaged Fluid Pressure
and Capillary Pressure . . . . . . . . . . . . . . . . . . . . . . . . 154
6.2.3 Thermodynamics of Porous Media with Phase Change . . . . . . 156
6.3 Capillary Pressure Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.3.1 Energy Approach to the Capillary Pressure Curve . . . . . . . . . 157
6.3.2 Capillary Pressure, Natural Imbibition and Isotherm of Sorption . 160
6.4 Unsaturated Thermoporoelastic Constitutive Equations . . . . . . . . . . . 162
6.4.1 Energy Separation and the Equivalent Pore Pressure Concept . . . 162
6.4.2 Equivalent Pore Pressure and Averaged Fluid Pressure . . . . . . 163
6.4.3 Equivalent Pore Pressure and Thermoporoelastic Constitutive
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.4.4 Equivalent Pore Pressure, Wetting and Free Swelling
ofMaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.5 Heat and Mass Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.5.1 Fourier’s Law, Thermal Equation and Phase Change . . . . . . . 167
6.5.2 Darcy’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.5.3 Fick’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
viii CONTENTS
6.6 Advanced Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.6.1 The Stress Partition Theorem in the Unsaturated Case . . . . . . . 171
6.6.2 Capillary Hysteresis. Porosimetry . . . . . . . . . . . . . . . . . . 174
6.6.3 Capillary Pressure Curve, Deformation and Equivalent
Pore Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.6.4 Isothermal Drying of Weakly Permeable Materials . . . . . . . . . 180
7 Penetration Fronts 189
7.1 Dissolution Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7.1.1 Mass Balance and Fick’s Law for the Solute . . . . . . . . . . . . 190
7.1.2 Instantaneous Dissolution and the Formation
of a Penetration Front . . . . . . . . . . . . . . . . . . . . . . . . 191
7.1.3 Stefan-like Problem . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.2 Solute Penetration with Non-linear Binding . . . . . . . . . . . . . . . . . 194
7.2.1 The Binding Process and the Formation of a Penetration Front . . 194
7.2.2 The Time Lag and the Diffusion Test . . . . . . . . . . . . . . . . 197
7.3 IonicMigration with Non-linear Binding . . . . . . . . . . . . . . . . . . 200
7.3.1 Ionic Migration in the Advection Approximation . . . . . . . . . . 200
7.3.2 The Travelling Wave Solution . . . . . . . . . . . . . . . . . . . . 202
7.3.3 The Time Lag and theMigration Test . . . . . . . . . . . . . . . 205
7.4 Advanced Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
7.4.1 Stefan-like Problem with Non-instantaneous Dissolution . . . . . . 209
7.4.2 Imbibition Front . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
7.4.3 Surfaces of Discontinuity and Wave Propagation . . . . . . . . . . 218
8 Poroplasticity 225
8.1 Poroplastic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
8.1.1 Plastic Strain and Plastic Porosity . . . . . . . . . . . . . . . . . . 225
8.1.2 Poroplastic State Equations for the Skeleton . . . . . . . . . . . . 226
8.1.3 Poroplastic State Equations for the Porous Material . . . . . . . . 227
8.1.4 Domain of Poroelasticity and the Loading Function.
Ideal and Hardening PoroplasticMaterial . . . . . . . . . . . . . . 229
8.2 Ideal Poroplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
8.2.1 The Flow Rule and the PlasticWork . . . . . . . . . . . . . . . . 230
8.2.2 Principle of Maximal Plastic Work and the Flow Rule. Standard
and Non-standardMaterials . . . . . . . . . . . . . . . . . . . . . 231
8.3 Hardening Poroplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
8.3.1 Hardening Variables and Trapped Energy . . . . . . . . . . . . . . 233
8.3.2 Flow Rule for the Hardening Material. Hardening Modulus . . . . 234
8.4 UsualModels of Poroplasticity . . . . . . . . . . . . . . . . . . . . . . . 237
8.4.1 Poroplastic Effective Stress . . . . . . . . . . . . . . . . . . . . . 237
8.4.2 Isotropic and Kinematic Hardening . . . . . . . . . . . . . . . . . 237
8.4.3 The Usual Cohesive–Frictional Poroplastic Model . . . . . . . . . 238
8.4.4 The Cam–ClayModel . . . . . . . . . . . . . . . . . . . . . . . . 244
8.5 Advanced Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
8.5.1 Uniqueness of Solution . . . . . . . . . . . . . . . . . . . . . . . 248
8.5.2 Limit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
CONTENTS ix
8.5.3 Thermal and Chemical Hardening . . . . . . . . . . . . . . . . . . 252
8.5.4 Localization of Deformation . . . . . . . . . . . . . . . . . . . . . 256
9 Poroviscoelasticity 261
9.1 Poroviscoelastic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . 261
9.1.1 Viscous Strain and Viscous Porosity . . . . . . . . . . . . . . . . 261
9.1.2 Poroviscoelastic State Equations for the Skeleton . . . . . . . . . 262
9.1.3 Complementary Evolution Laws . . . . . . . . . . . . . . . . . . . 263
9.2 Functional Approach to Linear Poroviscoelasticity . . . . . . . . . . . . . 263
9.2.1 Creep Test. Instantaneous and Relaxed Properties.
The Trapped Energy . . . . . . . . . . . . . . . . . . . . . . . . . 263
9.2.2 Creep and Relaxation Functions . . . . . . . . . . . . . . . . . . . 265
9.2.3 Poroviscoelastic Properties and Constituent Properties . . . . . . . 268
9.3 Primary and Secondary Consolidation . . . . . . . . . . . . . . . . . . . . 269
9.4 Advanced Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
9.4.1 Poroviscoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 272
9.4.2 Functional Approach to the Thermodynamics
of Poroviscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . 274
A Differential Operators 279
A.1 Orthonormal Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . 279
A.2 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
A.3 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Bibliography 285
Index 293
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