Advanced petroleum reservoir simulation towards developing reservoir emulators second edition

Advanced Petroleum Reservoir Simulation www.ebook3000.com Scrivener Publishing 100 Cummings Center, Suite 541J Beverly, MA 01915-6106 Publishers at Scrivener Martin Scrivener (martin@scrivenerpublishing.com) Phillip Carmical (pcarmical@scrivenerpublishing.com) www.ebook3000.com Advanced Petroleum Reservoir Simulation Towards Developing Reservoir Emulators Second Edition M R Islam, M E Hossain, S H Moussavizadegan, S Mustafiz, J H Abou-Kassem www.ebook3000.com Copyright © 2016 by Scrivener Publishing LLC All rights reserved Co-published by John Wiley & Sons, Inc Hoboken, New Jersey, and Scrivener Publishing LLC, Salem, Massachusetts 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/permission Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives or written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with a professional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages 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 For more information about Scrivener products please visit www.scrivenerpublishing.com Cover design by Kris Hackerott Library of Congress Cataloging-in-Publication Data: ISBN 978-1-119-03851-1 Printed in the United States of America 10 www.ebook3000.com Authors would like to dedicate this book to their teacher and 'grand teacher', Professor S.M Farouq Ali, Encana/Petroleum Society Chair Professor at the University of Calgary www.ebook3000.com Contents Preface xv Introduction 1.1 Summary 1.2 Opening Remarks 1.3 The Need for a Knowledge-Based Approach 1.4 Summary of Chapters Reservoir Simulation Background 2.1 Essence of Reservoir Simulation 2.2 Assumptions Behind Various Modeling Approaches 2.2.1 Material Balance Equation 2.2.2 Decline Curve 2.2.3 Statistical Method 2.2.4 Analytical Methods 2.2.5 Finite-Difference Methods 2.2.6 Darcy’s Law 2.3 Recent Advances in Reservoir Simulation 2.3.1 Speed and Accuracy 2.3.2 New Fluid-Flow Equations 2.3.3 Coupled Fluid Flow and Geo-Mechanical Stress Model 2.3.4 Fluid-Flow Modeling Under Thermal Stress 2.4 Memory Models 2.4.1 Thermal Hysteresis 2.4.2 Mathematical and Numerical Models 2.5 Future Challenges in Reservoir Simulation 2.5.1 Experimental Challenges 2.5.2 Numerical Challenges 2.5.2.1 Theory of Onset and Propagation of Fractures due to Thermal Stress 2.5.2.2 Viscous Fingering during Miscible Displacement www.ebook3000.com 1 2 10 11 12 13 15 16 19 19 19 21 26 29 31 32 32 33 33 35 35 36 vii viii Contents Reservoir Simulator-Input/Output 3.1 Input and Output Data 3.2 Geological and Geophysical Modeling 3.3 Reservoir Characterization 3.3.1 Representative Elementary Volume, REV 3.3.2 Fluid and Rock Properties 3.3.2.1 Fluid Properties 3.3.3 Rock Properties 3.4 Upscaling 3.4.1 Power Law Averaging Method 3.4.2 Pressure-Solver Method 3.4.3 Renormalization Technique 3.4.4 Multiphase Flow Upscaling 3.5 Pressure/Production Data 3.6 Phase Saturations Distribution 3.7 Reservoir Simulator Output 3.8 History Matching 3.8.1 History-Matching Formulation 3.8.2 Uncertainty Analysis 3.8.2.1 Measurement Uncertainty 3.8.2.2 Upscaling Uncertainty 3.8.2.3 Model Error 3.8.2.4 The Prediction Uncertainty 3.9 Real-Time Monitoring Reservoir Simulators: Problems, Shortcomings, and Some Solution Techniques 4.1 Multiple Solutions in Natural Phenomena 4.1.1 Knowledge Dimension 4.2 Adomian Decomposition 4.2.1 Governing Equations 4.2.2 Adomian Decomposition of Buckley-Leverett Equation 4.2.3 Results and Discussions 4.3 Some Remarks on Multiple Solutions Mathematical Formulation of Reservoir Simulation Problems 5.1 Black Oil Model and Compositional Model 5.2 General Purpose Compositional Model 5.2.1 Basic Definitions 5.2.2 Primary and Secondary Parameters and Model Variables www.ebook3000.com 39 40 42 45 46 49 49 54 58 59 60 62 63 65 66 68 70 72 75 76 78 79 80 81 85 87 90 104 106 108 111 114 117 119 120 120 122 Contents ix 5.2.3 Mass Conservation Equation 5.2.4 Energy Balance Equation 5.2.5 Volume Balance Equation 5.2.6 The Motion Equation in Porous Medium 5.2.7 The Compositional System of Equations and Model Variables 5.3 Simplification of the General Compositional Model 5.3.1 The Black Oil Model 5.3.2 The Water Oil Model 5.4 Some Examples in Application of the General Compositional Model 5.4.1 Isothermal Volatile Oil Reservoir 5.4.2 Steam Injection Inside a Dead Oil Reservoir 5.4.3 Steam Injection in Presence of Distillation and Solution Gas The Compositional Simulator Using Engineering Approach 6.1 Finite Control Volume Method 6.1.1 Reservoir Discretization in Rectangular Coordinates 6.1.2 Discretization of Governing Equations 6.1.2.1 Components Mass Conservation Equation 6.1.2.2 Energy Balance Equation 6.1.3 Discretization of Motion Equation 6.2 Uniform Temperature Reservoir Compositional Flow Equations in a 1-D Domain 6.3 Compositional Mass Balance Equation in a Multidimensional Domain 6.3.1 Implicit Formulation of Compositional Model in Multidimensional Domain 6.3.2 Reduced Equations of Implicit Compositional Model in Multidimensional Domain 6.3.3 Well Production and Injection Rate Terms 6.3.3.1 Production Wells 6.3.3.2 Injection Wells 6.3.4 Fictitious Well Rate Terms (Treatment of Boundary Conditions) 6.4 Variable Temperature Reservoir Compositional Flow Equations 6.4.1 Energy Balance Equation www.ebook3000.com 125 128 133 134 139 141 141 143 146 146 148 150 155 156 157 158 158 166 168 170 175 178 180 183 183 185 186 190 190 x Contents 6.4.2 Implicit Formulation of Variable Temperature Reservoir Compositional Flow Equations 6.5 Solution Method 6.5.1 Solution of Model Equations Using Newton’s Iteration 6.6 The Effects of Linearization 6.6.1 Case 1: Single Phase Flow of a Natural Gas 6.6.2 Effect of Interpolation Functions and Formulation 6.6.3 Effect of Time Interval 6.6.4 Effect of Permeability 6.6.5 Effect of Number of Gridblocks 6.6.6 Spatial and Transient Pressure Distribution Using Different Interpolation Functions 6.6.7 CPU Time 6.6.8 Case 2: An Oil/water Reservoir Development of a New Material Balance Equation for Oil Recovery 7.1 Summary 7.2 Introduction 7.3 Mathematical Model Development 7.3.1 Permeability Alteration 7.3 Porosity Alteration 7.4 Pore Volume Change 7.4.1 A Comprehensive MBE with Memory for Cumulative Oil Recovery 7.5 Numerical Simulation 7.5.1 Effects of Compressibilities on Dimensionless Parameters 7.4.2 Comparison of Dimensionless Parameters Based on Compressibility Factor 7.4.3 Effects of M on Dimensionless Parameter 7.4.4 Effects of Compressibility Factor with M Values 7.4.5 Comparison of Models Based on RF 7.4.6 Effects of M on MBE 7.5 Conclusions Appendix Chapter 7: Development of an MBE for a Compressible Undersaturated Oil Reservoir www.ebook3000.com 194 197 198 203 203 210 210 212 214 214 218 220 239 239 241 243 243 244 246 247 250 251 252 253 255 255 257 258 259 Contents xi State-of-the-art on Memory Formalism for Porous Media Applications 8.1 Summary 8.2 Introduction 8.3 Historical Development of Memory Concept 8.3.1 Constitutive Equations 8.3.2 Application of Memory in Diffusion in Porous Media 8.3.3 Definition of Memory 8.4 State-of-the-art Memory-Based Models 8.5 Basset Force: A History Term 8.6 Anomalous Diffusion: A memory Application 8.6.1 Fractional Order Transport Equations and Numerical Schemes 8.7 Future Trends 8.8 Conclusion Modeling Viscous Fingering During Miscible Displacement in a Reservoir 9.1 Improvement of the Numerical Scheme 9.1.1 The Governing Equation 9.1.2 Finite Difference Approximations 9.1.2.1 Barakat-Clark FTD Scheme 9.1.2.2 DuFort-Frankel Scheme 9.1.3 Proposed Barakat-Clark CTD Scheme 9.1.4 Accuracy and Truncation Errors 9.1.5 Some Results and Discussion 9.1.6 Influence of Boundary Conditions 9.2 Application of the New Numerical Scheme to Viscous Fingering 9.2.1 Stability Criterion and Onset of Fingering 9.2.2 Base Stable Case 9.2.3 Base Unstable Case 9.2.4 Parametric Study 9.2.4.1 Effect of Injection Pressure 9.2.4.2 Effect of Overall Porosity 9.2.4.3 Effect of Mobility Ratio 9.2.4.4 Effect of Longitudinal Dispersion 9.2.4.5 Effect of Transverse Dispersion 9.2.4.6 Effect of Aspect Ratio www.ebook3000.com 271 271 272 273 274 274 277 277 284 287 288 297 298 301 302 303 305 305 307 307 309 309 316 317 318 318 324 330 331 335 336 341 343 347 ... that can cover far beyond reservoir modeling or even petroleum engineering Advanced Petroleum Reservoir Simulation: Towards Developing Reservoir Emulators, Second Edition M R Islam, M E Hossain,... 3-phase reservoir simulation program This program is attached in the form of CD with the book www.ebook3000.com Advanced Petroleum Reservoir Simulation: Towards Developing Reservoir Emulators, Second. .. (pcarmical@scrivenerpublishing.com) www.ebook3000.com Advanced Petroleum Reservoir Simulation Towards Developing Reservoir Emulators Second Edition M R Islam, M E Hossain, S H Moussavizadegan,