ffffiirrss0011..iinndddd iiii 55//2277//22001111 1122::3355::5500 PPMM POROUS MEDIA TRANSPORT PHENOMENA ffffiirrss0011..iinndddd ii 55//2277//22001111 1122::3355::5500 PPMM ffffiirrss0011..iinndddd iiii 55//2277//22001111 1122::3355::5500 PPMM POROUS MEDIA TRANSPORT PHENOMENA FARUK CIVAN Mewbourne School of Petroleum and Geological Engineering The University of Oklahoma Norman, Oklahoma A JOHN WILEY & SONS, INC., PUBLICATION ffffiirrss0022..iinndddd iiiiii 55//2277//22001111 1122::3355::5511 PPMM Copyright © 2011 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. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions. 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For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Civan, Faruk. Porous media transport phenomena / Faruk Civan. p. cm. Includes index. ISBN 978-0-470-64995-4 (hardback) 1. Porous materials. I. Title. TA418.9.P6C58 2011 620.1'16--dc22 2011006414 Printed in Singapore eISBN 978-1-118-08643-8 oISBN 978-1-118-08681-0 ePub ISBN 978-1-118-08680-3 10 9 8 7 6 5 4 3 2 1 ffffiirrss0033..iinndddd iivv 55//2277//22001111 1122::3355::5511 PPMM Dedicated to my family with love and appreciation ffffiirrss0044..iinndddd vv 55//2277//22001111 22::5522::4488 PPMM ffffiirrss0044..iinndddd vvii 55//2277//22001111 22::5522::4488 PPMM CONTENTS PREFACE xv ABOUT THE AUTHOR xix CHAPTER 1 OVERVIEW 1 1.1 Introduction 1 1.2 Synopses of Topics Covered in Various Chapters 3 CHAPTER 2 TRANSPORT PROPERTIES OF POROUS MEDIA 7 2.1 Introduction 7 2.2 Permeability of Porous Media Based on the Bundle of Tortuous Leaky-Tube Model 10 2.2.1 Pore Structure 11 2.2.2 Equation of Permeability 13 2.2.3 Derivation of the Equation of Permeability 16 2.2.4 Pore Connectivity and Parametric Functions 20 2.2.5 Data Analysis and Correlation Method 24 2.2.6 Parametric Relationships of Typical Data 26 2.2.6.1 Example 1: Synthetic Spheres 26 2.2.6.2 Example 2: Dolomite 26 2.2.6.3 Example 3: Berea Sandstone 27 2.2.7 Correlation of Typical Permeability Data 29 2.2.7.1 Example 4: Synthetic Porous Media 29 2.2.7.2 Example 5: Glass Bead and Sand Packs 31 2.2.7.3 Example 6: Silty Soil 33 2.3 Permeability of Porous Media Undergoing Alteration by Scale Deposition 33 2.3.1 Permeability Alteration by Scale Deposition 36 2.3.2 Permeability Alteration in Thin Porous Disk by Scale Deposition 37 2.3.3 Data Analysis and Correlation Method 38 2.3.4 Correlation of Scale Effect on Permeability 39 2.3.4.1 Example 7: Scale Formation 39 2.3.4.2 Example 8: Acid Dissolution 40 2.3.4.3 Example 9: Wormhole Development 42 2.4 Temperature Effect on Permeability 44 2.4.1 The Modifi ed Kozeny–Carman Equation 46 2.4.2 The Vogel–Tammann–Fulcher (VTF) Equation 49 2.4.3 Data Analysis and Correlation 51 2.4.3.1 Example 10: Correlation Using the Modifi ed Kozeny–Carman Equation 51 2.4.3.2 Example 11: Correlation Using the VTF Equation 52 vii ffttoocc..iinndddd vviiii 55//2277//22001111 55::2200::0088 PPMM viii CONTENTS 2.5 Effects of Other Factors on Permeability 54 2.6 Exercises 54 CHAPTER 3 MACROSCOPIC TRANSPORT EQUATIONS 57 3.1 Introduction 57 3.2 REV 58 3.3 Volume-Averaging Rules 59 3.4 Mass-Weighted Volume-Averaging Rule 67 3.5 Surface Area Averaging Rules 68 3.6 Applications of Volume and Surface Averaging Rules 68 3.7 Double Decomposition for Turbulent Processes in Porous Media 70 3.8 Tortuosity Effect 73 3.9 Macroscopic Transport Equations by Control Volume Analysis 74 3.10 Generalized Volume-Averaged Transport Equations 76 3.11 Exercises 76 CHAPTER 4 SCALING AND CORRELATION OF TRANSPORT IN POROUS MEDIA 79 4.1 Introduction 79 4.2 Dimensional and Inspectional Analysis Methods 81 4.2.1 Dimensional Analysis 81 4.2.2 Inspectional Analysis 82 4.3 Scaling 84 4.3.1 Scaling as a Tool for Convenient Representation 84 4.3.2 Scaling as a Tool for Minimum Parametric Representation 84 4.3.3 Normalized Variables 86 4.3.4 Scaling Criteria and Options for Porous Media Processes 87 4.3.5 Scaling Immiscible Fluid Displacement in Laboratory Core Floods 89 4.4 Exercises 92 CHAPTER 5 FLUID MOTION IN POROUS MEDIA 97 5.1 Introduction 97 5.2 Flow Potential 98 5.3 Modifi cation of Darcy’s Law for Bulk- versus Fluid Volume Average Pressures 99 5.4 Macroscopic Equation of Motion from the Control Volume Approach and Dimensional Analysis 102 5.5 Modifi cation of Darcy’s Law for the Threshold Pressure Gradient 105 5.6 Convenient Formulations of the Forchheimer Equation 108 5.7 Determination of the Parameters of the Forchheimer Equation 111 5.8 Flow Demarcation Criteria 115 5.9 Entropy Generation in Porous Media 117 5.9.1 Flow through a Hydraulic Tube 118 5.9.2 Flow through Porous Media 120 5.10 Viscous Dissipation in Porous Media 123 5.11 Generalized Darcy’s Law by Control Volume Analysis 124 5.11.1 General Formulation 126 5.11.2 Simplifi ed Equations of Motion for Porous Media Flow 132 ffttoocc..iinndddd vviiiiii 55//2277//22001111 55::2200::0088 PPMM