Integrated Modeling of Reservoir Fluid Properties and Multiphase Flow in Offshore Production Systems (Petroleum Engineering) 3031398491, 9783031398490

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Petroleum Engineering

Tobias R. Gessner Jader R. Barbosa Jr.

Integrated Modeling of Reservoir Fluid Properties and Multiphase Flow in Offshore Production Systems

Petroleum Engineering Editor-in-Chief Gbenga Oluyemi, Robert Gordon University, Aberdeen, Aberdeenshire, UK Series Editors Amirmasoud Kalantari-Dahaghi, Department of Petroleum Engineering, West Virginia 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 dissemination of new research results and tutorial views in the field of exploration and production. The series contains monographs, lecture notes, and edited volumes. The subject focus is on upstream petroleum engineering, and coverage extends to all theoretical and applied aspects of the field. Material on traditional drilling and more modern methods such as fracking is of interest, as are topics including but not limited 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 the responsible publisher, Anthony Doyle at [email protected] or the Academic Series Editor, Dr. Gbenga Oluyemi [email protected].

Tobias R. Gessner · Jader R. Barbosa Jr.

Integrated Modeling of Reservoir Fluid Properties and Multiphase Flow in Offshore Production Systems

Tobias R. Gessner Deep Water/Reservoir and Artificial Lift Petrobras Vitória, Espírito Santo, Brazil

Jader R. Barbosa Jr. Department of Mechanical Engineering Universidade Federal de Santa Catarina Florianópolis, Santa Catarina, Brazil

ISSN 2366-2646 ISSN 2366-2654 (electronic) Petroleum Engineering ISBN 978-3-031-39849-0 ISBN 978-3-031-39850-6 (eBook) https://doi.org/10.1007/978-3-031-39850-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of 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 transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the 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 this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To our families: Viviane, Marcos and Marcelo Mariutzka, Emma and Esther

Preface

In the realm of offshore oil production, the accurate prediction of reservoir fluid properties and multiphase flow behavior is of utmost importance for efficient and safe operations. The complex and challenging nature of the Brazilian pre-salt reservoirs necessitates comprehensive modeling approaches that address the unique demands posed by these environments. This monograph presents a culmination of research and insights gained from extensive studies conducted in this domain, aimed at advancing our understanding of integrated modeling techniques for reservoir fluid properties and multiphase flow in offshore production systems. To accurately describe the thermodynamic properties of the oil-gas system, we adopted a compositional approach, illustrated by the widely-accepted Peng-Robinson and Soave-Redlich-Kwong cubic equations of state. Through this approach, we were able to reproduce data obtained from PVT tests conducted on actual samples. Our contributions in this area encompassed several significant advancements, including a new expression for the slope coefficient of the function α(T ) due to Soave (1972), applicable to molecules with high acentric factors. Furthermore, we devised a new methodology for characterizing SCN fractions valid up to C100 , and developed a multivariable fitting procedure based on the most relevant parameters of the model. Turning our attention to multiphase flow modeling, we embraced the mechanistic, or phenomenological, approach. Two state-of-the-art formulations, coined as the North Sea and Gulf of Mexico models, were implemented computationally, showcasing their potential application to pre-salt offshore production systems. To validate our newly developed computational tool, we relied on a comprehensive database consisting of 249 real operation records from six production systems. These systems, located in the southern region of the State of Espírito Santo, encompassed two pre-salt fields in the Campos Basin. Throughout our journey, we encountered intriguing insights that directed our focus toward key areas of improvement. By conducting a factorial design and assessing the effects of various parameters on pressure drop predictions, we discovered the need to refine the modelling of gas volume fraction in slug flow and the wall shear stress in bubble flow. Additionally, strong indications emerged regarding the levels of pipe wall roughness assumed in previous simulations. Delving deeper into these vii

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topics, we proposed modifications, leading to the development of a new flow model exclusively focused on the pre-salt production systems, aptly named Parque das Baleias after the producing area that contributed invaluable data to its formulation. In our endeavor to bridge theory and practice, this book accommodates different units of measurement to suit the context at hand. The International System of Units is utilized for presenting experimental results obtained in the laboratory, while the system traditionally used at Petrobras is employed for field readings. We believe this approach enhances accessibility and ensures that readers from both academic and practical backgrounds can engage with the material effectively. While we have diligently strived to produce an error-free text, we acknowledge the possibility of unintentional mistakes. We humbly request the reader’s understanding and assure you that any identified errors will be addressed in subsequent editions. Lastly, we would like to extend our heartfelt gratitude to our families and colleagues for their unwavering support throughout this endeavor. Their encouragement has been invaluable in bringing this book to fruition. We would also like to express our thanks to Dr. Alan Tihiro D. Nakashima for his assistance in preparing some of the figures and diagrams presented in the book. We hope that this comprehensive exploration of integrated modeling techniques in the Brazilian pre-salt reservoirs will contribute to the advancement of offshore oil production and inspire future research in this field. Vitória, ES, Brazil Florianópolis, SC, Brazil June 2023

Tobias R. Gessner Jader R. Barbosa Jr.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Oil Constituents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Classification of Reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Volumetric and Phase Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The Flow of Oil Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Well Flow Rate Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Scope and Objectives of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 5 9 10 15 22 25 27

2 Thermodynamics of Petroleum Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Equations of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Van der Waals Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Modern Equations of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Further Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Solution of Cubic Equations of State . . . . . . . . . . . . . . . . . . . . 2.2.5 Mixing Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Vapor-Liquid Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Types of Problems and Formulation . . . . . . . . . . . . . . . . . . . . 2.3.2 Chemical Potential and Fugacity of a Component . . . . . . . . . 2.3.3 Calculation of Equilibrium Constants . . . . . . . . . . . . . . . . . . . 2.3.4 Solving the Vapor-Liquid Equilibrium Problem . . . . . . . . . . . 2.3.5 Stability Analysis in Flash Calculations . . . . . . . . . . . . . . . . . 2.4 Obtaining the Thermodynamic Properties of the Mixture . . . . . . . . . 2.5 Characterization of Oil Reservoir Fluids . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Critical Properties, Acentric Factor and Specific Heat of SCN Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Splitting the Heavy Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Lumping Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Model Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 32 32 34 38 41 42 44 45 47 50 52 56 58 61 62 67 70 72

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2.6 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74 81

3 Developing a Fluid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Initial Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 New Expressions for the Angular Coefficient of the α(T ) Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Consistency Test and Extrapolation of Current Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Fitting of New Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 New Correlations for the Characterization of SCN Fractions . . . . . . 3.4.1 Consistency Test and Extrapolation of Current Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 New Fitting of Exponential Functions . . . . . . . . . . . . . . . . . . . 3.4.3 Results and Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Methodology for Fitting the Proposed Fluid Model . . . . . . . . . . . . . . 3.5.1 Introduction of Fitting Coefficients . . . . . . . . . . . . . . . . . . . . . 3.5.2 Definition of the Minimum Set of Variables . . . . . . . . . . . . . . 3.5.3 Composition of the Objective Function . . . . . . . . . . . . . . . . . . 3.5.4 Objective Function Optimization Using the Response Surface Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.5 Definition of Weights, Results and Comparison . . . . . . . . . . . 3.6 Empirical Correlations for the Calculation of Other Properties . . . . 3.6.1 Thermodynamic Properties of Formation Water . . . . . . . . . . 3.6.2 Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Results and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 86

123 128 132 132 135 138 149

4 Fluid Flow in Oil Production Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Stratified Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Interface Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Shear Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Annular Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Shear Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Liquid Entrained Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Bubbly and Dispersed Bubble Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Slip Law Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Wall Shear Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Slug and Elongated Bubble Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Slip Law Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Gas Volume Fraction in the Liquid Slug . . . . . . . . . . . . . . . . . 4.5.3 Wall Shear Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Flow Pattern Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Liquid Film Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 153 157 158 159 162 163 165 166 167 170 171 173 176 178 178 179

89 90 93 97 99 103 106 110 111 116 120

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4.6.2 Gas Bubbles Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Liquid Slug and Dumitrescu-Taylor Bubble Stability . . . . . . 4.7 Solution of Two-Phase Flows Using the Mechanistic Approach . . . 4.7.1 North Sea Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Gulf of Mexico Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181 183 184 185 190 193 200

5 Simulation of Offshore Production Systems . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Heat Transfer in Offshore Production Wells . . . . . . . . . . . . . . . . . . . . 5.4 Application of the Marching Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Data Collection and Model Construction . . . . . . . . . . . . . . . . . . . . . . . 5.6 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Qualitative Analysis of the Profiles Obtained . . . . . . . . . . . . . 5.6.2 Calculation of Pressure Drop in Offshore Production Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 205 208 208 209 212 213 214 217 219 220

6 Improving the Fluid Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Identification of the Most Relevant Parameters . . . . . . . . . . . . . . . . . . 6.2.1 Introduction of Fitting Coefficients . . . . . . . . . . . . . . . . . . . . . 6.2.2 Effect of Fitting Coefficients on Pressure Drop Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Survey of Alternatives to Current Calculation Methods . . . . . . . . . . . 6.3.1 Wall Shear Stress in Bubbly Flow . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Volume Fraction of Gas in the Liquid Slug for the Slug Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Review of Wall Roughness in Production Tubing and Pipelines . . . 6.5 Accuracy Test and Definition of the Best Set . . . . . . . . . . . . . . . . . . . 6.6 Results and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Flow Pattern, Volumetric Fraction and Pressure Gradient Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Calculation of Pressure Drop in Offshore Production Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3 Simulations with a Simplified Reservoir Model . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

229 229 231 231

224 227

233 235 235 240 246 252 254 254 256 258 263

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7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Modeling the Properties of Oil Reservoir Fluids in the Pre-salt Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Two-Phase Flow Modeling in Pre-salt Offshore Production Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Integration Between the Two Disciplines . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267 267 270 272 272

Appendix A: Obtaining the Roots of Cubic Equations of State . . . . . . . . . 275 Appendix B: Calculation of Departure Functions . . . . . . . . . . . . . . . . . . . . . 277 Appendix C: Weighting of Liquid Phase Properties . . . . . . . . . . . . . . . . . . . 283

Symbols and Abbreviations

Roman A a a* a˜ Ak Ar b b* Bg Bo c c c* C0 cP c˜ P cv c˜ v D d Dk e E0 f fˆi

Helmholtz free energy (J) First parameter of cubic equation of state (Pa · mol2 /m6 ) First parameter of cubic equation of state (dimensionless) (–) Molar Helmholtz free energy (J/mol) Cross-sectional area of the pipe occupied by phase k (m2 ) Archimedes number (–) Second parameter of cubic equation of state (m3 /mol) Second parameter of cubic equation of state (dimensionless) (–) Gas formation volume factor (m3 /m3std ) Oil formation volume factor (m3 /m3std ) Isentropic compressibility (1/Pa) Volume translation parameter of cubic equation of state (m3 /mol) Volume translation parameter of cubic equation of state (dimensionless) (–) Distribution parameter (–) Specific heat at constant pressure (J/(kg · K)) Molar specific heat at constant pressure (J/(mol · K)) Specific heat at constant volume (J/(kg · K)) Molar specific heat at constant volume (J/(mol · K)) Internal diameter of the pipe (m) Average particle diameter (m) Hydraulic diameter associated with phase k (m) Pipe wall roughness (m) Eotvos number (–) Fugacity (Pa) Fugacity of component i in the mixture (Pa)

xiii

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F(x) Fj fk fp (x) Fr G g g˜ GOR H h h˜ Ki kij L La m ˙ M Mw mape MPE N n NC P PA PI PSD ˙ Q q q R Rs Re S s˜ Sk T U u˜ U U0

Symbols and Abbreviations

Cumulative distribution function of variable x (–) Objective function in the search for parameter j Fanning friction factor of phase k with the pipe wall or at the interface (–) Probability density function of variable x (1/x) Froude number (–) Gibbs free energy (J) Acceleration due to gravity (9.81 m/s2 ) Molar Gibbs free energy (J/mol) Gas-oil ratio (m3std /m3std ) Enthalpy (J) Height or level (m) Molar enthalpy (J/mol) Equilibrium constant of component i in the mixture (–) Binary interaction parameter between components i and j (–) Length (m) Laplace number (–) Auxiliary variable of the cubic equation of state (–) Mass flow rate (kg/s) Molecular mass (g/mol) Mean Absolute Percentage Error (%) Mean Percentage Error (%) Number of system components (–) Total number of moles in the system (–) Number of carbon atoms in the molecule (–) Absolute pressure (Pa) Percentage accuracy (%) Reservoir productivity index (m3std /s · Pa) Percentual Standard Deviation (%) Volumetric flow rate at standard conditions (m3std /s) Heat transfer rate (W) Heat flux (W/m2 ) Universal gas constant (8.314 J/(mol · K)) Solubility ratio (m3std /m3std ) Reynolds number (–) Entropy (J/K) Molar entropy (J/(mol · K)) Wetted perimeter of phase k or length of the interface (m) Temperature (K) Internal energy (J) Molar internal energy (J/mol) Overall heat transfer coefficient (W/(m2 · K)) Slip velocity (m/s)

Symbols and Abbreviations

Uk U sk u2 V v˜ We xi yi z zi

xv

Average velocity of phase k in the cross-section (m/s) Superficial velocity of phase k (m/s) Temporal mean of velocity fluctuations (m/s) Volume (m3 ) Molar volume (m3 /mol) Weber number (–) Mole fraction of component i in the liquid phase (–) Mole fraction of component i in the vapor phase (–) Compressibility factor (–) Mole fraction of component i in the mixture (–)

Greek α α(T ) αk β β  U Ui UAPI  ρ ∈ εk ˜k η θ θf μ μi ν ρ σ τ φe φ2k φ2km ϕˆ i ψ ω

Shape parameter of the gamma probability distribution (–) Correction of the first parameter of the cubic equation of state (–) Volumetric fraction of phase k in the mixture (–) Scale parameter of the gamma probability distribution (–) Coefficient of thermal expansion (1/K) Gamma function (–) Relative density (–) Activity coefficient of component i in the mixture (–) API gravity (–) Fitting tolerance (–) Difference between the mass densities of the liquid and gas phases (kg/m3 ) Rate of energy dissipation per unit mass (m2 /s3 ) Mass fraction of phase k in the mixture (–) Mole fraction of phase k in the mixture (–) Third parameter of the gamma probability distribution (g/mol) Angle of inclination with respect to the horizontal (°) Liquid film contact angle (°) Dynamic viscosity (Pa · s) Chemical potential of component i in the mixture (J/mol) Kinematic viscosity (m2 /s) Mass density (kg/m3 ) Surface tension (J/m2 ) Shear stress (N/m2 ) Entrainment fraction (–) Two-phase multiplier of phase k based on its mass flow rate (–) Two-phase multiplier of phase k based on the mass flow rate of the mixture (–) Fugacity coefficient of component i in the mixture (–) Atomization/deposition rate (kg/(m2 · s)) Acentric factor (–)

xvi

Symbols and Abbreviations

Subscripts and Superscripts 

+ 0 a an atm b bh c Ci calc cell cond d dist DT e env evap exp f flat fric front g gd grav hor i ig ini is L l m max nb o od op PR pure

Derivative of the function Weight fraction of the sample Slip parameter Aqueous phase (formation water) Annular flow pattern Atmospheric conditions Bubble point Bottomhole conditions Critical point or gas-liquid core in annular SCN fraction with i carbon atoms Calculated value Unit cell of slug flow pattern Condensation Dew point Distributed arrangement Dumitrescu-Taylor bubble Entrained liquid Environment Evaporation Experimental value Liquid film Flat surface Friction or viscous friction term Front end of liquid slug Gas phase (natural gas) Gas dissolved in oil Gravitational term Horizontal direction Gas-liquid interface Ideal gas Initial Ideal solution Liquid state Liquid phase (combined water and oil) Mixture Maximum value Normal boiling point Oil phase Dead oil Operating condition Peng-Robinson equation of state Pure component

Symbols and Abbreviations

PVT r ref res salt sat SCN segr smooth SRK std sub surf tr V ver VPT w wk wave wavy

Pressure-Volume-Temperature (PVT) sample or test Reduced coordinate Reference value Reservoir Dissolved salts in the aqueous phase Liquid-vapor saturation SCN fraction Segregated arrangement Smooth surface Soave-Redlich-Kwong equation of state Standard conditions (1, atm and 15.56, °C) Subsaturation condition (single-phase liquid) Conditions at the Earth’s surface Transition Vapor state Vertical direction Valderrama-Patel-Teja equation of state Pipe wall Contact of pipe wall with phase k Interfacial wave Wavy surface

Abbreviations ANP API BSW IPR MCN OPW PDG PVT SCN SPU TBP TPR TPT WCT

Brazilian National Agency for Petroleum, Natural Gas and Biofuels American Petroleum Institute Basic Sediment and Water Inflow Performance Relationship Multiple Carbon Number Offshore Production Well Permanent Downhole Gauge Pressure-Volume-Temperature Single Carbon Number Stationary Production Unit True Boiling Point Tubing Performance Relationship Temperature and Pressure Transducer Wet Christmas Tree

xvii

Chapter 1

Introduction

1.1 Motivation Oil is an essential substance consumed by modern society due to its versatile applications. It powers various vehicles such as cars, tractors, trucks, aircraft, and ships, through the production of gasoline, kerosene, and diesel fuel. Additionally, it ensures the smooth operation of diverse machinery by providing greases and lubricants. Moreover, oil and natural gas play a crucial role in heating homes and commercial buildings, as well as in electricity generation. Furthermore, its derivatives serve as vital raw materials for manufacturing synthetic fibers used in clothing, plastics, paints, fertilizers, insecticides, soaps, and synthetic rubbers (Speight, 2014, p. 3). Despite its vital role in shaping the world as we know it, the emergence of the so-called oil industry, which encompasses both the oil and natural gas sectors, is relatively recent in historical terms. It all began with the drilling of the first modern well by Edwin L. Drake (1819–1880) in 1859. However, the global volume of oil extracted remained insignificant until the turn of the 20th century. According to Vassiliou (2018, p. 4), even by 1950, mineral oil accounted for only 30.3% of world consumption, with natural gas contributing just 9.9%, while coal dominated the primary energy sector with 55.5%. It wasn’t until 1964 that this scenario started to shift. In that year, oil’s share of energy consumption reached 40.8%, surpassing coal (39.7%) and natural gas (15.8%). The peak of oil usage, in terms of proportion, was observed in 1974, when it accounted for 49.4%. By comparison, coal stood at 25.6%, and natural gas at 19.5%. Subsequently, natural gas, along with nuclear and hydroelectric power, experienced continued growth. According to a survey conducted by BP in 2020, oil currently supplies 33.1% of the world’s energy demand. This significant share is maintained through a daily pro3 /d), which duction of approximately 95 million barrels (equivalent to 15 × 106 mstd has been growing at a rate of 1.4% per year over the last decade. Moreover, the current proved reserves amount to 1734 billion barrels, compared to 1532 billion barrels in 2009 and 1277 billion barrels in 1999. Notably, the volumes discovered in new © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. R. Gessner and J. R. Barbosa Jr., Integrated Modeling of Reservoir Fluid Properties and Multiphase Flow in Offshore Production Systems, Petroleum Engineering, https://doi.org/10.1007/978-3-031-39850-6_1

1

2

1 Introduction

accumulations have consistently outpaced the decline in mature fields. Considering these two factors, increased production and the expansion of reserves, it is projected that oil consumption will remain stable until the end of the 21st century (Speight, 2014, p. 47). It is well known that mineral resources are non-renewable sources of energy. As a result, as the shallowest reserves are depleted, oil exploration and production are moving towards increasingly hostile and difficult-to-access environments. Currently, a significant portion of these efforts is concentrated in oceanic areas, where water depths gradually increase. According to Vassiliou (2018, p. 7), in the early 1980s, the term deep water referred to a depth of approximately 250 m. The first major discoveries in the North Sea were made at depths below 150 m. After 25 years, depths below 450 m were considered shallow waters. Furthermore, exploration began in the so-called ultra-deep waters, which exceed 2100 m. Offshore oil has gained increasing importance in the global scenario. In 1972, offshore oil accounted for only 5% of the gross volume produced. By 1984, this proportion had risen to around 27%, and by 2000, it reached close to 35%. A similar percentage has been maintained since then. Obviously, the greater the difficulty in accessing the deposits, the higher the extraction cost. The calculation of reserves takes into account the commercial value of the product. During the 1990s, the price of Brent remained below US$30 per barrel, but then started to experience significant increases. The peak occurred from 2011 to 2013, when it exceeded the US$100 mark. There was a subsequent drop in the following years, with prices approaching US$40, and in 2019, the average barrel price was US$64 (BP, 2020). In the same year, Fortune magazine’s list of the 500 largest companies in the world1 revealed that oil companies occupied five of the top 10 positions: Sinopec Group (2nd), China National Petroleum (4th), Royal Dutch Shell (5th), Saudi Aramco (6th), and BP (8th). When their respective revenues are combined, the impressive figure of 1.75 trillion dollars per year is obtained. In Brazil, oil production in 2019 reached 2.88 million barrels per day, equivalent to 3.0% of the global share, ranking it 10th worldwide (BP, 2020). Over the past decade, there has been an average annual growth rate of 3.6%. National reserves experienced a significant increase from 8.2 billion barrels in 1999 to 12.9 billion in 2009, primarily due to the exploration of the pre-salt cluster. Since then, the reserve volume has remained relatively stable, with 12.7 billion barrels in 2019, positioning the country 15th in the global ranking. The Brazilian pre-salt region is a significant example of the oil industry’s ongoing offshore exploration for new assets. It covers a vast oceanic area stretching between the coastal states of Espírito Santo and Santa Catarina, approximately 300 km offshore (Costa Fraga et al., 2015). This area consists of water depths ranging from 1500 to 2200 m, categorized as deep and ultra-deep waters. To put it into perspective, the exclusive economic zone, which grants the country ownership of all water and subsoil resources, is defined by a line located 370 km from the coast.

1

Available for consultation at: https://fortune.com/global500/.

1.1 Motivation

3

The pre-salt layer, composed mainly of carbonate, is situated between 5000 and 7000 m below sea level. It lies beneath a saline formation with a thickness ranging from 300 to 2000 m. The oil accumulations in this region are colossal, with the Lula field alone estimated to have a recoverable volume of 8.3 billion barrels of oil, making it the largest field in Brazil. These oil reservoirs are known for their good quality, with densities ranging from 28 to 32 ◦ API2 (Vianna Filho et al., 2015). The first proof of the existence of hydrocarbons in the pre-salt occurred in the Parati area, in the Santos Basin, in 2006. Two years later, production began for commercial purposes in the Parque das Baleias region. According to data3 from the Brazilian National Agency for Petroleum (ANP), in July 2014, with the implementation of new projects, the extraction of pre-salt oil reached the mark of 500 thousand barrels per day. The level of 1 million barrels per day was reached soon after, in May 2016. Finally, in June 2017, the daily production of 1.35 million barrels surpassed the volume extracted from the post-salt for the first time in history. Building and equipping an offshore well is a truly challenging task due to geopressure and wall stability issues. As a result, the rocks cannot be drilled all at once, but in phases. Therefore, the installation of metallic casings becomes imperative intermittently, which still need to be cemented. With each new phase, the diameter of the drill passage is further reduced, leading to the final structure depicted in Fig. 1.1, which vaguely resembles a telescope (Thomas, 2004, Chap. 4). Another crucial piece of equipment, installed subsequently, is the production tubing, responsible for extracting the mixture from the reservoir through the well’s interior. Hence, it is manufactured using alloys that possess exceptional resistance to corrosion. This kilometer-long equipment, three times taller than the tallest skyscraper ever built, must be lowered in just two parts. Situated above it, on the seabed, is a collection of remote-activated valves, known as the Wet Christmas Tree (WCT). The WCT is responsible for halting the flow in the well during emergency situations, among other functions (Thomas, 2004, Chap. 6). The oil well and the subsea pipelines connected to it via the WCT form the Offshore Production Well (OPW). Among the equipment in this group, special mention goes to the production pipeline, which is a flexible structure consisting of multiple metallic and polymeric layers, as illustrated in Fig. 1.2. The section of the pipeline that lies completely on the seabed is known as the flowline, while the suspended section is called the riser. At the other end of the riser is the Stationary Production Unit (SPU), which is responsible for primary oil processing. A typical OPW found in pre-salt fields is depicted in Fig. 1.3. It is important to note that the lengths and depths shown in the figure may vary significantly, but they serve as a reference. Figure 1.3 also indicates the pressure and temperature sensors typically present in the well, which allow real-time monitoring of its operation. These sensors include the Temperature and Pressure Transducer (TPT) located inside the WCT and the Permanent Downhole Gauge (PDG) installed in the tubing near the top of the reservoir. 2

The API scale will be defined in Sect. 1.4. In general terms, the higher the value of the oil on the API scale, the better its quality. 3 Available for consultation at: https://www.presalpetroleo.gov.br.

4

1 Introduction

WCT

Seabed Conductor casing Cement Surface casing Drilling fluid Intermediate casing Completion fluid Production casing Production tubing Production liner Hydraulic packer

Reservoir

Fig. 1.1 Principal components and geometric features of an oil producing well

The rapid change in the scenario of oil production in Brazil can be attributed, in part, to the high production flows achieved per well in the pre-salt fields, aver3 /d). This can be attributed to aging around 20 thousand barrels per day (3180 mstd two main factors. Firstly, the reservoir rock exhibits high permeability, resulting in minimal fluid pressure loss along the porous medium. Secondly, the utilization of large diameter conduits in both the tubing and production pipeline helps to minimize frictional pressure losses during transportation to the surface. These two factors, combined with the immense static pressure of the formation, ensure a continuous flow of fluids to the surface, rendering the need for any supplementary sources of energy unnecessary.

1.2 Oil Constituents

5

Interlocked casing Internal pressure sheath Interlocked pressure armor Back up pressure armor Anti-wear layer First tensile armour Anti-wear layer

Second tensile armour

Outer sheath

Fig. 1.2 Material layers that constitute a flexible production pipeline (API, 2014)

1.2 Oil Constituents Petroleum is composed of a mixture of hydrocarbons with different molecule sizes, in varying amounts both geographically and in depth. Under Standard conditions4 , the lighter fractions tend to form a vapor phase known as natural gas, while the heavier fractions form a liquid mass referred to as dead oil or simply oil. In fluids extracted from the pre-salt layer, the presence of significant amounts of light fractions leads to 3 3 /mstd in certain situations (Arinelli a Gas-Oil Ratio (GOR5 ) that can reach 600 mstd et al., 2015). As it is not practical to produce only one of these phases, and in most cases, both phases flow to the surface, the distinction between oil wells and natural gas wells is solely based on the commercial interest of the field. Among all hydrocarbons, methane (CH4 ) has the simplest structure and is the most abundant substance in accumulations. Since it consists of only one carbon atom, it is commonly referred to as C1 . Similarly, ethane (C2 H6 ) is referred to as C2 , propane

4

Standard conditions, established by the American Petroleum Institute (API), are defined as 14.7 psia (101, 325 kPa) and 60 ◦ F (288.71 K). 5 The GOR represents the volumetric ratio between the gas and oil components of the mixture, measured under Standard conditions.

6

1 Introduction SPU

Surface

0m

Riser (2000 m)

Seabed

Flowline (3000 m)

WCT TPT

1400 m

Tubing (3000 m)

PDG

Reservoir

4400 m

Fig. 1.3 Offshore Production Well (OPW) typically found in pre-salt fields. Pipe lengths and depths are for illustrative purposes only

(C3 H8 ) as C3 , and so on. According to Pedersen et al. (2014, p. 1), hydrocarbons found in oil reservoirs can be classified into three categories: Paraffins: These are characterized by segments such as C, CH, CH2 , or CH3 , where the carbon atoms are connected by single bonds. Paraffins can be further divided into two types. In normal paraffins (or n-paraffins), the chains are open and linear. In iso-paraffins (or i-paraffins), the chains are open but contain at least one branch; Naphthenes: Substances belonging to this class are similar to paraffins as they are composed of the same types of segments. However, naphthenes have one or more cyclic structures in addition to the single bonds between segments; Aromatics: Similar to naphthenes, aromatics also possess one or more cyclic structures within their chains. The distinction lies in the carbon atom bonds, which alternate between double and single bonds in aromatics. The distribution of paraffinic (P), naphthenic (N), and aromatic (A) compounds in a fluid is commonly referred to as the PNA distribution. Oil reservoirs may also contain inorganic compounds such as nitrogen gas (N2 ), carbon dioxide (CO2 ), and hydrogen sulfide gas (H2 S). Water (H2 O) is also frequently present in these rock formations in the form of brine, but due to its low miscibility with the other constituents, most of it is concentrated in a separate zone below the oil

1.2 Oil Constituents

7

Table 1.1 Physicochemical properties of some petroleum constituents. Source Poling et al. (2000, Appendix A) Component Mw , g/mol Tc , K Pc , MPa ω N2 CO2 H2 S C1 C2 C3 iC4 nC4 iC5 nC5 nC6

28.01 44.01 34.08 16.04 30.07 44.10 58.12 58.12 72.15 72.15 86.18

126.2 304.2 373.4 190.6 305.3 369.8 407.8 425.1 460.4 469.7 507.6

3.40 7.37 8.96 4.60 4.87 4.25 3.64 3.80 3.38 3.37 3.02

0.037 0.225 0.090 0.011 0.099 0.152 0.186 0.200 0.229 0.252 0.300

(Pedersen et al., 2014, p. 1). The BSW6 of a well measures the ratio between its daily production of water and liquid. Some characteristics specific to the Brazilian pre-salt include the high mole fraction of CO2 , ranging from 1 to 20% in the Lula field and increasing to 44% in Libra and 79% in the Jupiter field (Rempto et al., 2018). Additionally, the connate water of the reservoir exhibits high salinity, exceeding 200,000 parts per million (Drexler et al., 2019). For comparison, the average salinity of seawater is 35,000 ppm. Even within the same hydrocarbon class, the physicochemical properties vary significantly from one substance to another. Take the paraffins as an example: when comparing methane and n-hexane, as listed in Table 1.1, one observes a considerable difference in their critical temperature (Tc ), ranging from 190.6 K to 507.6 K. Similarly, the critical pressure (Pc ) varies from 4.60 MPa to 3.02 MPa, and the acentric factor (ω) ranges from 0.011 to 0.300. The physical state of these substances is closely related to the chain length. Methane, ethane, propane, and n-butane, under Standard conditions, exist as gases, while paraffins with chain lengths between npentane (nC5) and n-hexadecane (nC16) are liquids. Chains with larger lengths, starting from n-heptadecane (nC17+ ), form solids. The properties of hydrocarbons with the same number of carbon atoms undergo remarkable variations due to the structure of their chains. Consider three hydrocarbons with six carbon atoms (C6 ): n-hexane (paraffin), methylcyclopentane (naphthene), and benzene (aromatic). According to Poling et al. (2000, appendix A), their critical temperatures are as follows: 507.4 K for n-hexane, 532.8 K for methylcyclopentane, and 562.0 K for benzene. Similarly, their critical pressures are 2.97, 3.78, and 4.89 MPa, respectively.

6

Basic Sediment and Water.

8

1 Introduction

Table 1.2 Number of possible isomers as a function of the amount of carbon atoms present in the hydrocarbon molecule. Source Speight (2014, p. 32) Number of carbon atoms Number of isomers 4 8 12 18

2 18 355 60523

Furthermore, it is observed that the mass density of n-hexane is lower than that of methylcyclopentane. Under Standard conditions, n-hexane has a mass density of 659 kg/m3 , while methylcyclopentane has a mass density of 754 kg/m3 . Both of these values are less than the mass density of benzene, which is 885 kg/m3 . Petroleum fluids can have fractions as high as C200 , or even higher. As the number of carbon atoms in the molecule increases, there is a simultaneous rapid increase in the number of isomers, which are substances with different structures but the same molecular formula (Table 1.2). Consequently, a sample of oil can contain millions of substances in its composition. However, conducting an individual analysis of each substance is impractical and economically unfeasible. Instead, it is customary to group them based on shared chemical characteristics. According to the widely used True Boiling Point (TBP) test, which provides typical results for this type of application, the components present in the oil mixture are classified into three categories (Pedersen et al., 2014, p. 105): Defined components: This group includes substances with known molecular structures. The critical properties and acentric factor of these substances are available in the literature. It is important to note that these parameters are crucial for the application of cubic equations of state. Table 1.1 lists the defined components, including contaminants such as N2 , CO2 , and H2 S, as well as hydrocarbons like C1 , C2 , C3 , iC4 , nC4 , iC5 , and nC5 . Some authors also include C6 , which, although having isomers, is typically treated as pure nC6 ; SCN Fractions: The increasing variety of structures that emerge as the number of carbon atoms exceeds C6 makes it more convenient to divide the C7+ portion into fractions known as Single Carbon Number (SCN). These fractions group together components with similar boiling points. However, even after separation from the original sample, SCN fractions remain mixtures with unknown critical properties and acentric factors; Heavy fraction: This group comprises substances with high boiling points that prevent their separation into SCN fractions. Considering that laboratory analyses typically cover only up to C19, the heavy fraction consists of C20+ molecules with undetermined structures and molar distributions.

1.3 Classification of Reservoirs

9

The SCN fractions correspond to the distillate content obtained by heating the mixture within specific temperature ranges, which adhere to the normal boiling points7 (Tnb ) of the respective n-paraffins. The TBP test also provides information on the relative density (γ) and molecular mass (Mw ) of each fraction. Alternatively, values compiled by Katz and Firoozabadi (1978) or empirical correlations extensively discussed in the literature can be used. These two properties serve as input parameters in calculating the critical pressure, temperature, and acentric factor. Finally, according to Ahmed (2016, p. 77), nearly all natural oil deposits contain unidentified heavy components to some extent. Representing the heavy fraction as a known sum of SCN fractions poses a significant challenge in thermodynamic modeling since, typically, only the mole fraction, density, and an estimated molecular mass of the heavy fraction are available, with an uncertainty of approximately 20% (Ahmed, 2016, p. 140).

1.3 Classification of Reservoirs As discussed previously, when examining the individual components of petroleum fluids separately, a wide range of values for various properties can be identified. However, considering them as a whole reveals even greater variations. The composition of petroleum fluids can either trigger significant events during the flow to the SPU or completely suppress them, leading to consequences for the design of the OPWs. Ultimately, this impacts the technical feasibility of the project itself. It is customary to attribute a significant portion of these discrepancies between different fields to the combined action of only two parameters: the reservoir temperature (Tr es ) and the critical temperature of the mixture contained therein. Consequently, reservoirs are automatically divided into four major groups, at least (Pedersen et al., 2014, p. 7): Natural gas reservoirs: In situations where Tr es  Tc , the flow remains singlephase throughout the tubing and production pipeline. No point in the system offers the necessary thermodynamic conditions for partial condensation (McCain, 1990, p. 148). If this behavior also prevails in the downstream separator, the gas is considered dry gas; otherwise, it is classified as wet gas; Condensate gas reservoirs: As the difference between Tr es and Tc decreases, the mixture eventually crosses the dew point curve, leading to the formation of a liquid phase called condensate. The presence of condensate gradually increases from that point onwards. However, due to pressure drop during flow, there is often partial or total vaporization of the condensate content before reaching the surface. This phenomenon, known as retrograde condensation, gives the formation fluid the name retrograde gas (Ahmed, 2016, p. 44); Near-critical oil reservoirs: When Tc exceeds Tr es by a small margin, the fluid enters the two-phase region through the bubble point curve, indicating its previous state 7

The normal boiling point of a substance refers to the temperature at which the liquid and vapor phases coexist at atmospheric pressure (Riazi, 2005, p. 31).

10

1 Introduction

as a liquid. Consequently, it should be treated as an oil. The proximity of these temperature values results in phases with very similar compositions and significant vaporization of the initial inventory as pressure drops. Some authors refer to this type of fluid as volatile oil; Black oil reservoirs: In high molecular weight hydrocarbon accumulations where Tc  Tr es , the stability of the liquid phase is increased due to the long hydrocarbon chains. This stability leads to a smaller amount of gas coming out of solution and gives the mixture a dark color. These fluids are categorized as light oils or heavy oils based on their density under standard conditions (Pedersen et al., 2014, p. 128).

1.4 Volumetric and Phase Behavior Oil reservoirs are subject to conditions that are very different from the standard. In the pre-salt layer, for example, the initial pressure8 easily exceeds 500 bar (Rempto et al., 2018), or 50 MPa, following a gradient close to 0.1 bar/m (Costa Fraga et al., 2015). On the other hand, temperature has a strong relationship with the thickness of the salt layer. When it is pronounced, the good thermal conductivity of the salt reduces the geothermal gradient and, consequently, the temperature of the reservoir (Gaffney, Cline & Associates, 2010, p. 8). When the layer is thin, the temperature increases with depth, reaching 120 ◦ C (393 K) in the perforated zone. Together, the tubing and production pipeline can reach lengths of up to 10 km (Carneiro et al., 2015). Both pressure and temperature gradually decrease along this path. Therefore, maximizing productivity in the fields requires a thorough understanding of how the fluid responds to different thermodynamic states. In this context, the term PVT property is used to refer to the dependence of the volume (V ) of a sample on the pressure (P) and temperature (T ) levels at which it is found (Pedersen et al., 2014, p. 47). This behavior can be better understood by conducting PVT experiments in the laboratory. The analyses to which the oil samples are subjected aim to quantify the mass density and mass fraction levels attained by the liquid and vapor phases for a fixed temperature and various pressures. However, the results are typically not expressed on a mass or molar basis. One of the peculiarities of the oil industry lies in the volumetric characterization of the reservoir fluid for a certain state P and T , followed by comparison with the readings under standard conditions. This rather unusual methodology yields properties that are uncommon in other areas of engineering, namely: Oil formation volume factor (B o ): This factor represents the ratio between the volume occupied by the oil under in situ and standard conditions, respectively. During transportation from the reservoir to the surface, a portion of the initial oil mass vaporizes due to the migration of light fractions into the vapor phase, resulting 8

The reservoir pressure tends to decrease with the advancement of production, which is called depletion.

1.4 Volumetric and Phase Behavior

11

Bo Bob Bo,ini

Bod Patm

Pb

Pini

P

Fig. 1.4 Behavior of the oil formation volume factor as a function of pressure

in oil shrinkage. Conversely, increasing pressure leads to the incorporation of light fractions by the liquid phase, causing the oil to appear swollen. Figure 1.4 illustrates this mechanism, which operates until the bubble point of the mixture, where the oil reaches its maximum volume. Beyond this point, the behavior of the single-phase fluid is determined solely by its isothermal compressibility, resulting in volume reduction with higher pressures; Natural gas formation volume factor (B g ): Similar to oil, Bg represents the ratio between the volume occupied by a certain portion of natural gas at P and T and its volume under standard conditions. However, the transfer of mass between phases is not considered, assuming that the composition remains constant in both states. Consequently, the volumetric behavior of natural gas is governed solely by its isothermal compressibility and expansion coefficient; Solubility ratio (R s ): This ratio quantifies the volume of gas that separates from the oil solution when the sample is taken from in situ to standard conditions. Therefore, a higher Rs indicates a greater presence of light fractions in the liquid, and it increases with pressure until the bubble point of the mixture. Figure 1.5 illustrates this behavior. Beyond the bubble point, the composition of the liquid no longer changes, and the solubility ratio becomes the gas-oil ratio (GOR). On the other hand, for an oil already in standard conditions, Rs is zero. Another commonly used parameter is the dead oil relative density (γo ), which is determined by comparing the mass density of the liquid phase under Standard conditions and the value observed for water at the same pressure and temperature level, which is equal to 999 kg/m3 (Riazi, 2005, p. 21). However, in some cases, it is preferable to use the American Petroleum Institute (API) scale, which is given by:

12

1 Introduction

Rs GOR

0 Patm

Pb

Pini

P

Fig. 1.5 Behavior of the solubility ratio as a function of pressure

γAP I =

141.5 − 131.5. γo

(1.1)

When calculating the relative density of natural gas (γg ), the comparison parameter is the density of air, which is equal to 1.225 kg/m3 . Naturally, each PVT experiment is better suited to certain types of reservoir fluids, and the choice of the set of tests to be performed is highly dependent on the phase behavior of the mixture. Also, as a result of this, the way data are organized undergoes adaptations. That being said, the routine analyses applied to pre-salt samples are: Differential liberation test: This test emulates the compositional and volumetric changes that occur in the reservoir throughout its productive life. At the beginning of the experiment, a predefined volume of single-phase sample is transferred into a PVT cell, which is maintained at a constant temperature (generally Tr es ) and an initial pressure close to Pr es . After the fluid stabilizes, the P–V pair obtained for these conditions is recorded. The cell is then expanded,9 which increases the volume of the sample and reduces the pressure. Once equilibrium is reestablished, the new P-V pair is recorded. This process is repeated until the bubble-point pressure of the mixture is extrapolated. From that point on, the vapor that arises in each stage of depressurization must be removed from the cell while keeping the pressure constant, and its volume and composition are measured under Standard conditions. The test proceeds until the atmospheric pressure is reached, and then the sample is cooled to the reference temperature. At the end of the experiment, the collected data enables the assessment of the values of Bo , Bg , Rs , mass density, 9

The internal volume of the PVT cell can be varied by moving a piston (Rosa et al., 2006, p. 74).

1.4 Volumetric and Phase Behavior

13

and vapor compressibility factor for each stage. The bubble pressure for the test temperature and the relative density of the dead oil are also available; Flash liberation test: Similar to the previous experiment, a known volume of singlephase sample is transferred to a PVT cell in this test. The cell is heated to a certain temperature (usually between Tr es and the expected surface temperature). The fluid is then stabilized at a pressure level above that found in the reservoir. Once this is done, the cell contents are directly depressurized to Patm without recording the phase behavior at the intermediate points. After separating and cooling both phases to the reference temperature, the relative density of dead oil and natural gas, and the gas-oil ratio (GOR) are calculated. It is also customary to determine the molar composition of each phase using gas chromatography techniques (for the vapor phase) and cryoscopy (for the liquid phase); Oil viscosity test: This experiment is conducted using a rolling ball viscometer, also known as a Stokes viscometer. In this test, the viscosity is correlated with the fall time of a metallic sphere in a cylinder filled with liquid, using the equipment calibration equation. The fluid sample should be maintained at a constant temperature, typically Tr es , and decreasing pressures. As shown in Fig. 1.6, the behavior presented by the single-phase mixture (i.e., above Pb ) is similar to that of any other substance: the reduction in pressure causes an expansion of the fluid, resulting in a decrease in viscosity. The point of minimum, which for pre-salt samples is around 1 × 10−3 Pa · s (Boyd et al., 2015), is reached at the bubble pressure. From then on, the liquid begins to undergo changes in its composition, gradually losing the lighter fractions to the newly formed vapor phase. Therefore, the viscosity of a hydrocarbon is closely associated with the size of its molecules. As a result, there is a significant increase in the property’s values as the pressure drops from Pb to atmospheric pressure. The PVT tests are essential tools; however, they are not sufficient for assessing the phase and volumetric behavior of a sample. Their inadvertent use would encounter, primarily, financial constraints due to the associated high costs. The time devoted to the task could also have a negative impact, depending on the number of points to be included. Furthermore, as the field develops, changes in the composition of the fluid leaving the reservoir would necessitate periodic studies to correct the phase behavior. Unfortunately, there is no prospect of foreseeing the effect of these changes on the OPW. Lastly, the mixing of streams from different formations, as observed in subsea production manifolds, would require an impractical quantity of samples to cover each of the potential scenarios. In this context, mathematical modeling emerges as a notably attractive alternative. Since the 1960s, the petroleum industry has been increasingly interested in numerical packages capable of representing the phase behavior of multicomponent mixtures (Pedersen et al., 2014, p. 83). It is worth noting that, with the processing speed available in today’s electronic devices, millions of calculations of thermodynamic properties can be performed in a matter of seconds. Behind these computational tools lie the cubic equations of state, discussed in Chap. 2. According to Goodwin et al. (2010, p. 54), the main advantages of this approach lie in its relative

14

1 Introduction

µo µod

µo,ini µob

Patm

Pb

Pini

P

Fig. 1.6 Behavior of the viscosity of the liquid phase as a function of pressure

simplicity in terms of mathematical formulation and the high accuracy of the obtained results. Additionally, cubic equations of state are versatile, being applicable to both homogeneous systems (i.e., single-phase and uniform at any sample point) and phase equilibrium systems (i.e., saturated or below the bubble pressure for mixtures). On the other hand, fluids originating from the pre-salt layer impose additional challenges in the calculation of volume and phase equilibrium, primarily due to their significant CO2 content. Dealing with a substance that deviates so significantly from the rest of the mixture requires a meticulous description of the physicochemical interactions among these components. The possibilities for adjusting the model based on experimental results from the sample are usually limited to modifications in the heavy fraction’s molecular mass, critical properties and acentric factor of the SCN fractions, as well as the volume translation parameter10 of each component (Pedersen et al., 2014, p. 198–199). As a result, efforts are focused on improving only a few parameters, such as the bubble point pressure, with no interventions being made in the partitioning parameters of the heavy fraction, which are crucial for the proper application of cubic equations of state. These facts highlight the limited degree of optimization in the currently available mathematical models while indicating potential future opportunities for improvement.

10

The volume translation, proposed by Peneloux et al. (1982), allows adjusting the volumetric behavior of a pure substance or mixture without affecting phase equilibrium calculations.

1.5 The Flow of Oil Mixtures

15

1.5 The Flow of Oil Mixtures The pressure drop experienced by the mixture on its way to the surface is governed by two main mechanisms. In vertical sections, it is mostly due to gravitational action, while in horizontal sections, viscous friction prevails. Thus, although the flow is sometimes single-phase in the region close to the bottom of the well, there is no way to avoid the appearance of a vapor phase from the point where P < Pb (Brill & Mukherjee, 1999, p. 1). Furthermore, the constant presence of water in the formation favors the entrainment of a portion of this content along with the hydrocarbons. Therefore, the flow of oil is essentially multiphase in nature. When two or more phases flow through the same pipeline, remarkably intricate interaction mechanisms arise. The forces induced by turbulence, which promote the mixing of different portions of the fluid, constantly compete with gravity, which attempts to separate the lighter from the heavier phases. The shear stress at the pipe wall exhibits significant heterogeneities, reflecting a non-uniform fluid viscosity distribution. The expansion of natural gas with decreasing pressure, considerably more pronounced than that of oil or water, results in a continuous increase in its in-situ volumetric flow rate. All of these factors lead to the development of uneven average velocities among the phases, a phenomenon known as slippage. In upward sections, natural gas, being less dense and viscous, will move more rapidly than the liquid phases. Conversely, in downward sections, the situation is reversed, and both oil and water will exhibit higher velocities (Brill & Mukherjee, 1999, p. 19). However, it should be noted that, compared to natural gas, the properties of the oil and water phases exhibit a much smaller contrast. As a result, their slip is typically minimal, except perhaps for horizontal flows with low flow rates. In the case of production pipelines of pre-salt reservoirs, although there may be long sections with slight inclinations on the seabed, the velocity of the mixture suppresses this effect. Furthermore, as a general rule, the reservoir water is dispersed in the oil in the form of small droplets, and the resulting emulsion further restricts their individual movement. Therefore, it is customary to treat oil and water as a single liquid phase with weighted properties, which, along with natural gas, acquires the characteristics of a two-phase flow. Figure 1.7 shows a schematic description of an idealized gas-liquid flow. Assuming that the mass flow rate of the mixture ( M˙ m ) is known, it is possible to quantify the value assumed by each of the phases through the following relation: M˙ k = εk M˙ m = ρk Uk Ak .

(1.2)

In Eq. (1.2), εk , ρk , Uk , and Ak represent the mass fraction, mass density, average in situ velocity, and the cross-sectional area occupied by phase k, respectively. Another commonly used concept is the superficial velocity (Usk ), which represents the velocity that would be obtained if phase k were to flow alone in the pipe, and is given by:

16

1 Introduction

Fig. 1.7 Main parameters of an idealized two-phase gas-liquid flow

Usk ≡

εk M˙ m M˙ k = . ρk A ρk A

(1.3)

Combining Eqs. (1.2) and (1.3), one has: Usk = αk Uk ,

(1.4)

where the volume fraction of the phase (αk ) is equivalent to: αk =

Ak . A

(1.5)

The superficial velocity of the mixture (Usm ), i.e., the velocity of the center of volume (Yadigaroglu & Hewitt, 2017, p. 26), results from the expression: Usm = Usg + Usl .

(1.6)

An important implication of the liquid-gas slip is its effect on the space occupied by these two fluids inside the pipe. According to Eq. (1.4), it can be observed that when phase k assumes a higher velocity for the same value of Usk , its volumetric fraction decreases. Conversely, the slower phase, usually the liquid phase, occupies a larger percentage of the pipe as the slip becomes more pronounced (Shoham, 2006, p. 8). Figure 1.7 represents other relevant parameters of this type of flow, such as the shear stresses at the pipe wall (τwk ) and at the interface (τi ), as well as the heat fluxes  ). from mixture to the external environment (qwk In theory, the main singularity of gas and liquid flows in relation to single-phase flows resides in the interface, a deformable surface highly subject to interactions involving the phases and the wall of the traversed pipeline. Consequently, the physical properties of the fluids, the fractions of liquid and gas, their superficial velocities, and the geometry of the pipe endow the flow with a certain macroscopic arrangement. The major challenge associated with two-phase flows, according to Yadigaroglu and Hewitt (2017, p. 7), lies in the fact that their topology is not known a priori but is part of the problem’s solution. In turn, Brennen (2005, p. 163) points out that research is still necessary to achieve a satisfactory understanding of all the

1.5 The Flow of Oil Mixtures

17

involved phenomena. The classification of flow patterns suggested by Barnea (1987) results in the following configurations, broadly divided into segregated (separated) and distributed (dispersed) flows: Stratified flow: This pattern is typically detected in horizontal or near-horizontal pipes with low liquid and gas flow rates. In this pattern, the two phases are separated by gravity, with the denser phase flowing at the bottom and the lighter phase at the top. The stratified pattern can be further subdivided into two types: smooth, where the interface is flat and undisturbed, and wavy, where Kelvin-Helmholtz instabilities cause small-scale distortions of the interface. The magnitude of these perturbations may either decay or grow, depending on the fluid properties and flow conditions; Annular flow: The occurrence of this pattern requires substantial gas flow rates. In the annular pattern, the liquid forms a thin film that coats the pipe wall and moves at a slower velocity than the gas. The film may be thicker at the bottom than at the top due to gravity, depending on the pipe inclination. In vertical flows, the film thickness remains approximately uniform, giving the arrangement an axisymmetric appearance. The higher speed of the gas phase creates interfacial waves, the crests of which are broken up into small droplets that become entrained in the core region. Due to several mechanisms, including turbulent fluctuations in the gas core, the droplets eventually return to the liquid film, giving rise to a mass interchange process known as droplet entrainment and deposition (Hewitt and Hall-Taylor, 1970); Slug flow: This intermittent flow pattern is characterized by the alternating flow of liquid and gas. The liquid flows in the form of slugs that fill the entire pipe crosssectional area and are interspersed with large gas bubbles separated from the wall by a liquid film. These large bubbles are also referred to as Dumitrescu-Taylor bubbles, named after pioneering studies independently conducted by Dumitrescu (1943) and Davies and Taylor (1950). In vertical flows, the Dumitrescu-Taylor bubbles take on a symmetric bullet-like shape, while in other cases, their features change significantly with the inclination and mixture velocity. For near-horizontal flows, the liquid film thickness around the long bubble is not uniform, resembling the stratified flow pattern. In every positive inclination, the liquid slug moves faster than the liquid film (which actually flows downward in vertical slug flow), generating a strong mixing region at the tail of the long bubble that entraps tiny bubbles into the liquid slug. Intermittent horizontal flows with weakly aerated slugs resulting from moderate gas rates are commonly referred to as the Elongated bubble regime, while the aerated slugs typical of higher gas flow rates give rise to what is known as the Slug flow regime; Bubbly flow: This pattern originates in vertical or sharply inclined pipes with low liquid and gas flows. In the bubbly flow pattern, the gas phase is fragmented into small bubbles that percolate through the continuous liquid, while manifesting a curious lateral zigzag movement. The distribution of bubbles is approximately homogeneous across the entire cross-section;

18

1 Introduction

Dispersed bubble flow: While this pattern resembles bubbly flow, the increased turbulence in the liquid phase disperses the gas into even smaller particles, causing them to no longer exhibit oscillatory movements. The slippage between the phases also experiences a significant reduction, allowing the arrangement to be sustainable at any angle of inclination, provided that the liquid flow is sufficiently high; Churn flow: This flow regime is typical of vertical or sharply inclined pipes and is intermediate between slug flow and annular flow. The transition from slug flow to churn flow is dictated by flooding in the liquid film around the Dumitrescu-Taylor bubble, leading to upward transport of liquid in this region and the destabilization of the liquid slugs between consecutive long bubbles (Jayanti and Hewitt, 1992). The resulting thick liquid film is periodically swept by large flooding-type waves, which are responsible for a net flow of liquid in the upward direction (Hewitt et al., 1985). The gas core is continuous, and there is a considerable amount of liquid entrained as droplets (Barbosa et al., 2002). As the gas flow rate increases, there is a gradual cessation of downwards film flow, ultimately giving rise to upwards, unidirectional annular flow. The flow patterns mentioned above are shown in Fig. 1.8 for horizontal or low inclination pipes, and in Fig. 1.9 for vertical or high inclination pipes. Their respective descriptions were taken from Shoham (2006, p. 10–11). According to Yadigaroglu and Hewitt (2017, p. 97), the identification of the arrangement assumed by the liquid and the gas in an experiment can be done by direct visualization, which includes photographs, X-ray images, or graphical representations of the volumetric fraction (based on instruments such as resistive or fiber optic probes, multi-beam gamma densitometers, electrode mesh sensors, etc.), or through indirect methods, such as X-ray beam attenuation. These methods still have a certain degree of subjectivity, which explains the existence in the literature of a series of patterns in addition to those already mentioned, such as wispy-annular (Yadigaroglu & Hewitt, 2017, p. 102), wavy annular (Shoham, 2006, p. 10), or mist flow (Beggs and Brill, 1973). A highly didactic approach to categorize all possible arrangements assumed by the gas-liquid system is achieved through the use of so-called pattern maps, which link the flow regime to the individual flow rates of its phases. These flow rates can be volumetric (equal, therefore, to Usk ), mass-based, or momentum-based, depending on the author’s preference. Examples of maps generated in laboratory settings for horizontal and vertical pipelines are presented in Figs. 1.10 and 1.11, respectively. It is important to emphasize that the boundaries established therein solely depict the apparatus geometry and the thermodynamic and transport properties of the selected fluids for that specific test, although their general arrangement remains independent of such parameters. The low level of generality of the experiments, coupled with the multitude of variables involved in the problem, underscores the need for mathematical models capable of making such predictions. Another important point, highlighted by Brennen (2005, p. 165), concerns the probabilistic nature of flow regime transitions, resulting from the large entrance lengths required for certain flow configurations to fully develop (sometimes

1.5 The Flow of Oil Mixtures

19

Fig. 1.8 Two-phase gas-liquid flow patterns in horizontal or low inclination pipes

surpassing the limit imposed by the experimental setup). This is not to mention that, in certain situations, the resulting pattern is the product of events that occur in this region (Bendiksen and Malnes, 1987). Here, an analogy can be drawn with the transition from laminar to turbulent regime commonly observed in single-phase flows. The shaded regions with indefinite thickness in Figs. 1.10 and 1.11, therefore, provide a more appropriate separation of the different arrangements than conventional solid lines would. Flow pattern determination based on physically-based transition criteria constitutes the first step towards the mathematical solution of the flow using the mechanistic (or phenomenological) approach. In segregated arrangements, which include the stratified, annular, and, in some approaches, the churn flow patterns (see Figs. 1.8 and 1.9), where two or more distinct fluids flow continuously, in the same direction or in counterflow, a force balance in the axial direction is performed to calculate the hydrodynamic equilibrium state. On the other hand, in distributed arrangements (bubble and slug-type flows), the most widely adopted strategy assumes a

20

1 Introduction

Fig. 1.9 Two-phase gas-liquid flow patterns in vertical or high inclination pipes

mixture-based approach, where the relative velocity of the dispersed phase is written in terms of its shape and diameter. Both approaches will be presented in detail in Chap. 4. The calculation of the hydrodynamic parameters of the flow ultimately allows determining the local convective heat transfer coefficients as well. This, combined with the overall thermal conductance of the pipe wall material and the conditions prevailing in the external environment, defines the magnitude of the heat flow leaving the mixture as a function of position along the OPW. Just as it has been observed for the mixture thermodynamic behavior, the modeling of oil flow in pre-salt wells and their respective subsea pipelines is also hindered by significant fractions of CO2 . In such a scenario, CO2 increases the equivalent molecular weight and density of the natural gas, making it similar to that of the liquid, even at moderate pressures (Carneiro et al., 2015). Until now, extensively validated governing equations to model such phenomena are not known. In fact, many reference works, such as those by Xiao et al. (1990), Gomez et al. (2000), and Zhang et al. (2003), are not unanimous in considering the influence of gas density in closure relationships for these flows. On the other hand, other authors (Hasan and Kabir, 1988; Bendiksen et al., 1991; Ansari et al., 1994; Petalas and Aziz, 1998)

1.5 The Flow of Oil Mixtures

21

Dispersed bubble

Usl , m/s

101

100

Elongated bubble

Slug

Annular

10−1

10−2

10−1

100 Usg , m/s

avy .w Str

Stratified smooth

10−2

101

102

Fig. 1.10 Two-phase flow pattern map for air and water at atmospheric pressure and temperature of 293.15 K in a 25.4-mm ID horizontal pipe (Mandhane et al., 1974)

102 Dispersed bubble

Usl [m/s]

101 100 Bubbly

Churn

10−1 Slug

Annular

10−2 10−3 −2 10

10−1

100 Usg [m/s]

101

102

Fig. 1.11 Two-phase flow pattern map for air and water at atmospheric pressure and temperature of 293.15 K in a 50-mm ID vertical pipe (Taitel et al., 1980)

22

1 Introduction

validated their approaches based on low-pressure synthetic flow-loop data or field data for shallower wells. In both cases, there is still a significant contrast between the liquid and gas phase properties.

1.6 Well Flow Rate Determination In any SPU, the rate at which fluids leave the formation that once held them to the surface results from the combined action of several factors. First, it is necessary to percolate the porous volume of the rock towards the openings installed at the bottom of the metallic casing of the well, known as the perforations. This movement corresponds to the recovery of oil. The narrower, more torturous the path is, the more difficult it will be to overcome it (Thomas, 2004, p. 170). Therefore, the permeability of the reservoir plays a key role in this process. In the case of the Brazilian pre-salt, the favorable permoporous characteristics, combined with the low viscosity of the mixture, generally result in small pressure drops between the undisturbed static volume (Pr es ) and the perforated ones. However, the exact number depends on the established daily production. A genuinely simple way to understand the pressure drop inherent to flow in a porous medium is based on tracing curves known as the Inflow Performance Relationship (IPR) or curves of available pressure. Figure 1.12 shows these curves, which graphically describe the behavior of the bottomhole pressure (Pbh ) as a function of the liquid flow under Standard conditions ( Q˙ l ) recovered from the reservoir. The IPR determination for a given OPW will not be discussed in detail, as there are already good commercial packages available for this purpose, such as IMEX, GEM, and STARS from CMG,11 as well as Schlumberger’s ECLIPSE.12 Qualitatively, it is known that Pbh is inversely proportional to Q˙ l . Above the bubble-point pressure, when the mixture of hydrocarbons has a single phase, this relationship is usually well represented by a straight line, attesting to the validity of the famous law proposed in 1856 by the French engineer Henry Darcy (1803-1858) for the flow of incompressible liquids. Below this point, the appearance of gas in the porous medium will lead the two phases (or three, considering the possible displacement of the connate water) to compete for the same space, intensifying the reduction of available pressure at the bottom of the well. In such contexts, the model proposed by Vogel (1968) is closer to the behavior observed in the field. After reaching the bottom of the well, the fluids must still retain enough energy to traverse the entire OPW and ascend to the SPU, in a process known as oil lift. The greater the difference in depth or friction with the pipe wall, the more energy is expended. While the first parameter remains fixed,13 the primary approach adopted 11

More information at: https://www.cmgl.ca/software. More information at: https://www.software.slb.com/products/eclipse. 13 This does not imply that the pressure drop due to gravity between these two points does not depend on the characteristics of the flow, as will be explained shortly. 12

1.6 Well Flow Rate Determination

23

Pbh Pres Pb

Darcy

Vogel Q˙ lb

Q˙ l

Fig. 1.12 Qualitative aspect of a typical IPR curve

in the pre-salt involves the installation of production tubings and pipelines with large diameters, ranging between 152 and 203 mm (Rempto et al., 2018), with the aim of minimizing the second parameter. A similar approach to the one previously used for the reservoir is also applicable to this new domain. This gives rise to the Tubing Performance Relationship (TPR) curves, or curves of required pressure, which graphically summarize the behavior of downhole pressure based on the liquid flow delivered to the SPU. Obtaining the TPR is directly linked to simulating the pressure and temperature profiles of the flow using one-dimensional mathematical models, which are discussed in Chap. 5. Figure 1.13 illustrates that the TPR consists of three main components: the pressure drops associated with viscous friction (Pfric ) and gravity (Pgrav ), and the prescribed value for fluid arrival at the surface (Psup ). Additionally, there is a fourth contribution related to the acceleration of the mixture along the flow path. However, this contribution is negligible for now. Naturally, the impact of the first component varies depending on the velocity of the fluids inside the pipe, as observed in singlephase flows where Pfric ∼ U 2 . In this case, however, the growth is less pronounced due to the increase in Q˙ l , which also raises the average mass density of the mixture, partially offsetting the effect on Pfric . Conversely, this mechanism leads to a subtle enlargement of the second component, particularly in certain parts of the curve. Conversely, for low flows, the phase slip is so dominant that any increase in liquid velocity results in a reduction of Pgrav . Considering all these contributions, it is evident that the TPR exhibits positive curvature, with a distinct point of minimum (Brill & Mukherjee, 1999, p. 79). The available and required pressure curves together form the basis of the socalled nodal analysis, formulated by Gilbert (1954). Although the bottom of the well

24 Fig. 1.13 Qualitative aspect of a typical TPR curve

1 Introduction

P Pbh

ΔPfric ΔPgrav

Psup Q˙ l

is the most intuitive place to position the node, given the clear transition in the flow regime, it is also possible to locate it in other regions of interest, such as the PDG or the WCT. The intersection of the IPR with the TPR, illustrated in Fig. 1.14, defines the operating point of the OPW, which represents the equilibrium flow resulting in a single pressure for the node. It should be noted that any change in the current conditions of the reservoir requires the design of a new IPR, and occasional changes in Psup or in the tubing or production pipeline geometry always require the creation of a new TPR. In both circumstances, the operating point will certainly shift to a different level. Simulations like this have two main applications in oil wells: (i) during the design phase, to provide support for the sizing of subsea equipment and estimate the flow curve throughout the productive life of the field, and (ii) during the operation phase, to aid in monitoring production, determining the individual flow rate per well14 and quickly diagnosing anomalous events. The success of these activities, in turn, depends on the models’ ability to accurately represent the observed behavior in practice. In the case of the pre-salt layer, as it is still a recent discovery and its economic viability has only been proven in Brazil, there are few studies in the areas of flow and thermodynamics that have focused on it. This stands in stark contrast to the mature fields of the North Sea and the Gulf of Mexico, for example, which have been in continuous production since the mid-20th century, and have contributed to the development of most of the models available in the literature today. On one hand, the growing physical foundation gradually incorporated into these studies strengthens the possibility of applying them to the present context. On the other hand, it is important 14

As a rule, a Single Point Umbilical (SPU) system has only two separator vessels, referred to as the “production” and “test” vessels, which enable the quantification of the total flow of oil reaching the surface and the individual contribution of only one well at a time.

1.7 Scope and Objectives of the Book

25

Pbh Pe

TPR

Pop

IPR

Q˙ l,op

Q˙ l

Fig. 1.14 Nodal analysis at the bottom of the well, indicating the operating pressure and flow rate of the OPW

to note that none of these models used actual pressure, temperature, and flow data from pre-salt wells during regression, nor were they validated using such information. The main limitations arise from the high values of pipe diameter, superficial velocity, and mass density of natural gas that are characteristic of this new scenario.

1.7 Scope and Objectives of the Book Given the aforementioned information, the main objective of this monograph is to present mathematical frameworks capable of describing the flow of petroleum fluids from the reservoir through their respective Offshore Production Wells. This will be achieved by utilizing one-dimensional, steady-state approaches and exemplified using field data from the Brazilian pre-salt cluster. A satisfactory modeling approach should accurately predict the thermodynamic and transport properties of the mixture, as well as the pressure drop it experiences until it reaches the surface. Additionally, it should provide a reasonable estimate of the heat transfer occurring along the flow path. In essence, we seek integrated solutions that employ similar tools across all modeling stages, consistently considering the interplay between them, as illustrated in the flowchart presented in Fig. 1.15. Subsequently, upon implementation into computational software, the resulting model should facilitate both the optimized sizing of subsea equipment during the design phase and the precise monitoring of production flows during the operational phase.

26

1 Introduction

Fig. 1.15 Flowchart illustrating the integrated approach to modeling reservoir fluid properties and multiphase flow proposed in this book

In order to achieve this objective, the following goals have been set: 1. Obtain a rigorous model for calculating thermodynamic and transport properties that is duly validated for oil samples from the pre-salt layer; 2. Propose a methodology for adjusting the various parameters present in the fluid model, based on a set of experimental data collected for a specific sample; 3. Identify one-dimensional two-phase gas-liquid flow models in steady state with potential application to OPWs in the pre-salt layer; 4. Develop a flow model for calculating the pressure and temperature profiles described by the fluid as it moves towards the surface, as well as the expected liquid flow rate, using simple and reliable numerical methods; 5. Propose a rigorous pressure drop model specifically designed for pre-salt OPWs, which minimizes simulation errors when compared to operational field data. It is believed that, in the future, the models discussed here can serve as a basis for comparison for a new generation of works originating from experimental data that reflect all the particularities related to the pre-salt OPWs. There is also the expectation that the unified approach to the problem can be adapted to other types of fields, such as natural gas wells, condensate gas, or heavier black oils.

References

27

References Ahmed, T. (2016). Equations of state and PVT analysis: Applications for improved reservoir modeling (2nd ed.). Gulf Professional Publishing. Ansari, A. M., Sylvester, N. D., Sarica, C., Shoham, O., & Brill, J. P. (1994). A comprehensive mechanistic model for upward two-phase flow in wellbores. SPE Production & Facilities, 9(2), 143–151. API. (2014). API recommended practice 17B (5th ed.). American Petroleum Institute. Arinelli, L. O., de Medeiros, J. L., & Araujo, O. Q. (2015). Performance analysis and comparison of membrane permeation versus supersonic separators for CO2 removal from a plausible natural gas of Libra field, Brazil. Paper no. 26164 presented at the Offshore Technology Conference, Rio de Janeiro, Brazil. Barbosa, J., Hewitt, G., König, G., & Richardson, S. (2002). Liquid entrainment, droplet concentration and pressure gradient at the onset of annular flow in a vertical pipe. International Journal of Multiphase Flow, 28(6), 943–961. Barnea, D. (1987). A unified model for predicting flow-pattern transitions for the whole range of pipe inclinations. International Journal of Multiphase Flow, 13(1), 1–12. Beggs, D. H., & Brill, J. P. (1973). A study of two-phase flow in inclined pipes. Journal of Petroleum Technology, 25(05), 607–617. Bendiksen, K. H., Maines, D., Moe, R., & Nuland, S. (1991). The dynamic two-fluid model OLGA: Theory and application. SPE Production Engineering, 6(02), 171–180. Bendiksen, K. H., & Malnes, D. (1987). Experimental data on inlet and outlet effects on the transition from stratified to slug flow in horizontal tubes. International Journal of Multiphase Flow, 13(1), 131–135. Boyd, A., Souza, A., Carneiro, G., Machado, V., Trevizan, W., Santos, B., Netto, P., Bagueira, R., Polinski, R., & Bertolini, A. (2015). Presalt carbonate evaluation for Santos basin, offshore Brazil. Petrophysics, 56(06), 577–591. BP. (2020). BP statistical review of world energy (69 ed.). British Petroleum Co. Brennen, C. (2005). Fundamentals of multiphase flow. Cambridge University Press. Brill, J., & Mukherjee, H. (1999). Multiphase flow in wells (Vol. 17). SPE monograph series. Society of Petroleum Engineers. Carneiro, J. N. E., Pasqualette, M. A., Reyes, J. F. R., Krogh, E., Johansen, S. T., Ciambelli, J. R. P., Rodrigues, H. T., & Fonseca, R. (2015). Numerical simulations of high CO2 content flows in production wells, flowlines and risers. Paper no. 26231 presented at the Offshore Technology Conference, Rio de Janeiro, Brazil. Costa Fraga, C. T., Capeleiro Pinto, A. C., Branco, C. C. M., de Sant’ Anna Pizarro, J. O., & da Silva Paulo, C. A. (2015). Brazilian pre-salt: An impressive journey from plans and challenges to concrete results. Paper no. 25710 presented at the Offshore Technology Conference, Houston, Texas. Davies, R. M. & Taylor, G. I. (1950). The mechanics of large bubbles rising through extended liquids and through liquids in tubes. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 200(1062), 375–390. Drexler, S., Silveira, T. M. G., Belli, G. D., & Couto, P. (2019). Experimental study of the effect of carbonated brine on wettability and oil displacement for EOR application in the Brazilian Pre-Salt reservoirs (pp. 1–15). Energy sources, Part A: Recovery, utilization, and environmental effects. Dumitrescu, D. T. (1943). Strömung an einer luftblase im senkrechten rohr. Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 23(3), 139–149. Gaffney, Cline & Associates. (2010). Exame e Avaliação de Dez Descobertas e Prospectos Selecionadas no Play do Pré-sal em Águas Profundas na Bacia de Santos. Agência Nacional do Petróleo.

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Gilbert, W. E. (1954). Flowing and gas-lift well performance (pp. 126–157). Division of Production, Los Angeles: Paper presented at the spring meeting of the Pacific Coast District. Gomez, L. E., Shoham, O., Schmidt, Z., Chokshi, R. N., & Northug, T. (2000). Unified mechanistic model for steady-state two-phase flow: Horizontal to vertical upward flow. SPE Journal, 5(3), 339–350. Goodwin, A., Sengers, J., & Peters, C. (2010). Applied Thermodynamics of Fluids. Royal Society of Chemistry. Hasan, A. R., & Kabir, C. S. (1988). A study of multiphase flow behavior in vertical wells. SPE Production Engineering, 3(02), 263–272. Hewitt, G., & Hall-Taylor, N. (1970). Annular two-phase flow. Pergamon Press. Hewitt, G., Martin, C., & Wilkes, N. (1985). Experimental and modelling studies of annular flow in the region between flow reversal and the pressure drop minimum. PhysicoChemical Hydrodynamics, 6, 69–86. Jayanti, S., & Hewitt, G. (1992). Prediction of the slug-to-churn flow transition in vertical two-phase flow. International Journal of Multiphase Flow, 18, 847–860. Katz, D. L., & Firoozabadi, A. (1978). Predicting phase behavior of condensate/crude-oil systems using methane interaction coefficients. Journal of Petroleum Technology, 30(11), 1649–1655. Mandhane, J., Gregory, G., & Aziz, K. (1974). A flow pattern map for gas-liquid flow in horizontal pipes. International Journal of Multiphase Flow, 1(4), 537–553. McCain, W. (1990). The properties of petroleum fluids (2nd ed.). PennWell Books. Pedersen, K., Christensen, P., & Shaikh, J. (2014). Phase behavior of petroleum reservoir fluids (2nd ed.). CRC Press. Peneloux, A., Rauzy, E., & Freze, R. (1982). A consistent correction for Redlich-Kwong-Soave volumes. Fluid Phase Equilibria, 8(1), 7–23. Petalas, N., & Aziz, K. (1998). A mechanistic model for multiphase flow in pipes. Paper no. 98-39 presented at the Annual Technical Meeting of The Petroleum Society, Calgary, Canada. Poling, B., Prausnitz, J., & O’Connell, J. (2000). The properties of gases and liquids (5th ed.). McGraw Hill professional. McGraw-Hill Education. Rempto, M. J., Pasqualette, M. A., Fontalvo, E. M. G., Carneiro, J. N. E., Fonseca, R., Ciambelli, J. R. P., Johansen, S. T., & Løvfall, B. T. (2018). High CO2 content effect on the flow of crude oils in production transient operations. In 11th North American Conference on Multiphase Production Technology. Riazi, M. R. (2005). Characterization and properties of petroleum fractions. ASTM manual series MNL 50. ASTM International. Rosa, A., de Souza Carvalho, R., & Xavier, J. (2006). Engenharia de Reservatórios de Petróleo. Interciência. Shoham, O. (2006). Mechanistic modeling of gas-liquid two-phase flow in pipes. Society of Petroleum Engineers. Speight, J. (2014). The chemistry and technology of petroleum (5th ed.). Chemical Industries. Taylor & Francis. Taitel, Y., Barnea, D., & Dukler, A. E. (1980). Modelling flow pattern transitions for steady upward gas-liquid flow in vertical tubes. AIChE Journal, 26(3), 345–354. Thomas, J. E. (2004). Fundamentos de engenharia de petróleo (2nd ed.). Interciência. Vassiliou, M. (2018). Historical dictionary of the petroleum industry (2nd ed.). Historical Dictionaries of Professions and Industries. Rowman & Littlefield Publishers. Vianna Filho, F. G. R., Naveiro, J. T., & de Oliveira, A. P. (2015). Developing mega projects simultaneously: The Brazilian pre-salt case. Paper no. 25896 presented at the Offshore Technology Conference, Houston, Texas. Vogel, J. V. (1968). Inflow performance relationships for solution-gas drive wells. Journal of Petroleum Technology, 20(01), 83–92. Xiao, J. J., Shoham, O., & Brill, J. P. (1990). A comprehensive mechanistic model for two-phase flow in pipelines. Paper no. 20631 presented at the SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana.

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Yadigaroglu, G., & Hewitt, G. (2017). Introduction to multiphase flow: Basic concepts. Applications and Modelling. Zurich lectures on multiphase flow. Springer International Publishing. Zhang, H.-Q., Wang, Q., Sarica, C., & Brill, J. P. (2003). Unified model for gas-liquid pipe flow via slug dynamics. Part 1: Model development. Journal of Energy Resources Technology, 125(4), 266–273.

Chapter 2

Thermodynamics of Petroleum Mixtures

2.1 Introduction During the 19th century, the study of thermodynamics witnessed remarkable advancements. For instance, in 1873, the Dutch physicist Johannes D. van der Waals (1837– 1923) introduced the first equation of state capable of simultaneously describing the volumetric behavior of liquids and gases. Initially applicable solely to pure substances, this work was later extended to multicomponent mixtures in 1890. In the field of equilibrium thermodynamics, significant contributions were made by Josiah W. Gibbs (1839–1903). Building upon concepts initially developed for heat engines (O’Connell & Haile, 2005, p. 4), this American physicist and mathematician established, in 1874, all the necessary criteria for the coexistence of two or more phases within a system. Additionally, the work of the physical chemist Gilbert N. Lewis (1875–1946), also born in the USA, is noteworthy. In 1901, Lewis introduced the concept of fugacity, which enabled the representation of vapor-liquid equilibrium in terms of a directly measurable property. A few decades later, in the 1940s, oil exploration and production began to delve into ever greater depths as the shallower accumulations were depleted. Consequently, there arose an urgent need to comprehend the phase behavior of mixtures under pressures and temperatures significantly higher than atmospheric conditions. However, despite the availability of the van der Waals equation of state for over 50 years, the initial models developed with this objective in mind followed a more pragmatic approach. In the method known as black oil, it is assumed that the composition of natural gas (and therefore its molar mass) remains constant for any P and T . It is also assumed that the dead oil is completely devoid of any trace of these light fractions. Consequently, the liquid phase encompasses a combination of these two macroconstituents (natural gas and dead oil), as well as their interaction. The works of Standing (1947), Lasater (1958), Glaso (1980), Vasquez and Beggs (1980), among others, have provided means to correlate the volumetric behavior of the fluid (Bo , Bg , and Rs ) with © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. R. Gessner and J. R. Barbosa Jr., Integrated Modeling of Reservoir Fluid Properties and Multiphase Flow in Offshore Production Systems, Petroleum Engineering, https://doi.org/10.1007/978-3-031-39850-6_2

31

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2 Thermodynamics of Petroleum Mixtures

properties such as γo , γg , GOR, and the prevailing pressure and temperature. However, it is important to note that these correlations are strongly empirical, and their applicability limits are quite evident. This scenario began to change years later, with the popularization of computers. It became possible to establish more elaborate thermodynamic models, which frequently resorted to iterative solution methods. Finally, in 1949, after the publication of the Redlich-Kwong equation, considered by many to be the first modern equation of state, it was definitively proven that theoretical models could adequately predict volumetric properties (Whitson & Brulé, 2000, p. 1.1). This was perhaps the moment when the thermodynamics of multicomponent systems and petroleum engineering converged. From that point forward, the ties between the two disciplines only grew closer. For example, when the Peng-Robinson equation of state was introduced in 1976, its primary focus was on the compositional simulation of petroleum reservoirs. Simultaneously, several methodologies were proposed to characterize these fluids and their mixtures. Eventually, in the 1980s, more robust numerical algorithms emerged, leading to computational tools that have been continuously improving up to the present day.

2.2 Equations of State An equation of state is a functional relationship between state variables, which, as a rule, include pressure, temperature, and molar volume (v). ˜ According to Goodwin et al. (2010, p. 53), hardly any other area of thermodynamics has been studied as intensively in the last 150 years, resulting in the emergence of hundreds of algebraic forms, without exaggeration. However, only a few have become industry standards, with most being limited to one-off applications. According to Elliott and Lira (2012, p. 223), a careful inspection of the interactions at the molecular scale reveals that, in general, two or three parameters, when well selected, are sufficient to characterize the thermodynamic state of any substance with reasonable engineering precision. In this sense, the van der Waals (1873) equation serves as an admirable example of the compromise between simplicity and physical foundation. Prior to his pioneering work, the vapor phase was associated with what is now called an ideal gas, and the liquid was treated as an entirely separate species. Van der Waals’ insightful analysis unified the two prevailing fluid conceptions until then, and this achievement earned him the Nobel Prize in Physics in 1910.

2.2.1 Van der Waals Equation To derive his equation, van der Waals (1873) utilized the phase behavior of a pure substance as a starting point. The P-v˜ diagram characteristic of a gas, for instance,

2.2 Equations of State

33

corresponds to a hyperbola, suggesting that its molar volume varies inversely with P. However, there exists a point beyond which the space occupied by the fluid molecules cannot be neglected, especially at high pressures. Consequently, v˜ reaches a finite value b greater than zero. At low temperatures, empirical evidence demonstrates the occurrence of phase change phenomena. Substances undergo a transition from the gaseous state, where the molecules are widely dispersed, to the liquid state, where they are much closer together, indicating the presence of a cohesive force. According to Newton’s Law of Gravitational Attraction, the attraction between any two masses is inversely proportional to the square of the distance between them. Recognizing that the average distance between the molecules of a substance depends on v˜ (smaller molar volume implies greater proximity), it was concluded that the magnitude of this force would be equal to v˜ −2 multiplied by a constant a. Therefore, the equation of state can be expressed as follows: P=

a RT − . v˜ − b v˜ 2

(2.1)

From a physical standpoint, the attraction term in Eq. (2.1) aims to bring the molecules closer together, consequently reducing the pressure. The initial part on the right-hand side represents the repulsion term arising from intermolecular elastic collisions, leading to an increase in P. The quantity of inter-particle collisions is determined by the average distance between them, denoted here as v˜ − b, as well as their kinetic energy, which is directly proportional to T . It is also worth noting that the parameters a and b of Eq. (2.1) acquire specific values for each type of substance, reflecting their critical properties. To evaluate these parameters, one begins by considering the volumetric behavior exhibited by the fluid at that particular point, where the following expressions hold true:   ∂P = 0, (2.2) ∂ v˜ Tc 

∂2 P ∂ v˜ 2

 = 0.

(2.3)

Tc

Using Eqs. (2.2) and (2.3) as boundary conditions in the solution for a, b, and the critical molar volume (v˜c ) yields: a=

27R 2 Tc 2 , 64Pc

(2.4)

RTc , 8Pc

(2.5)

3 RTc . 8 Pc

(2.6)

b=

v˜c =

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2 Thermodynamics of Petroleum Mixtures

It has been verified that Eq. (2.1) is applicable to any pure substance for which the values of Pc and Tc are known in advance. By expressing the equation in terms of molar volume, as stated by Pedersen et al. (2014, p. 85), one can arrive at:   ab RT a + b v˜ 2 + v˜ − = 0. (2.7) v˜ 3 − P P P As it constitutes a third-degree polynomial in v, ˜ the van der Waals equation, as well as all the others based on it, is called a cubic equation of state. It is also possible to express Eq. (2.7) as a function of the fluid compressibility factor (z) as follows:   z 3 − 1 + b∗ z 2 + a ∗ z − a ∗ b∗ = 0,

(2.8)

where: a∗ =

aP , R2 T 2

(2.9)

bP . RT

(2.10)

b∗ =

Although revolutionary, the van der Waals equation does not possess the accuracy required for engineering applications (Goodwin et al., 2010, p. 55). Instead, its predictions offer only a qualitative understanding of the volumetric behavior and vapor-liquid equilibrium of real fluids. Due to this limitation, the equation is now completely unused and is solely reserved for didactic purposes. However, the fundamental concept developed within it continues to influence all modern cubic equations of state.

2.2.2 Modern Equations of State In the subsequent works, it became increasingly evident that the parameter a in Eq. (2.1) should be temperature-dependent in the system. Interestingly, van der Waals himself was convinced that the two parameters of his equation of state were not necessarily constant, and even expressed the possibility of establishing this temperature dependence in the future. A captivating discussion on this matter can be found in van de Waals’ Nobel Prize lecture.1 Of the dozens of modifications introduced in the first half of the 20th century, the most important, according to Goodwin et al. (2010, p. 56), was the equation of Redlich and Kwong (1949). This expression proposed a new way of representing

1

Available at www.nobelprize.org.

2.2 Equations of State

35

the attraction term, significantly improving the characterization of the properties of several gases. The Redlich-Kwong equation of state determines that: P=

RT a(T ) − , v˜ − b v˜ (v˜ + b)

(2.11)

where: a(T ) = α(T )ac ,

(2.12)

α(T ) = Tr−1/2 .

(2.13)

Again, by leveraging the conditions imposed by the critical point, it is possible to demonstrate that the parameters ac and b in Eqs. (2.11) and (2.12), and v˜c are equivalent to: ac = 0.42748

b = 0.08664

v˜c =

R 2 Tc2 , Pc

(2.14)

RTc , Pc

(2.15)

1 RTc . 3 Pc

By rewriting Eq. (2.11) for the molar volume, one finds:   ab b RT RT 2 a v˜ + − − b2 v˜ − = 0. v˜ 3 − P P P P

(2.16)

(2.17)

Equation (2.17) can also be expressed as a function of z:   z 3 − z 2 + a ∗ − b∗ − b∗2 z − a ∗ b∗ = 0,

(2.18)

where the coefficients a ∗ and b∗ refer once again to Eqs. (2.9) and (2.10). The Redlich-Kwong equation of state was developed with the aim of predicting the behavior of small, nonpolar molecules in the vapor phase, which it accomplishes quite satisfactorily. However, its main limitation lies in its inability to accurately represent the thermodynamic properties of the liquid phase, thereby compromising calculations involving saturation and equilibrium as well (Goodwin et al., 2010, p. 56). As a result, in the subsequent years, further research endeavors sought means to enhance its capabilities. Soave (1972) proposed an important modification to the original cubic formulation. In an attempt to improve the vapor pressure estimate, an almost linear

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2 Thermodynamics of Petroleum Mixtures

relationship was found between the square roots of α(T ) = a(T )/ac and Tr = T /Tc . The curves for each substance are distinguished only by the slope since, for Tr = 1, α(T ) = 1 always holds. This behavior can be expressed mathematically as:   α(T )1/2 = 1 + m 1 − Tr 1/2 .

(2.19)

In order to correlate the different values of m obtained in the analysis, the acentric factor2 conceived by Pitzer (1955), which still did not exist at the time of publication of Redlich and Kwong (1949), was utilized. This is defined as:   Psat . (2.20) ω ≡ −1 − log Pc Tr =0.7 The regression proposed by Soave (1972) utilized a dataset covering 11 pure hydrocarbons within the range 0 ≤ ω ≤ 0.5, resulting in the following relationship: m = 0.480 + 1.574ω − 0.176ω2 .

(2.21)

Therefore, by simply replacing Eq. (2.13) with (2.19), the attraction term of the Redlich-Kwong equation of state is now dictated not only by temperature but also by the sphericity of the molecules, incorporated through the parameter ω. Simultaneously, both the expressions for ac , b, and v˜c , as well as the cubic form of the new equation of state, remain exactly the same as the previous ones, given by Eqs. (2.14)– (2.18). According to Goodwin et al. (2010, p. 57), the so-called Soave-Redlich-Kwong equation has received significant attention in academic and industrial circles, particularly in the oil and natural gas sector. However, despite its accuracy in vapor-liquid equilibrium calculations, the predictions of liquid phase density using this equation were generally too low. Peng and Robinson (1976) attributed this difficulty to the inadequate representation provided by this equation of state and all others developed up to that point for the critical compressibility factor (z c ) of a series of components. Consider the example of n-paraffins from C1 to C10, where z c should take values between 0.25 and 0.29, as indicated by the experimental data compiled by Poling et al. (2000). In the van der Waals equation, as seen in Eq. (2.6), this parameter remains fixed at 3/8 = 0.375, regardless of the substance. On the other hand, the arrangement proposed by Redlich and Kwong (1949) implies z c = 1/3 ≈ 0.333, as shown in Eq. (2.16), and the same holds true for the SRK equation of state. Peng and Robinson (1976) managed to reduce this difference through a subtle intervention in the attraction term, establishing that:

The term acentric factor, according to Pedersen et al. (2014, p. 5), suggests that ω represents a measure of the deviation of the molecule’s shape from that of a sphere. This reasoning is especially consistent with what is observed in n-paraffins, where an increase in the number of carbon atoms incurs longer and thinner chains, and ω grows continuously. The acentric factor also relates to the curvature of the Psat (T ) function of the substance.

2

2.2 Equations of State

37

P=

a(T ) RT − . v˜ − b v˜ (v˜ + b) + b (v˜ − b)

(2.22)

In this equation of state, the parameters a(T ) and α(T ) are again given by Eqs. (2.12) and (2.19). Therefore, applying the constraints imposed by the critical point, ac , b, and v˜c are given by: R 2 Tc2 , Pc

(2.23)

b = 0.07780

RTc , Pc

(2.24)

v˜c = 0.3074

RTc . Pc

(2.25)

ac = 0.45724

Based on Eq. (2.25), the new expression for the attraction term proposed by Peng and Robinson (1976) results in z c = 0.3074. Although this value is still higher than those presented by Poling et al. (2000), the deviation is certainly lower than in all previous works. On the other hand, the correlation for the slope m in Eq. (2.19) refers to a regression similar to that performed by Soave (1972), but with the use of a slightly larger set of substances, including non-hydrocarbons3 . The authors obtained the following expression: m = 0.37464 + 1.54226ω − 0.26992ω2 .

(2.26)

Equation (2.26) covers only the interval 0 ≤ ω ≤ 0.5. Two years later, Robinson and Peng (1978) published a new polynomial applicable to cases where 0.2 ≤ ω ≤ 2, which is given by: m = 0.379642 + 1.48503ω − 0.164423ω2 + 0.016666ω3 .

(2.27)

Equation (2.22) can also be expressed in its cubic form in terms of the molar volume:      ab b2 RT 2b RT RT a 2 2 3 − b v˜ + − − 3b v˜ − − −b =0 v˜ − P P P P P (2.28) Alternatively, it can be expressed in terms of the compressibility factor, resulting in the following cubic equation: 

3

3

As discussed in Sect. 1.2, nitrogen, carbon dioxide, and hydrogen sulfide are the most common contaminants in oil reservoirs.

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2 Thermodynamics of Petroleum Mixtures

      z 3 − 1 − b∗ z 2 + a ∗ − 2b∗ − 3b∗2 z − a ∗ b∗ − b∗2 − b∗3 = 0.

(2.29)

Goodwin et al. (2010, p. 57) points out the Peng-Robinson and Soave-RedlichKwong equations of state as the two most popular equations used nowadays. These equations have been employed to calculate the thermodynamic properties of pure substances, as well as binary, ternary, and multicomponent mixtures, addressing various problems such as liquid-vapor equilibrium at low or high pressure and liquid-liquid equilibrium. It is worth noting that since their publication over 40 years ago, several modifications have been suggested. However, only a few of these modifications have gained significant support, and an even smaller portion has had a substantial impact on the study of petroleum mixtures. The following sections will discuss these noteworthy modifications.

2.2.3 Further Improvements In the analysis conducted by Valderrama (2003), attempts to improve the performance of the Peng-Robinson and Soave-Redlich-Kwong equations are divided into three major fronts: 1. Proposing new calculation methodologies for α(T ) in Eq. (2.12); 2. Refining the dependence of the repulsion term on the specific molar volume of the substance in Eqs. (2.11) and (2.22); 3. Introducing a third parameter in the equation of state. The behavior of the attraction term with temperature is particularly relevant in the case of polar substances, where more sophisticated models lead to more accurate estimates of vapor pressure, specific heat, and enthalpy (Privat et al., 2015). In this context, an alternative to the function α(T ) proposed by Soave (1972), and which has gained good acceptance, was developed by Twu et al. (1995a, 1995b). Supported by a database containing Psat curves primarily for n-paraffins (with the inclusion of naphthenes and aromatics solely to verify the generality of the regression), the authors deemed it appropriate to formulate: α(T ) = α0 + ω (α1 − α0 ) ,

(2.30)

   α0 = TrN0 (M0 −1) exp L 0 1 − TrN0 M0 ,

(2.31)

   α1 = TrN1 (M1 −1) exp L 1 1 − TrN1 M1 .

(2.32)

where:

2.2 Equations of State

39

Table 2.1 Values recommended by Twu et al. (1995a, 1995b) for the coefficients of Eqs. (2.31) and (2.32) Coefficient Redlich-Kwong Peng-Robinson Tr ≤ 1 Tr > 1 Tr ≤ 1 Tr > 1 L0 L1 M0 M1 N0 N1

0.141599 0.500315 0.919422 0.799457 2.496441 3.291790

0.441411 0.032580 6.500018 1.289098 –0.200000 –8.000000

0.125283 0.511614 0.911807 0.784054 1.948150 2.812520

0.401219 0.024955 4.963070 1.248089 –0.200000 –8.000000

In Eq. (2.30), α0 and α1 represent the values of α(T ) for ω = 0 and ω = 1, respectively. Therefore, any estimate outside this range would be an extrapolation of the method. The coefficients of Eqs. (2.31) and (2.32) are presented in Table 2.1 for the Peng-Robinson and Redlich-Kwong equations of state. It is important to note that the same expression is applicable to supercritical fluids; the only requirement is the selection of the appropriate parameters. These parameters were obtained from experimental data on the solubility of methane and hydrogen in liquid hydrocarbons, ensuring the continuity of the function and its first-order derivative at Tr = 1. In turn, attempts to modify the repulsion term of the equation of state aim, in general, to improve the predictions of the liquid phase density. According to Pedersen et al. (2014, p. 92), in its early years, the Soave-Redlich-Kwong equation incurred deviations in this property that were so high that an external correlation calculation was often preferred. This strategy created new problems, especially near the critical point, where the distinction between the liquid and gas phases becomes quite difficult. The situation was satisfactorily reversed only when Peneloux et al. (1982) proposed the so-called volume translation, replacing the variable v˜ in the original equation with v˜ + c. Thus, the following equation was obtained (O’Connell & Haile, 2005, p. 60): P=

a(T ) RT − . v˜ + c − b (v˜ + c) (v˜ + c + b)

(2.33)

In Eq. (2.33), the parameter c assumes a distinct value for each substance, and although it can be estimated through correlations, experimental data undoubtedly yield more precise and reliable values. In the latter case, assuming c = 0 in the equation of state and after establishing a thermodynamic reference state, the volume translation is given by: c = v˜calc (P, T ) − v˜ex p (P, T ).

(2.34)

Even when introducing a constant that affects the results of both the liquid and vapor states, it is known that c is proportionally more relevant in the former than

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2 Thermodynamics of Petroleum Mixtures

in the latter. Hence, Peneloux et al. (1982) recommend the technique for adjusting the mass density of liquids at low pressures. Near the critical point, the deviations remain practically unchanged. It can be shown that the critical compressibility factor associated with Eq. (2.33) is given by: zc =

c Pc 1 − . 3 RTc

(2.35)

The volume translation adopted by Peneloux et al. (1982) for the Soave-RedlichKwong equation causes z c to become a function, albeit weakly, of the parameter c and the critical properties. Another notably positive characteristic of this approach is that the liquid-vapor equilibrium, discussed in Sect. 2.3, remains unchanged. It is important to note that this concept extends to all other cubic equations of state. For instance, in the case of the Peng-Robinson equation, one can observe (Jhaveri & Youngren, 1988): P=

a(T ) RT − . v˜ + c − b (v˜ + c) (v˜ + c + b) + b (v˜ + c − b)

(2.36)

Most likely, the difference between the specific masses calculated using Eqs. (2.22) and (2.36) will not be significant. On the other hand, there are no contraindications, and even a slight increase in accuracy would be beneficial. In this case, the volume translation implies: z c = 0.3074 −

c Pc . RTc

(2.37)

The reference values for the c parameter for components typically encountered in oil samples are presented in Table 2.2 for the Soave-Redlich-Kwong and PengRobinson equations of state. To conclude, the third and final front mentioned by Valderrama (2003) has motivated an entire new generation of equations of state. The so-called three-parameter equations stand out for their more faithful reproduction of the critical compressibility factor, thereby improving predictions in this region. Among them, the equation proposed by Patel and Teja (1982) deserves special mention, which establishes that: P=

a(T ) RT − . v˜ − b v˜ (v˜ + b) + cVPT (v˜ − b)

(2.38)

It is worth noting that when cVPT = 0, Eq. (2.38) reduces to the Redlich-Kwong equation and, for cVPT = b, the Peng-Robinson equation is obtained. However, the main obstacle to the application of the Patel-Teja equation lay in the treatment of this third parameter: the correlation provided by the authors, besides being limited to nonpolar substances, relied on the acentric factor and an artificial variable that resembled z c but was strictly mathematical in nature. In this context, the modifications suggested by Valderrama (1990) were of great value, as they began to use:

2.2 Equations of State

41

Table 2.2 Recommended values of the volume translation parameter c for some oil constituents, considering the Soave-Redlich-Kwong and Peng-Robinson equations of state. Source Whitson and Brulé (2000, p. 4.5) Component (c/b)SRK (c/b)PR N2 CO2 H2 S C1 C2 C3 iC4 nC4 iC5 nC5 nC6

–0.008 0.083 0.047 0.023 0.061 0.083 0.083 0.098 0.102 0.121 0.147

ac = (0.66121 − 0.76105z c )

b = (0.02207 + 0.20868z c )

–0.193 –0.082 –0.129 –0.160 –0.113 –0.086 –0.084 –0.068 –0.061 –0.039 –0.008

R 2 Tc2 , Pc

(2.39)

RTc , Pc

(2.40)

cVPT = (0.57765 − 1.87080z c )

RTc , Pc

m = 0.46283 + 3.58230 (ωz c ) + 8.19417 (ωz c )2 .

(2.41)

(2.42)

In Eq. (2.42), the coefficient m is the same as the one proposed by Soave (1972) and, therefore, is also intended for the function described in Eq. (2.19). According to Danesh et al. (1991), the Valderrama-Patel-Teja equation of state demonstrates excellent performance in predicting both the density and bubble pressure in natural gas and condensate systems. However, its accuracy deteriorates rapidly with increasing CO2 content in the mixture.

2.2.4 Solution of Cubic Equations of State Cubic equations of state provide an effective and convenient means of modeling the PVT behavior of a given substance. However, very few problems benefit from their explicit form given by P = P(T, v). ˜ In most cases, the functional relationship of

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2 Thermodynamics of Petroleum Mixtures

the form v˜ = v(P, ˜ T ) prevails. This requires the calculation of the roots of the cubic equation, expressed in terms of both v˜ and the compressibility factor.4 The details of this procedure are presented in Appendix A. In general terms, the cubic form of the equation of state will have a unique solution whenever the pressure is above the critical point. The same can occur for P < Pc , as long as it is sufficiently far from the saturation region. Otherwise, there will be three values of z capable of satisfying the equation of state: the largest corresponds to the compressibility factor of the vapor phase, and the smallest to that of the liquid phase. The intermediate value, however, violates the criterion of mechanical stability5 (O’Connell & Haile, 2005, p. 319), and therefore represents a solution without practical use.

2.2.5 Mixing Rules When an equation of state is applied to multicomponent systems, its basic form remains unchanged, and the properties of a mixture can be described by the same relationships used for pure substances. However, the parameters a and b must be adapted to characterize the fluid as a whole in terms of its individual constituents. The relationships developed for this purpose are called mixing rules. The explanation offered by Elliott and Lira (2012, p. 467) provides a highly persuasive account of the rule envisioned by van der Waals (1890) for the attraction parameter, using a binary mixture as an example. In this particular and simple circumstance, there are only three plausible types of interactions: between molecules of the same type, which are characteristic of the 1 + 1 and 2 + 2 interactions, or between molecules of different types, which constitutes the 1 + 2 interaction. Assuming a random distribution in the mixture, the probability of finding a molecule of type 1 is equal to its mole fraction.6 The 1 + 1 interaction is associated with the probability P11 of finding a second molecule of type 1 that exerts attraction and is attracted by the first molecule. For independent events, the conditional probability is calculated by multiplying the individual expectations, leading to P11 = x12 . Similarly, P22 = x22 , and with the same reasoning, the chances of interaction between molecules of different types are given by P12 = P21 = x1 x2 . Therefore, representing the proportionality constants of the attractive forces as a11 , a12 , and a22 , the parameter a corresponds to:

4

According to Elliott and Lira (2012, p. 265), the molar volume tends to infinity as the pressure approaches zero. The compressibility factor, on the other hand, always remains between 0 and slightly larger than 1, which facilitates the calculation of the roots of the equation of state, especially when a numerical method is being used. 5 For a fluid to be mechanically stable, it is necessary that (∂ P/∂ v; ˜ )T < 0. This criterion is not met when the solution of the equation of state lies between its two inflection points, known as spinodal points. 6 Throughout this section, x will designate the mole fraction of component i in any single-phase i mixture, which can be either in the liquid or vapor state.

2.2 Equations of State

43

a = x12 a11 + 2x1 x2 a12 + x22 a22 .

(2.43)

Repeating the aforementioned procedure for a mixture of N components yields:  xi x j ai j . (2.44) a= i

j

Naturally, in Eq. (2.44), when i = j, ai j corresponds to the pure component i. However, for the cross-parameters, where i = j, it is necessary to establish an averaging process between the two involved values. In such cases, van der Waals (1890) suggests assuming: ai j =

√ ai a j .

The mixing rule proposed for the parameter b is given by:  xi bi . b=

(2.45)

(2.46)

i

According to Elliott and Lira (2012, p. 467), Eq. (2.46) is justified by the fact that the minimum space occupied by each species is given by n i bi . Thus, assuming that their molecules present similar sizes, the minimum volume of the mixture will come simply from the summation of individual contributions. The same reasoning remains valid in the calculation of the translation parameter of Peneloux et al. (1982), such that:  xi ci . (2.47) c= i

The van der Waals mixing rules are effective for gas mixtures at low or moderate pressure (O’Connell & Haile, 2005, p. 173). According to Soave (1972), these rules are successful in combining non-polar fluids such as hydrocarbons, nitrogen, and carbon monoxide, but not carbon dioxide. The results are also realistic for mixtures of substances with significantly different volatilities. However, corrections are required in other cases. Soave (1972) suggested that the intermolecular attraction between two different species should be represented by: √  ai a j . ai j = 1 − ki j

(2.48)

The so-called binary interaction parameter (ki j ) is typically expressed as small positive numbers between 0 and 0.2. Naturally, ki j = 0 when i = j. The most accurate way to determine this parameter is through experimental data. Since it is a completely empirical quantity, its usage is limited to the equation of state used in the data regression. Reference values for the most common components in oil reservoirs

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Table 2.3 Examples of binary interaction parameters for the Soave-Redlich-Kwong and PengRobinson equations of state. Source Whitson and Brulé (2000, p. 4.3) Component Soave-Redlich-Kwong Peng-Robinson N2 CO2 H2 S N2 CO2 H2 S N2 CO2 H2 S C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+

– 0.000 0.120 0.020 0.060 0.080 0.080 0.080 0.080 0.080 0.080 0.080

0.000 – 0.120 0.120 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150

0.120 0.120 – 0.080 0.070 0.070 0.060 0.060 0.060 0.060 0.050 0.030

– 0.000 0.130 0.025 0.010 0.090 0.095 0.095 0.100 0.110 0.110 0.110

0.000 – 0.135 0.105 0.130 0.125 0.120 0.115 0.115 0.115 0.115 0.115

0.130 0.135 – 0.070 0.085 0.080 0.075 0.075 0.070 0.070 0.055 0.050

are listed in Table 2.3. The use of Eq. (2.48) as a combining rule7 in the SoaveRedlich-Kwong equation and subsequently in the Peng-Robinson equation has led to a significant improvement in the analysis of liquid-vapor equilibrium in multicomponent systems, which will be discussed further.

2.3 Vapor-Liquid Equilibrium A system is said to be in thermodynamic equilibrium when it does not have any possibility of performing work if it is isolated from its surroundings (Borgnakke & Sonntag, 2009, p. 672). By applying this postulate, the domain in question should be divided into two or more parts, checking if any hypothetical interaction between them would lead to such an outcome. Thus, it is observed, for example, that equilibrium presupposes the existence of a uniform temperature field, since otherwise a heat engine could operate. Similarly, all mechanical forces must be properly counterbalanced, which implies uniform pressure for all points, or else a turbine or a piston engine positioned between two other subsystems would again produce work. The first criterion constitutes the so-called thermal equilibrium, and the second is the mechanical equilibrium. There is also a third criterion, which states that the molecular structure of each component must remain the same over time. However, chemical 7

Many authors distinguish between a mixing rule and a combining rule. Unlike the former, the combining rule is independent of the mixture composition and is focused solely on representing the cross parameters.

2.3 Vapor-Liquid Equilibrium

45

reactions are beyond the scope of this book, and for this reason, chemical equilibrium will be left aside here. The Gibbs free energy, which refers to the maximum amount of work that can be extracted from a system in equilibrium with its surroundings, provides a very interesting perspective on the phase change process. The so-called phase equilibrium is achieved when G reaches its minimum level among all possible scenarios (Borgnakke & Sonntag, 2009, p. 674). Thus, by seeking to minimize it, the mixture will transition between the liquid and vapor states, or maintain both states.

2.3.1 Types of Problems and Formulation It is known that the P-T diagram of a mixture displays two saturation curves, corresponding to the bubble point and the dew point pressures, which in turn define the so-called phase envelope, i.e., a surface of pressure and temperature conditions where liquid and vapor coexist in a given proportion. If the objective of the analysis is to find a point located on one of these curves, there are four distinct ways to formulate the equilibrium problem (Elliott & Lira, 2012, p. 372): x ) and a temBubble-Point Pressure (Pb ): Starting from the liquid composition ( perature, the value of the mixture’s saturation pressure is sought; Dew-Point Pressure (Pd ): Starting from the vapor composition (y ) and a temperature, the value of the mixture’s saturation pressure is sought; Bubble-Point Temperature (Tb ): Starting from the liquid composition and a pressure, the value of the mixture’s saturation temperature is sought; Dew-Point Temperature (Td ): Starting from the vapor composition and a pressure, the value of the mixture’s saturation temperature is sought. If, by chance, the focus of the study is centered on the interior of the phase envelope, the fifth and final type of problem arises: Flash Calculation: Starting from the overall composition of the mixture (z ) and a given pressure and temperature condition, the composition of the liquid and vapor phases and the mole fraction (˜ε) of each phase are sought. The number of dimensions of the composition vectors will be equal to the number of components present in the mixture (N ), such that:   x = x1 x2 . . . x N −1 x N ,

(2.49)

  y = y1 y2 . . . y N −1 y N ,

(2.50)

  z = z 1 z 2 . . . z N −1 z N .

(2.51)

46

2 Thermodynamics of Petroleum Mixtures

Based on the global mass balance for a mixture containing n moles, where L are in the liquid state and V are in the vapor state, the mole fractions of the phases can be expressed as follows: ε˜ L =

L , n

(2.52)

ε˜ V =

V . n

(2.53)

Furthermore, the mass balance for component i establishes that: z i = xi ε˜ L + yi ε˜ V .

(2.54)

Another frequently used parameter in problems involving mixtures is the equilibrium constant, defined as (Elliott & Lira, 2012, p. 372): Ki ≡

yi . xi

(2.55)

Thus, combining Eqs. (2.54) and (2.55) yields: xi =

zi , 1 + (K i − 1) ε˜ V

(2.56)

yi =

K i zi . 1 + (K i − 1) ε˜ V

(2.57)

The calculation of liquid-vapor equilibrium is performed iteratively. This means that once the values of a certain set of variables are arbitrarily chosen, it is necessary to verify the deviation that they bring to the exact solution of the problem. The so-called objective function quantifies the error associated with that set for each iteration. Although the most appropriate formula depends on the type of problem, all of them are based on the mass balances of the components. For the calculation of bubble pressure or temperature, for example, where x is known beforehand, the exact solution is obtained when:   yi − 1 = K i xi − 1 = 0. (2.58) i

i

For dew point pressure or dew point temperature problems, y is considered an input data and the objective function is expressed as follows:   yi xi − 1 = − 1 = 0. (2.59) Ki i i

2.3 Vapor-Liquid Equilibrium

47

Table 2.4 Summary of the formulations for vapor-liquid equilibrium problems for mixtures. Source Elliott and Lira (2012, p. 373) Problem Input Unknowns Equations Bubble-point pressure

x T

y Psat

yi = K i xi i = 1, 2, . . . , N yi − 1 = 0

Dew-point pressure

y T

x Psat

xi = yi /K i i = 1, 2, . . . , N xi − 1 = 0

Bubble-point temperature

x P

y Tsat

yi = K i xi i = 1, 2, . . . , N yi − 1 = 0

Dew-point temperature

y P

x Tsat

xi = yi /K i i = 1, 2, . . . , N xi − 1 = 0

Flash calculation

z P T

x y ε˜ V

xi =  z i / 1 + ε˜ V (K i − 1) i = 1, 2, . . . , N xi i = 1, 2, . . . , N yi = K i yi − xi = 0

i

i

i

i

i

i

Finally, in flash calculations, which prescribe only z , it is customary to utilize the objective function of Rachford and Rice (1952), given by:  i

yi −

 i

xi =

 i

(K i − 1) z i = 0. 1 + (K i − 1) ε˜ V

(2.60)

In Table 2.4, the formulations related to each of the five aforementioned problem types are summarized. It should be noted that calculations of dew-point pressure and dew-point temperature will not be addressed here as they have little relevance in studies involving oil reservoirs.

2.3.2 Chemical Potential and Fugacity of a Component The Gibbs free energy plays a crucial role in understanding the phase transition process. However, when dealing with mixtures, it is necessary to have a separate relationship for each of its components, rather than just for the system as a whole. Otherwise, the problem becomes indeterminate, with the number of unknowns exceeding the number of equations available to solve them. It is known that the change in internal energy is equivalent to:

48

2 Thermodynamics of Petroleum Mixtures

 dU (S, V, n 1 , n 2 , . . . n i ) =

∂U ∂S



 dS + V,n



∂U ∂V

dV +

  ∂U 

S,n

i

∂n i

V,S, j=i

dn i .

(2.61) or from the thermodynamic definitions of temperature, pressure and chemical potential, one has:  μi dn i . (2.62) dU = T d S − Pd V + i

Equation (2.62) employs the chemical potential of a component i in the mixture (μi ) in the exact manner as defined by Gibbs (1874, p. 149) in formal terms. According to Gibbs, “If to any homogeneous mass, we suppose an infinitesimal quantity of any substance to be added, the mass remaining homogeneous and its entropy and volume remaining unchanged, the increase of energy of the mass divided by the quantity of the substance added is the potential for that substance in the mass considered.” In turn, the absolute enthalpy, Helmholtz and Gibbs free energies are given by: H ≡ U + P V,

(2.63)

A ≡ U − T S,

(2.64)

G ≡ U + P V − T S.

(2.65)

By differentiating both sides and substituting Eq. (2.62) into each of the expressions, the following result is obtained:  μi dn i , (2.66) dH = TdS + VdP + i

d A = −SdT − Pd V +



μi dn i ,

(2.67)

μi dn i .

(2.68)

i

dG = −SdT + V d P +

 i

Based on Eqs.(2.62), (2.66), (2.67), and (2.68), as well as the general form of Eq.(2.61), it can be concluded that μi can be defined simultaneously as:  μi ≡

∂U ∂n i



 V,S, j=i

=

∂H ∂n i



 P,S, j=i

=

∂A ∂n i



 V,T, j=i

=

∂G ∂n i

 P,T, j=i

. (2.69)

2.3 Vapor-Liquid Equilibrium

49

By assigning a chemical potential to each component, Gibbs (1874) laid the groundwork for the calculation of liquid-vapor equilibrium in any type of mixture. However, historically, a second thermodynamic property has proven to be even more advantageous for applications of this nature: fugacity, introduced by Lewis (1901). To better grasp this new concept, it is important to observe that if any phase containing a certain component comes into contact with another phase that does not possess it, a certain amount will transfer from the first phase to the second. Therefore, each species exhibits a tendency to escape from the phase in which it resides, which corresponds to its fugacity. This concept is based on two premises: 1. The tendency of an ideal gas to escape is mathematically equal to its partial pressure; 2. The fugacity of a component is the same in the liquid and vapor phases when they are in equilibrium. It is thus established that (Lewis & Randall, 1923, p. 205): dμi = RT d ln fˆi .

(2.70)

By integrating Eq. (2.70) throughout the phase change process and subsequently applying the limits, one arrives at the following expression: 

fˆiV μV − μiL . (2.71) = i ln RT fˆiL It is known, however, that the chemical potentials of component i in the saturated liquid and vapor phases are identical, which implies: fˆiL = fˆiV .

(2.72)

It becomes evident, therefore, that the phase equilibrium in a mixture is also reflected in the fugacity of its components: for each component, the tendencies of escape from the liquid and the vapor must be equal, resulting in a net mass transfer of zero. Another convenient way to represent these terms is given by: fˆiL = xi ϕˆiL P,

(2.73)

fˆiV = yi ϕˆiV P.

(2.74)

The fugacity coefficient quantifies the deviation exhibited by the substance with reference to the ideal gas behavior, reaching unity when this assumption is fully applicable. An expression for ϕˆi in terms of the non-ideality of the mixture is equivalent to (Goodwin et al., 2010, p. 22):

50

2 Thermodynamics of Petroleum Mixtures

V ln ϕˆi = ∞

1 1 − V RT



∂P ∂n i



 V,T, j=i

d V − ln z.

(2.75)

The analytical solution of the integral contained in Eq. (2.75) requires a pressureexplicit equation of state. It is important to note that V and z are properties of the mixture, which can exist in both liquid and vapor states. Thus, it can be inferred that the fugacity of a component arises not only from its molecular structure or fraction in the mixture, but also from the combined effect of all other components in that phase.

2.3.3 Calculation of Equilibrium Constants The iterative nature of liquid-vapor equilibrium problems for mixtures is largely due to the mutual dependence between K i , xi , and yi . However, in simplified analyses, it is possible to separate the influences of the pressure-temperature pair and the mole fraction on the fugacity of the component. For instance, the rule proposed by Lewis and Randall (1923, p. 222) for ideal solutions8 establishes that: fˆiL ,is (P, T, xi ) = xi f iL (P, T ),

(2.76)

fˆiV,is (P, T, yi ) = yi f iV (P, T ).

(2.77)

By substituting Eqs. (2.76) and (2.77) into (2.72) and invoking the ideal gas assumption for the vapor phase, the phase equilibrium is now given by: xi Psat,i (T ) = yi P.

(2.78)

Equation (2.78) became known as Raoult’s law, in honor of the French chemist François-Marie Raoult (1830–1901). This equation can still be written as: Ki ≡

yi Psat,i (T ) . = xi P

(2.79)

Equation (2.79) has the significant advantage of correlating K i solely as a function of P and T . However, based on the premises adopted throughout its derivation, Raoult’s law only applies to ideal gases in equilibrium with ideal solutions. There8

According to O’Connell and Haile (2005, p. 185), an ideal solution is a mixture of distinguishable species (having different masses or distinct structures, or both) where the intermolecular interactions are essentially the same. Therefore, ideal gas mixtures, whose molecules do not exert forces on each other, constitute a particular case of an ideal solution. In liquids, such behavior is favored in mixtures of components with similar molecules or in very dilute (or very concentrated) solutions.

2.3 Vapor-Liquid Equilibrium

51

fore, the equilibrium constants calculated from it are reasonable approximations as one moves further away from the critical point of the component in question, particularly at low pressures and high temperatures. Nevertheless, there is still an upper temperature limit to consider, as the saturation pressure ceases to exist for T > Tci . The severe limitations imposed by Eq. (2.79) motivated a series of research studies from the 1940s onwards, all with the aim of maintaining the calculation of equilibrium constants independent of the system’s composition. During this period, several empirical correlations were established. One such correlation, proposed by Wilson (1969), is given as follows:    1 1 (2.80) exp 5.37 (1 + ωi ) 1 − Ki = Pri Tri In this manner, in addition to P and T , Eq. (2.80) associates the equilibrium constant of the component with its critical properties and acentric factor. The saturation pressure is not an input parameter, enabling the calculation of K i for any temperature. On the other hand, in order to disregard the effect of composition, the ideal gas hypothesis, as already assumed in Raoult’s law, has been maintained. This renders the Wilson correlation applicable only to low-pressure systems. Michelsen and Mollerup (2007, p. 259) accept its use at higher pressures, but solely for hydrocarbons, with the caveat that predictions related to supercritical components will incur considerable errors. Nevertheless, the Wilson correlation serves perfectly as an initial estimate for phase equilibrium calculations (Pedersen et al., 2014, p. 143). On the other hand, it is known that phase equilibrium is intimately connected to the fugacity coefficients of the species present in the mixture. By substituting Eqs. (2.73) and (2.74) into (2.72), one arrives at: Ki ≡

ϕˆ L yi = iV . xi ϕˆi

(2.81)

Therefore, by solving Eq. (2.75) for each phase using an explicit equation for P, the equilibrium constants of the components can be directly obtained. Regarding the Peng-Robinson equation of state, Elliott and Lira (2012, p. 594) demonstrate that:



ln ϕˆi = bi∗ (z − 1) − ln z − b

 ∗

 ⎡ √  ∗⎤ z + 1 + 2 b   a   ⎦, − √ ai∗ − bi∗ ln ⎣ √ b∗ 8 z + 1 − 2 b∗ (2.82) ∗

where: ai∗L =

2  x j ai j , aL j

(2.83)

52

2 Thermodynamics of Petroleum Mixtures

2  y j ai j , aV j

(2.84)

bi∗L =

bi , bL

(2.85)

bi∗V =

bi . bV

(2.86)

ai∗V =

When exclusively considering the thermodynamic properties of the system, the applicability of Eq. (2.81) is independent of the pressure level. It relies solely on the equation of state and the mixing rules used in the calculation of ϕˆi . The molar volume of the liquid phase plays a crucial role in this type of calculation, which is why equations of state such as van der Waals or Redlich-Kwong are not recommended for this purpose. O’Connell and Haile (2005, p. 174) further emphasize that the fugacity coefficient of the liquid is significantly influenced by the binary interaction parameters, thus it is advisable to avoid significant modifications to these values.

2.3.4 Solving the Vapor-Liquid Equilibrium Problem Equation (2.81) underlies the entire calculation of equilibrium involving cubic equations of state. According to O’Connell and Haile (2005, p. 422), this methodology, known as phi-phi, was seldom used in the past, mainly because the available works at that time were unable to accurately describe the behavior of liquids and dense gases, not to mention the significant computational challenges that were imposed. As an alternative, the fugacity of the liquid phase can be represented in terms of the activity coefficient 9 of its components (γi ), with the fugacity coefficient restricted to the vapor phase. This approach is known as gamma-phi. However, the constant improvement of equations of state, coupled with the revolution in the field of computing that occurred a few decades ago, not only made the phi-phi models viable but also greatly popularized them. In general, the determination of the bubble pressure or bubble temperature of a mixture is divided into two iterative loops. In the inner loop, the equilibrium constants and the vapor phase composition are calculated for the current values of P and T using the successive substitution method. In the outer loop, the unknown variable is iterated until it satisfies the objective function of the problem, FPb (P) = 0 or FTb (T ) = 0. Here, the Newton-Raphson method is employed, which also requires the computation of F Pb (P) and F Tb (T ). As seen in Sect. 2.3.1, the objective function is the same in both cases and can be expressed as: 9

The activity coefficient measures the deviation exhibited by component i in relation to the ideal solution model, reaching unity when the latter is fully applicable. Thus, fˆiL (P, T, xi ) ≡ xi γi f iL (P, T ).

2.3 Vapor-Liquid Equilibrium

53

FPb (P) = FTb (T ) =



K i xi − 1 = 0.

(2.87)

i

Therefore, the derivative of Eq. (2.87) with respect to pressure is given by:   ∂ Ki  xi . (2.88) FP b (P) = ∂ P T,x ,y i Rewriting Eq. (2.81) for ln ϕˆi and replacing it in Eq. (2.88) yields: 

∂ Ki ∂P

 =

T, x ,y

∂ ∂P



eln ϕˆi

L

eln ϕˆi



 = Ki

V

T, x ,y

∂ ln ϕˆiL ∂P



 − T, x

∂ ln ϕˆiV ∂P

Thus, it is established that (Pedersen et al., 2014, p. 143):

      ∂ ln ϕˆiL ∂ ln ϕˆiV

K i xi − . FPb (P) = ∂P ∂P T, x T,y i





. T,y

(2.89)

(2.90)

Following the same procedure when differentiating Eq. (2.87) with respect to temperature, one arrives at:

      ∂ ln ϕˆiL ∂ ln ϕˆiV

K i xi − . (2.91) FTb (T ) = ∂T ∂T P, x P,y i The partial derivatives in Eqs. (2.90) and (2.91) can be calculated analytically, as demonstrated by Ritschel and Jørgensen (2017), albeit resulting in lengthy expressions that will not be reproduced here. Another option is to obtain them numerically, as suggested by Ahmed (2016, p. 551). By employing first-order schemes, this approach gives rise to approximations of the form: 



∂ ln ϕˆiL ∂P

∂ ln ϕˆiV ∂P



∂ ln ϕˆiL ∂T

 =

ln ϕˆiL (P + P, T, x) − ln ϕˆiL (P, T, x) + O(P), P

(2.92)

=

ln ϕˆiV (P, T, y) − ln ϕˆiV (P − P, T, y) + O(P), P

(2.93)

=

ln ϕˆiL (P, T, x) − ln ϕˆiL (P, T − T, x) + O(T ), T

(2.94)

T, x

 T,y

 P, x

54

2 Thermodynamics of Petroleum Mixtures



∂ ln ϕˆiV ∂T

 = P,y

ln ϕˆiV (P, T + T, y) − ln ϕˆiV (P, T, y) + O(T ). T

(2.95)

The selection of values for P and T in Eqs. (2.92)–(2.95) requires careful consideration. According to Ahmed (2016, p. 550), it is recommended to choose a pressure increment of approximately 50 kPa. As for temperature, increments of around 1 K tend to work well. Naturally, adopting a higher-order interpolation scheme would reduce truncation errors. However, it should be noted that increasing the number of points used in this task directly impacts the computational time of the simulations. Based on the aforementioned and the procedures mentioned by Pedersen et al. (2014, p. 143), Elliott and Lira (2012, p. 595), it is understood that both the calculation of bubble pressure and bubble temperature in a mixture should follow the steps below: 1. An initial value is assigned to the unknown variable, followed by the calculation of the fugacity coefficients of the components in the liquid phase using Eq. (2.82); 2. Using the Wilson correlation described by Eq. (2.80), the equilibrium constant for each component in the mixture is estimated; 3. The composition of the vapor phase is determined by setting Yi = K i xi , i = 1, 2, . . . , N . Subsequently, it is necessary to evaluate Y = Yi and normalize i

4. 5. 6. 7.

it as yi = Yi /Y ; The fugacity coefficients of the components in the vapor phase are calculated based on Eq. (2.82); The equilibrium constants are updated using Eq. (2.81); The composition of the vapor phase is determined again, as described in item 3; Return to item 4 until the mole fractions experience a negligible variation between two successive iterations. One way to achieve this is, for example, 2  Yi − Yiold /N < 1 × 10−8 . When this occurs, the equilibrium by adopting i

criterion imposed by Eq. (2.72) is satisfied for all components in the mixture. However, it is possible that the sum of the mole fractions is still incorrect; 8. Evaluate the objective function and its derivative given by Eqs. (2.87) and (2.90), or Eqs. (2.88) and (2.91); 9. Update the unknown variable by setting P = P old − FPb (P old )/F Pb (P old ), or T = T old − F Tb (T old )/F Tb (T old ). Then, calculate the fugacity coefficients of the components in the liquid phase according to Eq. (2.82); 10. Return to item 4 until convergence of the objective function. An option would be to set, for example, F Pb < 1 × 10−5 or FTb < 1 × 10−5 . Just as in all algorithms developed to solve liquid-vapor equilibrium problems, sensible initial estimates of the unknown variable and vapor phase composition need to be made. O’Connell and Haile (2005, p. 482) emphasize that if these values are too far from the final answer, the iterative process tends to converge to a trivial solution for all components. Additionally, poor performance is expected in the region near

2.3 Vapor-Liquid Equilibrium

55

the critical point of the mixture, where y ≈ x. Outside this region, the algorithm will work properly. The flash calculation is divided into two iterative loops. In the inner loop, the vapor mole fraction is iterated for the current values of K i until it satisfies the objective function represented by F f lash (˜ε V ) = 0. To accomplish this, the Newton-Raphson method is employed, which requires an expression for F f lash (˜ε V ). In the outer loop, the equilibrium constants and the composition of each phase are updated through successive substitution. The objective function established by Rachford and Rice (1952) is given by: F f lash (˜ε V ) =

 i

(K i − 1) z i = 0. 1 + (K i − 1) ε˜ V

(2.96)

In turn, Pedersen et al. (2014, p. 149) demonstrate that: F f lash (˜ε V ) = −

 i

(K i − 1)2 z i  2 . 1 + (K i − 1) ε˜ V

(2.97)

The computational algorithm for calculating the flash process in a mixture must, therefore, follow the sequence below: 1. Initialize the vapor mole fraction in the system; 2. Utilize the Wilson correlation, Eq. (2.80), to estimate the equilibrium constant for each component in the mixture; 3. Determine the composition of both phases by applying Eqs. (2.56) and (2.55) in that order, for i = 1, 2, . . . , N . The mole fractions will be represented by X i and X i and Y = Yi and normalize Yi . Subsequently, calculate the sums X = i

4. 5. 6.

7. 8. 9.

i

them as yi = Yi /Y and xi = X i / X ; Evaluate the objective function and its derivative using Eqs. (2.96) and (2.97); Update the vapor mole fraction by performing ε˜ V = ε˜ V,old − F f lash (˜ε V,old )/ F f lash (˜ε V,old ); Return to item 3 until the convergence of the objective function is achieved. One option would be to set, for example, F f lash < 1 × 10−5 . When this condition is met, the mass balance defined by Eq. (2.54) is satisfied for all components in the mixture, i.e., X = 1 and Y = 1. However, it is still necessary to verify the phase equilibrium of the mixture; Calculate the fugacity coefficients of the components in the liquid and vapor phases based on Eq. (2.82); Update the equilibrium constants using Eq. (2.81); Return to item 3 until the mole fractions of the components experience a negligible change between two successive iterations. One way to achieve this is by imposing, 2  2  xi − xiold + yi − yiold /2N < 1 × 10−8 . for example, i

56

2 Thermodynamics of Petroleum Mixtures

According to Michelsen and Mollerup (2007, p. 259), the main advantages of the above algorithm lie in its simplicity and ease of implementation, coupled with a robustness that makes it a very reliable means of obtaining the unknowns of the problem. On the other hand, solving the outer loop using the method of successive substitutions may require a considerable amount of time, especially in nearly critical mixtures where K i → 1 for all components. Under such circumstances, it is advisable to expedite the process by employing other procedures described by the same authors.

2.3.5 Stability Analysis in Flash Calculations Interestingly, one of the main challenges encountered in flash calculation lies not in solving the liquid-vapor equilibrium problem itself, but rather in determining whether the mixture will indeed separate into two or more phases. In this context, an insightful stability criterion is presented by Whitson and Brulé (2000, p. 4.9–13), which is based on the minimization of the system’s Gibbs free energy. Assuming that an initially homogeneous mixture is divided into two phases, namely a liquid and a vapor phase, the variation in this property can be expressed as:   δG = G L + G V − G.

(2.98)

If one considers starting from the liquid state, the parameter δG characterizes partial vaporization, which is equivalent to δn V . In this case, by truncating the Taylor series expansion at the first term, it is possible to calculate the Gibbs free energy of the remaining liquid by utilizing the approach proposed by Michelsen (1982):   ∂G   L V yi = G − δn V yi μi . (2.99) G = G − δn ∂n i P,T, j=i i i For the vapor phase, the following relationship holds:  yi μiV . G V = δn V

(2.100)

i

In this way, Eq. (2.98) can now be written as:      ∂G = yi μiV − μi . V ∂n P,T i

(2.101)

Therefore, it is concluded that the variation of the Gibbs free energy of the system caused by evaporation depends solely on the composition of the new phase and the differences in chemical potential between it and the homogeneous mixture that generated it. The phase transition process will only occur if it leads to a reduction in G. Otherwise, the initially conceived liquid state will prevail. By introducing a

2.3 Vapor-Liquid Equilibrium

57

function F as a result of the combination of Eqs. (2.71) and (2.101), the stability criterion of the homogeneous mixture is given by:

  fˆiV yi ln F(y ) = ≥ 0. (2.102) fˆi i

On the other hand, perhaps the system was initially in the vapor state, and δG resulted from the partial condensation of its content. Thus, one should bear in mind a second stability criterion which, following the same previous reasoning, is given by:

  fˆiL xi ln ≥ 0. (2.103) F( x) = fˆi i

In Eqs. (2.102) and (2.103), the compositions of the incipient phases remain undetermined. To calculate them, Michelsen (1982) proposed an algorithm similar to that used in flash calculations, but faster and safer. The Michelsen test unfolds in the following steps: 1. Assuming that the mixture is in the liquid state, the fugacity coefficients of the components are calculated according to Eq. (2.82); 2. The equilibrium constants of the components in the mixture are initialized using the Wilson correlation described by Eq. (2.80); of the incipi3. Setting Yi = K i z i or X i = z i /K i , i = 1, 2, . . . , N , the composition Xi ent phase is determined. Then, it is necessary to calculate Y = Yi or X = i

i

and perform a normalization as yi = Yi /Y or xi = X i / X ; 4. Using Eq. (2.82), the fugacity coefficients of the components in the incipient phase are calculated; 5. The equilibrium constants are updated using Eq. (2.81); 6. Return to step 3 until the mole fractions in the second phase show a negligible change between two successive iterations. One way to do this is to establish, 2 2   Yi − Yiold /N < 1 × 10−8 or X i − X iold /N < 1 × 10−8 . for example, i

i

When this occurs, the equilibrium criterion described in Eq. (2.72) is being respected for all components of the mixture; 7. Using the expression (ln K i )2 /N < 1 × 10−4 , it is verified whether a trivial i

solution to the problem has been found. If this is confirmed, the procedure is stopped; 8. Assuming now that the mixture is in the vapor state, the fugacity coefficients of the components are calculated according to Eq. (2.82); 9. Repeat steps 2–7 for this new configuration. At the end of the procedure, the single-phase mixture is considered stable when (i) the equilibrium calculations result in Y ≤ 1 and X ≤ 1, (ii) both converge to a trivial

58

2 Thermodynamics of Petroleum Mixtures

solution, or (iii) if Y ≤ 1 or X ≤ 1 and a trivial solution are obtained. According to Whitson and Brulé (2000, p. 4.13), from a mathematical perspective, it is not possible to reach such a conclusion with absolute certainty until all infinite possible compositions have been tested. However, it is customary to adopt the Michelsen test as a sufficient criterion. On the other hand, if a single equilibrium calculation indicates Y > 1 or X > 1, the mixture is declared unstable.

2.4 Obtaining the Thermodynamic Properties of the Mixture With the obtained phase composition and following the application of mixing rules pertaining to the equation of state, which facilitate the evaluation of its compressibility factor, all remaining thermodynamic properties become inherently interconnected. For instance, the density of the liquid or vapor can be determined as follows: ρ˜ =

P . z RT

Additionally, considering that v˜ = 1/ρ, ˜ it follows that:   ∂ ρ˜ −1 = 2 ∂P  . ∂P T v˜ ∂ v˜ T From the triple product rule:       ∂ ρ˜ ∂P ∂ ρ˜ =− . ∂T P ∂ P T ∂ T ρ˜

(2.104)

(2.105)

(2.106)

The variation experienced by any property M between the states (P1 , T1 ) and (P2 , T2 ) can also be expressed in terms of departure functions. These functions juxtapose the ideal gas hypothesis with the substance’s actual behavior. Therefore, the departure function of M is defined as: M R (P, T ) = M(P, T ) − M ig (P, T )

(2.107)

Furthermore, the change in property M between states 1 and 2 can be determined by:       ig ig ig ig (2.108) M = M2 − M2 + M2 − M1 − M1 − M1 . In Eq. (2.108), once the departure function M R is known for states 1 and 2, the term M is determined by the change in properties within the ideal gas domain. This process is considerably simpler than the conventional approach, as it involves

2.4 Obtaining the Thermodynamic Properties of the Mixture

59

integrals with straightforward solutions. By applying this concept to molar internal energy (u) ˜ and entropy (˜s ), one has:   ig ig (2.109) u˜ = u˜ 2 − u˜ 1 = u˜ 2R + u˜ 2 − u˜ 1 − u˜ 1R ,   ig ig ˜s = s˜2 − s˜1 = s˜2R + s˜2 − s˜1 − s˜1R .

(2.110)

In turn, as demonstrated by Borgnakke and Sonntag (2009, p. 148 and 294): T2 ig u˜ 2

−

ig u˜ 1

=

c˜vig dT ,

(2.111)

T1

T2 ig s˜2

ig s˜1

−

=

ig

c˜ P dT − R ln T



P2 P1

 ,

(2.112)

T1

and the respective departure functions are given by Elliott and Lira (2012, p. 305 and 308):   v˜   ∂P u˜ = T − P d v, ˜ ∂ T v˜ R

(2.113)

∞

v˜  s˜ = R

∞

∂P ∂T

 v˜

−

 R d v˜ + R ln z. v˜

(2.114)

It is evident that not all thermodynamic properties require this process. After ˜ calculating the values of P, T , v, ˜ u, ˜ and s˜ , quantities such as the molar enthalpy (h) and the Helmholtz (a) ˜ and Gibbs (g) ˜ free energies will arise from relationships of the form: h˜ ≡ u˜ + P v, ˜

(2.115)

a˜ ≡ u˜ − T s˜ ,

(2.116)

g˜ ≡ h˜ − T s˜ .

(2.117)

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2 Thermodynamics of Petroleum Mixtures

Table 2.5 Coefficients of Eq. (2.118) for some constituents of petroleum, valid within the range 50 ≤ T ≤ 1000, K. Source Poling et al. (2000) Component a0 a1 a2 a3 a4 N2 CO2 H2 S C1 C2 C3 iC4 nC4 iC5 nC5 nC6

3.539 3.259 4.266 4.568 4.178 3.847 3.351 5.547 1.959 7.554 8.831

–0.261 1.356 –3.438 –8.975 –4.427 5.131 17.883 5.536 38.191 –0.368 –0.166

0.007 1.502 1.319 3.631 5.660 6.011 5.477 8.057 2.434 11.846 14.302

0.157 –2.374 –1.331 –3.407 –6.651 –7.893 –8.099 –10.571 –5.175 –14.939 –18.314

–0.099 1.056 0.488 1.091 2.487 3.079 3.243 4.134 2.165 5.753 7.124

In Eq. (2.112), the molar specific heat at constant pressure (c˜ P ) of an ideal gas is determined by the weighted average of this property in each species, expressed as follows:  ig ig xi c˜ P,i . (2.118) c˜ P = i

The specific heat capacity of species i in the ideal gas state is assumed to obey the following expression: ig

c˜ P,i R

= a0,i + a1,i T + a2,i T 2 + a3,i T 3 + a4,i T 4 .

(2.119)

The coefficients in Eq. (2.119) corresponding to some constituents of petroleum are listed in Table 2.5. Moreover, the molar specific heat at constant volume (c˜v ) of an ideal gas, as shown in Eq. (2.111), is simply given by: ig

c˜vig = c˜ P − R.

(2.120)

The molar specific heat capacity at constant volume for a real fluid is obtained from the following equation: c˜v = c˜vig + c˜vR ,

(2.121)

2.5 Characterization of Oil Reservoir Fluids

where:

v˜ c˜vR

= ∞

61



∂2 P T ∂T 2

 v˜

d v. ˜

(2.122)

In conclusion, the molar specific heat at constant pressure for the real fluid is given by Elliott and Lira (2012, p. 244):     ∂P T ∂ ρ˜ . (2.123) c˜ P = c˜v − 2 ρ˜ ∂ T P ∂ T ρ˜ It is worth noting that the analytical solution of the integrals in Eqs. (2.113), (2.114), and (2.122) requires prior knowledge of a function of the form P = P(T, v). ˜ Not coincidentally, this is the case for the cubic equations of state listed in Sect. 2.2. However, considering that each of these equations prescribes a slightly different volumetric behavior for the same fluid, deviations from ideality will also differ. In Appendix B, one can find the algebraic formulas for the residual functions associated with the Redlich-Kwong (and hence Soave-Redlich-Kwong) and Peng-Robinson equations of state for u, ˜ s˜ , and c˜v . Similar expressions can be derived for any other explicit cubic equation in P, and they apply to both supercritical fluids and the liquid and vapor states, whether in the saturation region or outside it. It is sufficient to use the value of z corresponding to the phase in question.

2.5 Characterization of Oil Reservoir Fluids Each petroleum accumulation has unique chemical and molecular characteristics. According to Ahmed (2016, p. 71), the formidable quantity of substances involved, present in varying proportions, results in mixtures with significantly different properties. This has implications both for the volume percentage and the quality of its products, for example. Thus, depending on the characteristic blend of the field, commercial interest may be directed towards the production of LPG, fuels (gasoline, kerosene, diesel oil), or lubricants, among other options. Additionally, the knowledge of the fluid composition is crucial for the representation of its volumetric and phase behavior, as without it, the equations of state lose all usefulness. A serious question arises here: petroleum mixtures tend to be so complex from this point of view that the individual survey of each of their constituents becomes almost always impracticable. What should be done? In order to overcome this difficulty, specific methodologies have been developed, constituting the so-called characterization of the mixture (Pedersen et al., 2014, p. 105). In this context, the first step is to mitigate two major limitations of experimental tests. As mentioned in Sect. 1.2, it is known that the SCN fractions of the sample, despite grouping substances with similar boiling points, describe pseudocomponents with unknown critical properties, acentric factor, and specific heat.

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2 Thermodynamics of Petroleum Mixtures

In the heavy fraction, even the molar distribution remains unknown. Therefore, these gaps must be filled with empirical mathematical correlations, as well as statistical tools for result extrapolation. On the other hand, the computational effort involved in thermodynamic calculations cannot be neglected, as it grows rapidly according to the number of assumed constituents in the mixture. In such cases, techniques are employed to reduce this number without compromising the quality of predictions. Finally, it is common practice to adjust the model by making small modifications to its parameters in order to better reproduce the PVT tests of the sample.

2.5.1 Critical Properties, Acentric Factor and Specific Heat of SCN Fractions According to Whitson (1983), the correlation between the physical properties of SCN fractions and their values of density and average normal boiling temperature, as determined in laboratory experiments, dates back to the 1930s. Originally focused on process engineering applications, Watson et al. (1935) was the first to employ them in the study of the chemical composition of oil mixtures. Among the innovations introduced in this work, a rudimentary assessment of the PNA distribution present in the fraction was proposed based on the so-called Watson characterization factor (K W ), defined as: KW =

(1.8Tnb )1/3 . γ

(2.124)

The observation made by Watson et al. (1935) indicates that paraffins in pure hydrocarbons correspond to the range 12.5 < K W ≤ 13.5, while aromatics fall within 8.5 < K W ≤ 11. Naphthenes occupy the intermediate region. Subsequently, numerous methods for characterizing substances have been published, aiming to describe the properties of an increasingly diverse range of materials. Over the years, the presentation of these methods has evolved: the graphs and tables used in pioneering works have gradually been replaced by algebraic equations, which are better suited for implementation in computational algorithms. Despite these changes, Watson’s characterization factor has endured the test of time and continues to be utilized in some of these methods. Qualitatively, the trend displayed by the components listed in Table 1.1 suggests that both the critical temperature and the acentric factor of an SCN fraction increase monotonically and proportionally to its number of carbon atoms (NC ). Conversely, the critical pressure clearly decreases with increasing molecular mass. Riazi and Daubert (1980) found that the simplest equations capable of reproducing such behaviors have the following functional form: θ = aθ1b θ2c .

(2.125)

2.5 Characterization of Oil Reservoir Fluids

63

In Eq. (2.125), the symbol θ represents the desired property, while θ1 and θ2 denote the characteristic parameters of intermolecular forces and chemical structure of the component, respectively. The constants a, b, and c are determined through a regression analysis of experimental data. By employing this functional relationship, Whitson (1983) was able to mathematically represent the molecular masses reported by Katz and Firoozabadi (1978) within the range of C6 to C22 , as follows: 2.9223 −2.4750 γ Mw = 1.3829 × 10−6 Tnb

(2.126)

Later, Riazi and Daubert (1987) introduced an exponential term to Eq. (2.125), resulting in the following expression: θ = aθ1b θ2c exp (dθ1 + eθ2 + f θ1 θ2 )

(2.127)

Therefore, after adjusting the six constants based on the properties of 138 pure hydrocarbons from C1 to C20 , including paraffins, naphthenes, and aromatics with molecular masses ranging from 70 to 300 g/mol and average normal boiling temperatures between 300 and 620 K, several expressions were formulated to calculate the critical pressure and temperature of the SCN fractions as functions of θ1 and θ2 . By choosing Tnb and γ as parameters, these expressions can be written, respectively, as: −0.4844 4.0846 Pc = 3.1958 × 104 Tnb γ exp (−0.008505Tnb − 4.8014γ + 0.0057490Tnb γ ) ,

(2.128)

0.81067 γ 0.53691 exp (−0.00093145T − 0.54444γ + 0.00064791T γ ) . Tc = 9.5232Tnb nb nb

(2.129)

In Eq. (2.128), the pressure is expressed in MPa, and the same applies to all other models in this section. Riazi and Daubert (1987) reported an average absolute error of 2.7% in the correlation for Pc , and 0.5% for Tc . Once the critical properties of the fraction have been evaluated, their respective acentric factor can be obtained using the equation by Edmister (1958). In this equation, it is assumed that the saturation pressure of the substance is adequately described by an Antoine-type function of the form log Psat = A + B/T . By using the normal boiling point and the critical point as boundary conditions to calculate the constants A and B, the authors established that:     Pc Tnb 3 log − 1. (2.130) ω= 7 Tc − Tnb Patm One should note, however, that Eqs. (2.128) and (2.129) do not account for the C23+ fractions, and their inadvertent extrapolation would certainly have negative consequences. Even the premise implied in Eq. (2.130) ceases to be valid for

64

2 Thermodynamics of Petroleum Mixtures

hydrocarbon chains larger than C6 (Whitson & Brulé, 2000, p. 5.14). Such a limitation led Riazi and Al-Sahhaf (1996) to envision a new correlation given by:   θ = θ∞ − exp a − bθ1c .

(2.131)

Equation (2.131) establishes an upper limit for θ as θ1 → ∞, denoted as θ∞ . In order to validate it, the authors performed various adjustments of the constants a, b, and c using pure components from homologous series of n-alkanes (paraffins), n-alkylcyclopentanes (naphthenes), and n-alkylbenzenes (aromatics). Subsequently, the PNA distribution within the range between C6 and C22 was estimated based on the respective density and molecular mass values compiled by Katz and Firoozabadi (1978). This allowed for the calculation of the properties of the SCN fractions by setting θ = x P θ P + x N θ N + x A θ A , which were then subjected to a new regression analysis. Thus, Riazi and Al-Sahhaf (1996) concluded that the critical pressure and temperature, as well as the acentric factor of these pseudo-components, were related to their molecular mass according to:

Tnbr ≡

  Pc = exp 4.04233 − 0.7239Mw0.3 ,

(2.132)

  Tnb = 1.2 − exp −0.34742 − 0.02327Mw0.55 , Tc

(2.133)

  ω = −0.3 + exp −6.252 + 3.64457Mw0.1 .

(2.134)

A quick inspection of Eqs. (2.132) and (2.134) is sufficient to verify that Pc → 0 and ω → ∞ as Mw → ∞, a behavior supported by theoretical and experimental studies. Equation (2.133) establishes that, under the same limit, Tc → Tnb /1.2, which apparently still lacks confirmation. Regarding the critical temperature, another positive aspect of the work by Riazi and Al-Sahhaf (1996) deserves attention: by calculating the molecular mass of the SCN fraction where Pc = Patm and then substituting the result into the expression for Tnbr , one obtains Mw = 1379 g/mol and Tnb /Tc = 0.996. Considering that at this pressure level, the normal boiling temperature is defined as the substance’s critical temperature, which is nearly accurately represented in this case, the model demonstrates an internal consistency, serving as an indicator of its strong theoretical basis. Furthermore, according to Riazi et al., if a true boiling point (TBP) test is not available, the normal boiling temperature, relative density, and molecular mass of the fraction can be calculated using the following equations:   Tnb = 1080 − exp 6.97996 − 0.01964Mw2/3 ,

(2.135)

  γ = 1.07 − exp 3.56073 − 2.93886Mw0.1 ,

(2.136)

2.5 Characterization of Oil Reservoir Fluids

Mw = 14NC − 4.

65

(2.137)

Some authors, such as Søreide (1989), had previously suggested the existence of a limit for Tnb in hydrocarbons with notably large molecules. The formula for Mw was proposed by Pedersen et al. (1984) and is justified by the fact that each carbon atom (Mw = 12 g/mol) bonded to the chain is assumed to have two hydrogen atoms (Mw = 1 g/mol), resulting in a slope equal to 14 g/mol. The linear term, on the other hand, is based on an adjustment of the values compiled by Katz and Firoozabadi (1978). Another set of widely used empirical equations is presented in the work of Kesler and Lee (1976), where the critical pressure, critical temperature, and acentric factor are given by the following expressions:   ln Pc = 3.386 − 0.0566/γ − 0.43639 + 4.1216/γ + 0.21343/γ 2 × 10−3 Tnb   2 + 0.47579 + 1.182/γ + 0.15302/γ 2 × 10−6 Tnb   3 − 2.4505 + 9.9099/γ 2 × 10−10 Tnb , (2.138)

Tc = 189.8 + 450.6γ + (0.4244 + 0.1174γ ) Tnb + (0.1441 − 1.0069γ ) × 105 /Tnb ,

(2.139) 2 + 8.359T ω = −7.904 + 0.1352K W − 0.007465K W nbr + (1.408 − 0.01063K W ) /Tnbr .

(2.140)

According to Kesler and Lee (1976), Eq. (2.140) is only valid for Tnbr ≤ 0.8, which corresponds approximately to Mw ≤ 280 g/mol. In other cases, the following equation should be used: ω=

6 − ln (Pc /Patm ) − 5.92714 + 6.09648/Tnbr + 1.28862 ln Tnbr − 0.169347Tnbr 6 15.2518 − 15.6875/Tnbr − 13.4721 ln Tnbr + 0.43577Tnbr

.

(2.141)

Equation (2.141) was developed by Lee and Kesler (1975) using a methodology very similar to that of Edmister (1958), except for the vapor pressure adjustment function, which is now more refined and specifically focused on SCN fractions. This change resulted in a significantly improved performance. According to Riazi (2005, p. 65), the average absolute error of the acentric factor, which was around 3.5% when applying Eq. (2.130), has been reduced to 1.3%. Pedersen et al. (2004) preferred to adopt a different approach: instead of conventional regression, the authors devised an inference-based method, where the fitting of functions is determined by minimizing simulation errors involving differential liberation tests on a set of samples. Consequently, the expressions for Pc and Tc become dependent on the equation of state used in the analysis, which, to some

66

2 Thermodynamics of Petroleum Mixtures

Table 2.6 Values recommended by Pedersen et al. (2004) for the coefficients of Eqs. (2.142)– (2.144) Index Soave-Redlich-Kwong Peng-Robinson a b c a b c 1 2

−2.4367 2.5019

163.12 86.052

3

208.46

0.43475

4

−3987.2

−1877.4

5

1

−

0.74310 4.8122 × 10−3 9.6707 × 10−3 −3.7184 × 10−6 −

−2.2297 2.18811

73.4043 97.3562

163.910

0.618744

−4043.23

−2059.32

1/4

−

0.373765 5.49269 × 10−3 0.0117934 −4.93049 × 10−6 −

extent, diminishes the physical significance of the predictions. Another distinctive aspect of the work lies in the direct manipulation of the parameter m present in Eq. (2.19), circumventing the calculation of the acentric factor. Naturally, the latter can be recovered by inverting the correlation for m(ω) currently employed in the model. Pedersen et al. (2004) selected the following algebraic expressions for the critical pressure, critical temperature, and the slope coefficient of the function α(T ) proposed by Soave (1972): ln Pc = a1 + a2 γ a5 +

a3 a4 + 2, Mw Mw

Tc = b1 γ + b2 ln Mw + b3 Mw +

b4 , Mw

m = c1 + c2 Mw + c3 γ + c4 Mw2 .

(2.142)

(2.143)

(2.144)

The coefficients of Eqs. (2.142)–(2.144) related to the Soave-Redlich-Kwong and Peng-Robinson equations of state are presented in Table 2.6. ig Finally, regarding the estimation of c˜ P for the SCN fractions, Kesler and Lee (1976) recommends the following expression: ig

c˜ P = a0 + a1 T + a2 T 2 , Mw

(2.145)

where: a0 = (−1.41779 + 0.11828K W ) − b (1.09223 − 2.48245ω) ,

(2.146)

2.5 Characterization of Oil Reservoir Fluids

67

    2 + b (34.34 − 71.4ω) × 10−4 , a1 = − 6.99724 − 8.69326K W + 0.27715K W (2.147) a2 = [−22.582 + b (7.2661 − 9.2561ω)] × 10−7 ,  b=

(12.8 − K W ) (10 − K W ) 10ω

(2.148)

2 .

(2.149)

2.5.2 Splitting the Heavy Fraction When dealing with the heavy fraction, the idea of representing it as a single pseudocomponent with molecular mass and density values obtained from compositional analysis of the mixture inevitably arises. However, as pointed out by Ahmed (2016, p. 140), directly applying the residual portion obtained in the laboratory to a cubic equation of state can have disastrous effects, particularly in equilibrium calculations. A viable solution to this problem involves dividing the heavy fraction into a certain number of SCN fractions. It is important to note that some basic rules need to be followed throughout the process: 1. The sum of the individual mole fractions resulting from the splitting process must be equal to the value assigned to the original heavy fraction: 

NC,max

z Ci = z + ;

(2.150)

i=NC+

2. The combined molecular mass of the individual mole fractions must be equivalent to that of the original heavy fraction: 

NC,max

z Ci Mw,Ci = z + Mw+ ;

(2.151)

i=NC+

3. The average density of the individual mole fractions and the original heavy fraction must be compatible:  z C Mw,C z + Mw+ i i = . γCi γ+ i=N NC,max

(2.152)

C+

In the initial attempts to describe the composition of the residual fraction, it was common to attribute a continuous and asymptotic decay to the C7+ fractions, proportional to NC,i . For instance, Lohrenz et al. (1964) proposed an exponential distribution model. Recognizing the predominance of paraffins over other molecular

68

2 Thermodynamics of Petroleum Mixtures

structure types, it was claimed that the mole fraction for 7 ≤ NC ≤ 40 was adequately correlated with the carbon number of the fraction and the proportion of n-hexane present in the mixture, according to the following expression:   z Ci = z C6 exp A (i − 6)2 + B (i − 6) .

(2.153)

Equation (2.153) was designed to ensure continuity between the values of z C6 and z C7 . Lohrenz et al. (1964) provides an iterative calculation method for the constants A and B, transforming Eqs. (2.150) and (2.151) into objective functions with NC,max = 40. By doing so, the first two partitioning criteria are satisfied. The methodology developed by Pedersen et al. (1984) presents a similar approach. In this methodology, z Ci is given by:   i−A . (2.154) z Ci = exp B There are two ways to determine the constants A and B in Eq. (2.154). The first method involves a linear adjustment of ln z Ci as a function of NC,i for all experimentally obtained SCN fractions. The result of this regression is then extrapolated to the heavy fraction, incrementing NC,max periodically until the equality in Eq. (2.150) is confirmed. However, this strategy becomes unfeasible when the asymptotic limit of such a summation is lower than z + . On the other hand, the second option for calculating the constants is iterative in nature, similar to the one reported by Lohrenz et al. (1964). Pedersen et al. (1984) recommends fixing NC,max = 80 for light oil samples and NC,max = 200 for heavy oils. Despite the good robustness of the Lohrentz and Pedersen techniques, their application in petroleum fluids with high molecular mass can lead to inconsistent characterizations, as in such cases, the assumed exponential decay of the SCN fractions is no longer true. This even more challenging scenario motivated Whitson (1983) to create a method based on the so-called gamma probability distribution, denoted for any variable x as: f p (x) =

x α−1 exp (−x/β) β α (α)

(2.155)

In Eq. (2.155), f p (x) represents the probability density function of x. The parameters of shape and scale are denoted by α and β, respectively. The function refers to the gamma function.10 It is defined as follows: ∞

(α) =

x α−1 e−x d x.

(2.156)

0

The gamma function is an extension of the factorial to non-natural numbers. When α ∈ {1, 2 . . .}, we have (α) = (α − 1)!. Similar to the factorial, the gamma function also satisfies the recurrence formula (α + 1) = α (α) (Whitson & Brulé, 2000, p. 5.7).

10

2.5 Characterization of Oil Reservoir Fluids

69

According to Abramowitz and Stegun (1965, p. 257), (α) can be approximated by a polynomial of the form:

(α) = 1 +

8 

ci (α − 1)i ,

(2.157)

i=1

where c1 = −0.577191652, c2 = 0.988205891, c3 = −0.897056937, c4 = 0.918206857, c5 = −0.756704078, c6 = 0.482199394, c7 = −0.193527818, and c8 = 0.035868343 for 1 ≤ α ≤ 2. Outside this interval, the recurrence formula of the function is used. In order to apply the gamma distribution to represent the weight fraction, Whitson (1983) introduced a third parameter, η, which quantifies the minimum molecular mass attainable. Thus, by adopting x = Mw − η, one obtains: f p (Mw ) =

(Mw − η)α−1 exp [− (Mw − η) /β] . β α (α)

(2.158)

The cumulative distribution function (F) is calculated as: Mw F(Mw ) =

f p (Mw )d Mw = η

∞ yj e−y y α  · ,

(α) j=0 (α + j)!

(2.159)

where: y=

Mw − η . β

(2.160)

Naturally, it holds that F(η) = 0 and F(∞) = 1. A similar behavior occurs for the cumulative mole fraction of the mixture as a function of Mw , where z(η) = 0 and z(∞) = z + . The inherent proportionality of these two functions allows for the establishment of the following relationship:   z Ci = z + F(Mw,Ci+1/2 ) − F(Mw,Ci−1/2 ) .

(2.161)

The molecular weights used in Eq. (2.161) can be easily obtained from Eq. (2.137) for NC,i−1/2 and NC,i+1/2 . It is important to note that, following this approach, the first partitioning criterion will be fully satisfied only for Mw → ∞. On the other hand, it is known that including SCN fractions with an excessively high number of carbons not only does not improve the accuracy of the thermodynamic model but also results in increased computational effort during calculations. In this context, Li et al. (1985) advise disregarding the last 5% of the summation appearing in Eq. (2.150), which is equivalent to: F(Mw,Cmax ) ≈ 0.95.

(2.162)

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2 Thermodynamics of Petroleum Mixtures

The truncation of the sum necessitates a normalization of the values of F(Mw ) as given by Eq. (2.159). Consequently, the average molecular weight of fraction Ci is determined by the following expression: Mw,Ci = η + αβ

F(Mw,Ci+1/2 , α + 1) − F(Mw,Ci−1/2 , α + 1) . F(Mw,Ci+1/2 , α) − F(Mw,Ci−1/2 , α)

(2.163)

Extending Eq. (2.163) to the upper and lower limits of the heavy fraction, it is observed that: Mw+ = η + αβ

(2.164)

Therefore, in order to satisfy the second partitioning criterion, the scaling parameter must necessarily be equal to: β=

Mw+ − η α

(2.165)

The shape parameter provides Whitson’s method with a flexibility that is not seen in the works of Lohrenz et al. (1964) and Pedersen et al. (1984). As shown in Fig. 2.1, when α = 1, it results in an exponential distribution that is completely consistent with those authors. However, when α < 1, it intensifies the initial decay rate, and when α > 1, it shifts the peak of the probability density function to the right, gradually approaching a normal distribution. Finally, the third partitioning criterion is satisfied by introducing an adjustment coefficient ψγ in the calculation of the relative density of SCN fractions. This coefficient should be iteratively modified until the experimental value of γ+ is reproduced.

2.5.3 Lumping Components The splitting of the heavy fraction, combined with the calculation of critical properties and the acentric factor for each of the SCN fractions considered in the sample, enables the equation of state to make predictions about the volumetric and phase behavior of the mixture. However, it is known that the computation time in simulations of this type is closely linked to the number of components in the system (Ahmed 2016, p. 140). Therefore, in applications involving three-dimensional meshes and a considerable number of control volumes, such as those used to depict the flow of oil within a reservoir, a strict monitoring of this parameter is typically imposed. This is achieved by combining similar chemical species in order to establish a smaller number of pseudo-components, which, in the case of hydrocarbons, are also referred to as Multiple Carbon Number (MCN) groups. The representation of an MCN group follows the range of carbon numbers it encompasses. For example, the hypothetical

2.5 Characterization of Oil Reservoir Fluids

71

·10−2 α = 0.5 α = 1.0 α = 5.0

1.0

0.8

f

0.6

0.4

0.2

0.0 300

400

500 Mw , g/mol

600

700

Fig. 2.1 Gamma distributions for different values of the shape parameter, using Mw+ = 400 g/mol, η = 275 g/mol, and β given by Eq. (2.165)

combination of SCN fractions with 7 ≤ NC ≤ 10 results in the pseudo-component C7 -C10 . In theory, the successful accomplishment of this simplification task requires a three-pronged approach. The first aspect concerns the number of MCN groups that will constitute the C7+ fraction. The second aspect pertains to the criteria for allocating the SCN fractions to their respective groups. The final aspect is devoted to the weighting of critical properties and the acentric factor of these pseudocomponents, based on the previously assigned values of each constituent. The grouping of components in petroleum mixtures, known as lumping, has been the central topic of numerous studies conducted over the past 40 years. These studies have engendered a proportional number of discussions and currently find varying degrees of acceptance among different schools of thought that have emerged. It should be noted that all methodologies conceived to date are guided, to a greater or lesser extent, by essentially empirical criteria. Riazi (2005, p. 184) considers it customary to divide the C7+ fraction of condensate gas samples into only 3 pseudocomponents, a number that increases to 5 or 7 in the characterization of volatile oils and light black oils, and reaches 10 for heavy black oils. Whitson (1983) suggests delimiting the MCN groups within certain ranges of molecular mass, while Pedersen et al. (1985) argue that the best strategy to compose them is to attempt to produce

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approximately equal mass fractions. Li et al. (1985) assert that when dealing with phase equilibrium problems, the similarity between substances is predominantly dictated by the volatility associated with each of them. On the other hand, Behrens and Sandler (1988) seek to bring a certain elegance to the debate by applying the Gaussian quadrature rule in the definition of these groups. Regarding the mixing rules of pseudocomponents, Kay (1936) simply adopts an average of the values of its constituents weighted by the mole fraction, a practice discouraged by Pedersen et al. (1985), who prefer to do so on a mass basis. Wu and Batycky (1988) state that the best strategy certainly lies in a compromise between these two approaches, which would imply a hybrid method. Lee and Kesler (1975) employ distinct mixing rules within the same grouping, each developed for a specific property, which were later endorsed by Mehra et al. (1982) and Li et al. (1985). Furthermore, the technique of Leibovici (1993) deserves attention, as it begins with the determination of the parameters a I and b I associated with pseudocomponent I in the cubic equation of state to calculate Pc,I , Tc,I , and ω I . However, considering the low complexity of the computational mesh designed for representing offshore production wells (OPWs) in general, as well as the possibility of utilizing a preconceived fluid property table in such simulations (thus eliminating the need to solve phase equilibrium at each point along the system), the component lumping does not yield tangible benefits to the current study and will not be included. Furthermore, employing such schemes would entail the risk of obscuring the true accuracy of the model in future performance tests, potentially leading to decisions that could irreversibly affect the chosen course for this book.

2.5.4 Model Fitting According to Whitson and Brulé (2000, p. 4.19), in most cases, even the most diligent sample characterization, following all the steps described in the previous sections, will not result in a truly accurate thermodynamic model. It is common for calculation errors involving the bubble-point pressure of the mixture to be around 10%, and predictions regarding the liquid mass density may deviate by up to 5%. Additionally, important components, such as methane, may have their presence in one of the phases overestimated by a few percentage points. The solution or mitigation of such problems requires a delicate adjustment process. Firstly, according to Danesh (1998, p. 323), it is necessary to define which parameters will be modified and which will remain unchanged. The critical properties and acentric factor of SCN fractions emerge as natural candidates for this endeavor, largely due to being derived from empirical correlations rather than experimental observations. The majority of authors agree that adjusting Pc , Tc , and ω precedes any intervention in the parameters of binary interaction. For instance, Gani and Fredenslund (1987) consider the second alternative viable only when changes exceeding 10% are required in any of these three parameters. To avoid convergence issues and loss of reliability, Pedersen et al. (2014, p. 199) emphasize the importance of not

2.5 Characterization of Oil Reservoir Fluids

73

acting on the values individually but rather introducing adjustment factors into the expressions to which they refer. This ensures the natural tendencies of decreasing critical pressure and increasing critical temperature and acentric factor with chain length. Theoretically, the same effect could be achieved by modifying the input parameters of these correlations (typically limited to molecular weight and relative density of the SCN fraction). However, as mentioned by Whitson (1984), the low sensitivity observed on some occasions makes this approach inadvisable. Another parameter frequently manipulated is the molecular mass of the heavy fraction. Some authors, such as Christensen (1999), Zurita and McCain (2002), reserve it for the calibration of the mixture’s bubble-point pressure, although this purported function Pb (Mw+ ) proves to be much more intricate in practice. The adjustment of the oil’s mass density is entrusted to the volume translation parameters of the SCN fractions (Zuo & Zhang, 2000), where the cause-and-effect relationship is indisputable. Finally, Coats and Smart (1986) believe it is advantageous to directly intervene in the equation of state by proposing changes to the coefficients ac and b. However, such a strategy appears risky and strongly inclined towards excessively mathematizing a model intended to represent physical phenomena. After selecting the variables, efforts are directed towards creating the objective function. This function should be able to synthesize, in a single value, all the inconsistencies of the numerical predictions in relation to the actual behavior exhibited by the sample during the PVT testing. As a rule, the calculation of the mean absolute percentage errors (MAPE) for each of these properties is chosen, followed by weighting and averaging them. There is still no consensus on the best way to assign the weights, and sometimes a cumbersome trial-and-error process ensues. However, some guidelines persist, such as assigning a greater weight to the deviation in bubble pressure of the mixture compared to others. This decision is fully justifiable since it is a highly relevant parameter in the field of oil flow (as it delimits the end of singlephase flow and the beginning of two-phase flow) and significantly affects the trend described by other properties such as Rs , Bo , ρo , μo , etc. According to Danesh (1998, p. 323), mass density readings also deserve some primacy, followed by the volumes assumed by the oil and gas phases during differential liberation tests. The molar composition, on the other hand, is considered last due to the inherent experimental uncertainties involved in its determination. From there, the adjustment can be seen as a problem of nonlinear multivariate regression, aiming to iteratively minimize the average deviations by freely varying the pre-selected parameters. Methods such as steepest descent and Gauss-Newton prove particularly useful for this purpose. The steepest descent method calculates the gradient of the objective function based on the current combination of values to guide the next attempt, while the Gauss-Newton method examines the second-order partial derivatives to determine the location of the minimum point. The former is highly effective at the beginning of the search but loses speed as the gradient diminishes. The latter can approach the solution quickly but relies on a good initial guess to avoid divergence. The optimization algorithm should extract the best qualities from each of these methods, adapting to the various situations encountered during the task.

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When the process reaches its conclusion, there is always the risk that the obtained combination corresponds to a local minimum of the objective function instead of the global optimum. However, according to Agarwal et al. (1990), the adjustment of the fluid model should not be viewed as an unwavering pursuit of that particular point, but rather as a means to confine simulation errors within acceptable levels, thereby justifying any and all solutions that meet specific prerequisites.

2.6 Preliminary Results Based on the content presented thus far, it is now possible to perform some simulations of the thermodynamic properties of real petroleum mixtures from the pre-salt layer, albeit in a preliminary manner. Let us consider, for example, the samples PS01 and PS-02,11 which belong to a database whose details will be discussed later on. These two particular mixtures will serve as the basis for all the flow simulations in Chaps. 5 and 6 and, therefore, deserve greater attention. By applying Whitson’s methodology to partition the heavy fraction of each sample, with α = 1 and truncation at 95%, the profiles12 illustrated in Fig. 2.2 are obtained. In both cases, an exponential decay of z Ci with respect to Mw,Ci is observed. The differences lie in the number of SCN fractions generated by this method and their proportions. In the first sample, which has a higher molecular weight, the procedure reached the pseudocomponent C85 (Mw ≈ 1200 g/mol), while in the other sample, it did not even exceed C61 (Mw = 850 g/mol). It is worth noting that the difference in slope between the two curves in Fig. 2.2 arises from the need to satisfy the second partitioning criterion, taking into account the peculiarities found in each sample. Next, it is necessary to establish the critical properties and the acentric factor of the SCN fractions that make up these two mixtures. For this purpose, the correlations of Riazi and Daubert (1987) for Pc and Tc , and Kesler and Lee (1976) for ω were chosen, a combination that will prove particularly useful later on. In the absence of experimental data regarding the input parameters Mw , Tnb , and γ , the work of Riazi and Al-Sahhaf (1996) was consulted. The obtained values are presented in Table 2.7, with some omissions starting from pseudocomponent C30 , as the changes become increasingly subtle (it is worth noting that these are not MCN groups). Once the PS-01 and PS-02 samples have been characterized, they can be subjected to any type of calculation involving liquid-vapor equilibrium. Naturally, in order to assess the accuracy of the thermodynamic model at this early stage of development, there is a particular interest in simulating the flash and differential liberation tests conducted in the laboratory for each sample. The results13 regarding the 11

These are obviously fictitious designations. For confidentiality reasons, all sample compositions used in this study will be expressed in relative terms. 13 Due to confidentiality reasons, all graphs related to the differential liberation tests in this study have been dimensionless. 12

2.6 Preliminary Results

75

Table 2.7 Critical properties and acentric factors assigned to the SCN fractions in the characterization of samples PS-01 and PS-02 for the preliminary thermodynamic model Fraction Mw , g/mol Tnb , K γ Pc , MPa Tc , K ω C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21 C22 C23 C24 C25 C26 C27 C28 C29 C30 C35 C40 C45 C50 C55 C60 C65 C70 C75 C80 C85

Eq. (2.137)

Eq. (2.135)

Eq. (2.136)

Eq. (2.128)

Eq. (2.129)

Eq. (2.141)

94 108 122 136 150 164 178 192 206 220 234 248 262 276 290 304 318 332 346 360 374 388 402 416 486 556 626 696 766 836 906 976 1046 1116 1186

363.8 391.5 417.0 440.6 462.6 483.2 502.6 520.9 538.2 554.6 570.1 584.9 599.0 612.5 625.4 637.7 649.5 660.8 671.7 682.2 692.2 701.9 711.2 720.2 760.8 795.1 824.6 850.1 872.4 891.9 909.1 924.4 937.9 950.1 961.0

0.7271 0.7482 0.7659 0.7811 0.7943 0.8060 0.8163 0.8256 0.8341 0.8417 0.8487 0.8552 0.8611 0.8667 0.8718 0.8766 0.8811 0.8854 0.8894 0.8931 0.8967 0.9001 0.9033 0.9063 0.9196 0.9304 0.9394 0.9471 0.9536 0.9594 0.9645 0.9690 0.9731 0.9768 0.9801

3.157 2.879 2.645 2.447 2.278 2.131 2.004 1.892 1.793 1.705 1.626 1.555 1.492 1.434 1.381 1.333 1.289 1.248 1.211 1.176 1.144 1.114 1.087 1.061 0.954 0.874 0.814 0.766 0.728 0.697 0.672 0.651 0.633 0.619 0.606

544.3 575.6 603.6 628.9 651.9 673.0 692.5 710.6 727.4 743.1 757.9 771.7 784.8 797.2 808.9 820.0 830.5 840.5 850.1 859.2 868.0 876.3 884.4 892.0 926.2 954.7 978.7 999.4 1017 1033 1047 1059 1070 1079 1088

0.308 0.353 0.398 0.441 0.484 0.526 0.566 0.605 0.644 0.681 0.717 0.752 0.786 0.819 0.851 0.883 0.913 0.941 0.968 0.993 1.018 1.041 1.064 1.085 1.181 1.259 1.325 1.381 1.429 1.469 1.505 1.536 1.563 1.586 1.607

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z/z+

10−1

PS-01 PS-02

10−2

10−3 200

400

600 800 Mw , g/mol

1000

1200

Fig. 2.2 Results of the splitting of the heavy fraction of samples PS-01 and PS-02 for the preliminary thermodynamic model Table 2.8 Properties of dead oil and natural gas for samples PS-01 and PS-02. Comparison of the results obtained in the flash liberation test with those predicted by the preliminary thermodynamic model Sample Property Experimental Soave-Redlich-Kwong Peng-Robinson Calculated Error, % Calculated Error, % PS-01

PS-02

γod , ◦ API γg GOR, 3 /m 3 mstd std γod , ◦ API γg GOR, 3 /m 3 mstd std

28.5 0.81 243

73.0 0.83 192

156 3.5 –21

51.3 0.84 215

80 3.7 –11

29.2 0.80 233

72.5 0.83 187

148 4.0 –20

50.8 0.84 209

74 4.3 –10

Soave-Redlich-Kwong and Peng-Robinson equations of state in their respective original forms are presented in Table 2.8 and Figs. 2.3, 2.4, 2.5, 2.6, 2.7 and 2.8. It is observed that the latter provides more consistent predictions for the oil’s mass density and the gas-oil ratio (GOR), although still significantly deviating from the experimental values. In the calculation of gas relative density, the former appears to have a certain advantage, with deviations oscillating around 4%. Both formulations underestimate the effect of gas incorporation on the oil formation volume factor, leading to errors on the order of 20% in the bubble point pressure estimates.

2.6 Preliminary Results

77

1.0

(Bo − Bod ) / (Bob − Bod )

0.8

0.6

0.4

0.2 Experimental Soave-Redlich-Kwong Peng-Robinson

0.0 0.00

0.25

0.50 0.75 (P − Patm ) / (Pb − Patm )

1.00

1.25

Fig. 2.3 Variation of the oil formation volume factor as a function of pressure for sample PS-01. Comparison of the results obtained in the differential liberation test with those obtained by the preliminary thermodynamic model

The overall performance achieved in these initial simulations provides a comprehensive depiction of the challenges associated with representing the volumetric and phase behavior of oil reservoir fluids, while also indicating the path to be pursued in order to overcome them. It is believed that obtaining results that are more consistent with experimental observations necessitates improvements in three specific areas of the thermodynamic model. These areas are as follows: Cubic equations of state: It is necessary to ensure the correct functioning of the Soave-Redlich-Kwong and Peng-Robinson equations of state for the entire range of commonly encountered values of Pc , Tc , and ω in a hydrocarbon mixture, as well as subject their estimates to the volume translation technique by Peneloux et al. (1982). These measures aim to reduce deviations related to the calculation of oil density; Characterization of SCN fractions: Predictions of liquid-vapor equilibrium are strongly influenced by the sample characterization process, which brings forth two distinct situations. The Whitson method for splitting the heavy fraction, considered by many as the current state of the art, will remain unquestioned. On the

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1.0

Rs /GOR

0.8

0.6

0.4

0.2 Experimental Soave-Redlich-Kwong Peng-Robinson

0.0 0.00

0.25

0.50 0.75 (P − Patm ) / (Pb − Patm )

1.00

1.25

Fig. 2.4 Variation of gas solubility ratio as a function of pressure for sample PS-01. Comparison of the results obtained in the differential liberation test with those obtained by the preliminary thermodynamic model

other hand, correlations related to the critical properties and acentric factor of the SCN fractions result in values that vary significantly from author to author and need to be chosen with the utmost caution; Model adjustment: Even the most sophisticated thermodynamic models fail to reproduce certain nuances that make each oil accumulation truly unique because they are based on incomplete compositional surveys, subject to a series of practical limitations that detract valuable information. As a consequence, the final quality of the thermodynamic model will always be partially conditioned by the adoption of a good strategy for adjusting its input parameters.

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79

1.2

Experimental Soave-Redlich-Kwong Peng-Robinson

(ρo − ρob ) / (ρod − ρob )

0.8

0.4

0.0

−0.4

−0.8 0.00

0.25

0.50 0.75 (P − Patm ) / (Pb − Patm )

1.00

1.25

Fig. 2.5 Variation of oil density as a function of pressure for sample PS-01. Comparison of the results obtained in the differential liberation test with those obtained by the preliminary thermodynamic model 1.0

(Bo − Bod ) / (Bob − Bod )

0.8

0.6

0.4

0.2 Experimental Soave-Redlich-Kwong Peng-Robinson

0.0 0.00

0.25

0.75 0.50 (P − Patm ) / (Pb − Patm )

1.00

1.25

Fig. 2.6 Variation of the oil formation volume factor as a function of pressure for sample PS-02. Comparison of the results obtained in the differential liberation test with those obtained by the preliminary thermodynamic model

80

2 Thermodynamics of Petroleum Mixtures 1.0

Rs /GOR

0.8

0.6

0.4

0.2 Experimental Soave-Redlich-Kwong Peng-Robinson

0.0 0.00

0.25

0.50 0.75 (P − Patm ) / (Pb − Patm )

1.00

1.25

Fig. 2.7 Variation of gas solubility ratio as a function of pressure for sample PS-02. Comparison of the results obtained in the differential liberation test with those obtained by the preliminary thermodynamic model 1.2

Experimental Soave-Redlich-Kwong Peng-Robinson

(ρo − ρob ) / (ρod − ρob )

0.8

0.4

0.0

−0.4

−0.8 0.00

0.25

0.50 0.75 (P − Patm ) / (Pb − Patm )

1.00

1.25

Fig. 2.8 Variation of oil density as a function of pressure for sample PS-02. Comparison of the results obtained in the differential liberation test with those obtained by the preliminary thermodynamic model

References

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Chapter 3

Developing a Fluid Model

3.1 Introduction Simulations focused on oil production systems are always preceded by a thorough modeling of the thermodynamic and transport properties of the reservoir fluid. The reason for this lies primarily in the unilateral nature of certain phenomena involved: while the volumetric and phase behavior of the aforementioned mixture drastically affects the characteristics of the flow established between the bottom of the well and the surface, such as the macroscopic arrangement and the shear stresses acting on the pipe wall, it is very rare for the reverse to occur. The few exceptions mainly concern oils with non-Newtonian rheological behavior, where viscosity, as is well known, varies according to the rate of deformation. With this hierarchy defined, it becomes easy to foresee that an unfounded representation of the equilibrium mixture would compromise any prediction related to its flow. On the other hand, considering the wide diversity of substances present in the accumulations, each following a distinct distribution, it is natural for their respective models to emphasize certain aspects of the fluid while paying less attention to others. Reservoirs in the pre-salt layer distinguish themselves from others due to the high levels of pressure and temperature they are subjected to, the more pronounced presence of CO2 in their composition, and the extremely high salinity of the connate water, just to mention a few examples. These characteristics need to be taken into account when determining the best approach to the problem, as discussed in Sect. 3.2. Clearly, the content presented in Chap. 2 will guide all the stages of development of the new thermodynamic model. The envisioned methodology consists, in general, of (i) testing a certain set of works developed for other types of petroleum, (ii) identifying the most suitable ones for the current application, and (iii) when the opportunity arises, proposing improvements to them, always followed by subsequent validation. This does not mean that the contributions resulting from the process are limited to the pre-salt layer; on the contrary, in Sect. 3.3, the new calculation expressions for the angular coefficient of the function α(T ) by Soave (1972) synthesize an extensive © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. R. Gessner and J. R. Barbosa Jr., Integrated Modeling of Reservoir Fluid Properties and Multiphase Flow in Offshore Production Systems, Petroleum Engineering, https://doi.org/10.1007/978-3-031-39850-6_3

85

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database composed of over 1700 pure substances, with hydrocarbons accounting for only a fraction. Furthermore, in Sect. 3.4, which addresses the characterization of SCN fractions, the adjustment of exponential functions for Pc , Tc , and ω utilized values standardized by the API Technical Data Book (Daubert and Danner, 1997), which have been successfully employed in simulations involving numerous fields worldwide. The tuning procedure of the fluid model is described in detail in Sect. 3.5 so that it can be easily adapted to other scenarios. The three aforementioned topics collectively encompass the entire compositional aspect of the fluid model. To fill in the deliberately left empty spaces (the motivation behind this will become clear in the course of the chapter), the set of correlations listed in Sect. 3.6 was selected. These correlations were mainly extracted from established and widely disseminated publications that have withstood subsequent scrutiny and stood the test of time. Lastly, Sect. 3.7 is dedicated to the validation of the thermodynamic and transport properties obtained from this mixed formulation.

3.2 Initial Assumptions As a general rule, properties that have a recognized influence on the pressure and temperature profiles of the OPW, such as the mass density of the liquid phase, for instance, require a higher level of sophistication in their calculation, whereas those with a low impact (such as the viscosity of the gas, which is almost negligible when compared to that of the oil) can experience more expedited analyses. Even the routine of laboratory tests ends up guiding certain choices: empirical correlations, robust and less susceptible to small uncertainties in the input parameters, are useful in estimates that lack experimental data to support them, a situation in which compositional formulations run the risk of not functioning properly. Before embarking on the planning of the model, however, it is essential to have an overview of the representative properties that it should be able to emulate. For this study, a database consisting of ten real samples of pre-salt oil from the Campos Basin was selected, all originating from the southern region of Espírito Santo. The attributes of these samples are summarized in Table 3.1. The classification of the reservoirs presented here, within the categories mentioned in Sect. 1.3, is indeed a straightforward task. Despite their relatively high average temperature of approximately 402 K, these accumulations still remain well below the critical temperature of the mixtures contained within them, which is approximately 473 K.1 Therefore, these are not nearly critical oils, and a series of facts support this assertion. For instance, according to Ahmed (2016, p. 41), this latter large group is restricted to mixtures where the fractions between C2 and C6 account for at least 35% 3 3 /mstd . These of the overall molar composition and have a GOR greater than 530 mstd criteria represent more than twice the average values of the considered samples.

1

Simulated values, as there is no knowledge of experimental data in this regard.

3.2 Initial Assumptions

87

Table 3.1 Representative properties of the ten oil reservoir samples from the pre-salt layer selected for the fluid model development Property Min. Avg. Max. Reservoir temperature, K Bubble-point pressure at Tr es , MPa 3 Oil volume formation factor at Pb , m3 /mstd −3 Oil dynamic viscosity at Pb , 10 Pa · s 3 /m 3 Gas-oil ratio, mstd std Mole fraction of C2 -C6 in the mixture, % Dead oil density, ◦ API Dead oil molecular mass, g/mol Mole fraction of C20+ in the dead oil, % Relative density of the natural gas Mole fraction of CO2 in the natural gas, %

395 38 1.78 0.30 220 14.6 28.4 240 29.4 0.77 3.7

402 40.5 1.87 0.50 250 16.7 29.5 260 37.5 0.80 4.0

407 44 2.05 0.70 290 18.1 30.9 290 43.0 0.86 4.5

On the other hand, the mathematical treatment typically applied to black oils could result in an oversimplified depiction of the phase change phenomena involved. Table 3.1 reveals that the liquid content of such mixtures expands to twice its original volume under reservoir conditions compared to surface conditions. In other words, the release of gas in solution is so pronounced that, ultimately, the volume of oil accounts for only half of what it once was, imposing drastic changes to the composition of both phases. This behavior could never be faithfully reproduced by empirical correlations based on the assumption that the molecular mass of the volatile component remains constant throughout the entire process. Therefore, it is understood that the modeling of the thermodynamic properties of the oil-gas system demands a compositional approach. Among the equations of state discussed in Sect. 2.2, the Soave-Redlich-Kwong and Peng-Robinson equations prove to be naturally more suitable for this task for several reasons: they are the two most widely used approaches in the petroleum industry today, both easily implementable in computational algorithms, and capable of accurately describing phase equilibrium. On the other hand, the Valderrama-Patel-Teja equation of state is commonly associated with natural gas or gas condensate reservoirs (Danesh et al., 1991), where Tr es > Tc . After all, the inclusion of a third parameter in the formulation of black oil mixtures is justified more by the extra care in computing the density of the liquid phase, facilitated by the volume translation of Peneloux et al. (1982), rather than by the initiative to faithfully represent its critical point. The expression proposed by Twu et al. (1995) for the estimation of α(T ) also loses many of its virtues in this context, as simpler functions, such as the one proposed by Soave (1972), will perform almost identically. Additionally, the work of Twu et al. (1995) is hindered by the presence of a dangerous discontinuity in the second derivative of α(T ) under certain circumstances (Neau et al., 2009; Privat et al., 2015), which can impact specific heat predictions.

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The characterization of the mixture plays a central role in the development of the model. For the splitting of the heavy fraction, the method proposed by Whitson (1983) was chosen, which is known for its versatility and effectiveness. The correlations of the critical properties and acentric factor of the SCN fractions employed here should not only provide satisfactory results but also demonstrate unquestionable theoretical rigor, manifested in well-behaved and applicable trends over a wide range. These choices minimize the need for intervention in the binary interaction parameters, which correspond to the values suggested by Whitson and Brulé (2000, Sect. 4.3), as shown in Table 2.3. Regarding the modeling of the transport properties of the ten samples that constitute the scope of the present study, it is believed that a black-oil approach is the most suitable alternative. This can be observed, for instance, in the typical rheological behavior of these oils, which gives them a viscosity of approximately 5 × 10−4 Pa·s at reservoir pressure and temperature, and a value up to 200 times higher under Standard conditions. Compositional methodologies would face extreme difficulty in reproducing such variations, whereas certain empirical studies are capable of doing so without major difficulties. As for the viscosity of natural gas and the liquid-gas interfacial tension, these quantities have a lesser effect on flow simulations in OPWs, and perhaps due to this reason, experimental values for these parameters are rarely available. This combination of factors further supports the use of black oil correlations. The inclusion of formation water in the calculation of phase equilibrium in a mixture needs to be carefully considered, given the high level of sophistication that it typically demands from the compositional model. H2 O molecules have a notably angular geometry, which imparts significant polarity and establishes strong electrostatic interactions among themselves through so-called hydrogen bonds. When compared to methane gas, which has a molecular mass of 16.04 g/mol that differs little from its own mass of 18.02 g/mol, the critical temperature of pure water is three times higher, and the critical pressure is five times higher (Pedersen et al. 2014, p. 395). These numerous peculiarities, which are also reflected in the high values of specific heat, normal boiling temperature, and enthalpy of vaporization for water, are difficult to reproduce using cubic equations of state. Despite the existence of mixing rules, temperature-dependent functions α(T ), and specially designed binary interaction parameters for aqueous systems (Whitson and Brulé 2000, Sect. 9.9), the results are not always encouraging. When considering the multitude of salt ions in the massive proportions commonly found in pre-salt reservoirs, one can grasp the incredible effort involved in employing this strategy. On the other hand, even if it were successful, the complete compositional modeling of the fluid would hardly bring appreciable changes. Let us consider, for prac3 3 /mstd and BSW = 10%. Referring tical example, a sample with GOR = 250 mstd to the work of Culberson and McKetta (1951), the solubility of methane in water at 3 3 /mstd , resultPr es = 50 MPa and Tr es = 395 K is found to be approximately 5.6 mstd 3 3 ing in an approximate increase of 0.6 mstd /mstd in the original GOR. The same analysis, when repeated for ethane according to Culberson and McKetta (1950), incurs 3 3 /mstd . From propane onwards, the an even more subtle increment, equal to 0.2 mstd

3.3 New Expressions for the Angular Coefficient of the α(T ) Function

89

solubility of hydrocarbons becomes negligible. Among the contaminants, carbon dioxide demonstrates the greatest affinity with the aqueous phase, and, as shown in the experimental study by Takenouchi and Kennedy (1964), could induce a growth 3 3 /mstd in the GOR under the prevailing conditions. Adding up the three of 4.2 mstd 3 3 /mstd , a value that corresponds to only 2% of that obtained terms, one obtains 5.0 mstd in the laboratory for the dehydrated sample. Furthermore, the water vapor contained in the natural gas, once liquefied, would increase the BSW of the mixture to 10.13%, according to the correlation proposed by Bahadori et al. (2009). All these variations, being so small, end up blending with the inherent uncertainty of the measurement apparatus. Given the aforementioned, it is deemed reasonable to approach the behavior of water in the formation as that of an inert substance, which does not participate in the phase equilibrium of the system. Its thermodynamic and transport properties will be represented using empirical correlations.

3.3 New Expressions for the Angular Coefficient of the α(T ) Function In the literature review presented in Chap. 2, as well as in all the thermodynamics books consulted during its development, an important detail goes unnoticed. Let us consider the case of the pseudocomponent C100 , which often appears among the SCN fractions resulting from the characterization process of the petroleum sample. By substituting NC = 100 into Eqs. (2.134) and (2.137), both recommended by Riazi and Al-Sahhaf (1996), an acentric factor of 3.25 is obtained for the molecule. Now, this value significantly exceeds the validity range of the α(T ) functions for all the cubic equations of state discussed so far. What can be expected regarding their performance under these conditions? Undoubtedly, the exponential decay of the SCN fractions, which presumably constitute the residual portion of the sample, restricts the practical scope of the problem to a small mole fraction of the whole. Consequently, after applying the mixing rules, it is possible that the predictions of the fluid model would be minimally affected by it. On the other hand, the opportunity for improving the physical basis underlying the results, coupled with the expectation of providing a tangible contribution to future works, makes it a relevant object of study. Therefore, in the following, new expressions for the slope coefficient of the function α(T ) by Soave (1972) will be developed, which are applicable to the two most popular equations of state in the petroleum industry.

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3.3.1 Consistency Test and Extrapolation of Current Expressions After the publication of the Soave-Redlich-Kwong and Peng-Robinson equations, some researchers began to propose improvements in the expressions of the coefficient m. In addition to the one engendered by Robinson and Peng (1978), which also falls into this group, it is worth mentioning the study conducted by Graboski and Daubert (1978), based on a compilation of 4443 saturation pressures from the API Technical Databook covering both paraffins (up to the pseudocomponent C20 , for which ω ≈ 0.9) and naphthalene and aromatics (only up to C10 , where ω ≈ 0.5). Stryjek and Vera (1986) continued the topic in another important work, which included over 2300 experimental values of Psat for 90 pure substances, including hydrocarbons, inorganic compounds, alcohols, ketones, and ethers. In both cases, repeating the strategy that had already been seen in Sect. 2.2, the regression analysis employed polynomial functions. Examples like these highlight the need for a highly diverse database in order to better describe the relationship between m and the acentric factor. In this regard, the state of the art undoubtedly encompasses the research conducted by Pina-Martinez et al. (2019). Supported by an outstanding collection of saturation pressure values, vaporization enthalpies, and specific heat of saturated liquids encompassing 1721 species, the authors determined the optimal coefficient m for each of them, both in the Soave-Redlich-Kwong and Peng-Robinson equations of state, resulting in two new third-degree polynomials. However, more significant than the expression itself was the careful reproduction of all the results from this survey, presented in two 50-page tables. The initiative undertaken by Pina-Martinez et al. (2019) not only allows for the exact replication of the adjustments obtained in that work but also enables testing the behavior of the aforementioned functions with respect to the new dataset. The mean absolute percentage error (MAPE) was calculated for each of these functions in two distinct groups: one consisting of substances with low or moderate acentric factors (a total of 1111 components), and another where the parameter is more pronounced (607 components). The cutoff value, somewhat arbitrary, was set at ω = 0.5. Only three points were excluded from the analysis: the lower extreme and the two upper extremes, and the reasons for this will be discussed in Sect. 3.3.2. Consequently, as shown in Table 3.2, an improvement can be observed in the predictions of Graboski and Daubert (1978) for the Soave-Redlich-Kwong equation of state, as compared to the expression by Soave (1972), in the second group. On the other hand, for the polynomial by Stryjek and Vera (1986), dedicated to the Peng-Robinson equation, the deviations increase in the first region, which is better described by the original form proposed by Peng and Robinson (1976), and remain nearly unchanged for ω ≥ 0.5, where the modification by Robinson and Peng (1978) performs equally well. Clearly, deriving from the same database, the cubic functions presented by Pina-Martinez et al. (2019) better adapt to all scenarios. Extrapolating any of the equations listed in Table 3.2 requires great caution. As is well known, polynomial-based regressions can exhibit inconsistent trends when

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91

Table 3.2 Mean absolute percentage errors of the different expressions for calculating the coefficient m for the database compiled by Pina-Martinez et al. (2019) Expression Equation of state MAPE, % ω < 0.5 ω ≥ 0.5 Soave (1972) Graboski and Daubert (1978) Pina-Martinez et al. (2019) Peng and Robinson (1976) Robinson and Peng (1978) Stryjek and Vera (1986) Pina-Martinez et al. (2019)

Soave-Redlich-Kwong Soave-Redlich-Kwong Soave-Redlich-Kwong Peng-Robinson Peng-Robinson Peng-Robinson Peng-Robinson

0.90 0.90 0.89 1.33 1.41 1.40 1.04

0.71 0.68 0.53 2.37 0.55 0.54 0.48

Fig. 3.1 Behavior of the expressions for calculating the coefficient m in the Soave-Redlich-Kwong equation of state when extrapolated to ω = 3

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3 Developing a Fluid Model

Fig. 3.2 Behavior of the expressions for calculating the coefficient m in the Peng-Robinson equation of state when extrapolated to ω = 3

applied outside their validation range. It was decided to investigate this suspicion in the range given by 0 ≤ ω ≤ 3, which encompasses the vast majority of commonly encountered components (or assumed to be present) in reservoir fluids from the pre-salt layer. Analyzing Figs. 3.1 and 3.2, derived from this process, the following observations arise: 1. The quadratic expression developed by Peng and Robinson (1976) reaches a maximum at ω = 2.86, after which it starts to decrease. The same behavior is observed in the works of Soave (1972) and Graboski and Daubert (1978), but outside the region of interest. There is no practical basis for this reversal of trend in the coefficient m; 2. The functions suggested by Pina-Martinez et al. (2019), both of third degree, result in the growth of m with a negative second derivative up to a certain point, and then an increase in the rate thereafter. A similar trend is observed in the works of Robinson and Peng (1978) and Stryjek and Vera (1986), although it is almost imperceptible. Once again, this behavior lacks experimental confirmation. It becomes evident, therefore, that there is currently a lack of a set of equations capable of accurately reproducing the value of the angular coefficient of the function

3.3 New Expressions for the Angular Coefficient of the α(T ) Function

93

α(T ) for a wide range of substances. Furthermore, this set of equations should be able to be extrapolated to other conditions without compromising the quality of the estimates.

3.3.2 Fitting of New Functions The calculation sequence for the coefficient m, typically employed in thermodynamic simulators utilizing the Peng-Robinson equation of state, can be summarized as follows: Firstly, the function’s domain is divided into two parts using ω = 0.49 as the criterion. If the component under consideration belongs to the first part, that is, if its acentric factor is considered low or moderate, Eq. (2.26) proposed by Peng and Robinson (1976) is applied. Conversely, for higher values, Eq. (2.27) from Robinson and Peng (1978) becomes valid. Although trivial, this procedure supports the idea that, under certain circumstances, it is more advantageous to partition the phenomenon of interest and represent it through a set of equations rather than seeking a single intricate expression capable of describing the entire database.

Fig. 3.3 Fitting of coefficient m for the Soave-Redlich-Kwong equation of state in the interval 0 ≤ ω ≤ 0.5

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3 Developing a Fluid Model

Fig. 3.4 Fitting of coefficient m for the Peng-Robinson equation of state in the interval 0 ≤ ω ≤ 0.5

With this intention in mind, it was decided to initially perform the regression of the m values compiled by Pina-Martinez et al. (2019) only within the range 0 ≤ ω ≤ 0.5, which encompasses the C1 to nC6 paraffins, the contaminants N2 , CO2 , and H2 S, as well as the SCN fractions up to C10 in oil mixtures. The exotic element mercury (Hg), for which ω = −0.16, was excluded from the list. As shown in Figs. 3.3 and 3.4, the quadratic fits proved to be fully satisfactory for both the Soave-Redlich-Kwong and Peng-Robinson equations of state, making it unnecessary to resort to higher-degree polynomials. It is also worth noting that there will be no room for extrapolation of these functions, as the adjacent region falls within the scope of the second regression analysis. The attention is now focused on the behavior of m as the molecular eccentricity increases. Returning once again to the database of Pina-Martinez et al. (2019), the two substances2 with the highest property value were excluded from the list. In both cases, this was due to the lack of information regarding the enthalpy of vaporization and the specific heat of the saturated liquid (which reduces the reliability of the coefficient m estimated by the authors), combined with their deviation from the others on the 2

Hexacosamethyldodecasiloxane (C26 H78 O11 Si12 , ω = 1.31) and tricaprin (C33 H62 O6 , ω = 1.40).

3.3 New Expressions for the Angular Coefficient of the α(T ) Function

95

Fig. 3.5 Fitting of coefficient m for the Soave-Redlich-Kwong equation of state in the range 0.2 ≤ ω ≤ 1.24

abscissa axis (which would make the adjustment of ω highly susceptible to potential fluctuations in that region). After establishing the same lower limit adopted by Robinson and Peng (1978), the range 0.2 ≤ ω ≤ 1.24 was examined for the new regressions. Within this region, exponential functions enable convincing adjustments, both for the Soave-RedlichKwong equation of state (SRK) and the Peng-Robinson equation of state (PR), as they achieve determination coefficients identical to those obtained by third-degree polynomials, even with one fewer parameter. The results are shown in Figs. 3.5 and 3.6. Another highly favorable aspect of this type of expression lies in its monotonically increasing and asymptotic behavior for 0 ≤ ω ≤ ∞, which will undoubtedly result in more predictable and consistent extrapolations. As it may have been observed, the values between 0.2 ≤ ω ≤ 0.5 are common to both analyses, such that both the quadratic and exponential functions of each equation of state are applicable within this range. However, in order to avoid small discontinuities, the most suitable transition point from one function to another logically lies at an intersection point, such as the one found at ω = 0.42. Thus, the final form of the coefficient m calculation expressions for the Soave-Redlich-Kwong and Peng-Robinson equations are given, respectively, by:

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Fig. 3.6 Fitting of coefficient m for the Peng-Robinson equation of state in the range 0.2 ≤ ω ≤ 1.24

m SRK

 0.48671 + 1.5530ω − 0.18308ω 2 , ω ≤ 0.42 = 14.7 − exp (2.6533 − 0.10417ω) , ω > 0.42, 

m PR =

(3.1)

0.39485 + 1.4779ω − 0.20035ω 2 , ω ≤ 0.42 12.3 − exp (2.4759 − 0.11751ω) , ω > 0.42.

(3.2)

Repeating the testing methodology described in Sect. 3.3.1, it is observed that Eqs. (3.1) and (3.2) exhibit performance equivalent to that of the best expression in each category. The associated mean absolute percentage errors for each of them are

Table 3.3 Mean average percentage errors of Eqs. (3.1) and (3.2) in the calculation of the coefficient m for the database raised by Pina-Martinez et al. (2019) Expression Equation of state MAPE, % ω < 0.5 ω ≥ 0.5 Equation (3.1) Equation (3.2)

Soave-Redlich-Kwong Peng-Robinson

0.90 1.05

0.55 0.49

3.4 New Correlations for the Characterization of SCN Fractions

5

97

Database Eq. (3.1) Soave (1972)

4

mSRK

3

2

1

0 0.0

0.5

1.0

1.5 ω

2.0

2.5

3.0

Fig. 3.7 Behavior of the new expression for calculating the coefficient m in the Soave-RedlichKwong equation of state when extrapolated to ω = 3

provided in Table 3.3. Furthermore, the extrapolation up to ω = 3, as illustrated in Figs. 3.7 and 3.8, proved to be highly plausible in both cases, smoothly extending the predictions made for the known region and exhibiting a consistent behavior similar to that of the polynomials proposed by Soave (1972) and Robinson and Peng (1978).

3.4 New Correlations for the Characterization of SCN Fractions The selection of calculation expressions for the critical properties and acentric factor of SCN fractions that best suit a specific oil field represents a highly important task. After all, since these fractions originate solely from the similarity in the normal boiling temperature of their constituents, all the physicochemical characteristics they exhibit primarily depend on the PNA distribution established during the primary accumulation. Conducting experimental surveys would yield specific values for that particular sample. Furthermore, as is well known, the breaking of carbon-carbon bonds in the chains, triggered by an increase in temperature, a phenomenon known as cracking, would inevitably degrade the heavier molecules before their critical point could be reached. That is why the reported values in the literature rarely extend

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3 Developing a Fluid Model

4

Database Eq. (3.2) Robinson and Peng (1978)

mPR

3

2

1

0 0.0

0.5

1.0

1.5 ω

2.0

2.5

3.0

Fig. 3.8 Behavior of the new expression for calculating the coefficient m in the Peng-Robinson equation of state when extrapolated to ω = 3

beyond the C18 fraction (Riazi 2005, p. 33), thus necessitating the use of correlations from that point onward. Due to this practical limitation, the predictions provided by each author often diverge significantly when faced with longer hydrocarbon chains, leading to substantial differences in the characterization of higher molecular mass pseudocomponents. In the event of an incorrect tendency being assumed for Pc , Tc , and ω, the bubblepoint pressure curve of the mixture and the entire volumetric behavior of the oil-gas system calculated by the thermodynamic model will be irreparably compromised. However, what ultimately determines the suitability of a correlation over others? There is no way to answer such a question without first establishing an evaluation criterion. The most straightforward criterion consists of performing several characterizations of the same sample, each according to the recommendations of a given study, in order to find the one that best reproduces the laboratory experiments. This approach is adopted, for example, by Al-Meshari and McCain (2006). However, this type of test has three clear limitations: (i) its conclusions apply only to the analyzed sample and do not provide any information about what to expect for new mixtures, (ii) there is no evidence regarding the role played by the equation of state and whether the ranking of the best studies would remain the same after a possible change, and (iii) it allows the characteristic PNA distribution of the sample to favor or penalize one correlation over the others, when a simple adjustment of the curves would be sufficient to correct the deviations. Furthermore, the pursuit of greater

3.4 New Correlations for the Characterization of SCN Fractions

99

accuracy cannot be absolutized: considering that these methodologies are validated up to approximately component C20 , but are often extrapolated to C100 , it might be more sensible to prioritize those with a higher likelihood of maintaining the model’s good performance under conditions other than those tested experimentally. In the current scenario, numerous correlations remain available for the characterization of SCN fractions, while lacking means to validate them. It is based on this need that the transformation of the volume translation parameter c, calculated for the Soave-Redlich-Kwong and Peng-Robinson equations of state, was envisioned as a new indicator of the quality of the obtained predictions. This transformation will be further elaborated in the following discussion.

3.4.1 Consistency Test and Extrapolation of Current Correlations Let us consider the correlations by Cavett (1962), Kesler and Lee (1976), Riazi and Daubert (1987), Riazi and Al-Sahhaf (1996), Pedersen et al. (2004) (with the coefficients of the Peng-Robinson equation of state only3 ) and Bahadori et al. (2016). The first correlation, which is the oldest and highly empirical, was developed by the American Petroleum Institute to estimate the critical properties of SCN fractions up to C45 (Mw = 626 g/mol), approximately. On the other hand, the last correlation represents a promising alternative, although it is still not widely disseminated. The expressions of these correlations will not be presented here due to brevity. The remaining correlations have already been discussed in Sect. 2.5. By applying these six approaches simultaneously to calculate the critical pressure, the profiles illustrated in Fig. 3.9 are obtained. It is noticeable that the correlation of Cavett (1962) is the only one that violates the behavior commonly attributed to SCN fractions, which assumes a permanent tendency of the property value to decrease with increasing molecular mass. The remaining correlations, although consistent, yield predictions that deviate to such an extent that, in the pseudo-component C100 (Mw = 1396 g/mol), Pc varies between 0.10 and 1.12 MPa. A similar situation arises for the critical temperature, as shown in Fig. 3.10, where the results for the same fraction range from 986 to 1645 K. However, none of the studies can be deemed inconsistent in this case, as they all adhere to the expected trend of increasing Tc with Mw throughout the analyzed range. Figure 3.11, on the other hand, reveals a significant issue with the expression for the acentric factor proposed by Pedersen et al. (2004), which changes its trend starting from the C40 fraction (Mw = 556 g/mol) and defines a rather unusual profile. The remaining correlations all correctly indicate the directly proportional relationship between ω and Mw , although they once again exhibit noticeable divergences.

3

The adjustment focused on the Soave-Redlich-Kwong equation was also tested, but considering its minor influence on the results, it was decided to omit it from the text.

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4

Cavett (1962) Kesler and Lee (1976) Riazi and Daubert (1987) Riazi and Al-Sahhaf (1996) Pedersen et al. (2004) Bahadori et al. (2016)

Pc , M P a

3

2

1

0 0

300

600 900 Mw , g/mol

1200

1500

Fig. 3.9 Critical pressure of the SCN fraction as a function of its molecular mass, according to the different correlations

1400

1200

Tc , K

1000

800 Cavett (1962) Kesler and Lee (1976) Riazi and Daubert (1987) Riazi and Al-Sahhaf (1996) Pedersen et al. (2004) Bahadori et al. (2016)

600

400 0

300

600 900 Mw , g/mol

1200

1500

Fig. 3.10 Critical temperature of the SCN fraction as a function of its molecular mass, according to the different correlations

3.4 New Correlations for the Characterization of SCN Fractions

3.5

101

Kesler and Lee (1976) Riazi and Al-Sahhaf (1996) Pedersen et al. (2004) Bahadori et al. (2016)

3.0 2.5

ω

2.0 1.5 1.0 0.5 0.0 0

300

600 900 Mw , g/mol

1200

1500

Fig. 3.11 Acentric factor of the SCN fraction as a function of its molecular mass, according to the different correlations

Unfortunately, except in clearly incongruous cases, a simple inspection of Figs. 3.9, 3.10 and 3.11 does not allow one to clearly identify which correlations would have a higher probability of reproducing the real and true behavior of SCN fractions. It is in this context that the volume translation parameter comes into play once again: by applying it in such a way that the equation of state matches the density values calculated using Eq. (2.136) (or any other preferred equation) for each of the pseudocomponents, profiles of z c as a function of Mw emerge, whose physical consistency can be quickly verified. Obviously, it is not expected that Eqs. (2.35) and (2.37) used in this analysis faithfully reproduce the critical point of the substances in question, primarily because, as a rule, it is far removed from the Standard conditions where the adjustment is made. Nevertheless, it would be interesting if the results obtained in this way exhibited trends consistent with those described by Lee and Kesler (1975) and Pitzer et al. (1955), among other authors (Riazi 2005, p. 63), who argue that there is an inverse relationship between the critical compressibility factor of the SCN fraction and its molecular mass. As depicted in Fig. 3.12, the majority of the tested correlations4 attribute a plausible behavior to the Soave-Redlich-Kwong equation, but only initially. As heavier fractions are encountered, an inevitable increase in z c occurs. The sole exception is 4

In cases where no expression for the acentric factor was available, the correlation proposed by Kesler and Lee (1976) was employed.

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0.45

Cavett (1962) Kesler and Lee (1976) Riazi and Daubert (1987) Riazi and Al-Sahhaf (1996) Pedersen et al. (2004) Bahadori et al. (2016)

zc,SRK

0.40

0.35

0.30

0.25 0

300

600 900 Mw , g/mol

1200

1500

Fig. 3.12 Variation in the critical compressibility factor of the SCN fraction, resulting from the volume translation performed for the Soave-Redlich-Kwong equation of state, as a function of its molecular mass

found in the work by Riazi and Al-Sahhaf (1996), which predicts decreasing values throughout the entire considered range. By following the same procedure for the Peng-Robinson equation, the profiles illustrated in Fig. 3.13 are obtained, and the previous observations remain unchanged. This demonstrates that the aforementioned inconsistencies do not originate from the cubic equation of state but rather from the characterization process. Based on this information, the methodology developed by Riazi and Al-Sahhaf (1996) stands out as the best option among the tested alternatives for calculating the values of Pc , Tc , and ω for SCN fractions. However, it should be noted that this methodology also has certain deficiencies. For instance, Pan et al. (1997) argue that the generated critical pressure values are somewhat low, and the acentric factors for the aromatic portion are excessively high. They proposed modifications to overcome the weaknesses of these estimates, which were later adopted by the original authors (Riazi et al., 2004). Indeed, it is possible that the determination of the coefficients expressed in Eqs. (2.132)–(2.134) may have been compromised due to two difficulties: (i) inferring the PNA distribution of the fraction based solely on its density, and (ii) accurately and unambiguously weighting the critical properties and acentric factor assigned to each component class. As a result, it will be shown in Sect. 3.4.3 that simulating flash and differential liberation tests for the set of reservoir samples from the pre-salt layer using this characterization scheme is impractical.

3.4 New Correlations for the Characterization of SCN Fractions

0.40

Cavett (1962) Kesler and Lee (1976) Riazi and Daubert (1987) Riazi and Al-Sahhaf (1996) Pedersen et al. (2004) Bahadori et al. (2016)

0.35

zc,PR

103

0.30

0.25

0.20 0

300

600 900 Mw , g/mol

1200

1500

Fig. 3.13 Variation in the critical compressibility factor of the SCN fraction, resulting from the volume translation performed for the Peng-Robinson equation of state, as a function of its molecular mass

3.4.2 New Fitting of Exponential Functions A relatively straightforward approach to mitigate the aforementioned issues consists of performing new regressions for the coefficients a, b, and c of Eq. (2.131) for each of the properties in question. These regressions should be based on estimates obtained through other correlations, which would serve as a synthetic database.5 Maintaining the previously established values of θ∞ , as proposed by Riazi and AlSahhaf (1996) in Eqs. (2.132)–(2.134), is desirable in this case as it largely preserves the characterization of the heavier SCN fractions. On the other hand, there will also be little variation in the results towards the lower end, since all the expressions presented in Sect. 3.4.1 converge significantly as the molecular mass decreases. Therefore, the main influence of the new fitting will occur in the intermediate region of Mw . Therefore, the API Technical Data Book (Daubert and Danner, 1997) recommends the works of Riazi and Daubert (1987) for the calculation of critical pressure and temperature, and Kesler and Lee (1976) for the acentric factor within the C7-C22 range (Mw ≤ 300, g/mol). Upon a brief inspection of Figs. 3.12 and 3.13, it can be observed that the behavior of z c for this combination still exhibits physical validity, as 5

That is, a dataset composed of data that has not been obtained experimentally but rather derived from a mathematical model considered representative of the phenomenon.

104

3 Developing a Fluid Model

the minimum point is located at Mw ≈ 450 g/mol. These data were previously used in Chap. 2 and are presented in Table 2.7. The best fit is achieved by the following equations:   Pc = exp 9.4703 − 3.5668Mw0.115 ,

(3.3)

  Tnb = 1.2 − exp 0.52134 − 0.56281Mw0.157 , Tc

(3.4)

  ω = −0.3 + exp −5.8924 + 3.4234Mw0.1 .

(3.5)

Equations (3.3)–(3.5) incur average absolute percentage errors of 0.17% for the critical pressure, 0.11% for the critical temperature, and 0.25% for the acentric factor. It is important to emphasize that the internal consistency of the method, as demonstrated in Sect. 2.5 employing the original coefficients, has been rigorously maintained. Specifically, at a molecular weight of 4813 g/mol, one obtains both Pc = Patm and Tnb /Tc = 1. The quality of the fittings can also be observed in Figs. 3.14, 3.15 and 3.16. It can be seen that the new functions faithfully reproduce the values indicated by the correlations of Riazi and Daubert (1987) and Kesler and Lee (1976), as compiled in

3.5

Riazi and Al-Sahhaf (1996) Regression data Eq. (3.3)

3.0

Pc , MPa

2.5 2.0 1.5 1.0 0.5 0.0 0

300

600 900 Mw , g/mol

1200

1500

Fig. 3.14 Critical pressure of the SCN fraction as a function of its molecular mass, according to the correlation of Riazi and Al-Sahhaf (1996) and fitting proposed in Eq. (3.3)

3.4 New Correlations for the Characterization of SCN Fractions

105

1200

Tc , K

1000

800

600 Riazi and Al-Sahhaf (1996) Regression data Eq. (3.4)

400 0

300

600 900 Mw , g/mol

1200

1500

Fig. 3.15 Critical temperature of the SCN fraction as a function of its molecular mass, according to the correlation of Riazi and Al-Sahhaf (1996) and fitting proposed in Eq. (3.4)

3.5

Riazi and Al-Sahhaf (1996) Regression data Eq. (3.5)

3.0 2.5

ω

2.0 1.5 1.0 0.5 0.0 0

300

600 900 Mw , g/mol

1200

1500

Fig. 3.16 Acentric factor of the SCN fraction as a function of its molecular mass, according to the correlation of Riazi and Al-Sahhaf (1996) and fitting proposed in Eq. (3.5)

106

3 Developing a Fluid Model

0.33

Soave-Redlich-Kwong Peng-Robinson

0.32

zc

0.31

0.30

0.29

0.28 0

300

600 900 Mw , g/mol

1200

1500

Fig. 3.17 Variation in the critical compressibility factor of the SCN fraction, resulting from the volume translation performed for the Soave-Redlich-Kwong and Peng-Robinson equations of state and the use of Eqs. (3.3)–(3.5), as a function of its molecular mass

Table 2.7, up to Mw = 300 g/mol. Furthermore, as the molecular mass increases, they begin to exhibit increasingly similar trends to those presented in the work of Riazi and Al-Sahhaf (1996). This results in the critical compressibility factor of SCN fractions associated with volume translation maintaining the continuous and gradual decay as expected, whether in the Soave-Redlich-Kwong or Peng-Robinson equations of state, as illustrated in Fig. 3.17.

3.4.3 Results and Comparisons Despite all the aspects raised so far, it is still unknown, among the various available alternatives, which correlations have the greatest potential for application in reservoir fluids from the pre-salt layer. To identify them, laboratory test readings are available for each of the ten samples that make up the database used in this book. Of course, such a voluminous amount of information offers little room for visual comparisons, which would be subjective and, above all, terribly tedious. Therefore, it was chosen to base the calculations involving flash liberation tests on three scalar quantities:

3.4 New Correlations for the Characterization of SCN Fractions

107

Mass fraction of the released gas (g ): denotes the ratio between the mass of natural gas derived from depressurization and the initial mass of the sample; Mass density of the dead oil (ρod ): measured under Standard conditions, it is proportional to the relative density assumed by the respective phase in the test; Molecular weight of the released gas (Mwg ): has a direct relationship with the reported relative density of natural gas. The investigation of differential liberation tests will be limited to six quantities: Bubble-point pressure (Pb ): obtained at the temperature at which the test is conducted, usually equal to or close to reservoir conditions; Oil mass density at the bubble point (ρob ): always constitutes the lowest level of ρo observed during the experiment: below it, the loss of light fractions implies an increase in the mass density of the liquid phase, while above it, compressibility effects lead to the same outcome; Subsaturated oil isothermal compressibility (cosub ): the isothermal compressibility is given by c ≡ (1/ρ)(∂ρ/∂ P)T . Subsaturation, in turn, encompasses the entire region where the sample remains single-phase; Total mass fraction of the released gas (g ): denotes the ratio between the sum of the mass of natural gas removed at each depressurization stage and the initial mass of the sample. Its value can differ considerably from that determined in the flash liberation test, as it is the result of a succession of phase equilibrium conditions with different compositions; Mass density of the dead oil (ρod ): measured under Standard conditions, refers to the residual liquid phase that remains in the PVT cell after the last decompression stage; Thermal expansion coefficient of the dead oil (βod ): calculated based on the contraction exhibited by the oil phase when cooled from the test temperature to the Standard temperature. It is defined as β ≡ −(1/ρ)(∂ρ/∂T ) P . It is worth noting that, in both cases, the mass parameters (ρo , g ) were prioritized over their volumetric counterparts (Bo , Rs , etc.). The latter, although deeply ingrained in the mindset of petroleum engineers, stems from a combination of variables that could obscure important effects. In a manner analogous to the procedure described in Sect. 2.6, the ten samples underwent an initial splitting process of the heavy fraction, carried out according to the Whitson method, always with α = 1 and truncation at 95%. Subsequently, the characterization of the SCN fractions was performed using both the six sets of expressions from Sect. 3.4.1 and the new fitting presented in Sect. 3.4.2. From this combination of compositions and correlations, a total of 70 mixture representations were generated, each of which had their deviations computed in relation to the aforementioned tests. The results for the Soave-Redlich-Kwong equation of state are presented in Table 3.4. It is noteworthy that the work of Riazi and Al-Sahhaf (1996) provides noticeably less accurate predictions compared to the others. This can be attributed to errors associated with the determination of bubble-point pressure and isothermal

108

3 Developing a Fluid Model

Table 3.4 Mean absolute percentage errors for simulations of flash and differential liberation tests of ten pre-salt oil samples using the Soave-Redlich-Kwong equation of state and different correlations for critical properties and acentric factor of SCN fractions Correlation

MAPE flash, %

MAPE differential liberation, %

Avg

g

ρod

Mwg

Pb

ρob

cosub

g

ρod

βod

Cavett (1962)

4.13

0.29

3.68

17.8

0.70

36.5

2.75

0.59

25.0

10.2

Kesler and Lee (1976)

4.23

0.35

3.76

18.8

0.72

37.9

2.65

0.58

23.4

10.3

Riazi and Daubert (1987)

4.56

0.22

4.03

21.6

0.71

44.6

3.00

0.65

23.1

11.4

Riazi and Al-Sahhaf (1996)

6.08

2.05

5.21

45.4

4.48

106

6.16

1.39

12.0

20.9

Pedersen et al. (2004)

3.35

0.36

3.03

9.22

1.44

4.39

10.7

1.22

37.7

7.94

Bahadori et al. (2016)

4.75

0.19

4.17

17.4

0.67

35.7

3.20

0.67

25.2

10.2

Equations (3.3) to (3.5)

3.61

0.24

3.28

13.2

0.71

26.5

2.21

0.59

25.1

8.38

compressibility. On the other hand, the methodology proposed by Pedersen et al. (2004) and the newly adjusted exponential functions show the two best averages. The former possibly benefits from its close agreement with experimental readings of cosub , while the latter excels due to the regularity of its terms rather than any individual factor. Furthermore, it is evident that Eqs. (3.3)–(3.5) significantly outperform the performance achieved by Riazi and Daubert (1987), despite relying on his coefficients’ adjustments. Repeating the same procedure for the Peng-Robinson equation of state, the measured deviations are presented in Table 3.5. This time, the correlation proposed by Riazi and Al-Sahhaf (1996) yields mathematical models that are difficult to converge, thereby failing to simulate the differential liberation tests for two samples, which, in turn, prevents the calculation of their averages. The remaining studies, however, achieved values quite similar to the previous ones. Based on Tables 3.4 and 3.5, it is noteworthy to recognize the high precision demonstrated by the indirect adjustment proposed by Pedersen et al. (2004). Despite adopting somewhat disparate expressions for the critical properties and an unsupported approach for calculating the acentric factor of SCN fractions, this method appears to be the most suitable in this regard. On the other hand, Eqs. (3.3)–(3.5), proven to be highly effective in such simulations, exhibit a strong physical rigor. Among the methodologies tested herein, they represent the best alternative for characterizing samples of pre-salt petroleum reservoirs. Finally, it is also opportune to investigate the influence of the equation of state on the percentage errors listed in Tables 3.4 and 3.5. Considering that the formulas for the coefficient m employed in the simulations are based on a common database,

3.4 New Correlations for the Characterization of SCN Fractions

109

Table 3.5 Mean absolute percentage errors for simulations of flash and differential liberation tests of ten pre-salt oil samples using the Peng-Robinson equation of state and different correlations for critical properties and acentric factor of SCN fractions Correlation

MAPE flash, %

MAPE differential liberation, %

Avg

g

ρod

Mwg

Pb

ρob

cosub

g

ρod

βod

Cavett (1962)

4.37

0.25

3.91

18.1

0.64

21.8

3.12

0.62

37.4

10.0

Kesler and Lee (1976)

4.47

0.30

3.99

19.1

0.66

23.1

3.05

0.61

36.0

10.1

Riazi and Daubert (1987)

4.82

0.19

4.27

21.7

0.63

29.1

3.57

0.67

35.8

11.2

Riazi and Al-Sahhaf (1996)

6.79

1.69

5.78

–

–

–

–

–

–

–

Pedersen et al. (2004)

5.40

0.36

4.69

4.12

1.55

3.08

9.79

1.16

47.3

8.61

Bahadori et al. (2016)

4.98

0.18

4.39

17.6

0.65

20.9

3.81

0.69

37.5

10.1

Equations (3.3) to (3.5)

3.83

0.22

3.49

13.7

0.80

12.4

2.43

0.61

37.4

8.32

as discussed in Sect. 3.3, it can be inferred that the most empirical component of this type of model, which aims to describe the variation of the attraction term with temperature, now exhibits the same accuracy and reliability in both cases. Therefore, this can be considered one of the rare occasions in which the Soave-Redlich-Kwong and Peng-Robinson equations are compared, directly emphasizing the performance of their algebraic forms while disregarding the behavior of the function α(T ). It was deemed appropriate not to include the deviations calculated for the characterization of Pedersen et al. (2004), whose expressions for Pc , Tc , and ω of the SCN fractions vary according to the equation of state, thereby reintroducing experimental uncertainties. The same decision was made for the correlation of Riazi and Al-Sahhaf (1996) due to the previously reported convergence issues. The remaining works had their percentage errors combined by simple averaging, resulting in the values shown in Table 3.6. At first glance, one might presume, based solely on the overall values, that the two equations of state are equivalent. However, a closer inspection reveals, for instance, that the Peng-Robinson equation excels in calculating the isothermal compressibility, whereas the Soave-Redlich-Kwong equation is more effective in representing the thermal expansion coefficient. The latter also performs slightly better in parameters related to phase equilibrium, such as Pb , g , and Mwg . The volume translation, based on the results obtained for the dead oil in the flash liberation test, benefited both equations by aligning the predictions of mass density to such an extent that it becomes difficult to pinpoint the most suitable option among them. At present, it remains unknown which of the aforementioned parameters will have their deviations from experimental tests largely compensated for during the phase of adjusting the fluid models, and which will persist. It is even probable that the

110

3 Developing a Fluid Model

Table 3.6 Influence of the equation of state on the mean absolute percentage errors related to the simulations of flash and differential liberation tests of ten samples of pre-salt layer oil Equation of state

MAPE flash, %

MAPE differential liberation, %

Avg

g

ρod

Mwg

Pb

ρob

cosub

g

ρod

βod

Soave-RedlichKwong

4.26

0.26

3.78

17.7

0.70

36.2

2.76

0.62

24.4

10.1

Peng-Robinson

4.49

0.23

4.01

18.0

0.68

21.5

3.20

0.64

36.8

10.0

path taken in this iterative process is highly subject to the interdependencies existing among them, thereby favoring or hindering the search for an optimized model. The selection of the most suitable equation of state for simulating reservoir samples from the pre-salt layer of petroleum is therefore postponed to the next section, where the slight differences emphasized here will become clearer.

3.5 Methodology for Fitting the Proposed Fluid Model The process of sample characterization is an indispensable step in determining their thermodynamic properties; however, it is not sufficient. Its motivations, well established as early as the 1980s, arise from practical limitations imposed on laboratory tests: the significant difficulty in determining the critical properties and acentric factor of SCN fractions, and the complete ignorance of the species present in the heavy residual fraction. Thus, while empirical correlations and distribution models discussed in Sects. 2.5 and 3.4 have placed the petroleum industry on the map of compositional simulations, they are unable to translate all the nuances and peculiarities hidden behind such complex mixtures into values, and therefore, they will never correspond to the complete truth of the matter. Their predictions lack adjustments. In this context, the first question to address is: which parameters of the fluid model are truly capable of approximating the results of the simulations to the readings obtained from the flash and differential liberation tests? Undoubtedly, certain corrections will have an immediate effect on the calculation of important mixture properties, such as bubble-point pressure or mass density, while others might only increase the sluggishness of the optimization process. It becomes, therefore, extremely necessary to distinguish between them. Once selected, the second question arises: how can a multivariable fitting methodology be developed to gradually reduce the deviations presented by the thermodynamic model? The careful definition of the objective function itself is required, as it will decisively influence both the final solution and the path taken. These inquiries will be addressed, to the best of our ability, in the sections to follow.

3.5 Methodology for Fitting the Proposed Fluid Model

111

3.5.1 Introduction of Fitting Coefficients Among the various parameters that make up a model, those with lower reliability are always more subject to change. Therefore, when dealing with oil mixtures, any calculation involving SCN fractions deserves special attention. For instance, the critical pressure can be modified through a rearrangement of Eq. (3.3), resulting in the following expression:   Pc = exp a Pc − b Pc Mw0.115 ,

(3.6)

where: 

a Pc

ψ Pc − 1 = 9.4703 − 3.5668 ψ Pc b Pc =

 0.115 Mw,C , 7

3.5668 . ψ Pc

(3.7)

(3.8)

The manner in which the fitting coefficient ψ Pc is expressed in Eqs. (3.7) and (3.8) proves to be advantageous for several reasons: (i) for ψ Pc = 1, Eq. (3.6) recovers the parameters obtained through regression in Sect. 3.4.2, (ii) ψ Pc > 1 leads to a less pronounced decay of Pc with respect to Mw , while ψ Pc < 1 accentuates this trend, (iii) both versions yield the same value of 3.12 MPa for the pseudo-component C7 (Mw = 96 g/mol), which implies that ψ Pc exerts a greater influence on the heavy fractions, precisely where the uncertainty is greater, and (iv) in both cases, Pc → 0 as Mw → ∞, in absolute accordance with the original model proposed by Riazi and Al-Sahhaf (1996). The behavior of Eq. (3.6) within the interval 0.9 ≤ ψ Pc ≤ 1.1 is illustrated in Fig. 3.18. One must not overlook the fact that each newly assigned profile to the property corresponds to a distinct molecular mass range for which Pc = Patm . This corresponds to:  (Mw ) Pc =Patm =

a Pc − ln Patm b Pc

1/0.155 .

(3.9)

Following an identical reasoning to estimate the normal boiling temperature of SCN fractions, it is possible to rewrite Eq. (2.135) as:   Tnb = 1080 − exp aTnb − bTnb Mw2/3 ,

(3.10)

where, based on the same aforementioned assumptions, it is obtained that:   2/3 aTnb = 6.97996 + 0.01964 ψTnb − 1 Mw,C7 ,

(3.11)

bTnb = 0.01964ψTnb .

(3.12)

112

3 Developing a Fluid Model

3.5

ψPc = 0.9 ψPc = 1.0 ψPc = 1.1

3.0

Pc , MPa

2.5 2.0 1.5 1.0 0.5 0.0 0

300

600 900 Mw , g/mol

1200

1500

Fig. 3.18 Critical pressure of the SCN fraction as a function of its molecular mass and the fitting coefficient, according to Eq. (3.6)

The fitting imposed on Tnb directly affects the calculation of the critical temperature, given that Eq. (3.4) only describes the ratio between these quantities. However, if one wishes to preserve the value of Tc for the C7 fraction, as well as ensure the internal consistency of the model by achieving Tnb /Tc = 1 for Pc = Patm , the latter also needs to have its coefficients reassessed. Consequently, the following expression is obtained:   Tnb = 1.2 − exp aTc − bTc Mw0.157 . Tc

(3.13)

where: bTc =

0.157 2.13078 − 0.56281Mw,C 7 0.157 (Mw )0.157 Pc =Patm − Mw,C7

,

  0.157 aTc = 0.52134 + bTc − 0.56281 Mw,C . 7

(3.14)

(3.15)

Although it is derived through an indirect and hence more convoluted process, the modification of critical temperature profiles using Eqs. (3.10) and (3.13) operates in a predictable and highly secure manner, as demonstrated in Fig. 3.19 for the range 0.9 ≤ ψTnb ≤ 1.1.

3.5 Methodology for Fitting the Proposed Fluid Model

113

1200

Tc , K

1000

800

600 ψTnb = 0.9 ψTnb = 1.0 ψTnb = 1.1

400 0

300

600 900 Mw , g/mol

1200

1500

Fig. 3.19 Critical temperature of the SCN fraction as a function of its molecular mass and the fitting coefficient, according to Eqs. (3.10) and (3.13)

Repeating the same strategy for the acentric factor of the SCN fractions, Eq. (3.5) is now given by:   ω = −0.3 + exp −aω + bω Mw0.1 ,

(3.16)

0.1 , aω = 5.8924 + 3.4234 (ψω − 1) Mw,C 7

(3.17)

bω = 3.4234ψω .

(3.18)

where:

The effect of the fitting factor introduced in Eq. (3.16) is illustrated in Fig. 3.20 for the range 0.9 ≤ ψω ≤ 1.1. By utilizing it in conjunction with the recently added coefficients in the computation of critical pressure and temperature, it becomes feasible, ultimately, to adapt the characterization of SCN fractions for the different PNA distributions of petroleum mixtures, as well as to convincingly address a series of other particularities that may potentially arise. As mentioned in Chap. 1, the experimental determination of the molecular mass of the heavy fraction in a petroleum sample is commonly associated with uncertainties on the order of 20%. Consequently, it is reasonable to explore a wide range of values

114

3 Developing a Fluid Model

3.5

ψω = 0.9 ψω = 1.0 ψω = 1.1

3.0 2.5

ω

2.0 1.5 1.0 0.5 0.0 0

300

600 900 Mw , g/mol

1200

1500

Fig. 3.20 Acentric factor of the SCN fraction as a function of its molecular mass and the fitting coefficient, according to Eq. (3.16)

for the assigned model in order to improve predictions. The simplest approach to accomplish this is by establishing the following equation: Mw+ = ψ Mw+ Mw+,ex p .

(3.19)

Here, however, one must note that the composition of the mixture stated in the PVT test reports is originally obtained on a mass basis, so the mole fractions listed there already correspond to a secondary source of data (Christensen, 1999). Consequently, the correct procedure for adjusting Mw+ involves not one, but three steps. In the first step, starting from Mw+,ex p and the molecular masses of the pure substances and SCN fractions present, the observed mass fractions are recovered in the laboratory. Next, Eq. (3.19) is applied to recalculate the composition in the reverse direction, which will yield the new current molar distribution in the model. Another parameter that has been little explored so far, and which may prove capable of reducing deviations in the simulations, is the one that determines the shape of the gamma probability distribution adopted in the partitioning method of Whitson (1983), denoted in Chap. 2 as α. Moreover, since they are both dimensionless quantities with a central value of unity, there is no impediment to using them interchangeably, assuming: α = ψα .

(3.20)

3.5 Methodology for Fitting the Proposed Fluid Model

115

It is important to bear in mind that each modification in ψα requires a new fitting of the density of the SCN fractions present in the residual portion of the sample. To accomplish this, an expression based on Eq. (2.136) will be employed, which is given by:   γ = 1.07 − exp aγ − bγ Mw0.1 ,

(3.21)

  0.1 , aγ = 3.56073 + 2.93886 ψγ − 1 Mw,C 7

(3.22)

bγ = 2.93886ψγ .

(3.23)

where:

The behavior of Eq. (3.21) for the interval 0.9 ≤ ψγ ≤ 1.1, which is very similar to that of previous models, is depicted in Fig. 3.21. By means of this equation, a better representation of the density of the dead oil observed in the flash liberation test can also be achieved by introducing small alterations to the values of γ in the SCN fractions present therein. Thus, two distinct coefficients, ψγSCN and ψγ+ , are required. These adjustments prove to be quite straightforward in practice, given the availability of reliable experimental data in both cases, requiring only a few iterations

1.00

0.95

γ

0.90

0.85

0.80

0.75

ψγ = 0.9 ψγ = 1.0 ψγ = 1.1

0.70 0

300

600 900 Mw , g/mol

1200

1500

Fig. 3.21 Relative density of the SCN fraction as a function of its molecular mass and the fitting coefficient, according to Eq. (3.21)

116

3 Developing a Fluid Model

for a computational algorithm based on the Newton-Raphson method to find the optimal levels for each coefficient, free from interference by external factors. The involvement of the coefficients ψγ in the reproduction of the dead oil relative density does not hinder adjustments of mass density at other pressure and temperature levels. Such a role will be fulfilled by the volume translation parameter c of the equation of state, which, after being calibrated for standard conditions using Eq. (2.34), following the recommendation of Christensen (1999), can still undergo corrections. This implies performing, for each pseudocomponent:   1 1 . (3.24) − c = ψc ρ˜std,calc ρ˜std,ex p Naturally, the idea conveyed in Eq. (3.24) must be critically evaluated, considering its pros and cons, as the reduction in deviations of properties such as ρob will be partially offset (or even surpassed) by the increase in the error associated with ρod . In an attempt to enhance the effectiveness of the adjustment, it is proposed here to decompose it into two coefficients, ψcSCN and ψc pur e , allowing not only the manipulation of SCN fractions but also pure substances present in the mixture. For this second group, Eq. (3.24) is disregarded, and the multiplicative constant is directly applied to the values listed in Table 2.2.

3.5.2 Definition of the Minimum Set of Variables The incorporation of fitting coefficients into the thermodynamic model has introduced a total of nine additional variables. Considering that two of these variables, associated with the calculation of the density of the SCN fractions both outside and within the residual phase (ψγSCN and ψγ+ ), can be immediately quantified through a flash liberation test, there remain seven unknowns: four, once again, applicable to the SCN fractions (ψ Pc , ψTc , ψω , and ψcSCN ), two specific to the heavy fraction (ψ Mw+ and ψα ), and one designated for the defined components of the sample (ψc pur e ). It is now crucial to determine which properties of the mixture are influenced by each of these coefficients, and to what extent. Analyzing the task superficially, one might indeed believe that a simple sensitivity analysis in the well-known one variable at a time (OVAT) scheme (Antony, 2014, p. 1) would suffice to unveil such responses. However, in this case, while a particular parameter has its behavior tested, at what levels would the remaining parameters be maintained? The possibility of certain combinations altering (and even reversing) some trends cannot be neglected, leading to erroneous or, at the very least, incomplete conclusions. Thus, the so-called interaction effects must also be considered as part of the analysis. In light of this, it is understood that the application of the technique known as Design of Experiments (DoE) provides the best means of obtaining the desired responses. This technique was conceived by Sir Ronald A. Fisher (1890–1962) in

3.5 Methodology for Fitting the Proposed Fluid Model

117

the 1920s, as stated by Box et al. (2005, p. 397). At that time, Fisher was a temporary employee at the Rothamsted Experimental Station in England. As a young statistician, he sought to understand the influence of different types of fertilizers on crop productivity, which was highly susceptible to climatic factors such as average temperature and rainfall over each period. Fisher analyzed records spanning 90 years to investigate this relationship.6 Over the years, Fisher’s method for elucidating this problem has gained significant importance in engineering projects, both for the development of new products and the improvement of existing ones. It has paved the way for optimized solutions in terms of performance, reliability, and cost, while also providing great agility to the process. Today, it has expanded into areas such as marketing, market research, commerce, and services (Montgomery, 2012, p. 8). Although the primary purpose of this technique is readily apparent from its name (planning experiments in an optimized configuration), its versatility allows for its extension to certain “virtual experiments”, as proposed next. Let us consider the example of a property M of a substance, whose mathematical model employs algebraic expressions written in terms of variables A and B. In order to measure the effect exerted by these variables on the final result, a simple option would be to establish two levels for each variable (the upper level symbolized by the index “+” and the lower level by “−”), and create 22 = 4 test cases corresponding to all possible combinations of values. Thus, simulating the behavior of the property in these scenarios would allow us to estimate the main effects E A and E B by performing the following calculations: EA =

M(A+ , B + ) + M(A+ , B − ) − M(A− , B + ) − M(A− , B − ) , 2

(3.25)

EB =

M(A+ , B + ) + M(A− , B + ) − M(A+ , B − ) − M(A− , B − ) , 2

(3.26)

while the interaction effect, E AB , is given by:

E AB =

M(A+ , B + ) + M(A− , B − ) − M(A+ , B − ) − M(A− , B + ) . 2

(3.27)

It is worth noting that E A , E B , and E AB share the same unit as the property M. As these are quantitative parameters, the procedure described above bears resemblance to fitting a regression expression of the form (Montgomery, 2012, p. 185): M(xˆ1 , xˆ2 ) = β0 + β1 xˆ1 + β2 xˆ2 + β12 xˆ1 xˆ2 ,

(3.28)

where: 6

Further fascinating passages about Fisher’s life, which often intertwine with the history of modern statistics, are excellently recounted in Salsburg (2009).

118

3 Developing a Fluid Model

 + − A − A +A 2 xˆ1 =  A+ −A−  ,

(3.29)

2

 + − B − B +B 2 xˆ2 =  B + −B −  .

(3.30)

2

There is no doubt that Equation (3.28) will provide a more descriptive representation of the real phenomenon as the role played by non-linearities diminishes. However, this does not necessarily imply that it describes a planar surface: the term associated with β12 has the ability to twist the response surface, giving it a curved aspect (Montgomery, 2012, p. 186). Moreover, the introduction of normalized variables xˆ1 and xˆ2 (ranging from –1 at the lower level to unity at the upper level), as replacements for A and B, ensures that the coefficient β0 corresponds to the simple average of M. On the other hand, the remaining coefficients are half the values of the previously calculated effects, that is, β1 = E A /2, β2 = E B /2, and β12 = E AB /2. Observing the perfect equivalence between the number of available values of M and the unknowns of the function, one can conclude that the coefficient of determination for this fit will always be R 2 = 1. Despite requiring substantially more complex calculations, the computation of the fitting coefficient effects on the fluid model predictions follows the same philosophy as the previous case. The seven selected parameters and their respective assigned indices are listed in Table 3.7. It was decided to standardize the adjustment levels at ψ − = 0.9 and ψ + = 1.1, which, in practice, do not impose excessively severe oscillations on any of the properties. By adopting a method known as factorial design of combinations, which combines the two levels of all variables with each other, a total of 27 = 128 distinct scenarios are defined. These scenarios are subsequently applied in the simulation of flash and differential liberation experiments for the ten pre-salt reservoir samples presented here. This approach is justified by the observation that, from the perspective of representativeness in the analysis, it is more interesting to

Table 3.7 Fitting parameters selected for the DOE (Design of Experiments) analysis and their respective indices Parameter Description Index ψ Pc ψTc ψω ψ Mw+ ψα ψc pur e ψcSCN

Critical pressure of SCN fractions Critical temperature of SCN fractions Acentric factor of SCN fractions Molecular mass of the heavy fraction Molar distribution of the heavy fraction Volume translation of the pure components Volume translation of the SCN fractions

1 2 3 4 5 6 7

3.5 Methodology for Fitting the Proposed Fluid Model

119

Table 3.8 Primary effects of the fluid model fitting parameters on the average properties of the ten oil samples from the pre-salt layer. The values refer to the Soave-Redlich-Kwong equation of state βi

Flash liberation test g

Differential liberation test

ρod , kg/m3

Mwg , g/mol

Pb , MPa ρob , kg/m3

876

24.0

36.4

0

0.23

1

–3.1E-4

1.87

–0.03

5.83

2

–8.7E-4

0.35

–0.07

2.13

3

–3.3E-4

1.79

–0.03

4.01

4

7.4E-4

–0.06

0.05

1.38

5

1.2E-5

–0.11

9.9E-4

–2.1E-3

6.4E-16

0.07

1.5E-4

2.7E-14

2.9E-5

1.0E-14 12.9

6 7

–2.9E-16 33.1

cosub , 1/MPa

g

3.1E-3

19.6

–5.2E-4

4.1E-4

1.47

–6.7E-5

–2.2E-4

–5.1E-3

–0.29

–2.1E-5

–3.7E-4

–2.7E-4

1.61

–3.2E-5

2.58

–7.1E-5

1.9E-3

0.26

–2.2E-5

–0.25

7.1E-6

2.6E-4

–0.08

1.1E-6

0.62

3.2E-6

3.3E-13

0.04

2.5E-8

7.66

6.9E-5

886

βod , 1/K

625

13.3

0.27

ρod , kg/m3

–2.0E-13 34.8

5.6E-4

2.2E-5

Table 3.9 Primary effects of the fluid model fitting parameters on the average properties of the ten oil samples from the pre-salt layer. The values refer to the Peng-Robinson equation of state βi

Flash liberation test g

Differential liberation test

ρod , kg/m3

Mwg , g/mol

Pb , MPa ρob , kg/m3

876

24.1

35.9

0

0.23

1

–5.4E-4

1.72

–0.05

5.30

2

–9.4E-4

0.19

–0.08

1.96

3

–4.6E-4

1.43

–0.04

3.64

4

7.8E-4

–0.05

0.06

5

1.5E-5

–0.11

1.3E-3

6

6.6E-14 –0.04

–3.9E-4

7

–2.4E-14 20.5

–1.6E-6

cosub , 1/MPa

g

2.8E-3

17.5

–4.6E-4

7.6E-4

1.42

–2.0E-4

–5.1E-3

–0.44

–1.7E-5

–3.3E-4

8.9E-5

1.30

–2.6E-5 –1.9E-5

7.02

885

βod , 1/K

627

11.9

0.27

ρod , kg/m3

4.7E-4 –5.6E-5

1.26

2.06

–5.4E-5

2.3E-3

0.28

–0.03

–0.31

8.5E-6

3.0E-4

–0.08

1.2E-6

1.1E-13 –1.61

–7.3E-6

–3.2E-15 –0.02

–1.2E-8

3.9E-5

–1.8E-15 21.9

1.2E-5

–2.0E-14

7.93

work with the average values of all available experimental surveys than to restrict it to one or two mixtures. Another noteworthy aspect concerns the property M, which now takes the form of a nine-dimensional vector composed of the parameters described in Sect. 3.4.3. Taking into account that the expressions for β add little to the core of the question presented here, and are included, for example, in Chap. 6 of Montgomery (2012), they will not be detailed. Tables 3.8 and 3.9 list the seven primary effects of fluid models based on the Soave-Redlich-Kwong and Peng-Robinson equations of state, respectively. The subtle difference between the results of each formulation allows for drawing some general conclusions of utmost importance. First, it is observed that adjusting the bubble pressure in reservoir samples from the pre-salt layer would hardly be successful using, for instance, the methodologies proposed by Christensen (1999) or Zurita and McCain (2002), both mentioned in Sect. 2.5, which suggest

120

3 Developing a Fluid Model

an intervention in Mw+ to achieve the desired property value. Since the primary effect of the molecular weight of the heavy fraction is represented by β4 , it becomes clear that three other parameters, related to β1 , β2 , and β3 , should take precedence in this task. Furthermore, if one insisted on the strategy indicated by those authors, the inadvertent change in Mw+ would likely compromise the later calculation of cosub and βod . The previous conclusion becomes even clearer in Table 3.10, which normalizes the main effects obtained for the Soave-Redlich-Kwong equation by dividing them by β0 , and where the ratios greater than 1% are underlined. There, however, what truly catches attention is the absolute ineffectiveness of β5 and β6 in promoting minimally significant variations in any of the nine monitored properties. By excluding the fitting factors related to the molar distribution of the heavy fraction and the volume translation of the pure substances from the optimization process, it will now encompass only five parameters: ψ Pc , ψTc , ψω , ψ Mw+ , and ψcSCN . Additionally, it can be observed that in the flash liberation test, g and Mwg are minimally influenced by the modifications introduced in the model, given that the sum of all their effects corresponds to only 1% of their respective average levels. It is expected, therefore, that in these two aspects, good agreement with experimental data depends more on the characteristics and capabilities of the implemented equation of state than on the quality of the regression itself. Furthermore, the results reveal the prevalence of an intricate matrix of cause and effect, where each fitting parameter acts on more than one property, and each property is influenced by more than one fitting parameter. The same diagnosis remains valid for the Peng-Robinson equation of state, which, for the sake of conciseness, will not have its βi /β0 values reproduced.

3.5.3 Composition of the Objective Function Based on the investigation conducted thus far, which indicates the need for a simultaneous fitting of five parameters in the fluid model, whose impact on the simulations is both significant and unpredictable, it becomes evident that the objective function of the optimization problem needs to encompass as many experimental data points as possible. It is advisable, therefore, to compose it with the purpose of reflecting, to a greater or lesser extent, the deviations associated with all nine properties defined in Sect. 3.4.3, which were also employed in the calculation of effects in the previous section and will be referred to generically as Mi . Thus, it is established that: 1 FPVT (ψ Pc , ψTc , ψω , ψ Mw+ , ψcSCN ) = wi FPVT ,i , 9 i=1 9

(3.31)

where wi represents the so-called weights of the function, and the terms FPVT ,i are typically calculated as follows:

3.5 Methodology for Fitting the Proposed Fluid Model

121

Table 3.10 Main effects of the fluid model fitting parameters on the average properties of the ten oil samples from the pre-salt layer obtained from the respective flash and differential liberation tests. The values refer to the Soave-Redlich-Kwong equation of state β i /β0 Flash liberation test Differential liberation test g ρod Mwg Pb ρob cosub g ρod βod 1 2 3 4 5 6 7 12 13 14 15 16 17 23 24 25 26 27 34 35 36 37 45 46 47 56 57 67 Others Total

1.3E-3 3.7E-3 1.4E-3 3.2E-3 5.2E-5 2.7E-15 1.3E-15 6.2E-4 1.4E-3 2.4E-4 3.2E-5 2.4E-16 3.3E-15 5.0E-4 1.8E-4 6.8E-6 2.9E-15 1.1E-15 1.8E-4 2.3E-5 3.5E-15 4.9E-16 1.6E-5 2.8E-15 1.3E-15 6.4E-17 7.1E-15 1.3E-16 4.7E-4 0.01

2.1E-3 4.0E-4 2.0E-3 6.5E-5 1.2E-4 8.1E-5 0.04 1.0E-4 1.4E-4 3.3E-4 1.4E-4 1.7E-6 0.01 2.5E-5 1.4E-5 9.2E-6 2.3E-7 3.5E-3 3.4E-4 1.3E-4 9.1E-7 1.8E-4 2.5E-5 2.9E-6 4.8E-4 2.9E-8 4.1E-5 6.1E-6 1.7E-3 0.06

1.3E-3 3.0E-3 1.3E-3 2.3E-3 4.1E-5 6.2E-6 1.2E-6 4.7E-4 1.0E-3 1.9E-4 2.7E-5 6.0E-10 8.0E-8 3.8E-4 1.4E-4 4.8E-6 1.4E-8 5.7E-8 1.5E-4 2.0E-5 2.7E-9 4.0E-8 1.3E-5 1.9E-8 4.1E-8 2.3E-10 9.5E-10 1.5E-11 3.6E-4 0.01

0.16 0.06 0.11 0.04 5.5E-5 7.4E-16 2.8E-16 0.02 0.03 0.02 2.6E-3 7.9E-16 3.9E-17 0.01 2.1E-3 6.9E-4 2.4E-16 8.8E-16 0.02 3.0E-3 5.9E-16 1.1E-16 1.3E-4 4.3E-16 1.3E-16 3.9E-16 4.3E-16 3.5E-16 0.03 0.51

0.03 0.01 0.02 4.1E-3 3.9E-4 1.0E-3 0.02 4.3E-5 5.6E-4 2.7E-3 5.4E-4 6.2E-5 5.0E-3 2.3E-4 3.4E-4 1.3E-4 2.5E-5 2.5E-3 2.6E-3 5.8E-4 4.3E-5 7.0E-4 1.0E-4 8.5E-6 3.7E-4 7.1E-7 7.1E-7 4.1E-5 2.1E-3 0.11

0.17 0.07 0.12 0.02 2.3E-3 1.0E-3 0.02 4.7E-3 5.2E-3 0.01 2.5E-3 1.5E-4 9.4E-3 2.2E-3 2.4E-3 5.9E-4 6.2E-5 6.8E-4 0.01 2.8E-3 1.1E-4 2.4E-3 6.7E-4 2.1E-5 4.2E-5 1.8E-6 9.8E-5 4.5E-5 7.0E-3 0.47

1.5E-3 0.02 9.8E-4 7.0E-3 9.6E-4 1.2E-12 7.2E-13 8.0E-4 3.8E-3 7.0E-4 5.6E-4 1.0E-12 1.2E-12 1.3E-3 2.5E-3 2.5E-4 1.0E-12 1.2E-12 1.8E-4 5.2E-4 1.2E-12 8.2E-13 3.1E-4 1.1E-12 7.5E-13 1.0E-12 5.0E-13 9.2E-13 0.01 0.05

1.7E-3 3.2E-4 1.8E-3 3.0E-4 8.9E-5 4.6E-5 0.04 2.6E-5 1.6E-4 4.3E-4 1.6E-4 5.2E-7 0.01 1.1E-4 2.7E-5 1.4E-5 7.2E-7 3.6E-3 4.4E-4 1.5E-4 1.5E-7 2.1E-4 6.5E-6 2.0E-6 5.7E-4 3.7E-8 4.8E-5 3.6E-6 1.8E-3 0.06

0.12 0.04 0.06 0.04 2.0E-3 4.5E-5 0.04 4.2E-3 4.0E-3 4.0E-3 1.8E-3 5.7E-6 0.02 6.5E-4 3.0E-3 3.0E-4 8.6E-7 2.1E-3 5.0E-3 1.6E-3 2.6E-6 2.4E-3 4.3E-4 3.7E-6 7.7E-4 5.3E-8 1.7E-4 3.5E-6 4.3E-3 0.34

122

3 Developing a Fluid Model

 FPVT ,i =

Mi,calc − Mi,ex p Mi,ex p

2 .

(3.32)

Equation (3.32) gives the objective function the typical form of a second-degree polynomial, which, as will be discussed later, is crucial for the optimization process to be fast and efficient. Clearly, there are several other algebraic forms with similar characteristics that may offer certain benefits. Returning to the idea presented at the end of Sect. 2.5, for instance, it might be advantageous to impose a lower limit on the result in order to signal to the algorithm that deviations become less relevant as they fall within acceptable thresholds, and that it should focus its efforts on more problematic areas of the model. To achieve this, one can simply add an extra expression, associated with a parameter i , below which FPVT ,i = i2 = constant is incurred, while Eq. (3.32) remains valid in other situations. Applying the approach developed by Churchill and Usagi (1972), one obtains: 

FPVT ,i

n

 = (i )2n +

Mi,calc − Mi,ex p Mi,ex p

2n (3.33)

It is worth noting that the so-called asymptotic method not only provides a unified expression capable of synthesizing the two expected behaviors, but also promotes a smooth transition between them, which is primarily dictated by the constant n. The greater its value, the narrower this region becomes. For the present study, n = 3 was adopted, so that Eq. (3.33) can be rewritten as:

  Mi,calc − Mi,ex p 6 3 6 . (3.34) FPVT ,i = i + Mi,ex p In Fig. 3.22, the trends described by Eqs. (3.32) and (3.34) are illustrated. It is evident therein that the former is merely a particular case of the latter for i = 0, from which it follows that the proposed expression can also, at least locally, be represented by a quadratic function. Furthermore, it is observed that i acts as a kind of tolerance to the calculated deviation for property Mi , as FPVT ,i only undergoes a significant increase once it surpasses this deviation. Undoubtedly, determining the most appropriate values for i for each of the nine parameters that compose the objective function requires careful consideration. Considering the current capabilities of commercial software and maintaining an optimistic outlook on future advancements, it is desired to perform simulations with error margins of approximately 0.5% for the oil mass density, 1% for the bubble point pressure, 2% for the mass fractions and molecular weights of the phases, 10% for the compressibility of the single-phase mixture, and 30% for the thermal expansion coefficient. These values will be assigned to the parameter and compiled in Table 3.11.

3.5 Methodology for Fitting the Proposed Fluid Model

123

·10−3 2.5

Eq. (3.32) Eq. (3.34)

2.0

FPVT ,i

1.5

1.0

Δi = 0.03

0.5

Δi = 0.02 Δi = 0.01

0.0 0

1

2

3 (Mi,calc − Mi,exp )/Mi,exp

4

5 ·10−2

Fig. 3.22 Behavior of Eqs. (3.32) and (3.34) as a function of the relative error observed in the property Mi , for different values of i Table 3.11 Tolerances chosen for the nine properties monitored in the fluid model fitting Flash liberation test Differential liberation test g ρod Mwg Pb ρob cosub g ρod βod i

0.02

0.005

0.02

0.01

0.005

0.1

0.02

0.005

0.3

3.5.4 Objective Function Optimization Using the Response Surface Methodology The factorial design conducted in Sect. 3.5.2 has proven to be a notably useful tool as it has definitively elucidated the effects of the fitting parameters on the simulation results. However, it does not provide any guidance on how to operationalize these parameters in order to align the model predictions with the experimental readings obtained in the laboratory. This particular issue was an open topic of discussion in the 1920s, during the time of Ronald A. Fisher’s publications. It would take approximately another 30 years for a robust and comprehensive technique to be developed to address this problem.

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3 Developing a Fluid Model

When British statistician George E. P. Box (1919–2013) developed the now acclaimed Response Surface Methodology (RSM) (Box and Wilson, 1951), his aim was to optimize the chemical processes of the company that had hired him, Imperial Chemical Industry (ICI) (Salsburg, 2009). In a subsequent article, however, Box foresaw the expansion of these concepts to other areas of the industry and refined them further (Box, 1954). His technique takes advantage of the fact that, compared to the agricultural experiments that had previously supported the studies, the manufacturing sector provides faster responses to experiments and is subject to smaller uncertainties (Montgomery, 2012, p. 21). In this sense, the adjustment of mathematical models represents a significant advancement in the scenario envisioned by the author. Let us consider an arbitrary phenomenon, denoted by F, which is influenced by the variables x1 and x2 through mechanisms that are currently unknown. It is assumed that this phenomenon possesses an optimal point, although neither its value nor the underlying configuration leading to it are known. In an attempt to algebraically represent this phenomenon, the simplest suitable expression, despite its inherent imprecision, is a quadratic polynomial given by: F(x1 , x2 ) = β0 + β1 x1 + β2 x2 + β11 x12 + β12 x1 x2 + β22 x22 .

(3.35)

In Eq. (3.35), the presence of x12 and x22 provides the necessary curvature that the response surface of the model should exhibit in order to accommodate a stationary point, while the other terms were already included in the linear regression of Eq. (3.28). Therefore, at least six points are required to determine the coefficients β, whereas a factorial design like the one described in Sect. 3.5.2 could contribute with only 22 = 4 points. This deficiency is usually compensated by adding a star-shaped structure, defined by the average value of the variables x1 and x2 in the previous set, as well as by a pair of symmetrically placed points along each axis. This design, known as the central composite design, is illustrated in Fig. 3.23. In most cases, it is desirable for the orbital points to be equidistant from the center, thereby exhibiting rotatability (Box et al., 2005, p. 455). Thus, by quantifying the responses of F(x1 , x2 ) at the nine points indicated by the central composite design, a simple least squares fitting enables the determination of the optimal values of β for Eq. (3.35). Depending on the set of coefficients, the resulting stationary point may correspond to a maximum, as suggested by the example in Fig. 3.24, a minimum, or a saddle point (i.e., maximum in one direction and minimum in the other). To precisely locate it, it suffices to solve the system of equations formed by ∂ F/∂x1 = 0 and ∂ F/∂x2 = 0. Clearly, the quadratic response surface suggested here is merely an approximation of the true phenomenon. Therefore, if the correlation coefficient indicates a poorly effective fit or if the experimental design was conducted in a region far from the optimal combination, there is very little chance that the stationary point will have practical support. The first issue can be mitigated by reducing the scale of the design, resulting in a more localized representation of the phenomenon, but it also increases the risk of confusing any local optimum with the desired global optimum.

3.5 Methodology for Fitting the Proposed Fluid Model

125

Fig. 3.23 The central composite design for two and three variables (Montgomery, 2012)

Fig. 3.24 Response surface assigned to an arbitrary phenomenon F as a function of variables x1 and x2 (Montgomery, 2012)

126

3 Developing a Fluid Model

F

Fig. 3.25 Hypothetical behavior of the phenomenon F in the direction of maximum variation (Montgomery, 2012)

0

2

4

6 8 Experiments

10

12

The second issue favors the application of the so-called gradient method or steepest ascent/descent method, where a series of experiments (generally equidistant) are conducted in the direction of maximum variation indicated by the model, gradually stepping until the reading of F undergoes inflection. This procedure is exemplified in Fig. 3.25. Subsequently, the new values of β are calculated in the vicinity of this provisional maximum or minimum, the gradient is established once again, and the process continues in this manner. The iterative nature of optimization, however, is not limited to only the initial attempts. Even when the gradual improvement in results already allows for analytical inference of the location of the stationary point with a good degree of accuracy, constructing a second response surface around it will certainly reveal those coordinates in a more precise manner. This process can be repeated until any further gains become negligible, thereby confirming the convergence of the solution. Although the above example considers only x1 and x2 , the response surface methodology finds application in substantially more complex scenarios. By rewriting the second-degree polynomial of Eq. (3.35) for a set of n arbitrary variables, one obtains (Montgomery, 2012, p. 486): βi xi βi j xi x j , (3.36) F(x1 , x2 , . . . , xn ) = β0 + i

i

j≥i

or, using matrix notation: F(x1 , x2 , . . . , xn ) = β0 + x t b + x t Bx,

(3.37)

3.5 Methodology for Fitting the Proposed Fluid Model

127

where: ⎡ ⎤ x1 ⎢ x2 ⎥ ⎢ ⎥ x = ⎢ . ⎥, ⎣ .. ⎦ xn

⎡ ⎤ β1 ⎢β2 ⎥ ⎢ ⎥ b = ⎢ . ⎥, ⎣ .. ⎦ βn

⎡

β11 β12 /2 ⎢β12 /2 β22 ⎢ B=⎢ . .. ⎣ .. . β1n /2 β2n /2

⎤ . . . β1n /2 . . . β2n /2⎥ ⎥ .. ⎥ .. . . ⎦ . . . βnn .

(3.38)

The determination of the n (n + 3) /2 + 1 parameters listed in Eq. (3.36) involves a multivariable least squares fitting, the mathematical development of which is presented in Chap. 10 of Montgomery (2012). It is observed that the 2n + 2n + 1 points comprising the central composite design always yield a greater number of equations than the number of unknowns in the problem. In√ order to ensure the rotatability of this structure, it is recommended to use a spacing n times larger than the one chosen for the factorial experiments in the star points. At the end of the modeling process, the gradient of F at any point in the domain will be given by: ∇ F = b + 2Bx.

(3.39)

The coordinates of the stationary point, where ∇ F = 0, are given by: 1 x e = − B −1 b. 2

(3.40)

Once again, x e can take on three distinct forms. While visual classification becomes considerably more challenging, it is known that such information is embedded in the eigenvalues of the matrix B. If all eigenvalues have the same sign, positive results indicate a point of minimum, whereas negative eigenvalues represent a maximum. Otherwise, the response surface contains a saddle point. Based on the aforementioned, it is proposed that the optimization algorithm for the objective function FPVT adheres to the following steps: 1. The fitting coefficients of the model are initialized. Naturally, the intuitive choice is to set ψ Pc = ψTc = ψω = ψ Mw+ = ψcSCN = 1, although any other combination could be arbitrarily chosen; 2. A central composite design is outlined with 43 experiments around the most recent set of values, evaluating the objective function FPVT defined in Eq. (3.31) for each experiment. The lower and upper levels of the factorial combinations are defined by adjusting the current value by ±0.01, so that in the first iteration, they will all be equal to 0.99 and 1.01, √ respectively. In the star points, the distance from the mean should be ±0.01 5; 3. Using the method of least squares, the 21 coefficients appearing in Eq. (3.36) for n = 5 are adjusted. Once the response surface is known, it is necessary to calculate the gradient of FPVT at the central experiment, given by Eq. (3.39), as well as the coordinates of the stationary point expressed in Eq. (3.40);

128

3 Developing a Fluid Model

4. Ensure that the stationary point of the response surface does indeed correspond to a new minimum for the objective function. This requires inspecting the eigenvalues of the system of equations, followed by conducting an additional experiment at that location; 5. If the test in item 4 is successful, a straight trajectory is plotted between the current combination and the newly indicated point, with a series of equidistant experiments along it, such that the central coordinates of these experiments differ by a maximum of 0.01 for all five adjustment parameters. The smallest value obtained for FPVT will naturally be the optimum of the current iteration; 6. If the test in item 4 fails, position the experiments in the direction of the gradient of the objective function, again ensuring a maximum distance of 0.01 in all variables. These experiments should be taken sequentially until the inflection of FPVT is observed, confirming that the optimal value of the current iteration has been reached; 7. Calculate the progress made in minimizing the objective function. The search will converge when the difference between two successive iterations is less than 1 × 10−5 . Otherwise, return to item 2.

3.5.5 Definition of Weights, Results and Comparison The proper functioning of Eq. (3.31) relies heavily on the chosen levels for wi . By disproportionately emphasizing any of the nine listed properties, the method loses sight of the whole, undermining the chances of achieving a reasonable fit. However, it is not by chance that the definition of these weights has been postponed until now: they inevitably derive from a trial-and-error process that can only be conducted after the complete implementation of the optimization algorithm. In a first step, the objective is to find a combination of wi that enables the adjustment of the entire set of ten samples presented in Sect. 3.4.3. This adjustment aims to ensure that the MAPE of the simulated properties complies with the tolerances listed in Table 3.11. As shown in Table 3.12, without any adjustment, the fluid models based on the Soave-Redlich-Kwong equation of state are only able to fully satisfy one out of these nine targets, namely the one associated with the representation of ρod in the flash liberation test. On the other hand, adopting an equitable fitting with equal weights, all set to unity, leads to a clear change in the deviations, but not enough to bring them within the predetermined thresholds. Only by employing the specified distribution of wi was an adjustment achieved with good overall performance. Exceptions occur for ρob in the differential liberation test, where the MAPE in the order of 1% is still far from indicating a poor result, and for g and Mwg in the flash liberation test. In fact, these last two parameters could not be properly manipulated; their low sensitivity to fitting factors, as measured in Sect. 3.5.2, necessitates extreme interventions that, if implemented, would compromise the accuracy of all other estimates.

3.5 Methodology for Fitting the Proposed Fluid Model

129

Table 3.12 Derivation of the objective function weights for adjusting fluid models based on the Soave-Redlich-Kwong equation of state MAPE flash, % MAPE differential liberation, % g ρod Mwg Pb ρob cosub g ρod βod Goal No tuning Equal weights Final tuning wi,SRK

2.00 3.61

0.50 0.24

2.00 3.28

1.00 13.2

0.50 0.71

2.94

1.42

2.75

3.08

2.27

3.51

0.54

3.17

0.81

1

4

1

1

10.0 26.5

2.00 2.21

0.50 0.59

30.0 25.1

9.47

1.84

1.93

26.9

1.05

8.09

1.89

0.33

32.2

4

0.1

1

1

0

Table 3.13 Final results of the model fitting for the ten samples of pre-salt petroleum to the SoaveRedlich-Kwong equation of state Sample Fitting for Soave-Redlich-Kwong ψ Pc ψTc ψω ψ Mw+ ψcSCN PS-01 PS-02 PS-03 PS-04 PS-05 PS-06 PS-07 PS-08 PS-09 PS-10

1.029 1.069 1.063 1.071 1.047 1.039 1.061 1.080 1.008 1.033

1.010 1.029 1.048 1.044 1.086 1.019 1.025 1.008 0.997 1.031

1.021 1.046 1.061 1.054 1.045 1.028 1.040 1.057 1.053 1.025

1.004 1.014 1.007 1.028 0.919 1.003 1.014 1.056 1.031 0.995

1.001 0.996 0.965 0.981 0.968 0.999 0.970 0.987 0.997 0.971

It is evident, therefore, that the process of weight selection is highly empirical and even arbitrary at times. These characteristics give it a somewhat non-universal nature, leading to situations where a set of values that perform well in fitting a particular oil mixture may not necessarily achieve the same success in the subsequent attempt. Clearly, as a larger quantity of samples is monitored, the chances of this happening decrease. Furthermore, the combination of wi listed in Table 3.12 serves as an excellent starting point for promoting individual improvements. Another positive aspect worth highlighting, which also confirms the quality of the thermodynamic model, pertains to the observed adjustment factors at the end of the optimization process. Upon inspecting the numbers presented in Table 3.13, it is noted that the modifications made to characterize the ten samples from the pre-salt layer are subtle and all remain below 10%. This indicates that outside the pressure and temperature conditions covered by the experimental tests, the simulations are unlikely to exhibit significant changes in their results.

130

3 Developing a Fluid Model

Table 3.14 Derivation of the objective function weights for adjusting fluid models based on the Peng-Robinson equation of state MAPE flash, % MAPE differential liberation, % g ρod Mwg Pb ρob cosub g ρod βod Goal No tuning Equal weights Final tuning wi,PR

2.00 3.83

0.50 0.22

2.00 3.49

1.00 13.7

0.50 0.80

10.0 12.4

2.00 2.43

0.50 0.61

30.0 37.4

2.67

2.43

2.63

12.7

2.69

7.41

2.09

3.03

32.2

3.59

0.60

3.28

0.93

1.33

5.64

1.87

0.38

43.6

1

4

1

1

4

0

1

1

0

Table 3.15 Final results of the model fitting for the ten samples of pre-salt petroleum to the PengRobinson equation of state Sample Fitting for Peng-Robinson ψ Pc ψTc ψω ψ Mw+ ψcSCN PS-01 PS-02 PS-03 PS-04 PS-05 PS-06 PS-07 PS-08 PS-09 PS-10

0.991 1.074 1.065 1.073 1.052 1.019 1.059 1.078 1.028 1.036

0.997 1.033 1.063 1.045 1.090 1.066 1.022 1.021 1.009 1.080

1.073 1.049 1.055 1.060 1.047 1.047 1.048 1.061 1.018 1.031

1.035 1.016 1.032 1.048 0.922 0.944 1.034 1.066 1.005 0.920

0.997 0.997 0.940 0.965 0.947 0.988 0.934 0.973 1.001 0.947

The selection of weights for the objective function concerning the models based on the Peng-Robinson equation of state, therefore, followed the same steps described above, resulting in the values presented in Table 3.14. It can be observed that the final fitting was not as effective this time, and the deviations from the experimental data increased for seven out of the nine considered properties. These differences are, logically, small and limited to a certain set of samples. Furthermore, they still reflect a series of previous choices, such as the definition of binary interaction parameters and the calculation correlations of critical properties and acentric factor for SCN fractions. The fitting factors obtained for each optimization, shown in Table 3.15, once again highlight well-founded models that only require specific refinements. However, it is undeniable that the Soave-Redlich-Kwong equation of state exhibited superior performance within the scope of this study. Finally, it is worth highlighting the excellent compatibility demonstrated between the objective function and the response surface of the fitting. Together, Eqs. (3.31), (3.34), and (3.35) allowed for the precise mapping of simulation errors,

3.5 Methodology for Fitting the Proposed Fluid Model ·10−2

·10−3

1.0

R

2

3.5

= 0.9999

R2 = 0.9998

3.0

0.8

2.5

0.6

FPVT

FPVT

131

0.4

2.0 1.5

0.2

1.0

0.0 0.90

0.95

1.00

1.05

0.5 0.90

1.10

0.95

ψ Pc

1.10

·10−3

6.0

2.4

R2 = 0.9978

5.0

2.2

4.0

2.0

FPVT

FPVT

1.05

ψTc

·10−3

3.0

1.6

1.0

1.4

0.95

1.00

1.05

R2 = 0.9999

1.8

2.0

0.0 0.90

1.00

1.2 0.90

1.10

0.95

1.00

1.05

1.10

ψMw+

ψω ·10−3 2.8

R2 = 0.9991

2.6

FPVT

2.4 2.2 2.0 1.8 1.6 0.90

0.95

1.00

1.05

1.10

ψcSCN

Fig. 3.26 Behavior of the objective function during the parametric analysis of adjustment factors around the point ψ Pc = ψTc = ψω = ψ Mw+ = ψcSCN = 1. Values corresponding to the model based on the Soave-Redlich-Kwong equation of state for the PS-01 sample

resulting in a high convergence rate for the search algorithm described in Sect. 3.5.4: taking the average of ten samples, only 3.1 iterations were required for the models based on the Soave-Redlich-Kwong equation of state, and 3.3 iterations for PengRobinson. This fact can be confirmed through a parametric analysis, such as the one depicted in Fig. 3.26, which corresponds to the sample model PS-01 based on the

132

3 Developing a Fluid Model

Soave-Redlich-Kwong equation of state. The figure portrays the region near the starting point of the optimization process, when ψ Pc = ψTc = ψω = ψ Mw+ = ψcSCN = 1. The analysis reveals that, regardless of the direction examined, there always exists a second-degree polynomial capable of providing highly convincing fits, all with a correlation coefficient greater than R 2 = 0.99.

3.6 Empirical Correlations for the Calculation of Other Properties According to Whitson and Brulé (2000, Sect. 7.1), since the 1920s there has been an interest in determining, for each field, the portion of volatile hydrocarbons dissolved in the oil under reservoir conditions, as well as how much the oil would shrink upon reaching the surface. It was also well-known that the resulting natural gas would expand hundreds of times during its transport. These concerns led to the development of the volumetric-based set of properties mentioned in Sect. 1.4, which consists of the formation volumes of oil and natural gas, as well as their solubility ratio. During a time when computational resources were not yet available, which later revolutionized compositional modeling, engineers sought ways to interrelate these three quantities. The initial regressions included several other rather trivial parameters, such as phase densities under Standard conditions. The so-called black oil correlations reached their peak between 1960 and 1980, a period during which the main empirical models for calculating formation water properties were also published. However, due to their ease of implementation and the significant computational efficiency they provide, these correlations still have proponents worldwide and are subject to continuous improvement efforts, aiming to make them increasingly accurate and comprehensive.

3.6.1 Thermodynamic Properties of Formation Water A reliable representation of the volumetric behavior of formation water can be obtained through the correlation established by Rowe and Chou (1970) as given below: v = A(T ) − P · B(T ) − P 2 · C(T ) + εsalt · D(T ) + ε2salt · E(T ) − εsalt P · F(T ) −ε2salt P · G(T ) −

εsalt P 2 · H (T ). 2 (3.41)

3.6 Empirical Correlations for the Calculation of Other Properties

133

where εsalt denotes the mass fraction of salt dissolved in the aqueous medium and: A(T ) = 100.6741T −2 − 1.127522T −1 + 5.916365 × 10−3 − 1.035794T × 10−5 + 9.270048 × 10−9 T 2 , (3.42) B(T ) = 0.1042954T −2 − 1.1933737 × 10−3 T −1 + 5.307562 × 10−6 − 1.0688823 × 10−8 T + 8.492782 × 10−12 T 2 , C(T ) = 1.23269 × 10−11 − 6.861998 × 10−14 T,

(3.43) (3.44)

D(T ) = −2.5166 × 10−3 + 1.11766 × 10−5 T − 1.70552 × 10−8 T 2 ,

(3.45)

E(T ) = 2.84851 × 10−3 − 1.54305 × 10−5 T + 2.23982 × 10−8 T 2 ,

(3.46)

F(T ) = −1.5106 × 10−6 + 8.4605 × 10−9 T − 1.2715 × 10−11 T 2 ,

(3.47)

G(T ) = 2.7676 × 10−6 − 1.5695 × 10−8 T + 2.3102 × 10−11 T 2 ,

(3.48)

H (T ) = 6.4634 × 10−10 − 4.1671 × 10−12 T + 6.8600 × 10−15 T 2 .

(3.49)

According to its authors, when confronted with experimental data, the deviations from Eq. (3.41) consistently remain below 1.5% for a wide range of temperature values (295 to 450 K) and mass fraction of dissolved salt (0 to 0.25). Although the considered pressure range (0 to 34.3 MPa) falls slightly short of current requirements, Sutton (2009) claims that it is possible to extrapolate it to pressures up to eight times higher and still draw highly plausible predictions. The results presented in Fig. 3.27, which are well within the expected range for a typical brine found in pre-salt reservoirs, support this finding. It is worth mentioning that some of the coefficients in Eqs. (3.42)–(3.49) have been modified from the original publication in order to adapt the input parameters of the correlation to SI units. The same approach has been used in all the studies listed until the end of this chapter. In turn, the correlation presented by Jamieson et al. (1969) for the specific heat of the aqueous phase at the reference pressure states that: c P,Pref = A(εsalt ) + T · B(εsalt ) + T 2 · C(εsalt ) + T 3 · D(εsalt ),

(3.50)

where: A(εsalt ) = 5.328 − 97.6εsalt + 404ε2salt ,

(3.51)

B(εsalt ) = −6.913 × 10−3 + 0.7351εsalt − 3.15ε2salt ,

(3.52)

134

3 Developing a Fluid Model

1200 T = 293 K

1180

ρ, kg/m3

333 K

1160 373 K

1140 413 K

1120 1100 0

10

20 30 P , MPa

40

50

Fig. 3.27 Variation of the mass density of a brine typically found in pre-salt layer reservoirs as a function of pressure and temperature, predicted by the correlation of Rowe and Chou (1970)

C(εsalt ) = 9.6 × 10−6 − 1.927 × 10−3 εsalt + 8.23 × 10−3 ε2salt ,

(3.53)

D(εsalt ) = 2.5 × 10−9 + 1.666 × 10−6 εsalt − 7.125 × 10−6 ε2salt .

(3.54)

Equation (3.50) has been adjusted based on a database encompassing temperatures ranging from 270 to 450 K and salt mass fractions between 0 and 18%. Although the authors did not quantify the percentage deviation of these estimates, it is visually apparent that there is good agreement with experimental data. The trends of the model, as illustrated in Fig. 3.28, are supported by practical evidence. In fact, it is expected that the specific heat of the mixture will slightly increase with T , and exhibit a pronounced decrease with the growth of εsalt . Together, Eqs. (3.41) and (3.50) allow for the deduction of all other thermodynamic properties of formation water. For instance, specific enthalpy and entropy can be derived from the expressions (Elliott & Lira, 2012, p. 235): T h = h ref +

c P,Pref dT + Tref

   P  ∂v v−T d P, ∂T P

(3.55)

Pref

T s = sref + Tref

c P,Pref dT − T

P  Pref

∂v ∂T

 d P. P

(3.56)

3.6 Empirical Correlations for the Calculation of Other Properties

135

4400 εsalt = 0

cP,Pref , J/kg · K

4200 0.04

4000 0.08

3800 0.12

3600

0.16

3400 300

320

340

360 T, K

380

400

420

Fig. 3.28 Variation of the specific heat of a brine at the reference pressure as a function of salinity and temperature, predicted by the correlation of Jamieson et al. (1969)

The reference state in Eqs. (3.55) and (3.56) is often equated with the Standard conditions. Although the choice of values is arbitrary, it is typically assumed that h ref = 0 and sref = 0. Once the enthalpy is known, the specific internal energy of the phase can be obtained from Eq. (2.115). By differentiating Eq. (3.55) with respect to temperature and making the necessary simplifications, it follows that:  cP ≡

∂h ∂T





P = c P,Pref − P

T Pref

∂2v ∂T 2

 d P.

(3.57)

P

3.6.2 Transport Properties Among all the parameters governing the flow of oil in OPWs, the dynamic viscosity of the oil is undoubtedly one of the most important. Therefore, in an attempt to obtain better predictions, the black oil approach decomposes its computation into three sequential and complementary steps. The first step concerns the dead oil, in which the influence of temperature can be represented according to Bergman and Sutton (2007a) as follows: ln [ln (1000μod + 1)] = A + B ln (T − 83.15) .

(3.58)

136

3 Developing a Fluid Model

The evaluation of the constants A and B in Eq. (3.58) necessitates prior knowledge of μod at a minimum of two distinct temperature levels, preferably not in close proximity. When the rheology test of the sample is available, which includes a larger number of temperatures and experimental readings, it is advisable to determine these parameters through linear regression, ensuring the optimal curve fitting. Next, in cases where P > Pref , indicating that a portion of the natural gas is dissolved in the oil phase, the correlation proposed by Bergman and Sutton (2007b) can be employed: μo,sat =

C (1000μod ) D , 1000

C=



(3.59)

where:

1+

D = 0.3823 +

1 Rs 61.30ϕC

1+

0.8553 ,

1 − 0.3823 0.8193 . 

(3.60)

(3.61)

Rs 101.2ϕ D

Clearly, as Rs approaches  μo,sat tends toward μod , whereas in the opposite  zero, extreme, μob is equal to μo,sat Rs =GOR . In order to derive Eq. (3.59), the authors had at their disposal a comprehensive dataset consisting of 12,474 experimental readings, encompassing 1,849 oil samples. Thanks to this formidable database, the correlation is capable of operating within a wide range of solubility ratios (up to 3 3 /mstd ), temperatures (ranging from 290 to 390 K), and pressures (up to 1,160 mstd 66.4 MPa). Finally, the last step of the calculation is reserved for situations when P > Pb . In this zone of oil subsaturation, where the complete incorporation of natural gas renders the sample single-phase, Bergman and Sutton (2006) recommend adopting the following relationship:     P − Pb F μo,sub = exp E , (3.62) μob 689.48 where:   E = 6.56980 × 10−7 (ln μob )2 − 5.74459 × 10−6 ln μob + 1.56846 × 10−4 ϕ E , (3.63) F = 2.24623 × 10−2 ln μob + 1.02837.

(3.64)

3.6 Empirical Correlations for the Calculation of Other Properties

137

It is observed that μo,sub → μob as P → Pb , as expected. Similar to the previous work, the development of Eq. (3.62) relied on an extensive compilation of readings collected in the laboratory, totaling 10,248, corresponding to 1,399 distinct samples. This translates, once again, into very broad ranges of applicability: 273 to 490 K for temperature, 0.063 to 14,200 mPa · s for dynamic viscosity at the bubble point, and up to 172.4 MPa for the pressure differential. The viscosity of water present in the reservoir can also be estimated using empirical correlations, such as the one published by Kestin et al. (1978). However, it is important to note that in the pre-salt wells, water rarely appears as a continuous phase that flows independently, as it is usually emulsified in oil. An emulsion is defined as a colloidal system composed of two immiscible liquids, where droplets of one are suspended in the other. Its occurrence in oil reservoir mixtures is largely due to the action of natural surfactants that concentrate at the water-oil interface, significantly reducing surface tension and thereby inhibiting the coalescence of the dispersed phase (Tewari et al., 2018, p. 346). The dynamic viscosity of this type of emulsion, represented here by μl , increases with the volumetric fraction of water it contains and can exceed the value of the continuous oil phase by up to 100 times (Pedersen et al., 2014, p. 418). According to Pal and Rhodes (1989), this growth can be mathematically described by a function given as: 

μl = μo μl = μo

ψμl ψμl − 0.8415 × BSW



2.5

ψμl ψμl − 0.8415 × BSW

.

(3.65)

2.5 (3.66)

Naturally, as the water cut (BSW) approaches 0, the liquid viscosity (μl ) tends to the oil viscosity (μo ). Equation (3.66) is valid within the range 0 ≤ BSW ≤ 0.74, and its adjustment constant requires experimental results, once again. A widely used correlation for calculating the viscosity of natural gas, developed by Lee et al. (1966), remains popular to this day. According to this correlation, the viscosity (μg ) is given by:   C  , (3.67) μg = A exp B 1000ρg where: A=

(12.58 + 0.02156Mw ) × 10−7 T , 116.2 + 10.70Mw + T

B = 3.448 +

548.0 + 0.01009Mw , T

C = 2.447 − 0.2224B.

(3.68)

(3.69) (3.70)

138

3 Developing a Fluid Model

Equation (3.67) is derived from an investigation that encompassed several pure substances and four samples of natural gas reservoirs, subjected to pressures of up to 55 MPa and temperatures ranging from 310 to 445 K. The authors of this equation report an average error of merely 2.7%. For the representation of the gas-oil interfacial tension, a suitable approach is provided by Ramey (1973), which involves the following equation:     od ρo gd ρo ρg 1/4 + Pg . (3.71) − σgo = Po o Mwo o Mwo Mwg Equation (3.71) represents the model proposed by Weinaug and Katz (1943), which has been reformulated for a dead oil and natural gas system, or in other words, for the binary mixture of pseudocomponents that underlies the black oil approach. Ramey (1973) also provided correlations for estimating Po and Pg , known as the phase parachors. Nevertheless, more accurate values are generally obtained by adapting the work of Katz et al. (1943), which establishes that: Po = 35 + 2.38Mwod ,

(3.72)

Pg = 25 + 2.80Mwg .

(3.73)

It should be noted that Eq. (3.72) is valid for SCN fractions and blends of dead oil within the range of 180 ≤ Mw ≤ 360 g/mol, whereas Eq. (3.73) is applicable only to paraffins with a molecular mass lower than 120 g/mol. Finally, considering the objectives set for the present study, even though the water in the reservoir adds a third phase to the system, the calculation of σgw and σow is not necessary. In the first case, because once it is emulsified in the oil, there is certainly no well-defined surface between it and the gas phase; in the second case, due to the choice of not representing the relative motion between the oil and these droplets, the emulsion is treated as a single liquid phase.

3.7 Results and Validation By gathering all the information presented throughout this chapter and the previous one, one now possesses a highly comprehensive and versatile fluid model capable of accommodating any parameter required for the calculation of the mixture flow along its path towards the surface. However, it is still necessary to measure the accuracy of its estimations. This brings into focus once again the representation of the PS-01 and PS-02 samples, which was conducted in a preliminary manner in Sect. 2.6. Revisiting these samples not only allows one to confirm the correct functioning of the hybrid formulation (compositional and black oil) in two real situations, but also provides an opportunity to clarify the sequence of procedures that must be followed in case any doubts remain.

3.7 Results and Validation

139

Table 3.16 Sequence of weight optimization used in the fitting of the fluid model for sample PS-01 Flash liberation test Differential liberation test g ρod Mwg Pb ρob cosub g ρod βod Goal Deviation, % Original Deviation, % Attempt 1 wi Deviation, % Attempt 2 wi Deviation, % Attempt 3 wi Deviation, % Attempt 4 wi Deviation, %

2.00

0.50

2.00

1.00

0.50

10.0

2.00

0.50

30.0

2.83

0.32

2.77

6.82

1.01

22.0

0.72

0.33

30.0

1 2.77

4 0.15

1 2.71

1 1.22

4 0.42

0.1 11.2

1 0.74

1 0.47

0 33.7

1 2.44

2 0.25

1 2.45

1 1.15

4 0.31

0.1 9.73

1 0.64

1 0.32

0 32.5

1 2.47

2 0.31

1 2.48

1 1.18

2 0.36

0.1 9.77

1 0.52

1 0.26

0 32.6

1 2.53

2 0.31

1 2.52

2 1.02

2 0.40

0.1 10.3

1 0.48

1 0.27

0 33.2

To begin with, it is necessary to perform the adjustment of the thermodynamic model in order to approximate its estimates to the available experimental results. All simulations are based on the Soave-Redlich-Kwong equation of state, which, compared to the Peng-Robinson equation as discussed in Sect. 3.5.5, has achieved slightly superior performance. It has been previously mentioned that the adoption of customized values for wi to compose the objective function usually enhances the success of this endeavor. However, the major difficulty lies once again in the lack of an appropriate methodology to prescribe them. In order to minimally guide this process, the following guidelines have been established for the present study: (i) each new attempt can bring modifications to only one parameter, when compared to the previous combination, (ii) start by reducing the weights of the parameters that have met their goal with a good margin, and (iii) subsequently increase the levels of those parameters that have not satisfied the predefined goal. Thus, starting from the set listed in Table 3.12, the values shown in Tables 3.16 and 3.17 were assigned to the samples PS-01 and PS-02, respectively. It is observed that there is indeed a progressive improvement in the deviations presented by the model in relation to the laboratory tests, although subtle gains are obtained in most cases. Tables 3.16 and 3.17 reveal that, in general, the simulations aimed at the flash liberation tests were already sufficiently accurate prior to the adjustment of the thermodynamic models. Ultimately, a slight reduction in deviations involving the composition of natural gas and the phase proportions in equilibrium was observed, partially offset by the poorer estimation of the oil density. The results in volumetric basis, presented in Table 3.18, corroborate this statement. However, in the case of differential

140

3 Developing a Fluid Model

Table 3.17 Sequence of weight optimization used in the fitting of the fluid model for sample PS-02 Flash liberation test Differential liberation test g ρod Mwg Pb ρob cosub g ρod βod Goal Deviation, % Original Deviation, % Attempt 1 wi Deviation, % Attempt 2 wi Deviation, % Attempt 3 wi Deviation, % Attempt 4 wi Deviation, % Attempt 5 wi Deviation, %

2.00

0.50

2.00

1.00

0.50

10.0

2.00

0.50

30.0

3.63

0.20

3.32

15.8

0.00

28.7

1.94

0.15

24.4

1 3.58

4 0.05

1 3.24

1 0.43

4 1.03

0.1 7.24

1 1.77

1 0.23

0 32.6

1 3.45

2 0.67

1 3.14

1 0.51

4 0.72

0.1 6.43

1 1.28

1 0.42

0 32.6

1 3.45

2 0.63

1 3.14

0.5 0.87

4 0.72

0.1 6.81

1 1.32

1 0.38

0 32.2

1 3.46

2 0.75

1 3.15

0.5 0.94

6 0.64

0.1 6.86

1 1.34

1 0.50

0 32.3

1 3.45

3 0.68

1 3.14

0.5 0.98

6 0.68

0.1 6.90

1 1.31

1 0.43

0 32.2

Table 3.18 Properties of dead oil and natural gas regarding samples PS-01 and PS-02. Comparison of results obtained in the flash liberation test with those predicted by the thermodynamic model before and after adjustment Sample Property Test Original model Adjusted model Value Dev, % Value Dev, % PS-01

PS-02

γod , ◦ API γg 3 /m 3 GOR, mstd std ◦ γod , API γg 3 /m 3 GOR, mstd std

28.5 0.81 243 29.2 0.80 233

29.0 0.83 244 29.6 0.83 235

1.8 2.8 0.5 1.4 3.3 1.1

29.0 0.83 244 30.3 0.84 234

1.8 2.5 0.4 3.8 3.1 0.5

3.7 Results and Validation

141

1.0

(Bo − Bod ) / (Bob − Bod )

0.8

0.6

0.4

0.2 Experimental Original model Adjusted model

0.0 0.00

0.25

0.50 0.75 (P − Patm ) / (Pb − Patm )

1.00

1.25

Fig. 3.29 Variation of the oil formation volume factor as a function of pressure for sample PS-01. Comparison of the results obtained from the differential liberation test with those obtained from the thermodynamic model before and after adjustment

liberation tests, the benefits conferred by the adjustment process are unequivocal. As illustrated in Figs. 3.29, 3.30 and 3.31 for sample PS-01, and Figs. 3.32, 3.33 and 3.34 for sample PS-02, the modifications introduced in determining the critical properties and acentric factor of the SCN fractions, the volume translation of each pseudocomponent, and the molecular mass of the heavy fraction, always result in a more convincing representation of the laboratory-determined values of Bo , Rs , and ρo . In both circumstances, it is evident that the bubble-point pressure had been underestimated by the original model, which affected the phase equilibrium calculation in all subsequent expansion stages. The estimates in the single-phase liquid region remained practically unchanged. The fitting alone was unable to attenuate the somewhat exacerbated increase in oil density at the final pressure levels of each test. The next step involves portraying the combined effect of pressure and gas incorporation on the viscosity of the oil. This is a phenomenon that exhibits significant sample-to-sample variation and is of great importance for numerous studies. Therefore, it is often investigated concurrently with volumetric and phase behavior in the actual differential liberation test. For each stage of depressurization to which a certain mixture is subjected, an experimental viscosity reading of the remaining liquid

142

3 Developing a Fluid Model

1.0

Rs /GOR

0.8

0.6

0.4

0.2 Experimental Original model Adjusted model

0.0 0.00

0.25

0.50 0.75 (P − Patm ) / (Pb − Patm )

1.00

1.25

Fig. 3.30 Variation of the gas solubility ratio in oil as a function of pressure for sample PS-01. Comparison of the results obtained from the differential liberation experiment with those obtained from the thermodynamic model before and after adjustment

phase is obtained using a rolling ball viscometer. This set of values provides a precise adjustment of the coefficients ϕC , ϕ D , and ϕ E appearing in Eqs. (3.60), (3.61), and (3.63). The results of the regressions conducted for samples PS-01 and PS-02 are presented in Figs. 3.35 and 3.36. Upon examination, it is evident that, in both cases, the black oil approach proved perfectly suitable for calculating the property throughout the entire range of considered pressures. Finally, attention is directed towards modeling the viscosity of dead oil and waterin-oil emulsion, which involves adjusting the constants A, B, and ψμl in Eqs. (3.58) and (3.66) based on the readings from rheological tests conducted for each sample. The results, presented in Figs. 3.37 and 3.38, once again support the high capability of the black oil approach to reproduce experimental values, fully justifying the choice made in Sect. 3.2. It is evident that a series of thermodynamic and transport properties of the reservoir fluid often lack experimental results to assist in the validation of their respective calculation models. In such cases, the only option is to compare the estimates obtained with the predictions of another simulator, identifying any points for improvement based on the inconsistencies that may arise. The PVTsim software developed by

3.7 Results and Validation

143

1.2

Experimental Original model Adjusted model

(ρo − ρob ) / (ρod − ρob )

1.0 0.8 0.6 0.4 0.2 0.0 0.00

0.25

0.50 0.75 (P − Patm ) / (Pb − Patm )

1.00

1.25

Fig. 3.31 Variation of the oil density as a function of pressure for sample PS-01. Comparison of the results obtained from the differential liberation test with those obtained from the thermodynamic model before and after adjustment

Calsep,7 for instance, fulfills all the necessary requirements for determining such values. It is a commercial package that has been present in the industry for over 30 years and is one of the market leaders, offering various solutions for the representation of hydrocarbon mixtures mainly from the North Sea. Certainly, the main disadvantage of conducting a sample characterization in proprietary software lies in the limited number of options offered to the user throughout the activity. PVTsim allows the use of the Soave-Redlich-Kwong equation of state with volume translation, the same employed in all simulations thus far, but it provides only one method for splitting the heavy fraction, namely the method proposed by Pedersen et al. (2004), and a single set of correlations for the critical properties and acentric factor of the SCN fractions, published by the same author (see Sect. 2.5). Due to this fact alone, discrepancies on the order of 2% can be expected between the two tools. Furthermore, it was found that the viscosity adjustment of the dead oil implemented in that package introduced significant distortions in the property value as the pressure increased and had to be disregarded.

7

More information at: http://www.calsep.com/about/pvtsim-nova.html.

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3 Developing a Fluid Model

(Bo − Bod ) / (Bob − Bod )

1.0

0.8

0.6

0.4

0.2 Experimental Original model Adjusted model

0.0 0.00

0.25

0.50 0.75 (P − Patm ) / (Pb − Patm )

1.00

1.25

Fig. 3.32 Variation of the oil formation volume factor as a function of pressure for sample PS-02. Comparison of the results obtained from the differential liberation test with those obtained from the thermodynamic model before and after adjustment

1.0

Rs /GOR

0.8

0.6

0.4

0.2 Experimental Original model Adjusted model

0.0 0.00

0.25

0.50 0.75 (P − Patm ) / (Pb − Patm )

1.00

1.25

Fig. 3.33 Variation of the gas solubility ratio in oil as a function of pressure for sample PS-02. Comparison of the results obtained from the differential liberation test with those obtained from the thermodynamic model before and after adjustment

3.7 Results and Validation

145

1.2

Experimental Original model Adjusted model

(ρo − ρob ) / (ρod − ρob )

1.0 0.8 0.6 0.4 0.2 0.0 0.00

0.25

0.50 0.75 (P − Patm ) / (Pb − Patm )

1.00

1.25

Fig. 3.34 Variation of the oil density as a function of pressure for sample PS-02. Comparison of the results obtained in the differential liberation experiment with those obtained by the thermodynamic model before and after adjustment

1.0

Experimental Adjusted model

(μo − μob ) / (μod − μob )

0.8

0.6

0.4

0.2

0.0 0.00

0.25

0.50 0.75 (P − Patm ) / (Pb − Patm )

1.00

1.25

Fig. 3.35 Variation of the dynamic oil viscosity as a function of pressure for sample PS-01. Comparison of the results obtained from the differential liberation test with those obtained from the black oil model

146

3 Developing a Fluid Model

1.0

Experimental Adjusted model

(μo − μob ) / (μod − μob )

0.8

0.6

0.4

0.2

0.0 0.00

0.25

0.50 0.75 (P − Patm ) / (Pb − Patm )

1.00

1.25

Fig. 3.36 Variation of the dynamic oil viscosity as a function of pressure for sample PS-02. Comparison of the results obtained from the differential liberation test with those obtained from the black oil model 5

50

Experimental Adjusted model

40

μl /μo

μod /μod,333K

4

3

2

30 20 10 0

1 290

Experimental Adjusted model

300

310

T, K

320

330

0

20

40

60

BSW, %

Fig. 3.37 Variation of the dynamic viscosity of dead oil as a function of temperature and of the emulsion as a function of BSW for sample PS-01. Comparison of the results obtained from the rheology test with those obtained from the black oil model

3.7 Results and Validation 70

Experimental Adjusted model

Experimental Adjusted model

60

4

50

μl /μo

μod /μod,333K

5

147

3

40 30 20

2

10 1 290

0 300

310

T, K

320

330

0

20

40

60

BSW, %

Fig. 3.38 Variation of the dynamic viscosity of dead oil as a function of temperature and of the emulsion as a function of BSW for sample PS-02. Comparison of the results obtained from the rheology test with those obtained from the black oil model

Despite the setbacks, PVTsim was able to establish reasonably acceptable compositional models for the PS-01 and PS-02 samples. Clearly, in both cases, the standardized approach imposed by the software, which disregards the particularities of pre-salt reservoir fluids, brought certain disadvantages to the representation of volumetric and phase behavior observed in the laboratory. However, overall, the deviations remained small and will not be detailed here for the sake of conciseness. The comparison between the commercial simulator and the work developed here is presented in Tables 3.19 and 3.20, which provide the predicted properties for the liquid and gas phases of the PS-01 and PS-02 samples at typical pressure and temperature levels associated with flow at the SPU, WCT, and PDG. As it encompasses both oil and formation water, the liquid results are derived from weighting the values stipulated for each of these major constituents using the equations provided in Appendix C. The choice of a 10% BSW in the example was arbitrary. According to Tables 3.19 and 3.20, as a general rule, both tools produce very similar estimates. There is a more pronounced discrepancy in the gas properties for P = 30 MPa and T = 393 K, where the duplication of characterization schemes becomes more evident and leads the flash calculation to different outcomes. If it were possible to use a single set of equations, these deviations would certainly be reduced (as well as all the others, it should be noted). It is worth mentioning that the viscosity fitting of the oil in PVTsim showed inconsistencies, and the differences that arise in this property as the temperature decreases are a result of this. On the other hand, the deviations related to surface tension are within the expected 15% range stated by Ramey (1973) for their black oil correlation. Lastly, a pronounced disparity is observed between the estimates of the thermal expansion coefficient of the oil, especially for P = 4 MPa and T = 333 K. It is impossible to determine which one is more accurate in each case, and considering the numbers in Table 3.4,

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3 Developing a Fluid Model

Table 3.19 Thermodynamic and transport properties of the liquid and gas phases related to the PS-01 sample with 10% BSW, at pressure and temperature levels typically encountered at the surface, WCT, and PDG of pre-salt OPWs. Comparison of the values obtained by the fluid model with those of the commercial simulator PVTsim Location Property Gas Liquid Model PVTsim Dev, % Model PVTsim Dev, % SPU P = 4 MPa T = 333 K

WCT P = 14 MPa T = 373 K

PDG P = 30 MPa T = 393 K

ρ, kg/m3

32.9

32.9

0.1

852

850

0.2

, % c, 10−7 /Pa β, 10−3 /K c P , J/(kg · K) μ, 10−3 Pa · s σ, 10−3 J/m2 ρ, kg/m3

14.2 2.61 3.99 2303 0.013 16.6 107

14.2 2.62 3.99 2321 0.013 16.4 108

–0.3 –0.1 –0.2 –0.8 –4.2 1.2 –0.8

85.8 0.013 0.66 2155 3.27 16.6 792

85.8 0.012 0.41 2230 4.92 16.4 789

0.0 5.7 62 –3.3 –34 1.2 0.4

, % c, 10−7 /Pa β, 10−3 /K c P , J/(kg · K) μ, 10−3 Pa · s σ, 10−3 J/m2 ρ, kg/m3

11.5 0.72 4.41 2826 0.021 7.25 214

11.6 0.72 4.46 2871 0.018 6.40 228

–1.3 –0.4 –1.3 –1.6 14 13 –6.2

88.5 0.020 0.87 2305 1.25 7.25 723

88.4 0.020 0.71 2338 1.58 6.40 719

0.2 3.3 21 –1.4 –21 13 0.5

, % c, 10−7 /Pa β, 10−3 /K c P , J/(kg · K) μ, 10−3 Pa · s σ, 10−3 J/m2

5.68 0.24 3.65 3031 0.035 1.61

6.25 0.23 3.71 3082 0.029 1.11

–9.2 3.9 –1.6 –1.6 20 45

94.3 0.026 1.04 2410 0.82 1.61

93.7 0.025 0.94 2408 0.83 1.11

0.6 0.7 11 0.1 –0.6 45

it is highly likely that both of them deviate from the truth at some point. However, it is important to note that this parameter results in minimal variations in the density of the respective phase and will, therefore, have little impact on the flow simulations discussed in the following chapters.

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149

Table 3.20 Thermodynamic and transport properties of the liquid and gas phases related to the PS-02 sample with 10% BSW, at pressure and temperature levels typically encountered at the surface, WCT, and PDG of pre-salt OPWs. Comparison of the values obtained by the fluid model with those of the commercial simulator PVTsim Location Property Gas Liquid Model PVTsim Dev, % Model PVTsim Dev, % SPU P = 4 MPa T = 333 K

WCT P = 14 MPa T = 373 K

PDG P = 30 MPa T = 393 K

ρ, kg/m3

32.7

32.6

0.1

847

847

0.0

, % c, 10−7 /Pa β, 10−3 /K c P , J/(kg · K) μ, 10−3 Pa · s σ, 10−3 J/m2 ρ, kg/m3

13.6 2.61 3.97 2306 0.013 16.5 106

13.7 2.61 3.98 2323 0.013 16.6 107

–0.1 –0.1 –0.1 –0.8 –4.3 –0.5 –0.6

86.4 0.012 0.66 2150 3.11 16.5 789

86.3 0.012 0.41 2250 4.08 16.6 788

0.0 3.8 61 –4.5 –24 –0.5 0.2

, % c, 10−7 /Pa β, 10−3 /K c P , J/(kg · K) μ, 10−3 Pa · s σ, 10−3 J/m2 ρ, kg/m3

11.0 0.72 4.39 2826 0.021 7.34 212

11.1 0.72 4.44 2871 0.018 6.60 223

–0.9 –0.4 –1.1 –1.6 15 11 –4.7

89.0 0.019 0.86 2299 1.36 7.34 723

88.9 0.019 0.70 2355 1.70 6.60 724

0.1 4.1 24 –2.4 –20 11 –0.1

, % c, 10−7 /Pa β, 10−3 /K c P , J/(kg · K) μ, 10−3 Pa · s σ, 10−3 J/m2

5.33 0.24 3.65 3033 0.036 1.69

5.94 0.24 3.71 3090 0.029 1.26

–10 2.9 –1.6 –1.9 25 35

94.7 0.024 1.03 2403 0.95 1.69

94.1 0.023 0.90 2419 0.95 1.26

0.7 4.5 15 –0.7 0.0 35

References Ahmed, T. (2016). Equations of state and PVT analysis: Applications for improved reservoir modeling (2nd ed.). Cambridge, UK: Gulf Professional Publishing. Al-Meshari, A. A., & McCain, W. D. (2006). An accurate set of correlations for calculating critical properties and acentric factors for single carbon number groups. In Paper no. 106338 presented at the SPE technical symposium of Saudi Arabia Section, Dhahran. Antony, J. (2014). Design of experiments for engineers and scientists. Elsevier Insights: Elsevier Science. Bahadori, A., Mahmood, T., Lee, M., & Phung, L. T. K. (2016). Prediction of physical properties of hydrocarbon compounds using empirical correlations. Petroleum Science and Technology, 34(19), 1631–1635. Bahadori, A., Vuthaluru, H., & Mokhatab, S. (2009). Rapid estimation of water content of sour natural gases. Journal of the Japan Petroleum Institute, 52(5), 270–274.

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Bergman, D. F., & Sutton, R. P. (2006). Undersaturated oil viscosity correlation for adverse conditions. In Paper no. 103144 presented at the SPE annual technical conference and exhibition, San Antonio, Texas. Bergman, D. F., & Sutton, R. P. (2007a). A consistent and accurate dead-oil-viscosity method. In Paper no. 110194 presented at the SPE annual technical conference and exhibition, Anaheim, California. Bergman, D. F., & Sutton, R. P. (2007b). An update to viscosity correlations for gas-saturated crude oils. In Paper no. 110195 presented at the SPE annual technical conference and exhibition, Anaheim, California. Box, G., Hunter, J., & Hunter, W. (2005). Statistics for experimenters: Design, innovation, and discovery. Wiley, Hoboken, NJ: Wiley Series in Probability and Statistics. Box, G. E. P. (1954). The exploration and exploitation of response surfaces: Some general considerations and examples. Biometrics, 10(1), 16–60. Box, G. E. P., & Wilson, K. B. (1951). On the experimental attainment of optimum conditions. Journal of the Royal Statistical Society. Series B (Methodological), 13(1), 1–45. Cavett, R. H. (1962). Physical data for distillation calculations, vapor-liquid equilibria. In Proceedings of the 27th API meeting (Vol. 42, pp. 351–366). American Petroleum Institute. Christensen, P. L. (1999). Regression to experimental PVT data. Journal of Canadian Petroleum Technology, 38(13). Churchill, S. W., & Usagi, R. (1972). A general expression for the correlation of rates of transfer and other phenomena. AIChE Journal, 18(6), 1121–1128. Culberson, O. L., & McKetta, J. J. (1950). Phase equilibria in hydrocarbon-water systems II - the solubility of ethane in water at pressures to 10,000 psi. Journal of Petroleum Technology, 2(11), 319–322. Culberson, O. L., & McKetta, J. J. (1951). Phase equilibria in hydrocarbon-water systems III the solubility of methane in water at pressures to 10,000 psia. Journal of Petroleum Technology, 3(08), 223–226. Danesh, A., Xu, D.-H., & Todd, A. C. (1991). Comparative study of cubic equations of state for predicting phase behaviour and volumetric properties of injection gas-reservoir oil systems. Fluid Phase Equilibria, 63(3), 259–278. Daubert, T. E., & Danner, R. P. (Eds.). (1997). API technical data book - petroleum refining (6th ed.). Washington, DC: American Petroleum Institute. Elliott, J., & Lira, C. (2012). Introductory chemical engineering thermodynamics (2nd ed.). Upper Saddle River, NJ: Prentice-Hall. Graboski, M. S., & Daubert, T. E. (1978). A modified Soave equation of state for phase equilibrium calculations. 1. Hydrocarbon systems. Industrial and Engineering Chemistry Process Design and Development, 17(4), 443–448. Jamieson, D., Tudhope, J., Morris, R., & Cartwright, G. (1969). Physical properties of sea water solutions: heat capacity. Desalination, 7(1), 23–30. Katz, D. L., Monroe, R. R., & Trainer, R. P. (1943). Surface tension of crude oils containing dissolved gases. Petroleum Technology, 6(5), 1–10. Kesler, M. G., & Lee, B. I. (1976). Improve prediction of enthalpy fractions. Hydrocarbon Processing, 55(3), 153–158. Kestin, J., Khalifa, H. E., Abe, Y., Grimes, C. E., Sookiazian, H., & Wakeham, W. A. (1978). Effect of pressure on the viscosity of aqueous sodium chloride solutions in the temperature range 20–150◦ C. Journal of Chemical and Engineering Data, 23(4), 328–336. Lee, A. L., Gonzalez, M. H., & Eakin, B. E. (1966). The viscosity of natural gases. Journal of Petroleum Technology, 18(8), 997–1000. Lee, B. I., & Kesler, M. G. (1975). A generalized thermodynamic correlation based on threeparameter corresponding states. AIChE Journal, 21(3), 510–527. Montgomery, D. (2012). Design and Analysis of Experiments (8th ed.). Hoboken, NJ: Wiley, Incorporated.

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Neau, E., Hernández-Garduza, O., Escandell, J., Nicolas, C., & Raspo, I. (2009). The Soave, Twu and Boston-Mathias alpha functions in cubic equations of state: Part I. Theoretical analysis of their variations according to temperature. Fluid Phase Equilibria, 276(2), 87–93. Pal, R., & Rhodes, E. (1989). Viscosity/concentration relationships for emulsions. Journal of Rheology, 33(7), 1021–1045. Pan, H., Firoozabadi, A., & Fotland, P. (1997). Pressure and composition effect on wax precipitation: Experimental data and model results. SPE Production and Facilities, 12(4), 250–258. Pedersen, K., Christensen, P., & Shaikh, J. (2014). Phase behavior of petroleum reservoir fluids (2nd ed.). Boca Raton, FL: CRC Press. Pedersen, K. S., Milter, J., & Sørensen, H. (2004). Cubic equations of state applied to HT/HP and highly aromatic fluids. SPE Journal, 9(2), 186–192. Peneloux, A., Rauzy, E., & Freze, R. (1982). A consistent correction for Redlich-Kwong-Soave volumes. Fluid Phase Equilibria, 8(1), 7–23. Peng, D.-Y., & Robinson, D. B. (1976). A new two-constant equation of state. Industrial and Engineering Chemistry Fundamentals, 15(1), 59–64. Pina-Martinez, A., Privat, R., Jaubert, J.-N., & Peng, D.-Y. (2019). Updated versions of the generalized Soave α-function suitable for the Redlich-Kwong and Peng-Robinson equations of state. Fluid Phase Equilibria, 485, 264–269. Pitzer, K. S., Lippmann, D. Z., Curl Jr., R. F., Huggins, C. M., & Petersen, D. E. (1955). The volumetric and thermodynamic properties of fluids. II. Compressibility factor, vapor pressure and entropy of vaporization. Journal of the American Chemical Society, 77(13), 3433–3440. Privat, R., Visconte, M., Zazoua-Khames, A., Jaubert, J.-N., & Gani, R. (2015). Analysis and prediction of the alpha-function parameters used in cubic equations of state. Chemical Engineering Science, 126, 584–603. Ramey, H. J. J. (1973). Correlations of surface and interfacial tensions of reservoir fluids. In Paper 4429 available from SPE, Richardson, Texas. Riazi, M., Al-Adwani, H., & Bishara, A. (2004). The impact of characterization methods on properties of reservoir fluids and crude oils: Options and restrictions. Journal of Petroleum Science and Engineering, 42(2), 195–207. Riazi, M. R. (2005). Characterization and properties of petroleum fractions. ASTM manual series MNL 50. Philadelphia, PA: ASTM International. Riazi, M. R., & Al-Sahhaf, T. A. (1996). Physical properties of heavy petroleum fractions and crude oils. Fluid Phase Equilibria, 117(1), 217–224. Riazi, M. R., & Daubert, T. E. (1987). Characterization parameters for petroleum fractions. Industrial and Engineering Chemistry Research, 26(4), 755–759. Robinson, D., & Peng, D. (1978). The characterization of the Heptanes and Heavier fractions for the GPA Peng-Robinson programs. Tulsa, OK: Gas Processors Association. Rowe, A. M., & Chou, J. C. S. (1970). Pressure-volume-temperature-concentration relation of aqueous sodium chloride solutions. Journal of Chemical and Engineering Data, 15(1), 61–66. Salsburg, D. (2009). Uma senhora toma chá: como a estatística revolucionou a ciência do século XX. Rio de Janeiro: Zahar. Soave, G. (1972). Equilibrium constants from a modified Redlich-Kwong equation of state. Chemical Engineering Science, 27(6), 1197–1203. Stryjek, R., & Vera, J. H. (1986). PRSV: An improved Peng-Robinson equation of state for pure compounds and mixtures. The Canadian Journal of Chemical Engineering, 64(2), 323–333. Sutton, R. P. (2009). An improved model for water-hydrocarbon surface tension at reservoir conditions. In Paper no. 124968 presented at the SPE annual technical conference and exhibition, New Orleans, Louisiana. Takenouchi, S., & Kennedy, G. (1964). The binary system of H2 O-CO2 at high temperatures and pressures. American Journal of Science, 262(9), 1055–1074. Tewari, R., Dandekar, A., & Ortiz, J. (2018). Petroleum fluid phase behavior: Characterization, processes, and applications. Emerging trends and technologies in petroleum engineering. Boca Raton, FL: CRC Press.

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Chapter 4

Fluid Flow in Oil Production Systems

4.1 Introduction The study of liquid and gas flow within petroleum production systems began in the 1950s, with a focus on applications in vertical wells. During that time, the data primarily relied on operational readings obtained in the field. These included parameters such as volumetric flow rate, thermophysical properties of each phase, pressure inside the pipeline, internal diameter, and inclination angle. Based on this modest set of information, works like those of Poettman and Carpenter (1952), Baxendell and Thomas (1961) emerged. In both studies, the inability to measure the velocity of the liquid and gas or determine the flow pattern led to the neglect of such effects. Hagedorn and Brow (1965) attempted to address this limitation by proposing an empirical model to calculate phase slip, which resulted in a rudimentary estimate of the volume occupied by each phase. However, the data used in the regression analysis were not measured experimentally but retrocalculated from the pressure gradient. Later, the development of dedicated test facilities allowed for the observation of flow patterns through a visualization section, as well as the estimation of the volumetric liquid fraction by installing quick-closing valves (Brill & Arirachakaran, 1992). This is how, for example, the correlation by Eaton et al. (1967) emerged as one of the pioneers in modeling horizontal oil flows (θ = 0◦ ). The work of Duns and Ros (1963) for vertical production wells (θ = 90◦ ) was also based on a considerable amount of experimental data. For the first time, the problem solution began with the determination of the flow pattern using an empirical map created for this purpose. Specific correlations for each type of arrangement also provided a more accurate evaluation of the other parameters. This strategy proved to be perfectly suitable, as it was repeated by almost all subsequent authors. Orkiszewski (1967) compared a significant number of models published until then and suggested several modifications aimed at improving the accuracy of the analyses in relation to readings from actual wells. In turn, Beggs and Brill (1973), having a test section made of acrylic material capable of being rotated in any direction, developed a unified modeling of flow in © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. R. Gessner and J. R. Barbosa Jr., Integrated Modeling of Reservoir Fluid Properties and Multiphase Flow in Offshore Production Systems, Petroleum Engineering, https://doi.org/10.1007/978-3-031-39850-6_4

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4 Fluid Flow in Oil Production Systems

wellbores and production pipelines, which quickly gained popularity and still has fervent supporters to this day. In the 1980s, according to Brill and Arirachakaran (1992), with the widespread adoption of computers in large companies, the aforementioned empirical correlations became part of numerical algorithms for simulating pressure profiles and flow rates in wells and pipelines. This initiative, on the one hand, brought about extraordinary improvements in design tools and production monitoring, but on the other hand, it revealed a series of problems associated with flow models. Flow regime transitions, for example, previously attributed primarily to volumetric flow rates, were found to be highly sensitive to other parameters, such as the inclination angle, and their representation was still inadequate. Discontinuities between correlations related to different flow patterns made the method unstable, while older models that did not make such distinctions were overly simplistic and imprecise. It was thus established that, regardless of the amount of data collected from experimental setups or field installations, the quality of predictions could not be improved without first conducting a more in-depth investigation of the underlying phenomena. Simultaneously, the nuclear industry celebrated significant progress in this field. Zuber and Findlay (1965), supported by a series of excellent works by other authors, proposed mechanistic models for bubble and slug flow patterns, while a detailed description of annular flow was provided by Hewitt and Hall-Taylor (1970). In this latter work, several pattern transition criteria were also established, similar to what Wallis (1969) had done shortly before. Although the fluids used in all these studies, water and steam, were found to be trivial compared to those encountered in the oil industry, the modeling of their flow in vertical pipelines was evidently well-founded. The first adaptation for this new context was likely performed by Aziz et al. (1972). Taitel and Dukler (1976) not only reformulated the mathematical treatment devoted to the stratified pattern in horizontal or low-inclination pipelines but also established all their stability criteria. The slug flow in such a configuration received a thorough analysis by Dukler and Hubbard (1975). Later, Taitel et al. (1980) revisited the transition mechanisms involved in vertical flow, culminating in the unified model proposed by Barnea (1987). It is worth emphasizing that, according to Taitel (1994), the term mechanistic modeling carries the idea of identifying and representing the dominant mechanisms of a specific process, while neglecting secondary aspects that either overly complicate the problem’s solution or yield marginal gains. In the case of oil flow, such an approach assumes a thorough understanding of the assumed topology of the phases involved and the key phenomena associated with it, while also being simple enough to ensure that the computational effort required for the solution remains within acceptable limits. Considering that they stem from one-dimensional formulations, mechanistic models rely on closure relationships capable of replacing the various information lost in the process of averaging the conservation equations over the cross-section of the duct. Friction factors at the pipe wall and gas-liquid interface, the fraction of entrained liquid in the gas core, and the degree of aeration of the liquid slug are some of the flow parameters evaluated through correlations, often involving a certain degree of

4.1 Introduction

155

empiricism. According to Shippen and Bailey (2012), as a result, the major future contributions in this field will not come in the form of new models, but rather as new comprehensive and physically-based closure relationships. Returning to the historical context (Brill & Arirachakaran, 1992), the 1980s were also marked by the growth of oil exploration and production in offshore fields, including the Arctic region. The high implementation costs of these projects prompted multimillion-dollar investments in new research, fostered by consortia in the United States, Norway, France, and the United Kingdom. Consequently, test facilities were constructed, incorporating the latest and most sophisticated experimental techniques for measuring the extensive array of flow variables. The use of densitometers, ultrasonic and capacitance sensors, laser Doppler anemometers, and highspeed photographic techniques proliferated. Computerized data acquisition systems enabled the collection of a large quantity of high-quality information. Simultaneously, researchers agreed that understanding the interaction mechanisms between phases required a combined experimental and theoretical approach. Together, all these factors culminated in the significant enhancement of petroleum flow models witnessed in the subsequent years. Xiao et al. (1990) proposed one of the early mechanistic models referred to as comprehensive models.1 In order to represent horizontal or slightly inclined flows, they assembled a set of equations for predicting the flow pattern, and specific calculation models for the volumetric fraction and shear stress for each arrangement. The validation of their work was conducted using a mixed database consisting of experimental readings conducted in the laboratory and pressure records from actual operating wells, totaling 426 data points covering diameters from 50.8 to 660.4 mm. It was concluded that the model demonstrated superior performance compared to all four tested empirical correlations, yielding a mean absolute percentage error (MAPE) of 30.5%. Bendiksen et al. (1991) presented the studies conducted in the laboratories of SINTEF over the course of a decade. These investigations gave rise to the transient simulator OLGA,2 which continues to be commercially available and holds a leading market position. The conservation equations used, as well as the closure equations related to each pattern (not all are explicitly mentioned in the text), cover inclination angles between −15 ≤ θ ≤ 90◦ and a wide range of diameters (25–200 mm) and pressure levels (up to 10 MPa). Ultimately, a good agreement was demonstrated between the simulations of pressure drop, volume fractions, and flow patterns, and the experimental data presented by them. Petalas and Aziz (1994) developed a model for vertical or highly inclined flows, employing a methodology similar to that used by Xiao et al. (1990). Their validation was based on a dataset comprising 1,712 well operation records, encompassing fields ranging from heavy oil to gas condensate mixtures, flowing through pipes with 1

Comprehensive models are composed of correlations for determining the flow pattern, and a set of independent mechanistic models for calculating the volumetric fractions and shear stress in each pattern. 2 More information at: https://www.software.slb.com/products/olga.

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4 Fluid Flow in Oil Production Systems

diameters ranging from 25 to 203 mm at flow rates of up to 27,000 barrels per day. Although the authors claim that their estimates outperformed the mechanistic model by Hasan and Kabir (1988) and six other empirical correlations, the results did not clearly indicate such superiority, and it is possible that the work of Hagedorn and Brown (1965) performed better in comparison. Petalas and Aziz (1998) were pioneers in the creation of a unified mechanistic model capable of representing flows with any inclination angle (−90 ≤ θ ≤ 90◦ ). In addition to utilizing closure relationships already employed in previous works, such as Xiao et al. (1990) and Ansari et al. (1994), the authors introduced new expressions for calculating the interfacial friction factors in the stratified and annular flow patterns, the liquid phase friction factor in stratified flow, and the Taylor bubble velocity and relative slug length in slug flow. The validation was based on a set of 5,951 experimental readings of pressure drop, volumetric fractions, all obtained in a laboratory setup. The volumetric liquid fraction was accurately predicted within ±15% in 3,663 instances (62% of the total), while the pressure drop remained within this range in 2,567 cases (43%). None of the four evaluated empirical correlations achieved such values. Gomez et al. (2000b) generated another unified model, this time solely for 0 ≤ θ ≤ 90◦ . By adopting the same transition criteria suggested by Barnea (1987) and formulations from various authors, such as Taitel and Dukler (1976) for stratified flow, Alves et al. (1991) for annular flow, Taitel and Dukler (1990) for slug flow, and Hasan and Kabir (1988) for bubbly flow, the study focused on selecting the best closure equations available at that time. To validate the assigned flow pattern set, the authors relied on 260 laboratory readings, encompassing pipe diameters ranging from 50 to 100 mm subjected to pressures up to 12 MPa. At this stage, depending on the phase topology, the MAPE reached 9.8% for pressure drop calculation and 9.6% for the volumetric liquid fraction. Subsequently, by utilizing the database of Ansari et al. (1994), the authors achieved a MAPE of 12.6%. This value was only surpassed by the empirical correlation of Hagedorn and Brown (1965), which registered 9.3%. Finally, Zhang et al. (2003b, c) based their model on the premise that slug flow shares boundaries with all other flow patterns. The authors improved the momentum conservation equation for slug flow and established a new correlation for calculating the gas volume fraction in the slug (Zhang et al., 2003a). The slug flow model, applicable for the entire range of −90 ≤ θ ≤ 90◦ , aimed not only to calculate the flow properties in this configuration but also to characterize its transitions. Additionally, closure equations were selected to eliminate any discontinuities in the results. The flow pattern determination was validated using laboratory data from four different authors, covering pipe diameters ranging from 25 to 73 mm and pressures up to 3 MPa. On the other hand, the estimates for liquid volume fraction and pressure drop were supported by an even larger database, with diameters ranging from 23 to 73 mm and pressures not exceeding 600 kPa. However, for unknown reasons, the authors did not determine the overall deviation of the model, which was only addressed graphically, nor did they employ any other correlation to assess its accuracy. The one-dimensional approach is discouraged by Bonizzi et al. (2009) in situations where transient phenomena predominate, such as those observed in churn flow.

4.2 Stratified Flow

157

Multidimensional models are better suited for such purposes. Nonetheless, there are several practically relevant problems that can be addressed using the former approach, although the success of the endeavor depends on its accurate representation. In this context, the following sections will provide detailed explanations of the most prominent phenomenological aspects behind the following flow patterns: stratified flow (Sect. 4.2), annular flow (Sect. 4.3), bubbly or dispersed bubble flow (Sect. 4.4), and slug or elongated bubble flow (Sect. 4.5). The main aspects related to flow pattern transitions are described in Sect. 4.6. Finally, in Sect. 4.7, two formulations for solving two-phase flows in everyday engineering applications are presented.

4.2 Stratified Flow In stratified flow, the space occupied by each of the phases is governed by two main effects. Firstly, gravitational attraction, which has a greater influence on the heavier fluid, acts to slow down the movement of liquid in upward pipes and accelerate it in downward pipes. The second effect arises from the shear stresses generated at the pipe wall, which hinder the movement of the liquid and gas, in addition to the interfacial shear stress that reduces the apparent slip between them. Notice that interfacial shear gives rise to a breakup of the continuous liquid into droplets carried downstream by the gas. However, the occurrence of significant droplet entrainment and deposition is an exception rather than the rule in this type of flow arrangement (Fabre, 2003b, p. 79), and will be disregarded for now. Thus, based on the assumptions of fully developed flow and uniform stress distribution within it, as proposed by Taitel and Dukler (1976), the force balance exerted on the liquid and gas phases establishes that:  − Al  − Ag

dP dL

dP dL

 − τwl Sl + τi Si − ρl Al g sin θ = 0,

(4.1)

− τwg Sg − τi Si − ρg Ag g sin θ = 0.

(4.2)

l

 g

where Sl , Sg and Si are the liquid and gas wetted perimeters and the interfacial perimeter (area per unit pipe length). By equating the pressure gradients, one gets: τwl Sl τwg Sg − + τi Si Ag Al



1 1 + Ag Al



  − ρl − ρg g sin θ = 0.

(4.3)

Equation (4.3) does not have a direct solution. Instead, it represents a sort of objective function of the flow model, which, when satisfied through the application of some iterative method, ends up fixing all the other parameters involved (Xiao et al., 1990; Gomez et al., 2000b). Many authors prefer to express it in terms of the liquid level in the pipe (h l ), although there are other alternatives available.

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4 Fluid Flow in Oil Production Systems

4.2.1 Interface Shape The perimeters Sl , Sg , and Si appearing in Eqs. (4.1)–(4.3) exhibit a strong interdependence. When the gas attains sufficiently high velocities within the pipe, for instance, it is possible for the interface to adopt a concave shape due to the spreading of the liquid phase towards the wall, resulting in an increase in the values of Si and Sl , while reducing Sg . Conversely, gravity-dominated systems with a high Eotvos number3 feature a practically flat interface (Ullmann & Brauner, 2006). This curvature effect is further pronounced in flows with a low volumetric fraction of liquid, as observed in the production of condensate gas (Chen et al., 1997). The Eotvos number of the system can be expressed as: ρg D 2 , σ

(4.4)

ρ = ρl − ρg .

(4.5)

Eo = where:

The mechanisms responsible for the morphology of the interface have not yet been fully understood. According to Zhang and Sarica (2011), historically, the main hypotheses raised regarding this matter have included the effects of surface tension, ripple action, gas secondary flow, and droplet deposition on the pipe wall. However, for the purpose of modeling, it is customary to correlate the potential energy gain of the liquid phase (due to the increase in its center of gravity) with its kinetic energy, as suggested by Hart et al. (1989). This correlation leads to functional relationships of the following form: (4.6) θ f = θ f (θ f,flat , Frl ). where θ f and θ f,flat are, respectively, the wetting angle of the liquid film with and without the curvature effect and Frl is the Froude4 number of the liquid phase, given by: Uk Frk =  . (4.7) ρ g D cos θ ρk Other authors, such as Zhang and Sarica (2011), have added to their regressions the gas phase Froude number, Frg . Grolman and Fortuin (1997) went further, considering the influence of surface tension, through the parameter σ/σref .

3 4

Named in honor of the Hungarian physicist Loránd Eötvös (1848–1919). In honor of the English engineer William Froude (1810–1879).

4.2 Stratified Flow

159

Fig. 4.1 Geometric parameters of the stratified flow pattern assuming a flat interface

In the idealized scenario of a perfectly flat interface, as illustrated in Fig. 4.1, Sl , Sg , Si , Ag , Al and h l are calculated in terms of θ f,flat using the following trigonometric relationships: Dθ f,flat , (4.8) Sl = 2 Sg = π D − Sl ,

(4.9)

 D 2 − 2 cos θ f,flat , Si = 2   D 2 θ f,flat − sin θ f,flat Al = , 8 A g = A − Al , hl =

 θ D 1 − cos f,flat 2 2

(4.10)

(4.11) (4.12)

.

(4.13)

In the event that one chooses to consider the curvature of the gas-liquid interface, the double circle model by Chen et al. (1997) is a good starting point.

4.2.2 Shear Stresses The characterization of stratified flows also requires a highly accurate estimation of the wall and interfacial shear stresses. Regarding this matter, Fabre (2003b, p. 79)

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4 Fluid Flow in Oil Production Systems

highlights that considerable velocity differences observed in some cases could mistakenly suggest that the transfer of momentum between phases is ineffective. However, one must not forget that greater interfacial friction leads to a higher mechanical energy loss by the gas phase. Therefore, considering that the pressure drop is the same across both media, the term d P/d L represents an important driving force for the liquid. This observation demonstrates that friction plays a central role in this type of arrangement. The simplest approach for predicting the values of τwl , τwg , and τi , which has been replicated by numerous researchers since the work of Taitel and Dukler (1976), is based on the calculation of friction factors.5 Thus: ρk |Uk | Uk , 2

  ρg Ug − Ul Ug − Ul τi = f i . 2 τwk = f k

(4.14)

(4.15)

The liquid and gas wall friction factors ( f k ) are written in terms of the Reynolds number6 of the respective phase (Rek ) and the pipe roughness (e), as follows: f k = f k (Rek , e/Dk ).

(4.16)

where Dk is the hydraulic diameter of the cross sectional area occupied by phase k and: ρk Uk Dk . (4.17) Rek = μk Once the values of τwl and τwg are known, the average wall shear stress is given by: τwg Sg + τwl Sl . (4.18) τw = πD The representation of the interfacial friction factor ( f i ) is significantly more challenging, primarily due to the deformability of the contact region between the liquid and gas phases. The presence of interfacial waves, for instance, amplifies the momentum transfer several times compared to that induced by a smooth surface, resulting in an increased pressure drop and a reduced liquid volumetric fraction. According to Tzotzi and Andritsos (2013), despite its undeniable importance in flow solutions, the modeling of this parameter still lacks refinement. The authors also suggest that the interface of the stratified arrangement can assume four distinct forms: 1. A smooth surface, free from any oscillation, observable only at low flow rates; 2. A surface with two-dimensional, regular, short-length disturbances, also called Jeffreys waves (Jeffreys, 1925), caused by resonance with the pressure fluctu5 6

In this study, the Fanning formulation was chosen. However, some authors prefer the Darcy model. In honor of the English physicist Osborne Reynolds (1842–1912).

4.2 Stratified Flow

161

ations traveling through the pipe. The two-dimensional waves are reasonably periodic and uniform, although their amplitude and length increase as they move downstream; 3. A region with irregular and high amplitude disturbances referred to as roll waves resulting from Kelvin-Helmholtz instabilities, as discussed in Sect. 4.6. These three-dimensional waves exhibit a tendency to alter their shape while propagating through the pipe; 4. An atomization region, where liquid droplets are extracted from the crests of the roll waves to deposit on the pipe wall. This phenomenon bears resemblance to the annular flow of viscous liquids in large-diameter pipes, where the liquid film predominantly occupies the lower region. In the first configuration, the gas velocity profile in the interface region closely approximates that near the pipe wall (Taitel & Dukler, 1976), leading to the approximation f i ≈ f g . The absence of oscillations allows for the use of correlations originally developed for single-phase flow in smooth tubes, such as the Blasius equation as follows: (4.19) f i,smooth = f i,smooth (Rei ), where:

  ρg Ug − Ul Dg . Rei = μg

(4.20)

Several studies in the literature, however, adopt Rei = Reg . As the gas flow rate increases and the resulting Jeffreys waves emerge, the value of f i starts to rise rapidly, surpassing f g (Fabre, 2003b, p. 110). From then on, the shear at the interface is strongly influenced by the geometric characteristics of such structures, which, roughly speaking, become the main contributors to the roughness of the location. It is known, for example, that the height of the waves correlates with the dimensionless liquid film thickness normalized by the pipe diameter, which is proportional to αl . Therefore, it is common to represent the interfacial friction factor of the second and third aforementioned configurations through expressions of the following form: f i,wavy = f i,wavy ( f i,smooth , αl , Usg /Usg,ref ).

(4.21)

Some authors, such as Xiao et al. (1990) and Grolman and Fortuin (1997), have taken a different approach and developed correlations for the apparent roughness of the interface, which can be directly used in the Moody diagram, along with the value of Reg for the flow. Andreussi and Persen (1987), on the other hand, substituted the velocity ratio in Eq. (4.21) with Frg − Frg,ref , another valid way to consider the influence of the gas phase flow rate. These reference values determine the transition between the smooth and wavy stratified flow patterns. Although two-dimensional waves should not be confused with roll waves, as a general rule, correlations for f i,wavy do not differentiate between them and purportedly apply to both. In one of the few studies where the three-dimensional configura-

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4 Fluid Flow in Oil Production Systems

tion was separately analyzed, Tzotzi and Andritsos (2013) deemed it appropriate to incorporate the viscosity effect of the liquid, resulting in the following equation: f i,wavy = f i,wavy ( f i,smooth , αl , Usg /Usg,ref , μl /μref ).

(4.22)

Finally, according to Andritsos (1992), although the effects of gas flow rate growth are felt, the overall appearance of the waves does not change during the transition from roll waves to the atomization region. This fact suggests that the Kelvin-Helmholtz instability is primarily responsible for the break up of droplets from the liquid film when the flow’s kinetic energy allows it. Consequently, all correlations of f i associated with either of these configurations would also remain valid for the other.

4.3 Annular Flow Annular flow, as depicted in Fig. 4.2, exhibits mechanisms similar to those of the stratified pattern. Here, once again, the hydrodynamic equilibrium between the phases is governed by the mutual action of gravity and shear stresses at the pipe wall and interface. However, this time, a portion of the liquid content is dispersed in the form of small droplets that are carried by the gas, constituting the so-called core of the flow. Behind this effect lie the highly influential phenomena of atomization (entrainment) and deposition (Hewitt & Hall-Taylor, 1970). Repeating the force balance demonstrated in Sect. 4.2 for the liquid film and core regions, while maintaining the same assumptions, yields:

Fig. 4.2 Geometry parameters of the annular flow assuming constant (axissymmetric) film thickness

4.3 Annular Flow

 − Af

dP dL

163



  − τw f S f + τi Si − ρl A f g sin θ + ψSi Uc − U f = 0,

(4.23)

f

   dP     − τi Si − ρg Ag + ρl Ae g sin θ − ψSi Uc − U f = 0. − Ag + Ae dL c (4.24) Equating the pressure gradients gives: 

   τw f S f + τi + ψ Uc − U f Si − Af



1 1 + Ac Af

 − (ρl − ρc ) g sin θ = 0, (4.25)

where: Ac = Ag + Ae , ρc =

ρg Ag + ρl Ae . Ac

(4.26) (4.27)

However, the representation of momentum transfer at the interface often combines the influence of shear and the atomization and deposition of droplets into a single term (Ambrosini et al., 1991). By doing so, the final form of the expression proposed by Alves et al. (1991) for calculating the phase volumetric fractions is: τw f S f + τi Si − Af



1 1 + Ac Af

 − (ρl − ρc ) g sin θ = 0.

(4.28)

Since the liquid film covers the entire inner surface of the pipe, S f = π D. Assuming a circular-shaped interface, its perimeter will be given, regardless of the film thickness distribution, by:  (4.29) Si = π D 1 − α f .

4.3.1 Shear Stresses Alves et al. (1991) developed adapted forms of the expressions proposed by Taitel and Dukler (1976) for calculating shear stresses in stratified flows. They suggest adopting the following equations:



ρl U f U f , τw f = f f 2

  ρc Uc − U f Uc − U f τi = f i . 2

(4.30)

(4.31)

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4 Fluid Flow in Oil Production Systems

In Eq. (4.31), it is assumed that the velocity difference between the gas phase and the entrained droplets is negligible, such that Ug ≈ Ue ≈ Uc . The velocities of the film and the core can be determined by the following expressions: Uf =

(1 − φe ) Usl , αf

(4.32)

Uc =

Usg + φe Usl , αc

(4.33)

where the entrained liquid fraction (φe ) corresponds to: φe =

Use . Usl

(4.34)

The friction factor between the liquid film and the pipe wall is usually determined using functional relationships similar to Eq. (4.16). Although annular flow is a highly turbulent flow configuration, the continuous liquid flow is often characterized by low flow rates, resulting in a locally laminar regime (Hewitt & Hall-Taylor 1970, p. 80). Therefore, the influence of pipe roughness on f f is considered negligible by some researchers. The shear at the interface undergoes the combined action of two mechanisms. The first mechanism, arises from the continuous entrainment of droplets, which are accelerated from U f to Uc . This significantly alters the radial velocity profile observed in the gas core: the more uniform average velocity distribution, characteristic of turbulent flow, gives way to a distribution with more pronounced rates in the central region (Gill et al., 1964). The second mechanism pertains to the occurrence of waves, which once again contribute to the increased apparent roughness of the interface. As stated by Asali et al. (1985), these waves now manifest in two forms: 1. Capillary waves, also known as ripples, are formed at low liquid flow rates. They have a long crest and a steep front, and they move at a reduced speed. The film surface between these waves is approximately smooth; 2. At higher liquid flow rates, roll waves are formed around the pipe circumference and develop considerable velocities. They have a structure similar to that found in stratified flow; however, their shape is preserved for longer distances. The transition between the ripple and roll-wave regimes is accompanied by the onset of liquid breakup from the crests of the roll waves (Asali et al., 1985). Although the wave height is correlated with the film thickness, it is usually several times greater. As a result, Wallis (1969, p. 321) observed that the apparent roughness of the interface is always about four times the average film thickness. Thus, the respective friction factor can be expressed as follows: f i,wavy = f i,wavy ( f i,smooth , α f ), where f i,smooth is given by Eq. (4.19).

(4.35)

4.3 Annular Flow

165

The modeling of this parameter, however, is still far from achieving standardization and encompasses different levels of complexity. Oliemans et al. (1986), for instance, introduced the influence of surface tension through the Weber number.7 Asali et al. (1985) substituted α f with Re f and added Reg to their analysis. Henstock and Hanratty (1976) went further, suggesting the inclusion of μg /μl and ρg /ρl . Ambrosini et al. (1991) adopted all the previously selected dimensionless numbers by Oliemans et al. (1986), plus Reg and ρg /ρl . In more recent studies, f i,smooth has been omitted. Fore et al. (2000) replaced it with Reg . Assuming fully turbulent flow in the core, Belt et al. (2009) arrived at an expression dependent solely on α f . The same approach had previously been taken by Fukano and Furukawa (1998), who also considered the ratio νg /νl .

4.3.2 Liquid Entrained Fraction Liquid entrainment in the gas core is influenced by several factors, such as pipe diameter and orientation, phase velocities, and fluid properties (Govan, 1990). Most of the works in the literature are exclusively focused on vertical flows, assuming uniform film thickness. In horizontal flows, however, gravity not only promotes a heterogeneous distribution of the liquid around the wall but also affects droplet deposition, making the modeling of the phenomenon more complex. In comparison, fewer studies have been conducted on inclined pipelines, and the impact of inclination on droplet behavior is still an open topic (Magrini et al., 2012). Inspecting the various approaches that have been proposed in recent decades, a certain consensus emerges regarding the direct influence of the parameters Weg and Rel on entrainment. The Weber number quantifies the ratio between the gas inertia force and the forces acting at the interface, indicating the propensity for droplet formation. On the other hand, the Reynolds number establishes the relationship between the inertia and viscous forces in the film, reflecting its local turbulence intensity. Other less significant parameters are often included in the analysis in an attempt to enhance correlation accuracy. Therefore, in general, it can be stated that: φe = φe (Weg , Reg , Rel , ρg /ρl ), where: Wek =

ρk Uk2 D . σ

(4.36)

(4.37)

However, due to the high degree of uncertainty in experimental measurements, it is possible to arrive at many other combinations of dimensionless groups. Paleev and Filippovich (1966), for instance, preferred to substitute Rel with the Laplace

7

In homage to the German engineer Moritz Weber (1871–1951).

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4 Fluid Flow in Oil Production Systems

number,8 denoted as Lal , for the liquid phase, given by Lal = Rel2 /Wel . Oliemans et al. (1986) included the effect of the term μg /μl on entrainment, a decision later repeated by Pan and Hanratty (2002a). These latter authors also considered the existence of a minimum value of Rel for the onset of atomization, as previously suggested by Azzopardi and Whalley (1980) and Owen (1986), and later endorsed in the work of Sawant et al. (2008). Pan and Hanratty (2002b) entertained a similar idea for the gas phase as well.

4.4 Bubbly and Dispersed Bubble Flows In the context of these flow patterns, two main effects govern the average velocity of the gas phase. The first effect concerns the relative motion between the gas phase and the liquid, while the second effect relates to the influence of the flow rate and its distribution along the cross-sectional area of the pipe. Although there is naturally some interdependence between these effects, it is generally agreed that representing each one separately brings benefits to the analysis of the phenomenon. In this case, an option is to consider the following equation: Ug = C0 Usm + U0 .

(4.38)

The so-called slip law, expressed in Eq. (4.38) and illustrated in Fig. 4.3, is commonly attributed to Zuber and Findlay (1965). This attribution recognizes the wellfounded and comprehensive approach proposed in their work. However, the authors themselves claim that its development dates back to at least the 1930s. If the parameters C0 and U0 remain constant within a certain range of flow conditions, the slip law demonstrates a linear relationship between Ug and Usm . This behavior is observed not only in the two flow patterns discussed here but also in all other dispersed flow configurations (Azzopardi & Hills, 2003, p. 169). It is worth noting that for C0 = 1 and U0 = 0, the model corresponds to the hypothesis of homogeneous flow, where any difference in velocity between the liquid and gas is disregarded. Combining Eqs. (1.4) and (4.38) gives: αg =

8

Usg . C0 Usm + U0

(4.39)

Named after the French mathematician Pierre-Simon Laplace (1749–1827), one of the pioneers in the study of capillarity.

4.4 Bubbly and Dispersed Bubble Flows

167

3.0 2.5

Ug , m/s

2.0 1.5 m

1.0 0.5

Ug

=

+

s 6U 1.3

2 0 .2 Ho

mo

g

ou ene

ow s fl

0.0 0.0

0.5

1.0 Usm , m/s

1.5

2.0

Fig. 4.3 Gas bubble velocity as a function of the mixture superficial velocity in a water-steam system pressurized at 4 MPa, flowing in a 50.8-mm diameter pipe (Zuber & Findlay, 1965)

4.4.1 Slip Law Parameters It is known that the slip velocity (U0 ) of the dispersed phase is strongly dependent on its size and shape. In the context of fluid particles moving through stationary media, according to Clift et al. (1978, p. 23), both gas bubbles and liquid droplets can be divided into three categories: Spherical: When interfacial or viscous forces surpass the order of magnitude of buoyancy, the particle tends to adopt a spherical shape, causing the difference between its major and minor diameters to not exceed 10%. Gas bubbles of this type have generally small volumes and undergo a uniform and rectilinear upward motion; Ellipsoidal: As the fluid particle size increases, the growing effect of buoyancy force imparts an oblate shape to the particle, flattened in the direction of flow. Here, bubbles experience periodic expansions and contractions, making it challenging to represent their geometry. They exhibit characteristic movements ranging from lateral zigzag to nearly perfect helix. These oscillations are believed to arise from the intermittent behavior of the wake they create (Saffman, 1956); Spherical cap: Large fluid particles often develop a flat base, losing symmetry between their upper and lower portions. These large bubbles resemble truncated or slightly elongated spheroids. In the case of bubbles, a thin downstream gas film called skirt may form. The wake structure for this configuration is more stable, causing the bubbles to exhibit a constant and rectilinear motion once again.

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4 Fluid Flow in Oil Production Systems

Starting from the condition of a static liquid, the upward velocity that a group of gas bubbles can achieve within a vertical pipe is determined by a balance of forces originating from various mechanisms. The final value of this velocity is dictated by the weight and geometry of the particles, as well as the properties of the fluid that constitutes the continuous medium. Other considerations include the interactions among the particles themselves and with the pipe wall. Therefore, for the present case, the aim is to establish a relationship of the following form: U0 = U0 (ρl , μl , d, gρ, σ, μg , D, αg ),

(4.40)

where the first three terms are necessary for calculating the drag force acting on a solid sphere in a fluid flow. The next three terms are required to quantify the terminal velocity of a gas bubble in an unbounded medium, where gρ represents the buoyancy forces, σ denotes the resistance to changes in geometry, and μg influences the development of streamlines within the fluid particle. The presence of a pipe diameter, D, implies a confined medium, while αg is associated with the number of bubbles per unit volume and their average spacing. To accurately model this phenomenon using dimensional analysis, which involves the nine parameters and three primary dimensions (mass, length, and time), six dimensionless numbers are required. One possible combination is as follows: Fr0 = Fr0 (Ar, Eo, μg /μl , d/D, αg )

(4.41)

where d is the average bubble diameter, and: Fr0 = 

U0 ρ gd ρl

,

(4.42)

Ar =

gρd 3 , ρl νl2

(4.43)

Eo =

gρd 2 . σ

(4.44)

The parameter selection expressed in Eq. (4.41) proves advantageous as it positions the unknown of the problem solely on the left-hand side of the equation. If additional dimensionless numbers, such as Re and We, were included in this listing, the slip velocity would also move to the right along with the input data of the function, resulting in recursive solutions. In Eq. (4.41), the Archimedes number9 of the ascending bubble (Ar ) expresses the relationship between the buoyancy forces and the viscous forces to which it is subjected. Therefore, a higher value of Ar will result in a more pronounced Fr0 . 9

In honor of the famous scientist from ancient Greece.

4.4 Bubbly and Dispersed Bubble Flows

169

The Eotvos number encompasses the effects of particle shape change, starting as a perfect sphere for Eo → 0 and transforming into a cap for Eo → ∞. The viscosity ratio of the continuous and dispersed phases accounts for the velocity distribution at the interface, allowing differentiation between bubble displacement in a liquid medium and droplet displacement in a gas medium. The diameter ratio measures the influence of the pipe wall, which imposes increasing resistance to flow as d/D → 1. Finally, the increase in αg brings bubbles closer to each other and to the wake left by the preceding bubbles, restricting their movement. However, when extrapolating a certain threshold of Ar , the bubble rise velocity ceases to be a function of its diameter. This phenomenon occurs for isolated particles in an infinite medium, which implies d/D → 0 and αg → 0, and it applies exclusively to ellipsoidal shapes. Under such circumstances, disregarding the contribution of phase viscosity, Harmathy (1960) suggests the following approach: Fr0 (∞, Eo, 0, 0, 0) = c1 Eo−1/4 ,

(4.45)

where c1 = 1.53 is an empirical constant. In dimensional form, Eq. (4.45) can be rewritten as:   σgρ 1/4 U0 (∞, Eo, 0, 0, 0) = c1 . (4.46) ρl2 As a rule, the results obtained for an isolated particle serve as a basis for determining the slip velocity of a group of bubbles. Zuber and Findlay (1965), for example, proposed the following expression:  c U0 (∞, Eo, 0, 0, αg ) = U0 (∞, Eo, 0, 0, 0) · 1 − αg 2 ,

(4.47)

where c2 = 3 for d < 0.5 mm, and decreases by half in the range 0.5 ≤ d ≤ 20 mm. According to Wallis (1969, p. 252), c2 = 2 is suggested as the initial approximation, although it is acknowledged that this value may be overestimated in certain cases. When the liquid ceases to be stagnant and starts to flow along with the gas phase, it becomes possible to consider the effect of pipe orientation. The most widely accepted and adopted solution consists of multiplying U0 in Eq. (4.47) by sin θ. Consequently, horizontal flows will result in U0 = 0. It is also customary to assign a negligible slip velocity to the small spherical bubbles typically found in the dispersed pattern. These bubbles are usually associated with very small Archimedes numbers, which inherently reduce the parameter value. Additionally, they arise from flows with high flow rates, where Usm  U0 . It is worth noting, however, that even if there is no relative motion between the phases, the gas can still move at a different rate than Usm . Considering that the velocity profile developed in the cross-sectional area of the duct has its maximum point in the central region of the flow, it is observed that Ug will be greater the further away the bubbles are from the wall (Ishii, 1977, p. 26). The distribution parameter (C0 ) of the discrete phase seeks to describe this behavior. If the particles preferably occupy the central axis of the pipeline, then C0 > 1. Otherwise, a higher concentration in the

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4 Fluid Flow in Oil Production Systems

peripheral region will likely result in C0 < 1. In the case where they are uniformly distributed throughout the cross-sectional area, which is commonly assumed in the dispersed bubble pattern, C0 is equal to 1. The distribution of bubbles in the flow is governed by two mechanisms. The first arises from the velocity profile encountered in the cross-sectional area of the duct, which induces the appearance of radial forces that drag the bubbles towards the central axis. The second originates from the interaction among the particles, which tends to disperse them. This delicate equilibrium is influenced by the same parameters previously discussed for U0 , as well as the superficial velocity of the mixture. However, historically, its representation has been approached in a simplified manner, primarily because the resulting values do not vary as drastically as those associated with slip velocity. Therefore, a relationship of the following form is sought: C0 = C0 (Usm , ρl , μl , D, ρg , αg ).

(4.48)

By conducting a dimensional analysis, it is observed that the seven parameters in Eq. (4.48) utilize three primary dimensions and, therefore, require four dimensionless numbers for the complete representation of the phenomenon. One of these combinations is provided by Hibiki and Ishii (2003) as follows: C0 = C0 (Rem , ρg /ρl , αg ), where: Rem =

ρl Usm D . μl

(4.49)

(4.50)

Considering a hypothetical scenario of an isolated and lightweight fluid particle, which implies ρg /ρl → 0 and αg → 0, the distribution parameter becomes dependent solely on the relationship between the maximum and average velocities recorded in the cross-sectional area of the duct. Consequently, for the laminar regime, C0 ≈ 2, while for the turbulent regime, C0 ≈ 1.2 (Ishii 1977, p. 29). As the flow incorporates a greater number of bubbles, the aforementioned scattering mechanism will transport some of them from the center to the peripheral region, thereby reducing these values. Conversely, the radial drag force is directly proportional to the ratio of the mass density of the phases, such that C0 → 1 as ρg /ρl → 1.

4.4.2 Wall Shear Stress The modeling of τw can be approached using at least two equally plausible methods. In one approach, the calculation is performed under the assumption that the properties of the mixture are identical to those of the liquid phase, followed by the application 2 . In the other approach, of the correction imposed by the two-phase multiplier φlm it is assumed that only the liquid flows inside the pipe, and the shear stress on the

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171

wall associated with this condition is related to the true value through the multiplier φl2 . The expressions obtained for each case correspond, respectively, to (Yadigaroglu and Hewitt (2017, p. 178)): τw = f w (Relm , e/D)

ρl |Ulm | Ulm 2 φlm , 2

(4.51)

ρl |Usl | Usl 2 φl , 2

(4.52)

τw = f w (Resl , e/D) where Ukm =

4 M˙ m , π D 2 ρk

(4.53)

Rekm =

ρk Ukm D , μk

(4.54)

Resk =

ρk Usk D . μk

(4.55)

Equation (4.52) is contraindicated for flows with phase change because its base velocity is no longer constant in such circumstances. Also, by ignoring the presence of the gas phase, its correction is systematically higher than that of Eq. (4.51), i.e., 2 . However, both multipliers are directly proportional to αg and tend to φl2 > φlm unity as αg → 0. Both multipliers also lose relevance in situations where ρg → ρl , showing the effect of pressure, or for M˙ m → ∞ (Azzopardi & Hills 2003, p. 173). According to Bendiksen et al. (1996), several of such aspects are strongly related to the upward bubble movement, which is the main mechanism responsible for the increase in flow turbulence and the most evident reason, therefore, for the existence 2 or φl2 will increase as more particles are of two-phase multipliers. In this view, φlm present in the pipe cross-section and the faster they rise relative to the continuous phases. Thus, in mathematical terms: 2 2 = φlm (αg , U0 /Ulm ), φlm

(4.56)

φl2 = φl2 (αg , U0 /Usl ).

(4.57)

4.5 Slug and Elongated Bubble Flow The principal difficulty associated with modeling slug flow lies in its intermittent nature. When observing a fixed position in the pipeline where this arrangement is present, an observer will perceive the gas phase alternately as small particles entrained by the liquid slug and as a Dumitrescu-Taylor bubble that occupies a significant

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4 Fluid Flow in Oil Production Systems

Unit cell: Lcell , αk,cell , τw

Dumitrescu-Taylor bubble region: UDT , Uk,segr , αk,segr , Lsegr , τw,segr Liquid slug region: Uk,dist , αk,dist , Ldist , τw,dist

Fig. 4.4 Geometry parameters of the slug flow pattern

portion of the cross-sectional area. Such fluctuation in αg induces changes in all other flow parameters for this reference frame, even when the flow rates of both phases are kept constant at the inlet (Fabre, 2003a, p. 117). However, when one follows the movement of a Dumitrescu-Taylor bubble, it becomes evident that its shape, length, and distance to neighboring bubbles remain practically unchanged. The volume of liquid slugs between these bubbles also remains constant, as does their gas content. This sudden simplification of phenomena, achieved through a mere change of reference frame, served as the starting point for the concept of the unit cell proposed by Dukler and Hubbard (1975). In this concept, it is argued that the slug flow pattern can be represented as a succession of identical structures moving at a uniform velocity. As depicted in Fig. 4.4, each of these cells results from the superposition of two types of flow arrangement: segregated (stratified or annular, but without liquid entrainment) in the region of the Dumitrescu-Taylor bubble, and distributed (discrete bubbles or dispersed bubbles) in the liquid slug. Naturally, in the transition between these regions, as well as at the boundaries between adjacent structures, mass and momentum transport occur. Assuming fully developed flow in both regions, the conservation of mass for the gas phase states that, for a control volume that encompasses the entire DumitrescuTaylor bubble and moves at the velocity U DT :     αg,dist U DT − Ug,dist = αg,segr U DT − Ug,segr ,

(4.58)

In Eq. (4.58), the term on the left-hand side accounts for the gas incorporation into the bubble nose, originating from the slug ahead, while the term on the right-

4.5 Slug and Elongated Bubble Flow

173

hand side represents the formation of bubbles in its tail, which become part of the slug immediately behind. Considering that U DT − Ug,dist > 0, it can be observed that U DT generally surpasses the gas phase’s own motion at the location, Ug,segr . Conversely, when αg,dist → 0, it follows that U DT → Ug,segr . The average superficial velocity and volumetric fraction of the gas phase in the unit cell are given, respectively, by (Shoham (2006, p. 240)): Usg =

    αg Ug L dist + αg Ug L segr

αg,cell =

L cell     αg L dist + αg L segr L cell

,

.

(4.59)

(4.60)

Given that the length of the unit cell (L cell ) is obtained by summing the values associated with the liquid slug (L dist ) and Dumitrescu-Taylor bubble (L segr ) regions, and utilizing Eqs. (4.58) and (4.59), Eq. (4.60) can be reformulated as follows: αg,cell

  αg,dist U DT − Ug,dist + Usg = . U DT

(4.61)

The principal advantage of the methodology introduced by Dukler and Hubbard (1975) is thus revealed, which enables the expression of average volumetric fractions of the phases solely in terms of Usg , U DT , Ug,dist , and αg,dist , making them independent of factors such as the shape and length of the Dumitrescu-Taylor bubble and the slug length (Taitel & Barnea 1990, p. 88). From a modeling perspective, it is customary to associate the elongated bubble pattern with cases where αg,dist → 0 (Barnea, 1987). For such situations, Eq. (4.61) reduces to αg,cell = Usg /U DT .

4.5.1 Slip Law Parameters The slip law governs two important phenomena within the unit cell. One of them pertains to the velocity of gas bubbles in the liquid slug region, where the parameters C0,dist and U0,dist are equal to those discussed in Sect. 4.4. The other deals with the motion of the Dumitrescu-Taylor bubble, approximated by the following equation: U DT = C0,DT Usm + U0,DT .

(4.62)

Theoretical considerations suggest that the few differences between one structure and another are related to their geometry. The latter structures have a diameter very close to that of the confining duct, typically resulting in d = D and a length equal to L DT . Additionally, considering the effect of flow orientation, one seeks a relationship for U0,DT of the form:

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4 Fluid Flow in Oil Production Systems

U0,DT = U0,DT (ρl , μl , D, ρg, σ, μg , L DT , θ).

(4.63)

After performing a dimensional analysis, Eq. (4.63) can be rewritten as: Fr0,DT = Fr0,DT (Ar, Eo, μg /μl , D/L DT , θ).

(4.64)

In Eq. (4.64), Fr0,DT , Ar , and Eo are defined by Eqs. (4.42)–(4.44), with d = D. Their physical interpretation remains the same for bubbly flow. The same holds for the viscosity ratio, which will be neglected due to the lack of in-depth studies. On the other hand, classical experimental results, such as those by Harmathy (1960), suggest that the rise velocity of the Dumitrescu-Taylor bubble has little or no dependence on its volume, rendering the influence of D/L DT negligible. These simplifications are discussed in Fabre (2003a, p. 128): Fr0,DT = Fr0,DT (Ar, Eo, θ).

(4.65)

The pipe inclination angle plays a key role in the Dumitrescu-Taylor bubble motion. Setting aside other effects, Dumitrescu (1943) found the following relation for vertical flows: (4.66) Fr0,DT (∞, ∞, 90◦ ) = 0.35. On the other hand, the existence of slip velocity in horizontal pipes may, at first glance, seem paradoxical. After all, in this configuration, the buoyancy force acting in the direction normal to the flow should cause mere bubble deformation instead of displacement. However, when studying the emptying of liquid-filled tubes into the atmosphere, Benjamin (1968) observed that the driving force sustaining the entire process originates from the height difference between the film moving towards the outlet and the static volume of undisturbed fluid. According to the proposed analytical model, the penetration speed of the air bubble into the tube would be equivalent to: Fr0,DT (∞, ∞, 0◦ ) = 0.54.

(4.67)

The theoretical results of Benjamin (1968) deviate only by 2.2% from a series of experimental values, proving that U0,DT not only exists in horizontal flows but is also surprisingly more pronounced than in the vertical direction. For intermediate inclination angles, the complex shape change experienced by the Dumitrescu-Taylor bubbles has consequences that defy common sense once again. Figure 4.5, based on the experimental results of Zukoski (1966), reveals that the maximum slip velocity is reached for θ between 40◦ and 50◦ , resulting from the most efficient combination of the two mechanisms (buoyancy and liquid level difference). Zukoski (1966) also measured the influence of liquid viscosity and surface tension on bubble motion. In the first case, it was verified that the penalty imposed on Fr0,DT

4.5 Slug and Elongated Bubble Flow

175

0.8

Eo = 4000 Eo = 62.5 Eo = 14.9 Eo = 8.33

F r0,DT

0.6

0.4

0.2

0.0 0

15

30

45 θ

60

75

90

Fig. 4.5 Variation of the Dumitrescu-Taylor bubble slip velocity as a function of Eotvos number and pipe inclination angle for Ar → ∞ (Zukoski, 1966)

due to the decrease in Ar is independent of the other parameters and has an asymptotic behavior so that its effect is no longer perceptible for Ar ≥ 2 × 105 (Wallis 1969, p. 289). In the second case, it was noted that the reduction of Eo implies an increase in the radius of curvature of the bubble nose, which slows down bubble motion or prevents it completely, as predicted by the so-called Jamin effect.10 However, the magnitude of this restriction depends on the pipeline orientation. In general terms, it appears from Fig. 4.5 that the impact of surface tension is greater in horizontal flow than in vertical flow. In turn, by employing the same reasoning applied to bubbly flow, it is customary to express the distribution parameter of the Dumitrescu-Taylor bubble through the following equation: C0,DT = C0,DT (Usm , ρl , μl , D, ρg, σ, θ).

(4.68)

By dimensionless normalization of the parameters in Eq. (4.68), it is possible to arrive at (Bendiksen, 1984): C0,DT = C0,DT (Rem , Frm , Eo, θ),

(4.69)

where the Froude number of the mixture (Frm ) represents the ratio of buoyancy forces (which, for θ = 90◦ , aim to displace the gas mass towards the upper wall of the pipe) to inertial forces (which promote flow centralization), and is given by: 10

A tribute to the pioneering work of French physicist Jules Jamin (1818–1886).

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4 Fluid Flow in Oil Production Systems

Frm = 

Usm ρ gD ρl

.

(4.70)

As observed in bubbly flow, there is a clear relationship between C0,DT and the maximum velocity in the cross-sectional area of the duct. For high inclination angles, where the gas mass naturally aligns with the longitudinal axis of the duct, it is found that in the laminar regime, with Eo → ∞ imposed, C0,DT is approximately 2 (Collins et al., 1978), and in the turbulent regime, C0,DT is approximately 1.2 (Nicklin et al., 1962). The exact values are solely dependent on Rem . On the other hand, for small inclination angles, Bendiksen (1984) argues that the distribution parameter is also influenced by the Froude number of the mixture. When the velocity is high, corresponding to Frm ≥ 3.5, the inertia forces of the flow are able to maintain the gas mass in the same position as in the vertical orientation, thus preserving C0,DT . In such cases, U0,DT experiences a drastic reduction since the mechanism responsible for its emergence practically ceases to exist. When Frm < 3.5, buoyancy forces prevail and displace the bubble toward the periphery of the duct, resulting in C0,DT ≈ 1.05 for horizontal flows in turbulent regime.

4.5.2 Gas Volume Fraction in the Liquid Slug Another important characteristic of slug flow is the entrainment of small bubbles originating from the Dumitrescu-Taylor bubble into the liquid slug, creating a twophase mixture. According to Fabre (2003a, p. 150), the average gas volume fraction in this region is influenced by three primary mechanisms: the generation of a population of bubbles during the transition between the liquid film and the slug immediately behind it, the trajectories they follow during the initial stages of the mixing process, and the subsequent development of the dispersed pattern. In reality, the gas entry into the liquid slug begins to take shape well in advance. During the formation of the Dumitrescu-Taylor bubble nose, the liquid resulting from phase segregation gradually takes on its characteristic appearance of a ring. The strong acceleration present in that region causes its thickness to be drastically reduced over a short distance, on the order of the tube diameter. Subsequently, the film rapidly moves toward the liquid slug until it impinges upon it. The wall jet and resulting shear layer generate considerable turbulence, which entrains a certain amount of gas. Upon reaching the mixing zone, which corresponds to the region furthest upstream of the liquid slug, the phases are highly turbulent and subjected to the action of centrifugal, interfacial tension, and buoyancy forces. As a result, a portion of the total gas volume contained in that zone is reabsorbed in the Dumitrescu-Taylor bubble from which it detached. On the other hand, another portion will continue to enter the liquid slug as small bubbles with a specific diameter and distribution, forming a type of bubbly flow within the slug body.

4.5 Slug and Elongated Bubble Flow

177

The swirling in the mixing zone is proportional to the transfer of momentum caused by the incidence of the liquid film and depends, therefore, on ρl and the velocity difference given by U = Usm − Ul,segr . Added to this effect is the agitation caused by the slug at Usm . The viscous dissipation, which hinders the penetration of the gas volume into the liquid medium, is associated with μl and D. On the other hand, the reincorporation of the fragmented bubbles back into the Dumitrescu-Taylor bubble is influenced by the size of the particles (where σ is relevant) and the buoyancy effect (expressed by gρ). Based on the aforementioned, it is observed that αg,dist must obey a relationship of the form: αg,dist = αg,dist (ρl , U, Usm , μl , D, gρ, σ)

(4.71)

The parameters of Eq. (4.71) can be written in dimensionless form to obtain: αg,dist = αg,dist (Wem , Frm , Rem , U/Usm )

(4.72)

where the mixture Weber number (Wem ) is given by: Wem =

ρl U 2 D σ

(4.73)

In Eq. (4.72), Wem scales with the gas flow injected into the liquid slug, Frm is inversely proportional to the portion brought back into the Dumitrescu-Taylor bubble, and Rem indicates the turbulence level of the flow within the slug body. Considering that these three dimensionless groups will enhance the confinement of the gas phase as they become higher, it can be observed that there is a close relationship between αg,dist and the superficial velocity of the mixture, which has been explored since the publication of the pioneering model by Gregory et al. (1978). Another common aspect in most studies conducted in this field is the prescription of a certain value of U below which αg,dist = 0, particularly observed in horizontal flows. In such cases, the inertia of the liquid film is unable to overcome the resistance imposed by surface tension, which occurs when Wem ≈ O(1) (Brauner & Ullmann, 2004). As the pipe inclination angle increases, the gravitational force promotes a rapid growth in U , accentuating the drag of particles to the same extent. Consequently, laboratory experiments indicate that αg,dist can be up to 5 times higher in vertical pipes compared to the results obtained for θ = 0◦ (Fabre 2003a, p. 150). It should be noted, furthermore, that the presence of the liquid film velocity in Eq. (4.72) renders the calculation of the gas volume fraction in the liquid slug recursive. This approach, which is computationally more costly, has been adopted by several authors, including Zhang et al. (2003a), Brauner and Ullmann (2004), and Al-Safran (2009), among others. On the other hand, other authors such as Malnes (1983), Andreussi and Bendiksen (1989), and Gomez et al. (2000a), aiming for a more expedient analysis, assumed U ≈ Usm , replacing the parameter U/Usm with empirical expressions in terms of θ.

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4.5.3 Wall Shear Stress Due to the assumption of fully developed flow, which is intrinsic to the unit cell model and applicable to both the distributed and segregated flow configurations, it can be observed that Ul,segr will result from a force balance identical to that of Eq. (4.3) in the case of stratified films or Eq. (4.25) for annular films. In both situations, it is possible to rewrite Eq. (4.58) to obtain: Ug,segr = U DT −

 αg,dist  U DT − Ug,dist . αg,segr

On the other hand, the conservation of mass establishes that:   Usm − αg Ug segr Ul,segr = . 1 − αg,segr

(4.74)

(4.75)

After mathematically establishing the hydrodynamic equilibrium of the phases in the segregated region, the average wall shear stress can be determined using the following equation:  τw = τw,dist

L dist L cell



  L dist + τw,segr 1 − , L cell

(4.76)

where, for annular shaped films, τw,segr = τw f . Otherwise, the wall shear stress is given by: τwg Sg + τwl Sl . (4.77) τw,segr = πD Algebraically manipulating Eq. (4.60), one arrives at: L dist αg,segr − αg,cell = . L cell αg,segr − αg,dist

(4.78)

4.6 Flow Pattern Determination Flow pattern transitions are, fundamentally, natural responses of the flow to the excessive growth of certain instabilities that originate within it (Brennen, 2005, p. 164). As a result, each of the boundaries depicted in Figs. 1.10 and 1.11 for horizontal and vertical pipes, as well as any others that occur within the range of inclination angles, portray an equivalence between the mechanisms that promote the maintenance of the current flow configuration and those attempting to disrupt it. These conflicts are particularly evident in the liquid film of stratified and annular patterns (Sect. 4.6.1), the

4.6 Flow Pattern Determination

179

fluid particles of dispersed flow patterns (Sect. 4.6.2), and also in the liquid slug and Dumitrescu-Taylor bubble of the slug and elongated bubble patterns (Sect. 4.6.3).

4.6.1 Liquid Film Stability Firstly, let us consider the forces acting at the interface of the stratified flow pattern in the longitudinal flow direction. Assuming that any pressure fluctuations in the pipe give rise to a small two-dimensional wave with height h wave and velocity Uwave , this wave will propagate only if the energy transferred by the gas phase, proportional to ρg Ug2 , is equal to or greater than the viscous dissipation that occurs due to its displacement above the liquid film, on the order of μl Uwave / h wave . Therefore, it is established that: Uwave . (4.79) ρg Ug2 ∼ μl h wave Assuming the existence of a relationship between wave kinetic and potential energies, one obtains: 2 ∼ ρg cos θh wave . (4.80) ρl Uwave Equations (4.79) and (4.80), when combined, enable the expression of the Jeffreys wave stability criterion as follows:

Ug ≥ c3

νl ρg cos θ . ρg Uwave

(4.81)

In the event that the gas velocity becomes sufficiently high, a transition from stratified smooth to stratified wavy flow will occur. Otherwise, disturbances at the interface will be suppressed, and the first condition will prevail. Taitel and Dukler (1976) suggest using c3 = 20, a value that approximates the model predictions based on laboratory data. Furthermore, relying on such evidence, the authors made the assumption that Uwave = Ul , greatly simplifying the calculation. Analyzing now the forces that manifest in the direction transverse to the flow, it is observed that the action of gravity has a stabilizing effect, as the minimum potential energy of the disturbance occurs for h wave → 0. Conversely, the acceleration of the gas caused by the reduction of its cross-sectional area leads to a decrease in pressure immediately above the wave crest, generating a force that opposes the previous one. The Bernoulli equation satisfactorily describes the relationship established between these two mechanisms, particularly because the influence of viscosity and surface tension tends to be small in this type of situation. Therefore, it is inferred that the supposed wave will acquire a pronounced amplitude only when: ρg Ug δUg ∼ ρg cos θh wave .

(4.82)

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4 Fluid Flow in Oil Production Systems

The change in velocity experienced by the gas is obtained from its mass conservation equation as follows: δUg =

Ug δ A g Ug Si = h wave Ag Ag

(4.83)

Substituting Eq. (4.83) in Eq. (4.82), it is verified that the Kelvin-Helmholtz instability, responsible for the emergence of roll waves, will take place when:

Ug ≥ c4

ρg cos θ Ag . ρg Si

(4.84)

Equation (4.84) establishes an important criterion for the transition from stratified to other flow patterns. If the roll waves do not develop, the liquid will be unable to block the passage of gas (which would initiate the slug or elongated bubble patterns), and the detachment of droplets from their crests will not be observed (essential for maintaining the annular pattern). Taitel and Dukler (1976) propose adopting c4 = 1 − h l /D, where h l is obtained from Eq. (4.13). Interestingly, the mechanism responsible for the gradual degradation of stratified flow, on the other hand, favors the stabilization of the annular flow pattern. As evidence, it should be noted that the atomization of the liquid phase occurs due to the predominance of the gas’s inertia force over those acting at the interface. The ripples present there, in turn, decisively contribute to the process, as they simultaneously mitigate the effect of surface tension and force an increase in velocity equal to δUg just above them. It is estimated that the entrainment of droplets will initiate when:  2 ρg Ug + δUg ∼

σ . h wave

(4.85)

Using Eq. (4.83) and making the necessary simplifications, one arrives at: 

 2Si c5 σ 1+ h wave h wave = Ag ρg Ug2

(4.86)

Equation (4.86) establishes an upper limit for the height of waves: any roll wave that surpasses this threshold will eventually be disrupted by the gas flow, causing the excess liquid to form the dispersed phase. Additionally, it is observed that as the gas velocity, Ug , approaches infinity, the wave height, h wave , tends to zero. Conversely, at lower flow rates, it is possible for these waves to grow until they reach the upper wall of the pipe, leading to the disruption of the existing arrangement (Brennen, 2005, p. 195). Apart from preventing gas blockage, the atomization phenomenon plays a role in transporting the liquid along the tube, particularly in vertical flows, and in homogenizing the film, thereby counteracting the effect of gravity in horizontal flows. Criteria for stability of the annular flow pattern, considering these two aspects, have been developed by Taitel et al. (1980) and Baik and Hanratty (2003).

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181

Moreover, according to Barnea (1986), it is believed that the preservation of the liquid film is achievable only when there is no recirculation in the region near the wall. In practical terms, this implies asserting that: 

∂τi ∂h f

 ≤ 0.

(4.87)

Usg ,Usl

4.6.2 Gas Bubbles Stability As mentioned in Sect. 1.5, the dispersed bubble pattern depends on high levels of turbulence in the flow to maintain their peculiar characteristics. By comparing the magnitudes of forces of this nature and those acting on the bubble surfaces, it is observed that the maximum diameter of the bubbles will be reached when: ρl u  ∼ 2

σ . dmax

(4.88)

If it is assumed that the velocity fluctuations (u  2 ) do not vary with position or direction, it is permissible to associate them, as stated by Hinze (1955), with the local rate of energy dissipation per unit mass ( ), through the following relationship: u  ∼ ( dmax )2/3 ,

(4.89)



4τw Usm

,

=

ρm D

(4.90)

ρm = αg ρg + αl ρl .

(4.91)

2

where:

Substituting Eq. (4.89) in Eq. (4.88) yields:  dmax = c6

σ ρl

3/5

−2/5 .

(4.92)

Supported by a set of experimental data concerning liquid-liquid systems, Hinze (1955) determined the fitting constant of his theoretical model as c6 = 0.725. Subsequently, Barnea et al. (1982) incorporated the particle-particle interaction effect √ into the analysis, resulting in a correction given by c6 = 0.725 + 4.15 αg . The diameter of gas bubbles also determines the type of trajectory they describe. It is known that the flow will assume a dispersed form only when the upward motion of the bubbles is rectilinear, which, in practice, requires the particles to have a spherical shape. This characteristic is linked, according to the content of Sect. 4.4, to the condition that the Eotvos number does not exceed a certain reference value (Eoref ).

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4 Fluid Flow in Oil Production Systems

It becomes evident, therefore, that the stability of the arrangement will be true only for: (4.93) dmax ≤ dref , where the reference diameter (dref ) that ensures sphericity is equivalent to:

dref =

Eoref σ . ρg

(4.94)

Therefore, if particle diameter exceeds this limit, by reducing the liquid flow rate in the pipe (and, consequently, the flow turbulence), for example, the bubbles will develop an ellipsoidal shape, resulting in zigzag or helix movements that define the transition to bubbly pattern. Barnea et al. (1982) placed such a boundary at Eoref = 1.6, although the exact value is subjective. Nonetheless, there is good agreement with the value commonly associated with the change in the bubble rise mechanism in stagnant liquid, which is Eo = 0.4 (Clift et al. 1978, p. 172). The absence of oscillations in the direction perpendicular to the flow contributes to the maintenance of the dispersed bubble pattern. By following straight trajectories, the particles minimize any possibility of collision with each other, rendering the coalescence process notably ineffective. However, as the free distance between them decreases due to an increase in αg , the frequency of these collisions grows exponentially (Azzopardi & Hills 2003, p. 24). Under certain circumstances, irreversible agglomeration of the gas mass occurs, leading to rapid deterioration of the flow pattern. The critical value, as indicated by Taitel et al. (1980), is reached at αg,max = 0.52, which corresponds to the packing of a set of perfect spheres in a regular cubic arrangement. Song et al. (1995) prefer to associate it with the size of the bubbles, using the following expression: αg,max = 0.55 − 2.37

dmax . D

(4.95)

By taking the limit as dmax → 0 in Eq. (4.95), the result remains highly consistent with that of Taitel et al. (1980). However, in the case of the bubble regime, where dmax > dref , the correlation proposed by Song et al. (1995) suggests an intensification of coalescence until the point where αg,max → 0. This type of flow easily disintegrates as the inclination angle decreases, causing the gas to accumulate on the upper wall of the pipe. Experimental evidence indicates that a phenomenon known as creaming occurs at θ = 60◦ , or even earlier. In an attempt to define its boundaries, Barnea (1986) equated the force originating from velocity fluctuations with the force resulting from the particle’s buoyancy: ρl u  ∼ ρg cos θdmax . 2

Thus:

(4.96)

4.6 Flow Pattern Determination

183



cos θmin

3D = min 8dmax



τw 1 ρ U2 2 m sm



 Frm2 , 1

.

(4.97)

The influence of pipe orientation is also clearly evident in the churn flow pattern. In this case, slight deviations from the vertical, on the order of 10–20◦ , as reported by Barnea et al. (1985), enhance the agglomeration of the gas phase to the extent of completely suppressing its inherent chaotic behavior.

4.6.3 Liquid Slug and Dumitrescu-Taylor Bubble Stability The transition criteria for the slug and elongated bubble flow patterns are determined by various instabilities that may affect the Dumitrescu-Taylor bubble or the liquid slug. One of these instabilities refers to the events that occur when the gas phase is blocked by a roll wave, which coincidentally reaches the upper wall of the pipe when the flow is still stratified or annular. According to Bendiksen and Espedal (1992), the liquid slug will only be able to develop when the initial propagation velocity of its front (Ufront ) is greater than that of the bubble that begins to form behind it. Mathematically, this can be expressed as: Ufront > U DT .

(4.98)

By applying the gas mass conservation equation within a control volume located at the front of the slug and moving concurrently with it, one obtains the following expression:     (4.99) αg,segr Ufront − Ug,segr = αg,dist Ufront − Ug,dist . In Eq. (4.99), the term on the left-hand side quantifies the mass flow rate entering the control volume from the undisturbed region, where the segregated arrangement still prevails, while the right-hand side the gas phase outlet in the form   represents of bubbles. Given that, by definition, αg Ug segr = Usg , it is possible to rewrite this relationship as follows:   Usg − αg Ug dist . (4.100) Ufront = αg,segr − αg,dist Therefore, by substituting Eqs. (4.61) and (4.100) into Eq. (4.98), it can be established that:     Usg − αg Ug dist Usg − αg Ug dist > . (4.101) αg,segr − αg,dist αg,cell − αg,dist which is satisfied only for: αg,segr < αg,cell .

(4.102)

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Equations (4.98)–(4.102) demonstrate the so-called minimum slip criterion, briefly mentioned by Wallis (1969, p. 348) and further elaborated by Bendiksen et al. (1991). This criterion stipulates that the slug flow pattern will disintegrate whenever stratified or annular flows cause a deceleration of the gas phase, resulting in an increase in the value of αg,segr compared to αg,cell . The liquid slug region can also disintegrate due to excessive aeration. Analogously to what occurs in bubbly flow, as the number of entrained bubbles increases and the distance between them decreases, it becomes more difficult to suppress the effects of coalescence, leading to the emergence of other types of structures. This limit was once again determined by Brauner and Barnea (1986) as αg,dist = 0.52. Simultaneously, under such circumstances, it is highly likely that the Dumitrescu-Taylor bubbles are so close to each other that the wake created by one begins to distort the nose of the succeeding bubble, destabilizing it. According to Chen and Brill (1997), based on the experimental results presented by van Hout et al. (1992), this phenomenon becomes evident when L dist /L cell ≤ 0.15. Conversely, at the other extreme, as αg,dist approaches zero, the transition from the slug flow pattern to elongated bubbles occurs (Barnea, 1987). Finally, according to Bendiksen et al. (1991), the Dumitrescu-Taylor bubble will collapse when its length in the unit cell becomes negligible. By inspecting Eq. (4.78), it is easy to conclude that this occurs only when αg,dist → αg,cell . The physical meaning behind this result points to somewhat disparate responses between the bubble and slug regions in the face of changes in flow conditions. On one hand, an increase in liquid flow, which would already reduce the value of αg,cell , further amplifies the turbulence in the mixing zone, leading to an increase in αg,dist . On the other hand, a decrease in gas flow would cause an immediate decrease in αg,cell , but without affecting the bubble trapping by the liquid slug to the same extent. Both situations may result in L segr → 0.

4.7 Solution of Two-Phase Flows Using the Mechanistic Approach Upon careful analysis of the various unified models available in the literature, it is readily apparent that none of them fully adhere to the theoretical aspects discussed in Sects. 4.2–4.6. This can be partly attributed to practical limitations, such as the acceptable time interval for completing a flow simulation, the stability of the computational code, its sensitivity to input parameters, and so on. The remaining portion consists of gaps left by previous studies, which, once filled in the future, will enable even more comprehensive and physically based analyses. It is also worth noting how each author emphasizes the detailed treatment of a specific aspect of the model: while some prioritize the accurate identification of the flow pattern, others focus on ensuring the continuity of volumetric fraction and shear stress values during the transition between them. Naturally, the subject of study itself guides certain choices: systems

4.7 Solution of Two-Phase Flows Using the Mechanistic Approach

185

producing condensate gas typically require a faithful representation of annular flow, while oil wells are more prone to bubble and slug flow patterns, whereas production pipelines tend to favor phase stratification. Two widely disseminated mechanistic models in the oil industry for solving twophase flow will be presented below, originating from two distinct working philosophies. We shall refer to them as the North Sea and Gulf of Mexico models, both in reference to the origin of the resources that funded each model and the field data employed in their validation. It is believed that both models have the potential to serve as a starting point for calculating pressure drop in OPWs which is the subject of Chap. 5.

4.7.1 North Sea Model This model primarily refers to the work of Bendiksen et al. (1991), utilizing additional references only when necessary. According to the authors, it is a general-purpose tool for solving problems involving the simultaneous flow of oil and natural gas, both in steady-state and transient regimes. However, the influence of the temporal variable on other flow parameters will not be considered here. The mechanistic model proposed by Bendiksen et al. (1991) laid the foundation for the initial versions of the OLGA simulator, which later became the industry standard and has remained so ever since. Although several improvements have likely been implemented throughout its successful trajectory (with little or no documentation available, as it is proprietary software with commercial purposes), user experience indicates the preservation of its most important original characteristics. Starting with the stratified pattern, Bendiksen et al. (1991) recommends calculating the hydraulic diameters of the gas and liquid by employing the following equations: 4 Ag , (4.103) Dg = Sg + Si Dl =

4 Al . Sl

(4.104)

The parameters on the right-hand side of Eqs. (4.103) and (4.104) will be evaluated, for simplicity, assuming a flat interface. After substituting the results into Eq. (4.17), the friction factor of each phase with the pipe wall is determined using the equation proposed by Haaland (1983), which is given as follows: 1 √ = −3.6 log fk



e/Dk 3.7

1.11

 6.9 + . Rek

(4.105)

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4 Fluid Flow in Oil Production Systems

Equation (4.105) considers only turbulent flows, when Rek ≥ 2900. In the laminar regime, where Rek ≤ 2400, one has: fk =

16 Rek

(4.106)

In the region defined by 2400 < Rek < 2900, a useful approach is to employ linear interpolation to determine the friction factors associated with these two extremes. The calculation of the interfacial friction factor takes into account the possible presence of waves. Bendiksen et al. (1991) suggests estimating the average height of such disturbances through a balance of the inertia, gravitational, and surface tension forces acting at the liquid-gas interface. This yields the equation:

h wave

⎧ ⎫     2  2 2 ⎪ ⎪ ⎨ ⎬  ρg Ug − Ul 1 ρg Ug − Ul 4σ  + . = − ⎪ 2ρg cos θ 2⎩ 2ρg cos θ ρg cos θ ⎪ ⎭

(4.107)

When the radicand of Eq. (4.107) becomes negative, h wave becomes zero, resulting in a smooth interface. Here, the Blasius equation establishes that: −1/4

f i,smooth = 0.079Rei

.

(4.108)

On the other hand, if the gas-liquid interface is covered by ripples the calculation of f i is performed based on h wave and Dg , while adhering to the maximum limit determined by the corresponding value for the annular flow pattern. The interfacial friction factor can thus be expressed as follows:  h wave = min , f i,an . 4Dg 

f i,wavy

(4.109)

In the annular flow regime, D f is evaluated using Eq. (4.104), and f k is determined using Eqs. (4.105) and (4.106), following the same procedure as for the stratified pattern. Bendiksen et al. (1991) chose to employ the correlation by Wallis (1969, p. 320) for f i,an in vertical pipes. Since no other alternative was provided by them to cover other situations, one can extend it to all inclination angles. Therefore:   f i,an = 0.005 1 + 75α f .

(4.110)

Regarding the liquid entrainment in the core, the models adopted by Bendiksen et al. (1991) are not widely spread in the industry and rely on a very limited database. Considering the significant advancements in theoretical and experimental studies in this area in recent years, it was deemed more appropriate to replace them with correlations that have a stronger physical basis and are more widely known. The choice was made to use the works of Pan and Hanratty (2002a) (for horizontal

4.7 Solution of Two-Phase Flows Using the Mechanistic Approach

187

pipelines) and Pan and Hanratty (2002b) (for vertical pipelines). However, for the sake of brevity, they will not be reproduced here. For other inclination angles, the entrained liquid fraction is interpolated by using the following equation: φe = φe,hor cos2 θ + φe,ver sin2 θ.

(4.111)

Contrary to what is commonly found in works of this nature, Bendiksen et al. (1991) devised a unified model for bubble and dispersed bubble patterns by combining the slip velocity of the former with the distribution parameter of the latter. These are represented by the following equations: C0 = 1, 

σρg U0 = 1.18 ρl2

1/4



1 − αg

(4.112) 3/2 √

sin θ.

(4.113)

According to Bendiksen et al. (1996), the two-phase shear stress multiplier on the pipe wall for such an arrangement is given by the expression:   ! αg U0 1 1 + 15.3 . (4.114) φl2 = 1 − αg 1 − αg Usl Regarding the slug flow pattern, Bendiksen (1985) proposed an equation to correct the slip velocity of the Dumitrescu-Taylor bubble in vertical flows, taking into account the effects of surface tension. This equation is given by: Fr0,DT (∞, Eo, 90◦ ) = Fr0,DT (∞, ∞, 90◦ )

   6.8 1 − 0.96 exp (−0.0165Eo) 20 1− 1+  3/2 . Eo Eo 1 − 0.52 exp (−0.0165Eo)

(4.115) Equation (4.115) relies on the experimental data provided by Zukoski (1966). Similarly, the correlation presented by Weber (1981) for horizontal pipelines is expressed as follows: Fr0,DT (∞, Eo, 0◦ ) = Fr0,DT (∞, ∞, 0◦ ) − 1.76Eo−0.56 .

(4.116)

In order to address the influence of viscous forces on Fr0,DT , the correction proposed by Wallis (1969, p. 289) will be applied, which establishes that:  √ Fr0,DT (Ar ≥ 4, Eo, θ) = 1 − exp −0.01 Ar . Fr0,DT (∞, Eo, θ)

(4.117)

When the Dumitrescu-Taylor bubble is located at the center of the pipe crosssection, a configuration referred to as Type I, the viscosity of the liquid phase also plays a significant role in calculating the distribution parameter. As stated by

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4 Fluid Flow in Oil Production Systems

Bendiksen (1985), the drastic change observed in the velocity profile necessitates the use of two expressions: one for laminar flow and another for turbulent flow. Therefore, in vertical flows, the following expressions are adopted, respectively: I (Rem ≤ 2400, ∞, 90◦ ) = 2.29, C0,DT I (Rem ≥ 2900, ∞, 90◦ ) = C0,DT

log Rem + 0.309 . log Rem − 0.743

(4.118) (4.119)

Equations (4.118) and (4.119) form the foundation for incorporating the effect of surface tension. It is recommended to perform the following steps: I C0,DT (Rem ≤ 2400, Eo, 90◦ ) I C0,DT (Rem , ∞, 90◦ ) I C0,DT (Rem ≥ 2900, Eo, 90◦ )

=1−

20  1 − exp (−0.0125Eo) , Eo

(4.120)

2  3 − log Rem exp (−0.025Eo) . Eo (4.121) In the laminar-turbulent flow transition region, the values obtained through Eqs. (4.120) and (4.121) can be linearly interpolated. In turn, the Type II configuration assumes that the bubble is closer to the upper wall of the pipe. This loss of symmetry significantly complicates the modeling of the distribution parameter, which, for horizontal flows, is represented in a simplified manner as: I C0,DT (Rem , ∞, 90◦ )

=1−

II (0◦ ) = 1.05. C0,DT

(4.122)

Therefore, in the event that the forces of inertia prevail, Bendiksen et al. (1991) postulate that: I I (Rem , Eo, θ) = C0,DT (Rem , Eo, 90◦ ), C0,DT

(4.123)

I Fr0,DT (Ar, Eo, θ) = Fr0,DT (Ar, Eo, 90◦ ) sin θ.

(4.124)

On the other hand, if the buoyancy forces are relevant, the following relationships are recommended: II II I C0,DT (Rem , Eo, θ) = C0,DT (0◦ ) cos2 θ + C0,DT (Rem , Eo, 90◦ ) sin2 θ,

(4.125)

II Fr0,DT (Ar, Eo, θ) = Fr0,DT (Ar, Eo, 0◦ ) cos θ + Fr0,DT (Ar, Eo, 90◦ ) sin θ. (4.126)

4.7 Solution of Two-Phase Flows Using the Mechanistic Approach

189

The transition from the first (inertia-dominated) to the second configuration (buoyancy-dominated) is assumed to take place at Frm = 3.5. Bendiksen et al. (1991) do not provide any correlation for calculating the volumetric gas fraction in the liquid slug. In order to address this gap, a combined form of the models proposed by Malnes (1983) and Andreussi and Bendiksen (1989) was chosen to be utilized. For vertical pipes, the calculation of αg,dist is performed as follows: αg,dist (Frm , Eo, 90◦ ) =

Frm . Frm + 83Eo−1/4

(4.127)

For horizontal pipes, the following expression is proposed: αg,dist (Frm , Eo, 0◦ ) =

Frm − Frmin , Frm + 2400Eo−3/4

(4.128)

where: 

Frmin

D2 = 2.6 1 − 2 02 D

 .

(4.129)

Equation (4.129) was formulated by Andreussi and Bendiksen (1989) with D0 = 25.4 mm. For other orientations, it is deemed reasonable to perform an interpolation of the following nature: αg,dist (θ) = αg,dist (0◦ ) cos2 θ + αg,dist (90◦ ) sin2 θ.

(4.130)

It is also necessary to ensure that αg,dist (θ) ≥ 0. Bendiksen et al. (1991) acknowledge the existence of only four flow patterns: stratified, annular, bubbly and slug. The transitions between these patterns are strongly governed by the minimum slip criterion. However, there are only two exceptions: the transition from bubbly to slug flow, which is dictated by the volumetric gas fraction dispersed in the liquid, and the transition from the stratified to annular flow, attributed to the growth of waves at the liquid-gas interface. Subsequently, Bendiksen and Espedal (1992) proposed a stability analysis for the transition between the stratified and non-stratified flow, complementing the existing rule. However, the final equation obtained by the authors relies on the determination of Uwave . As this parameter is unknown (according to them, “a complicated function of shear stresses and the perimeters occupied by each phase”), it can be approximated by the velocity of the liquid phase, in which case the criterion introduced by Bendiksen and Espedal (1992) becomes identical to that of Taitel and Dukler (1976) The calculation procedure for the pattern has not been detailed by Bendiksen et al. (1991). Thus, the following steps are proposed here: 1. Verify the transition from stratified to annular flow, governed by the growth of waves at the liquid-gas interface as stated in Eq. (4.107), until the limit where

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4 Fluid Flow in Oil Production Systems

h l + h w = D. The resulting pattern will constitute the segregated arrangement mentioned in item 3; 2. Test the transition from bubbly to slug flow, dictated by the volumetric gas fraction in the liquid slug. The flow pattern obtained in this way will represent the distributed arrangement mentioned in item 3; 3. By using the minimum slip criterion, examine the transition from segregated to distributed pattern. If the former is stratified, it is necessary to further apply the Kelvin-Helmholtz stability criterion to confirm it conclusively.

4.7.2 Gulf of Mexico Model The University of Tulsa has been responsible for numerous advancements in the understanding of multiphase flow in petroleum systems. Through the research consortium titled Tulsa University Fluid Flow Projects (TUFFP), which began in 1973, empirical correlations by Beggs and Brill (1973), Mukherjee and Brill (1985) were developed, as well as mechanistic models by Xiao et al. (1990), Ansari et al. (1994), Gomez et al. (2000b), and Zhang et al. (2003b), among others. The latter two works, in particular, encapsulate the current level of development in their projects: the highly robust flow pattern calculation by Gomez et al. (2000b) and the carefully selected closure equations by Zhang et al. (2003b) form a combination of great effectiveness, as will be discussed later. The models from the University of Tulsa are supported by an extensive experimental database built over the past decades, encompassing both laboratory results and real pressure and temperature measurements from wells operating in the field. In stratified flow, the calculation of the hydraulic diameter and the wall friction factor for each phase remains identical to that of Sect. 4.7.1 (see Eqs. 4.103–4.106), while f i is now subject to the correlation proposed by Andritsos and Hanratty (1987). Therefore: "   f i,wavy Usg hl = 1 + 15 −1 , (4.131) fg D Usg,tr where:

" ρg,atm . Usg,tr = 5 ρg

(4.132)

The parameter Usg,tr in Eq. (4.131) represents the minimum superficial gas velocity required for the onset of disturbances at the interface. It is assumed that these disturbances depend solely on the phase’s density values under flow conditions and atmospheric pressure (ρg,atm ). Implicitly, this criterion also signifies the transition between smooth and wavy stratified patterns. If Usg < Usg,tr , Andritsos & Hanratty (1987) recommend setting f i,smooth = f g .

4.7 Solution of Two-Phase Flows Using the Mechanistic Approach

191

To estimate the interfacial friction factor in annular flow, Zhang et al. (2003b) used the correlation developed by Ambrosini et al. (1991), expressed by:  "  f i,an ρg −0.6 Re , (4.133) = 1 + 13.8We0.2 Re − 200 τ i g g f i,smooth ρl where Reg , Weg and f i,smooth were defined in Eqs. (4.17), (4.37) and (4.108), respectively, and the interfacial friction Reynolds number (Reτi ) is given by:

Reτi =

ρg



τi h ρg l

μg

.

(4.134)

The entrained liquid fraction is computed from the Oliemans et al. (1986) correlation given by: φe −1.24 −0.92 = 0.00302We1.8 Resl0.7 Frsg sg Resg 1 − φe



ρl ρg

0.38 

μl μg

0.97 (4.135)

where Resk is evaluated from Eq. (4.55) and: Wesk =

2 D ρk Usk , σ

Usk Frsk = √ . gD

(4.136) (4.137)

Regarding the bubbly flow pattern, Zhang et al. (2003b) adopt a unitary distribution parameter, as stated in Eq. (4.112), and a slip velocity dictated by the correlation proposed by Harmathy (1960), given by: 

σρg U0 = 1.53 ρl2

1/4

 sin θ 1 − αg

(4.138)

It is noteworthy that neither the model proposed by Zhang et al. (2003b) nor the one presented by Gomez et al. (2000b) includes the prescription of a two-phase multiplier as a means to correct the shear stress on the wall for the upward motion of gas bubbles. Instead, it is assumed that: τw = f w (Resm , e/D)

ρm |Usm | Usm , 2

(4.139)

where ρm is defined in Eq. (4.91) and: Resm =

ρm Usm D . μl

(4.140)

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4 Fluid Flow in Oil Production Systems

Naturally, the treatment of the dispersed bubble pattern follows a similar approach to the previous case. The only distinction lies in the modeling of the slip velocity, where U0 = 0 is assigned, imparting it with the properties of homogeneous flow. The upwards motion of Dumitrescu-Taylor bubbles in the slug flow pattern is described with remarkable simplicity by Gomez et al. (2000b) and Zhang et al. (2003b). Neglecting the influence of buoyancy, surface tension, and inclination angle on the region predominantly occupied by the gas phase, the relationships for calculating the distribution parameter can be summarized as follows: C0,DT (Rem ≤ 2400) = 2,

(4.141)

C0,DT (Rem ≥ 2900) = 1.2.

(4.142)

The modeling of slip velocity is equally concise and does not consider the effect of viscous forces or surface tension. Only the following equation is utilized to represent this phenomenon: Fr0,DT = 0.54 cos θ + 0.35 sin θ.

(4.143)

In order to portray the phenomenon of bubble entrapment within the liquid slug, Zhang et al. (2003a) devised a model that balances the turbulent kinetic energy present in the liquid phase with the surface free energy of the dispersed gas particles. The former is based on the shear stress at the duct wall and the momentum transfer caused by the collision of the film against the liquid slug, while the latter is proportional, as known, to σ/d. According to the authors, the term related to momentum conservation varies significantly depending on the flow orientation and allows for an accurate estimation of αg,dist across the entire range of 0 ≤ θ ≤ 90◦ , given by: αg,dist =

τdist

τdist , √ + 3.16 σρg

(4.144)

where:     αl,segr ρl U DT − Ul,segr Usm − Ul,segr 2 τdist = τw,dist + . 2.5 − sin θ 4L dist /D (4.145) It is noteworthy that Eq. (4.145) necessitates an estimation of the piston length. Based on the investigations conducted by Barnea and Brauner (1985) in horizontal flow and by Taitel et al. (1980) in vertical ducts, Zhang et al. (2003b) make the following assumption: L dist = 32 cos2 θ + 16 sin2 θ. D

(4.146)

4.8 Preliminary Results

193

Finally, the procedure for determining the flow pattern employed by Gomez et al. (2000b) presents several common aspects to the unified methodology of Barnea (1987), consisting of the following steps: 1. Verify the stability of the dispersed bubble pattern, which implies dmax ≤ dref in Eqs. (4.92) to (4.94) and θ ≥ θmin in Eq. (4.97). Additionally, it is necessary to consider the maximum possible packing for the arrangement, such that αg ≤ 0.52; 2. If the conditions presented in Step 1 are not satisfied, test the stability of stratified flow, according to the criterion established in Eq. (4.84); 3. If the criterion set in Step 2 is not met, examine the stability of the annular flow pattern. For this purpose, it is required that α f ≤ 0.24 (to prevent obstruction  of the cross-sectional area of the pipe by roll waves) and ∂τi /∂h f Usg ,Usl ≤ 0 (preventing the occurrence of liquid flow reversal near the pipe wall); 4. If the conditions presented in Step 3 are not satisfied, inspect the stability of the bubble pattern. This requires that θ ≥ θmin in Eq. (4.97) and αg ≤ 0.25; 5. If the criterion of Step 4 is not met, it can be concluded, by elimination, that slug flow occurs.

4.8 Preliminary Results Upon acquiring the knowledge presented throughout this chapter, it becomes possible to establish an initial overview of the changes that occur in the flow of reservoir fluids as they are brought to the surface. Let us consider a typical OPW from pre-salt fields, as illustrated in Fig. 1.3. As a general rule, its riser has an internal diameter of 6 in (152.4 mm) and, following a catenary shape, reaches the SPU at an inclination angle of approximately 83◦ . On the other hand, the flowline has a larger diameter of 8 in (203.2 mm) and rests almost horizontally, as allowed by the seabed relief undulations. Its production tubing is vertical and initially has a nominal diameter of 6 5/8 in (150.4 mm internal). It is worth noting that these last two components communicate directly with each other via the WCT, causing the pressure and temperature conditions downstream to be equivalent to those upstream. As the well gains depth, the production tubing needs to conform to the telescopic shape of the well, which is why its nominal diameter at the PDG is usually reduced to only 5 1/2 in (118.6 mm internal). Consider, furthermore, that the flowing mixture inside the pipeline exhibits composition and volumetric and phase behavior compatible with those of sample PS01, and that the daily liquid flow rate reaches approximately 18.9 thousand barrels 3 /d), with a BSW content of 10%. Its thermodynamic and transport prop(3000 mstd erties have already been successfully represented by the fluid model developed in the previous chapter, and are listed in Table 3.19 for three pairs of P and T . By associating each set of values with the corresponding geometry within the four configurations described above, it is possible to gather a series of details regarding the

194

4 Fluid Flow in Oil Production Systems

two-phase flow that occurs throughout the OPW, at representative points along the entire trajectory. The flow pattern maps generated by the North Sea and Gulf of Mexico mechanistic models for each scenario are presented in Figs. 4.6 and 4.7. It can be observed that, in the latter case, the various flow configurations are arranged similarly to those suggested by experiments with low-pressure air and water, as illustrated in Figs. 1.10 and 1.11. In the former case, some significant discrepancies arise, but discussing them here is currently beyond the scope of the book. Setting that aside, it is evident that both approaches provide equally plausible predictions, supported by physical foundations, within the regions of true interest. Upon arrival at the SPU, for instance, where the oil has already lost a significant portion of the light fractions that were once dissolved in it, there is a high likelihood that the flow will exhibit the slug flow pattern or even annular flow, as indicated by the maps. Downstream of the WCT, in the flowline, slightly upward sections favor the emergence of slug flow, while in the downward sections, phase stratification is more likely to occur. Upstream of it, in the production tubing, the options are limited to slug flow or dispersed bubbles patterns. Meanwhile, in the PDG, due to the still

Fig. 4.6 Two-phase flow pattern maps corresponding to the SPU, WCT (downstream and upstream) and PDG of a typical pre-salt OPW, predicted by North Sea mechanistic model

4.8 Preliminary Results

195

Fig. 4.7 Two-phase flow pattern maps corresponding to the SPU, WCT (downstream and upstream) and PDG of a typical pre-salt OPW, predicted by Gulf of Mexico mechanistic model

low presence of gas, the flow assumes a dispersed bubble arrangement (as previously mentioned, the North Sea model does not differentiate between the two subtypes, simply referring to them as bubbles). It is also worth noting that transitions constitute partly probabilistic phenomena and should be understood as less defined solid lines, but rather as gray areas where there is no clear prevalence of a pattern (see Sect. 1.5 for more details). On the other hand, it is known that these differences observed early in the calculation stage lead to implications for a variety of other parameters, such as the volumetric phase fractions and average wall shear stress, which directly depend on the configuration assigned to the flow. Another significant quantity with practical importance is the pressure gradient in the pipe, obtained by summing Eqs. (4.1) and (4.2), yielding: −A

  dP − τwg Sg − τwl Sl − ρg Ag + ρl Al g sin θ = 0. dL

(4.147)

Rearranging the terms and making the necessary simplifications, one arrives at: dP 4τw = −ρm g sin θ − . dL D

(4.148)

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4 Fluid Flow in Oil Production Systems

Fig. 4.8 Behavior of the liquid volumetric fraction as a function of the superficial velocities of the phases in the SPU, WCT (downstream and upstream) and PDG of a typical pre-salt OPW, according to North Sea mechanistic model

Equation (4.148) is flow pattern-independent, despite including the force balance for stratified flow in its brief demonstration. The three-dimensional representations of αl and d P/d L as functions of Usg and Usl are shown in Figs. 4.8, 4.9, 4.10 and 4.11, revealing a series of discontinuities in the flow pattern transitions of the Gulf of Mexico model. If confirmed, these sudden variations will greatly hinder the determination of pressure and temperature profiles in the flow within OPWs, which is the subject of Chap. 5. In the North Sea model, however, the changes appear to be much smoother, except for the boundary between slug and annular flows at low liquid flow rates, where the increased gas phase velocity imposes a rapid alteration in the force equilibrium described in Eq. (4.28). To better understand the behavior of the results, it is sufficient to analyze the residual behavior of the function described by Eq. (4.28) for the PDG flow conditions, for example. Figure 4.12 shows the existence of three possible solutions for the liquid film volumetric fraction at Usg = 5.6 m/s (and its vicinity) and Usl = 1 × 10−3 m/s. In such situations, the choice always falls on the smallest value, which represents a theoretically more stable flow configuration. By increasing the gas velocity to Usg = 6.6 m/s, αl f decreases from 0.009 to 0.006, which is quite plausible. However, when it is decreased such that Usg = 4.6 m/s, the force balance solution abruptly jumps to αl f = 0.27. After applying the other transition criteria, the slug flow pattern will prevail at this point, leaving behind the discontinuity observed in Figs. 4.8 and 4.10. It is worth noting, on the other hand, that this event is limited to

4.8 Preliminary Results

197

Fig. 4.9 Behavior of the liquid volumetric fraction as a function of the superficial velocities of the phases in the SPU, WCT (downstream and upstream) and PDG of a typical pre-salt OPW, according to Gulf of Mexico mechanistic model

Fig. 4.10 Behavior of the pressure gradient in the pipe as a function of the superficial velocities of the phases in the SPU, WCT (downstream and upstream) and PDG of a typical pre-salt OPW, according to North Sea mechanistic model

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4 Fluid Flow in Oil Production Systems

Fig. 4.11 Behavior of the pressure gradient in the pipe as a function of the superficial velocities of the phases in the SPU, WCT (downstream and upstream) and PDG of a typical pre-salt OPW, according to Gulf of Mexico mechanistic model

Residual of Eq.(4.28), Pa/m

8000

4000

0

−4000 Usg = 4.6 m/s Usg = 5.6 m/s Usg = 6.6 m/s

−8000 0.00

0.05

0.10

0.15 αlf

0.20

0.25

0.30

Fig. 4.12 Behavior of the force balance for annular flow as a function of the liquid film volumetric fraction and gas superficial velocity, for (Usl = 1 × 10−3 m/s), according to the North Sea mechanistic model.

4.8 Preliminary Results

199

Usl levels below 0.1 m/s, which renders the chances of its occurrence in simulations of a real petroleum OPW virtually nonexistent. In conclusion, the results obtained at the four points of the OPW are presented in Table 4.1 and reveal two important trends: when compared to the North Sea, the Gulf of Mexico model (i) consistently predicts higher values for αg and (ii) underestimates τw upon arrival at the SPU. Consequently, its values of d P/d L become somewhat modest on two occasions. It is evident that, being a hypothetical problem, there is no means of determining the most accurate methodology among them. However, indications suggest that the Gulf of Mexico model will estimate pressure drop in the riser significantly lower than the North Sea model, and this behavior will persist throughout the entire production tubing, albeit to a lesser extent. As for the flowline, it is expected that both approaches will yield similar profiles. The remaining question is which one will perform better in the actual tests conducted in the field.

Table 4.1 Two-phase flow parameters associated with the SPU, WCT (downstream and upstream) and PDG of a typical pre-salt OPW, predicted by the two mechanistic models Position Parameter Predicted value North Sea Gulf of Mexico SPU P = 4 MPa T = 333 K

WCT P = 14 MPa T = 373 K

WCT P = 14 MPa T = 373 K

PDG P = 30 MPa T = 393 K

D = 152.4 mm

Flow pattern

Slug

Annular

θ= Usg = 9.13 m/s Usl = 2.13 m/s D = 203.2 mm

αg τw , N/m2 d P/d L, Pa/m Flow pattern

0.747 74.0 −4275 Strat. wavy

0.808 39.8 −2897 Slug

θ = 0◦ Usg = 1.28 m/s Usl = 1.33 m/s D = 150.4 mm

αg τw , N/m2 d P/d L, Pa/m Flow pattern

0.375 6.28 −124 Slug

0.436 7.24 −143 Disp. bubbles

θ = 90◦ Usg = 2.34 m/s Usl = 2.43 m/s D = 118.6 mm

αg τw , N/m2 d P/d L, Pa/m Flow pattern

0.452 23.1 −5345 Bubbly

0.491 21.7 −5042 Disp. bubbles

θ = 90◦ Usg = 0.93 m/s Usl = 4.56 m/s

αg τw , N/m2 d P/d L, Pa/m

0.166 39.6 −7594

0.169 40.2 −7594

83◦

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Bendiksen, K., & Espedal, M. (1992). Onset of slugging in horizontal gas-liquid pipe flow. International Journal of Multiphase Flow, 18(2), 237–247. Bendiksen, K. H., Maines, D., Moe, R., & Nuland, S. (1991). The dynamic two-fluid model OLGA: Theory and application. SPE Production Engineering, 6(02), 171–180. Bendiksen, K. H., Malnes, D., & Nydal, O. J. (1996). On the modelling of slug flow. Chemical Engineering Communications, 141–142(1), 71–102. Benjamin, T. B. (1968). Gravity currents and related phenomena. Journal of Fluid Mechanics, 31(2), 209–248. Bonizzi, M., Andreussi, P., & Banerjee, S. (2009). Flow regime independent, high resolution multifield modelling of near-horizontal gas-liquid flows in pipelines. International Journal of Multiphase Flow, 35(1), 34–46. Brauner, N., & Barnea, D. (1986). Slug/churn transition in upward gas-liquid flow. Chemical Engineering Science, 41(1), 159–163. Brauner, N., & Ullmann, A. (2004). Modelling of gas entrainment from taylor bubbles. Part A: Slug flow. International Journal of Multiphase Flow, 30(3), 239–272. Brennen, C. (2005). Fundamentals of multiphase flow. Cambridge, UK: Cambridge University Press. Brill, J. P., & Arirachakaran, S. J. (1992). State of the art in multiphase flow. Journal of Petroleum Technology, 44(05), 538–541. Chen, X., & Brill, J. (1997). Slug to churn transition in upward vertical two-phase flow. Chemical Engineering Science, 52(23), 4269–4272. Chen, X. T., Cal, X. D., & Brill, J. P. (1997). Gas-liquid stratified-wavy flow in horizontal pipelines. Journal of Energy Resources Technology, 119(4), 209–216. Clift, R., Grace, J. R., & Weber, M. E. (1978). Bubbles, Drops, and Particles. New York, NY: Academic Press. Collins, R., Moraes, F. F. D., Davidson, J. F., & Harrison, D. (1978). The motion of a large gas bubble rising through liquid flowing in a tube. Journal of Fluid Mechanics, 89(3), 497–514. Dukler, A. E., & Hubbard, M. G. (1975). A model for gas-liquid slug flow in horizontal and near horizontal tubes. Industrial & Engineering Chemistry Fundamentals, 14(4), 337–347. Dumitrescu, D. T. (1943). Strömung an einer luftblase im senkrechten rohr. Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 23(3), 139–149. Duns, H. J., & Ros, N. C. J. (1963). Vertical flow of gas and liquid mixtures in wells. In Paper no. 10132 presented at the World Petroleum Congress, Frankfurt am Main, Germany (pp. 451–465). Eaton, B. A., Knowles, C. R., & Silberbrg, I. H. (1967). The prediction of flow patterns, liquid holdup and pressure losses occurring during continuous two-phase flow in horizontal pipelines. Journal of Petroleum Technology, 19(06), 815–828. Fabre, J. (2003). Gas-liquid slug flow. In V. Bertola (Ed.), Modelling and Experimentation in TwoPhase Flow (pp. 117–156). Vienna: Springer. Fabre, J. (2003). Modelling of stratified gas-liquid flow. In V. Bertola (Ed.), Modelling and Experimentation in Two-Phase Flow (Vol. 450, pp. 79–116). Vienna: Springer. Fore, L., Beus, S., & Bauer, R. (2000). Interfacial friction in gas-liquid annular flow: Analogies to full and transition roughness. International Journal of Multiphase Flow, 26(11), 1755–1769. Fukano, T., & Furukawa, T. (1998). Prediction of the effects of liquid viscosity on interfacial shear stress and frictional pressure drop in vertical upward gas-liquid annular flow. International Journal of Multiphase Flow, 24(4), 587–603. Gill, L., Hewitt, G., & Lacey, P. (1964). Sampling probe studies of the gas core in annular two-phase flow – II: Studies of the effect of phase flow rates on phase and velocity distribution. Chemical Engineering Science, 19(9), 665–682. Gomez, L., Shoham, O., & Taitel, Y. (2000). Prediction of slug liquid holdup: Horizontal to upward vertical flow. International Journal of Multiphase Flow, 26(3), 517–521.

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Gomez, L. E., Shoham, O., Schmidt, Z., Chokshi, R. N., & Northug, T. (2000). Unified mechanistic model for steady-state two-phase flow: Horizontal to vertical upward flow. SPE Journal, 5(3), 339–350. Govan, A. H. (1990). Modelling of Vertical Annular and Dispersed Two-Phase Flows. Ph.D. thesis, Imperial College, University of London. Gregory, G., Nicholson, M., & Aziz, K. (1978). Correlation of the liquid volume fraction in the slug for horizontal gas-liquid slug flow. International Journal of Multiphase Flow, 4(1), 33–39. Grolman, E., & Fortuin, J. M. (1997). Gas-liquid flow in slightly inclined pipes. Chemical Engineering Science, 52(24), 4461–4471. Haaland, S. E. (1983). Simple and explicit formulas for the friction factor in turbulent pipe flow. Journal of Fluids Engineering, 105(1), 89–90. Hagedorn, A. R., & Brown, K. E. (1965). Experimental study of pressure gradients occurring during continuous two-phase flow in small-diameter vertical conduits. Journal of Petroleum Technology, 17(04), 475–484. Harmathy, T. Z. (1960). Velocity of large drops and bubbles in media of infinite or restricted extent. AIChE Journal, 6(2), 281–288. Hart, J., Hamersma, P., & Fortuin, J. (1989). Correlations predicting frictional pressure drop and liquid holdup during horizontal gas-liquid pipe flow with a small liquid holdup. International Journal of Multiphase Flow, 15(6), 947–964. Hasan, A. R., & Kabir, C. S. (1988). A study of multiphase flow behavior in vertical wells. SPE Production Engineering, 3(02), 263–272. Henstock, W. H., & Hanratty, T. J. (1976). The interfacial drag and the height of the wall layer in annular flows. AIChE Journal, 22(6), 990–1000. Hewitt, G., & Hall-Taylor, N. (1970). Annular two-phase flow. Oxford, UK: Pergamon Press. Hibiki, T., & Ishii, M. (2003). One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase flow regimes. International Journal of Heat and Mass Transfer, 46(25), 4935–4948. Hinze, J. O. (1955). Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes. AIChE Journal, 1(3), 289–295. Ishii, M. (1977). One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two phase flow regimes. ANL-77-47. Argonne National Laboratory, DuPage, IL. Jeffreys, H. (1925). On the formation of water waves by wind. Proceedings of the Royal Society of London, 107(742), 189–206. Magrini, K. L., Sarica, C., Al-Sarkhi, A., & Zhang, H.-Q. (2012). Liquid entrainment in annular gas/liquid flow in inclined pipes. SPE Journal, 17(02), 617–630. Malnes, D. (1983). Slug Flow in Vertical, Horizontal and Inclined Pipes. IFE KR E.: Institutt for Energiteknikk. Institute for Energy Technology, Kjeller, Norway. Mukherjee, H., & Brill, J. P. (1985). Pressure drop correlations for inclined two-phase flow. Journal of Energy Resources Technology, 107(4), 549–554. Nicklin, D. J., Wilkes, J. O., & Davidson, J. F. (1962). Two-phase flow in vertical tubes. Transactions of the Institution of Chemical Engineers, 40, 61–67. Oliemans, R., Pots, B., & Trompé, N. (1986). Modelling of annular dispersed two-phase flow in vertical pipes. International Journal of Multiphase Flow, 12(5), 711–732. Orkiszewski, J. (1967). Predicting two-phase pressure drops in vertical pipe. Journal of Petroleum Technology, 19(06), 829–838. Owen, D. G. (1986). An Experimental and Theoretical Analysis of Equilibrium Annular Flow. Ph.D. thesis, University of Birmingham. Paleev, I., & Filippovich, B. (1966). Phenomena of liquid transfer in two-phase dispersed annular flow. International Journal of Heat and Mass Transfer, 9(10), 1089–1093. Pan, L., & Hanratty, T. J. (2002). Correlation of entrainment for annular flow in horizontal pipes. International Journal of Multiphase Flow, 28(3), 385–408.

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Chapter 5

Simulation of Offshore Production Systems

5.1 Introduction The advent of computer science, which took place in the 1980s, represented a significant breakthrough for the modeling of oil production systems. Prior to its development, predictions were the result of a laborious process that involved dividing the well into segments with constant diameter, inclination angle, mass flow rate, and temperature. These segments were then graphically estimated for pressure drop using master curves. These curves were typically plotted by overlaying different values of the Gas-Liquid Ratio (GLR1 ) and individually described the results of various possible combinations of the aforementioned parameters. The complete set of master curves also covered a certain range of densities for oil, water, and natural gas, encompassing the entire BSW spectrum, from 0 to 100%. In comparison to this multitude of curves, the numerical approach, based on mathematical algorithms for calculating pressure and temperature profiles, as well as operating flow rates, resulted in notably faster and more precise analyses. In the subsequent years, as the exploration and production of oil shifted towards offshore areas with increasing water depth, where the temperature at the seabed reaches a minimum value of approximately 4 ◦ C, computational simulation of OPWs began to provide important insights for the newly established discipline of flow assurance. This discipline brought together initiatives aimed at ensuring uninterrupted flow of hydrocarbons from the wellbore to the SPU within safe operating limits. Currently, studies on the steady-state flow of both liquid and gas phases are vital for preventing blockages caused by organic deposits (such as paraffins and asphaltenes) or inorganic deposits (such as sulfate scales), as well as mitigating erosion caused by particulate matter that may detach from the reservoir rock. 1

GLR corresponds to the volumetric ratio between the gas and liquid components of the mixture, measured under Standard conditions.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. R. Gessner and J. R. Barbosa Jr., Integrated Modeling of Reservoir Fluid Properties and Multiphase Flow in Offshore Production Systems, Petroleum Engineering, https://doi.org/10.1007/978-3-031-39850-6_5

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On the other hand, transient models, developed since the 1990s, are largely focused on simulating cleaning operations using Pipeline Inspection Gauges (PIGs). PIGs are cylindrical devices that move inside pipelines by applying a pressure differential between their ends, effectively scraping the pipeline wall as they travel. Transient models are also used to address issues related to the cooling of the mixture in the pipeline during production shutdowns (e.g., gas hydrate blockage, increased yield stress due to wax deposition, etc.). Additionally, they account for the effects of system fluctuations introduced by severe slugging or liquid loading. Both approaches, steady-state and transient, are applicable to both design and production monitoring. Later, in the 2000s, the concept of integrated modeling became popular, giving rise to coupling tools between reservoir models and the subsea wells and pipelines installed therein (Ozdogan et al., 2008). However, due to the disparity in intrinsic scales between each of these domains, it was decided to consider the influence of the temporal variable only in the former. Thus, the OPWs were represented as a sequence of steady states. Despite this simplification, the environments created for this purpose served as a foundation for collaborative work among various disciplines, particularly in the design phase, greatly expediting the process of estimating project flow curves, for example. During this same period, the real-time acquisition of data from subsurface sensors and the SPU, through DCS/SCADA systems, enabled the automation of flow simulations, generating new results for each change in boundary conditions. Among the products thus derived are virtual multiphase meters, capable of inferring the instantaneous individual flow rate of wells in a wide range of situations, reliably and accurately (Haldipur and Metcalf, 2008). More comprehensive tools, such as the one presented by Havre et al. (2009), provide detailed information about the field’s operation at that moment (real-time mode), while also allowing for the anticipation of potential difficulties (look-ahead mode) and testing the system’s performance under different scenarios (what-if mode). Another notable area of work involves the online optimization of production within the constraints imposed by the reservoir, wells, pipelines, and crude oil processing (Bieker et al., 2007). Regardless of the purpose, architecture, or formulation used, however, all simulations aimed at OPWs are based on solving computational versions of the conservation equations for mass, momentum, and energy of the phases. These equations will be presented in Sect. 5.2 for steady-state applications. It is evident that beneath this straightforward and objective strategy, several other intricate details also require representation, namely: Fluid Properties: Compositional models (such as the one discussed in Chap. 3) or black oil models are responsible for determining, based on a pressure-temperature or pressure-enthalpy pair, the number of phases present and their proportions, as well as their respective thermodynamic and transport properties. These calculations can be performed during the simulation or beforehand by defining a mesh in pressure-temperature (P, T ) or pressure-enthalpy (P, h m ) space, typically composed of 50 × 50 equidistant points. The model is then applied at each node, and the results are presented in a table or fluid file. In the latter case, the simulation

5.1 Introduction

207

gains efficiency as the intricate determination of liquid-vapor equilibrium in the mixture is reduced to a simple two-dimensional interpolation. Consistent fluid properties are essential for successful modeling as they constitute important input parameters in all the aforementioned conservation equations; Pipe Flow: Empirical correlations or mechanistic models (such as the ones presented in Chap. 4) are necessary to quantify the volumetric fractions and average shear stress of the flow along the production tubing and pipeline. These values serve as closure relations for the momentum conservation equation of the mixture; Flow in Singularities: All equipment in the multiphase production system that affects the composition or thermodynamic state of the fluid must also be represented. Valves such as choke valves and gas-lift valves, centrifugal pumps, and subsea manifolds are examples of singularities with their own algebraic equations, which, depending on the desired level of fidelity in the simulation, can be as complex as the previously mentioned models; Heat Transfer: In the flowline and riser, heat transfer involves calculating convection on the inner walls (caused by the flow of the hydrocarbon mixture) and on the outer walls (caused by the ocean current), as well as heat conduction within the pipelines, which consist of multiple layers with distinct properties and functions. In the well, the last two mechanisms are replaced by forced or natural convection of the fluid in the annular space between the tubing and the production casing, and transient radial conduction in the rocks above the reservoir. The overall heat transfer coefficient (U) accounts for the influence of each of these components on the magnitude of the heat flux leaving the flow, affecting the energy conservation equation of the mixture. In expedited analyses, it is common to assume prescribed values of U instead of calculating them. An up-to-date analysis of heat transfer in wellbores is provided by Hasan and Kabir (2020); Reservoir Productivity: The simplest way to represent the rate at which the mixture reaches the perforations as a function of the local pressure is through the use of IPR curves, introduced in Sect. 1.6 and further discussed in Chap. 6. These curves need to be regularly updated due to the tendency of increasing water cut and gas-oil ratio (GOR) levels and decreasing static pressure over time. In this context, integrated models adopt a clearly more predictive approach, although they require significant computational effort. Among the five topics mentioned above, only the flow in singularities will not be included in the scope of this book, as there are simple ways to circumvent it and still fulfill the goals established in the initial chapter. Section 5.3 is dedicated to the simplified modeling of heat transfer in the well and subsea pipelines. In Sect. 5.4, the main methodology for calculating the flow rate, pressure profiles, and temperature profiles used in this work is presented. The preparation of computational versions for a set of real OPWs from the pre-salt region, along with the simultaneous creation of a database with field readings, is the subject of Sect. 5.5. Finally, in Sect. 5.6, a discussion is conducted regarding the initial results, which were obtained on a preliminary basis.

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5.2 Conservation Equations In their general concept, conservation equations establish that certain properties of an isolated system do not vary in time and space, regardless of the physical phenomenon occurring within it. However, in cases where interaction with the surroundings occurs, these equations ensure that the contribution of each possible exchange mechanism on the aforementioned quantity can be expressed on a common basis and calculated accurately. Fluid flow is governed by three types of conservation equations: mass (also known as the continuity equation), linear momentum (based on Newton’s second law), and energy (equivalent to the first law of thermodynamics). Additionally, for completeness, it should be mentioned that the conservation equation of angular momentum and the second law of thermodynamics are essential for certain analyses (White 2016, Chap. 3). However, there is no unique way to express these equations; the most appropriate formulation depends on the problem at hand. In the present case, a one-dimensional representation in steady-state flow of a mixture with two continuous phases within a control volume has been chosen. The omission of further details regarding the movement of discrete particles, such as gas bubbles or liquid droplets, greatly simplifies the derivation of the expressions, as demonstrated subsequently, with little to no detriment to the simulations that will be conducted in the future.

5.2.1 Mass Conservation As famously stated by Lavoisier,2 “in nature, nothing is created, nothing is lost, everything is transformed”. Let us consider the example of the pipe section illustrated in Fig. 5.1, which has a circular cross-section with a diameter D and an area A, a length L, and initial and final ends denoted by 0 and 1, respectively. By placing a control volume in the portion occupied solely by the liquid phase, it can be observed that: M˙ l,1 − M˙ l,0 = M˙ cond ,

(5.1)

where M˙ cond denotes a condensation mass flow rate, since the possibility of phase change along the path between 0 and 1 must be considered. This same balance, when applied to natural gas, results in: M˙ g,1 − M˙ g,0 = M˙ evap ,

2

Antoine-Laurent de Lavoisier (1743-1794), French chemist.

(5.2)

5.2 Conservation Equations

209

1

ΔL

˙ l ,1 M

0 ˙ cond M θ

˙ l ,0 M

Fig. 5.1 Mass balance in the control volume

where M˙ evap is the evaporation mass flow rate through the phase interface. It is also known that: M˙ cond = − M˙ evap .

(5.3)

Adding Eqs. (5.1) and (5.2) yields: 

M˙ g + M˙ l

 1

  − M˙ g + M˙ l 0 = 0.

(5.4)

Therefore, the mass conservation equation of the mixture is equivalent to: M˙ m,1 = M˙ m,0 = M˙ m .

(5.5)

Despite its simplicity, Eq. (5.5) forms the basis for calculating the superficial velocities of the liquid and gas phases at both ends of the pipe segment. By associating the parameters εk and ρk with the local pressure and mixture enthalpy, whose variations will be quantified shortly, one can simply apply Eq. (1.3) with fixed values of A and M˙ m to determine Usk .

5.2.2 Momentum Conservation The rate at which the product of mass and velocity of a system varies with respect to time is solely determined by the resultant of the external forces acting on it. Therefore, for the control volume depicted in Fig. 5.2, the momentum balance in the flow direction results in:

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5 Simulation of Offshore Production Systems

1   ˙ Ul l M

ΔL

(P

1

A l) 1

Ui ond

0

˙c M F wl Fi

  ˙ Ul 0 Ml l) 0 (P A

θ

Fig. 5.2 Momentum balance in the control volume



   M˙ l Ul 1 − M˙ l Ul 0 = (P Al )0 − (P Al )1 − Fwl −

1

ρl Al g sin θ d L + Fi + M˙ cond Ui ,

0

(5.6) where the term contained in the integral represents the force of gravity, which opposes the upward movement of the fluid, and Ui denotes the velocity at the interface. Following an identical procedure for natural gas, one has: 

M˙ g Ug





− M˙ g Ug 1

 0



= P Ag

 0



− P Ag

 1

1 − Fwg −

ρg A g g sin θd L − Fi + M˙ evap Ui .

0

(5.7) When added together, Eqs. (5.6) and (5.7) stipulate that:   M˙ m Um,1 − Um,0 = (P A)0 − (P A)1 − Fw −

1 ρm Ag sin θ d L

(5.8)

0

where: Um = εg Ug + εl Ul ,

(5.9)

Fw = Fwg + Fwl .

(5.10)

5.2 Conservation Equations

211

Assuming that the cross-sectional area remains constant, Eq. (5.8) can be rewritten as:

 Fw M˙ m  Um,1 − Um,0 = P0 − P1 − − ρ¯m g sin θ L , A A

(5.11)

where ρ¯m represents the average density of the mixture between the two ends: 1 ρ¯m = L

1 ρm d L.

(5.12)

0

Therefore, the pressure drop in the segment is equivalent to: P1 − P0 = −ρ¯m g sin θ L −

 M˙ m  Fw − Um,1 − Um,0 . A A

(5.13)

Equation (5.13) indicates the existence of three mechanisms that directly influence the flow pressure profile. The first mechanism originates from gravitational action, accounting for 50 to 80% of the total pressure drop in pre-salt OPWs. The second mechanism is related to viscous friction, representing approximately 20 to 50% of the total pressure drop. Finally, there is the effect of mixture acceleration, primarily caused by the expansion of the gas phase, which typically constitutes less than 1% of the total pressure drop. Regarding the calculation of ρ¯m , it is essential within the current scope to employ a simple method (without compromising the computational time of the simulations) and robust approach (that avoids the occurrence of numerical instabilities). By assuming a linear profile for the mixture’s density along the control volume, Eq. (5.12) leads to the following expression: ρ¯m =

ρm,0 + ρm,1 . 2

(5.14)

In the case of the average shear force on the pipe wall, it is important to consider the need for a formulation that can be applied to all flow regimes, in addition to the stratified flow pattern chosen as an example in Fig. 5.2. In this regard, it is understood that the best alternative would be to employ the estimates of τw provided by the flow models in Sect. 4.7, assuming once again a uniform variation between sections 0 and 1. Consequently, the following expression is obtained:   τw,0 + τw,1 (5.15) Fw = π DL 2 The limitations of this linear weighting of flow properties, which led to Eqs. (5.14) and (5.15), will be discussed in Sect. 5.4.

212

5 Simulation of Offshore Production Systems

5.2.3 Energy Conservation The sum of the heat transfer rates crossing the system and all the work done by (or on) it determine the rate of change of its total energy with respect to time. By applying this principle to the control volume illustrated in Fig. 5.3, the energy balance takes the following form: 

     Ul2 Ul2 ˙ ˙ − Ml h l + Ml h l + 2 2 1 0   1 Ui2 ˙ . = −qwl − ρl Ul Al g sin θ d L + qi + Fi Ui + Mcond h i + 2 0

(5.16) In Eq. (5.16), the definite integral between 0 and 1 represents the calculation of the work done by the body forces (resulting from gravitational action), while qwl and qi denote, respectively, the heat transfer rate from the liquid phase to the external environment and at the interface. By applying the same analysis to natural gas, one obtains:



  Ug2 Ug2 − M˙ g h g + M˙ g h g + 2 2 1

1 = −qwg −

0

  Ui2 ˙ . ρg Ug A g g sin θ d L − qi − Fi Ui + Mevap h i + 2

0

(5.17)

1

ΔL

 ˙ cond M

0 F iU i 

 ˙ hl Ml

2

qi



Ul + 2

0

θ

Fig. 5.3 Energy balance in the control volume

q wl

2

hi



Ui + 2



 ˙ hl Ml

2



Ul + 2

1

5.3 Heat Transfer in Offshore Production Wells

213

The sum of Eqs. (5.16) and (5.17) results in: 

M˙ m



U2 hm + m 2





− M˙ m 1



U2 hm + m 2



1 = −qw −

0

ρm Um Ag sin θ d L, 0

(5.18) where: h m = εg h g + εl h l ,

(5.19)

Um2 = εg Ug2 + εl Ul2 ,

(5.20)

qw = qwg + qwl .

(5.21)

Considering that ρm Um A = M˙ m , a parameter that, according to the mass conservation equation, represents a constant of the flow under analysis, Eq. (5.18) can be expressed as follows:     U2 U2 − M˙ m h m + m = −qw − M˙ m g sin θ L . M˙ m h m + m 2 1 2 0

(5.22)

Thus, the mixture enthalpy change between the two ends will be equivalent to: h m,1 − h m,0 = −

 qw 1 2 2 Um,1 − Um,0 . − g sin θ L − 2 M˙ m

(5.23)

In a similar manner to what occurred for ρ¯m and Fw , the average rate at which the mixture within the control volume transfers heat to the external environment is determined by the following calculation:     qw,0 + qw,1 . (5.24) qw = π DL 2

5.3 Heat Transfer in Offshore Production Wells Although it has a broader usage beyond the representation of convection phenomena, the overall heat transfer coefficient is closely linked to Newton’s cooling law, as it also establishes a linear relationship between heat flow and temperature difference. Thus: qw = U (T − Tenv ) .

(5.25)

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5 Simulation of Offshore Production Systems

Table 5.1 Reference values for the overall heat transfer coefficient in the main components of an OPW Structure U , W/(m2 · K) Min Max 6 in production riser 8 in production flowline, no insulation 8 in production flowline, insulated Production tubing, any diameter

6.0 5.0 2.5 2.5

8.0 6.0 3.0 6.0

The OPW traverses two significantly distinct environments. In the production pipeline, Tenv represents the sea temperature, which ranges from 15–25 ◦ C at the surface to approximately 4 ◦ C at depths equal to or greater than 1000 m. The exact temperature profile in a specific region can be determined through meteoceanographic studies. U primarily refers to radial heat conduction, which accounts for no less than 90% of the total thermal resistance in the system. In several designs, the flowline incorporates one or two additional insulation layers to further mitigate heat loss to the external environment. However, this approach is not applicable to the riser due to weight restrictions as it is suspended. Typical values of U for each case are listed in Table 5.1. In the well, Tenv represents the temperature of the rock due to the exclusive action of the geothermal gradient, which, despite the different thermal properties of the various layers between the reservoir and the seabed, will be assumed constant. This simplification has little impact on the simulated profiles, as there are more pronounced uncertainties in the estimation of U. The latter parameter depends not only on the drilling sequence, cement type, casing, diameter and material of the production tubing, and the characteristics of the fluid contained in each of the annular spaces, but also on the effects of transient heat conduction during the productive life of the system. Given these numerous factors, it is natural for U to fall within a significantly wide range of values, as shown again in Table 5.1.

5.4 Application of the Marching Algorithm There is no doubt that the approximate calculation of ρ¯m , Fw , and qw proposed in Sect. 5.2 has significant practical utility. However, as the distance between the endpoints 0 and 1 increases, it is highly likely that it will begin to produce erroneous values, which quickly compromise the proper functioning of the conservation equations. The gas desolubilization previously contained in the oil, as well as the intensification of shear and thermal exchange resulting from it, are nonlinear phenomena, implying that the profiles of these three parameters bear little resemblance to a straight line.

5.4 Application of the Marching Algorithm

215

M˙

m

Fig. 5.4 Multiple segments within a straight pipe of length L

In order to overcome this difficulty, the so-called marching method proposes dividing the length L of the pipeline into n smaller segments, where Eqs. (5.14), (5.15), and (5.22) apply fully. Consequently, as depicted in Fig. 5.4, after prescribing known values at one end, the calculation of P and h m will proceed iteratively towards the other end (Brill and Mukherjee 1999, p. 26). The solution procedure comprises the following steps: 1. Starting from the pre-established values of pressure, enthalpy, and mixture mass flow rate at the inlet of the pipeline segment, the fluid properties (temperature, phase mass fractions, density, viscosity, surface tension), the phase superficial velocities, and all other local flow parameters (volumetric fractions, in situ velocities, wall shear stress) are calculated, along with the heat transfer rate to the external environments. The following steps will provide details on each of these calculations; 2. All the parameters obtained previously are transferred to the next segment as an initialization step; 3. The enthalpy of the mixture in the segment is updated using Eq. (5.22). Subsequently, it is necessary to calculate the thermodynamic and transport properties of the fluid for the new P-h m pair; 4. The superficial velocities of the phases are evaluated using Eq. (1.3); 5. By applying one of the mechanistic models presented in Sect. 4.7, all the local flow parameters are determined; 6. The magnitude of the heat flux in the segment and the average heat transfer rate to the external environment are calculated, according to Eqs. (5.25) and (5.24);

216

5 Simulation of Offshore Production Systems ˙m ΔM

˙m M

Fig. 5.5 Handling of pipeline sections with different diameters, inclination angles and inlet and outlet points along the OPW

7. The calculation process returns to step 3 until qw exhibits a negligible variation between two successive iterations. At the end of this loop, the mass and energy conservation equations of the mixture will be satisfied at the specified pressure; 8. Through Eqs. (5.14), (5.15), and (5.13), the average mixture density, the wall shear force, and the flow pressure at the analyzed segment are updated in that order. Once this is done, it is necessary to recalculate the fluid properties for the new pair P-h m ; 9. Return to step 3 until the convergence of ρ¯m and Fw is achieved. When this outer iterative loop is completed, the momentum conservation equation of the mixture will be satisfied; 10. For all subsequent segments, repeat the same procedure starting from step 2 until reaching the opposite end of the pipeline. Naturally, one could arrange the calculation procedure in different ways, resulting in slightly distinct but equally valid implementations. The algorithm must also be prepared to handle potential mass inlet or outlet points, as well as variations in diameter and angle of inclination, as illustrated in Fig. 5.5. Consequently, the computational model of a specific OPW is composed of a series of m uniformly geometric pipeline sections (Shoham 2006, p. 22). An important aspect to consider concerns the positioning of boundary conditions employed in simulations of this kind. It is customary to prescribe the flow pressure at the SPU inlet, while the mass flow rate and temperature are related to conditions at the bottom of the well. Thus, there is a mismatch in the calculation direction of the values of P, which moves towards the subsurface, and T and M˙ m , which align with the flow direction itself. This fact calls for some adaptations in the previously proposed methodology. For instance, the pressure solution process is no longer required to apply the energy conservation equation or update the mixture enthalpy along the way. When reaching the other end of the OPW, where a reservoir productivity model such as an IPR curve is present, the new well operating flow rate must be calculated. The direction of the solution process then reverses, free from observing the momen-

5.5 Data Collection and Model Construction

217

tum conservation equation and modifying the pressures encountered along the path to the SPU. This procedure is repeated until the values of P, T , and M˙ m at the opposite ends of the boundary conditions undergo sufficiently small changes between two successive iterations (e.g., 1 × 103 Pa, 1 × 10−2 K, and 1 × 10−3 kg/s), thus confirming the convergence of the method.

5.5 Data Collection and Model Construction Periodically, each OPW must undergo the so-called production test. After aligning the mixture flowing from the OPW to the test separator and allowing sufficient time for the flow to stabilize, the test consists of generating a report that records various operational parameters of the system. This includes readings of pressure and temperature obtained from sensors, as well as the average flow rates of oil, natural gas, and water. Originally, this test is a legal requirement imposed by regulatory agencies to provide concrete evidence for the daily allocation of oil production among the interconnected wells through the SPU. However, from an experimental perspective, such a report captures a snapshot of the flow of oil from the reservoir to the surface. By accumulating a sufficient number of these reports, whether by retrieving past operational conditions, incorporating data from other systems within the same field, or both, it becomes possible to establish a reliable and comprehensive flow database. This database is particularly useful for validating the two calculation methodologies introduced in Sect. 4.7. The selection of production tests requires certain considerations. Firstly, it is futile to represent a large number of OPWs if their production history is very recent. To develop such a model, the engineer relies on assumptions that do not always yield the desired effect. It is normal, therefore, for updates to occur along the way. Time and experience bring a clearer understanding of the phenomena involved, allowing, for example, the distinction between a legitimate operating point of the well and a potential sensor reading problem. Furthermore, if one wishes to conduct a truly comprehensive performance analysis, it would be interesting to consider different combinations of diameter, length, and depth, at varying levels of pressure and flow rate. Systems that are very similar to each other and produce under almost identical conditions have little to contribute in such circumstances. Based on these guidelines, a total of six OPWs were selected for the present study, all located in the southern region of the State of Espírito Santo, in two pre-salt fields of the Campos Basin. Their main attributes are presented in Table 5.2. It is worth noting that the letter g following the pressure unit designates its gauge value. Although they are geographically close to each other, significant differences can be observed in the length of the pipelines as well as in the water depth separating them from the surface. In fact, except for the fixed diameter of the risers and flowlines, there is always a noticeable range between the minimum and maximum values of each listed parameter. The highest pressure levels in the SPU usually occur at the beginning of the well’s productive life, when the primary reservoir energy shows no

218

5 Simulation of Offshore Production Systems

Table 5.2 Constructive and operational aspects of the six OPWs selected for analysis Parameter Min Avg Max Water depth, m Production riser length, m Production riser internal diameter, mm Production flowline length, m Production flowline internal diameter, mm PDG depth, m Tubing internal diameter, mm SPU pressure, barg 3 /d Liquid flow rate, mstd BSW, %

1330 2000 –

1400 2100 152.4

1450 2200 –

2000 –

3600 203.2

5800 –

3500 76.2 15 1500 0

3750 118.6 65 2900 4

3900 150.4 140 3900 20

Table 5.3 Representative samples, subsea arrangements and number of production tests of the six OPWs selected for analysis Well Sample Arrangement Production tests Pipeline Tubing OPW-01 OPW-02 OPW-03 OPW-04 OPW-05 OPW-06

PS-01 PS-01 PS-01 PS-01 PS-02 PS-02

Satellite Satellite Satellite Manifold Satellite Satellite Total

33 19 10 27 27 33 149

16 11 10 27 27 9 100

signs of decline and the BSW content is close to zero. Apart from that, there is a certain correlation between these pressure levels and the resulting liquid flow rate: the higher Psur f , the lower Q˙ l . Table 5.3 presents additional relevant information. It can be observed that oil sample PS-01 represents the volumetric and phase behavior of the mixture produced by the first four OPWs. The oil from this field is, therefore, heavier and contains more dissolved gas compared to the other sample, represented by PS-02, where the last two OPWs are located. Regarding the subsea arrangement, the predominant scheme is known as the satellite arrangement, in which there is a dedicated production pipeline for each well (see Fig. 1.3). On the other hand, in the manifold arrangement, the flowlines are connected to a homonymous equipment that centralizes production, directing it to a single riser. This particularity of OPW-04 allows for a more effective investigation of the flow in its horizontal section, as both the inlet and outlet of the manifold are equipped with pressure and temperature sensors. Clearly, due

5.6 Preliminary Results

219

to the harsh environment to which they are subjected, such instruments may experience malfunctions. Failures in the PDG or in the cable transmitting its signal to the WCT are the most common, and they ultimately reduce the quantity of valid records of pressure and temperature drop in the production reports. It seems that the temperature and pressure transmitter (TPT) is less affected by this type of problem, thus providing a considerably larger amount of information regarding the flow in the subsea pipelines. Considering that the production test always occurs at a constant and known flow rate, a computational model aiming to represent it does not necessarily need to employ IPR curves. In this situation, it is more appropriate to position a mass source at the reservoir location, assigning it the same value as Tr es 3 , while Q˙ l and BSW correspond to the specific production test report. Furthermore, the representation of the surfaceinstalled choke valve becomes redundant: it suffices to utilize the upstream reading as the boundary condition for Psur f . These measures not only facilitate the construction and subsequent resolution of the model but also result in more accurate pressure and temperature profiles, which will be incorporated into all simulations performed in this chapter. Finally, if the focus of the analysis is on a specific section of the OPW, it may be advantageous to divide it into subsystems. The arrangement of its sensors allows, for example, representing only the production tubing. This section maintains the same mass source as the complete model at its lower end, while prescribing the pressure reading related to the TPT at the other end. Similarly, there would be no issue in relocating the mass source to the WCT and matching its temperature value again to the one indicated by the TPT, thus creating a separate representation of the flow in the production pipeline. This level of detail becomes even more significant in systems such as the SMP-04, where the instrumentation present in the manifold enables the separation of the riser from the flowline, if necessary. Compared to the complete OPW model, these sectorized calculations obviously encompass a smaller number of sections, which greatly accelerates the convergence of the marching method.

5.6 Preliminary Results The development of a computational tool focused on the flow of oil in OPWs undoubtedly constitutes an extensive task, filled with intricacies and subtleties. However, after putting into practice all the concepts discussed throughout this chapter, another significant stage has been successfully completed. Now, it remains to be seen whether the various profiles stipulated by the marching method have a solid physical basis and how accurate their pressure drop estimates are for the different sections that make up the system. The following section addresses these two questions.

3

Flow models in multiphase systems often neglect potential variations in reservoir temperature during field exploitation.

220

5 Simulation of Offshore Production Systems

5.6.1 Qualitative Analysis of the Profiles Obtained Even though it requires a certain level of experience to be carried out, a simple inspection of parameters such as pressure, wall shear stress, temperature, volumetric fractions, and phase velocities is capable of revealing important aspects of the flow, highlighting the mechanisms at play in each situation, and exposing any potential physical inconsistencies in the simulated values. This diagnostic becomes clearer in comprehensive models, which depict the profiles experienced by the mixture throughout its path to the surface in a continuous manner, and where the phenomena occurring in the production tubing can be predicted to have an impact on the pipeline, and vice versa. A typical pressure profile during the early production life of SMP-01 is illustrated in Fig. 5.6, while Fig. 5.7 depicts the wall shear stress profile for the same operating condition. In the first 1000 m of the production tubing, where the nominal diameter is only 3 1/2 in (89 mm), the simulator indicates a significant pressure drop due to both gravitational effects and friction. This initiates the gas desolubilization process, which manifests as small bubbles. Subsequently, as the flow area increases in the shallower part of the well, reducing viscous dissipation, the pressure gradient

500

Flow pattern Single phase Disp. bubbles Slug

400

P , barg

300 PDG

200

TPT

100

SPU

0 Tubing

0

2

Riser

Flowline

4

6

8

10

L, km Fig. 5.6 Pressure profile calculated for the early production life of OPW-01, highlighting the different flow patterns developing inside the production tubing, flowline and riser. Comparison with the sensor readings in the well and in the SPU

5.6 Preliminary Results

221

250

200

τw , N/m2

150

100

50

0 Tubing

0

2

Riser

Flowline

4

6

8

10

L, km Fig. 5.7 Average wall shear stress profile calculated for the early production life of OPW-01

decreases, but not enough to prevent a change in flow pattern to slug flow. Once in the flowline, the gravitational component practically vanishes, and the friction with the larger-diameter pipe wall experiences a drastic reduction. This situation persists until reaching the production riser, where the mixture resumes an upward trajectory in a more confined space, further accentuating the pressure drop along the path. Figure 5.6 also reveals an excellent agreement between the calculated values and field measurements. Similarly, the resulting temperature profile of this simulation, depicted in Fig. 5.8, exhibits practical validation. Its slightly curved aspect along the production tubing is attributed to the gradual increase in temperature difference between the flow and the rock formations as the mixture approaches the seabed. Upon closer inspection, a slight heating can be observed in the initial segment, reflecting the high viscous dissipation imposed there. Figure 5.9 provides a clearer depiction of this phenomenon. In the flowline, the external conditions remain constant, resulting in a nearly linear temperature decline. It is noticeable that the change in gradient occurring in the riser coincides with an intensification of the pressure drop, suggesting that its origin is more closely linked to the gas expansion in that section rather than the heat transfer mechanism itself. Overall, the temperature evolution in the system does not fully align with the readings from the sensors, which are also presented in Figs. 5.8 and 5.9, but it falls within the tolerance range established for this and all other simulations, namely ±2 ◦ C for the TPT and ±4 ◦ C upon arrival at the SPU.

222

5 Simulation of Offshore Production Systems

160

Flow Environment PDG

120

T , °C

TPT

80 SPU

40

0 Tubing

0

Riser

Flowline

2

4

6

8

10

L, km Fig. 5.8 Internal (flow) and external (environment) temperature profiles calculated for the early production life of OPW-01. Comparison with the sensor readings in the well and SPU

130

PDG

T , °C

125

120

115

TPT

110 0.0

0.5

1.0

1.5 2.0 L, km

2.5

3.0

3.5

Fig. 5.9 Tubing temperature profile calculated for the first years of production of OPW-01. Comparison with the sensor readings in the well

5.6 Preliminary Results

223

1.0

αl

0.8

0.6

0.4

Tubing

0

2

Riser

Flowline

0.2 4

6

8

10

L, km Fig. 5.10 Liquid volumetric fraction profile calculated for the first years of production of OPW-01

The volumetric fraction and average velocity profiles calculated for OPW-01 are presented in Figs. 5.10 and 5.11, respectively. It can be observed that the pressure drop in the well causes a rapid vaporization of the mixture, so that upon reaching the WCT, natural gas already accounts for the majority of the displaced volume. As it expands along the path, its density decreases, inducing an acceleration of the overall flow. It is also noticeable that the velocity developed by the two phases depends much more on the production tubing diameter than on the slip between them. In the flowline, on the other hand, this relative motion gains strength, and the liquid component, being slower, increases its accommodation space. The transition to the riser imposes a new configuration to the flow, increasing Ug and Ul , and returning αl to its previous level. From that point onwards, the depressurization of the mixture intensifies both the gas release and its velocity gain. Given the discussion above, there is no doubt that the simulation conducted for OPW-01 provides a fairly accurate depiction of reality. It is worth mentioning that the profiles in Figs. 5.6, 5.7, 5.8, 5.9 and 5.11 correspond to the North Sea flow model. The Gulf of Mexico model exhibited identical trends to the previous case, differing only in absolute values. The discontinuities in αl or τw were so small compared to the minimum and maximum values found in each case that it is not worthwhile to reproduce them here. As for the performance of the marching method, no convergence issues were observed at any point. The thermal calculation also deserves special mention, particularly when considering the simplistic approach taken in modeling

224

5 Simulation of Offshore Production Systems

16

Gas Liquid

U , m/s

12

8

4

Tubing

0

2

Riser

Flowline

0 4

6

8

10

L, km Fig. 5.11 Average velocity profiles of the liquid and gas phases calculated for the first years of production of OPW-01

heat transfer throughout the system. Finally, it can be concluded that the formulation of the conservation equations was well-suited to the objectives of the present analysis, enabling the capture of certain phenomena that defy common sense, such as fluid heating in the narrowest section of the production tubing and cooling due to expansion in the riser.

5.6.2 Calculation of Pressure Drop in Offshore Production Wells Physical basis is a requirement that no mathematical model can do without, although it does not guarantee the quality of the estimates by itself. Therefore, in order to assess the accuracy of the pressure drop calculations in the previously selected six OPWs, all the available operational conditions were replicated in the simulator. Subsystems are more suitable for this purpose for two reasons. Firstly, by adopting a continuous representation of the flow, the disparity in the number of readings from the sensors would result in a total of 49 profiles without any reference parameters in the production tubing, rendering it an futile effort. Secondly, it is necessary to prevent

5.6 Preliminary Results

225

Table 5.4 Summary of the fluid flow database collected for the analysis Equipment Flow orientation Structures Fluids Production pipeline Production flowline Production tubing

Horizontal and inclined Horizontal Vertical

Readings

5

2

122

1 6

1 2

27 100

+

20

%

200

ΔPcalc , bar

160

%

-20

120

80

OPW-01 OPW-02 OPW-03 OPW-04 OPW-05 OPW-06

40

0 0

40

80

120 ΔPmeas , bar

160

200

Fig. 5.12 Tubing and pipeline pressure drops in the six OPWs. Comparison of field data with values calculated by the North Sea model

deviations originating from the pipeline from interfering with the flow predictions in the well, or vice versa, which could compromise the conclusiveness of the analysis. In the specific case of OPW-04, greater attention is focused on the flowline section, which holds valuable evidence regarding the mechanism of viscous dissipation, whereas its riser contributes little in terms of new information and will be disregarded. These decisions allow for the rewriting of the operational data table, as shown in Table 5.3, in the form presented in Table 5.4. By performing 249 simulations for the North Sea flow model and repeating them for the Gulf of Mexico version, the resulting values are shown in Figs. 5.12 and 5.13, respectively. It is evident that the greatest difference between them lies in an intermediate region, ranging from 40 to 120 bar, corresponding to the pressure drop in the five production pipelines. This behavior had already been anticipated, as known

226

5 Simulation of Offshore Production Systems

+ 20 %

200

ΔPcalc , bar

160

%

-20

120

80

OPW-01 OPW-02 OPW-03 OPW-04 OPW-05 OPW-06

40

0 0

40

80

120 ΔPmeas , bar

160

200

Fig. 5.13 Tubing and pipeline pressure drops in the six OPWs. Comparison of field data with values calculated by the Gulf of Mexico model

from the preliminary calculations in Sect. 4.8: by underestimating the volumetric liquid fraction and the average wall shear stress, the Gulf of Mexico model consistently provides lower estimates in this range. A similar trend is also observed in the range 0 < P < 40 bar, which pertains to the OPW-04 flowline. As for the six production tubing, which account for the third and most prominent level of pressure drop (120 < P < 200 bar), both approaches performed similarly, thus confirming the findings outlined in Sect. 4.8. After visually comparing the overall results, the impression is formed that the North Sea model performed better in representing the flow database of the present study. However, a definitive verdict necessitates the utilization of statistical parameters. In this regard, five parameters will be monitored. The Mean Percentage Error (MPE) quantifies the systematic error of the simulations and is defined as follows: MPE( X ) =

N 1 X i,calc − X i,ex p × 100. N i=1 X i,ex p

(5.26)

The Percentage Standard Deviation (PSD) seeks to depict the dispersion of results, which constitutes the random component of the error. It can be obtained by:

References

227

Table 5.5 Statistical parameters associated with the pressure drop predictions for production tubing and pipelines according to the two flow models Model Statistical parameters, % MPE PSD MAPE PA10% PA20% North Sea Gulf of Mexico

–1.47 –7.80

   PSD( X ) = 

6.95 8.69

5.59 10.1

83.5 53.0

99.6 94.0

2 N  1 X i,calc − X i,ex p × 100 − MPE( X ) . N − 1 i=1 X i,ex p

(5.27)

The MAPE, utilized since Chap. 3, possesses mixed characteristics between the MPE and the PSD, making it a comprehensive parameter on its own. It is given by:  N  1  X i,calc − X i,ex p   MAPE( X ) =  × 100. N i=1  X i,ex p

(5.28)

Finally, the Percentage of Accuracy (PA) indicates the model’s ability to generate predictions within a certain tolerance range, that is: Number of simulations with deviation within the range ± Y % PAY % ( X ) = . Total number of simulations (5.29) The results of the statistical analysis are presented in Table 5.5, and they confer a compelling superiority to the North Sea flow model in all aspects. Its pressure drop predictions, simultaneously neutral in terms of trend and with low dispersion, reproduced field data with deviations below 10% on 215 occasions, surpassing 20% only once. These values are consistent with those of a good commercial simulator and become even more promising when considering the improvement opportunities that will be investigated in the next chapter.

References Bieker, H. P., Slupphaug, O., & Johansen, T. A. (2007). Real-time production optimization of oil and gas production systems: A technology survey. SPE Production and Operations, 22(04), 382–391. Brill, J., & Mukherjee, H. (1999). Multiphase flow in wells. SPE monograph series (Vol. 17). Richardson, TX: Society of Petroleum Engineers. Haldipur, P., & Metcalf, G. D. (2008). Virtual metering technology field experience examples. In Paper no. 19525 presented at the offshore technology conference, Houston, Texas.

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Hasan, A., & Kabir, C. (2020). Fluid flow and heat transfer in wellbores (2nd ed.). Richardson, TX: Society of Petroleum Engineers. Havre, K., Trudvang, C., & Kjørrefjord, G. (2009). Practical experiences from development of online simulators for operational support. In14th international conference on multiphase production technology. Ozdogan, U., Keating, J. F., Knobles, M. M., Chawathe, A., & Seren, D. (2008). Recent advances and practical applications of integrated production modeling. In Paper no. 113904 presented at the SPE Europec/EAGE conference and exhibition, Rome, Italy. Shoham, O. (2006). Mechanistic modeling of gas-liquid two-phase flow in pipes. Richardson, TX: Society of Petroleum Engineers. White, F. (2016). Fluid mechanics (8th ed.). New York, NY: McGraw-Hill Education.

Chapter 6

Improving the Fluid Flow Model

6.1 Introduction Due to its inherently phenomenological character and the ease with which it is typically implemented, the mechanistic approach has greatly expanded its range of successful applications. During this widespread dissemination movement that has been taking place in recent decades, it is not uncommon for idealized models developed for a specific area to serve as a starting point for others. The development of the commercial simulator OLGA, for instance, originated in the nuclear industry. In the 1970s, the IFE institute was working on mathematical models focused on the cooling circuit of Boiling Water Reactors (BWRs), where water and steam coexist in certain sections (Bendiksen et al. 1991). However, it was soon realized that these models could also be used to represent the simultaneous flow of oil and natural gas in pipelines and production tubing. Subsequently, Statoil, (now Equinor), sponsored the new line of research. In 1983, the first version of the simulator was delivered, and due to the urgent need for a larger experimental database, a cooperation agreement involving several companies in the industry was established (Bendiksen et al. 1991). Another curious case of gas-liquid two-phase flow, which receives a treatment similar to that described in Chap. 4, comes from geosciences. It is well known that the degassing process of magma plays a crucial role in volcanic eruptions, determining both the volume of lava carried to the surface and the intensity of the explosions. According to Pering and McGonigle (2018), in events where the viscosity of the molten material remains sufficiently low, which depends on its temperature and composition (typically, basaltic lavas are less viscous than others), the gas bubbles formed not only flow more rapidly but also can coalesce and create long Dumitrescu-Taylor bubbles. By reusing classical concepts established by Davies and Taylor (1950) and Wallis (1969), as well as some correlations derived from important works such as Joseph (2003) and Viana et al. (2003), the authors were able to construct maps containing five different eruption regimes for three active volcanoes: Stromboli, Etna (both in Italy), and Yasur (located in Vanuatu, Melanesia). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. R. Gessner and J. R. Barbosa Jr., Integrated Modeling of Reservoir Fluid Properties and Multiphase Flow in Offshore Production Systems, Petroleum Engineering, https://doi.org/10.1007/978-3-031-39850-6_6

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6 Improving the Fluid Flow Model

Leaving the depths of the Earth’s globe to look in the opposite direction, flows of similar nature are encountered within Advanced Life Support (ALS) systems responsible for the revitalization of air, water recovery, waste processing, and thermal comfort during manned space missions. Balasubramaniam et al. (2006) elaborated a technical report aiming to depict the state of the art in modeling pressure drop and pattern prediction under microgravity or partial gravity conditions, such as those experienced on the Moon (g = 1.62 m/s2 ) and Mars (g = 3.72 m/s2 ). Their ultimate goal was to create analytical and numerical tools for the design of this new generation of space systems, as well as to predict their in-situ performance based on data obtained on the ground. Special interest was shown in the annular arrangement, which had a series of experimental readings conducted during parabolic flights on NASA aircraft (Bousman et al. 1996). Compared to previous examples, the task of improving an already validated model for oil flows may seem trivial. However, this is a misconception: without a clear understanding of the parameters to intervene in and lacking absolute control over the discontinuities that may be introduced in this manner, there is a high likelihood that the modifications will have the opposite effect of what is desired. Moreover, the available options in the literature for this specific sector of the industry are numerous, forming a kind of labyrinth from which one can only escape with the assistance of highly pragmatic selection criteria. There is no doubt that the key to designing a good mechanistic model lies in the judicious selection of closure relationships. With the theory regarding the phenomena governing the simultaneous flow of two phases already well established, the accuracy of the estimates is closely tied to the proper functioning of the empirical correlations incorporated therein. As stated by Shippen and Bailey (2012), it is clear that the way this type of approach has become popular has, on the other hand, brought a serious disadvantage: by focusing efforts on the characterization of specific phenomena, various initiatives have proposed or relied upon closure relations applicable only within certain ranges. This point had already been raised by Hanratty et al. (2003), who asserted that, in order to advance more promisingly, “the research emphasis in this area should change from a strictly engineering perspective (which has had limited success in developing general approaches) to a science-oriented-one”. Despite the progress that has followed since then, the development of a unique and definitive calculation methodology seems to remain a distant prospect. In view of the aforementioned, it is understood that the adaptation of the North Sea mechanistic model (described in Sect. 4.7 and tested in Sects. 4.8 and 5.6) for pre-salt OPWs should be undertaken as follows: firstly, it is necessary to identify, among the various parameters that compose its formulation, those that truly impact pressure drop estimates, which is the subject of Sect. 6.2. Those parameters found to be less useful in this regard will remain unchanged, while the more significant ones will have their correlations revisited in Sect. 6.3, which presents alternative studies applicable to the present scope. In addition to the inherent uncertainties in two-phase flow, the assumed roughness levels for the production tubing and pipelines also appear to exert a significant influence on these results, hence the topic of discussion in Sect. 6.4. Subsequently, the various modeling options must be individually compared

6.2 Identification of the Most Relevant Parameters

231

with the flow database from the previous chapter, which is carried out in Sect. 6.5. The combination with the lowest deviation will then undergo thorough validation in Sect. 6.6.

6.2 Identification of the Most Relevant Parameters A highly effective approach to infer the level of uncertainty associated with the representation of a given phenomenon is by examining the quantity of methods devised for this purpose. Such reasoning, albeit picturesque, enables us to make relevant observations. For instance, in the modeling of stratified flow, there exists a true profusion of correlations for calculating shear stresses, all of which converge to the force balance proposed by Taitel and Dukler (1976). A similar situation arises in the case of annular flow. When flow occurs in the form of bubbles, several expressions emerge for predicting the velocities of the phases, but they all stem from the slip law attributed to Zuber and Findlay (1965). In the case of slug flow, each author holds a slightly discordant opinion regarding the velocity attained by the Dumitrescu-Taylor bubble and the volumetric fraction of gas entrapped in the liquid slug, yet none of them dispute the validity of the unit cell proposed by Dukler and Hubbard (1975). Another unresolved discussion concerns the mechanisms underlying pattern transitions. However, as indicated in Sect. 4.8, the criteria used in the North Sea approach appear to have been optimized in an effort to minimize the discontinuities of αg and τw in these boundary regions, which are computationally undesirable. Any interference in these regions would pose a risk of reintroducing such discontinuities. Furthermore, these criteria are composed of straightforward expressions with reasonable physical basis, and there is no evidence to suggest that they may be compromising the quality of pressure drop estimates in OPWs. All these facts corroborate the idea that the most relevant parameters of the flow model in question are invariably found in its closure relationships. To identify them, it will be necessary to repeat the same procedure conceived in Chap. 3, which consists of equipping them with fitting coefficients in order to temporarily and reversibly modify their behavior. Subsequently, the effects that each parameter exerts on the simulations of the database in Sect. 5.6 can be determined.

6.2.1 Introduction of Fitting Coefficients In stratified flow, subtle changes in shear stresses can have significant consequences for local force equilibrium. In this case, it is deemed appropriate to adjust τwg and τwl in the simplest manner possible by introducing the multiplication factors ψτwg and ψτwl directly into the values calculated by the correlations. This leads to the following expression:

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6 Improving the Fluid Flow Model

τwg = ψτwg τwg,calc ,

(6.1)

τwl = ψτwl τwl,calc .

(6.2)

The shear stress at the interface requires additional attention, as it is seldom the case that τi < τwg . Therefore, it is more appropriate to consider the following approach:   τi = τwg + ψτi τi,calc − τwg .

(6.3)

Although a direct manifestation in the analyzed OPWs is unlikely, the annular pattern lends itself to the solution of the film velocity in the Dumitrescu-Taylor bubble region in pipelines with high inclination angles. Additionally, it is partially subject to the action of fitting coefficients. The exception lies in the liquid entrainment mechanism, which does not persist under such circumstances. By adapting Eqs. (6.2) and (6.3) to the new flow regime, one arrives at: τw f = ψτwl τw f,calc ,

(6.4)

τi = ψτi τi,calc .

(6.5)

It is worth noting that Eqs. (6.2) and (6.4) use the same coefficient ψτwl , which also happens for ψτi in Eqs. (6.3) and (6.5). It is known that the modeling of bubble flows revolves around the assumed velocity of the gas phase and the average wall shear stress. In this case, it is essential to ensure that the action on Ug does not reverse its relative motion in the mixture, thereby ruining the physical meaning of the predictions. Therefore, it is established that:   Ug = Usm + ψUg,bolhas Ug,calc − Usm .

(6.6)

The fitting of τw will also begin from a reference value, that is:   τw = τw,ref + ψτw,bubbly τw,calc − τw,ref ,

(6.7)

where: τw,ref = f w (Relm , e/D)

ρl |Ulm | Ulm . 2

(6.8)

The combination of Eqs. (6.7) and (6.8) prevents the wall shear stress τw from reaching values lower than those observed in a hypothetical flow where all properties of the mixture are identical to those of the liquid phase. In regard to slug flow, the fourth and final flow pattern discussed in the North Sea approach, its key parameters pertain to the representation of the Dumitrescu-Taylor bubble velocity and the gas volume fraction in the slug. Once again, it is crucial that

6.2 Identification of the Most Relevant Parameters

233

modifications to U DT are based on the average displacement of the mixture in order to maintain the consistency of the estimates. By applying the same technique used for Ug in Eq. 6.6, one obtains:   U DT = Usm + ψU DT U DT,calc − Usm .

(6.9)

Finally, αg,dist will be directly linked to a multiplication factor, resulting in the following expression: αg,dist = ψαg,dist αg,dist,calc .

(6.10)

6.2.2 Effect of Fitting Coefficients on Pressure Drop Estimates After its valuable contribution to the development of the thermodynamic package discussed in Chap. 3, factorial design will also play a prominent role in calculating the effects of the newly introduced fitting coefficients in the North Sea mechanistic model on the estimates of P. The description, indices, and chosen minimum and maximum levels for each of the seven parameters are presented in Table 6.1. In principle, there would be no disadvantage in standardizing these values to 1.0 ± 0.1, for example. However, it was deemed sensible to introduce some distinction between closure relationships that are based solely on measurable flow quantities and those that depend on the results of auxiliary expressions. The second group includes τi , τw , and αg,dist , for which a larger variation of ±0.2 was considered appropriate. By factorial combination of the adjustment coefficients, a total of 27 = 128 possible configurations are defined. These configurations are subsequently used in simulating the database in Sect. 5.6, from which two average pressure drop values are obtained: one for the pipelines (including the OPW-04 flowline) and another for the production tubing. After all, considering the different conditions of P and T to which they are subjected, along with the unique geometry of each section, it would be naive to assume that they are governed by the same mechanisms. The primary effects resulting from this process are shown in Table 6.2. In that table, β0 represents the average value of the monitored variable and is of little importance at the moment. The remaining coefficients reveal trends in the model that are in absolute agreement with practical experience. For instance, since β1 , β2 , β3 , and β5 are positive numbers, it is confirmed that pressure drop is directly related to the shear stresses observed at the pipe wall and interface. The same applies to β4 and β6 , as the acceleration of gas bubbles leads to an increase in the mixture’s mass density, which in turn affects the gravitational component of P. On the other hand, β7 has the opposite sign, indicating that the higher the gas volume fraction in the piston, the lower the presence of liquid in the unit cell (see Eq. 4.61), resulting in a reduction in ρm and, consequently, P.

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6 Improving the Fluid Flow Model

Table 6.1 List of parameters selected for the factorial design applied to the flow model, their indices and minimum and maximum levels Parameter Description Index Levels ψτwg ψτwl ψτi ψUg,bubbly ψτw,bubbly ψU DT ψαg,dist

τwg fitting in stratified flow τwl and τw f fitting in stratified and annular flows τi fitting in stratified and annular flows Ug fitting in bubbly flow τw fitting in bubbly flow U DT fitting in slug flow αg,dist fitting in slug flow

1 2 3 4 5 6 7

1.0 ± 0.1 1.0 ± 0.1 1.0 ± 0.2 1.0 ± 0.1 1.0 ± 0.2 1.0 ± 0.1 1.0 ± 0.2

Table 6.2 Primary effects of the fitting parameters of the North Sea mechanistic model on the average pressure drop in pipelines and tubing of six pre-salt layer OPWs βi Parameter DP, bar pipeline P, bar tubing 0 1 2 3 4 5 6 7

ψτwg ψτwl ψτi ψUg,bubbly ψτw,bubbly ψU DT ψαg,dist

71.46 0.106 0.263 0.437 0.001 1.555 0.277 −1.422

146.6 0.001 0.080 0.116 0.037 1.435 0.091 −1.120

Regarding the differences between flow in pipelines and production tubing, it is perfectly understandable that ψUg,bubbly has a negligible effect on the former, where slug flow predominates, while ψτwg , ψτwl , ψτi , and ψU DT prove to be less effective in the latter, which is more prone to bubbly flow. It is also evident that ψτw,bubbly and ψαg,dist emerge as the most influential parameters in both situations. The previous conclusion becomes even more explicit in Table 6.3, which normalizes the main effects by dividing them by β0 . Results larger than 0.5% are underlined. Once again, the simulations proved to be more sensitive to the fitting of the gas volumetric fraction in the liquid slug, followed by the wall shear stress for bubbly flow. The influence of these two parameters is so superior to the others that even their cross-effect (denoted by β57 ) deserves highlighting. Therefore, these will be the closure relationships investigated in more depth in the next section. The calculation of interfacial shear in stratified and annular arrangements, ranking third, offers a rather modest margin of gain and is limited to production pipelines, thus exempting it from further analysis.

6.3 Survey of Alternatives to Current Calculation Methods

235

Table 6.3 Main effects of the fitting parameters of the North Sea mechanistic model on the average pressure drop in pipelines and tubings of six pre-salt layer OPWs βi /β0 Avg. P, bar βi /β0 Avg. P, bar βi /β0 Avg. P, bar Pipeline Tubing Pipeline Tubing Pipeline Tubing 1 2 3 4 5 6 7 12 13 14

0.001 0.004 0.006 1.4E-5 0.022 0.004 0.020 1.1E-4 4.2E-5 7.6E-8

7.1E-6 5.5E-4 7.9E-4 2.5E-4 0.010 6.2E-4 0.008 6.6E-6 6.1E-6 6.1E-6

15 16 17 23 24 25 26 27 34 35

3.8E-5 3.5E-5 1.9E-4 5.8E-4 7.4E-8 4.5E-5 8.0E-4 0.001 8.2E-8 5.7E-4

6.1E-6 3.2E-8 4.8E-6 1.0E-4 6.1E-6 5.0E-8 4.9E-5 5.1E-4 5.9E-6 5.7E-5

36 37 45 46 47 56 57 67 Others Total

9.5E-5 0.002 1.1E-7 5.3E-7 1.4E-5 3.2E-4 0.006 0.002 0.001 0.073

2.0E-5 7.1E-4 1.9E-5 1.4E-7 1.1E-4 4.2E-5 0.001 5.4E-4 5.5E-4 0.024

6.3 Survey of Alternatives to Current Calculation Methods Any attempt to describe a specific physical phenomenon is inherently constrained by the theoretical knowledge and available technology of its time. Regarding the closure relations present in the North Sea model, the origins can be primarily traced back to the 1990s when the foundations of the mechanistic approach had already been established, although there were still limited experimental studies conducted on oil mixtures. This situation has significantly changed over the past 25–30 years. Today, with the advancements in measurement instruments and the development of increasingly versatile experimental rigs, a much larger number of tests can be conducted encompassing various fluids, superficial velocities, pressures, diameters, and orientations. Consequently, this has led to noticeably more precise and comprehensive correlations. Therefore, it is appropriate to provide an overview of the main methodologies developed during this period for the calculation of τw in bubbly patterns and αg,dist in slug flows, which are the two predominant parameters in flow simulations in pre-salt OPWs.

6.3.1 Wall Shear Stress in Bubbly Flow The methods of representing the magnitude of friction originating from the pipe wall due to the simultaneous flow of gas bubbles and the continuous liquid medium vary substantially, both in terms of complexity and physical basis. It is highly likely that Zhang et al. (2003b) and Gomez et al. (2000b) have opted for the simplest of these methods: an expression for τw (see Eq. 4.139) without a two-phase multiplier, and

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6 Improving the Fluid Flow Model

whose Reynolds number (defined in Eq. 4.140) takes into account the individual contribution of each phase in calculating the resulting inertia force, as it assumes 2 , but does not repeat the procedure for viscous forces by using μ = μl . ρU 2 = ρm Usm Petalas and Aziz (1998) are more consistent in this regard, relating an dimensionless group to Eq. (4.139), given by: Resm =

ρm Usm D , μm

(6.11)

where the apparent viscosity of the mixture (μm ) is equivalent to: μm = αg μg + αl μl ,

(6.12)

Despite their practicality for everyday use, the two aforementioned alternatives have the disadvantage of not having undergone thorough validation, at least in the publications to which they refer. Both alternatives were endorsed based on the overall performance of their respective models in estimating αg and d P/d L, and received little to no additional attention. The same can be said about the correlation by Bendiksen et al. (1996), where vital information regarding the dataset used in the regression, application intervals, etc., is lacking. In this regard, the work of Shannak (2008) is an exception. Building upon an impressive dataset of 16,000 experimental data points previously collected by Friedel (1979), encompassing horizontal and vertical flows of water and steam, refrigerants, and binary systems composed of air, nitrogen, methane, synthetic oil, and water, among others, for diameters up to 257 mm, maximum pressure of 21.2 MPa, mass fluxes of up to 1 × 104 kg/(m2 · s), and surface tensions ranging from 0.002 to 0.080 J/m2 , the author proposed a novel expression for τw , wherein it is established that: τw = f w (Rem , e/D)

ρm |Usm | Usm . 2

(6.13)

The first distinction of Eq. (6.13) from its counterparts lies in the definition of the Reynolds number of the mixture, which is now expressed as follows: Rem =

2 ρg Usg + ρl Usl2

μg

Usg D

+ μl UDsl

,

or by rearranging the terms:     2 ρg Usg ρl Usl2 1 1 1 = + . 2 2 2 2 Rem ρg Usg + ρl Usl Resg ρg Usg + ρl Usl Resl

(6.14)

(6.15)

A quick inspection of Eqs. (6.14) and (6.15) is enough to verify that Rem → Resg when Usl → 0, and that Rem → Resl to Usg → 0.

6.3 Survey of Alternatives to Current Calculation Methods

237

In turn, the mass density of the mixture must be calculated by invoking the hypothesis of homogeneous flow, in which the volumetric fraction of each phase simply corresponds to αk = Usk /Usm . Therefore: ρm =

ρg Usg + ρl Usl . Usm

(6.16)

The correlation proposed by Shannak (2008) provides estimations with PSD of 35% in relation to the database Friedel (1979), a value that drops to 2% when considering only the tests undertaken by the author himself, using air and water in pipes ducts with inclinations of 0◦ and 90◦ . This is, of course, an excellent achievement, but not significantly superior to the original model of Friedel (1979), which recorded 40% and 3.2% in the two comparisons. This alternative approach describes the wall shear stress through Eq. (4.51) and a two-phase multiplier of the form:  2 φlm = εl2 + ε2g

ρl ρg



0.24 3.24ε0.78 f w (Regm , e/D) g εl + 0.035 0.09 f w (Relm , e/D) Frm Wem



ρl ρg

0.9 

μg μl

0.2   μg 0.7 , 1− μl

(6.17)

where ρm is again linked to Eq. (6.16) and: Usm Frm = √ , gD Wem =

2 D ρm Usm . σ

(6.18)

(6.19)

Yadigaroglu and Hewitt (2017, p. 181) indicate the model of Friedel (1979) for systems in which μg /μl > 0.001, and highlight the positive fact that he recognizes the effect of surface tension on τw . Clearly, the works mentioned thus far constitute a small sample of the myriad of existing research. Ghajar and Bhagwat (2013), for instance, compiled no less than 14 expressions for the apparent mixture viscosity from various authors. Their justifications range from mere intuition to analogies with heat transfer in porous media (Awad & Muzychka, 2008). However, the decision has been made to restrict the scope of the current survey to the most widely used methodologies in the petroleum industry and those supported by the strongest experimental evidence. It is almost certain that at least one of these methods will yield benefits for future pressure drop simulations in the pre-salt deep-water fields. In the meantime, it is worth subjecting them to initial testing as a means of anticipating behaviors and, perhaps, identifying any anomalies. Let us consider the correlations of Friedel (1979), Bendiksen et al. (1996), which is used in the North Sea model, Petalas and Aziz (1998), Zhang et al. (2003b), and Shannak (2008). By simultaneously applying these correlations in the calculation of the wall shear stress, and considering the geometries and fluid properties typically

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6 Improving the Fluid Flow Model

1.25

1.20

Friedel (1979) Bendiksen et al. (1996) Petalas and Aziz (1998) Zhang et al. (2003b) Shannak (2008)

φ2lm

1.15

1.10

1.05

1.00 0.00

0.05

0.10 αg

0.15

0.20

Fig. 6.1 Behavior of the two-phase multiplier as a function of the gas volumetric fraction at the PDG of a typical pre-salt OPW, according to the various correlations

encountered in the PDG, WCT, and SPU (further details in Sect. 4.8), as well as the assumption that the flow always occurs in the bubbly pattern, the profiles illustrated in Figs. 6.1, 6.2, 6.3 and 6.4 are obtained. It is worth noting that the chosen maximum value of αg on the abscissa axis for each figure is merely an attempt to represent the degassing process that occurs along the path, and the use of the two-phase multiplier 2 instead of τw on the ordinate axis aims to facilitate the comparison of these φlm different scenarios. Since the mixture mass flow rate is kept constant in all sections, Relm and Ulm become locally constant, establishing a direct relationship between these two quantities. Thus, even if certain correlations do not explicitly manipulate 2 , its value can be easily obtained through Eq. (4.56). φlm The most evident observation that can be derived from Figs. 6.1, 6.2, 6.3 and 6.4 2 with αg , correctly captured by all the expresconcerns the increasing trend of φlm sions. As the mixture progressively forms a more significant gas phase, one expects both an elevation in the average flow velocity and an intensification of turbulence levels, thereby enhancing the viscous friction at the pipe wall. Additionally, it is noticeable that the correlation proposed by Friedel (1979) describes a consistently steeper curve compared to the others until αg ≈ 0.1, where it loses strength. Another noteworthy aspect is found in the predictions obtained from the work of Bendiksen et al. (1996), which simultaneously represent the minimum growth limit in the PDG and the maximum limit in the SPU. The explanation behind this is straightforward: by relating the wall shear stress to the gas bubble rise velocity, the model becomes

6.3 Survey of Alternatives to Current Calculation Methods

1.8

φ2lm

1.6

239

Friedel (1979) Bendiksen et al. (1996) Petalas and Aziz (1998) Zhang et al. (2003b) Shannak (2008)

1.4

1.2

1.0 0.0

0.1

0.2 αg

0.3

0.4

Fig. 6.2 Behavior of the two-phase multiplier as a function of the gas volumetric fraction upstream of the WCT of a typical pre-salt OPW, according to the various correlations

2.0

1.8

Friedel (1979) Bendiksen et al. (1996) Petalas and Aziz (1998) Zhang et al. (2003b) Shannak (2008)

φ2lm

1.6

1.4

1.2

1.0 0.0

0.1

0.2 αg

0.3

0.4

Fig. 6.3 Behavior of the two-phase multiplier as a function of the gas volumetric fraction downstream of the WCT of a typical pre-salt OPW, according to the various correlations

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6 Improving the Fluid Flow Model

3.0

2 φlm

2.5

Friedel (1979) Bendiksen et al. (1996) Petalas and Aziz (1998) Zhang et al. (2003b) Shannak (2008)

2.0

1.5

1.0 0.0

0.1

0.2

0.3 αg

0.4

0.5

0.6

Fig. 6.4 Behavior of the two-phase multiplier as a function of the gas volumetric fraction as it reaches the SPU of a typical pre-salt OPW, according to the various correlations

susceptible to variations in the mass density difference between the phases, leading to progressively more severe estimates as pressure decreases. On the other hand, the correlations of Petalas and Aziz (1998), Zhang et al. (2003b), Shannak (2008) exhibited similar performance despite vehement disagreement regarding the calculation of the two-phase flow Reynolds number. The first correlation consistently emphasizes 2 than the second one, while the third correlation oscillates slightly less growth of φlm within the range encompassed by the other two. Furthermore, it is evident that none of the five models for calculating τw demonstrates any signs of physical inconsistency in their outcomes, and all of them present themselves as viable alternatives for the accuracy test described in Sect. 6.5.

6.3.2 Volume Fraction of Gas in the Liquid Slug for the Slug Pattern To gain a deeper understanding of the significance of αg,dist in modeling the slug flow pattern, it is necessary to refer back to Chap. 4, where the concept of the unit cell introduced by Dukler and Hubbard (1975) was discussed. The following demonstration, which is seldom found in textbooks or academic articles, rearranges the terms of Eq. (4.61) as follows:

6.3 Survey of Alternatives to Current Calculation Methods

    Usg = αg Ug dist + αg,uc − αg,dist U DT . Dividing now both sides by αg,uc yields:     αg,dist αg,dist Ug,dist + 1 − U DT , Ug = αg,uc αg,uc

241

(6.20)

(6.21)

where: U DT αg,dist .  = Usg αg,uc U DT + αg,dist − Ug,dist

(6.22)

Therefore, in cases where αg,dist → 0, the average velocity of the gas phase is solely dictated by the Dumitrescu-Taylor bubble, and at the opposite extreme, αg,dist → αg,cell , Ug reflects only the motion of the bubbles contained in the liquid slug. All other possibilities theoretically originate from a combination of these two flow regimes. Considering that both regimes adhere to the slip law proposed by Zuber and Findlay (1965), Eq. (6.21) can also be expressed in the following form: Ug = C0,uc Usm + U0,uc ,

(6.23)

where:  C0,uc =  U0,uc =

αg,dist αg,uc αg,dist αg,uc



  αg,dist C0,DT , + 1− αg,uc

(6.24)

  αg,dist U0,dist + 1 − U0,DT . αg,uc

(6.25)

C0,dist 

These results not only demonstrate that the unit cell possesses a characteristic distribution parameter and slip velocity, but also indicate that they are largely determined by the amount of gas retained in the liquid slug. In the early attemps to represent it, Gregory et al. (1978) conducted 167 experiments at atmospheric pressure for horizontal pipes with diameters of 25.4 and 50.8 mm, using air and light oil with superficial velocities of up to 15 m/s. Subsequently, their results were incorporated into the correlation proposed by Malnes (1983), who, through insightful dimensional analysis, managed to reasonably anticipate the effect of surface tension on the phenomenon. Conversely, the geometric attributes of the pipeline were left out. Andreussi and Bendiksen (1989) made a significant contribution in this regard by also associating αg,dist with the values of D and θ. The study was based on the regression of 75 readings obtained in the laboratory for the low-pressure flow of air and water in pipes with diameters of 50 and 90 mm, and inclination angles varying between ±7◦ . In turn, based on a comprehensive set of 283 experimental data points, compiled by six independent authors for two-phase mixtures of air, freon, oxygen, kerosene,

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6 Improving the Fluid Flow Model

light oil, diesel, and water, with diameters ranging from 51 to 203 mm and pressures generally below 600 kPa, Gomez et al. (2000a) developed a unified model for the entire range 0 ≤ θ ≤ 90◦ . In this model, it is established that:   αg,dist = 1 − exp −7.85 × 10−3 θ − 2.48 × 10−6 Rel,dist ,

(6.26)

where: Rel,dist =

ρl Usm D . μl

(6.27)

Equation (6.26) disregards the impact of σ, but aims to depict the downward trend exhibited by αg,dist as the viscosity of the liquid increases. It is unfortunate that the investigated fluids were closely related in this aspect, undoubtedly diminishing the conclusiveness of the results. Zhang et al. (2003a) did not concur with such an approach, opting once again to consider surface tension over μl in their correlation valid for −90 ≤ θ ≤ 90◦ . Interestingly, Gomez et al. (2000a) and Zhang et al. (2003a) share three authors in their respective databases, which, on the other hand, highlights the immense difficulty in experimentally capturing multiple simultaneous action mechanisms. Several years later, with the increasing interest in heavy oil reservoir production, the debate regarding viscosity resurfaced. Laboratory experiments were conducted by Kora et al. (2011) to measure the gas volumetric fraction in the liquid slug under 144 operating conditions. They utilized a mixture of air and mineral oil, with a viscosity range of 0.18 ≤ μl ≤ 0.59 Pa · s (adjusted through heating), flowing at superficial velocities up to 4.4 m/s in a horizontal pipe with a fixed diameter of 50.8 mm, all at atmospheric pressure (Patm ). Based on these findings, Al-Safran et al. (2015) proposed the following correlation: 





−0.2 Fr 1.4 − 0.89 − 0.057 αg,dist = 0.15 + 0.075 Rem m



2 −0.2 1.4 − 0.89 + 2.27. Rem Frm

(6.28) It is worth noting that the “dimensionless viscosity number” introduced by the authors is nothing more than the ratio Frm2 /Rem , and that the term Rem−0.2 Frm1.4 in Eq. (6.28) also appears in the study conducted by Kora et al. (2011). After replicating the readings from an external source that had not participated in the regression, AlSafran et al. (2015) found that their methodology exhibited superior performance compared to all 10 expressions tested on that occasion. The same successful strategy was employed by Al-Ruhaimani et al. (2017) in the modeling of vertical flows. Equipped with 67 readings related to a two-phase system consisting of air and lubricating oil with viscosities ranging from 0.13 to 0.59 Pa · s, which developed superficial velocities up to 1.8 m/s in a pipeline with the same diameter of 50.8 mm, the authors derived a correlation of the following form:

6.3 Survey of Alternatives to Current Calculation Methods

αg,dist = 0.088 − √

0.266 . Rem Frm

243

(6.29)

In its original form, Eq. (6.29) features the “inverse viscosity number”, which corresponds to the ratio Rem /Frm (Al-Ruhaimani et al. 2017). The authors also applied the new method to a database consisting of 523 data points from four researchers. The agreement was found to be highly satisfactory, but only for systems with viscous liquids. In the case of fluids such as kerosene (μl ≈ 1.5 × 10−3 Pa · s), the results fell short of expectations. Another notable work in the field was conducted by Abdul-Majeed and Al-Mashat (2018), who analyzed the flow of an air and lubricating oil mixture with 0.20 ≤ μl ≤ 0.80 Pa · s viscosity and superficial velocities up to 4 m/s in pipes with diameters of 80 and 100 mm and inclinations ranging from 0 to 90◦ , amounting to a total of 180 combinations, all under atmospheric pressure. Their correlation suggests that:   αg,dist = 0.016 + 6.11 × 10−4 θ + 0.0195 − 1.24 × 10−4 θ Rem0.2 Frm0.6 . (6.30) In addition to providing a highly consistent representation of the aforementioned readings, achieving MAPEs in the range of 0.8% for 0 ≤ θ ≤ 60◦ and 0.2% for θ = 90◦ , Eq. (6.30) reproduces the observations of Kora et al. (2011) and Al-Ruhaimani et al. (2017) with deviations of 1.8% and 0.5%, respectively, which is an admirable achievement. It can be observed, in summary, that there still does not exist a unique and completely secure way of representing αg,dist . Although some progress has been made in this direction, all the correlations discussed here disregard at least one of the seven parameters considered relevant for the phenomenon in Sect. 4.5, and overlook the experimental validation of others. Judging by the overall panorama of these seven studies, as summarized in Table 6.4, the most persistent difficulties lie in gathering data at high pressure (in order to vary the term ρg in the regressions) and in understanding the effects of surface tension on the entrainment of gas bubbles. Repeating the same practical test as in the previous section for the αg,dist models, the profiles illustrated in Figs. 6.5, 6.6 and 6.7 are obtained. This time, it is the superficial velocity of the mixture that lends its values to the abscissa axis, always with locally constant gas and liquid mass fractions, establishing a direct link between Usm and M˙ m . The PDG did not participate in the analysis due to the simple fact that there were never any slugs present, regardless of the assumed parameters. In the remaining sections, the flow pattern determination was temporarily suppressed in order to impose the slug flow regime for all simulations. The correlations for vertical flow were adopted upstream of the WCT and at the arrival to the SPU (although θ = 83◦ in this latter case, but rounding is forgivable). The horizontal flow correlations were applied downstream of the WCT, and the comprehensive ones were adopted at all three points of the fictitious OPW. A quick inspection of Figs. 6.5, 6.6 and 6.7 is enough to infer that all the expressions accurately capture the increasing trend of αg,dist with Usm , except for the one

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6 Improving the Fluid Flow Model

Table 6.4 Parameters used by different models for calculating the gas volumetric fraction in the liquid slug Model Parameters U or θ ρl Usm μl D ρg σ Malnes (1983) Andreussi and Bendiksen (1989) Gomez et al. (2000a) Zhang et al. (2003a) Al-Safran et al. (2015) Al-Ruhaimani et al. (2017) Abdul-Majeed and Al-Mashat (2018)

X X X

X

X X

X X

X X X X X

X X X X X

X∗ X X X

X

X∗

X X X∗ X∗ X

X∗ X∗ X∗ X∗

X∗ X∗

X∗

X: parameter used in the model, validated against experimental data X∗ parameter used in the model, lacking experimental validation

1.0

αg,dist

0.8 0.6 0.4 0.2 0.0 2

3

4

5 Usm , m/s

Malnes (1983) Zhang et al. (2003a) Abdul-Majeed and Al-Mashat (2018)

6

7

8

Gomez et al. (2000a) Al-Ruhaimani et al. (2017)

Fig. 6.5 Behavior of the gas volumetric fraction in the liquid slug as a function of the mixture superficial velocity upstream of the WCT of a typical pre-salt OPW, according to the various correlations

6.3 Survey of Alternatives to Current Calculation Methods

245

1.0

αg,dist

0.8 0.6 0.4 0.2 0.0 2

3

4

5 Usm , m/s

Andreussi and Bendiksen (1989) Zhang et al. (2003a) Abdul-Majeed and Al-Mashat (2018)

6

7

8

Gomez et al. (2000a) Al-Safran et al. (2015)

Fig. 6.6 Behavior of the gas volumetric fraction in the liquid slug as a function of the mixture superficial velocity downstream of the WCT of a typical pre-salt OPW, according to the various correlations

developed by Al-Ruhaimani et al. (2017). The poor performance of the latter is partly due to the low viscosity of the oils from the pre-salt layer, which is far below the lower limit for which it was validated, but also to a serious oversight in its regression, which set a minimum threshold of 0.088 for the variable without any plausible justification. Similarly, the correlation of Al-Safran et al. (2015) does not seem to have adapted well to this lighter fluid, providing values that are almost unrelated to the superficial velocity of the mixture. Both of these expressions cannot be included in the accuracy test of Sect. 6.5 as they lack physical basis. Therefore, it is observed that the work of Gomez et al. (2000a) represents a kind of upper limit of the estimates. Their work and that of Abdul-Majeed and Al-Mashat (2018) were the only ones to reach the cut-off value of 0.9 implemented in the present computational code, which serves as a safeguard against potential numerical issues. The methodology of Zhang et al. (2003a) can also be considered optimistic in vertical flow situations, as well as that of Abdul-Majeed and Al-Mashat (2018) for the horizontal flowline. It should be noted that the larger the gas volume in the slug, the less stable it becomes (Brauner and Barnea 1986), and for αg,dist ≈ 0.52, the North Sea model is already highly susceptible to a flow pattern transition. Therefore, in the simulations that will be performed, these three calculation alternatives will certainly

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6 Improving the Fluid Flow Model

1.0

αg,dist

0.8 0.6 0.4 0.2 0.0 2

3

4

5 Usm , m/s

Malnes (1983) Zhang et al. (2003a) Abdul-Majeed and Al-Mashat (2018)

6

7

8

Gomez et al. (2000a) Al-Ruhaimani et al. (2017)

Fig. 6.7 Behavior of the gas volumetric fraction in the liquid slug as a function of the mixture superficial velocity as it reaches the SPU of a typical pre-salt OPW, according to the various correlations

lead to the “disappearance” of gas slugs in specific sections of the system. Moreover, according to Fig. 6.6, the expression proposed by Andreussi and Bendiksen (1989) is the only one that stipulates a minimum velocity for the beginning of the aeration process, while Figs. 6.5 and 6.7 highlight the remarkable agreement between the predictions of Malnes (1983), Abdul-Majeed and Al-Mashat (2018), obtained from two completely different approaches.

6.4 Review of Wall Roughness in Production Tubing and Pipelines Despite indicating a clear direction for the improvement of the flow model, it is undeniable that the performance of the adjustment coefficients on the pressure drop estimates in Sect. 6.2 fell somewhat below expectations. After all, as indicated in Table 6.3, the sum of all 128 effects associated with the seven parameters analyzed accounts for only 7.3% of the average P in the production pipelines. In the tubing, where the gas is still predominantly dissolved in the oil and the mixture is initially single-phase, the proportion becomes three times lower, reaching 2.4%.

6.4 Review of Wall Roughness in Production Tubing and Pipelines

247

Is it really true that there is so little room for improvement in simulations? Perhaps this is due to the minimum and maximum levels set during the factorial design. In this case, the only option would be to continue with the outlined roadmap at the beginning of the chapter, hoping that the different closure relationships themselves would be capable of yielding significant gains in the quality of the results. Alternatively, the current limitations may not solely stem from uncertainties surrounding the twophase flow, but also from the assumptions made regarding the representation of the geometry of the two structures. It is well known that oil production in the pre-salt oilfields generally involves higher values of Ug and Ul compared to conventional fields. This reality increases the contribution of viscous friction to the total pressure drop experienced by the mixture as it travels to the SPU, thereby highlighting the significance of the quantities associated with it. Considering that the evolution of the density and viscosity of each phase is well described by the fluid model, and that the internal diameter classifications for tubing and pipelines leave no room for doubt, it is also necessary to investigate the influence exerted by surface roughness on these calculations. For the production tubing, a value of e = 50 µm is commonly assigned. All flow simulations conducted so far have been based on this value, which is the same as that for commercial steel. However, it is possible that the high percentage of chromium present in its alloy (up to 26% in super duplex steel) offers a superior finish, although it also depends on the manufacturing process and the material used. White (2016, p. 363), Cengel and Cimbala (2017, p. 371) assume an exceptionally low roughness for stainless steels in general, on the order of 2 µm. Unfortunately, there are few studies in the literature dedicated to this topic. Smith et al. (1954) conducted a reverse calculation based on pressure records from 14 natural gas wells with simple metallurgy and diameters ranging from 35 to 162 mm. It was found that wall friction was adequately represented within the range 5.67 × 105 ≤ Re ≤ 4.48 × 106 assuming e = 17 µm. In a subsequent study, Cullender and Smith (1956) reduced this value to 15 µm. Both studies disregard the cumulative effect of natural degradation mechanisms, such as chemical corrosion (related to CO2 and H2 S contents in the mixture), erosion (common in sandstone reservoirs), and organic (paraffin and asphaltene) and inorganic (deposition of salts such as BaSO4 , CaSO4 , SrSO4 , and CaCO3 ) scale formation, which increase surface roughness over time. In the case of production pipelines, the issue becomes even more complex as their cross-sectional area is not strictly circular. The interlocked casing, the most internal layer of this type of structure, results from the mechanical shaping of a continuous metal strip into a profile such that, when rolled into a helix, its edges become trapped between the edges of the adjacent turns. Thus, the inner wall of the pipeline acquires a corrugated appearance, similar to that shown in Fig. 6.8. The resulting grooves are detrimental to the flow of oil, leading to the development of recirculation zones, as illustrated in Fig. 6.9. These zones, in turn, cause an increase in local turbulence levels and alterations in the velocity distribution along the cross-section. The impact of these factors on viscous dissipation can be addressed in various ways. Stel et al. (2012), for instance, proposed a friction factor formula specifically tailored to such geometries,

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6 Improving the Fluid Flow Model

Fig. 6.8 Interlocked casing geometry of subsea production pipelines (Neto et al. 2009)

w Recirculation zones

D Flow direction

Fig. 6.9 Representation of oil flow inside the interlocked casing (API 2014)

which bears a close resemblance to that of Colebrook (1939), but replaces e/D with w/D (where w represents the depth of the grooves) in the modified coefficient adjustment. On the other hand, Naidek et al. (2017) preferred to apply a correction to the value obtained for conventional pipes, also in terms of w/D. However, the simplest and perhaps the most widely adopted approach to date is to introduce an equivalent roughness, which is empirically determined. According to the Recommended Practice 17B of the American Petroleum Institute (API) (API 2014), the flow resistance of the interlocked casing is equivalent to that of a circular pipe with a constant passage area and a roughness equal to D/250.

6.4 Review of Wall Roughness in Production Tubing and Pipelines

249

1.08

API (2014) practice

dP/dL variation

1.04

Min. value of Bernardo et al. (2016)

1.00

Reference value Max. value of Bernardo et al. (2016)

0.96

0.92 0

200

400

600

800

1000

e, μm Fig. 6.10 Behavior of the pressure gradient as a function of the wall roughness of the production pipeline at the SPU of a typical pre-salt OPW

Therefore, for 8-inch flowlines and 6-inch risers, the roughness values would be approximately 600 µm and 800 µm, respectively. These are exaggerated numbers, no doubt, but they summarize a deeply ingrained culture in the oil industry of always “err on the side of safety” in production development projects. More reasonable estimates can be found in the important study conducted by Bernardo et al. (2016), which gathered 104 operational conditions related to three systems of desulfated 3 . Since this seawater injection, with flow rates ranging from 3100 to 9300 mstd/d concerns the single-phase flow of a very simple fluid in compositional terms, there was no other significant uncertainty in the models other than the pipe roughness. Bernardo et al. (2016) found that the average deviation of the simulations reached its minimum value for e = 210 µm. This single value for flowlines and risers served as the basis for all calculations performed in this book so far. When analyzed separately for the three systems, the parameter ranged between 150 and 260 µm. Figures 6.10, 6.11, 6.12, 6.13 and 6.13 represent the results of a consistent test procedure outlined in Sect. 6.3. These figures demonstrate the impact of surface roughness on the pressure gradient of fluid flow in a typical pre-salt OPW. The observed variations depend on the specific section being considered.

250

6 Improving the Fluid Flow Model

1.8

dP/dL variation

1.6

API (2014) practice

1.4

1.2 Max. value of Bernardo et al. (2016)

1.0

Reference value Min. value of Bernardo et al. (2016)

0.8

0.6 0

200

400

600

800

1000

e, μm Fig. 6.11 Behavior of the pressure gradient as a function of the wall roughness of the production pipeline downstream of the WCT of a typical pre-salt OPW

Upon reaching the SPU (Fig. 6.10), the pressure gradient prediction can decrease by 1.1% if the lowest level indicated by Bernardo et al. (2016) is employed or increase by 4.8% if the methodology proposed by the American Petroleum Institute (API 2014) is utilized. Moving downstream of the WCT (Fig. 6.11), where the gravitational component is negligible, these values become −7.1% and 54%, respectively. It is noteworthy to mention the occurrence of a discontinuity at approximately e ≈ 400 µm due to the transition from the stratified (left) to the slug (right) flow regimes. Upstream of the WCT (Fig. 6.12), the variation in d P/d L diminishes significantly: a decrease of 1.1% is observed when following the recommendation provided by Smith et al. (1954), and a decrease of 1.8% is obtained when using the typical roughness value for stainless steel. At the PDG (Fig. 6.13), characterized by a smaller internal diameter and accentuated frictional effects, the variations are more pronounced, with decreases of 3.2% and 5.5%, respectively. With such results, it becomes evident that a careful selection of roughness in pipelines and production tubing can play a fundamental role in the successful outcome of simulations focused on oil flow. Therefore, this will be the third and final parameter to participate in the accuracy test undertaken subsequently.

6.4 Review of Wall Roughness in Production Tubing and Pipelines

251

1.005

Reference value

Variac¸a ˜o dP/dL

1.000

0.995

0.990 Smith et al. (1954) Cullender and Smith (1956)

0.985 Stainless steel

0.980 0

10

20

30 e, μm

40

50

60

Fig. 6.12 Behavior of the pressure gradient as a function of the wall roughness of the production pipeline upstream of the WCT of a typical pre-salt OPW

1.02

Reference value

Variac¸a˜o dP/dL

1.00

0.98

Smith et al. (1954) Cullender and Smith (1956)

0.96

Stainless steel

0.94 0

10

20

30 e, μm

40

50

60

Fig. 6.13 Behavior of the pressure gradient as a function of the wall roughness of the production pipeline in the PDG of a typical pre-salt OPW

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6 Improving the Fluid Flow Model

6.5 Accuracy Test and Definition of the Best Set Reaching the culmination of the adaptation of the North Sea model, which will establish two new calculation correlations (or confirm the ones already implemented, naturally) and, in addition, determine the most appropriate values of pipe wall roughness for pre-salt OPWs, it is worthwhile to briefly recapitulate the advancements achieved in each workstream up to this point. Here is a concise list of the key advancements made thus far: Calculation of wall shear stress in bubbly flow: Five distinct methodologies have been presented, all of which result in plausible estimates, at least seemingly; Calculation of gas volume fraction in slug flow:Out of the seven tested expressions, two exhibited serious inconsistencies and had to be disregarded. The correlations proposed by Malnes (1983) and Andreussi and Bendiksen (1989) form, in practice, a single model weighted according to Eq. (4.130), and for simplicity, it will be temporarily referred to as the “North Sea” model. Consequently, there are four viable calculation options; Pipe wall roughness: There are valid justifications for considering the wall roughness of production pipelines between 150 and 800 µm. Additionally, including the values of 210 µm, 250 µm, and subsequent increments of 50 µm, a total of 14 possible values are established. As for tubing, the minimum and maximum limits correspond to 2 and 50 µm, respectively. Furthermore, including the values of 5 µm, 10 µm, and subsequent increments of 5 µm, a total of 11 values are obtained. The concept behind the accuracy test is quite straightforward and involves freely associating these three parameters with each other in the search for the set that best represents the compiled flow database in Chap. 5. However, its execution entails a considerable computational effort, as the 149 pressure drop records in the production pipelines must be reproduced 5 × 4 × 14 = 280 times, and the 100 readings in the tubing, 5 × 4 × 11 = 220 times. Another inherent difficulty that cannot be overlooked relates to convergence issues in certain arrangements of correlation, roughness, and operational condition. Fortunately, perhaps due to the numerous simulations that had already been carried out up to this point and that required continuous improvement in both the flow model and the marching algorithm codes, such incidents proved to be extremely rare and were quickly resolved. Of the five statistical parameters introduced at the end of the previous chapter, it is understood that only two should guide the choice of a particular combination: the Mean Absolute Percentage Error (MAPE), which conveniently summarizes systematic and random errors into a single indicator, and the 10% Percentage Accuracy (PA10% ), aiming to constrain the predictions of P within the specified tolerance range. This is because, when considering them in greater numbers, the gains achieved by one parameter are usually counterbalanced by the worsening performance of the others, resulting in a more or less equal standing among all scenarios. On the other hand, if both MAPE and PA10% can be optimized, there are excellent chances that the MPE, PSD, and PA20% of the simulations will also reach convincing levels.

6.5 Accuracy Test and Definition of the Best Set

253

The objective function designed for this task, denoted as FP , assumes that a 1% reduction in the MAPE is as desirable as a 10% increase in the PA10% , and applies the necessary correction in the denominator to ensure that it becomes a percentage measure itself. Thus:  MAPE( X ) + 0.1 100 − PA10% ( X ) . (6.31) FP = 1.1 In Table 6.5, all the combinations between the calculation alternatives for τw and αg,dist are listed, along with the optimal roughness values and the minimum FP obtained in each case. It can be observed that the three most effective sets in the P simulations performed similarly in objective terms, and all of them are based on

Table 6.5 Minimum values of the pressure drop objective function simulations using the different model combinations for τw and αg,dist and the optimized roughness values for the production tubing and pipelines obtained in each case Model for pipeline τw Model for tubing αg,dist e, µm FP , % Shannak (2008) Petalas and Aziz (1998) Zhang et al. (2003b) Bendiksen et al. (1996) Shannak (2008) Shannak (2008) Zhang et al. (2003b) Petalas and Aziz (1998) Zhang et al. (2003b) Petalas and Aziz (1998) Friedel (1979) Bendiksen et al. (1996) Bendiksen et al. (1996) Bendiksen et al. (1996) Petalas and Aziz (1998) Shannak (2008) Zhang et al. (2003b) Friedel (1979) Friedel (1979) Friedel (1979)

Abdul-Majeed and Al-Mashat (2018) Abdul-Majeed and Al-Mashat (2018) Abdul-Majeed and Al-Mashat (2018) Abdul-Majeed and Al-Mashat (2018) Gomez et al. (2000a) Zhang et al. (2003a) Gomez et al. (2000a) Gomez et al. (2000a) Zhang et al. (2003a) Zhang et al. (2003a) Gomez et al. (2000a) Zhang et al. (2003a) Gomez et al. (2000a) North Sea North Sea North Sea North Sea Zhang et al. (2003a) Abdul-Majeed and Al-Mashat (2018) North Sea

350

15

4.91

400

20

4.97

350

15

4.99

250

20

5.28

500 500 500 500 500 500 400 300 300 300 350 400 350 400 250

20 20 20 25 20 25 15 25 25 25 25 15 15 15 15

5.30 5.31 5.31 5.32 5.37 5.38 5.38 5.50 5.64 5.65 5.71 5.73 5.73 5.80 6.44

150

10

9.17

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6 Improving the Fluid Flow Model

the correlation proposed by Abdul-Majeed and Al-Mashat (2018) to predict the gas volumetric fraction in the liquid slug. As a tiebreaker criterion, the expression for wall friction presented by Shannak (2008) has significantly stronger experimental support compared to the formulas suggested by Petalas and Aziz (1998), Zhang et al. (2003b), and thus should prevail in the final selection. Automatically associated with it are the roughness values of 350 µm for the internal surface of the production pipelines and 15 µm for the production tubing. This new two-phase flow model, a version developed based on the North Sea mechanistic approach with exclusive interest in the pre-salt OPWs, will be referred to as the Parque das Baleias (Whale Park), in reference to the producing area1 that provided its data for its development.

6.6 Results and Validation As intriguing as the discovery of such a promising set of closure equations may be, it must be approached with utmost caution. After all, considering the large number of involved parameters and the exponential amount of interaction effects to which they are subject, some of which are completely obscure, one must never disregard the possibility of the model malfunctioning in certain scenarios. However, the initial results are encouraging: upon repeating the calculation related to the start of the productive life of OPW-01 using the same methodology as described in Sect. 5.6, pressure profiles, wall shear stress, temperature, volumetric fraction, and phase velocity were obtained that were nearly identical to those illustrated in Figs. 5.6, 5.7, 5.8, 5.9, 5.10 and 5.11, such that they will not be repeated here. It is now necessary to assess the quality of the flow pattern maps, αl and d P/d L (discussed in Sect. 6.6.1), verify the spread of pressure drop estimates in the six OPWs (Sect. 6.6.2), and conduct a series of simulations with varying flow rate (Sect. 6.6.3), which will be carried out next.

6.6.1 Flow Pattern, Volumetric Fraction and Pressure Gradient Maps Despite no changes having been made to the representation of the flow pattern transition mechanisms originating from the North Sea approach, it is natural that the new behavior exhibited by the calculation expressions for τw and αg,dist will also have an impact in this area. The criteria of minimum slip (which distinguishes segregated and distributed flow configurations) and for gas dispersion in the liquid slug (which

1 The Parque das Baleias comprises the Baleia Anã, Cachalote, Caxaréu, Mangangá, Pirambu, and Jubarte fields in the southern coast of the State of Espírito Santo, where Petrobras currently operates four FPSO-type platforms.

6.6 Results and Validation

255

Fig. 6.14 Two-phase flow pattern maps for the SPU, WCT (downstream and upstream) and PDG of a typical pre-salt OPW, predicted by the Parque das Baleias mechanistic model

establishes the boundary between the bubbly and slug flow patterns), both inherited by the Parque das Baleias mechanistic model, favor this type of occurrence. Indeed, returning to the fictitious OPW and reconstructing the flow maps pertaining to SPU, downstream and upstream of the WCT, and at the PDG, a noticeable series of differences compared to the original version becomes apparent, as illustrated in Fig. 6.14. In vertical sections or those with a high inclination, the bubbly flow pattern has occupied a portion of the zone previously filled by slug flow, which, in turn, resisted the transition to annular flow. In the horizontal direction, the stratified and annular flow patterns have relinquished a considerable region to the slug and bubbly flow patterns. However, it remains uncertain whether the maps depicted in Fig. 6.14 are more or less accurate than those presented in Fig. 4.6, given the absence of experimental observations under minimally compatible conditions of diameter, flow rate, and pressure. dP , it can be Regarding the presence of discontinuities in the results of αl and dL observed from Figs. 6.15 and 6.16 that the Parque das Baleias mechanistic model maintains the pre-existing characteristic of promoting smooth transitions, except once again at the boundary between slug and annular flow at low liquid flow rates.

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6 Improving the Fluid Flow Model

Fig. 6.15 Behavior of the liquid volumetric fraction as a function of the phase superficial velocities in the SPU, WCT (downstream and upstream) and PDG of a typical pre-salt OPW, according to the Parque das Baleias mechanistic model

However, as previously discussed in Sect. 4.8, this singular occurrence remains distant from the region of practical interest, and there is no risk of its impact being felt in simulations of a real OPW.

6.6.2 Calculation of Pressure Drop in Offshore Production Wells Once the solid physical basis of the new model has been demonstrated, it is now time to delve into the issues related to the precision of its estimations. Figure 6.17 reproduces the analysis conducted for the first time in Sect. 5.6, where the 249 simulated pressure drop values are juxtaposed with their corresponding field measurements from the flow database. It is readily apparent that the set of data points is distributed in an approximately equal manner above and below the continuous straight line, indicating a neutral trend. Furthermore, the spread around this line seems to be significantly smaller than in previous cases, particularly in the range between the WCT and the PDG (120 < P < 200 bar).

6.6 Results and Validation

257

Fig. 6.16 Behavior of the pressure gradient in the pipeline as a function of the phase superficial velocities in the SPU, WCT (downstream and upstream) and PDG of a typical pre-salt OPW, according to the Parque das Baleias mechanistic model

Clearly, part of the credit is due to the optimization of pipe wall roughness, which undoubtedly also brought benefits to the North Sea flow model, and a fair verdict must take this into account. Hence, the original formulation achieved its best results for e = 300 µm in the pipelines and 25 µm in the production tubing (these numbers are shown in the 14th line of Table 6.5). Upon visual inspection through Fig. 6.18, it is evident that there was a noticeable improvement compared to the performance depicted in Fig. 5.12, but within the range of 60 < P < 80 bar, the estimates still fall short. Thus, it is indicated that the enhancement of the flow model represents a step forward in calculating pressure drop in the pre-salt OPWs. The initial overall impression gains a definitive status in Table 6.6, which presents the statistical parameters related to both approaches. Except for a slightly worse Mean Percentage Error (MPE), which is already low, it can be observed that the Parque das Baleias model outperforms the original North Sea version in all other four criteria. The introduced modifications not only significantly reduced data dispersion but also ensured that nearly 90% of the simulations fall within the tolerance range of ±10%, with only one simulation lying outside the ±20% range (specifically, the lowest pressure drop reading of P in the OPW-04 flowline, with an absolute error of 2 bar).

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6 Improving the Fluid Flow Model

+ 20

%

200

ΔPcalc , bar

160

%

-20

120

80

OPW-01 OPW-02 OPW-03 OPW-04 OPW-05 OPW-06

40

0 0

40

80

120 ΔPmeas , bar

160

200

Fig. 6.17 Pressure drop in production tubing and pipelines of the six OPWs. Comparison of field data with the values calculated by the Parque das Baleias model, using the optimized wall roughness values

6.6.3 Simulations with a Simplified Reservoir Model To conclude the validation of the new flow model, it was deemed appropriate to perform several simulations with variable flow rate, following the same nodal analysis approach as Gilbert (1954), using a simplified reservoir representation that still sufficiently captures its main mechanisms. The linear IPR (Inflow Performance Relationship) assumes an incompressible fluid and a porous medium with constant permeability, where the flow towards the perforations is radial and in steady-state (Rosa et al. 2006, p. 203). Under this specific circumstance, according to Darcy’s law, there is a direct relationship between the liquid flow entering the well and the pressure drop experienced along the path, such that the so-called productivity index (PI) remains constant. Thus, it is established that: Pbh = Pr es −

Q˙ l . PI

(6.32)

6.6 Results and Validation

259

+ 20

%

200

ΔPcalc , bar

160

%

-20

120

80

OPW-01 OPW-02 OPW-03 OPW-04 OPW-05 OPW-06

40

0 0

40

80

120 ΔPmeas , bar

160

200

Fig. 6.18 Pressure drop in production tubing and pipelines of the six OPWs. Comparison of field data with the values calculated by the North Sea model, using the optimized wall roughness values Table 6.6 Statistical parameters of the pressure drop simulations in pipelines and production tubing according to the North Sea and Parque das Baleias mechanistic models, using the optimized wall roughness values obtained in each case Model Statistical parameters, % MPE PSD MAPE PA10% PA20% North Sea Parque das Baleias

−0.15 −1.22

6.68 6.05

4.77 4.23

85.5 88.4

98.8 99.6

It is important to emphasize that the applicability of Eq. (6.32) is limited to the range Pr es ≤ Pbh ≤ Pb . In addition to the permeability characteristics, the PI also depends on the formation thickness, the chosen drill bit diameter for the final drilling stage, any damages incurred during the process, the density and penetration of the perforations, the water content in the mixture, and so on. This is a quantity that exhibits certain temporal variation and should therefore be periodically measured, preferably after each new production test.

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6 Improving the Fluid Flow Model

As it is widely known, nodal analysis dates back to the era of graphical methods, prior to the widespread use of computational algorithms. However, its automation does not pose a particularly challenging task. It can be carried out in the following manner: 3 1. The well operating flow rate is initialized, assuming Q˙ l,op = 3000 mstd /d. This is the value most commonly found in the fields of southern Espírito Santo; 2. Solve the flow in the OPW as a function of the prescribed liquid flow rate, verifying TPR ) to displace the mixture to the pressure required at the bottom of the well (Pbh the surface; IPR ) under such conditions 3. Based on Eq. (6.32), the available reservoir pressure (Pbh is determined; 4. The objective function associated with the downhole pressure is updated, given TPR IPR − Pbh . The positive threshold closest to zero obtained thus far is by FPbh = Pbh + designated FPbh , while the negative threshold is denoted FP−bh . It is also necessary + − to store the corresponding operation flow rates, Q˙ l,op and Q˙ l,op ;

600

TPR

550

Pbh , barg

500 Initial guess 1

450

Iter. 3 and 4

Pop = 443.3 barg

IPR

Iter. 2

400 Q˙ l,op = 2833 m3std /d

350

300 0

1000

2000 3000 3 ˙ Ql , mstd /d

4000

5000

Fig. 6.19 Result of bottom-hole nodal analysis of OPW-01 for Psur f = 60 barg, BSW = 0%, 3 /(d · bar). Using the iterative method, the flow and pressure Pr es = 500 barg and PI = 50 mstd converge on the fourth iteration

6.6 Results and Validation

261

3200

3000

Q˙ l , m3std /d

2800

2600

2400

2200

2000 20

40

60 80 Psurf, barg

100

120

Fig. 6.20 Liquid flow rate behavior calculated for OPW-01 as a function of the surface pressure for BSW = 0% and Pr es = 500 barg

5. If only FP+bh or FP−bh is known, the new liquid flow will result from the expression old Q˙ l,op = (1 − 0.1FPbh /|FPbh |) × Q˙ l,op . Otherwise, a similarity of triangles must be + + − FP+bh ( Q˙ l,op − applied between these two extremes, resulting in Q˙ l,op = Q˙ l,op − + ˙ Q l,op )/(FPbh − FPbh− ). 6. Return to item 2 until |FPbh | ≤ 0.01 bar. Taking as an example the beginning of the productive life of OPW-01, when the 3 PI was around 50 mstd /(d · bar), the static pressure corresponded to approximately 500 barg, and it had BSW = 0% and Psup = 60 barg, the initial estimate of the oper3 /d, the objective ating flow rate leads to FPbh = 22.3 bar. By reducing it to 2700 mstd function becomes −17.1 bar. On the third attempt, the similarity of triangles results in 3 /d and FPbh = −0.4 bar. The iterative process converges shortly Q˙ l,op = 2830 mstd 3 /d, Pbh = 443.3 bar, and FPbh = −0.009 bar. In after, obtaining Q˙ l,op = 2833 mstd fact, it is precisely at this point that the crossing of the IPR and TPR curves, plotted in Fig. 6.19, occurs. The implementation of the simplified reservoir model paves the way for a series of estimations of utmost importance in monitoring oil production in the field. In Fig. 6.20, one can observe the influence of downstream pressure on the liquid flow rate ( Q˙ l ) of OPW-01. Such a curve allows for accurate determination of the new operating point of the well, in case it becomes necessary to restrict or relieve its

262

6 Improving the Fluid Flow Model

2850

3 /d Q˙ l , mstd

2800

2750

2700

2650

2600 0

5

10 BSW, %

15

20

Fig. 6.21 Liquid flow rate behavior calculated for OPW-01 as a function of the BSW for Psur f = 60 barg and Pr es = 500 barg

production choke between two successive tests, or even to define its daily potential if it cannot be tested under identical conditions as those imposed during alignment with the main separator. On the other hand, Fig. 6.21 describes the decrease in Q˙ l resulting from an increase in the water content of the mixture, assuming a fixed oil rate. It is worth noting that the water density is substantially higher than that of oil, exerting an immediate effect on the gravitational component of pressure drop. In such situations, the computational tool assists in the search for a new level of Psurf capable of maintaining a constant liquid flow rate. Lastly, Fig. 6.22 illustrates the impact of reservoir depletion on Q˙ l . This type of analysis serves, for instance, to quantify production losses caused by a potential failure in the saltwater injection system. Here, however, more important than the numbers themselves is the observation that the three curves obtained in the simulator exhibit perfect physical coherence and are devoid of any discontinuity or reversal of trends that could indicate incorrect behavior. This demonstrates the perfect integration between the fluid package, the marching method, and the new two-phase flow model.

References

263

3000

2800

3 /d Q˙ l , mstd

2600

2400

2200

2000

1800 0

20

40 60 Depletion, barg

80

100

Fig. 6.22 Liquid flow rate behavior calculated for OPW-01 as a function of the reservoir depletion for Psur f = 60 barg and BSW = 0%

References Abdul-Majeed, G. H., & Al-Mashat, A. M. (2018). A unified correlation for predicting slug liquid holdup in viscous two-phase flow for pipe inclination from horizontal to vertical. SN Applied Sciences, 1, 71. Al-Ruhaimani, F., Pereyra, E., Sarica, C., Al-Safran, E., & Torres, C. (2017). Prediction of slugliquid holdup for high-viscosity oils in upward gas/liquid vertical-pipe flow. SPE Production & Operations, 33(2), 281–299. Al-Safran, E., Kora, C., & Sarica, C. (2015). Prediction of slug liquid holdup in high viscosity liquid and gas two-phase flow in horizontal pipes. Journal of Petroleum Science and Engineering, 133, 566–575. Andreussi, P., & Bendiksen, K. (1989). An investigation of void fraction in liquid slugs for horizontal and inclined gas-liquid pipe flow. International Journal of Multiphase Flow, 15(6), 937–946. API (2014). API Recommended Practice 17B, 5th edn. Washington, DC: American Petroleum Institute. Awad, M. M., & Muzychka, Y. S. (2008). Effective property models for homogeneous two-phase flows. Experimental Thermal and Fluid Science, 33(1), 106–113. Balasubramaniam, R., Ramé, E., Kizito, J., & Kassemi, M. (2006). Two phase flow modeling: Summary of flow regimes and pressure drop correlations in reduced and partial gravity. Technical Report NASA/CR—2006-214085, National Aeronautics and Space Administration (NASA). Cleveland, OH: Glenn Research Center. Bendiksen, K. H., Maines, D., Moe, R., & Nuland, S. (1991). The dynamic two-fluid model OLGA: Theory and application. SPE Production Engineering, 6(02), 171–180.

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Bendiksen, K. H., Malnes, D., & Nydal, O. J. (1996). On the modelling of slug flow. Chemical Engineering Communications, 141–142(1), 71–102. Bernardo, L. A., Andreolli, I., Tocantins, M. W., & Santos, A. R. (2016). Roughness analysis within flexible water injection pipes in petroleum production projects. Journal of Petroleum Science and Engineering, 140, 64–72. Bousman, W. S., McQuillen, J. B., & Witte, L. C. (1996). Gas-liquid flow patterns in microgravity: Effects of tube diameter, liquid viscosity and surface tension. International Journal of Multiphase Flow, 22(6), 1035–1053. Brauner, N., & Barnea, D. (1986). Slug/churn transition in upward gas-liquid flow. Chemical Engineering Science, 41(1), 159–163. Cengel, Y. & Cimbala, J. (2017). Fluid mechanics: Fundamentals and applications, 4th edn. New York, NY: McGraw-Hill Education. Colebrook, C. F. (1939). Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws. Journal of the Institution of Civil Engineers, 11(4), 133–156. Cullender, M. H., & Smith, R. V. (1956). Practical solution of gas-flow equations for wells and pipelines with large temperature gradients. Transactions of AIME, 207(1), 281–287. Davies, R. M., & Taylor, G. I. (1950). The mechanics of large bubbles rising through extended liquids and through liquids in tubes. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 200(1062), 375–390. Dukler, A. E., & Hubbard, M. G. (1975). A model for gas-liquid slug flow in horizontal and near horizontal tubes. Industrial & Engineering Chemistry Fundamentals, 14(4), 337–347. Friedel, L. (1979). Improved friction pressure drop correlation for horizontal and vertical two-phase pipe flow. In Paper E2 Presented at the European Two-phase Flow Group Meeting, Ispra, Italy. Ghajar, A. J., & Bhagwat, S. M. (2013). Effect of void fraction and two-phase dynamic viscosity models on prediction of hydrostatic and frictional pressure drop in vertical upward gas-liquid two-phase flow. Heat Transfer Engineering, 34(13), 1044–1059. Gilbert, W. E. (1954). Flowing and gas-lift well performance (pp. 126–157). Division of Production, Los Angeles: Paper presented at the spring meeting of the Pacific Coast District. Gomez, L., Shoham, O., & Taitel, Y. (2000). Prediction of slug liquid holdup: Horizontal to upward vertical flow. International Journal of Multiphase Flow, 26(3), 517–521. Gomez, L. E., Shoham, O., Schmidt, Z., Chokshi, R. N., & Northug, T. (2000). Unified mechanistic model for steady-state two-phase flow: Horizontal to vertical upward flow. SPE Journal, 5(3), 339–350. Gregory, G., Nicholson, M., & Aziz, K. (1978). Correlation of the liquid volume fraction in the slug for horizontal gas-liquid slug flow. International Journal of Multiphase Flow, 4(1), 33–39. Hanratty, T. J., Theofanous, T., Delhaye, J.-M., Eaton, J., McLaughlin, J., Prosperetti, A., Sundaresan, S., & Tryggvason, G. (2003). Workshop on scientific issues in multiphase flow. International Journal of Multiphase Flow, 29(7), 1047–1059. Joseph, D. D. (2003). Rise velocity of a spherical cap bubble. Journal of Fluid Mechanics, 488, 213–223. Kora, C., Sarica, C., Zhang, H.-q., Al-Sarkhi, A., & Al-Safran, E. (2011). Effects of high oil viscosity on slug liquid holdup in horizontal pipes. In Paper no. 146954 presented at the Canadian Unconventional Resources Conference, Calgary, Canada. Malnes, D. (1983). Slug Flow in Vertical, Horizontal and Inclined Pipes. IFE KR E.: Institutt for Energiteknikk. Institute for Energy Technology, Kjeller, Norway. Naidek, B. P., Kashiwakura, L. Y., Alves, R. F., Bassani, C. L., Stel, H., & Morales, R. E. M. (2017). Experimental analysis of horizontal liquid-gas slug flow pressure drop in d-type corrugated pipes. Experimental Thermal and Fluid Science, 81, 234–243. Neto, A. G., Martins, C. A., Pesce, C. P., & Ferreira, T. B. (2009). A numerical simulation of crushing in flexible pipes. In Paper presented at the 20th International Congress of Mechanical Engineering, Gramado, Brazil.

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Pering, T. D., & McGonigle, A. J. S. (2018). Combining spherical-cap and Taylor bubble fluid dynamics with plume measurements to characterize basaltic degassing. Geosciences, 8(42). Petalas, N., & Aziz, K. (1998). A mechanistic model for multiphase flow in pipes. In Paper no. 98-39 presented at the Annual Technical Meeting of The Petroleum Society, Calgary, Canada. Rosa, A., de Souza Carvalho, R., & Xavier, J. (2006). Engenharia de Reservatórios de Petróleo. Rio de Janeiro: Interciência. Shannak, B. A. (2008). Frictional pressure drop of gas liquid two-phase flow in pipes. Nuclear Engineering and Design, 238(12), 3277–3284. Shippen, M., & Bailey, W. J. (2012). Steady-state multiphase flow-past, present, and future, with a perspective on flow assurance. Energy Fuels, 26(7), 4145–4157. Smith, R. V., Williams, R. H., & Dewees, E. J. (1954). Measurement of resistance to flow of fluids in natural gas wells. Journal of Petroleum Technology, 6(11), 41–48. Stel, H., Franco, A. T., Junqueira, S. L. M., Erthal, R. H., Mendes, R., Gonçalves, M. A. L., & Morales, R. E. M. (2012). Turbulent flow in d-type corrugated pipes: Flow pattern and friction factor. Journal of Fluids Engineering, 134(12), 121202. Taitel, Y., & Dukler, A. E. (1976). A model for predicting flow regime transitions in horizontal and near horizontal gas-liquid flow. AIChE Journal, 22(1), 47–55. Viana, F., Pardo, R., Yánez, R., Trallero, J. L., & Joseph, D. D. (2003). Universal correlation for the rise velocity of long gas bubbles in round pipes. Journal of Fluid Mechanics, 494, 379–398. Wallis, G. (1969). One-dimensional two-phase flow. New York, NY: McGraw-Hill. White, F. (2016). Fluid mechanics, 8th edn. New York, NY: McGraw-Hill Education. Yadigaroglu, G., & Hewitt, G. (2017). Introduction to multiphase flow: Basic concepts, applications and modelling. Zurich Lectures on Multiphase FlowZurich, Switzerland: Springer International Publishing. Zhang, H.-Q., Wang, Q., Sarica, C., & Brill, J. P. (2003). A unified mechanistic model for slug liquid holdup and transition between slug and dispersed bubble flows. International Journal of Multiphase Flow, 29(1), 97–107. Zhang, H.-Q., Wang, Q., Sarica, C., & Brill, J. P. (2003b). Unified model for gas-liquid pipe flow via slug dynamics. Part 1: Model development. Journal of Energy Resources Technology, 125(4), 266–273. Zuber, N., & Findlay, J. A. (1965). Average volumetric concentration in two-phase flow systems. Journal of Heat Transfer, 87(4), 453–468.

Chapter 7

Concluding Remarks

This monograph has been dedicated to the presentation and implementation of the necessary steps for the development of a mathematical model capable of depicting the flow of reservoir fluids in the pre-salt layer along their respective offshore production systems, using a one-dimensional approach in steady-state conditions. As we reach its conclusion, it remains to perform an evaluation of the initial expectations regarding the topic, what has effectively been achieved, and what will be reserved for future studies. The aspects related to the calculation of thermodynamic and transport properties of the mixture are discussed in Sect. 7.1. In Sect. 7.2, attention is directed towards estimating the pressure and temperature drops experienced by the fluid as it travels to the surface. Finally, it was deemed pertinent to provide some comments regarding the connections established between these two disciplines, as discussed in Sect. 7.3.

7.1 Modeling the Properties of Oil Reservoir Fluids in the Pre-salt Cluster The fluid model was developed based on a hybrid formulation. For the thermodynamic properties of the oil-gas system, a compositional approach was chosen, relying on cubic equations of state. Initially, the results were not promising: in the first attempts to reproduce the readings from PVT tests conducted in the laboratory for two real samples, as discussed in Chap. 2, a series of inconsistencies and accuracy problems in representing volumetric and phase behavior were observed. Only with the improvements made in Chap. 3 was it possible to achieve a more accurate description of the phenomena involved. These improvements consisted primarily of (i) proposing a new expression for the angular coefficient of the function α(T ) by Soave (1972), applicable to molecules with high acentric factors, (ii) developing a © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. R. Gessner and J. R. Barbosa Jr., Integrated Modeling of Reservoir Fluid Properties and Multiphase Flow in Offshore Production Systems, Petroleum Engineering, https://doi.org/10.1007/978-3-031-39850-6_7

267

268

7 Concluding Remarks

new methodology for characterizing SCN fractions valid up to C100 , and (iii) devising a multivariable fitting procedure based on the most relevant parameters of the model. As for the transport properties, the black oil approach was adopted, which offered a robust set of empirical correlations that were easy to implement. The same approach was taken for formation water, which, being treated as an inert substance, was excluded from the phase equilibrium calculation. The adjustment of the angular coefficient in the function α(T ) by Soave (1972) encompassed both the Soave-Redlich-Kwong and Peng-Robinson equations of state, relying on a survey that covered 1721 species, as published by Pina-Martinez et al. (2019). It was deemed appropriate to separate this database into two groups: one with low ω, which was assigned a quadratic function, and another with high or moderate ω, for which an exponential function was applied. The resulting expressions exhibited performance equivalent to that of the best tested works in each region. Their extrapolation up to ω = 3 proved to be highly plausible in both cases, smoothly extending the predictions made within the regression interval and displaying a coherent aspect akin to the renowned polynomials by Soave (1972), Robinson and Peng (1978). The new methodology for characterizing SCN fractions owes its origin to the effect of the volume translation parameter by Peneloux et al. (1982) on the assumed critical compressibility factor for the substance. This constitutes a strategy that is likely unprecedented at the present time. By applying it in a manner that equated the relative density values assigned to each of the pseudo-components, profiles of z c as a function of Mw emerged, and their physical consistency could be quickly verified. Among six alternative calculation approaches considered, the correlations proposed by Riazi and Al-Sahhaf (1996) stood out as the best from this perspective, although the coefficients established therein yielded poor estimates in some cases. This motivated a new regression analysis, based on synthetic data within the C7 -C22 range, in which care was taken to preserve the so-called internal consistency of the method, which predicts Tnb /Tc = 1 for Pc = Patm . The resulting expressions were then tested in the characterization of a set of ten actual petroleum samples from the Campos Basin pre-salt cluster, all originating from the southern region of the State of Espírito Santo. It was found that they provided the model with the smallest deviations compared to available flash and differential liberation tests, along with the work by Pedersen et al. (2004), while possessing a much more evident physical rigor than the latter. In regard to the adjustment of the compositional model, initially, it was necessary to define the minimal set of variables that would be involved in the process. By applying the technique known as Design of Experiments, it was found that out of the seven tested parameters, only five had a considerable effect on the predictions: critical pressure and temperature, acentric factor of SCN fractions, volume translation parameter of the cubic equation of state, and molecular weight of the heavy fraction. The objective function of the problem was composed of nine terms, each associated with the average deviation of the simulation for a specific property and the desired target value, combined through the asymptotic method. This allowed the algorithm to focus its efforts on the most deficient results, relegating the estimates that were already good enough to a secondary position. The minimization of the objective

7.1 Modeling the Properties of Oil Reservoir Fluids in the Pre-salt Cluster

269

function employed the Response Surface Methodology. It was quickly realized that the success of this task depended largely on the weights assigned to its terms and that, in the absence of a well-established procedure, such values would be determined through trial and error. The validation stage was conducted, once again, based on the aforementioned petroleum samples. In all of them, the algorithm demonstrated the capability to determine adjustments that, while not necessarily corresponding to the global optimum in each case (which is difficult to assert categorically), noticeably enhance the accuracy of the simulations. Another positive aspect worth highlighting pertains to the minimal interventions required for the parameters in all ten instances, indicating the excellent quality of the model. This also implies that, beyond the conditions established in the laboratory experiments, the simulations are unlikely to exhibit significant variations in their results. In the comparison between the equations of state of Soave-Redlich-Kwong and Peng-Robinson, it was observed that the former incurred smaller deviations for seven out of nine monitored variables, which made it the standard choice for the compositional model. The average error obtained from it in the flash liberation tests of the ten samples from the pre-salt was 3.59% for the mass fraction of released gas, 0.60% for the mass density of dead oil, and 3.28% for the molecular mass of the released gas. As for the respective differential liberation tests, the average errors were 0.93% for the bubble point pressure of the mixture, 1.33% for the density of oil at the bubble point, 5.64% for the isothermal compressibility of the undersaturated oil, 1.87% for the mass fraction of the total released gas, 0.38% for the density of dead oil, and 43.6% for the thermal expansion coefficient of dead oil. On the other hand, the black oil approach encountered no difficulty in reproducing the rheological behavior of the mixture, providing accurate viscosity estimates throughout the entire range of pressure and temperature considered. Properties without experimental confirmation were compared with estimates from the commercial software PVTsim, and the discrepancies were found to be negligible. It is important to emphasize that, although some pre-salt reservoir fluids exhibit high levels of CO2 , ranging from 1 to 20% in the Lula field, jumping to 44% in Libra and 79% in Jupiter, the mole fraction of contaminants in the analyzed samples did not exceed 4.5%. Therefore, if there is a future desire to extend the current model to accumulations with such characteristics, it will certainly be necessary to revise the values assigned to the binary interaction parameters, which vary considerably depending on the consulted source. As they gain relevance in the overall performance of the simulations, it may also be worth assigning them a specific fitting factor and including them in the optimization algorithm. A third line of work could attempt to describe them as a function of temperature in the system, thus improving the representation of the oil’s thermal expansion coefficient, among other benefits. Finally, depending on the unfolding of events, there is a possibility to revisit the implemented mixing rules, replacing the current classical van der Waals (van der Waals 1890) and Soave (Soave 1972) scheme with a more suitable one.

270

7 Concluding Remarks

7.2 Two-Phase Flow Modeling in Pre-salt Offshore Production Wells The two-phase flow modeling followed the mechanistic, or phenomenological, approach. After a thorough analysis of the main works available in the literature, carried out in Chap. 4, two formulations with potential application to pre-salt OPWs were identified: the first, referred to as the North Sea model, in reference to both the origin of the resources that financed it and the field data used for its validation, primarily draws on the studies by Bendiksen et al. (1991); the second, named the Gulf of Mexico model following the same criteria, combines the pattern calculation by Gomez et al. (2000) with the closure relationships of Zhang et al. (2003). It was observed that the North Sea model was less prone to discontinuities in the estimates of volumetric fraction and dP/dL, while its Gulf of Mexico counterpart was better suited to identify the different regimes assumed by the two-phase gas-liquid flow. However, in order to choose the best of the two alternatives, it was first necessary to obtain the pressure and temperature profiles of the flow along the pipelines and production tubing, which was done in Chapter 5. The so-called marching method divided the length of the pipeline into several smaller segments, where the discretized equations of mass, momentum, and energy conservation of the mixture were fully applicable. Thus, after prescribing known values at one end, the calculation of pressure and enthalpy could proceed iteratively towards the other end without major obstacles. On the other hand, the representation of heat exchange along the way was treated in a simplified manner, based on pre-established values for the overall heat transfer coefficient in the two environments traversed by the OPW. A preliminary qualitative analysis of the predicted P and T values demonstrated that the results not only had practical support but also captured some phenomena that went beyond common sense, such as the heating of the fluid in the narrowest section of the production tubing and the cooling due to expansion in the riser. The validation of this new computational tool was supported by a database consisting of 249 real operation records from six OPWs (149 for pipelines and 100 for production tubing), all located in the southern region of the State of Espírito Santo, in two pre-salt fields of the Campos Basin. Through this validation, it became evident that the major difference between the pressure drop estimates provided by the North Sea and Gulf of Mexico flow models lied within the interval 40 ≤ P ≤ 120 bar. The latter model underestimated both the volumetric liquid fraction and the average wall shear stress, leading to erroneously low values in that range. On the other hand, the former model demonstrated a clear superiority in all statistical aspects. Its results, while being neutral in terms of trend and exhibiting low spread, reproduced sensor readings with deviations below 10% in 215 instances, surpassing the 20% threshold only once. This performance was perfectly compatible with that of a good commercial simulator. Clearly, despite its excellent beginnings, the North Sea mechanistic model still had room for improvement. By conducting a new factorial design, this time dedicated to assessing the effects of the various parameters that constituted its equations on

7.2 Two-Phase Flow Modeling in Pre-salt Offshore Production Wells

271

the predictions of pressure drop, it was observed in Chap. 6 that efforts should focus on modeling (i) the gas volume fraction in slug flow and (ii) the wall shear stress in bubbly flow. Additionally, there were strong indications that the assumed levels of pipe wall roughness for the simulations up until that point needed to be revised. After conducting a more thorough investigation into these three topics, modifications were proposed, culminating in a new flow model with exclusive focus on the pre-salt OPWs, named Parque das Baleias, in reference to the producing area that provided data for its development. In the final evaluation, it was observed that the changes introduced in the set of equations brought about certain implications for the calculation of the flow pattern. However, due to the absence of experimental observations under similar conditions of diameter, flow rate, and pressure, it was unfortunately not possible to delve into the merits of the issue. The Parque das Baleias mechanistic model maintained the pre-existing characteristic of providing smooth transitions in the results of volumetric dP , except again at the boundary between slug and annular flow for low fraction and dL liquid flow rates, in a region where there was no practical interest. When confronted with the same aforementioned database, its predictions yielded a MAPE (Mean Absolute Percentage Error) of 4.23%, a significant improvement compared to the 5.59% obtained by the North Sea model using the original roughness values, with 88.4% of them falling within the tolerance range of ±10%, and only one data point falling outside the ±20% range. Subsequently, the simulations with varying flow rates made it possible to generate curves of great importance for monitoring oil production in the field. These curves demonstrated perfect physical coherence, free from discontinuities or any other anomalous events, thereby confirming the seamless integration between the fluid package, the marching algorithm, and the two-phase flow model. Despite numerous advancements, it is undeniable that certain topics intentionally excluded from the present scope will require further investigation in the future. Heat transfer in the well, for instance, heavily relies on the thermophysical properties of the substance occupying the annular space between the tubing and the production casing, the natural convection mechanism triggered therein, and the transient conduction heating of the surrounding rocks. Fixed values for the overall coefficient U will never fully reflect the truth of the matter. This realization also applies to pipelines, where the exacerbation of internal convection due to accelerated mixing, as well as variations in the velocity of the marine current impacting its external surface, can intensify the temperature drop experienced along the path to the SPU. Regarding the possibility of extending this new simulation tool to other fields in the Brazilian pre-salt region, it appears that such an endeavor would require additional studies, particularly in accumulations with high GOR. This is because the increased presence of gas in the system theoretically favors the occurrence of annular flow in the production riser, amplifying the effects of interfacial shear stress on pressure drop. A flow database that includes wells with such characteristics would certainly elevate the position of τi in the ranking of the model’s key parameters, and a refinement of its calculation expression would become mandatory. Perhaps this measure would also help resolve (or mitigate) the discontinuities observed in the transition to slug flow.

272

7 Concluding Remarks

In addition to GOR, production systems with significantly different concepts from the one presented here, or operating at different flow rates, are also fully capable of reshuffling the priority list and thereby opening new opportunities for improvement.

7.3 Integration Between the Two Disciplines There is no doubt that a significant portion of the achievements attained in this study can be attributed to the initial realization that the flow of oil reservoir fluids through pipelines and production tubing constitutes a singular problem, rather than two independent topics. Indeed, it was due to the peculiarities of hydrocarbon flow in the pre-salt OPWs that a hybrid formulation (compositional and black oil) was chosen for the thermodynamic and transport properties of the mixture. Conversely, it would never have been possible to enhance the gas-liquid two-phase flow model without full confidence in the results obtained in the previous stage. Furthermore, the integrated approach allowed for the utilization of similar tools on both fronts, expediting the development of solutions and giving them a more universal character. It is worth noting that all the research reported here was dedicated to the production 3 3 /mstd from accumulations with of light black oils with GOR between 220 and 290 mstd static pressure and average temperature of 50 MPa and 402 K, respectively. These accumulations were produced through pipelines with an internal diameter ranging 3 /d. It from 76.2 to 203.2 mm and a liquid flow rate of approximately 2900 mstd should be noted that this same methodology can be applied to other scenarios, always considering their intrinsic characteristics.

References Bendiksen, K. H., Maines, D., Moe, R., & Nuland, S. (1991). The dynamic two-fluid model OLGA: Theory and application. SPE Production Engineering, 6(02), 171–180. Gomez, L. E., Shoham, O., Schmidt, Z., Chokshi, R. N., & Northug, T. (2000). Unified mechanistic model for steady-state two-phase flow: Horizontal to vertical upward flow. SPE Journal, 5(3), 339–350. Pedersen, K. S., Milter, J., & Sørensen, H. (2004). Cubic equations of state applied to HT/HP and highly aromatic fluids. SPE Journal, 9(2), 186–192. Peneloux, A., Rauzy, E., & Freze, R. (1982). A consistent correction for Redlich-Kwong-Soave volumes. Fluid Phase Equilibria, 8(1), 7–23. Pina-Martinez, A., Privat, R., Jaubert, J.-N., & Peng, D.-Y. (2019). Updated versions of the generalized Soave α-function suitable for the Redlich-Kwong and Peng-Robinson equations of state. Fluid Phase Equilibria, 485, 264–269. Riazi, M. R., & Al-Sahhaf, T. A. (1996). Physical properties of heavy petroleum fractions and crude oils. Fluid Phase Equilibria, 117(1), 217–224. Robinson, D., & Peng, D. (1978). The Characterization of the Heptanes and Heavier Fractions for the GPA Peng-Robinson Programs. Tulsa, OK: Gas Processors Association. Soave, G. (1972). Equilibrium constants from a modified Redlich-Kwong equation of state. Chemical Engineering Science, 27(6), 1197–1203.

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van der Waals, J. (1890). Molekulartheorie eines körpers, der aus zwei verschiedenen stoffen besteht (molecular theory of a substance composed of two different species). Zeitschrift für Physikalische Chemie, 133–173. Zhang, H.-Q., Wang, Q., Sarica, C., & Brill, J. P. (2003). Unified model for gas-liquid pipe flow via slug dynamics. Part 1: Model development. Journal of Energy Resources Technology, 125(4), 266–273.

Appendix A

Obtaining the Roots of Cubic Equations of State

Consider a third-degree polynomial with real coefficients given by: z 3 + a2 z 2 + a1 z + a0 = 0.

(A.1)

The coefficient a3 of z 3 will be assumed to be equal to 1 without loss of generality, by dividing the entire Eq. A.1 by a3 if necessary. Furthermore, it is possible to eliminate the quadratic term of the polynomial by rewriting it in terms of a new variable x, where z and x are related as described by Elliott and Lira (2012, p. 822): z=x−

a2 , 3

(A.2)

In this manner, one obtains: x 3 + 3 px + 2q = 0, where:

a22 − 3a1 , 9

(A.4)

2a23 − 9a2 a1 + 27a0 . 54

(A.5)

p= q=

(A.3)

The number of real roots of the polynomial is associated with the value of its discriminant D, which is given by: D = q 2 − p3 .

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. R. Gessner and J. R. Barbosa Jr., Integrated Modeling of Reservoir Fluid Properties and Multiphase Flow in Offshore Production Systems, Petroleum Engineering, https://doi.org/10.1007/978-3-031-39850-6

(A.6)

275

276

Appendix A: Obtaining the Roots of Cubic Equations of State

Thus, if D > 0, the algebraic solution leads to a single real root, expressed as:  √ 1/3  √ 1/3 + −q − D . x1 = −q + D

(A.7)

On the other hand, for D ≤ 0, the trigonometric solution results in three real roots, given by:   θ √ , (A.8) x1 = −2 p cos 3   θ + 2π √ x2 = −2 p cos , 3

(A.9)

 θ + 4π , x3 = −2 p cos 3

(A.10)



√



where: θ = cos

−1

q  p3

 .

(A.11)

The roots of the function z = z(P, T ), in turn, are computed based on the values of x obtained in Eqs. A.7–A.10, simply by substituting them into Eq. A.2. Reference Elliott, J., & Lira, C. (2012). Introductory Chemical Engineering Thermodynamics, 2nd edn. Upper Saddle River, NJ: Prentice-Hall.

Appendix B

Calculation of Departure Functions

B.1

Redlich-Kwong Equation of State

The function P = P(T, v) ˜ proposed by Redlich and Kwong (1949) is expressed in Eq. 2.11. Taking its derivative with respect to temperature yields: 

∂P ∂T

 v˜

=

da/dT R − . v˜ − b v˜ (v˜ + b)

In this case, one has:  

∂P 1 da T − a(T ) . −P=− T ∂T v˜ dT v˜ (v˜ + b)

(B.1)

(B.2)

Therefore, by substituting Eq. B.2 into Eq. 2.113 and performing the integration, one arrives at the molar internal energy departure function for the mentioned equation of state, given by:

  da v˜ + b 1 u˜ R = T − a(T ) ln . (B.3) b dT v˜ Alternatively, rewriting it in terms of z gives:

  A T da z+B u˜ R = − 1 ln . RT B a(T ) dT z

(B.4)

Based on Eq. B.1, it is also observed that: 

∂P ∂T

 v˜

−

bR da/dT R = − . v˜ v˜ (v˜ − b) v˜ (v˜ + b)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. R. Gessner and J. R. Barbosa Jr., Integrated Modeling of Reservoir Fluid Properties and Multiphase Flow in Offshore Production Systems, Petroleum Engineering, https://doi.org/10.1007/978-3-031-39850-6

(B.5)

277

278

Appendix B: Calculation of Departure Functions

Hence, by substituting Eq. B.5 into 2.114 and solving the integral, one obtains the molar entropy departure function for the equation of state under analysis, expressed as:     1 da v˜ + b v˜ − b + ln + R ln z (B.6) s˜ R = R ln v˜ b dT v˜ Alternatively:

  s˜ R A T da z+B = ln (z − B) + ln . R B a(T ) dT z

(B.7)

Finally, differentiating Eq. B.1 with respect to temperature gives: 

∂2 P ∂T 2

 v˜

=−

d 2 a/dT 2 . v˜ (v˜ + b)

(B.8)

Subsequently, substituting Eq. B.8 into Eq. 2.122 and performing the integration, one arrives at the molar heat capacity departure function at constant volume associated with this equation of state, given by: c˜vR

  T d 2a v˜ + b . = ln b dT 2 v˜

(B.9)

However, it can be demonstrated that:



T da T da d 2a a(T ) − 1 . = dT 2 2T 2 a(T ) dT a(T ) dT

(B.10)

Thus, Eq. B.10 can be rewritten as: c˜vR





  T da T da v˜ + b a(T ) − 1 ln . = 2bT a(T ) dT a(T ) dT v˜

(B.11)

Alternatively, this can be expressed as: A c˜vR = R 2B



T da a(T ) dT



  T da z+B − 1 ln . a(T ) dT z

(B.12)

It should be noted that the same mixing rule applied to the parameter a(T ) remains valid in the calculation of da/dT , that is:  dai j da = , xi x j dT dT i j

(B.13)

Appendix B: Calculation of Departure Functions

279

where, for ki j constant with respect to temperature: dai j 1 − ki j = √ dT 2 ai a j

  da j dai ai + aj , dT dT

(B.14)

and, for the function α(T ) described in Eq. 2.19: √ dai aci m i αi (T )Tri =− . dT T

B.2

(B.15)

Peng-Robinson Equation of State

The function P = P(T, v) ˜ proposed by Peng and Robinson (1976) is expressed in Eq. 2.22. Differentiating it with respect to temperature yields: 

∂P ∂T

 v˜

=

da/dT R − . v˜ − b v˜ (v˜ + b) + b (v˜ − b)

In this case, it follows that:  

∂P 1 da T − a(T ) . −P=− T ∂T v˜ dT v˜ (v˜ + b) + b (v˜ − b)

(B.16)

(B.17)

Therefore, substituting Eq. B.17 into Eq. 2.113 and integrating, one obtains the departure function for the molar internal energy for the Peng-Robinson equation of state, given by:  ⎡ √  ⎤

v˜ + 1 + 2 b da 1  u˜ R = √ T − a(T ) ln ⎣ √  ⎦, dT b 8 v˜ + 1 − 2 b

(B.18)

or alternatively, expressing it in terms of z:  ⎡ √  ⎤

z + 1 + 2 B A T da u˜   ⎦. = − 1 ln ⎣ √ RT B a(T ) dT z+ 1− 2 B

R

(B.19)

Also, based on Eq. B.16, it can be observed that: 

∂P ∂T

 v˜

−

bR da/dT R = − . v˜ v˜ (v˜ − b) v˜ (v˜ + b) + b (v˜ − b)

(B.20)

280

Appendix B: Calculation of Departure Functions

Hence, by substituting Eq. B.20 into Eq. 2.114 and solving the integral, one obtains the molar entropy departure function for the Peng-Robinson equation of state, expressed as:  s˜ R = R ln

v˜ − b v˜



 √  ⎤ v ˜ + 1 + 2 b 1 da ⎣   ⎦ + R ln z, + √ ln √ b 8 dT v˜ + 1 − 2 b ⎡

(B.21)

or alternatively:  √  ⎤ z + 1 + 2 B s˜ T da A   ⎦. = ln (z − B) + √ ln ⎣ √ R B 8 a(T ) dT z+ 1− 2 B

R



⎡

(B.22)

Finally, by differentiating equation B.16 with respect to temperature, one obtains: 

∂2 P ∂T 2

 v˜

=−

d 2 a/dT 2 . v˜ (v˜ + b) + b (v˜ − b)

(B.23)

Subsequently, substituting Eq. B.23 into Eq. 2.122 and performing integration, yields the departure function of molar heat capacity at constant volume associated with this equation of state, given by:  ⎡ √  ⎤ 2 v˜ + 1 + 2 b d a T  c˜vR = √ ln ⎣ √  ⎦. b 8 dT 2 v˜ + 1 − 2 b

(B.24)

However, it is possible to demonstrate that:



d 2a T da T da a(T ) −1 . = dT 2 2T 2 a(T ) dT a(T ) dT

(B.25)

Thus, Eq. B.25 can be rewritten as:

c˜vR =

T da a(T ) √ a(T ) dT 2bT 8



 ⎡ √  ⎤

1 + v ˜ + 2 b T da   ⎦, − 1 ln ⎣ √ a(T ) dT v˜ + 1 − 2 b

(B.26)

Appendix B: Calculation of Departure Functions

281

or, alternatively: c˜vR R



=

T da A √ a(T ) dT 2B 8



 ⎡ √  ⎤

z + 1 + 2 B T da   ⎦. − 1 ln ⎣ √ a(T ) dT z+ 1− 2 B

(B.27)

It should be noted that the same mixing rule applied to the parameter a(T ) remains valid in the calculation of da/dT , that is:  dai j da = , xi x j dT dT i j

(B.28)

where, for temperature-independent ki j : dai j 1 − ki j = √ dT 2 ai a j

  da j dai ai + aj , dT dT

(B.29)

and, for the function α(T ) described in Eq. 2.19: √ dai aci m i αi (T )Tri =− . dT T

(B.30)

References Peng, D.-Y., & Robinson, D. B. (1976). A new two-constant equation of state. Industrial & Engineering Chemistry Fundamentals, 15(1), 59–64. Redlich, O., & Kwong, J. N. S. (1949). On the thermodynamics of solutions. v. an equation of state. fugacities of gaseous solutions. Chemical Reviews, 44(1), 233–244.

Appendix C

Weighting of Liquid Phase Properties

Consider the liquid phase of a certain petroleum mixture. By knowing the thermophysical properties of the oil and the accompanying water, as well as their proportions, it is possible to determine the mass density by employing the following equation:  ρl = εl

εo εa + ρo ρa

−1

,

(C.1)

where: εl = εo + εa .

(C.2)

By differentiating equation C.1 with respect to pressure, and after making the necessary simplifications, one arrives at:     

 ρl2 εo ∂ρo εa ∂ρa ∂ρl . (C.3) = + ∂P T εl ρ2o ∂ P T ρa2 ∂ P T It is worth noting that the mass fractions of oil and water do not vary with P and T since a system with constant composition is assumed. Together, Eqs. C.1 and C.3 allow one to determine the isothermal compressibility of the liquid, which is defined as:   1 ∂ρl . (C.4) cl ≡ ρl ∂ P T Similar to what was observed for pressure, the derivative of Eq. C.1 with respect to temperature leads to:

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. R. Gessner and J. R. Barbosa Jr., Integrated Modeling of Reservoir Fluid Properties and Multiphase Flow in Offshore Production Systems, Petroleum Engineering, https://doi.org/10.1007/978-3-031-39850-6

283

284

Appendix C: Weighting of Liquid Phase Properties



∂ρl ∂T

 P

ρ2 = l εl



εo ρ2o



∂ρo ∂T

 P

εa + 2 ρa



∂ρa ∂T



.

(C.5)

P

Equations C.1 and C.5 also define the value of the thermal expansion coefficient of the phase, expressed as:   1 ∂ρl βl ≡ − . (C.6) ρl ∂T P Finally, as it is an intensive property, the specific enthalpy of the liquid is given by: hl =

εo h o + εa h a . εl

Differentiating equation C.7 with respect to temperature yields:   ∂h l εo c Po + εa c Pa = . c Pl ≡ ∂T P εl

(C.7)

(C.8)