Modelling and Applications of Transport Phenomena in Porous Media

393 Pages·1991·22.312 MB·English

by  J. Bear (auth.), Jacob Bear, J-M. Buchlin (eds.)

Modellingand Applications ofTransportPhenomena in Porous Media
Theory and Applications ofTransportinPorousMedia
SeriesEditor:
JacobBear,Technion -IsraelInstitute ofTechnology, Haifa, Israel
Volume 5
The titlespublishedinthisseriesarelistedattheendo/thisvolume.
Modelling and
Applications of
Transport Phenomena
in Porous Media
Edited by
Jacob Bear
Technion - Israel Institute of Technology,
Haifa, Israel
and
J-M. Buchlin
Von Karman Institute for Fluid Dynamics,
Rhode-Saint-Genese, Belgium
Lecture Series presented at the
von Karman Institute for Fluid Dynamics
Rhode-Saint-Genese, Belgium, Nov. 30-Dec. 4, 1987.
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Modelling and applications of transport phenomena in porous media I
edited by Jacob Bear. J.M. Buchlin.
p. cm. -- (Theory and applications of transport in porous
media; v. 5)
"Lecture series presented at the von Karman Institute for Fluid
Dynamics. Rhode-Saint-Genesee. Belgium. Nov. 3D-Dec. 4. 1987."
Includes bibliographical references and index.
ISBN 978-94-010-5163-7 ISBN 978-94-011-2632-8 (eBook)
DOI 10.1007/978-94-011-2632-8
1. Porous materials--Permeability--Mathematical models.
2. Transport theory--Mathematical models. I. Bear. Jacob.
II. Buchlin. J-M. III. Von Karman Institute for Fluid Dynamics.
IV. Series.
TA418.9.P6M65 1991
620.1' 16--dc20 91-30660
ISBN 978-94-010-5163-7
Printed on acid·free paper
All Rights Reserved
© 1991 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1991
Softcover reprint of the hardcover 1s t edition 1991
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
PREFACE xi
1 EIGHT LECTURES ON MATHEMATICAL
MODELLING OF TRANSPORT IN POROUS MEDIA
J. BEAR
Albert and Anne Mansfield
Chair in Water Resources
Technion - Israel Institute of Technology
Haifa, Israel 32000. 1
1.1 Lecture One: Introduction 1
1.1.1 Porous medium 2
1.1.2 Modelling process 4
1.1.3 Selecting the size of an REV 11
1.2 Lecture Two: Microscopic
Balance Equations 22
1.2.1 Velocity and flux 22
1.2.2 The general balance equation 24
1.2.3 Particular balance equations 28
1.2.4 Averaging rules 35
1.3 Lecture Three: Macroscopic
Balance Equations 44
1.3.1 General balance equation 44
1.3.2 Particular cases 45
1.3.3 Stress in porous media 51
v
vi
1.4 Lecture Four: Advective Flux 59
1.4.1 Advectiveflux ofa singlefluid that occupies the entire
void space. 59
1.4.2 Particular cases 70
1.4.3 Multiphase flow 74
1.5 Lecture Five: Complete
Transport Model 78
1.5.1 Boundary conditions 78
1.5.2 Content of a complete model 101
1.6 Lecture Six: Modelling Mass
Transport of a Single Fluid
Phase Under Isothermal
Conditions 105
1.6.1 Basic mass balance equations 105
1.6.2 Stationary nondeformable solid skeleton 109
1.6.3 Deformable porous medium 114
1.6.4 Boundary conditions 125
1.6.5 Complete mathematical model 135
1.7 Lecture Seven: Diffusive Flux 137
1.7.1 Diffusive mass flux 138
1.7.2 Diffusive heat flux 142
1.8 Lecture Eight: Modelling
Contaminant Transport 145
1.8.1 The Phenomenon of dispersion 146
1.8.2 Fluxes 152
1.8.3 Sources and Sinks 158
1.8.4 Mass balance equation for a single component 167
1.8.5 Balance equations with immobile liquid 171
1.8.6 Balance equations for radionuclide decay chain 176
1.8.7 Two multicomponent phases 177
1.8.8 Boundary Conditions 180
1.8.9 Complete Mathematical Model 185
References 187
List ofMainSymbols 190
vii
2 MULTIPHASE FLOW IN POROUS MEDIA
Th. DRACOS
Swiss Federal Inst. of Technology (E.T.H.)
Zurich, Switzerland. 195
2.1 Capillary Pressure 195
2.1.1 Interfacial tension, contact angle and wettability 195
2.1.2 Interfacial curvature and capillary pressure 197
2.1.3 Equilibrium between a liquid and its vapor 198
2.1.4 Microscopic domain 199
2.1.5 Macroscopic space 199
2.1.6 Phase distribution in the pore space 201
2.2 Flow Equations for Immiscible Fluids 203
2.3 Mass Balance Equations 205
2.4 Simultaneous Flow of Two Fluids Having a Small Density
Difference 207
2.5 Measurement ofthe relations PcoASa'), and kr,a;(Sa,) 209
2.6 Mathematical descripton of the relations between Pc,w, Sw,
and kr,w 212
2.7 Complete Statement of Multiphase Flow Problems 215
2.8 Solute transport in multiphase flow through porous media 218
References 219
ListofMainSymbols 220
3 PHASE CHANGE PHENOMENA AT
LIQUID SATURATED SELF HEATED
PARTICULATE BEDS
J-M. BUCHLIN and A. STUBOS
von Karman Institute for Fluid Dynamics
Rhode Saint Genese B-1640, Belgium. 221
3.1 Introduction 221
3.2 Preboiling Phenomenology 222
3.3 Boiling regime and dryout heat flux 226
3.4 Constitutive Relationships-Bed Disturbances 238
3.4.1 Introduction 238
3.4.2 Bed permeability 239
3.4.3 Relative permeabilities and passabilities 240
3.4.4 Capillary pressure 250
VIU
3.4.5 Bed structural changes 258
3.5 Conclusions 263
Appendix 266
A. Zero-Dimensional Model 266
B. Fractional downward heat flux by conduction 269
C. Subcooled zone thickness at the top ofthe bed 269
References 271
List ofMain Symbols 276
4 HEAT TRANSFER IN SELF-HEATED
PARTICLE BEDS
SUBMERGED IN LIQUID COOLANT
KENT MEHR and JORGEN WURTZ
Commission of the European Communities
Joint Research Centre, Ispra, Italy. 277
4.1 The PARR Scenario 277
4.2 Specific PAHR Phenomena 278
4.2.1 Bed characteristics 278
4.2.2 Heat conduction 279
4.2.3 Boiling debris beds. 281
4.2.4 Dryout 284
4.2.5 Downward boiling 285
4.2.6 Unsteady state 285
4.2.7 Channeling 286
4.3 PAHR-2D 289
4.3.1 Basic equations 289
4.3.2 Spatial discretisation 290
4.3.3 Boundary conditions 294
4.3.4 Time integration 295
4.3.5 Solution procedure 295
4.3.6 DI0 Post test calculation 296
4.3.7 Steep power ramp 300
4.4 In-pile experiments 302
4.4.1 Boiling 305
4.4.2 Dryout 306
4.4.3 Bed disturbance 306
References 308
List ofMainSymbols 309
ix
5 PHYSICAL MECHANISMS DURING THE
DRYING OF A POROUS MEDIUM
CH. MOYNE, CH. BASILICO,
J. CH. BATSALE and A. DEGIOVANNI.
Laboratoire d'Energetique et de Mecanique
Theorique et Appliquee V.A. C.N.R.S. 875,
Ecoles des Mines, Nancy, France. 311
5.1 General Aspects ofthe Drying Process 312
5.1.1 The three drying periods 312
5.1.2 The characteristic drying curve concept 313
5.1.3 Conclusion 317
5.2 A General Model for Simultaneous Heat and Mass Transfer
in a Porous Medium 318
5.2.1 The fundamental hypotheses ofthe model 318
5.2.2 Phenomenological laws 319
5.2.3 Conservation laws 322
5.2.4 System to be solved 324
5.2.5 Numerical solution 326
5.3 Application to Drying 326
5.3.1 High temperature convective drying 327
5.3.2 Low temperature convective drying 332
5.3.3 The diffusion model 332
5.3.4 Receding drying front 334
5.4 Conclusions 338
References 339
ListofMainSymbols 340
6 STOCHASTIC DESCRIPTION
OF POROUS MEDIA
G. DE MARSILY
Ecole des Mines de Paris,
I'Universite Pierre et Marie Curie
Paris, France. 343
6.1 Definition of Properties of Porous Media: The Example of
Porosity 344
6.2 Stochastic Approach to Permeability
and Spatial Variability 349
x
6.3 Stochastic Partial Differential Equations 351
6.3.1 Properties ofstochastic partial differential equations 352
6.3.2 Spectral methods 353
6.3.3 The method of perturbations 355
6.3.4 Simulation method (Monte-Carlo) 357
6.4 Example ofstochastic solution to the
transport equation 359
6.5 The problem ofestimation ofa RF by kriging 360
6.6 The intrinsic hypothesis: definition of the variogram 362
6.6.1 The intrinsic hypothesis 363
6.6.2 Determination ofthe vario~ram 365
6.6.3 Behaviour of the variogram for large h 367
6.6.4 Behaviour close to the origin 367
6.6.5 Anisotropy in the variogram 370
6.7 Conclusions 370
References 371
List ofMainSymbols 375
Index 377