tan nguyen artificial lift methods
TRANSCRIPT
Petroleum Engineering
Tan Nguyen
Artificial Lift MethodsDesign, Practices, and Applications
Petroleum Engineering
Editor-in-Chief
Gbenga Oluyemi, Robert Gordon University, Aberdeen, Aberdeenshire, UK
Series Editors
Amirmasoud Kalantari-Dahaghi, Department of Petroleum Engineering, WestVirginia University, Morgantown, WV, USA
Alireza Shahkarami, Department of Engineering, Saint Francis University, Loretto,PA, USA
Martin Fernø, Department of Physics and Technology, University of Bergen,Bergen, Norway
The Springer series in Petroleum Engineering promotes and expedites the dissem-ination of new research results and tutorial views in the field of explorationand production. The series contains monographs, lecture notes, and edited volumes.The subject focus is on upstream petroleum engineering, and coverage extends toall theoretical and applied aspects of the field. Material on traditional drilling andmore modern methods such as fracking is of interest, as are topics including but notlimited to:
• Exploration• Formation evaluation (well logging)• Drilling• Economics• Reservoir simulation• Reservoir engineering• Well engineering• Artificial lift systems• Facilities engineering
Contributions to the series can be made by submitting a proposal to theresponsible Springer Editor, Charlotte Cross at [email protected] or theAcademic Series Editor, Dr. Gbenga Oluyemi [email protected].
More information about this series at http://www.springer.com/series/15095
Tan Nguyen
Artificial Lift MethodsDesign, Practices, and Applications
123
Tan NguyenPetroleum DepartmentNew Mexico TechSocorro, NM, USA
ISSN 2366-2646 ISSN 2366-2654 (electronic)Petroleum EngineeringISBN 978-3-030-40719-3 ISBN 978-3-030-40720-9 (eBook)https://doi.org/10.1007/978-3-030-40720-9
© Springer Nature Switzerland AG 2020This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, expressed or implied, with respect to the material containedherein or for any errors or omissions that may have been made. The publisher remains neutral with regardto jurisdictional claims in published maps and institutional affiliations.
This Springer imprint is published by the registered company Springer Nature Switzerland AGThe registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface
This book titled Artificial Lift Methods—Design, Practices, and Applicationsconsists of seven chapters: (1) review, (2) gas lift, (3) electrical submersible pump,(4) progressing cavity pump, (5) sucker rod pump, (6) plunger lift, and (7) artificiallift selection methodology for vertical and horizontal wells in conventional andunconventional reservoirs. Each chapter or each artificial lift method is unique andalmost independent of other chapters except Chap. 1 and Chap. 7. In Chap. 1, theauthor reviews some basics of math, physics, fluid properties, flow inside reser-voirs, flow inside a tubing, and multiphase flow in a tubing which are directlyand/or indirectly related to other chapters in the book. Therefore, readers arestrongly encouraged to review Chap. 1 before reading other chapters. In Chap. 7,the author begins with reviewing the advantages and disadvantages of each com-mon lift method. Next, the artificial lift selection methodology for vertical wells inconventional reservoirs is presented. Finally, the author presents the most recenttrends of artificial lift selection methodology for horizontal wells in unconventionalplays. Readers do not need to have a deep understanding of all the artificial liftmethods to read Chap. 7. In other words, it could be reasonable for readers to getstarted reading Chap. 7 to get an idea of what the artificial lift selection method-ology looks like before reading other chapters.
Each artificial lift chapter (from Chap. 2 to Chap. 6) begins with the funda-mentals of the lift method where the author reviews the heart of the lift system andhow it works. The lift system is explained in the concept of the flow in the reservoir,the flow in the tubing, and how the external energy helps to lift the liquid. Next, thechapter focuses on the uniqueness of each lift method and then the detailed design.The author then closes each chapter with examples so readers know how to applythe presented concepts into practical applications.
All of the equations in this book are labeled with three digits “(X.X.X)”. The firstdigit represents the number of the chapter. The second digit represents the numberof the section. And the last digit is for the order of the equations in the chapter. Forexample, Eq. (3.2.5) can be interpreted as Chap. 3, section 2, and equation number 5.
v
The page number is labeled as the following format “X-XX”. The first digit is forthe chapter number following with a dash. The digits following the dash are thepage number in each chapter. The page number in each chapter always begins withpage number 1. For example, page 2.5 represents for Chap. 2 and page number 5;page 4.5 can be found in Chap. 4 page number 5.
Socorro, USA Tan Nguyen
vi Preface
Acknowledgements
I would like to first give thanks to my God whom I serve and worship with all of myheart. Next, I would like to thank New Mexico Tech and the Petroleum and NaturalGas Engineering Department for allowing me to take one semester sabbatical leaveand for giving me the service time so I can initiate, continue, and complete writingthis book. I would also like to thank Sebastian Pivnicka for reviewing the sucker rodpump and the plunger lift chapters. I also want to thank my students who contributedto this book while taking the artificial lift class with me in Spring 2019: KienNguyen, Benjamin Adu-Gyamfi, Carson Healy, and Tonya Ross. I would also like tothank my wife for her continued and tireless support of my family and to my life. Mywife also helped me to debug the spellings, labels of equations and figures, andprepared the Table of Contents. Finally, I want to thank my two boys, Ryan andStephen, for being such a good boys and for letting daddy focus on writing this book.
vii
Contents
1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Review of Math and Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Review of Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Review of Conservation of Motion (Momentum Equation) . . . . . . 41.4 Review of Basic Thermodynamic Properties of Liquid
and Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4.1 Specific Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4.2 Bubble Point Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4.3 Solution Gas-Oil Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4.4 Oil Formation Volume Factor . . . . . . . . . . . . . . . . . . . . . 161.4.5 Oil Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4.6 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.4.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Review of Inflow Performance Relationship (IPR) . . . . . . . . . . . . 261.5.1 IPR for Undersaturated Oil Reservoirs (Pr > Pb) . . . . . . . 261.5.2 IPR for Saturated Oil Reservoirs (Pr < Pb) . . . . . . . . . . . . 291.5.3 IPR for Gas Reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.6 Review of Outflow Performance Relationship (OPR) . . . . . . . . . . 311.6.1 Single Phase—Incompressible Fluid OPR . . . . . . . . . . . . 311.6.2 Single Phase—Compressible Fluid OPR . . . . . . . . . . . . . 321.6.3 Two-Phase Mixture OPR . . . . . . . . . . . . . . . . . . . . . . . . 331.6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2 Gas Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.1 Fundamentals of Gas Lift System . . . . . . . . . . . . . . . . . . . . . . . . 41
2.1.1 Continuous Gas Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.1.2 Intermittent Gas Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
ix
2.2 Gas Lift Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.2.1 Surface Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.2.2 Subsurface Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3 Gas Lift Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.3.1 Open Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.3.2 Semi-closed Installation . . . . . . . . . . . . . . . . . . . . . . . . . 532.3.3 Closed Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.3.4 Chamber Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.3.5 Tubing Flow Installation . . . . . . . . . . . . . . . . . . . . . . . . . 562.3.6 Annular Flow Installation . . . . . . . . . . . . . . . . . . . . . . . . 572.3.7 Dual Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.3.8 Coiled Tubing Installation . . . . . . . . . . . . . . . . . . . . . . . 582.3.9 Macaroni Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.4 Gas Lift Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.4.1 Gas Lift Valve Classification . . . . . . . . . . . . . . . . . . . . . 612.4.2 Gas Lift Valve Performance . . . . . . . . . . . . . . . . . . . . . . 642.4.3 Operation of a Gas Lift Valve . . . . . . . . . . . . . . . . . . . . 712.4.4 Test Rack Opening Pressure . . . . . . . . . . . . . . . . . . . . . . 77
2.5 Multiple Valve Gas Lift Unloading . . . . . . . . . . . . . . . . . . . . . . . 782.5.1 Fundamentals of Unloading Process . . . . . . . . . . . . . . . . 782.5.2 Description of Unloading Process . . . . . . . . . . . . . . . . . . 80
2.6 Continuous Gas Lift Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842.6.1 Determination of IPR and OPR. . . . . . . . . . . . . . . . . . . . 852.6.2 Determination of Operating Valve Location . . . . . . . . . . . 872.6.3 Determination of Unloading Valve Location . . . . . . . . . . 892.6.4 Determination of Injection Gas Rate . . . . . . . . . . . . . . . . 892.6.5 Selection of Gas Lift Valves . . . . . . . . . . . . . . . . . . . . . . 902.6.6 Discussion on Valve Spacing . . . . . . . . . . . . . . . . . . . . . 92
2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3 Electrical Submersible Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.1 Fundamentals of Electrical Submersible Pump . . . . . . . . . . . . . . . 107
3.1.1 Introduction and History of ESPs . . . . . . . . . . . . . . . . . . 1073.1.2 A Basic ESP System . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093.1.3 Working Principle of an ESP . . . . . . . . . . . . . . . . . . . . . 1113.1.4 ESP Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.1.5 Deploying ESP System . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.2 Theoretical Performance of a Centrifugal Pump . . . . . . . . . . . . . . 1133.2.1 Pump Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.2.2 Specific Speed Number . . . . . . . . . . . . . . . . . . . . . . . . . 1203.2.3 Affinity Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.3 Actual Pump Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
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3.3.1 Hydraulic Horsepower . . . . . . . . . . . . . . . . . . . . . . . . . . 1253.3.2 Brake Horsepower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253.3.3 Pump Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253.3.4 Actual Pump Performance Curves . . . . . . . . . . . . . . . . . . 126
3.4 Viscous Effect on Pump Performance . . . . . . . . . . . . . . . . . . . . . 1273.4.1 Stepanoff Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283.4.2 Hydraulic Institute Method . . . . . . . . . . . . . . . . . . . . . . . 1303.4.3 Turzo et al. Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1323.4.4 Evdocia Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.5 Gas Effect on Pump Performance . . . . . . . . . . . . . . . . . . . . . . . . 1333.5.1 Homogeneous Flow Modeling . . . . . . . . . . . . . . . . . . . . 1343.5.2 Empirical Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . 1353.5.3 Experimental Study on Two-Phase Centrifugal
Pump Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373.6 Pump Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.6.1 Impeller Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393.6.2 Pump Shaft Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403.6.3 Total Pump Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403.6.4 Discussion on Pump Trust . . . . . . . . . . . . . . . . . . . . . . . 142
3.7 ESP Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1443.7.1 Collection of Basic Data . . . . . . . . . . . . . . . . . . . . . . . . . 1443.7.2 Selection of Pump and Motor Diameter . . . . . . . . . . . . . . 1453.7.3 Selection of Pump Depth . . . . . . . . . . . . . . . . . . . . . . . . 1453.7.4 Analyzing Well Flow Capacity . . . . . . . . . . . . . . . . . . . . 1473.7.5 Selection of ESP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1483.7.6 Consideration of the Effect of Gas and Viscosity
on Pump Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.7.7 Selection of Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1513.7.8 Selection of Cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1533.7.9 Selection of Gas Separator . . . . . . . . . . . . . . . . . . . . . . . 1543.7.10 Selection of Surface Equipment . . . . . . . . . . . . . . . . . . . 154
3.8 ESP Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1593.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
4 Progressing Cavity Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1814.1 Fundamentals of Progressing Cavity Pump (PCP) . . . . . . . . . . . . . 1814.2 History of PCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1854.3 Applications of PCPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
4.3.1 Application of PCP to Pump Heavy Oiland Bitumen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
4.3.2 Application of PCP to Pump High Solid ContentsFluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Contents xi
4.4 PCP System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1884.4.1 Drive Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1884.4.2 Rotating Stuffing Box . . . . . . . . . . . . . . . . . . . . . . . . . . . 1894.4.3 Polished Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1904.4.4 Pony Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1924.4.5 Rod Centralizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1924.4.6 Centralized Torque Anchor . . . . . . . . . . . . . . . . . . . . . . . 192
4.5 Pump Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1934.5.1 Review of 3-D Vector Theory . . . . . . . . . . . . . . . . . . . . 1944.5.2 Review of Hypocycloid Theory . . . . . . . . . . . . . . . . . . . 1954.5.3 Modeling the Design and Theoretical Performance
of a Multi-lobe PCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1964.5.4 Estimation of Actual Multi-lobe PCP Performance . . . . . . 2054.5.5 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2084.5.6 Modeling Actual Multi-lobe PCP Performance
Using Dimensionless Approach . . . . . . . . . . . . . . . . . . . . 2104.5.7 Extension of This Model to Predict Performance
of Positive Displacement Motors . . . . . . . . . . . . . . . . . . . 2124.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
5 Sucker Rod Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2275.1 Fundamentals of Sucker Rod Pump . . . . . . . . . . . . . . . . . . . . . . . 227
5.1.1 Introduction and Main Principles of SuckerRod Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
5.1.2 API Rod Pump Classification . . . . . . . . . . . . . . . . . . . . . 2305.1.3 Rod String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
5.2 Pumping Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2355.2.1 Simple Harmonic Motion (SHM) . . . . . . . . . . . . . . . . . . 2355.2.2 Crank and Pitman Motion (CPM) . . . . . . . . . . . . . . . . . . 236
5.3 Basic Rod Pump Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2375.3.1 Theoretical and Actual Pump Displacement
(Downhole Pump Rate) . . . . . . . . . . . . . . . . . . . . . . . . . 2375.3.2 Calculation of Polished Rod Loads . . . . . . . . . . . . . . . . . 2385.3.3 Calculations of Counterbalance . . . . . . . . . . . . . . . . . . . . 2425.3.4 Surface Torque Calculation . . . . . . . . . . . . . . . . . . . . . . . 2435.3.5 Calculation of Nameplate Motor Horsepower . . . . . . . . . . 246
5.4 API Recommended Design Procedure . . . . . . . . . . . . . . . . . . . . . 2475.5 Viscous Effect on Rod Pump Performance . . . . . . . . . . . . . . . . . . 2555.6 Common Sucker-Rod Pump Failures . . . . . . . . . . . . . . . . . . . . . . 258
5.6.1 Rod String Failures and Design . . . . . . . . . . . . . . . . . . . 2585.6.2 Pump Barrel Failures or Improper Operations . . . . . . . . . 260
5.7 Dynamometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
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5.8 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
6 Plunger Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2796.1 Fundamentals of Plunger Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
6.1.1 Introduction and Main Principles of Plunger Lift . . . . . . . 2796.1.2 Surface and Subsurface Plunger Lift Equipment . . . . . . . . 2856.1.3 Plunger Lift Applications . . . . . . . . . . . . . . . . . . . . . . . . 292
6.2 Review of Inflow Performance of Gas Wells . . . . . . . . . . . . . . . . 2946.3 Analytical Modeling of Plunger Lift System . . . . . . . . . . . . . . . . 297
6.3.1 Modeling the Buildup Stage . . . . . . . . . . . . . . . . . . . . . . 2976.3.2 Modeling the Upstroke Stage . . . . . . . . . . . . . . . . . . . . . 2986.3.3 Model the Blowdown Stage . . . . . . . . . . . . . . . . . . . . . . 3026.3.4 Modeling the Downtroke Stage . . . . . . . . . . . . . . . . . . . . 303
6.4 Approximation Modeling of Plunger Lift System . . . . . . . . . . . . . 3046.4.1 Foss and Gaul Approximation Model . . . . . . . . . . . . . . . 3046.4.2 Lea Approximation Model . . . . . . . . . . . . . . . . . . . . . . . 306
6.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
7 Artificial Lift Selection Methodology for Vertical and HorizontalWells in Conventional and Unconventional Reservoirs . . . . . . . . . . . 3177.1 Characteristics of Common Artificial Lift Methods . . . . . . . . . . . . 318
7.1.1 Gas Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3187.1.2 Electrical Submersible Pump . . . . . . . . . . . . . . . . . . . . . 3197.1.3 Sucker Rod Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3207.1.4 Plunger Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3217.1.5 Progressive Cavity Pump (PCP) . . . . . . . . . . . . . . . . . . . 322
7.2 Artificial Lift Selection for Vertical Wells in ConventionalReservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3227.2.1 Important Factors Impacting AL Selection . . . . . . . . . . . . 3237.2.2 Selection of Artificial Lift Method for Vertical
Wells in Conventional Reservoirs . . . . . . . . . . . . . . . . . . 3247.3 AL Selection for Wells in Heavy Oil Reservoirs . . . . . . . . . . . . . 3307.4 AL Selection for Horizontal Wells in Unconventional
Reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3337.4.1 Basic Concept and Challenges of AL for Horizontal
Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3337.4.2 AL Selection Methodology for Horizontal
Wells in Unconventional Reservoirs . . . . . . . . . . . . . . . . 338References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
Contents xiii
Chapter 1Review
1.1 Review of Math and Physics
This section gives a quick review of definitions of common terms used throughoutthis book.
Scalar is a quantity described fully by a magnitude.Vector is a quantity described by both a magnitude and a direction.Dot product of two vectors ~a and ~b is a scalar and defined as c ¼ ak k bk kcosh
where h is the angle formed by the two vectors ~a and ~b.Cross product between two vectors ~a and ~b is a vector and defined as:
~c ¼ ak k bk ksinhf g~n.where ~n is the normal vector of the plane formed by the two vectors ~a and ~b.Momentum, M, is a vector and defined as follows:
~M ¼ m~u ð1:1:1Þ
Force, F, is defined as rate of change of momentum respect to time.
~F ¼ d~Mdt
¼ d m~uð Þdt
ð1:1:2Þ
For incompressible fluids (constant density), Eq. (1.1.2) becomes:
~F ¼ d~Mdt
¼ d m~uð Þdt
¼ md~udt
¼ m~a ð1:1:3Þ
Pressure, P, is a scalar and defined as the potential energy stored per unit volume
P ¼ EnergyðJÞVolume m3ð Þ ¼
WorkðNmÞVolume
¼~F:~S~A:~S
ð1:1:4Þ
© Springer Nature Switzerland AG 2020T. Nguyen, Artificial Lift Methods, Petroleum Engineering,https://doi.org/10.1007/978-3-030-40720-9_1
1
Therefore
~F ¼ P~A ð1:1:5Þ
Note that the area, A, is a vector and defined using the cross product as Shown inFig. 1.1b.
The gradient of a scalar pressure, which is a function of three directions x, y, andz, is a vector and can be expressed mathematically as:
rP ¼ @P@x
~iþ @P@y
~jþ @P@z
~k ð1:1:6Þ
where~i;~j;~k are the three unit vectors in the Cartesian coordinate system.The divergence of a velocity vector, which is a function of three directions x, y,
and z, is a vector operator that produces a scalar quantity of a vector field and givenas:
r:~u ¼ @ux@x
þ @uy@y
þ @uz@z
ð1:1:7Þ
Partial time and total time derivative
Let consider temperature of an object at a fixed locationLet Q be a property of a fluid such as temperature, velocity, density, etc. In
general, Q is a function of time, t, and space in three directions: x, y, and z. Then Qcan be written as Q = Q(t, x, y, z). The total differential change of Q is expressedas:
dQ ¼ @Q@t
dtþ @Q@x
dxþ @Q@y
dyþ @Q@z
dz ð1:1:8Þ
Divide both sides of Eq. (1.1.8) by dt yields the total time derivative of Q:
θ
(a) Dot product
θ
(b) Cross product
Fig. 1.1 Dot and cross product between two vectors
2 1 Review
dQdt
¼ @Q@t
þ @Q@x
dxdt
þ @Q@y
dydt
þ @Q@z
dzdt
ð1:1:9Þ
In Eq. (1.1.9), @Q@t and dQ
dt are the partial and total time derivative, respectively.If Q is the temperature of an object at a fixed location, the temperature change ofthis object respect to time in the absence of any motion is defined as the partial timederivative (local time rate of temperature change). If this object moves into otherregions of temperature, the temperature change of this object is defined as the totalderivative or convective temperature change. Note that dx/dt = ux, dy/dt = uy, anddz/dt = uz are the three components of the local fluid velocity u. Equation (1.1.9)can be written as the substantial derivative:
DQDt
¼ @Q@t
þ @Q@x
ux þ @Q@y
uy þ @Q@z
uz ð1:1:10Þ
1.2 Review of Conservation of Mass
The conservation of mass states that the rate of mass accumulation in a controlvolume equals to the rate of mass out subtracted from the rate of mass. In otherwords, the rate of density increase in a control volume is equal to the net rate ofmass flux per unit volume.
@q@t
¼ � @qux@x
þ @quy@y
þ @quz@z
� �¼ � r:q~uð Þ ð1:2:1Þ
where q~u is the mass flux in the unit of kg=sm2 that is the mass rate across an unit area
of the control volume. Equation (1.2.1) can also be written as follows:
@q@t
þ ux@q@x
þ uy@q@y
þ uz@q@z
¼ �q@ux@x
þ @uy@y
þ @uz@z
� �ð1:2:2Þ
In cylindrical coordinates (r, h, z):
@q@t
þ 1r@
@rqrurð Þþ 1
r@
@hquhð Þþ @
@zquzð Þ ¼ 0 ð1:2:2aÞ
Note that the left hand side of Eq. (1.2.2) is the substantial time derivative ofdensity and hence this equation can be expressed as:
1.1 Review of Math and Physics 3
DqDt
¼ �q r:~uð Þ ð1:2:3Þ
If the fluid is incompressible then Eq. (1.2.3) is reduced to a much simpler form:
r:u ¼ @ux@x
þ @uy@y
þ @uz@z
¼ 0 ð1:2:4Þ
1.3 Review of Conservation of Motion (MomentumEquation)
According to the second Newton’s law, the rate of momentum change of a controlvolume equals to sum of all forces acting on that control volume.
D m~uð ÞDt
¼X
~F ð1:3:1Þ
D q~uð ÞDt
¼ 1V
X~F ð1:3:2Þ
Neglecting the electrical and magnetic forces, there will be three main forcesacting on the control volume including pressure force, viscous force, and gravita-tional force. Equation (1.3.2) can be re-written for the control volume per unitvolume as follows:
D q~uð ÞDt
¼ �rP�r:sþ q~g ð1:3:3Þ
where D q~uð ÞDt is the substantial time derivative of the mass flux defined in Eq. (1.1.10)
and rP is the gradient of the scalar pressure defined in Eq. (1.1.6). Note that s is astress tensor which has nine components. For a Newtonian fluid (viscosity, l, is aconstant and independent to shear rate), a viscous stress tensor is defined as:
sij ¼ 2leij ¼ l@ui@xj
þ @uj@xi
� �ð1:3:4Þ
where eij ¼ 12
@ui@xj
þ @uj@xi
� �is the rate of deformation. Therefore, the stress tensor is
expressed as:
4 1 Review
sij ¼sxx sxy sxzsyx syy syzszx szy szz
0@
1A ¼
2l @ux@x l @uy
@x þ @ux@y
� �l @uz
@x þ @ux@z
� �l @ux
@y þ @uy@x
� �2l @uy
@y l @uz@y þ @uy
@z
� �l @ux
@z þ @uz@x
� �l @uy
@z þ @uz@y
� �2l @uz
@z
0BBB@
1CCCA
ð1:3:5Þ
In general, the momentum equation in rectangular coordinates in three directions(x, y, and z) can be given as follows:
D quxð ÞDt
¼ � @P@x
þ l@2ux@x2
þ @2ux@y2
þ @2ux@z2
� �þ qgx ð1:3:6Þ
D quy� �Dt
¼ � @P@y
þ l@2uy@x2
þ @2uy@y2
þ @2uy@z2
� �þ qgy ð1:3:7Þ
D quzð ÞDt
¼ � @P@z
þ l@2uz@x2
þ @2uz@y2
þ @2uz@z2
� �þ qgz ð1:3:8Þ
If the fluid is incompressible (constant density) then Eqs. (1.3.6), (1.3.7), and(1.3.8) becomes:
q@ux@t
þ ux@ux@x
þ uy@ux@y
þ uz@ux@z
� �¼ � @P
@xþ l
@2ux@x2
þ @2ux@y2
þ @2ux@z2
� �þ qgx
ð1:3:9Þ
q@uy@t
þ ux@uy@x
þ uy@uy@y
þ uz@uy@z
� �¼ � @P
@yþ l
@2uy@x2
þ @2uy@y2
þ @2uy@z2
� �þ qgy
ð1:3:10Þ
q@uz@t
þ ux@uz@x
þ uy@uz@y
þ uz@uz@z
� �¼ � @P
@zþ l
@2uz@x2
þ @2uz@y2
þ @2uz@z2
� �þ qgz
ð1:3:11Þ
In cylindrical coordinates (r, h, z), the momentum equation for an incompress-ible and Newtonian fluid is given as [7]:
q@ur@t
þ ur@ur@r
þ uhr@ur@h
� u2hrþ uz
@ur@z
� �
¼ � @P@r
þ l@
@r1r@
@rrurð Þ
� �þ 1
r2@2ur@h2
� 2r2@uh@h
þ @2ur@z2
� þ qgr
ð1:3:12Þ
1.3 Review of Conservation of Motion (Momentum Equation) 5
q@uh@t
þ ur@uh@r
þ uhr@uh@h
þ uruhr
þ uz@uh@z
� �
¼ � 1r@P@h
þ l@
@r1r@
@rruhð Þ
� �þ 1
r2@2uh@h2
þ 2r2@ur@h
þ @2uh@z2
� þ qgh
ð1:3:13Þ
q@uz@t
þ ur@uz@r
þ uhr@uz@h
þ uz@uz@z
� �
¼ � @P@z
þ l1r@
@rr@uz@r
� �þ 1
r2@2uz@h2
þ @2uz@z2
� þ qgz
ð1:3:14Þ
Example 1.1 Please use the momentum equation to develop equations for calcu-lating the hydrostatic pressure of a homogeneous liquid and a homogeneous gas in atube with a length of L and a true vertical depth of h as shown in Fig. 1.2.
SolutionUnder static conditions, the left hand side of Eq. (1.3.3) is equal to zero. In addi-tion, the shear force (viscous force) also equals to zero. Therefore, Eq. (1.3.3) canbe reduces to:
rP ¼ q~g
This is a one dimensional problem. Assuming that the z-direction is the same asthat of the gravitational force, ~g. Equation (1.3.14) becomes:
@P@z
¼ qgzsinh or @P ¼ qgzsinh@z ð1:3:15Þ
Integrating this Eq. gives:
D ¼ gzsinhZ2
1
q P; Tð Þ@z ð1:3:16Þ
h
L
θ
ρ 1
2
Fig. 1.2 Fluid in an inclinedcolumn
6 1 Review
If the fluid is incompressible such as liquid with a density of qL then Eq. (1.3.16)becomes:
DP ¼ qLgzLsinh ¼ qLgzh ð1:3:17Þ
If the unit of density is in kg/m3, g = 9.81 m/s2, and h is in m then the unit of DPin Eq. (1.3.17) is Pa or N/m2.
In oil field unit, Eq. (1.3.17) becomes:
DP ¼ 0:052qLh ¼ 0:052qLTVD ð1:3:18Þ
where TVD is true vertical depth in ft and density is in pounds per gallon (ppg).If the fluid is compressible such as gas with a density of qg, Eq. (1.3.15)
becomes:
Z2
1
@P ¼ gzsinhZ2
1
PMzRT
@z ð1:3:19Þ
where M is the molecular weight of gas; z is the compressibility factor, R is the gasconstant, and T is the fluid temperature of the control volume. Assuming the fluidtemperature is constant and equals to the average fluid temperature, Tave, at the topand bottom of the fluid column. Rearrange Eq. (1.3.19) gives:
Z2
1
@PP
¼ MgzsinhzRTave
Z2
1
@z ð1:3:20Þ
P2 ¼ P1eMgzLsinhzRTave ¼ P1e
MgzhzRTave ð1:3:21Þ
In SI unit, the molecular weight, M, is in kg/mol, gravitational accelerationfactor, g, is in m/s2, height, h, is in meter, gas constant, R, is in Pa�m3
mol�K, and tem-perature, T, is in K, then the unit of P2 is the same as that of P1.
In oil field unit, Eq. (1.3.21) becomes:
P2 ¼ P1e0:01877cgh
zTave
� �ð1:3:22Þ
where cg is the specific gravity of gas, h is depth or height of the liquid column in ft,and T is average temperature in Rankin, oR. The pressures P1 and P2 are in the unitsof psi.
Example 1.2 Using the momentum equation, please derive an equation to calculatethe total pressure loss of an incompressible Newtonian fluid in pipe flow (Fig. 1.3).
1.3 Review of Conservation of Motion (Momentum Equation) 7
SolutionThis is again a one dimensional problem. Let z be the direction of the flow. Understeady state conditions and neglecting the acceleration, Eq. (1.3.14) becomes:
@P@z
Total
¼ @P@z
friction
þ @P@z
gravity
@P@z
Total
¼ l1r@
@rr@uz@r
� �� þ qgzsinh
ð1:3:23Þ
Equation (1.3.23) tells us that the total pressure loss of an incompressibleNewtonian fluid flowing in an inclined pipe under laminar steady state conditionsand neglecting acceleration consists of two components: pressure loss due to fric-tion and pressure loss due to gravity.
The pressure loss due to friction per unit length (frictional pressure lossgradient):
@P@z
f¼ l
1r@
@rr@uz@r
� �� ð1:3:24Þ
@ r@uz@r
� �¼ 1
l@P@z
f
r@r ð1:3:25Þ
Integrating both side of Eq. (1.3.25) gives:
r@uz@r
¼ 12l
@P@z
f
r2 þC1 ð1:3:26Þ
h
L
θ
ρ 1
2
z
r
Fig. 1.3 Flow in an inclinedpipe
8 1 Review
@uz ¼ 12l
@P@z
f
r@rþ C1
r@r ð1:3:27Þ
At the center of the pipe: @r ¼ 0; @uz ¼ 0 hence C1 ¼ 0
uz ¼ 14l
@P@z
f
r2 þC2 ð1:3:28Þ
No slip boundary condition at the pipe wall:@r ¼ R; uz ¼ 0 hence C2 ¼ � 1
4l@P@z
fR2. Therefore:
uz ¼ 14l
@P@z
f
r2 � R2� � ð1:3:29Þ
Equation (1.3.29) reveals that the velocity profile of an incompressibleNewtonian fluid in pipe flow under laminar flow conditions is a parabolic.
The liquid flow rate in pipe flow is defined as:
Q ¼ ZR
0
uzdA ¼ ZR
0
uz2prdr ð1:3:30Þ
Combining Eqs. (1.3.29) and (1.3.30) gives:
Q ¼ p2l
@P@z
f
ZR
0
r2 � R2� �rdr ð1:3:31Þ
Q ¼ p8l
@P@z
fR4 ð1:3:32Þ
Using the definition of average liquid velocity, �u, in pipe flow, the flow rate canbe written as follows:
Q ¼ p�uR2 ð1:3:33Þ
Combining Eqs. (1.3.32) and (1.3.33) yields an equation to calculate for thefrictional pressure loss in pipe flow under steady state and laminar flow conditions:
@P@z
f
¼ 8l�uR2 ¼ 32l�u
D2 ð1:3:34Þ
If the unit offluid viscosity, l, is in Pas, average fluid velocity in pipe, �u, is in m/s,and pipe diameter is in m, the unit of @P
@z
fis in Pa/m.
1.3 Review of Conservation of Motion (Momentum Equation) 9
In oil field unit, Eq. (1.3.34) becomes:
@P@z
f
¼ 8l�uR2 ¼ l�u
1500D2 ð1:3:35Þ
where l is in cp, �u is in ft/s, D is the inner pipe diameter in inch, and @P@z
fis the
frictional pressure drop gradient in psi/ft.The pressure drop due to gravity is shown in Eq. (1.3.17).The total pressure drop of an incompressible Newtonian fluid in an inclined pipe
under steady state and laminar flow conditions without acceleration is given:
@P@z
Total
¼ @P@z
friction
þ @P@z
gravity
¼ 32l�uD2 þ qLgsinh ð1:3:36Þ
Equation (1.3.36) indicates that the frictional pressure drop is independent tofluid density and inclination angle. It depends on fluid viscosity, pipe geometry, andliquid flow rate. The pressure drop due to gravity is a function of fluid density andinclination angle.
Example 1.3 Using the momentum equation, please derive an equation for cal-culating the pressure drop of a fluid through an orifice. The pipe and orificediameters are dp and do as shown in Fig. 1.4. Note that this example is a funda-mental for showing how to predict the performance of an orifice gas lift valve.
Solution
Assumptions:
• Steady state flow, @uz@t ¼ 0.
• One dimensional flow; z direction. ur ¼ 0 and uh ¼ 0.• Viscosity is small and hence the viscous force is neglected.• Pressure drop due to gravity is much smaller than that due to convection.
With these assumptions, the pressure drop of the fluid through an orifice ismainly due to the change of the local velocity along the flow. Note that the fluidvelocity not only changes its magnitude but also its direction when the fluid flowsthrough the orifice. Equation (1.3.14) becomes:
Flow
1 3
z
2
Fig. 1.4 Flow through anorifice
10 1 Review
q uz@uz@z
� �¼ � @P
@zð1:3:37Þ
At location 1 and 3:
DP ¼ P1 � P3 ¼ Z3
1
quz@uz ð1:3:38Þ
If the fluid is incompressible, Eq. (1.3.38) can be expressed as:
DP ¼ P1 � P3 ¼q u23 � u21� �
2ð1:3:39Þ
Applying the continuity equation, Eq. (1.2.4), for this incompressible fluidyields:
@uz@z
¼ 0 ð1:3:40Þ
Equation (1.3.40) tells us that if the pipe diameter is uniform, the velocity of anincompressible fluid along the pipe is the same. Therefore, if points 1 and 3 are faraway from the orifice, then u1 = u3 and the total pressure drop through an orifice is,DP = 0. This means the total pressure lost upstream of the orifice is the same as thetotal pressure gained downstream of the orifice.
The total pressure lost upstream of the orifice:
DPlost ¼ P1 � P2 ¼q u22 � u21� �
2ð1:3:41Þ
The total pressure gained downstream of the orifice:
DPgained ¼ P2 � P3 ¼q u23 � u22� �
2ð1:3:42Þ
In reality, when the fluid flows through an orifice, there are permanent pressurelosses due to vortices, contraction and expansion before and after the orifice, anddead zones (no flow). Therefore, the pressure at point 1 is always greater than thepressure at point 3. In other words, the pressure gained downstream of the orifice isalways less than the pressure lost upstream of the orifice.
If the fluid is compressible such as gas, Eq. (1.3.38) becomes:
DP ¼ P1 � P2 ¼ 12q2u
22 �
12q1u
21 ð1:3:43Þ
1.3 Review of Conservation of Motion (Momentum Equation) 11
Applying the continuity equation gives:
@
@zquzð Þ ¼ 0 ð1:3:44Þ
Converting Eq. (1.3.44) to mass flux yields:
Qm ¼ q1u1A1 ¼ q2u2A2 ð1:3:45Þ
Combining Eqs. (1.3.43) and (1.3.45) gives:
DP ¼ P1 � P2 ¼ 12
q1q2
A1
A2
� �2
�1
" #q1u
21 ð1:3:46Þ
Equation (1.3.46) can be used to calculate for the total theoretical pressure dropof an incompressible fluid through an orifice.
In terms of volumetric flow rate upstream of the orifice, combining Eqs. (1.3.45)and (1.3.46) yields:
qv ¼ qmassq1
¼ pd2p4
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 P1 � P2ð Þ
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2
q21d4pd4o� q2
q1
h isð1:3:47Þ
Let b ¼ dodp, Eq. (1.3.47) becomes:
qv ¼ Cpd2p4
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 P1 � P2ð Þ
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
q1 1� b4� �
sð1:3:48Þ
where C is the discharge coefficient factor, which takes care of the permanentlosses. Equation (1.3.48) can be used to predict the performance of gas flowingthrough an orifice.
Example 1.4 Using the momentum equation, please derive an equation for cal-culating the pressure gain when an incompressible is rotated at a constant angularvelocity in the impeller of a centrifugal pump (Fig. 1.5).
ds
dθ
ω
Fig. 1.5 Tangential andangular velocities
12 1 Review
Solution
Assumptions:
• Because of the constant angular velocity of the object, the motion is understeady state conditions
• ur and uz equal to zero.• @uh
@h ¼ 0, @uh@r ¼ 0, and @uh
@z ¼ 0• Neglecting the viscous and gravitational forces.
Equation (1.3.12) can now be reduced to:
u2hr¼ 1
q@P@r
ð1:3:49Þ
Equation (1.3.49) states that the centrifugal forceu2hr
� �acting radially outwards
on the fluid element is balanced by the pressure gradient force directed intwards.Where uh is the tangential velocity component and the relationship between thetangential velocity and angular velocity is expressed as:
uh ¼ rx ð1:3:50Þ
Combining Eqs. (1.3.49) and (1.3.50) and integrating both sides of the equationgives:
ZPd
Pi
@P ¼ ZR2
R1
qx2r@r ð1:3:51Þ
DP ¼ Pd � Pi ¼ qx2
2R22 � R2
1
� � ¼ q2
u22 � u21� � ð1:3:52Þ
Pump head is defined as:
H ¼ DPqg
¼ u22 � u21� �
2gð1:3:53Þ
where Pi and Pd are the impeller intake and impeller discharge pressures, respec-tively. u1 and u2 are the tangential fluid velocities at the intake and discharge of thepump impeller. Equation (1.3.53) tells us that the theoretical pump head or theo-retical pressure gain across the impeller of a centrifugal pump is mainly due to thecentrifugal force. Equation (1.3.53) can also be interpreted that the pressure gainedacross the impeller is due to the gain in kinetic energy of the fluid received from therotation of the pump impeller.
1.3 Review of Conservation of Motion (Momentum Equation) 13
1.4 Review of Basic Thermodynamic Properties of Liquidand Gas
1.4.1 Specific Gravity
The specific gravity of gas, cg, is defined as the ratio between gas density underactual conditions and air density at standard conditions.
Petroleum industry defines standard conditions at which the temperature is 15.5 °C(60 °F) and the pressure is 1 atm (14.7 psi). Density of air under standard conditionsis 1.225 kg/m3 = 0.001225 g/cm3 = 0.0765 lbm/ft3.
The specific gravity of liquid such as oil, is defined similar to that of gas butusing water under standard conditions as a reference liquid instead of using air. Thedensity of water under standard conditions is 1.0 g/mL = 1000 kg/m3 = 62.4 lbm/ft3 = 8.3 lbm/gal.
The petroleum industry uses API gravity with a unit of oAPI to measure of howheavy a petroleum fluid is compared to water. If the API gravity is smaller than 10,the petroleum fluid is heavier than water and it sinks. If the API gravity is greaterthan 10, the petroleum fluid is considered to be lighter than water and it floats. Therelationship between the liquid specific gravity and API gravity is given as:
cl ¼141:5
131:5þ �APIð1:4:1Þ
1.4.2 Bubble Point Pressure
Under original reservoir conditions, formation oils often contain some dissolved gasin solution. If the reservoir is produced for a period of time, the reservoir pressurewill decline to a value that the dissolved gas begins to come out of solution andform bubbles. This pressure is known as the bubble point pressure, Pb. If thereservoir pressure is smaller than the bubble point pressure, the formation fluid willexhibit two phase: gas and liquid.
Bubble point pressure (Pb in psi) is a function of bubble point solution gas oilratio (Rsb in scf/STBO), dissolved oil and gas specific gravity (co, cg), and tem-perature (T in °F). Table 1.1 summarizes the common available correlations from1947 to 1988 for predicting the bubble point pressure.
14 1 Review
1.4.3 Solution Gas-Oil Ratio
Solution gas-oil ratio (Rs) is the amount of gas dissolved in one Stock Tank Barrelof Oil (STBO) at a specific pressure and temperature. Since oil volume is measuredat atmospheric conditions STBO, the unit of Rs is scf/STB.
RsSCFSTBO
� �¼ Vg;scðdissolvedÞ
Vo;scðproducedÞ ð1:4:5Þ
As pressure increases, more gas is dissolved into oil and hence Rs increasesapproximately linearly with pressure. Oil under these conditions is said to be sat-urated. When the pressure reaches the bubble point pressure, no more gas is dis-solved into oil (undersaturated oil) and hence Rs is a constant. This relationship isshown in Fig. 1.6. Below the bubble point pressure, Rs is a function of the reservoirpressure. Above the bubble point pressure, Rs is constant as Rsb called the solutiongas-oil ratio at the bubble point pressure.
For practical applications, the solution gas oil ratio at the bubble point pressure isthe value of interest. As mentioned, above the bubble pressure, the solution gas oilratio remains the same. Solution gas-oil ratio (Rs in SCF/STBO) is a function of
Table 1.1 Correlations for predicting bubble point pressure [2], [13], and [15]
Author Origin Corelation Year Eq.
Standing CaliforniaPb ¼ 18:2 Rs
cg
� �0:83�100:00091T�0:0125API � 1:4
� [17] (1.4.2)
Glaso North Sea Pb ¼ 10y [12] (1.4.3)
y ¼ 1:7669þ 1:7447logX � 0:3022 logXð Þ2 (1.4.3a)
X ¼ Rscg
� �0:816T0:172
API0:989
� � (1.4.3b)
Al-Marhoun Middle East Pb ¼ 5:38�10�3R0:715s c3:144o T þ 460ð Þ1:326
c1:878g
1988 (1.4.4)
Reservoir Pressure, P
Form
atio
n G
as-O
il R
atio
, Rs
Pb
Undersaturated oil.No more free gas is dissolved into oil
Saturated oil.More free gas is dissolved into oil
R = f(P)s
R constantsb
Fig. 1.6 Relationshipbetween solution gas-oil ratioand pressure
1.4 Review of Basic Thermodynamic Properties of Liquid and Gas 15
dissolved gas and oil specific gravity (co, cg), temperature (T in °F), and pressure(P in psi). It is determined by rearranging the bubble point pressure equations.Table 1.2 summarizes the common available correlations from 1947 to 1998 forpredicting the solution gas-oil ratio. Note that the produced GOR, defined as theratio between produced gas and produced oil, and the Rs will be the same if thereservoir pressure is higher than the bubble point pressure. If the reservoir pressure(or the flowing bottomhole pressure) is less than the bubble point pressure, theproduced GOR will normally be higher than the Rs under this reservoir pressure(Pr < Pb). However, the produced GOR will be smaller than then initial Rs or thebubble point formation gas-oil ration Rsb.
Note that Vasquez and Beggs suggested that separator conditions affect the gasgravity and hence the new gas gravity, cgc, should be adjusted to a separator pressureof 114.7 psia. If the separator conditions is not available, temperature and pressure atthe separator conditions are assumed to be 60 °F and 14.7 psia, respectively.
All the equations presented in Table 1.2 were derived from the bubble pointpressure equations to obtain the bubble point formation gas-oil ratio.
1.4.4 Oil Formation Volume Factor
Oil formation volume factor, Bo, is defined as the ratio of the volume of oil atreservoir conditions to the volume of oil at surface conditions.
Table 1.2 Correlations for predicting solution gas-oil ratio [1–3], [5], [13], [17–21]
Author Origin Corelation Year Eq.
Standing California Rs ¼ cgP
18:2 þ 1:4� �
10x� 1:2048 [17] (1.4.6)
x ¼ 0:0125API � 0:00091T (1.4.6a)
Vasquez andBeggs
WorldwideFor API� 30 : Rs ¼ 0:0362cgcP
1:094e25:72APIT þ 460ð Þ
� �1977 (1.4.7)
For API[ 30 : Rs ¼ 0:0178cgcP1:187e
23:93APIT þ 460ð Þ
� �(1.4.8)
cgc ¼ cg 1þ 5:91� 10�5API � TsplogPsp
114:7
� �h iThe subscript “sp” is for separator conditions
(1.4.8a)
Glasso North SeaRs ¼ cg 10x � API0:989
T0:172
� �1:2255 [12] (1.4.9)
x ¼ 2:8869� 14:1811� 3:309logPð Þ0:5 (1.4.9a)
Al-Marhoun MiddleEast
Rs ¼ A1:3984 1985 (1.4.10)
A ¼ 185:483Pc1:8778g c�3:1437o T þ 460ð Þ�1:3266 (1.4.10a)
Petrosky andFarshad
Gulf ofMexico
Rs ¼ P112:727 þ 12:34� �
c0:8439g 10xh i1:7318 [16] (1.4.11)
x ¼ 7:916� 10�4API1:541 � 4:561� 10�5T1:391 (1.4.11a)
16 1 Review
Bo ¼ Vo;reservoir
Vo;surfaceð1:4:12Þ
As oil travels up to surface, the fluid pressure reduces leading to more gascoming out of the solution. Therefore, the oil formation volume factor is normallygreater than one. The oil formation volume factor increases as the reservoir pressureis increased until the reservoir pressure is the same as the bubble point pressure.This is because more gas is dissolved into the oil as the reservoir pressure isincreased causing the volume of oil under reservoir conditions to swell. If thereservoir pressure is higher than the bubble point pressure, the oil formation volumefactor decreases slightly because no more free gas is dissolved into the solution andthe oil is compressed.
The formation volume factor for gas, Bg, is defined as the ratio of volume of onemole of gas at a given pressure and temperature to the volume of one mole gas atstandard conditions.
Bg ¼ VVsc
¼ zpscTpTsc
ð1:4:13Þ
For oil field unit, where Bg is in the unit of bbl/SCF, T is in °F, and p is in psia,Bg is given as:
Bg ¼ 0:005z T þ 460ð Þ
pð1:4:13aÞ
The two-phase formation volume factor Bt is calculated as follows:
Bt ¼ Bo þBg Rsi � Rsð Þ ð1:4:14Þ
The subscript “i” is for the initial reservoir conditions (Fig. 1.7).Oil formation volume factor (Bo in bbl/STBO) is a function of dissolved gas and
oil specific gravity (co, cg), temperature (T in °F), and pressure (P in psi). Table 1.3summarizes the common available correlations from 1947 to 1998 for predictingthe oil formation volume factor.
1.4.5 Oil Viscosity
For Newtonian fluids, viscosity, l, is a constant at a constant temperature anddefined as the ratio between shear stress and shear rate. Viscosity alone is enough tocharacterize a Newtonian fluid. However, viscosity is not enough to characterize anon-Newtonian fluid because the viscosity of this fluid is a function of shear rate.Depending on the relationship between shear stress, s, and shear rate, c, anon-Newtonian fluid can be characterized as Bingham model, s ¼ sy þ lpc, Power
1.4 Review of Basic Thermodynamic Properties of Liquid and Gas 17