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Poromechanics Poromechanics Poromechanics Olivier Coussy Laboratoire Central des Ponts et Chauss´ees Institut Navier, Marne-la-Vall´ee, France Copyright c 2004 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wileyeurope.com or www.wiley.com 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 Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to [email protected], or faxed to (+44) 1243 770620. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-470-84920-7 Produced from LaTeX files supplied by the author and processed by Laserwords Private Limited, Chennai, India Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire 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. 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 1.2TheSkeletonDeformation.......................... 2 1.2.1 Deformation Gradient and Transport Formulae . 2 1.2.2 Eulerian and Lagrangian Porosities. Void Ratio . 5 1.2.3 StrainTensor............................. 6 1.2.4 Infinitesimal Transformation and the Linearized Strain Tensor . 7 1.3Kinematics.................................. 8 1.3.1 ParticleDerivative.......................... 8 1.3.2 StrainRates.............................. 10 1.4MassBalance................................. 12 1.4.1 EquationofContinuity........................ 12 1.4.2 The Relative Flow Vector of a Fluid Mass. Filtration Vector. Fluid MassContent............................. 12 1.5AdvancedAnalysis.............................. 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 2.1MomentumBalance............................. 19 2.1.1 The Hypothesis of Local Forces . 19 2.1.2 TheMomentumBalance....................... 20 2.1.3 TheDynamicTheorem........................ 21 2.2TheStressTensor............................... 21 2.2.1 Action–ReactionLaw........................ 21 2.2.2 The Tetrahedron Lemma and the Cauchy Stress Tensor . 22 vi CONTENTS 2.3EquationofMotion.............................. 24 2.3.1 TheLocalDynamicResultantTheorem............... 24 2.3.2 The Dynamic Moment Theorem and the Symmetry oftheStressTensor......................... 25 2.3.3 PartialStressTensor......................... 26 2.4KineticEnergyTheorem........................... 27 2.4.1 StrainWorkRates.......................... 27 2.4.2 Piola–Kirchhoff Stress Tensor . 29 2.4.3 KineticEnergyTheorem....................... 29 2.5AdvancedAnalysis.............................. 30 2.5.1 The Stress Partition Theorem . 30 2.5.2 Momentum Balance and the Double Porosity Network . 32 2.5.3 TheTortuosityEffect......................... 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 PostulateofLocalState....................... 40 3.2.2 TheFirstLaw............................ 40 3.2.3 TheSecondLaw........................... 42 3.3 Conduction Laws . 45 3.3.1 Darcy’sLaw............................. 45 3.3.2 Fourier’sLaw............................ 49 3.4 Constitutive Equations of the Skeleton . 51 3.4.1 StateEquationsoftheSkeleton................... 51 3.4.2 ComplementaryEvolutionLaws................... 54 3.5 Recapitulating the Laws . 57 3.6AdvancedAnalysis.............................. 59 3.6.1 FluidParticleHead.BernoulliTheorem............... 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 AnisotropicPoroelasticity...................... 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 4.4AdvancedAnalysis.............................. 87 4.4.1 Non-linearIsotropicPoroelasticity................. 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 5.1LinearizedPoroelasticityProblems.....................113 5.1.1 The Hypothesis of Small Perturbations . 113 5.1.2 Field Equations and Boundary Conditions . 115 5.1.3 TheDiffusionEquation.......................116 5.2SolvedProblemsofPoroelasticity......................117 5.2.1 InjectionofaFluid..........................117 5.2.2 ConsolidationofaSoilLayer....................125 5.2.3 Drilling of a Borehole . 129 5.3 Thermoporoelasticity Problems . 133 5.3.1 FieldEquations............................133 5.3.2 Half-space Subjected to a Change in Temperature . 135 5.4AdvancedAnalysis..............................137 5.4.1 UniquenessofSolution.......................137 5.4.2 TheBeltrami–MichellEquations..................140 5.4.3 Mandel’sProblem..........................142 5.4.4 Non-linearSedimentation......................145 6 Unsaturated Thermoporoelasticity 151 6.1MassandMomentumBalance........................151 6.1.1 Partial Porosities and Degree of Saturation . 151 6.1.2 MassandMomentumBalance....................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’sLaw.............................168 6.5.3 Fick’sLaw..............................168 viii CONTENTS 6.6AdvancedAnalysis..............................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 PorePressure.............................178 6.6.4 Isothermal Drying of Weakly Permeable Materials . 180 7 Penetration Fronts 189 7.1DissolutionFronts..............................189 7.1.1 Mass Balance and Fick’s Law for the Solute . 190 7.1.2 Instantaneous Dissolution and the Formation ofaPenetrationFront........................191 7.1.3 Stefan-likeProblem.........................192 7.2SolutePenetrationwithNon-linearBinding.................194 7.2.1 The Binding Process and the Formation of a Penetration Front . 194 7.2.2 TheTimeLagandtheDiffusionTest................197
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