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Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1998[/DP]End of Citation title: author: publisher: isbn10 | asin: print isbn13: ebook isbn13: language: subject publication date: lcc: ddc: subject:
The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media Lewis, R. W. 0471928097 9780471928096 9780585294810
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The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media Second Edition R W. Lewis University of Wales Swansea, UK B. A. Schrefler University of Padua, Italy
Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1998[/DP]End of Citation
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Copyright © 1998 by John Wiley & Sons Ltd, Baffins Lane, Chichester, West Sussex P019 IUD, England National 01243 779777 International (+44) 1243 779777 e-mail (for orders and customer service enquiries):
[email protected] Visit our Home Page on http://www.wiley.co.uk or http://www.wiley.com 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. Other Wiley Editorial Offices John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012. USA Wiley-VCH Verlag GmbH, Pappelallee 3, D-69469 Weinheim, Germany Jacaranda Wiley Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01. Jin Xing Distripark, Singapore 0512 John Wiley & Sons (Canada) Ltd, 22 Worcester Road, Rexdale, Ontario M9W ILI. Canada 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
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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 Delhi Printed 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. Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1998[/DP]End of Citation
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For Celia and Chantal Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1998[/DP]End of Citation
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Contents Preface
xv
1 Introduction References 2 Mechanics of Saturated and Partially Saturated Porous Media
1 5 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 2.5.3.1 Enthalpy Balance Equation for the Multiphase Medium 2.5.4 Summary of Governing Equations
50 53 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 3.1 Introduction
75 75
3.2 Coupled Solution: Saturated One-Phase Flow in a Deforming Porous Medium 75 3.2.1 Governing Equations
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 3.8.1 Governing Equations
87
3.8.2 Initial and Boundary Conditions
88
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
3.11.1 Governing Equations
93
3.11.2 Initial and Boundary Conditions
94
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 4.1 Introduction
99 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 6.1 Introduction
177 177
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 6.3 Far-Field Boundary Conditions 6.3.1 Infinite Elements
188 192 193
6.4 A Coupled Solution for the Settlement above Gas Reservoirs
197
6.5 Single-Aquifer Withdrawal
201
6.5.1 Isolated Aquifer
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6.5.2 Embedded Aquifer
206
6.6 Conclusions
208
References
209
7 Modelling Subsidence: Case Studies
213
7.1 Introduction
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
7.3.1 Background
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
7.4.1 Background
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
7.5.1 Background
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
8.1 Introduction
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
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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
9.1 Introduction
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
10.1 Introduction
341
10.2 Non-Isothermal Fully Saturated Consolidation
344
10.2.1 Governing Equations
344
10.2.2 Initial and Boundary Conditions
345
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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 10.5 Non-Isothermal Airflow and Water Flow in a Deforming Porous Medium
352 354
10.5.1 Governing Equations
354
10.5.2 Initial and Boundary Conditions
356
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
11.1 Introduction
397
11.2 Formulation of Secondary Consolidation
398
11.3 Application of the Creep Model
400
11.3.1 Background
400
11.3.2 Mathematical Modelling
403
11.3.3 Results
405
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References
408
12 Soil-Structure Interaction
409
12.1 Introduction
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
12.5 Conclusions
425
References
425
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13 Back Analysis in Consolidation
427
13.1 Introduction
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 13.5 Case Study
434 436
13.5.1 Background
436
13.5.2 Hypothetical Case Study
437
13.6 Summary
442
References
443
14 Large-Strain Quasi-Static and Dynamic Soil Behaviour
445
14.1 Introduction
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
14.8 Conclusions
473
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. LEWIS BERNARD A. SCHREFLER SWANSEA/ PADUA JULY 1997 Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1998[/DP]End of Citation
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