ESSENTIALS OF MULTIPHASE FLOW AND TRANSPORT IN POROUS MEDIA ESSENTIALS OF MULTIPHASE FLOW AND TRANSPORT IN POROUS MEDIA George F. Pinder University of Vermont William G. Gray University of North Carolina at Chapel Hill A JOHN WILEY & SONS, INC., PUBLICATION Copyright © 2008 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. 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 as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. 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Library of Congress Cataloging-in-Publication Data: ISBN 978-0-470-31762-4 Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1 To the Future: Jasmine, Ryan, Charlotte, Benjamin, and Lucy CONTENTS Preface xi Acknowledgments xiii 1 Setting the Stage 1 1.1 Introduction / 1 1.2 Phases and Porous Media / 2 1.3 Grain and Pore Size Distributions / 6 1.4 The Concept of Saturation / 12 1.5 The Concept of Pressure / 13 1.6 Surface Tension Considerations / 16 1.7 Concept of Concentration / 30 1.8 Summary / 32 1.9 Exercises / 32 Bibliography / 33 2 Mass Conservation Equations 35 2.1 Introduction / 35 2.2 Microscale Mass Conservation / 38 2.3 Integral Forms of Mass Conservation / 39 2.4 Integral Theorems / 44 2.4.1 Divergence Theorem / 45 2.4.2 Transport Theorem / 45 2.5 Point Forms of Mass Conservation / 46 2.6 The Macroscale Perspective / 48 2.6.1 The Representative Elementary Volume / 49 2.6.2 Global and Local Coordinate Systems / 50 2.6.3 Macroscopic Variables / 53 2.6.4 Defi nitions of Macroscale Quantities / 56 2.6.5 Summary of Macroscale Quantities / 62 2.7 The Averaging Theorems / 63 2.7.1 Spatial Averaging Theorem / 64 2.7.2 Temporal Averaging Theorem / 66 2.8 Macroscale Mass Conservation / 67 2.8.1 Macroscale Point Forms / 67 2.8.2 Integral Forms / 71 vii viii CONTENTS 2.9 Applications / 73 2.9.1 Integral Analysis / 74 2.9.2 Point Analysis / 76 2.10 Summary / 79 2.11 Exercises / 79 Bibliography / 81 3 Flow Equations 83 3.1 Introduction / 83 3.2 Darcy’s Experiments / 85 3.3 Fluid Properties / 88 3.4 Equations of State for Fluids / 89 3.4.1 Mass Fraction / 89 3.4.2 Mass Density and Pressure / 90 3.4.3 Fluid Viscosity / 92 3.5 Hydraulic Potential / 93 3.5.1 Hydrostatic Force and Hydraulic Head / 93 3.5.2 Derivatives of Hydraulic Head / 97 3.6 Single-Phase Fluid Flow / 98 3.6.1 Darcy’s Law / 99 3.6.2 Hydraulic Conductivity and Permeability / 102 3.6.3 Derivation of Groundwater Flow Equation / 106 3.6.4 Recapitulation of the Derivation / 111 3.6.5 Initial and Boundary Conditions / 113 3.6.6 Two-Dimensional Flow / 116 3.7 Two-Phase Immiscible Flow / 121 3.7.1 Derivation of Flow Equations / 121 3.7.2 Observations on the pc-sw Relationship / 127 3.7.3 Formulas for the pc-sw Relationship / 135 3.7.4 Observations of the kα -sw Relationship / 143 rel 3.7.5 Formulas for the kα -sw Relation / 146 rel 3.7.6 Special Cases of Multiphase Flow / 149 3.8 The Buckley-Leverett Analysis / 155 3.8.1 Fractional Flow / 155 3.8.2 Derivation of the Buckley-Leverett Equation / 157 3.8.3 Solution of the Buckley-Leverett Equation / 158 3.9 Summary / 160 3.10 Exercises / 161 Bibliography / 162 4 Mass Transport Equations 165 4.1 Introduction / 165 4.2 Velocity in the Species Transport Equations / 167 4.2.1 Direct Approach / 168 4.2.2 Rigorous Approach / 169 CONTENTS ix 4.2.3 Distribution Approach / 172 4.2.4 Summary / 175 4.3 Closure Relations for the Dispersion Vector / 176 4.4 Chemical Reaction Rates / 180 4.5 Interphase Transfer Terms / 182 4.5.1 Kinetic Formulation / 183 4.5.2 Equilibrium Formulation / 187 4.5.3 Summary: Kinetic vs. Equilibrium Formulations / 194 4.6 Initial and Boundary Conditions / 195 4.7 Conclusion / 196 4.8 Exercises / 197 Bibliography / 198 5 Simulation 199 5.1 1-D Simulation of Air-Water Flow / 199 5.1.1 Drainage in a Homogeneous Soil / 201 5.1.2 Drainage in a Heterogeneous Soil / 205 5.1.3 Imbibition in Homogeneous Soil / 206 5.2 1-D Simulation of DNAPL-Water Flow / 207 5.2.1 Primary DNAPL Imbibition in Homogeneous Soil / 208 5.2.2 Density Effect / 208 5.2.3 DNAPL Drainage in Homogeneous Soil / 209 5.2.4 Secondary Imbibition of DNAPL in Homogeneous Soil / 210 5.2.5 Secondary Drainage in Homogeneous Soil / 211 5.2.6 Primary Imbibition in Heterogeneous Soil / 212 5.3 2-D Simulation of DNAPL-Water Flow / 213 5.3.1 DNAPL Descent into a Water-Saturated Reservoir / 213 5.4 Simulation of Multiphase Flow and Transport / 216 5.4.1 1-D Two-Phase Flow and Transport / 217 5.4.2 2-D Two-Phase Flow and Transport / 218 5.5 2-D Single-Phase Flow and Transport / 224 5.5.1 Base Case / 228 5.5.2 Effect of Infl ow / 228 5.5.3 Impact of Well Discharge / 230 5.5.4 Effect of Adsorption / 231 5.5.5 Effect of a Low Transmissivity Region / 232 5.5.6 Effect of a High Transmissivity Region / 234 5.5.7 Effect of Rate of Reaction / 235 5.6 3-D Single-Phase Flow and Transport / 236 5.7 2-D Three-Phase Flow / 239 5.8 Summary / 244 Bibliography / 245 Select Symbols 247 Index 253