Lewis, R. W. and Schrefler, B. a. - The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, 2nd Edition

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Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1998[/DP]End of Citation

title: The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media

author: Lewis, R. W.publisher:

isbn10 | asin: 0471928097print isbn13: 9780471928096

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The Finite Element Method in the Static and Dynamic Deformation and Consolidation of PorousMedia

Second Edition

R W. Lewis

University of Wales Swansea, UK

B. A. Schrefler

University of Padua, Italy

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Copyright © 1998 by John Wiley & Sons Ltd,Baffins Lane, Chichester, West Sussex P019 IUD, England

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Reprinted January 2000

All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright. Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency, 90 Tottenham Court Road. London WIP 9HE. UK, without the permission in writing of the Publisher.

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Library of Congress Cataloging-in-Publication Data

Lewis, R. W. (Roland Wynne) The finite element method in the static and dynamic deformation and consolidation of porous media / R. W. Lewis, B. A. Schrefler. 2nd ed. p. cm. Rev. ed. of: The finite element in the deformation and consolidation of porous media / Roland W. Lewis, Bernard A. Schrefler. 1987. Includes bibliographical references and index. ISBN 0-471-92809-7 1. Porous materials-Mathematical models. 2. Finite element method. 3. Multiphase flow-Mathematical models. I. Schrefler, B. A.II. Lewis, R. W. (Roland Wynne). Finite element method in the deformation and consolidation of porous media. III. Title.

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TA418.9.P6L49 1998 624.1 '5136-dc21 98-12080 CIP

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN 0 471 92809 7

Typeset on 10/12pt Times by Thomson Press (India) Ltd., New DelhiPrinted and bound in Great Britain by Bookcraft (Bath) Ltd This book is printed on acid-free paper responsibly manufactured from sustainable forestry, in which at least two trees are planted for each one used for paper production.

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Page v

For Celia and Chantal

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Page vii

Contents

Preface xv

1 Introduction 1

References 5

2 Mechanics of Saturated and Partially Saturated Porous Media 9

2.1 Introduction 9

2.2 Averaging Principles 9

2.2.1 Averaging Process 11

2.2.2 Microscopic Balance Equations 13

2.2.3 Macroscopic Balance Equations 14

2.3 Macroscopic Balance Equations for a Non-Isothermal Partially Saturated Porous Material 18

2.3.1 Kinematic Equations 18

2.3.2 Mass Balance Equations 21

2.3.2.1 Solid Phase 21

2.3.2.2 Liquid Phase: Water 22

2.3.2.3 Gaseous Phases: Dry Air and Vapour 22

2.3.3 Linear Momentum Balance Equation 24

2.3.4 Angular Momentum Balance Equation 28

2.3.5 Balance of Energy Equation 29

2.3.6 Entropy Inequality 32

2.4 Constitutive Equations 34

2.4.1 Stress Tensor in the Fluid Phases 35

2.4.2 Gaseous Mixture of Dry Air and Water Vapour 35

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2.4.3 Sorption Equilibrium 35

2.4.4 Clausius-Clapeyron Equation 36

2.4.5 Pore Size Distribution 37

2.4.6 Equation of State for Water 37

2.4.7 Darcy's Law 39

2.4.8 Fick's Law 40

2.4.9 Stress Tensor in the Solid Phase and Total Stress 41

2.4.10 Solid Density 43

2.4.11 Fourier's Law 44

2.5 General Field Equations 44

2.5.1 Mass Balance Equation 44

2.5.2 Linear Momentum Balance Equation 48

2.5.2.1 Fluids 48

2.5.2.2 Solid Phase 49

2.5.2.3 Multiphase Medium 49

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2.5.3 Energy Balance Equation 50

2.5.3.1 Enthalpy Balance Equation for the Multiphase Medium 53

2.5.4 Summary of Governing Equations 54

2.5.4.1 Mass Balance Equations or Continuity Equations 54

2.5.4.2 Linear Momentum Balance Equations 56

2.5.4.3 Enthalpy Balance: Multiphase Medium 56

2.6 Physical Approach: Extended Biot Theory 56

2.6.1 The Physical Model 57

2.6.2 Constitutive Equations 61

2.6.3 Governing Equations 63

2.6.3.1 Linear Momentum Balance Equation: Multiphase Medium 63

2.6.3.2 Mass Balance Equations 64

2.6.3.3 Energy Balance Equation 66

References 68

Appendix 2A 71

Appendix 2B 72

Appendix 2C 72

3 Numerical Solution for Isothermal Consolidation 75

3.1 Introduction 75

3.2 Coupled Solution: Saturated One-Phase Flow in a Deforming Porous Medium 75


3.2.2 Initial and Boundary Conditions 77

3.3 Solution of the Boundary Value Problem 77

3.4 Application of the Finite Element Method 79

3.5 Choice of Elements 83

3.6 Discretisation in Time 84

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3.7 Numerical Properties of the Time Discretisation 85

3.8 Saturated-Unsaturated Flow in a Deforming Porous Medium: One-Phase Flow 86



3.9 Discretisation of the Governing Equations for the Consolidation of Partially Saturated Soils 88

3.10 Stability, Convergence and Consistency in the Non-Linear Case 89

3.11 Airflow and Water Flow in a Deforming Porous Medium 93



3.12 Discretisation of the Governing Equations for Air and Water Flow in Deforming Porous Media 95

References 97

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4 Solid-Phase Constitutive Relationships, Variable Permeabilities and Solution Procedures 99

4.1 Introduction 99

4.2 Stress Invariants 100

4.3 Linear Elastic Analysis 102

4.4 Variable Elastic Analysis 103

4.4.1 Bilinear Models 104

4.4.2 Variable Elastic Model 105

4.4.2.1 Hyperbolic Model 105

4.4.2.2 E-v and K-G Variable Elastic Models 107

4.4.2.3 Spline functions 107

4.4.3 Thermo-Elastic Behaviour 107

4.4.4 Solution Procedures 109

4.5 Elastoplastic Models 112

4.5.1 Constitutive Law 112

4.5.2 Mohr-Coulomb Yield Surface 114

4.5.3 Critical State Model 118

4.5.3.1 Modified Cam Clay Model 118

4.5.3.2 p-q-θ Critical State Model 122

4.5.4 Corners of Yield and Potential Surfaces 124

4.5.5 Generalised Plasticity 124

4.5.6 Thermo-Plastic Behaviour 126

4.5.7 Solution Procedures 129

4.5.7.1 Explicit Algorithms 129

4.5.7.2 Implicit Algorithms 130

4.5.7.3 Consistent Stiffness Matrix 131

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4.6 Partially Saturated Models 131

4.6.1 Elastic Behaviour 133

4.6.2 Plastic Behaviour 134

4.7 Variation of Permeability 136

4.8 Conclusions 139

References 139

5 Verification of Elastic and Elastoplastic Consolidation Programs 145

5.1 Introduction 145

5.2 Elastic Solutions for Drained and Undrained Conditions 146

5.2.1 Plane Strain, Uniform Loading 147

5.2.2 Radially Symmetric, Uniform Loading 149

5.3 Elastic Analysis of Consolidation under Strip and Circular Uniform Loading 152

5.4 Elastoplastic Solutions 154

5.4.1 Undrained Triaxial Tests On Normally Consolidated Soil 154

5.4.1.1 Mohr-Coulomb Analysis 154

5.4.1.2 Critical State Ellipse Analysis 155

5.4.2 Drained and Undrained Analyses of Strip Loading 156

5.4.2.1 Mohr-Coulomb Analysis 156

5.4.2.2 Critical State Ellipse Analysis 157

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5.4.2.3 Analysis by Critical State Ellipse, with Mohr-Coulomb Cut-off (c = 0) 158

5.5 Elastoplastic Analysis of Consolidation under Uniform Loading and Strip Loading 158

5.5.1 One-Dimensional Consolidation 159

5.5.2 Two-Dimensional Consolidation 161

5.6 Linear and Non-Linear Elastic Consolidation with Variable Permeability 163

5.6.1 Consolidation of Swansea Blue Clay in a Rowe Consolidation Cell 163

5.6.2 Consolidation of Kaolin in a Rowe Consolidation Cell 165

5.7 Multiphase Flow in Porous Media: A Benchmark Problem for Non-Saturated Flow 167

5.8 Conclusions 174

References 175

6 Modelling Subsidence: Numerical Aspects and Problems of Regional Scale 177


6.1.1 More about Coupling and Staggered Procedures 181

6.2 Problems of Regional Scale: Vertically Averaged Models 184

6.2.1 Spatially Averaged Quantities: The Megascopic Level 184

6.2.2 Macrolevel Governing Equations 186

6.2.2.1 Equilibrium Equation for the Two-Phase Medium 186

6.2.2.2 Fluid-Phase Behaviour 187

6.2.3 Implementation of the Numerical Model 188

6.3 Far-Field Boundary Conditions 192

6.3.1 Infinite Elements 193

6.4 A Coupled Solution for the Settlement above Gas Reservoirs 197

6.5 Single-Aquifer Withdrawal 201

6.5.1 Isolated Aquifer 202

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6.5.2 Embedded Aquifer 206

6.6 Conclusions 208

References 209

7 Modelling Subsidence: Case Studies 213


7.2 The Subsidence of Venice 214

7.2.1 Background 214

7.2.2 The Mathematical Model 220

7.2.3 Results 224

7.3 Subsidence in the Po Delta and the Polesine 230


7.3.2 The Contarina Model 232

7.3.2.1 Available Data 232

7.3.2.2 Parametric Investigation 235

7.3.2.3 Subsidence Rebound 245

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7.4 Subsidence above Gas Reservoirs: The Ravenna Case 248


7.4.2 Results 251

7.4.3 Comparison with a Volumetric Reservoir 257

7.4.4 New Results for the Ravenna Field 262

7.5 Subsidence of Abano Terme 268


7.5.2 The Mathematical Model 273

7.6 Conclusions 276

References 277

8 Modelling Three-Phase Flow in Deforming Saturated Oil Reservoirs 281


8.2 Development of the Governing Equations 282

8.2.1 The Equilibrium Equation for a Three-Phase System 282

8.2.2 Three-Phase Flow Equations 283

8.3 Application of the Finite Element Method 284

8.4 Numerical Procedures 287

8.4.1 Treatment of Fluid Non-Linear Terms 287

8.4.2 Stability Analysis 288

8.4.3 Mass Balance and Convergence Checks 289

8.4.4 Computational Procedures 290

8.5 Validation and Applications 290

8.5.1 Non-Linear Soil Column Analysis 291

8.5.2 Reservoir Compaction Problems 293

8.5.3 Surface Subsidence Analyses 298

8.5.3.1 Effect of Reservoir Parameters 300

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8.5.3.2 Effect of Water Injection Schemes On Subsidence Analysis 302

8.6 Conclusions 304

References 305

9 Fractured Reservoir Simulation 307


9.2 Description of the Model 308

9.3 Development of the Governing Equations 310

9.4 A special Case: Single-Phase Flow in a Deforming Fractured Porous Medium 312

9.5 Discretisation in Space 313

9.6 Validation of the Model 322

References 338

10 Heat and Fluid Flow in Deforming Porous Media 341


10.2 Non-Isothermal Fully Saturated Consolidation 344



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10.3 Discretisation for Non-Isothermal Consolidation of Saturated Porous Media 346

10.4 Solution Procedures 348

10.4.1 Monolithic Augmentation Approach 349

10.4.2 Partitioned Solution Procedures 349

10.4.2.1 Numerical Properties of Partitioned Procedures 352

10.5 Non-Isothermal Airflow and Water Flow in a Deforming Porous Medium 354



10.6 Discretisation for Airflow and Water Flow in a Deforming Porous Medium 358

10.7 Numerical Examples 364

10.7.1 Thermo-Elastic Consolidation 364

10.7.2 Thermo-Elastoplastic Consolidation 368

10.7.3 Thermo-Elastic Consolidation around a Cylindrical Heat Source 370

10.7.4 Non-Isothermal Consolidation 376

10.7.5 Thermo-Elastic Consolidation of Partially Saturated Clay 379

10.8 Conclusions 386

References 386

Appendix 10A 389

Appendix 10B 394

11 Secondary Consolidation Creep in Solids 397


11.2 Formulation of Secondary Consolidation 398

11.3 Application of the Creep Model 400


11.3.2 Mathematical Modelling 403

11.3.3 Results 405

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References 408

12 Soil-Structure Interaction 409


12.2 Governing Equations 410

12.3 Material Models 412

12.3.1 Interface Behaviour 412

12.3.2 Soil Behaviour 413

12.4 Applications 414

12.4.1 Test 1: Shallow Foundation 414

12.4.2 Test 2: Pile-Soil Interaction 418

12.4.3 Test 3: Frame on Soft Soil 421

12.4.3.1 Interaction of Two Adjacent Footings 424

12.4.3.2 Eccentric Loading 425

12.4.3.3 Horizontal Loading 425


References 425

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13 Back Analysis in Consolidation 427


13.2 Definition of Back Analysis 428

13.3 Methodology 429

13.3.1 Direct Method 429

13.3.2 Indirect Method 430

13.3.3 Probabilistic Approaches 431

13.3.4 Alternative Methods 431

13.4 Parameter Identification 432

13.4.1 Optimisation Methods 433

13.4.1.1 Simplex Method 433

13.4.1.2 Rosenbrock's Algorithm 434

13.4.1.3 Levenberg-Marquardt Method 434

13.4.2 Sensitivity Analysis 434

13.5 Case Study 436


13.5.2 Hypothetical Case Study 437

13.6 Summary 442

References 443

14 Large-Strain Quasi-Static and Dynamic Soil Behaviour 445


14.2 Kinematic Equations 445

14.3 Constitutive Equations 448

14.4 Governing Equations and Their Weak Form 451

14.5 The Rate Form of Stress Power 453

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14.6 Finite Element Discretisation 453

14.6.1 Spatial Discretisation 454

14.6.2 Discretisation in Time and Solution Procedure 456

14.7 Examples 459

14.7.1 Finite-Strain and Small-Strain Fully Saturated Consolidation 459

14.7.2 Finite-Strain and Small-Strain Partially Saturated Consolidation 461

14.7.3 Slope under Seismic Behaviour: Finite Strains 463

14.7.4 Dynamic Strain Localisation 464


References 474

Subject Index 477

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Preface

Our first text on this subject 'The Finite Element Method in the Deformation and Consolidation of Porous Media', was published ten years ago and has been out of print for much of the past decade. It was the first book of its kind, despite the many available texts on groundwater flow through deforming porous media. The topic has been covered, albeit briefly, in many texts on geomechanics, petroleum engineering and finite element methods. However, there still exists no other book which covers all the mechanical and numerical aspects of flow in porous media in such detail.

In the intervening period there was a rapid expansion in the research and practical applications of these types of problem, which has prompted us to write this new and thoroughly updated version. It contains not only the results of research carried out at our two institutions but also reports on the work done under various European research programmes, e.g. Science (Greco Geomateriaux), TEMPUS PHARE (with the Technical University of Lodz and the Polish Academy of Sciences IPPT-PAN), and in particular Human Capital and Mobility, where an Alliance of Laboratories in Europe for Research and Technology (ALERT) was created, concentrating on research in geomaterials (soil, rock and concrete). Both our institutions were partners in this network, and the scientific exchanges proved to be extremely fruitful. Also, collaborative work carried out with the Norwegian Geotechnical Institute, under the BRINORD agreement, contributed to a better understanding of petroleum reservoir subsidence.

The chapters from the previous edition have been extensively updated and several new chapters have been added to give a much broader coverage of recent research interests. The theoretical part of the book is completely new: it now incorporates both phenomenological and averaging approaches.

We are indebted to many of our coworkers and in particular we thank Drs N. Abd. Rahman, P. Baggio, G. Bolzon, D. Gawin, H.R. Ghafouri, C.E. Majorana, E.A. Meroi, R.S. Ransing, V. Salomoni, L. Sanavia, L. Simoni, Y. Sukirman. D.V. Tran, E. Turska, X. Wang, X. Zhan, H.W. Zhang and Y. Zheng, who over the years have contributed to the work, Also, many thanks to Drs S.M. Hassanizadeh and D. Pigozzi for their advice on the theoretical chapter.

Finally, we would like to dedicate this book to the two ladies in our lives, Celia and Chantal, without whom it might never have been completed.

ROLAND W. LEWISBERNARD A. SCHREFLERSWANSEA/ PADUA JULY 1997

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Lewis, R. W. and Schrefler, B. a. - The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, 2nd Edition