Multi Phase Flow in Well

Multiphase Flow in WellsJames P. Brill Floyd M. Stevenson Endowed Presidential Chair in Petroleum Engineering Executive Director; Fluid Flow Projects U . of Tulsa and Hemanta Mukherjee Manager; Production Enhancement, West and South Africa Schlumberger Oilfield Services First Printing Henry L. Doherty Memorial Fund of AIME Society of Petroleum Engineers Inc. Richardson, Texas 1999 Table of Contents Chapter 1-Introduction 1.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Objectives of Monograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Organization of Monograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Nomenclature and Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................................................. 1 1 1 1 2 4 Chapter 2-Single-Phase-Flow Concepts 5 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Pressure-Gradient Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.5 Flow in an Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I I 2.6 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Chapter 3-Multiphase-Flow Concepts 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 PhaseBehavior ...................................................... 3.3 Definition of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Pressure Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Flow Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Liquid Holdup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Pressure-Traverse Computing Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................................... ............................................... 19 19 19 20 23 23 24 26 27 28 28 56 58 66 67 70 70 70 76 78 78 80 86 91 92 94 95 Chapter 4-Multiphase-Flow Pressure-Gradient Prediction 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Pressure-Gradient Prediction ........................................... 4.3 Evaluation of Wellbore Pressure-Gradient-PredictionMethods . . . . . . . . . . . . . . . . 4.4 Pressure-Gradient Prediction in Annuli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Evaluation of Annulus Liquid-Holdup and Pressure-Gradient-PredictionMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 General Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 5-Flow 5.1 5.2 5.3 5.4 Chapter 6-Well 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 Through Restrictions and Piping Components Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description of Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Through Chokes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Through Piping Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .............................. 28 .......................... 70 Design Applications Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vertical-Flow Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inflow Performance .................................................. Production-Systems Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Artificial Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas-Well Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Erosional Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . and SI Metric Conversion Factors Equilibrium and Rock Properties ................................................. 78 Appendix A-Nomenclature Appendix B-Fluid Appendix C-Vapor-/Liquid-Phase .......................................135 Appendix D-Tubing and Casing Properties .......................................... 145 AuthorIndex .................................................................... 149 Subject Index ........................... 99 ............................................. 102 .................................................................... 155 Chapter 1 Introduction 1.1 Scope The accurate design of oil and gas well tubing strings requires the ability to predict flow behavior in the wells. Wells normally produce a mixture of gas and liquids, regardless of whether they are classified as oil wells or gas wells. This multiphase flow is significantly more complex than single-phase flow. However, the technology to predict multiphase-flow behavior has improved dramatically in the past decade. It is now possible to select tubing sizes, predict pressure drops, and calculate flow rates in wells with acceptable engineering accuracy. This chapter sets the stage for the monograph by describing the nature and occurrence of multiphase flow, and by presenting important historical events that have impacted on the development of modern multiphase-flow concepts. The common occurrence of multiphase flow in wells can be discussed with the simplified production system shown in Fig. 1.1. Fluids entering the wellbore from the reservoir can range from an undersaturated oil to a single-phase gas. Free water can a&ompany the fluids as a result of water coning, water flooding, or production of interstitial water. Alternatively, a free gas saturation in an oil reservoir can result in a gasAiquid mixture entering the well. Retrograde condensation can result in hydrocarbon liquids condensing in a gas condensate reservoir so that a gasfliquid mixture again enters the wellbore. Even when single-phase gas or liquid flow exists near the bottom of a well, multiphase flow can occur throughout most of the wellbore. This is a result of evolution of gas from oil or condensation of gas with reduction of pressure and temperature as the fluids flow up the well. Although many of the wells drilled on land tend to be nearly vertical, wells drilled offshore and in other hostile environments, such as the Arctic, are normally directional or deviated. Inclination angles can vary from vertical near the surface to horizontal near the production zone. Flow rates of gas, oil, and water vary widely. Pipe diarneters can be as small as 1 in. and as large as 9 in. Flow also can occur in a casingltubing annulus. Depths can range from a few hundred feet to more than 20,000 ft. Pressures can be as low as a few atmospheres or as high as 20,000 psia. Temperatures can be above 400°F or approach the freezing point of water. Oil viscosities in wellbores can be less than 1 cp or greater than 10,000 cp. Fluids entering the wellbore often flow through a complicated well completion region consisting of perforations, fractures-,gravel pack, and other such items. The effect of this region must be included when coupling the well to the reservoir through inflow performance relationship (IPR)procedures. Most wells contain some type of well-control device that requires produced fluids to flow through a restriction. This can vary from a bottomhole choke to a INTRODUCTION subsurface safety valve or a remotely controlled, variable-sized surface choke. Wells can be produced by artificial-lift mechanisms involving a submersible pump or gas injection. The broad changes in flow variables encountered in producing wells have made the development of prediction methods much more difficult. Techniques that work for gas condensate wells do not necessarily work for oil wells. Assumptions that are valid for some wells are totally invalid for others. 1.2 Objectives of Monograph Multiphase flow in pipes is not unique to the petroleum industry. The simultaneous flow of two or three phases in a single conduit is encountered in petrochemical plants, refineries, steam generators, automobile engines, storm sewers, nuclear reactors, and many other areas. Consequently, the technical literature in chemical, civil, mechanical, petroleum, and nuclear engineering all contain valuable contributions to multiphase flow in pipes. Unfortunately, each discipline uses different terminology, units, and nomenclature and is interested in different ranges of variables. An important objective of this monograph is to bring together the pertinent technology of all relevant disciplines into a single book that uses the nomenclature and units standardized by the Society of Petroleum Engineers. Numerous methods have been proposed to predict multiphase flow in wells. Some are very general and others are applicable over a very narrow range of flow variables. Some are very empirical in nature and others make an attempt to model the basic phenomena encountered. It is vitally important that persons engaged in designing wells be aware of the limitations of all methods. Thus, a second objective of this monograph is to present the important methods to predict flow behavier in wells and to discuss their limitations and ranges of applicability. Finally, engineers engaged in the design of wells should have a thorough understanding of the calculation procedures necessary to optimize their design calculations. This includes coupling the wells to both surface facilities and reservoirs. It also includes understanding computer algorithms, fluid physical properties, and mass transfer for multicomponent mixtures. A significant part of this monograph is devoted to these concepts. 1.3 Organization of Monograph Predicting flow behavior when multiphase flow occurs in weIls requires an understanding of concepts that are not a part of the curriculum in most engineering disciplines. Before multiphase-flow technology can be mastered, one must have an adequate knowledge of not only single-phase-flow fluid mechanics, thermodynamics, Chap. and bean or choke performance. Also included is a coverage of flow through piping components such as valves. 2 through 5 . the methods for describing the performance of each category have been vastly improved. gas and liquid phases were permitted to travel at different velocities. 1. resulting in extensive use of mixture Reynolds numbers. often based on dimensionless groups. with slippage effects being accounted for through empirical liquid-holdup correlations. gas lift. 2 through 5. Applications include flowing. Some landmark publications are identified that have had a lasting influence on the understanding of the total system involved in flowing fluids from the reservoir to surface storage and processing facilities. A detailed description of flow patterns for both circular pipes and annuli is given. and pumping wells. ~ i n expanded and clarified Gilbert's concepts. such as predicting erosional velocities and unloading gas wells..NQQECOCATlON I 2 3 4 5 6 7 SEPARATOR SURFACE CHOKE WELLHEAD SAFETY VALVE RESTRICTION A++ m s Pr 8 tA 18 GAS SALES STOCKTANK Fig. Chap. Production problems. 6 is devoted to the use of the material from Chaps. 3 contains a description of the variables encountered in multiphase flow in pipes. steadystate-flow concepts for both Newtonian and non-Newtonian fluids and for gas. Finally. Steady-state. a significant part of this monograph will be devoted to a review of these important topics. physical properties of each phase. 1 clearly states the objectives of this monograph and describes the common occurrence of multiphase flow during the production and transportation of oil and gas. unit conversion factors. However. The importance of liquid holdup and methods to measure this parameter are discussed. Some investigators also used an empiriMULTIPHASE FLOW IN WELLS . Procedures are given for the use of blackoil and compositional models to predict mass transfer and in-situ volumetric flow rates. thermodynamic principles. Chap. Although thrs monograph addresses only multiphase flow in wells.l-Multiphase production system. and through surface pipe. 2 concentrates on a review of important ~ i n ~ l e . Included are brief reviews of conservation of mass and momentum and their use in the prediction of pressure gradients in both circular pipes and in annuli. the reader is prepared to address the many important practical applications encountered in multiphase flow in wells. Techniques are presented to determine values for mixture variables. It is interesting to note that this same procedure is still followed today under the names of production systems analysis or NODAL^ analysis. Appendix A provides acomplete Nomenclature and a list of conversion factors. Most early investigators used data obtained from laboratory test facilities. The concepts of production systems analysis or NODAL^^ analysis are thoroughly described with examples. vertical-lift performance in the well. Empirical flow-pattern maps were used. After mastering the material in Chaps. pipe diameter and inclination angle. also are given. flow pattern was observed and liquid holdup was measured by use of quick-closing ball valves. a short discussion on nomenclature. These data usually included volumetric flow rates of gas and liquid.4 Historical Background Gilbert2 developed many of the concepts used today to analyze the well performance of flowing and gas-lift wells. 1. The standard computing algorithm for generating pressure and temperature vs. and tees. equipment. and the SI (metric) unit system is included. length traverses is presented.' and heat transfer. Chap. much of the material also is applicable to multiphase flow in near-horizontal flowlines and pipelines.1 The Empirical Period (1950-75). with a few using field data. Gilbert also presented a clear description of the unsteady flow behavior or "heading" that can exist in a well and showed how to minimize or eliminate the phenomenon. Graphical techniques were presented for coupling these categories to permit analysis of individual well problems. 4 contains a detailed descriptionland evaluation of all corn*)used correlations and mechanistic models to predict flow patterns and pressure gradients in wells. He divided the production system into three distinct categories: inflow performance from the reservoir. However. Chap. Brill and Arirachakaran4 divided this history into three partially overlapping periods. 5 contains a detailed description of critical and subcritical flow of single-phase and multiphase mixtures through chokes. Frictional pressure losses relied on single-phase-flow equations. and pressures at the inlet and outlet of the pipe. Chap. d ~ The history of attempts to improve the prediction of vertical-lift performance for wells is especially interesting. Thus. Basic theoretical concepts are presented in Chaps. Most wells contain some type of restriction for controlling flow rates and protecting downstria. 2 through 5 in various well design applications. 1. elbows. Fluids were treated as homogeneous mixtures.4. Also included is the use of conservation of energy. and heat transfer to predict fluid temperature behavior when hot fluids are produced from a reservoir. but also vaporlliquid equilibrium (VLE) and fluid physical properties for multicomponent hydrocarbon systems.~ h a s e . 2 . up the well. In some cases. Appendices B and C contain methods to predict fluid physical properties and mass transfer for black-oil and compositional systems. pressure-gradient equations were developed that were based on conservation of momentum and mass principles applied to the homogeneous mixtures. fraction of liquid entrained in the gas core for annular flow. to be very sensitive to other parameters.cal multiplicative factor to represent the increased friction resulting from the presence of a second phase. momentum.was the work to predict flow-pattern transitions for all inclination merically integrating the pressure gradient from one end of a pipe to 6 . ultrasonics.K.phases.3 T h e Modeling Period (1980-Present). it was quickly recognized that there were many the modeling approach is more accurate and precise than empirical problems with the methods available. Ozon er rates for wells and pipelines. including burst pipe and leak detection. and knowledge-based nient. and simulations can require large amounts of computer time. The empirical correlations nificant improvements have made transient simulators more userused to predict pressure gradient. equations were much more advanced.a Thisopened the door for improved models for each the other were well understood. that could be used with confidence. Improved correlations for these parameters became possible as a result of the experimental research being conducted. progress in this area had already been made by the complex and require specialized training to understand and use. France. Two classic papers dealing with multiphase flow in horizontal pipes by Dukler and ~ u b b a r d ~ and Taitel and ~ u k l e r 6 clearly show that mechanistic models for slug flow and flow-pattern prediction already were available. easier to use. Procedures to connect wells to reservoirs al. Dramatic changes also are being made in user-friendly computer conservation of mass. Empirical expert systems. This has been the modeling period." Hasan and Kabir. affordable.. and flow in pipelines with low velocities. and virtually every major producing of the flow patterns. and new high-speed photographic techniques months when Gilbert2 first proposed the basic well-performanceabounded. laser Doppler an engineer can perform in minutes analyses that would have taken anemometers. and more readily available. An important improvement in steady-state mechanistic models personal computer (PC) in the early 1980's. The improved technology also canies an additional without the introduction of more basic physical mechanisms. data integration.^-^^ This ap.of graphical user interfaces. correlations and sir. and energy. Both the transient simulators and the mechanistic models are Fortunately.ity of fast computations and data transfer. Flow-pattern transitions. computers have data. The magnitude and quality of required input data are much greater. Even when these mechanisms were partially included.of PC's and workstations has completed this evolution until today clear densitometers. Various simplifications were found to be conve. proach involves writing separate equations for each phase to describe absorption of gamma rays and neutrons. The 1970's saw a trend in the petroleum industry to adopt some basic physical mechanisms already in use in other industries.ent values for such items as electrical capacitance and resistance. PC-based data-acquisition hardware and software im. previously thought to be ment of these mechanistic models as experimental research is condependent mostly on flow rates (or superficial velocities).programs that rapidly analyze a broad variety of well characteristics cal approach. It became clear that no matter how much data were nomena that occur.S. There have been significant improvements in the design ulation techniques. pioneered by the nuclear industry. These include changes in inlet. terrain slugging. permitting acquisition of large amounts of higher-quality In addition to accelerated computing speeds. mass storage devices. As computers became more powerful. capacitance sensors.programs. their investigation was severely hampered by the unavailability of sufficiently accurate instrumentation and real-time data-acquisition systems. proved. transient simulators have been much more difficult to use than steady-state programs. they gradually displaced ~ n v e s t i ~ a t orecognized that improved understanding of multi. sonic velocity. sig1. the empirical period resulted in a collection of empirical correlations whose accuracy was limited by the lack of inclusion of basic physical mechanisms. Furthermore. basic physical mechanisms that occur during multiphase flow in At the same time that improved experimental research was being pipes. including the possibility of linking the various company had a computer program to predict pressure drop or flow models through unified flow-pattern-transition criteria. These meaproach. and they will enjoy increased use in the future. Methods are available today to measure such variables as loconducted. Many companies developed versatile computer rs phase flow in pipes required a combined experimental and theoreti. interfacial ~henomenabetween methods.surements take advantage of how gas and liquid phases yield differsient codes applicable to petroleum industry problem^. such as interfacial friction factors.evaluation procedures. but appear to be capable of simulating a variety of applications that are time-dependent. the current state of the art in multiphase flow in pipes is the An empirical liquid-holdup correlation for each flow pattern was emergence of both two-fluid transient simulators and steady-state equally inadequate.2 The Awakening Years (1970-85). Thus.. they challenges in the 1980's required a much better understanding of made it possible to develop generalized pressure-gradient curves multiphase-flow technology. Millions of dollars were invested in multi. and the U. Although the fluids used for these studies (steam pretation of results is bettercanied out by engineers with a specialized and water) were trivial by comparison to those encountered in the background who are fully aware of any simplifying assumptions or petroleum industry. Empirical flow-pattern maps correlations. were found ducted on the basic mechanisms of multiphase flow.curves.or outlet-flow rates or pressures for pipelines and wellbores. Norway. the accuracy of the predictions could not improve gence problems.I9 Xiao et a1. and small changes in pressure and temperature. This under. for the deveopment of tran. Use of nu. Internuclear industry.4.4. and liquid holdup in a slug body. Petroleum industry cumbersome to analyze individual well problems.. Analysis of the data improved understanding of the complex played an equally important role in data-acquisition. This was especially true in the prediction of flow patterns and gas bubblerise velocities in liquid columns. The petroleum industry adopted the two-fluid modeling ap. Numerical instabilities are common.gradient curves.4. Until recently. coupled with the introduction of the friendly. The availability new instrumentation for measuring important variables.~plifiedclosure relationships were still necessary for some parameters. pipe blowdown. and optics. lift installations. Gradient curves were especially useful in the design of gasphase flow through research consortiums in the U.. Techniques for nu.20 and Chokthrough simple IPR techniques abounded. In general. The resulting transient codes are still being evaluated. efforts were expended to develop improved theoretical cal velocities and void fractions. The result has been the emergence of commercial INTRODUCTION . The availabildynamic mechanisms that exist during multiphase flow. Transient simulators have the capability to anagathered in either laboratory test facilities or from carefully tested lyze complex time-dependent problems but often suffer from converfield installations. such as use of a single mixture-energy equation. 1. The assumption of a homogeneous mixture was mechanistic models that more accurately describe the physical pheoversimplified. ~ r o w n 'presented the true shi21 published combined or "comprehensive" mechanistic models. and standing was transformed into improved mechanistic models to bet. The resulting sixequation problem must be solved simultaneously with numerical sim. results are more difficult to interpret than for steady-state flow. dramatically improved practical tools available to petroleum engineers. provided well characteristics The increased cost of Arctic and offshore developments has justi.vastly improved instrumentation has permitted vital research on the ter describe the physical phenomena occuning.4 Role of Computers. In addition. cost. concept of NODAL^^ or production system analysis.I8 Ansari et a1. Sophisticated test facilities were constructed that used along with the performance of all wells in a field. The early computers were too slow and 1. However.were not significantly different than those used to develop the fied increased spending. the methods used to formulate conservation limitations which have been included in the developments. pigging of pipelines. especially inclination angle. it is now possible to continue improvewere inadequate. However. Their attempts to evaluate the models with field data confirm that Unfortunately. : "State of the Art in Multiphase Flow. SPE. 18-22 January.337. Trans. AIME. J. Eng. 1.: "Prediction of Pressure Drop and Liquid Holdup in Vertical Two-Phase Flow Through Large Diameter Tubing. The SI Metric System of Units andSPE Metric Standard. 21.. 8.733. A. AIME.: "Flow Pattern Transition for Vertical DownwardTwo-Phase Flow.R. Barnea. Ansari. 19. Y.. Oklahoma (1994). J.. et al.: "Flow Pattern Transition for .computer programs that are not only easy to use. Hasan.: "Gas Liquid Flow in Inclined Tubes: Flow Pattern Transition for Upward Flow. Fund. 23. France (June 1993) 29.: "Studying Transient Multi-Phase Flow Using the O Pipeline Analysis Code (PLAC). et al.M." PhD dissertation. Well Logging and Formation Evaluation.J.. Multiphase Flow (1987) 13. 14. Ferschneider. 0 . Brill." paper SPE 2063 1 presented at the 1990 SPE AnnualTechnical Conference and Exhibition. and Dukler. C. Chokshi.M. and Arirachakaran. of Tulsa. K. L . as well as throughout this monograph (except in Examples 4. Energy Res.: "A New Multiphase Flow Model Predicts Pressure and Temperature Profiles in Wells. D. Soon it will be possible to access and cross-reference these databases with a well identification and to input data automatically into a multiphase-flow design program. compositional-fluid-model approach for more precise calculations of mass transfer and fluid properties. E. Trans. D.E. A.. Consequently. 0." Proc.: 'Transition From Annular Flow and From Dispersed Bubble Flow-Unified Models for the Whole Range of Pipe Inclinations. Shoham.P. Barnea. (1980) 26. which deal with mechanistic models that were developed on the basis of SI units). D. 293. 11. 17. S. 1... & Prod. 131. 20. Nind.E. Taitel. 12. 5. Also emerging is a trend for large standardized database-s-containing information ranging from drilling and completion records to well-test and production data. References 1. U.. Sci. A. Appendix A presents customary units.. er al. SPE Letter and Computer Symbols Standardfor Economics. This approach gives better pressure and temperature predictions and is more robust than the standard black-oil approach. Occasionally. D. Brown. (March 1990) 112.. Taitel. there will be some confusion in nomenclature because many of the important publications have different nomenclature and terminology. 3b." JPT (May 1992) 538. Sci. Eng. in all languages." AIChE J. Natural Gas Engineering. and Brill.E. W. 6.735. 23-26 September.47. A. (1976) 22. Sci (1982) 37.. (1954) 126. Richardson.J. G. Bamea." SPEPF (May 1994) 143. Houston. J. Sixth International Conference on Multi-Phase Production. A. Y. and Chwetzoff. 285. Chem. ~ ~ SI is the official abbreviation. thus dramatically reducing manual input requirements. Oklahoma (1980) 2a. (1985) 40. Trans.and Taitel. Inc. Richard- son. J. Shoham." SPEPE(May 199 I) 17 1 . 291. (1975) 14. and Brown. Petroleum f Publishing Co.E. et al..: "A Model for Predicting Flow Regime Transitions in Horizontal and Near Horizontal Gas-Liquid Flow.P.G. 2. 3a. AIME. P. MULTIPHASE n o w IN WELLS . 0. P." J." Chem..5 Nomenclature and Units Much of the technology for multiphase flow in pipes was developed outside the petroleum industry.: "A Comprehensive Mechanistic Model for Two-Phase Flow in Pipelines. 1. New York C ~ t y 4. Horizontal to Vertical. Y. Xiao.: "A Comprehensive Mechanistic Model for TwoPhase Flow. Dukler. Shoham.E. Eng. J. As much as possible. 18. 297. .: "Application of Producin tion System Analysis to Determine Completion Sens~tivity Gas Well Production. this monograph uses the standard symbols adopted by the Society of Petroleum ~ n ~ i n e e r s .741. Black. and Kabir. Downward Inclined Two-Phase Flow..: "Flowing and Gas-Lift Well Performance. . Barnea. AIME.M.: "The Dynamic Two-Fluid Model OLGA: Theory and Application. The faster computing speeds on PC's and engineering workstations also make possible a truly interactive. 5.H. D. for the International System of Units (les Systeme International d'unites). McGraw-Hill Book f (1964).345.. D. Prac. 16. and PerroIeum reservoir Engineering. A.: "A Model for Gas-Liquid Slug Flow in Horizontal and Near Horizontal Tubes." SPEPE (Mav 1988) 263:.25.10." Intl.: The Technology o Artificial Lifr Methods. Co.: Principles o Oil Well Production. 8-1 1 September. K. and Hubbard. it will be necessary to define new symbols or deviate from those recommended by the Society of Petroleum Engineers. Mach. er al. Ozon." paper presented at the 1981 ASME Energy Sources Technical Conference and Exhibition.9 and 4. Pauchon. Gilbert. T. customary units23still are used frequently in many parts of the world. .: "A Study of Multiphase Flow Behavior in Vertical Wells.E. R. Tech.: "A Comprehensive Mechanistic Model for U p ward Two-Phase Flow in Wellbores. Aberdeen.: "A Unified Model for Predicting Flow Pattern Transitions for the Whole Range of Pipe Inclinations. but also incorporate the very latest technology in a friendly. Bendiksen. M.M." AIChE I.. and 13. and 4.E. 9. 22. K. New Orleans. Barnea. Cannes." Ind. C.: "Modeling Flow Pattern Transitions for Steady Upward Gas-Liquid Flow in Vertical Tubes." Intl." paper SPE 16535 presented at the 1987 SPE Offshore Europe Conference. However. Y. No. SPE. along with factors to convert to the SI metric system of units. Taitel..S.S." Drill. 15.. 3. (1982) 37. Promo.A. 7. Barnea. Texas (1984).N." Chem. No. Eng.W." Chem. Texas (1986). Tulsa. and Dukler. Tulsa. unintimidating manner. Multiphase Flow (1986) 12. Trans. J. . 11) v dX Fig. .. These studies . Eq. .7 can be solved for shear stress pvd (2. . . 2.. . . Fanning friction factors are retained to preserve original equations. . .. The procedure first requires an evaluation of whether the flow is laminar or turbulent. . .... . . A logical approach to definine friction factors for turbulent tlow is to begin with the simplest case (i. 2. . which is the ~ o i s e u i l l equation... Smooth pipe is seldom encountered in oil and gas production. ..2-Moody 6 diagram. .. Because the pressure gradient in Eq 2. . . The equations most often used. which can approach smooth pipe.4... ...1 Laminar Flow. .. . . ... . . .12) N~e' r = fv2. pressure gradlent I S ldentlcal to Eq. . The velocity profile for laminar flow can be integrated to yield the pressure gradient. the frictional component of the pressure-gradient equation becomes nd2/4 which is often called the Darcy-Weisbach equation. . . . . .. .. .4... 2. 2... . 2. . .. . Numerous empirical equations have been proposed to predict friction factors for smooth pipe. . . . (2.5 f = 0.11.0056 + 0. e~ .dZ 2.. Our ability to predict flow behavior under turbulent-flow conditions is a direct result of extensive experimental studies of velocity profiles and pressure gradients. (2.3 Calculation of frictional pressure gradients requires the determination of values for friction factors. . ..11 results only from wall the shear stresses or fnct~on. an analytical expression can be derived for the friction factor.. ..9.e.. ... 2. have shown that both velocity profile and pressure gradient are very sensitive to characteristics of the pipe wall...13) Fig.. This was done for flow in horizontal capillary tubes to give Eq.. .~ .9) Substituting Eq. laboratory studies often are conducted with glass or plastic pipes. .. . . . . . . . . 2. . .000. .2 M1'I TIPHASE FLOW IN WELLS . The Reynolds number is defined as 2. Comblnlng these equations glves 6 4 = 64 f = . . (2. (2.. .. . ... .5N2". . .. . .5.2 Turbulent Flow. However. For laminar flow. ..8) 8 . .8 into Eq. . . . Newtonian fluids and flow through annuli. .. ... .1-Control volume. . . ..... the smooth-wall pipe).. . .. and their suggested ranges of are those of D~~~ er a1. .. then proceed to the partially rough wall and finally to the fully rough wall... Laminar flow is considered to exist if the Reynolds number is less than 2.. . . . . 2. are still accepted values. length. tightly packed sand grains that would give the same pressure-gradient behavior as the actual pipe. In turbulent flow. the pressure gradient. It has become the basis for modem friction-factor charts. Wall roughness is a function of the pipe material. In turbulent flow. Zigrang and Sylvesterlohave given one of the most accurate and simple to use. A common value used to generate pressure-gradient curves is 0. E .00005 ft. is the mean protruding height of uniformly distributed and sized.00015 ft. 2. If the laminar sublayer that exists within the boundary layer is sufficiently thick.17 forf requires a trial-and-error process. His correlation for fully rough wall pipe is given as Eq. Until it is again updated.17 can be expressed as9 Values off are estimated.17 degenerates to Eq. However. Rather. 2. these values should not be considered inviolate and could change significantly as a result of paraffin deposition.6 where Although the Blasius equation is considered less accurate. or from explicit approximations to the Colebrook equation.2 The region in whichf varies both with Reynolds number and relative roughness is called the transition region or partially rough wall. The absolute roughness of a pipe.17. Numerous explicit approximations to the Colebrook equation have been proposed. the behavior is similar to a smooth pipe. the age of the pipe. the effect of wall roughness has been found to depend on both the relative roughness and the Reynolds number. Dimensional analysis suggests that the effect of roughness does not result from its absolute dimensions. given in Fig. Note that Eq. approximate values for absolute roughness normally are sufficient.3. 2.19 can be used in lieu of Eq. hydrates. 2.2 is a graph that shows the variations of friction factors with Reynolds number and relative roughness based on Eqs. Individual protrusions and indentations vary in height.. and the environment to which it is exposed. 2. &Id. The recommended value for new tubing is E = 0. SINGLE-PHASE In most cases. Consequently. the method of manufacture. 2. but rather from its dimensions relative to the inside diameter of the pipe. Eq. this value of ~ l d should be used for future predictions. thus. Fig. Fig.. erosion. width. The sublayer thickness is directly related to the Reynolds number. the roughness can have a significant effect on the friction factor and. the inside wall of a pipe is not smooth. From a microscopic sense.00075 ft. until they agree to within an acceptable tolerance. Eq. 2. It is important to emphasize that E is not a physically measured property. A direct substitution procedure that uses the calculated value as the next assumed value results in convergence in only two or three iterations.3-Pipe roughness. If measured pressure gradients are available. wall roughness is not uniform. 2.f. it receives greater use because of its similarity to the laminar frictionfactor equation.16 and is still the best one available.12 and 2. The only way this can be done is to compare the behavior of a normal pipe with one that is sand-roughened. Nikuradse's7 famous sand-grain experiments formed the basis for friction-factor data from rough pipes. Both can be expressed in the form Normally. and then calculated. "very dirty" pipe can have roughness values of 0. Reynolds number relationship can be established and an effective relative roughness obtained from Fig. shape. bloody2 has done this and his results. a friction factor vs. 2. Colebrook8proposed an empirical equation to describe the variation off in the transition region. n o w CONCEPTS .16 for large Reynolds numbers corresponding to fully developed turbulent or rough pipe flow. Initial values of roughness often are needed for design calculations. the friction component of the pressure gradient is small compared with the potential energy component.2. and distribution. fc. it is the sand-grain roughness that would result in the same friction factor.where and ~lasius.17. Solving Eq. The initial assumption can be obtained from one of the explicit smooth-pipe equations.ll For most wells.. 2. 2. or corrosion. For tubing exposed to an environment that causes significant changes in roughness. . Note that the pressure change consists of a loss owing to friction of .17.181.. the density. one for each half of the well. The resulting equation is Pwf .. Calculate the pressure gradient from Eqs.000 ft qw = 20. Calculate the pressure change in a water ~njectionwell. (2.3. the trapezoidal rule for numerical integration can be applied by determining the value of I for each of any number of increments in p between pf and p..TZ/Tscp.75ygL = (T)(C + 41mf + I.2 Ibm WM : I sect.0 In..P. or Fig. 18. flows in a well.28) Use of the Cullender and Smith method to calculate flowing bottomhole pressures in gas wells can best be explained with an example problem.9.4333)(8. 2. f = 0.5 and 2.. and L = ft. + 2. calculate the flowing bottomhole pressure in a gas well using the Cullender and Smith method with two increments.00006 ft The average velocity in the pipe is Eq.* L = 8. Q = 32.where Example 2. neglecting acceleration effects = = . Eq.1-Single-Phase Liquid Pressure Drop.284.0.10. and B.22 cannot be integrated analytically without making assumptions about 2. 2.126 psflft .. the gravitationalconversion constant.0.466. Eq. v = q / A . 2.5 can be written as While this method can be used with any number of steps. pmf.. is considered-Eq.4333 = 0. The following data are known. 2.400 = 59. With the following data...4 = 3. consequently..5 psi. 2. The right side of Eq. When a compressible fluid. P = pMIZRT.B.2 1 is applicable for any consistent set of units. = 0.43 Single-Phase Gas Flow.000) Lower half: - 181.= 110°F E = 0.000 B/D 0 = -90" pw = 62.. = p. d = in.000 ft lj= 245°F q = 2. = 4. = MMscf/D. the pressure gradient all vary with pressure. flow is turbulent.2. Assuming the well can be divided into two halves for integration purposes-where only the intermediate value of pressure. The relative roughness for the pipe is and p = psia.466.75 L = 10.. q.0155. Cullender and Smith12 developed the most widely used method to calculate flowing bottomhole pressure in gas wells. 2.. appears when problems are solved w~th customary unlts 8 . pw = 1.19.9 psi and a gain from an elevation change of 3. Substituting field units and integrating the left side of Eq. Cullender and Smith1* demonstrated that the equivalent of four-step accuracy can be obtained with a two-step calculation if the Simpson's rule13 numerical-integration approach is used. .9 + 3.0 cp E = 0. y..2-Single-Phase Gas Pressure Drop. Upper half: The pressure change is then Ap = ( .000. 2. that at the mid-depth. 2.0227 + 0.4 psi. 2. the Reynolds number is Because NRe> 2.9 15 MMscfD MULTIPHASE EOW IN WELLS 'For examples In this monograph.2 1 gives Pwf where From Eq. and. For gases.2.4 1bm/ft3 d= 5.4106 psilft. T andf.. However. 2. q.000 psia T.). velocity.20 and separating variables gives Example 2.25 can be separated into two expressions. T = OR.274 + 62.22 may be expressed as From Eqs.. Neglecting kinetic energy. Combining these expressions with Eq. 2. q =q.441 in.0227 = + 0. such as gas.00007 ft d = 2. 001)(4.OW CONCFPTS . and Imf = 4.60 = 2. Estimate pif (Second Trial): Select pLf= 2.00279 + = + 2. Estimate p (Third Trial): f : Select pif= 2.001)(4.015 and Id = 4. The value of p. 2.41 and This is close enough to the previously calculated value of 2.23): At pLl= 2.796.4. 2. and Z=0. if the more accurate Simpson's rule numerical-integration approach is used.000 + (18. SINGLE-PHASE FI. the flowing bottomhole pressure is predicted to be Calculate p d (Eq..41.676 psia.f= 2.26): and = 2.00279 + Estimate pf (Second Trial): : Select pLf= 2.378 (0. 2. and many oiVwater mixtures.26): pml = 2. = 2. Therefore. (0. 4.16 and 2.28. If flow is not completely turbulent. polymers injected during EOR projects..250 psia.. 2. The material presented previously is valid only for Newtonian fluids.74 1 psia (not close enough to PC/). f=0. and Z = 0.: p = p.868. Estimate f .744 psia.41 + 181. Calculate p.4 Non-Newtonian Fluids. Calculate p.00279 + Calculate pmf (Eq.f is now calculated. Fig.479)' + 0. 1 = .27): pwf = 2. 2.000 + 379 = 2.000) 189.425 = 197.71.26): 2.371 psia. Estimate first Trial): and = 2. 2.75)(0.00 1)(4. 196. T= 24S°F.24.' ~ / 2 sin 0) 199. Calculate Id (Eq.27): Calculate Imf (Eq.75)(0. (Eq.. depending on such factors as shear rate. and fluid composition. Calculate f assuming a completely turbulent flow (fully rough wall).379 psia (not close enough to pLf).797.379 psia.796.39. and spacers used during well-completion activities. (Eq.001)(4. and Z=0.867.23): Calculate Imf At pLf= 2.684 = 189.23): Calculate Imf At pif= 2.379 psia. a Reynolds number calculated at surface conditions can be assumed to exist throughout the well. crude oils at temperatures approaching the pour point.4.744 psia. Lower Halfof Well. (0. 2. 2. From Eq. = 2. fluids.75)(10. Calculate p d (Eq.741 psia.741 psla.37 1 psia (not close enough to pif).000 + 37 1 + 2. The design of piping systems for non-Newtonian fluids becomes complicated because the use of conventional friction-factorcorrelations is not directly applicable.684)~ 0.378)~ 0.23): At p. Therefore.00.676 psia. However.250 psia.00279 = 196. 2.These non-Newtonian fluids may assume any type of rheological behavior. and Z=0. These include many drilling muds. T= l l 0 + 67SoF.~(Eq. Calculate Itf (Eq. T= 110°F. 2. and Z = 0. from = 2.425)~ 0. temperature.479 (0.f (Eq. and Z=0.81. T = 178 OF.379 psia. such as cement slurries.5 x 1 0 . the pressure at the mid-point of the well is 2. From Eqs.and Preliminary Calculations. fluids encountered in the petroleum industry often act as non-Newtonian fluids.000 psia.37 1 psia.( l first Trial): and Imf = 4.379 + 362 = 2.00 + 189. 2. 2.23): At pLf= 2. Calculate I g (Eq. fracturing fluids.23): At pil= 2.000) = 2.741 psia. 2.379 psia.379 + (18. T= 17g°F.75)(10. Upper Half o Well. the flowing bottomhole pressure is 2. 2. T = 245OF. mixing speed). The rheological behavior of the oiYwater dispersion system is suitably described by the Ostwald-de Waele power-law model. f' = . The second method treats the mixture as a nonNewtonian fluid and is based on the following ass~mptions'~: 1. It is also possible for an oiYwater mixture to haven' greater than unity. This method is covered in Chap.. From Metzner and ~ e e d . Generalized Reynolds Number. The type of behavior that a fluid system will follow normally is unknown but can be determined from laboratory experiments with an appropriate viscometer. 2. resulting in dilatant (shear thickening) behavior. Non-Newtonian Fricfion Factor. and droplet size distribution of the dispersed phase (or. Govier and ~ z i z suggested ' ~ this friction factor for power-law. and f. For a typical oiYwater mixture. For non-Newtonian flow.' Knudsen and ~ a t z . From the same plot. while the quantity. and 7 = apparent =the viscosity.29 will describe pseudoplastic (shear thinning) behavior. 3 because it can involve the combining of the viscosities from each phase to obtain a mixture viscosity.33 and 2.corresponds to the Newtonian viscosity. isalso the true shear rate at the pipe wall for a Newtonian fluid.describes the types of rheological behavior that ] can be encountered. Eqs.DlLATU(T FUIY WwM VIELO WUE SHEAR RATE Fig 2. The power-law stress-strain relationship can be expressed as where NRcM. Ifn' is unity. pseudoplastic fluids flowing in rough pipes: . the frictional pressure gradients for pipe flow in this particular oiYwater dispersed system readily can be determined.4-Rheological models. The first method treats the mixture as a Newtonian fluid with an apparent viscosity that can vary with water fraction. the relationship of the tangential line to the laminar-flow curve can be written as Combining Eqs. the Fanning friction '~ factor for non-Newtonian laminar flow can be written as Turbulenf Flow-Smooth Pipes. Newtonian flow Fanning friction factor for rough pipe.30 and 2. calculated at the same generalized Reynolds number. 2. Svld. =Newtonian MULTIPHASE FLOW IN WELLS . 2.15 where n' =the slope of a logarithmic plot of r. the procedure for calculating friction factors for non-Newtonian fluids is similar to that for Newtonian fluids. where z is the laminar wall shear stress. Metzner and ~ e e dintroduced l ~ the concept of a "generalized Reynolds number" for non-Newtonian flow. The mixture is homogeneous. indirectly. vs. The usual Reynolds number definition is given in Eq. Laminar Flow. the ratio. 8v/d.29 will describe Newtonian behavior.10.34 clearly reduce to the normal Reynolds number and the Newtonian viscosity for a Newtonian fluid. 3.32 gives and Power-Law Model..36. which also can be written as Turbulent Flow-Rough Pipes. in-situ holdups are the same as their respective input volume fractions. the relationship between the true shear rate and the apparent shear rate can be expressed as where jM-R =the friction factor calculated from Eq.p. temperature. These tests must be conducted for a specified set of operating conditions: input water fraction. 2.R generalized Reynolds number. 2. Slippage between phases is neglected. In general. Eq. often called the flow-behavior index. and K' will be equal to the constant viscosity. p. 2. [ s W / ( 8 v / d ) ] . when n' = 1 and K' = p . 2. Dodge and ~ e t z n e r " proposed this implicit friction-factor equation. Once the fluid system is correctly characterized. n' usually is less than unity. Thus. and Eq. Two methods commonly are used to design piping systems for the transport of oiuwater mixtures. Thus. . OillWater Mixture Pressure Example 2. An annulus is characterized by the existence of two circular pipes. is determined for the particular system. l ~ Fig. .74 1brn/ft3 pw= 62 23 Ibm/ft3 n'= 0. 2.5 Flow in an Annulus For Newtonian fluids.fw = 55. The fluids are pumped upward through the tubing-rod string annulus. = pz L + 2f'pm2v.36.J is given by Eq. e= z & ~ I ( ~ & q) - where Concentric Fully Ecwntric osc = 0 e=o e= 0. To remove or "unload" undesirable liquids that can accumulate at the bottom of these wells. The hydraulic diameter is four times the area for flow divided by the wetted perimeter.1 Friction Factor for Single-Phase Flow. 2. . To determine appropriate characteristic dimensions. this concept is better suited for high degrees of turbulence.0. Once the Fanning friction factor. the frictional pressure gradient can be calculated from K' and n' are clearly the two important parameters that will affect the frictional pressure-gradient calculation for a dispersion system. multiple completions. The follow~ng of also are given: q. and for eccentricities of 0. and also requires a trial. . This trend is dictated by economics. These two parameters can be determined experimentally with an appropriate viscometer. a clear understanding of flow in an annulus must be achieved. .37.38 is analogous to the Colebrook equation for Newtonian fluids given by Eq. K.^^ Calculate the pipe inlet pressure. dh = d. The permanency of the siphon tube in the tubing string requires the fluids to flow upward through the tubing-stringlsiphon-tube annulus.5--Annuli configuration^.000 ft po = 53. However.9 psia.3-Non-Newtonian mlxture at atmoDrop.R by Eq. p l . Eccentricity.l& and the degree of eccentricity.0. 2. . a siphon tube often is installed inside the tubing string. compute the friction factor for turbulent flow of pseudoplastic fluids in rough pipes. = pJo p. =outer diameter of the inner pipe (tubing).38 were used. .5. the hydraulic diameter is not always the most representative characteristic dimension for flow in an annulus. . calculate the generalized Reynolds number. = 30 ft3/sec d = 20 In.17.8589 K'= 7 1475 x Ibf secnlft2 Calculate the mlxture density.00676.flow Fanning friction factor for smooth pipe. Fully Rough Wall Turbulence. a rigorous treatment of the flow field is possible for any annulus configuration.P H A T F FI OW c O V P F l T < 2.0. . .4 Ibml ft3. many oil wells with high production rates produce through the casing/tubing annulus. annuli have been treated based on the hydraulic diameter concept..and-errorsolution procedure. . . Note that if Eq. However. For noncircular conduits. Fig. is given where jhl. . . . . Combining the developments by ~ e ~ and Snyder ~ d a ~ and . f'. For laminar flow. Using Eq. and& is given by Eq. .d. calculated at the same generalized Reynolds number. In the past.500. . . TIYGI F . a rod string is instdlled inside the tubing string to connect the prime mover unit on the surface to the pump at the bottom of the well. where the flow occurs through the area bounded by the outer pipe inner wall and the inner pipe outer wall. 2. Bird er al. these "casing flow" wells account for a significant part of the world's oil production. . For the glven flow rate and water fraction. .17.(2.43) + However. In the petroleum industry. Szilas er al. .14. For annulus configurations. Govier and ~ z i zrecommended. . .5. Annuli can have eccentricity values varying from zero to one. Although few in number when compared with all producing wells. . In sucker-rod pumping wells. The degree of eccentricity accounts for the displacement of the inner pipe center from the outer pipe center and is expressed by The criterion for turbulent flow is considered to be NReM > 1. For this example the flow regime is turbulent. . A horizontal pipe discharges an o~Ywater spheric pressure Determ~ne Inlet pressure required to malntaln the a constant volumetr~c flow rate of the dispersion for an ~nput-water fract~on 20%. flow in wells normally occurs in a tubing string. p . and DBc = distance between the pipe centers. and 1. . . single-phase-flow friction factors often are determined through the application of the hydraulic-diameter concept. 2. where d. the trial-and-error procedure would have given a f ' value of 0. & = 0001 ft L = 10. . . and regulated production rates. . ..40 essentially reduces to the von Karman equation. 2. = 205. 2.18 Frictional Pressure Gradient. Other casing-flow applications are found in wells under various types of artificial lift. Two geometrical parameters identify these configurations: the annulus pipe-diameter ratio. 2. Eq. gave analytical solutions for both the velocity profile and friction factor for a concentric annulus. 2. 2. K = d. l9 obtained a similar result for pseudoplastic fluids from the following equation. using p. Casing flow also can occur in gas well production. 4 =inner diameter of the outer pipe (casing).5 shows cross sections of annuli with the same pipe-diameter-ratio value.5 Dec=(4-4P e= 1 Note that Eq. 2. various semiempirical and analytical models have been used successfully to predict the velocity distribution and pressure gradient. k~ 1 Newtonian Laminar Flow. The resulting friction-factor correlations normally take the form of the Blasius-type expression.15 where C and n are determined empirically. used the same technique. fb . Through use of a bipolar coordinate system. Fig. the equation of motion. it can be shown that factor becomes independent of the FcIFNc ratio.' and ~ o d ~Winkler23 e . These solutions are presented in a later section.6-Friction-geometry parameter in annuli and circular Semiempirical approaches involve the use of experimental data for pipe (laminar single-phase How at same Reynolds n~mber). 2.Thus. at high Reynolds numbers. ~ ~ 0. Their solution was based on previous developments by and the circular and noncircular configurations. resulting in FEA = 16. However. The Gunn and ~ a r l i n proce.8 1. as given in Eq. The Fanning equation can be written as where and the Reynolds number is given by The Fanning friction factor in circular pipes. Apthe ratio FCIFNc.6 0. . 2. At low ~ e ~ d a ? O El-Saden. f& is the Fanning friction factor and F E is the friction-geometry parame~ ter for an eccentric annulus. axial flow.6 presents a plot of Eq. The friction factor in laminar flow is determined from solution of the continuity equation.53. fully developed. .*' analytical solutions also can be found for these flow 28 parameters for an eccentric annulus. having a constant value 6 of 1 for pipe flow. . As before. semi-empirical correlations. .0 successfully used the Dodge correlation for flow in an annulus.e2)+ ( 1 2 Ke where Fp is a friction geometry parameter. For a concentric annulust5 + e2). 2. . Snyder and Goldstein2' presented an analytical solution for laminar flow in an eccenwhere F. they showed that for turbulent flow in noncircular sections. the formulation of an analytical model is even more complicated.is given by and cosh q0 = K(1 . K= 414 however. Examples include 0. and FNC are the so-called friction geometry parameters for tric annulus. the function given in Eq. (2. such as friction factor.52. .the number of terms can be truncated after several terms. GUMand Darling concluded that the similarity existing between friction factors for circular and noncircular configurations in the laminar region is also accompanied by a similarity in the turbulent region. and the Fanning equation. . he noted that this procedure did not take into account the annulus pipe-diameter ratio. .~B turbulent flow in combination with characteristics of laminar flow in g~~ the same noncircular configuration.ls Redberger and Charles. respectively. Empirical correlations involve the application of curve-fitting techniques to experimental data to predict an overall flow quantity. dure is an example of this prediction category. the friction plying the Fanning equation. the friction factor is inversely proportional to a numerical solution for the problem.4 0. In noncircular configurations.45 is established from suitable experimental data involving many noncircular and types of configurations. the mechanisms of turbulence are by no means fully understood. Note that K = 0 for pipe flow. This also corresponds to the analytical value for single-phase laminar MULTIPHASE FLOW IN WELLS . . The solution is obtained for Newtonian. and application of universal velocity distributions. 2. the following functional dependency for friction factor exists. even for circular pipes.and that for fully concentric pipes (e = 0 ) .Goldstein.55) Although an infinite-series term is involved in Eq. where the transport phenomena are intrinsically more complex than for circular pipes. .26 who presented Reynolds numbers. . 2. . 2. At intermediate values of Reynolds numbers. ECCENTRICITY In turbulent flow. . Three ways have been used to predict the flow behavior of a turbulent-flow field in an annulus: empirical correlations. Using dimensional analysis.However. .FEA approaches 24 as K approaches 1.2 0. . . Fig.O.553 I I I I the correlations of Knudsen and Katz. steady-state. where fGA is the Fanning friction factor and FCA is the friction-geometry parameter for a concentric annulus. Annulus Pipe Diameter Ratio. 6 Conservation of Energy Application of energy conservation to fluid flow in pipes requires that in a given pipe segment the energy in.62.3. 2. an approach analogous to regular pipe flow can be postulated.flow through parallel plates. Also. Concentric Annulus. which represents the expected friction-geometry parameter values (K= 0. minus the energy out. 2. consequently. Then.l4 > 0. 2.6. 2. plus the heat energy transferred to or from the surroundings must equal the rate of energy acc~mulation. To predict the frictional pressure gradient in an eccentric annulus. which is valid for eccentricities from 0 to 0. 2. the friction factor for concentric and eccentric annuli are predicted. Eq. The generalized Reynolds number concept also can be applied for an annular-flow geometry.64. for a high degree of eccentricity. and flow-behavior indices. the Blasius equation can be used to calculate the frictional pressure gradient.65 reduces to where. and This correlation has a claimed accuracy of f 5%. f = Fanning friction factor and F = laminar' flow friction-geometry parameter presented earlier. = 0. Then. 2. For a fixed-diameter ratio.63) developed for laminar flow with a specific value of d. the error involved in predicting friction-factor values in annulus configurations by applying only the hydraulic-diameter concept and neglecting eccentricity can vary between 40 to SO%. one might try to use the R correlation (Eq. Finally. The generalized Reynolds number for a concentric pipe can be used in the non-Newtonian pipe flow friction-factor correlations. 2. A correlating parameter. was defined as the ratio of frictional pressure losses in an eccentric annulus to those in aconcentric annulus. 2. recommended the Gunn and Darling24 procedure for turbulent flow because of its simplicity and good performance. n' = flow-behavior index. ~ sure gradient then can be obtained by use of the modified R for turbulent flow and a concentric annulus in Eq.553) for the experimental facility used by Caetano et aL2' where K' is a parameter related to the consistency index and is expressed by Eq.9. the friction-geometry parameter and.62.J a w o r s kThe~frictional pres~. respectively. The corrected value for an eccentric annulus is then determined from In these equations.29 The slot-flow approximation works well as long as d. with Eq. the friction factor decrease with an increase in the degree of eccentricity. This is demonstrated by the dashed line in Fig.3 to 0. n'. with a large amount of turbulent-flow data for many non-circular configurations. as suggested by ~ a s . Combining the Gunn and Darling developments into a Nikuradse-type expression.0. The relationship between true shear rate and apparent shear rate at the wall is expressed as For steady-state flow. The concentric annular flow of power-law fluids can be approximated closely by use of the less complex flow equations for a narrow slot. The generalized Reynolds number is The parameter J is the mechanical equivalent of heat and is necessary when dealing with customary units where mechanical energy and thermal energy have different units. pipe diameter ratios of 0. the frictional pressure gradient in a concentric annulus is first calculated with Eq. Caetano et al.lrl.01. Non-Newtonian Turbulent Flow. from Therefore. 2. the frictional pressure gradient for a concentric annulus can be calculated as Eccentric Annulus. consequently.95. For an eccentric annulus. of 0.45. is given by . The approach combines the behavior for friction factor in noncircular configurations. friction factor can be written as Newtonian Turbulent Flow.4 to 1. the frictiongeometry parameter and.60 for an annular geometry. There is no available exact analytical solution for non-Newtonian laminar flow in an annulus. depending on the annulus pipe-diameter ratio and the degree of eccentricity. as given by Eq. R. developed a numerical model to Haciislamoglu and ~ a n g l i n a i s ~ ~ analyze the flow behavior of yieldlpower-law fluids in concentric and eccentric annuli. .'~ Non-Newtonian Laminar Flow. the friction factor are always smaller than for a circular pipe. The empirical correlation. There is no documented model to predict non-Newtonian turbulent flow in an annulus. However. Haciislamoglu and Langlinais28 developed a correlation to predict frictional pressure losses of power-law fluids in an eccentric annulus. for a horizontal pipe. . is defined in terms of overall heat-transfer coefficient and temperature difference between the fluids and the surroundings.7. The surrounding rock gradually heats up.68 with Eq. dL dh dL .6. 2. heat transfer in this region normally can be considered steady state. Eq. Because of the high thermal conductivity and relatively small radial distance between the flowing fluids and the borehole wall. An axial cross section of a typical wellbore is shown in Fig. 2. . . . Heat transfer through the tubing and casing walls and through a cement-filled annulus between the casing and borehole wall primarily results from conduction.72 is used to determine temperature change when fluids flow through pipes. When hot reservoir fluids enter a wellbore and begin to flow to the surface. an increase in fluid enthalpy equals the heat transferred to the fluid from the surroundings. (2. . if no heat transfer occurs. 2.2 from conservation of mass principles yields where q is the amount of heat flowing radially through a solid with thermal conductivity.67 and 2. 2. 2. for a constantmass flow rate. Prediction of fluid temperatures in the wellbore as a function of depth and time is necessary to determine the fluid's physical properties and calculate pressure gradients.67. k. 2. Heat transfer resulting from conduction can be described by Fourier's equation in radial coordinateslS Fig. the kinetic energy term is negligible.1 Wellbore Heat Transfer.76 gives MULTIPHASE FLOW IN WELLS . where Because of the strong dependence of enthalpy and heat transfer on temperature. e is the intrinsic specific energy and is defined by Combining Eqs. the earth surrounding the well reaches a steady-state temperature distribution. Eq.66 yields pve + ddL( PlJ) + ( e+- d-b v ) p : l ) dL -end -- A In Eqs.. Q. 2. Heat transfer within the tubing and in a fluid-filled annulus is primarily a result of convection.~ permission of the McGraw-Hill Cos.J pvv + -g.72 clearly shows that the steady-state enthalpy-gradient equation is made up of three components. solving for the enthalpy gradient yields The heat flux. Integration of Eq. All heat lost by the fluids instantaneously flows through the wellbore and into the surrounding rock. Thus. 2. 2. . they immediately begin losing heat to the cooler surrounding rock. . 2.Qnd A . .Because specific enthalpy is defined as Eq.69 can be expressed as p ~ g sin8 g.J -+pv--- d..71) Finally. Also. The following description of steadystate heat transfer in a wellbore would have to be modified for other types of completions.66 and 2. Therefore. an increase in elevation causes a decrease in enthalpy and a corresponding decrease in temperature. 2. Eventually. .) Expanding the left side of Eq. reducing the temperature difference and the heat transfer between the fluids and the rock. Normally.7--Cross section of typical ~ e l l b o r e(Reproducedwith . Thus. ...90 and 2. . . The viscous force working against the buoyancy generates a circular motion of the fluid in the annulus. . requires application of conservation of mass. . . .(?) For conduction through the cement in the casinghorehole annulus.. . .95) A simple expression for the total heat loss from the fluids in the tubing can be estimated from Newton's law of cooling... .. 4 T.. . ..' "... . .35 Alves et presented approximate analytical solutions.. The product of /3 (coefficient of thermal expansion) and the temperature difference is a measure of the density difference.0025 q = 2nr. . . . . . Prediction of temperatures in wells and f ( t ) = 0.... .91 as simplified equations that.. . . . . For convection in the tubing. . . For convection through the casingltubing annulus. .. .405 where + 0. solution is valid for times greater than 1 week.. creating a buoyancy force.. (2... ..78 as follows. . 4 1 T . P ( ...T." """ . ..TT..rro) ~ P ~ . efficients..... = -2nhL ... the temperature difference across the annulus is usually small and convective (natural) heat transfer becomes important. but a numerical-solution procedure is given in Chap.. .. \ where T. (2..88) Hasan and ~ a b ishowed that for typical reservoirs Eq.. . (2.. 4 Tc. . q is constant. . A summary of the Alves et .. .....) T.... are valid for all times.. . (2.. .6.. Unfortunately. . r I..31The infinite-reservoir. . Their soluy the tion degenerates to equations presented by ~ a m e for ~ ~case of for injection wells and by Coulter and ~ a r d o n ~ ' the case of horizontal pipelines. . . 2. is a measure of the interaction between the hydrodynamic boundary layer and the thermal boundary layer and is defined as Heat transfer into the surrounding rock is by heat conduction and is a transient process. ...86) To monitor temperatures at the wellbore... .96) where U = an overall heat-transfer coefficient.. 2... . . However. The density of the heated fluid next to the tubing wall is less than the fluid next to the casing. . 2. . formulation was for flow in pipes at any inclination angle. The Prandtl number.5. . . ... .h. They presented Eqs. 2. ..hAT.... .72. it should yield more accurate predictions. . Sagar et a1.. .. ..95 and 2. .. .h.. .. 5 1. .. The transient radial-heat-conduction equation is identical to the diffusivity equation encountered in transient welltest analysis...Heat transfer resulting from radial convection can be described by l5 q = 2nrAL. . NG~ = (rc... .84 determines the total temperature change between the fluids and the undisturbed geothermal temperature of the surrounding rock..77 and 2.Tco = -. At). it is evident that (r.~ P :n 3 .5 In(?. 2. 0-is the bracketed term in Eq... .... N G . . 2nAL kc I... and Hasan and ~ a b i s ~ 3. . (2. the logarithmic approximation to the E. Expressions for temperature change through the wellbore can be developed from Eqs.. .96.. primarily pertaining to steam injection or producing wells. ... line-source solution is Combining Eqs. .. for x < 0. .. when used together. . in most cases of oil producr ~ ~ tion.2 Temperature Prediction. 2. .. ...32 Thus. the literature reports no work on natural convection in vertical annular geometry. and energy principles.. . .. The complexity of these equations prevents a rigorous analytical solution.. .. .. .. Np.. . .2.... . is given by L .. .95. .. . Comparing Eqs... .. ....79 through 2.. . Because the Alves et al. Although numerous other publications have appeared. . If steady-state heat transfer occurs in the wellbore.78) Hasan and ~ a b i also stated that.. 2. .. = --. .T. . . .... . . . .81) 2nAL r. . ... . (2.. .Tl0 = -&u k."" .). . For conduction through the tubing wall. 9 1 Tf . . . (2. .. . .. reflects the extent of motion of the annulus fluid resulting from natural convection.. (2... .85) and a =the earth thermal diffusivity defined as . . (2.&UAT.5 and 2. =the undisturbed geothermal earth temperature.. ...88 can r ~ ~ cause significant errors if applied to times less than 250 hours. Hasan and Kabir recommended using where h = local convective-film coefficient.80) where the Grashof number. .. .93) For conduction through the casing..82) . . . If t. The Alves et al. . This can be accomplished by coupling the pressuregradient and enthalpy-gradient equations given by Eqs. all are modifications to the Ramey method or to the calculation of heat-transfer cosolution follows.. . momentum...... ..15 k a =2 PC ' . method involves fewer restrictive assumptions. . (2. which is equivalent to the Ramey expression for incompressibleliquid flow.. s i n e ~ ( 1 e-LIA). h = h(p.. . 2.. dL Consider an isenthalpic process so that dh = 0 = CpdT + () "T dp. 2. .106 and 2... and neglecting acceleration effects. . The resulting solution is where w is in pounds per second. Eq.. typically varies from approximately 1. 2. ..0 to 2..108 is equivalent to the Coulter and E3ardon3' equation to predict temperatures in horizontal pipelines..96. where 9 = 0°.104 yields a generalized differential equation that incorporates both the enthalpy.. and neglecting acceleration effects.. and p~ is in pounds per cubic foot.... 2.0°F/100 ft of vertical depth. Thus.e -LIA).105 can be integrated assuming constant values for U.106 degenerates to Tf = (T. is in inches. depending on the thickness of the earth crust. Cp. It is first necessary to define several physical properties in terms of mixtures. gc.2.Tei)e-LIA sin 8 JgcCp .11 3 shows that the Alves et solution is actually the Ramey equation for single-phase liquid. 2. dt.Tei)e-LIA + 7 -A(I .97 and 2. Comparison of Eqs.109) . and where and Neglecting friction. 2.. (2. presence of volcanic activity. . TI = T.106 degenerates to () "T dp. Combining Eqs. g. Eq.103 simplifies to pg sin 0 where q = the Joule-Thompson coefficient and represents isenthalpic cooling (or heating) by expansion. . .99 and 2. 2. .O.106 degenerates to TI = (T. . Calculation of flowing temperatures as a function of depth and time can be very tedious because of the complexity of the overall heat-transfer coefficient in Eq.g.100 can be simplified to this differential equation.. a change in enthalpy can be calculated by considering effects of temperature and pressure separately. 2.. L sine) + (Ti .. .g G L sin 8) + (T. .T).. v. . 2. For the case of horizontal flow. . e ~ ~ ~ For the case of an incompressible liquid. 2. $.~ I ~ J If the surrounding temperature varies linearly with depth. and dp/dL..q..103 simplifies to = CpdT + and Eq. The correction term is a function of the total pressure gradient and the dimensionless parameter. q = 0. where T. ) ~ .. . 2. . .L ~ ( l - which is equivalent to the ~ a r n equation for an ideal gas. . . Analysis of this dimensionless coefficient can show when consideration of the correction term becomes important.T .. . Combining Eqs.106 is recommended when calculating temperature changes in wells flowing multiphase mixtures. The resulting equation is independent of time Eq. These mixture properties are discussed in Chap.99) .. . 2.Because enthalpy is a state property.and pressure-gradient equations with no limiting assumptions.. plus a correction term..e-LIA). Eq. .. Eq. . . . . Combining Eqs. The geothermal temperature gradient..Cfldp. Shiu and Beggs3* proposed an empirical correlation for A that was developed from a broad set of flowing-temperature surveys. and Eq. 9 =0. Eq. 2.. 2. and Eq. . d v / a . MULTIPHASE n o w INWELLS . and other such factors.106 degenerates to all the more restrictive approximate analytical expressions to predict temperatures of fluids flowing in pipes. For an ideal gas. .72 gives + g. . (2.. dp + (T. 3. Eq. . =surrounding temperature at the inlet of the pipe and is often taken as the reservoir temperature.. ..101 and 2. Thus.98 gives dh = CpdT . 488 ft. L at tDw = 7. Moody. Reynolds Number: = 19. = 0..144. (1908) 56. New York City (1975).4 Btu/hr-ft-OF 16 = 0.961 and 1041. . References I.534)(0. Determine heat-transfer coefficient using Eq. . L. McGraw-Hill Book Co.= 0. > 2.015) sin 75"(4.8)~ "*(54.. Drew. i200.000) sin7S0] +0 + (0..000. and Katz. . the Prandtl number is = 168.9 + 113.L.89. Determine A From Eq.F. Jr.000/4. = 0. -..829) = 4. McGraw-Hill Book Co. 3. r.0 . 15. Inc..488)(1 . 5.87 gives Example 2.O cp = 6. From Ref.783) i 1 - and hf = (482'3)(0'08) = 94.Because .: Fluid Dynamics and Heat Transfer.2.892112) 2. 3. = l ..112.72 X 104 Ibdft .783 ft. 1 . 2.. 67 1.015)(10. H.4"F.10.631)(2. Nikuradse.(0. 2.08 Btuhr-ft-OF Cp.0112)~ - 'w = 120. Inc.1 14).65 Btullbm-OF a. (4.: Forschungsheff (1933) 301. x = 1 = 4t. Koo.750 in. rw = 4. +0 + (0.O15"F/ft YAPI = 30°API L = 10. From Shiu-Beggs (Eq.0 149)(17.: Z Math. --.L. .783)(1 .8 gc = O. and McAdams.r < 0.04 ft2/hr go = 5.000)sin 75'1 = Therefore. and T. y..015) sin 75O(19.488) = 200 .892) 'W4(30)0 2608 X -' (0.: "Friction Factors for Pipe Flow. D. = [200. flow is turbulent. T.0149)(4.0 in. AIChE (1930) 28.G. calculate the wellhead temperature after the well has been flowing for 2 weeks.: Compte Rendus (1840) 11. Eq. ASME (1944) 66. 1 4( 120. 2.373)(123.e .. R. 2. 2. T. J.113 and the Sh~u-Beggs correlation for A. k.'0. Knudsen. and Ditsworth. 7.lrf t 2 0 ~ . Determine reservoir time function.0025.= j (0. 2. = 0. A singlephase oil flows up a tubing string that is cemented from top to bottom.113 and 2.96. Determine Twh from Eq. 2.04 ft2/hr)(2 weeks)(l68 hr/wk) (4.4-Temperature Prediction Comparison. T. = 2. = 200°F p..0021.0021) + 0. Uslng Eq. 6. W.428)(0. 4.C. Determine hf: The Dittus and Boelter equation39gives an acceptable correlation for the Nusselt number for turbulent flow (NRr lo4) in the tubing when cooling is taking place > f(t) = - 1 [ln(0. The following is given.: Trans..3 L Because NR.42 Btuihr-ft-OF k.6 Btufl. 4.102.794.015)(10. From Shiu-Beggs (Eas. = 25 Btuhr-ft-OF = 1. 5. Poiseuille.56. (1841) 12. E. Blasius. = 0..000 ft =t 75" ' 1.e .446 in..57721 = 2.L.96) - SINGLE-PHASE FLOW CONCEPTS 17 .H. 2.(0.000/19.77)' 5253(4.000 bbl/D TR= Tbh = c.. New York City (1958). Phys.0 . J. from Eq. A = (0.95. J.1 13. 8. No. f(t).sec r = 2.7)' 9303 = (0. Allen.: Fluid Mechanics.1 14).B." Trans. : "Fluid Flow and Energy Losses in Non Circular Conduits. 39.. Berkeley. Burington..F. of California. (1959) 5.: "Determination of Turbulent Pressure Loss of Non-Newtonian Oil Flow in Rough Pipes. 163. Hasan. Dallas." JPT(Apri1 1962) 427.D. Alhanati. 27.: Pub.. Texas (1967) 1. 1. and Reed. SPE. er al. Sagar." AlChE J. New York City (1973). Austin. Houston. Taylor Bubble-Rise Velocity and Flow-Pattern Prediction.8. 30. Coulter. 35. 11: "Coil Tubing Operations and Services-Part 4..: Handbook of Marhemafical Tables and Formulas.P. M.R. AIME. Brill. Trans. Chem. and Lightfoot. 10." Cdn.. I. Energy Res." AIChE J. C.2. A. (March 1980) 102.443.N.H.510. Hear Transfer (1961) 83. A. and Metzner.25.and Two-Phase Vertical Flow Through 0. and Smith." Oil & Gas J. Haciislamoglu. M.: "Axial Laminar Flow in a Circular Pipe Containing a Fixed Eccentric Core.J.: Pressure Buildup and Flow Tests in Wells.: "Non-Newtonian Flow inEccentric Annuli. third edition.: The Flow of Complex Mixtures in Pipes. AIME.B. F. and Goldstein. Szilas. A." PhD dissertation. A.. Winkler. Hasan. and Russell.. M. Jr. Snyder. Monograph Series. (June 1990) 112.: "Predicting Temperatures in Flowing Wells. Eng. and Boelter." J.F. U. of Texas.P.: "Predicting Temperature Profiles in a Flowing Well." World Oil (March 1992) 7 1. AIChE (1963) 41. Dittus. Richardson.A.. Shoham.S. Martinez. New York City (1960). 31. New York City (1972)." SPEPE (November 1992) 363.: "Flow of Non-Newtonian Fluids-<orrelation of the Laminar.T.and Kabir. and Kabir. Colebrook.J.K. 23. Govier.A. C.: "Turbulent Flow in Pipes With Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws. and Langlinais. K. andCharles.M. 13.280. Inc. C. W. Bourgoyne. and Beggs. D. Sas-Jaworsky. et al.. A.. Pan I: Single-Phase Friction Factor. 15. Oklahoma (I 99 1). M.625-Inch Fully Eccentric Plain Annular Configurations. Shiu. 20. U . L. 0. Cullender. R." J.E. fifth edition. J. El-Saden. Alves." J. 29." Trans. D.: "Heat Conduction in an Eccentrically Hollow. 38. H.S. Calgary (1975). P. H. 22." Rheologica Acta (1981) 20. and Bardon. 32.: "Practical Solution of Gas-Flow Equations for Wells and Pipelines With Large Temperature Gradients. N. 2-5 October. 207. Textbook Series.E.: Transport Phenomena. and Aziz. and Darling. Metzner. Trans. D. Texas (1968). (1965) 11. E." AIChE J.F. Eng. Texas (1991) 2.W. 24.R. John Wiley & Sons. Energy Res. 37. Tech.J. 133." J. Matthews.: "A Green's Function Solution for the Case of Laminar Incompressible Flow Between Non-Concentric Circular Cylinders. and Navratil.D. 17. Civil Eng. Transition. G. Caetano. 9.C. 225. Redberger. A. Bobok.. A. 21. 19.W. 18. J.: "Prediction of Dispersion Viscosity of O i W a ter Mixture Flow in Horizontal Pipes. 33." SPEPF (August 1994) 21 1. G. 189.996x0. D. and Schmidt. Tech.: Two-Phase Flow in Pipes. D. 12. and Brill. Franklin Inst.G. (26 February 1979) 107. 14. Theory and Practice of the Testing of Gas Wells. (1955 ) 1. (January 1959) 267.434. Tech. J.W: "Single. J.S.B. Infinitely Long Cylinder with Internal Heat Generation. 148. Energy Res.: Applied Drilling Engineering. 25.: "An Analysis of Fully Developed Laminar Flow in an Accentric. 5. Inst.S. No. 28. Dodge. Tulsa.. Gunn. Z.: '"Turbulent Flow of Non-Newtonian Systems. C.E.D. ( 1962) 40.W." SPEPE (November 199 1) 44 1 . MULTIPHASE now IN WELLS . Energy Resources Conservation Board. (1939) 11. Ramey. SPE. Dodge. 163. N." paper SPE 18221 presented at the 1988SPE AnnualTechnical Conference and Exhibition. Richardson. Zigrang. W.: "Upward VerticalTwo-Phase Flow Through an Annulus. McGraw-Hill Book Co. Jr. Bird. 11.W. Tech. 34. H. Doty. (June 1985) 107.' 36.C.: "Heat Transfer During Two-Phase Flow in Wellbores: Pan I-Formation Temperature. of Tulsa. Energy Res.J. R. Van Nostrand Reinhold Co.: "A Review of Explicit Friction Factor Equations.:"A Unified Model for Predicting Flowing Temperature Distribution in Wellbores and Pipelines." J.A: "Friction Losses in Annular Flow.: "Aspects of Wellbore Heat Transfer During Two-Phase Flow. California (1930) 2.B.R. 16.J.W. and Turbulent-Flow Regions. E.R. K.: "Revised Equation Improves Flowing Gas Temperature Prediction. J. E. L. R." JPT (December 1956) 281.." ASME PN (1964) 63-WA-11. J. 26." paper SPE 22866 presented at the 1991 SPE Annual Technical Conference and Exhibition. C. D." J.F. Heyda. and Sylvester. and Shoham.S. A. U. Stewart. (March 1992) 114. 6-9 October. R.M.. 0..V. F. M.P.462. and Beggs.: "Wellbore Heat Transmission. H...N." J. bounded by the dewpoint curve above and a curve below formed by connecting the maximum temperature for each liquid volume percent. During multiphase flow through pipes. friction factors.2 Phase Behavior A complex mixture of hydrocarbon compounds or components can exist as a single-phase liquid. For black oils with associated gas.. This parameter. Some were reviewed in Chap. and surface-tension forces vary significantly with flow rates. The pros and cons are presented for the use of empirical correlations based on dimensional analysis and dynamic similarity. Especially important is the significant variation in pressure gradient with flow pattern. Typical oil reservoirs have temperatures below the critical temperature of the hydrocarbon mixture. the accuracy of prediction was excellent. These have resulted in new predictive methods that rely much less on empirical correlations. Fig.e. Flow patterns are described in detail. However. Appendix B gives a summary of available correlations. 3. Several different flow patterns can exist in a given well as a result of the large pressure and temperature changes the fluids encounter. as for most gases. mass transfer occurs continuously between the gas and the liquid phases within the twophase envelope of Fig. the liquid often flows faster than the gas. a simplified parameter has been defined to account for gas that dissolves (condenses) or evolves (boils) from solution in the oil. inclination angle. The increased cornplexity of multiphase flow logically resulted in a higher degree of empiricism for predicting flow behavior. depending on the pressure. The shapes and ranges of pressure and temperature for actual envelopes vary widely with composition. a characteristic known as flow pattern or flow regime. and fluid properties of the phases. slippage between phases. Unlike a single component or compound. such as hydrocarbons produced from oil reservoirs. 3.1 Introduction 3. the gas and liquid phases normally do not travel at the same velocity in the pipe.1 gives a typical phase diagram for a multicomponent hydrocarbon system. However. turbulence. Analytical solutions are available for many single-phase-flow problems. The term black oil is a misnomer and refers to any liquid phase that contains dissolved gas.Chapter 3 Multiphase-Flow Concepts 3. or as a two-phase mixture. Even when empirical correlations were necessary (i. for turbulent-flow friction factors). inertia. 2. Fig. a phenomena in which condensation occurs during pressure reduction rather than with pressure increase.. Thus. MULTIPHASE FLOW CONCEITS 3. and other such parameters for multiphase flow in pipes. Expansion of the highly compressible gas phase with decreasing pressure increases the insitu volumetric flow rate of the gas. causing a phenomenon known as slippage. Perhaps the most distinguishing aspect of multiphase flow is the variation in the physical distribution of the phases in the flow conduit. the less dense. the flow behavior is much more complex than for single-phase flow. For upward flow. As pressures and temperatures change. since the mid-1970's. This chapter introduces and discusses basic definitions for flow parameters unique to multiphase flow in pipes. have gravities less than 40°API. more compressible. less viscous gas phase tends to flow at a higher velocity than the liquid phase.1 permits a qualitative classification of the types of reservoirs encountered in oil and gas systems. Dry gas reservoirs have temperatures above the cricondentherm. These oils are typically dark in color. such as water or carbon dioxide. This abnormal or retrograde behavior occurs in a region between the critical point and the cricondentherm. pipe diameter. Buoyancy. Because the black-oil model cannot predict retrograde condensation . Many condensate fluids exhibit retrograde condensation. for downward flow. temperature. - - l + - + When two or more phases flow simultaneously in pipes.1 Black-Oil Model. when two phases exist simultaneously a multicomponent mixture will exhibit an envelope rather than a single line on a pressurettemperature diagram.1.2. All attempts to describe mass transfer assume that equilibrium exists between the phases. 3. the ability to predict flow pattern as a function of the flow parameters is of primary concern. Virtually all the existing standard design methods rely on these empirical correlations. including methods available to predict their occurrence. and the composition of the mixture. Two approaches have been used to simulate mass transfer for hydrocarbon systems: the "black-oil" or constant-composition model and the (variable) compositional model. As a result. can be measured in the laboratory or determined from empirical conelations. Each is described in the following sections. a single-phase gas. R. Volatile oil and condensate reservoirs normally have temperatures between the critical temperature and the cricondentherm for the hydrocarbon mixture. dramatic advances have taken place that improve understanding of the fundamental mechanisms that govern multiphase flow. The phases tend to separate because of differences in density. the flow pattern that exists depends on the relative magnitudes of the forces that act on the fluids. and undergo relatively small changes in composition within the two-phase envelope. Many empirical correlations have been developed to predict flow pattern. A better description of the fluid system is a constant-composition model. Shear stresses at the pipe wall are different for each phase as a result of their different densities and viscosities. 000 Ibm/D. The following sections discuss each parameter. . With Eq. .7. single-phase-flow equations often are modified to account for the presence of a second phase.3) where Bg is derived from the engineering equation of state to be Bg = psrZT/pZrcTrc. &. .1 Weighting Factors. 323 Volumetric Flow Rates. 3.4) Appendix B gives methods to predict gas compressibility.6. The amount of gas that can be dissolved in water and the corresponding possible changes in water volume are much smaller than for gasloil systems. . If a detailed composition is available for a gasloil system. However. also has been defined to describe the shrinkage or expansion of the oil phase. From Eq. Appendix C provides a description of VLE calculations. = 5. this results in areMULTIPHASE FLOW IN WELLS . ." a VLE calculation will determine the amount of the feed that exists in the vapor and liquid phases and the composition of each phase. . Fig... c ~ c~ - W . 33. it should not be used for temperatures approaching the cntical-point temperature.. A second parameter.. For thecompositional model.1-Compositional-Model Flow Rates.q o . This is a result of the lower density and viscosity of the gas. When water also is present. . Appendix B gives methods to predict these properties. . . .. . When gas and liquid flow simultaneously up a well. oil density and other physical properties of the two phases can be calculated.. Under steady-state conditions. the higher mobility of the gas phase tends to make the gas travel faster than the liquid. (3. = 20( Ibm mole vapor ' )' ) and p .and qr = (q*. the nearly constant composition that results for the liquid phase and the increased computation requirements make the black-oil model more attractive for nonvolatile oils. At a given location in the pipe. Oil formation volume factor can be measured in the laboratory or predicted with empirical correlations given in Appendix B.. solution gaslwater ratio.5. This involves defining mixture expressions for velocities and fluid properties that use weighting factors based on either volume or mass fractions. R. it is possible to calculate the in-situ volumetric flow rates of each phase. . it also is possible to calculate the interfacial tension and densities. 3. 3.. If free water exists when the compositional model is used. they also are much more difficult to perform. called the oil formation volume factor. The choice of variables and weighting factors often depends on the predicted flow pattern. . it is possible to determine the quality or mass fraction of gas in the mixture..1-Typical phase diagram. For volatile oils and condensate . For the black-oil model. Appendix B also gives correlations for these parameters and physical properties of the water.792 Ibm vapor/lbm mixture. B. and water formation volume factor. .. can be defined. . completed.. O F where xg is the no-slip quality or gas mass fraction and is obtained from the results of a VLE calculation as follows. 3... Once the composition of each phase is known.. . - Example 3. the water flow rate must be added to q~ to account for all the liquid. Given the composition of a fluid mixture or "feed. volumetric flow rates are determined from 3 3 Definition of Variables . . From these results. enthalpies. a VLE calculation is performed on the gas composition. Dissolved gas is by far the most important factor that causes volume change.' phenomena. . . fluids. = 0. and viscosities of each phase. ... it is possible to generate black-oil parameters from VLE calculations.0 Ibm/ft3. ~ R ~ ) B ~ . volumetric flow rates are calculated from and Reservoir Temperature. VLE calculations are considered more rigorous than black-oilmodel parameters to describe mass transfer. 3 2 2 Compositional Model. Appendix C also gives methods to predict these properties. A gas-condensate well is flowing at a rate of 500. Once the black-oil-model parameters are known.. yielding mole liquid = 0'05( mole feed ML = 100( mole liquid Ibm ) ) mole vapor = 0'95( mole feed M. . . Oil volume changes occur as a result of changes in dissolved gas and because of the compressibility and thermal expansion of the oil. After mass transfer calculations are . Z. vaporAiquid equilibrium (VLE) or "flash" calculations are more accurate to describe mass transfer than black-oil-model parameters. . When performing multiphase-flow calculations.. . With Eq. (3.. . . . = 50 lbm/ft3 p. However.. The in-situ volume fraction of liquid. . The water cut.3. 3. Thus. or no-slip conditions. it is possible for slippage to occur between the oil and water phases.duced area fraction for the gas phase and an expanded area fraction for the liquid phase.83 ft/sec With Eq. When oil and water flow simultaneously in pipes. MULTIPHASE FLOW CONCEPTS . . . where q~is the sum of the oil and water flow rates for the black-oil model. Thus. With Eq. The difference in velocities of the gas and liquid phases can be estimated from a variety of equations developed from experimental data for a limited range of flow conditions. For the black-oil-model case. = 28 l scfISTBO d = 6. f based on in-situ rather than stock-tank flow rates. . . 3. For the case of equal phase velocities. Because of the slip between phases. A more common procedure is to develop empirical correlations to predict liquid holdup. . Example 3. The following is known from the pressure/volume/temperature (PVT) example problem given in Appendix B.14) I . Thus. .and space-averaged velocities for each phase can be calculated from a knowledge of the time.86 = 7. For all other cases. while the gas flows at a velocity greater than the mixture velocity. . .HL A slip velocity can be defined as the difference between the actual phase velocities. . . which assume a given phase occupies the entire pipe area. or is given by Eq.and space-averaged liquid holdup obtained from the empirical correlations. As described previously.8. in which the fluids exist as a homogeneous mixture. = 0. At a location in the tubing where the pressure and temperature are 1. with or without gas. . . .009 1 ft3/scf R. 3. B. With Eq. 3. 3. the individual phase velocities normally are quite different.10. . . .700 psia and 180°F.1.3. the water flow rate must be added to the oil or condensate flow rate to account for all the liquid. Because the no-slip liquid holdup can be determined rigorously. (3. is simply I -f. . 3. = 3. . slippage of gas past liquid results in larger in-situ liquid volume fractions than would be present if the two phases flowed at the same velocity. c i a p . Under steady-state-flow conditions. An oil well I S flowing 10. OiUWater Mirture. This type of slippage is normally small compared with the slippage that can occur between gas and liquid. it commonly is used as a correlating parameter to predict other multiphase-flow parameters.2 Velocities. are the phase velocities essentially equal. the liquid typically flows at a velocity less than the mixture velocity.3 Fluid Properties. 3.0 in. v. However. HL.3. Superficial velocities. 4 describes these in detail. the volume fraction of liquid in the pipe can be calculated analytically from a knowledge of the in-situ volumetric flow rates given in the previous section.000 scfISTBO or a gas-production rate of 10 MMscfm. . and A total or mixture velocity then can be defined as If there were no slip between phases. slippage will result in a disproportionate amount of the slower phase being present at any given location in the well. .97 + 3. . it is possible to calculate the density and viscosity of each phase (oil. If free water exists when the compositional model is used. = vsL + v.5 for the compositional model. and vg = "sg -' . slippage can be important when velocities are low. significant slippage can occur between the gas and liquid. . the oil fraction in a liquid phase is calculated from With Eq. . Liquid holdup can be defined as the fraction of a pipe cross section or volume increment that is occupied by the liquid phase. . A variety of methods have been used to define mixture fluid properties. = 1. . With Eq.197 bbVSTBO B. HL..000 STBOJD with a producing gasloil ratio of 1. Regardless of whether the black-oil or the compositional model is used. Thus. calculate the in-situ volumetric flow rates and superficial velocities of the liquid and gas phases. = 0.and slug-flow patterns. 3. oil and water properties are combined under the assumption that there is no slippage between the oil and water. gas) and the gasloil and gas/ water surface tensions.. Only for the cases of the highly turbulent. and for the compositional-model case when free water exists. especially in horizontal wells where stratified flow can occur. Assuming no slippage. such as HL. annular-flow pattern.2-Superficial Velocities: Black-Oil Model. both the gas and liquid would flow at the mixture velocity. Also calculate the mixture velocity and the no-slip liquid holdup. . . . .778 ft3/sec. for a broad range of flow conditions.12. must be determined from empirical correlations or mechanistic models. are important correlating parameters. A variety of other velocities are encountered in multiphase flow that pertain to flow mechanisms in specific flow Examples are bubbles that rise velocities of small bubbles and larger bullet-sha~ed often occur in bubble. dispersed-bubble-flow pattern and high-velocity. Time. . . water. Thus. MULTIPHASEWW IN WELLS . Although Eq. with inversion taking place at lower water fractions when oil viscosities are high. (3..2 to 0. Eq.. . 3. because this is the phase that predominates at the pipe wall where most of the friction losses occur. . .18 often is not valid for the viscosity of two immiscible liquids. such as oil and water. such as the dispersed-phase viscosity and the dropletsize distribution of the dispersed phase.. depending on whether HLorAL is used as the volumetric weighting factor. . by the use of surfactants. Al other empirical correlations use Eq. beginning with the classical pa. . The inversion point of an oiVwater mixture occurs at water fractions ranging from 0.. .3-Effect of oil viscosity on emulsion inversion.20 is used by the Hagedom and Brown9 correlation l described in Chap.3 then could be used to estimate the water fraction at which inversion takes place..2. 3. inversion can be accomplished at even lower water fractions. .. .23) Eq.18 is the most common way to treat the apparent viscosity of an oivwater mixture. Other factors. ..^-^ Furthermore.5. (3.H3. as shown in Fig 3. Numerous equations have been proposed to describe the physical properties of gasAiquid mixtures..2-Effect of water on emulsion viscosity.~ per by W ~ e l f i nwhich predicted an exponential increase in mixture viscosity with increasing water fraction. and . 4.. In general... when a Newtonian-fluid flow model is used. .. Gas/Liquid Mirture. 3. will yield an accurate friction factor from the Moody diagram shown in Fig. a > 1 0 WATER FRACTION Fig 3. .2 and Input Water Fraction Required To Invert Mixture Fig 3. and an energy loss requirement to invert the mixture that is not accounted for in the conservation equations used. .40 C 1 I I I I 1 - E 'r Laminar-Flow Regime Line \ =t 0 Q.21. An alternative approach is to treat the fluid as a Newtonian mixture with an apparent viscosity that....3 gives a plot that demonstrates this for experimental data from a variety of source^. 3. . The following expressions have been used to calculate multiphase-flow mixture densities8 pI = pLHL+ p g ( l . these equations are referred to as "slip" or "no-slip" properties. for the case of two-phase viscosity. momentum-conservation equation.. Fig. The viscosity of a dispersion or emulsion has been found to depend mainly on the determination of which phase is continuous. 3. .3 Studies have shown that Eq.22) p n = pLAL+ p B ( l . the viscosity of the liquid mixture can be several times greater than the oil viscosity when the continuous phase is oil but the water fraction is approaching the point where an inversion of the dispersion or emulsion will occur..19 has never been used in multiphase-flow design correlations. An even better alternative is to conduct flow tests on actual crudes and water to determine the rheological characteristics and the probable inversion point.24 contains the subscript k because it appears in the kinetic energy term for the specific case of a homogeneous-mixture.2. a more accurate method is to use the oil viscosity when oil is the continuous phase and the water viscosity when water is the continuous phase.8 and Although Eq. the energy loss appears as an additional pressure loss that is seen as an increase in apparent viscosity. .. . . ..3.. 3.A d . when used in the traditional Reynolds number expression. .3 For some oiUwater systems.. 2. also are important. Thus. Chap.3 This increase has been attributed to two different causes: a deviation from Newtonian behavior near the inversion point. Numerous studies have been published following this approach. . .3 Fig. The apparent liquid viscosity then will be governed primarily by the viscosity of the continuous phase. 2 presented a treatment of oiVwater mixtures as a non-Newtonian fluid. 3. . . In bubbly flow. Bubble flow is characterized by a uniformly distributed gas phase and discrete bubbles in a continuous liquid phase. (3. .Bubble Flow DlspersedBubbk. both as a thin film along the pipe wall and as dispersed droplets in the core. leaving a very thin liquid film flowing along the wall. chum flow. . . relatively fewer and larger bubbles move faster than the liquid phase because of slippage. it is necessary to predict the enthalpy of the multiphase mixture. Each unit is composed of a gas pocket called a Taylor b ~ b b l e . . . Annular Flow. For vertical flow.) phases and with a single liquid phase. The pressure-drop component caused by acceleration is normally negligible and is considered only for cases of high flow velocities. . The liquid slug. They differ in the manner used to calculate the three components of the total pressure gradient. Thus. The pressure drop caused by elevation change depends on the density of the two-phase mixture which is usually calculated with Eq. An oscillatory or alternating direction of motion in the liquid phase is typical of chum flow. A consensus exists on how to classify flow patterns. .. or for wells producing oil and water or crude oils with foaming tendencies.4. flowing-pressure gradients. and the equation can be written as ( ) ( )+ ( ) + ( ) = acc . .5 Flow Patterns Predicting the flow pattern that occurs at a given location in a well is extremely important. Some i n v e ~ t i ~ a t o r s lhave ~named annular l-~ flow as mist or annular-mist flow. . . shown schematically in Fig. Slug and chum flow are sometimes combined into a flow pattern called intermittent flow. Annular flow is characterized by the axial continuity of the gas phase in a central core with the liquid flowing upward. where the definitions for pfand vfcan vary with different investigators. . . . ~ ~ . Chum flow is a chaotic flow of gas and liquid in which the shape of both the Taylor bubbles and the liquid slugs are distorted.5.. causing no relative motion between the two phases.15 Fig 3. Neither phase appears to be continuous. + h. most investigatorslOnow recognize the existence of four flow patterns: bubble flow. The pressure-drop component caused by friction losses requires evaluation of a two-phase friction factor. Except for conditions of high velocity. In dispersedbubble flow. numerous tiny bubbles are transported by the liquid phase. and a film of liquid around the Taylor bubble flowing downward relative to the Taylor bubble. Churn Flow. and annular flow. Chap. 3. Bubble Flow.1 Flow-Pattern Classification in Wells. high-temperature wells.3 for single-phase flow can be modified for multiphase flow by considering the fluids to be a homogeneous mixture. Slug flow is characterized by a series of slug units. It is common to introduce a transition between slug flow and annular flow that incorporates churn flow. . 3. Many methods have been developed to predict two-phase. Slug Flow.1° (Reproduced with permission of American Inst. The continuity of the liquid in the slug is repeatedly destroyed by a high local gas concentration.4-Upward-vertical-flow patterns.x. the enthalpy of a multiphase mixture can be calculated from h = hL(l . Most VLE calculation methods include a provision to predict the enthalpies of the gas and liquid phases.x. of Chemical Engineers. . are described next. Consequently.) When performing temperature-change calculations for multiphase flow in wells.. .4 Pressure Gradient The pressure-gradient equation derived in Sec. . 3.. slug flow. . .25) 3.5--Flow patterns in upward vertical flowthrough a concentric annulus. carrying distributed gas bubbles. most of the pressure drop in vertical flow is caused by this component. At high gas flow rates more liquid becomes dispersed in the core. The Taylor bubble is an axially symmetrical. bridges the pipe and separates two consecutive Taylor bubbles. 3. bubble flow is further classified into bubbly and dispersedbubble flows. 6 = 90°. 4 describes these methods. . . bullet-shaped gas pocket that occupies almost the entire cross-sectional area of the pipe. Flow Front Back Slug Flow Churn Flow Annular Flow ~ubbld~low slug flow Churn Flows ~nnular Row Fig 3. Brill and ~ e summarized numerous ~investigations that have de~ ~ s scribed flow patterns in wells and that made attempts to predict when they occur. ~ ~ a plug of liquid called a slug.22. These flow patterns. . dL = dZ. Essentially all flow-pattern predictions are based on data from low-pressure systems. these predictions may be inadequate for high-pressure. . The empirical correlation or mechanistic model used to predict flow behavior varies with flow pattern. . . For upward multiphase flow of gas and liquid. 2. If enthalpies are expressed per unit mass. .. The interfacial shear stress acting at the corelfilm interface In the and the amount of entrained l ~ q u ~ d core are important parameters in annular flow.sin 6 = 1. . Based on the presence or absence of slippage between the two phases. with negligible mass transfer between the . which characterizes the bubble-flow pattern. on the order of 3 to 5 mrn diameter. resulting in gas expansion and the liberation of more solution gas from the oil phase. 3. further decrease in pressure and temperature occurs. Eqs. 3. Liquid accumulation near the pipe-wall contact point is an additional characteristic of annular flow in a fully eccentric annulus. followed by liquid slugs that bridge the entire cross-sectional area and contain small spherically distributed gas bubbles. This creates a chaotic two-phase flow.6-Flow patterns in upward vertical flow through a fully eccentric annulus. Comparison among flow patterns occumng in upward vertical flow in a pipe and in an annulus reveals that the existence of an inner pipe in the annulus changes the characteristics of slug flow and annular flow. This requires a thorough understanding of the mechanisms of each flow pattern. 3 6 Liquid Holdup . The large gas bubbles.3 discussed the interrelationships between gas and liquid phases traveling at different velocities (slippage). In a fully eccentric annulus. Further continuation of the upward motion of the flow into the region of lower pressure causes the expansion of Taylor bubbles along with the liberation of more gas from liquid slugs. The spherical bubbles are very small. and through the preferential channel. The liquid phase flows backward in the form of films. having a preferential liquid flow channel through which most of the liquid phase is shed backward.4 shows this progressive change of flow patterns. There is no change in flow characteristics observed with the annulus configuration. 3.6 show flow patterns in concentric and fully eccentric annuli.5 and 3. This causes a higher local void fraction relative to the cross-sectional average void fraction.2 Flow-Pattern Classification in Annuli. Figs. whereas the cap bubbles follow a straight path with a faster rise velocity. respectively. and 3. The outer film that wets the casing wall is always thicker than the inner film flowing on the tubing wall. Contrary to the concentric annulus case. The experimental data collected by Caetano et al. it experiences pressure decline resulting in the liberation of some of the gas dissolved in the liquid phase. forming an approximately homogeneous flow through the annulus cross-sectional area. They are discussed in Chap. This characterizes the existence of annular flow.5. The slug-flow pattern thus occurs. Chap. no symmetry is observed for the Taylor bubble with respect to either vertical or horizontal planes.19. one flowing around the tubing wall and one around the casing wall. regardless of the annulus geometry. The liquid flows upward. The second method to predict flow patterns considers the basic mechanisms that are important in causing a flow-pattern change. Sec. After identifying the different flow patterns. compared with the annulus-cap bubbles. tools are needed to predict their occurrence and flow behavior. 3.13 and 3. 3. The characteristics of chum flow are similar to those of pipe flow. and the values of various flow parthe transition between flow patterns were determined. for a fully eccentric annulus (see Fig. As the liquid flows upward. wetting both the tubing and the casing walls. Because of the presence of this channel. namely spherical bubbles and cap bubbles. Chap. Churn Flow. At high liquid velocities. Methods to predict the occurrence of the various flow patterns in wells have fallen into two categories.14 show the importance of liquid holdup in predicting velocities of each phase. 3.5.15 reveal that. The coalescence creates larger Taylor bubbles separated by the continuous liquid phase. partially as wavy films around the tubing and casing walls and partially in the form of tiny spherical droplets entrained in the gas core. 4. the flow pattern havingacontinuous liquid phase at the bottomhole can be completely transformed into a flow pattern with a continuous gas phase at the wellhead. Starting with a flowing bottomhole pressure above the bubblepoint pressure. defined earlier as chum flow. Thus. which are relatively larger but still always smaller than half of the configuration hydraulic diameter. 3. temperature. around the Taylor bubble. Because of continuous changes in pressure. This tends to create a high turbulent region behind the Taylor bubble.22. the mixture appears to flow at the same velocity with no slippage between the phases. The discrete bubbles occur in two different shapes. The Taylor bubbles do not occupy the total cross-sectional area because they have a preferential channel through which most of the liquid ahead of the bubble flows backward (see Fig. This creates more and larger bubbles that start coalescing with each other. These flow modifications seem to be a function of the pipediameter ratio and the eccentricity of the annulus.20. are similar to the ones occumng in pipe flow and also are ternled Taylor bubbles. The following is a description of a typical sequence of flow patterns in an oil wellbore. which probably happens as a result of the low local gas velocities. The gas is a continuous phase flowing in the core of the annulus cross-sectional area. Slug Flow.15 3. The upward movement of the small spherical bubbles follows a zig-zag path. Two films exist in the annular-flow pattern. Empirical flow-pattern maps were drawn that could be used to predict the transitions. 4 presents some of the more successful maps developed with this approach. Bubble Flow. there is a tendency for the small bubbles and the cap bubbles to migrate into the widest gap of the annulus cross-sectional area. it is essential to define the flow patterns in these configurations. This preferential channel exists from the top to the bottom of the bubble and from the tubing wall to the casing wall. Most early attempts were based on conducting experimental tests in small-diameter pipes at low pressures with air and water. The liberated gas appears as small bubbles in the continuous liquid phase.3 Flow-Pattern Occurrence.5). Fig. This flow is characterized by large cap bubbles of gas moving upward. Flow pat&metersat terns were observed. which occupy almost the entire cross-sectional area of the annulus.3. their characteristics can be substantially different. As the flow continues upward. 3. only a liquid phase exists at the bottom.Flow Bubble Flow Flow Anhular Flow Annular Flow. 4 presents mechanistic models for flow-pattern prediction.24 show the importance of liquid MULTIPHASE FLOW IN WELLS .5. This accumulation results from the merging of the casing and tubing liquid films.6) the preferential liquid channel always is located where the pipe walls are in contact. 3. The Taylor bubbles In an annulus are not symmetric. The chum flow continues to exist until the gas flow rate is sufficiently high to push the liquid against the pipe wall. The gas phase is dispersed into small discrete bubbles in a continuous liquid phase. 3.4 Flow-Pattern Prediction. although the same flow patterns described for wellbores occur in annuli. Fig 3.I6 used the best models for flow-pattern prediction and for flowpattern behavior. This approach is not restricted to a narrow range of flow parameters and has proved to be highly successful. Eqs. Ansari et al. and mass transfer between the two phases. where the liquid phase can be a hydrocarbon mixture. When quick-closing valves are used to measure liquid holdup. 3. These include laser-Doppler methods (light). they are unsuitable for use in field operations. 4 will show that Eq. it often becomes necessary to obtain and average many measurements to ensure accuracy and repeatability. Vertical upflow of a multiphase mixture is known to stabilize fairly rapidly. 180' out of phase.Fig 3. However. gamma ray or neutron absorption (nuclear). 3. such as Kerosene. and capacitance or resistance sensors (electricity). holdup in predicting the physical properties of gasfliquid mixtures. depending on the pipe diameter.7--Capacitance sensor. 3. The difference in capacitance between a pipe full of air or natural gas vs. one full of a hydrocarbon liquid is .3 discussed how the elevation component of the pressure-gradient equation for single-phase flow accounts for 80 to 95% of the total pressure gradient.22 is very important in predicting multiphase-flow pressure gradients. This approach prohibits the use of a pipe or fluid that will conduct electricity. especially for larger-diameter pipe. In general.6. The most accurate method for measuring liquid holdup is to trap a representative sample of gas and liquid in a section of pipe and physically measure the volume fraction occupied by the liquid phase.8-Nuclear densitometer. Gregory and attar'^ were the first to attempt to use capacitance several designs and concluded that use of sensors. and where the presence of water can be prevented. so a meaningful measurement can be obtained after the mixture has flowed about 100 pipe diameters from the pipe inlet. They also recommended specific dimensions and angles for the electrodes. especially for downward-inclined flow. many different methods have been tried. The same will be true for rnultiphase flow in vertical pipes. They eval~~ated two helical capacitor plates. This can result in a very timeconsuming process. Consequently. the minimum distance between valves is dependent on the flow pattern. an accurate prediction of liquid holdup is normally the most important parameter in calculating multiphase-tlow pressure drops in wells. The center ring was the positive electrode. Thus. Fig 3. Chap. the electronics required to process the capacitance sensor poses difficulties.20 further improved on the sensor and electronics design for capacitance sensors. Two of the more commonly used methods are discussed next.1 Capacitance Sensors. A representative sample is much larger for intermittent flow (slug and chum) than for other flow patterns because of the nonuniform configuration of each phase. 2. ped Kouba er al. They showed that measurements with this design were as accurate as those with the helical sensors. and the outer rings were ground-referenced. acrylic. Fig. the use of quick-closing valves in the field is not feasible. capacitance sensors cannot incorporate steel pipe or water as the liquid phase. As a result. Chap. Set. I X and ~ o u b a ~ ~ d e v e l o a different design in which the electrodes were three equally-spaced rings. ultrasonics (sound). For this reason. or glass pipe can be used. This is commonly done by use of two quick-closing ball valves that can be operated simultaneously. worked best. and that the overall length of the sensor was much shorter. the instantaneous liquid-holdup measurement would not be averaged over as long a pipe section. Capacitance sensors require the use of capacitor plates or rings to measure the dielectric constant of the two-phase mixture. Liquid-holdup measurements are required to develop and evaluate empirical correlations for predicting this important parameter. Butler et al. the entrance length for other pipe-inclination angles is often much longer. all of which rely on basic physics. Also. Because all slugs and Taylor bubbles have different lengths. i t is necessary to trap several of each to obtain representative averages. Thus. Thus.7 is a photograph of a capacitance sensor. They have been used extensively in laboratory multiphase-flow experiments where PVC. 4 presents several empirical correlations and mechanistic models to predict liquid holdup. to measure liquid holdup continuously. very small.26 varies throughout the pipe length. This is true for multiphase mixtures and even for crude oils above the bubblepoint. and interfacial-surface tensions at the guessed average pressure and temperature in the increment. 3. As aresult. Thus. Kouba et al. In addition. Brill et used nuclear densitometers or gamma-ray absorption sensors to determine multiphaseflow behavior in large-diameter pipelines in the Prudhoe Bay field of Alaska. Gamma rays are potentially dangerous. 3. it is necessary to divide the pipe into calculation increments within which the pressure gradient can be considered constant. 2. typically on the order of 10 Hz.9 could be sufficiently long that there is a significant change in pressure gradient within a segment because of changes in such variables as densities. Segments in Fig. often CS'~'. All computer prog.2 Nuclear Densitometers. ordinary differential equations in the variables p and T. They pass through the pipe wall. MULTIPHASE FLOW IN WELLS . where a pump is located. or even flow pattern.105 constitute an initial-value problem with two coupled.This limitation can be rectified partially by the use of multiple-beam gamma densitometers. the measurement of liquid holdup can be sensitive to the location of the liquid in the pipe. Thus. it is necessary to integrate the pressure gradient over the total length of the wellbore. the measured lengths of rapidly moving slugs and bubbles may not always be as accurate as desired.6. 3. oressure-traverse calculations use what is often called a "marching algorithm. Appendices B and C provide procedures to perform these calculations. For oil and gas wells. and inclination angle. 3. with a new segment resulting whenever a significant change occurs in inclination angle.72) be coupled by use of a numerical algorithm.7 Pressure-Traverse Computing Algorithm When performing pressure-drop calculations. Thus. 3. and determination of the enthalpy of each phase anithemixture enthalpy at the end of the increment. showed that dynamic calibration of the sensors was necessary because the capacitance measurement also was dependent on the location of the liquid phase within the sensor. Each of these media will absorb some of the gamma rays. the total pressure drop is \\ n = number of segments m = number of calculat~on Increments tn a segment Fig.for a guessed value of pi+]. Nuclear densitometers suffer from several disadvantages. 3. In spite of these deficiencies. 3. Note that the numerical integration of the enthalpy-gradient equation involves a trial-and-error solution for z+. or nodes. 26 For compressible and slightly compressible fluids. such things as changes in humidity or minute concentrations of water in the liquid phase can cause significant measurement errors. through the fluids in the pipe. They used a pair of densitometers to measure the liquid holdup and the lengths of slugs and bubbles. or where the pipe diameter or configuration changes. numerically integrating the pressure-gradient and enthalpygradient equations over short increments of pipe. Fig. Fig. even for a pipe segment. start at either the top or bottom of a well and "march" to the other end. A change in mass flow rate would occur when a well is gaslifted or when production from different zones is commingled. superficial and in-situ velocities.ams for performing steady-state.is contained in a lead-shielded cylinder for safety reasons to prevent the escape of gamma rays. It also involves calculation of an average enthalpy gradient. the electrical response is directly proportional to the liquid holdup. depending on the size and type of radioactive source and the method used to count the gamma rays.6. Because the major variable involved is the amount of liquid in the cone of gamma rays at a given time and the liquid phase absorbs more gamma rays than the less-dense gas phase. these devices have been used to acquire data in many different field and laboratory tests. Eqs. multivhase-flow. the pressure gradient from Eq. The response time of the electronics is fairly long.5 and 2. The PVT-calculation box involves the determination of mass transfer between the phases and the densities. The number of gamma rays that successfully pass through all media can be counted and converted into an electrical signal. in part because the temperature also varies throughout the wellbore. The source. New segments. 2. Approximate analytical solutions for integrating the enthalpy-gradient equation normally are adequate and simplify the calculations significantly. although adequate measurements can be obtained with relatively small sources of radioactive material. nuclear densitometers have experienced increased use in the field. temperature.9-Segmenting typical wellbore. Pressure-drop or pressure-traverse calculations require that both the pressure-gradient equation and the enthalpy-gradient equation (Eq. 2. depending in part on the density of the material. a cone of gamma rays is emitted. Nuclear densitometers use a radioactive source that emits gamma rays. Since then. The approximate analytical solutions to the enthalpy-gradient equation shown in Fig. 3.^ and &reto be calculated. When the top of the cylinder is removed.8 shows a nuclear densitometer. and through any insulation or other coatings around the pipe.9 shows a typical deviated wellbore that has been divided into five pipe segments. viscosities. also would be required where there is a change in mass flow rate. A complete traverse is calculated by sequentially marching through all increments in all segments of the well. respectively. The pressure drop then is determined by Fig. Also.10 were covered in Sec." To solve this initial-value problem. 3. Perhaps their greatest drawbacks are safety and governmental regulation.10 gives a flow chart of the "marching algorithm" for a single pipe calculation increment for which pi and 5 are known and^^. They are a nonintrusive instrument that can be attached to the outside of steel pipe and provide fairly accurate measurements of the volume fractions and flow patterns of the fluids inside the p~pe. the pressure gradient throughout the piping system will vary with pressure. " MS thesis. J. Charles. Englewood Cliffs. and Reynolds numbers.A. Cdn.: "Analysis ofTwo-Phase Flow Tests in Large-Diameter Flow Lines in Prudhoe Bay Field.48.: "Water Layer Speeds Heavy-Crude Flow. 11.24 w h e n designing laboratory experiments. Aziz. Kouba. and Taylor. U. B.68." Drill.: FluidMechnnics.: "Dynamic Calibration of Two Types of Capacitance Sensors Used in Measuring Liquid Holdup in Two-Phase Intermittent Flow. Brill. and Brill.443. Per. (April-June 1973) 12. of Tulsa.. Prac.: "Ratio-Arm Bridge Capacitance Transducer for Two-Phase Flow Measurements." J. er 01. Bamea. Jr.E. G. (1939) 11. Dumitrescu. K. D.: "Pressure Drop in Wells Producing Oil and Gas..D. and Fogarasi. Chem. 22. 375. (26 August 1961) 59. A. J. Trans.: "A Comprehensive Mechanistic Model for TwoPhase Row in Wellbores. (1942) 148. U. Woelfin." paper presented at the 1995 ISA Intl. K. Inc.and Brill. 18. (1943) 23.E." SPEJ (June 1981) 363.. Russell. Prentice-Hall Book Co. & Prod. Dittus. G.M. 2. Taylor Bubble-Rise Velocity and Flow-Pattern Prediction. second edition.: "Horizontal Pipeline Flow of Mixtures of Oil and Water."J.: "Turbulent Flow in Pipes With Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws. E. Berkeley. 16.1&Marching algorithm for a calculation increment.. Marh. 4 gives methods for calculating the pressure gradient at the guessed average pressure and temperature in the increment.L THAT FROM FIND T.8 Dimensional Analysis The concept of using dimensional analysis and dynamic similarity has been very successful in designing experimental test facilities and analyzing data. H.D. Chen.P. (July-September 1972) 11.E. 4. Tulsa. Trans. Oklahoma (1979). Instrumentation Symposium.: 'The Viscosity of Crude-Oil Emulsions. 1. Tech..P.F." SPEPE (November 1990) 373. McGraw-Hill Book Co. 13.R. involves a trial-and-error calculation that will match pressure and temperature conditions at a given node from both the upstream and the downstream system components. ---(GUESS \I .W. Tokyo (1963) 451." Cdn. and Ditsworth. U.W.. 6 gives a detailed analysis of production-design calculations. Insr. 12.. h L ( l I)+ hpx 1 CALCULATE dp1dL PI + 1 =PI * (dP/dL)L Fig. G.. Hagedom. X." Proc. 17.F. 133. and Water Fraction on Horizontal Oil-Water Flow.345.M. Many of the accepted equations used in singlephase fluid flow and in heat-transfer calculations have been develk~~ oped with these concepts.CNCUUTE T..+. Eng. 3. Mixture Velocity. Colorado.K. and liquid holdup.: "An Experimental Study ofTwo-Phase Oil-Water Flow in Horizontal Pipes. J. (1980) 26.. Eng. K.M. often called NODAL^^ analysis. it is not surprising that dimensional analysis also has been used extensively to develop predictive methods for multiphase flow. Butler.Seattle (May 1986) 479. of California. Sixth World Pet. I I BY APPROXIMATE SOLUTloNS " dL +m. (March 1992) 114. J. This is certainly true for multiphase flow in wells. . U. Part I: Single-Phase Friction Factor. Colebrook.-h * * T . 234.+. 15. MULTIPHASE FLOW CONCEPTS References I . A. 14. Energy Res. 4 also includes mechanistic models based on more fundamental concepts. R.E. h = .I. 5. Per. Craft. of Tulsa. dimensionless analysis is used when there is insufficient understanding of the basic fundamentals that govern physical phenomena. Tech. Because of the increased complexity of multiphase flow over single-phase flow. R.: "A Nonintrusive Flowmetering Method for Two-Phase Intermittent Flow in Horizontal Pipes.." Oil & Gas J.. Gregory.: 'The Mechanics of Large Bubbles Rising Through Extended Liquids andThrough Liquids in Tubes. Shoham. A.: AppliedPerroleurnReservoir Engineering.. T... G.9." MS thesis. W. Hodgson. 7-1 1 May.W. Guzhov. L. The process involves preparing a list of all variables that might be important and using them to develop a collection of dimensionless groups following a procedure known as the Buckingham pi theorem. Duns and Rosll performed a dimensional analysis of multiphase flow through wells that is presented in Chap.: "Modeling Flow Pattern Transitions for Steady Upward Gas-Liquid Flow in Vertical Tubes. Duns. Arirachakaran. AIME. CALCULATE d d d L and dNdl n. Prandtl. er al. Jr. and Brown. 10. Chap..M.T. Chap. The Dittus and Boelter23 equation for convective film coefficient uses the Nusselt. InstrumentationSymposium." ZAgnew. J. Oklahoma (1983).: "Stromumg an Einer Luftblase im Senkrechten Rohr. and Boelter. eral. Y.P. and Hawkins. Caetano.C. J. M.C. Tulsa. 19. Depending on the analysis performed.W. 0.. F. Aurora. 8 . and Brill. AIME. 20. 139. 6. One example is the C o ~ e b r o o equation for determining friction factor. and Govier. and Mattar. Ansari. Tech." J . L.F.. . 4. MATCHES h. H. 297. Civil Eng.. 4 includes many of the early and widely accepted empirical correlations to predict flow pattern. 23.L. Chap. New York City (1975)..P. G.: "Upward Vertical Two-Phase Flow Through an Annulus.. Oklahoma (1991). which uses Reynolds number and relative roughness as dimensionless groups.W. Davies. London (1949) 2OOA. It is not uncommon for initial predictive methods to rely heavily on empirical correlations based on dimensionless groups." J. 9. The dimensionless groups they developed have been used extensively by many other investigators when developing empirical correlations." SPEPF (May 1994) 143. friction factors. 0.F. Commercial computer programs that perform production-systems-analysis calculations link the "marching algorithm" with reservoirs by use of inflow performance relationships that also account for various well-completion parameters. G.: "An In-Situ Volume Fraction Sensor for Two-Phase Rows of Non-Electrolytes. and Beggs. New Jersey (1991) 7. Cong.38." AIChE J. Mech. Inc." Proc. 3. N.: "Vertical Flow of Gas and Liquid Mixtures in Wells.E. Cdn. Chap." Proc. use of dimensionless groups helps ensure that small-scale tests can "scale up" to field applications.J. 7. 3.58 (in Russian). Brill.: Tie-Phase Flow in Pipes.J." JPT (April 1965) 475. California (1930) 2. T.: "Emulsion Formation During the Flow of Two Liquids in a Pipe. calculations may be required in both upstream and downstream directions..: "An Experimental Study on the Effects of Oil Viscosity. Kouba. Taitel. D. C.T. R. 32nd ISA Intl. and Dukler. 24. of Tulsa.: Pub. G. Tulsa..: "Experimental Study of Pressure Gradients Occurring During Continuous Two-Phase Flow in Small-Diameter Vertical Conduits. Oglesby. 21. In general." NeJi Khoz (August 1973) 8. M.. Shoham. only to be replaced by more fundamental equations as technology improves.. A. (1957) 37. M. S. Govier. Royal Soc. and Ros. Allen. Production systems analysis.P. . a computer program based on a method that incorporates some basic mechanisms but also permits the adjustment of empirical parameters to fit available pressure measurements.5) = 105.. Determine friction factor: p.35 = 27. Calculate the Vertical. Given: p.500-ft-deep vertical experimental well.001 I 1 . and 110 cp.2. ' Liquid velocity number: = 27..5 in.0068. Fig.1-i ." Three methods are presented in this category. Hagedorn and Brown used four dimensionless groups proposed by Duns and ~ 0 s . and 3.61 Ibm/ft3 and p. and ? Fancher and Browa3 Because the numerator of the Reynolds number is not dimensionless... Ibrn/ft-sec Fig. = 47.. Rather. and four different liquids were used: water and crude oils with viscosities of about 10. nominal diameters were used..500. A liquid-holdup value must be determined to calculate the pressure-gradient component that results from a change in elevation.1 shows the friction-factor correlations developed for the methods of Poettmann and Carpenter.O. 2. Thus.83)(0.v. = 5. the three curves for gaslliquid ratio in the Fancher and Brown correlation can be considered to represent gadliquid ratios of 1. Fancher and Brown3 GLR > 3.04)(7.2-Hagedorn and Brown4 correlation for IfL&. A no-slip condition characterizes the dispersed-bubble-flow pattern.. Tubing with 1.. permitted the calculation of pseudo liquid-holdup values for each test to match measured pressure gradients...25..1--Category "a" friction-factor correlations.. 4. 00 ..250.500 0. 1 0 100 lo o0 4 Fig.88 lbm/ft3. The Asheim6 method uses MONA.000 scf1STBL. f = 0. Determine no-slip mixture density: a specialized one developed for use with vertical gas wells that also produce condensate fluids and/or free water.. The Gray method5 is These groups can be modified to include constants that make them dimensionally consistent when common oilfield units are used. . 2. 4.86 lbm/(ft-sec) Gas velocity number: From Fig.. These data represent some of the most extensive large-scale tests ever reported..500 < GLR c 3..1-Using the Poettmann and Carpenter Method. Example 4.1 0..C b .. after assuming a friction-factor correlation. They can yield satisfactory results only for high-flow-rate wells for which the flow pattern would be dispersed-bubble flow. However. units must be specified for the abscissa in Fig.39 psflft Liquid viscosity number: Category 'b.86 to 110 cp . Fancher and Brown3 3 0 .. To correlate the pseudo liquid-holdup values.04 lbm/ft3.. Fancher and Brown3 1..1. 4. Category "a" methods no longer should be used to predict multiphase-flow pressure gradients in wells. Hagedorn and ~ r o w is ~ generalized method developed for a broad n a range of vertical two-phase-flow conditions.10 - Poenman and Carpenter1 Correlatron based on vlscoslties of 0.. 1. Multiphase-Flow Pressure Gradient for Example 3.C I: .000 GLR < 1.d = (27. Thus..' Baxendell and ~ h o m a s . The Hagedorn and Brown4 method is based on data obtained from a 1.. pvd. 3. Determine total pressure gradient: Pipe diameter number: = 27. Hagedorn and Brown developed this presssure-gradient equation for vertical multiphase flow Liquid-Holdup Prediction.000 0 .. they developed a pressure-gradient equation that.30. 1. 4. .. Air was the gas phase. the values used to develop a liquid-holdup correlation were not true measures of the portion of pipe occupied by liquid. For interpolation purposes. those units are in IbmJft-sec. 4. Baxendell and Thomas2 1 -. and 1.04 + 0. it is important to recognize that Hagedorn and Brown did not measure liquid holdup. Hagedorn and Brown Method.1. ( ~ ~ ~ l $ 380)/@14 Fig. 4.3-Hagedorn and Brown4 correlation for N L ~ . Fig. 4.4-Hagedorn and Brown4 correlation for p N,, = 1 . 9 3 8 ~ ~ ~ &, hL and where VSL is in feet per second, vsg is in feet per second, pL is in pounds per cubic foot, o i s in dynes per centimeter,^^ centipoise, and d is in feet. Fig. 4.2 shows the correlation for liquid holdup divided by a secondary correction factor, q. The abscissa requires a value of NLc, which is correlated with NL in Fig. 4.3. Fig. 4.4 shows the secondary correction-factor correlation. Once a value for liquid holdup has been determined from Figs. 4.2 through 4.4, the slip density can be calculated from Eq. 3.22. Friction-Factor Prediction. Hagedorn and Brown assumed that two-phase friction factors could be predicted in the same way as single-phase friction factors. Thus, f is obtained from the singlephase Moody diagram in Fig. 2.2 for a given relative roughness and a two-phase Reynolds number defined as where& is defined in Eq. 3.20. Accelerarion Term. The pressure gradient resulting from acceleration is given by Modificarions. Over the years, several modifications have been proposed to improve the Hagedorn and Brown pressure-gradient predictions. In conversations with Hagedorn, he suggested the first modification when he found that the correlation did not predict accurate pressure gradients for bubble flow. He suggested that, if the Griffith and wallisI3 criteria predicted the occurrence of bubble flow, the Griffithl%buble-flow method should be used to predict pressure gradient. This approach is part of the Orkiszewski8 method, presented later. The second modification is much more significant and is possibly a result of Hagedorn and Brown correlating pseudo liquid-holdup values rather than measured ones. Liquid-holdup values obtained from Figs. 4.2 through 4.4 are often less than no-slip holdup values. This is physically impossible because, for upward multiphase flow, liquid cannot travel faster than the gas phase. When this incorrect prediction occurs, the liquid-holdup value must be replaced with one that is physically possible. The approach normally used is to proceed with the no-slip liquid holdup. No attempt has been made to investigate whether this problem is peculiar to a specific range of variables. The third common modification is to use the Duns and ROS' rnethod, if their mist- or transition-flow patterns are predicted to occur. There is no physical justification for this modification. It is used primarily to make the entire Hagedom and Brown method a Category "c" method. Because the Duns and Ros method is presented later in this section, it is not described here. A fourth modification often used is to replace the acceleration presssure-gradient component given by Eq. 4.8 with that proposed by Duns and Ros. Eq. 4.8 was not used by Hagedorn and Brown to analyze their data and is known to overpredict acceleration affects. In summary, the modified Hagedom and Brown method is a combination of several methods. The original Hagedorn and Brown method is used only for the slug region, as predicted by Griftirh and Wallis and Duns and Ros, and then only if a physically possible liquid-holdup value is found. where Example 4.2-Using the Modified Hagedorn and Brown Method, Calculate the Vertical, Multiphase-Flow Pressure Gradient for Example 3.2. Given: ,uo = 0.97 cp, uo = 8.41 dyneslcm, ,ug = 0.016 cp, and E = 0.00006 ft. 1. Determine Duns and Ros dimensionless groups: and 1, 2 designate downstream and upstream ends of a calculation increment, respectively. If Ek is defined as the total pressure gradient can be calculated from MITLTIPHASE FLOW IN WFI 1.S 5. Determine friction factor: From Eq. 3.19, N,, = 1 . 9 3 8(PL)% ~~~ = 0.13 cp. From Eq. 4.7, and and FromFig. 2.2 or Eq. 2.17. f=0.0135. 6. Determine pressure gradient, neglecting kinetic energy effects: From Eq. 4.2, 2. Determine liquid holdup: From Fig. 4.3, NLC = 0.0024. From Fig 4.2, the abscissa, a, is 7. If calculations had been made with the incorrect liquid-holdup value of 0.3, the pressure gradient would have been 0.14 psilft. Gray Method. Gray developed a method to determine the pressure gradient in a vertical gas well that also produces condensate fluids or watec5 A total of 108 well-test-data sets were used to develop the empirical correlations. Of these data sets, 88 were obtained on wells reportedly producing free liquids. The authors cautioned use of the method for velocities higher than 50 ft/sec, nominal diameters greater than 3.5 in., condensateAiquid loadings above 50 bbl/MMscf, and waterniquid loadings above 5 bbl/MMscf. Gray proposed this equation to predict the pressure gradient for two-phase flow in vertical gas wells. From Fig. 4.4, the abscissa, a. is Liquid-Holdup Prediction. From a dimensional analysis and selected laboratory tests, three dimensionless groups of variables were selected to correlate liquid holdup. :.* and = 1.0 and 3. Check validity of HL: Because HL <AL, set HL =AL = 0.507. 4. Calculate slip density: Pr = PLHL + P J ~ HJ - The resulting correlation developed to predict liquid holdup was given as MULTIPHASE-FLOW PRESSURE-GRADIENT PREDICTION where Eq. 4.25 gives the functional relationship assumed between the gas- and liquid-phase velocities. in Gray suggested this expression f o r u ~ Eqs. 4.12 and 4.13 when both condensate and water are present. Using Eqs. 4.22 and 4.25 to eliminate the gas- and liquid-phase velocities in Eq. 4.24 gives [ ( v *+ ~ ' V S L - a,) 2 0.5 +Q,U?V~~] Gray stated that the liquid holdup in condensate wells often tends to be smaller than in oil wells producing at comparable gas~liquid ratios. This probably arises from natural time lags in the flashing and condensing processes, and also from the lower average interfacial surface tension in condensing systems compared with flashing Two limiting cases must be defined to make the holdup expresblack-oil systems. The formulation of Eq. 4.15 is such that the liquid sion complete. When the constant-slip term, a>,(buoyancy) is zero, holdup is ordinarily very near to the no-slip holdup. the holdup becomes Friction-Factor Prediction. The effect of liquid holdup on friction loss can be interpreted as a variation in wall roughness from that ordinarily experienced in single-phase, dry-gas flow. Gray proposed that friction factors for wet-gas wells were wholly dependent When the liquid superficial velocity approaches zero, the flow situaon a pseudo-wall-roughness factor because flow is normally in the tion may correspond to gas bubbling through stagnant liquid. The fully-developed turbulent region. Friction factors can be obtained holdup for this case is found by rearranging Eq. 4.26 to obtain from the Moody diagram (Fig. 2.2) or from Eq. 2.16 for completely turbulent flow. The pseudo-wall-roughness, E , was correlated with a modified Weber number, similar to that used in the Duns and ~ 0 s ~ method for mist flow. Defining a roughness variable, Friction-Factor Determination. This two-phase friction-factor correlation was used in Eq. 4.21. then. for R r 0.007, where p, is defined in Eq. 3.24. The wall friction factor,&, is determined from Fig 2.2 for a no-slip Reynolds number defined by and, for R < 0.007, E~ is the absolute wall with the restriction that E 22.77 x roughness for single-phase gas flow. It is noteworthy that the correlation finally adopted includes both the effects of unusually high apparent roughness observed in certain laboratory test conditions and the effects of low apparent roughness in certain field systems. Parameter Values. For slug flow, the normal values of the three Iparameters are a1 = 1.2, a? =0.35 dgd, and a3 = 1.0. For homogeneous flow the normal values of the three parameters are a1 = 1.0, a2 = 0, and a3 = 1 .O. Asheim showed that, by selecting either slug flow or homogeneous flow, the three parameters can be adjusted for an optimized fit of field data. Category "c." The methods considered in this category differ in how they predict flow pattern and how, for each flow pattem, they predict liquid holdup and the friction and acceleration pressure-gradient components. For vertical flow of a homogeneous slip mixture, Eq. 3.26 can be expressed as Asheim Method. The ~ s h e i m ~ method, or MONA, is independent of flow pattem, but does permit the selection of three empirical parameters for either the bubble- or slug-flow patterns. Once selected, however, these parameters must be used for all multiphaseflow calculations. The method can be used in conjunction with a history-matching procedure to adjust the three empirical parameters and minimize computation errors. The pressure gradient is determined from Liquid-Holdup Determination. The gas and liquid velocities can be determined from and Eqs. 4.22 and 4.23 can be combined to give Duns and Ros Method. The Duns and Ros7 method is a result of an extensive laboratory study in which liquid holdup and pressure gradients were measured. About 4,000 two-phase-flow tests were conducted in a 185-ft-high vertical-flow loop. Pipe diameters ranged from 1.26 to 5.60 in. and included two annulus configurations. Most of the tests were at near-atmospheric conditions with air for the gas phase and liquid hydrocarbons or water as the liquid phase. Liquid holdup was measured by use of a radioactive-tracer technique. A transparent section permitted the observation of flow pattern. For each of three flow patterns observed, correlations were developed for friction factor and sllp velocity, from which liquid holdup can be calculated. Duns and Ros performed the first dimensional analysis of twophase flow in pipes. They identified 12 variables that were potentially ~mportant the prediction of pressure gradient. Performing a diin mensional analysis of these variables resulted in nine dimensionless groups, or n terms. Through a process of elimination, four of the MULTIPHASE FLOW IN WELLS 32 6. (4'36) MULTIPHASE-FLOW PRESSURE-GRADIENT PREDICTION Fig. as shown in Fig. . The correlations for S are different for each flow pattern and are given later.34 for the slip velocity. 4. Bubble flow exists if Ngv < NSvBIS. . . Calculate the elevation component of the pressure gradient. rather than for liquid holdup. Flg. Bubble Flow. The slip velocity was defined by Eq. 4. 4. . . 4. Slug/transition boundary: N8 = 50 + 36NL. we will refer to Regions I through III as bubble. Calculate the slip density from Eq: 3. 3. 4.6-Duns and Ros7bubblelslug transition parameters.5 illustrates the actual flow patterns existing in each region. Duns and Ros also identified the heading region as a fifth region. . 4. F. S. and mist flow. . yields a quadratic equation for obtaining HL. groups were identified as being important and were used to select the range of variables in the experimental program.4.15 and. can be obtained from Transitionlmist boundary: where F3 and F4 also are obtained from Fig.13 and 3. 4.32b) where F I and F2 are given in Fig. The flow-pattern transition boundaries are defined as functions of the dimensionless groups Ng.36. . S is defined in a similar way to the gasand liquid-velocity numbers given by Eqs.22. 4. They are functions of the liquid velocity number.7. . They identified four separate regions for computation purposes.7.'"gv Fig.. these transition boundFor aries. Duns and Ros chose to develop empirical correlations for a dimensionless slip-velocity number.5 shows the flow-pattern map developed by Duns and Ros. when combined with Eqs. 33 . using the appropriate correlation. Fig. . NL. but this is now considered part of Region II.5-Duns and Ros7flow-pattern map. . slug. Solve Eq.3 and 4. 2. 4. and NLV. S. 4. For the dimensionless slip-velocity number is given by where L1 and L2 are functions of Nd. Eqs. (4. The friction pressure-gradient component for bubble flow is given by Liquid-Holdup Prediction. . . Regions I through I11 and a transition region. . Flow-Partem Prediction. . 1. .3 1. . respectively. . 4. or. 4. . in field units. bubble flow. as given in Eq. 4.35 or 4. Calculate the dimensionless slip velocity.6 gave these four groups. . 5. .3 through 4. . Duns and Ros proposed these equations. 3. Calculate the liquid holdup from Eq. . Thus. Bubblelslug boundary: The following procedure is used to calculate the elevation component of the pressure gradient. . In this monograph. 3.14.33 or 4. ". . . (4. At the transition to slug flow. Thus. ... Friction in the mist-flow pattern originates from the shear stress between the gas and the pipe wall..8..2) as a function of a Reynolds number for the liquid phase.9 as functions of the liquid viscosity number. Mist flow exists if N. experimental data.. v.. 4. 2.... = I. The value of roughness may be very small. Then &Idapproaches 0... 34 As the wave height of the film on the pipe wall increases. . . = 0 .. the waviness of the film may become large. . = P .. the friction factor is obtained from a Moody diagram (Fig.. Thus.... 3..029Nd + F .46) F ... a function of a dimensionless number containing liquid viscosity..2 1 1 1111*... .. . > NgVfil. Between these limits. It becomes important for kinematic viscosities greater than approximately 50 cSt and is given by This influence was accounted for by making Nw. Duns and Ros noted that the wall roughness for mist flow is the thickness of the liquid film that covers the pipe wall. NL. 4. The factorf3 is considered by Duns and Ros as a second-order correction factor for both liquid viscosity and insitu gadliquid ratio. where the coordinates are Nwe vs. 4. . 0.44) and The friction pressure-gradient component for slug flow is calculated exactly the same way as for bubble flow. . 4. Waves on the film cause an increased shear stress between the gas and the film that. 4. 4. The result is nearly a no-slip condition between the phases. ... . (4. the dimensionless slip-velocity number is Fig. Slug Flow.30.. but the relative roughness never becomes smaller than the value for the pipe itself.. and F..40 is a correction for the in-situ gashquid ratio and is given in Fig.. . where F 5 . &Idcan be obtained from equations developed from Fig. is in pounds per cubic foot.. . the liquid 1s transported mainly as small droplets...40) f2 nefriction factor is governed mainly by f..t. The mixture density for use in the elevation component of the pressure gradient then is calculated from Eq. friction-factor parameter. Fig. Nwe Np . . ... the acceleration component for slug flow is considered negligible. . and UL is in dynes per centimeter. .. ....vs$l lI r -. p. ...sllp-velocity parameters.... . ...10. .7-Duns and Ros7bubble-flow.. = 0..2) as a function of a Reynolds number for the gas phase.. with the crests of opposite waves touching and forming liquid bridges. This process is affected by liquid viscosity and also is governed by a form of the Weber number The factorf2in Eq. . which is obtained from a Moody diagram (Fig. and H .. Mist Flow.Duns and Ros assumed that. 1 1 1 t 1 l Illfill I I IItIIII 1 L lot 10-2 10-I 1 10 Fig. ...23. 4. Values off for the mist-flow pattern can be found for &Id> 0. ...10 shows the functional relationship... F6. at high gas flow rates.. .. the friction component of the pressure gradient is determined from where d is in feet. These waves result from the drag of the gas deforming the film in opposition to the surface tension. S = 0 .. 2.8-Duns and Ros7 bubble-flow. . in turn. Duns and Ros considered the acceleration component of the pressure gradient to be negligible for bubble flow. . . . Because there is no slip. (4. the actual area available for gas flow decreases because the diameter open to flow of gas is now d .Duns and Ros suggested that the friction component of the pressure gradient could be refined by replacing d MULTIPHASE n o w IN WELLS . can cause the greatest part of the pressure gradient. . Also... .. vsg is in feet per second.05 from this extrapolation of the Moody diagram.. D~~~ and R~~ developed this equation forf: ~ ~ f = fIj.40. and F7 are given in Fig. .. Slug flow exists if NgVBIS NgY< NBYSITi slug < For flow.5. film-thickness correlation.3-Using the Duns and Ros Method. The MMSM method includes liquidholdup correlations derived from the data for bubble and slug flow that are simpler in form than those used in the original Duns and Ros method. 4.38. 4.9-Duns and Ros7 slug-flow.35. 4. Given: p.24.. Verification.36.is related to the sonic k velocity of a two-phase mixture. . s and Ros7 mist-flow. In mist flow. Multiphase-Flow Pressure Gradient for Example 3.4. .as Modifications.E and vxg with vsgd2/(dthroughout the calculations. where = 4.. to obtain the pressure gradient.O. 2.. Duns and Ros suggested linear interpolation between the flow-pattern boundaries.54 < NBYBIS :. The pressure gradient in the transition region then is calculated from < N.87) = 14. . The transition region exists if N g v S . F3=1.52. pg =0. F2=0. 4. . Increased accuracy was claimed in the transition region if the gas density used in the mist-flow pressure-gradient calculation was modified to be From Eq.. =0. . F l = 1. and N. the total pressure gradient can be calculated from The dimensionless kinetic energy term. 3. .5. Two proprietary modifications of the Duns and Ros method have been developed but are not available in the literature. . with d .5.. =gas density calculated at the given conditions of pressure and temperature. .. slip-velocity parameters.2.0.3. This modification accounts for some of the liquid being entrained in the gas.O.. Calculate the Vertical. acceleration often cannot be neglected as it was in bubble and slug flow. In ajoint Mobil-Shell study undertaken between 1974 and 1976. involved modifications based on carefully obtained data from 17 high-GOR vertical oil wells. and F4=26.55) From Eq. NgVs. Eq.bubble flow exists. Eq. 4. ( A ) slug 1 A ) mtsr . . including the 17 used in the Ros field method. = 8. E is similar to the Mach number for compressible flow.. Ek. 4 . = 1.O. If we define a dimensionless kinetic energy.06 and N.6. Determine liquid holdup: FromFig. < N. l M u n ..53 can incorrectly yield Ek values greater than 1. and 21 directional wells were selected as the basis for the modifications. a.12.946. L2 = I. When this occurs in a pipe. Possible discontinuities at flow-pattern-transition boundaries also were removed. a "shock" is established.1 1 1 1 1 1 1 1 1 1 1 Illl 4 6 8 I 1 1 1 1 I I 1 1 10-4 2 4 6 8 10-3 2 4 6 8 10-2 2 2 4 6 8 10-1 2 10" 4 6 8 11 0 Fig. Fig. 4.54 will predict the same result if Ek becomes 1. Region I (froth or bubble). 4. a modification resulted in the Moreland-Mobil-Shell method (MMSM). The first. 40 vertical oil wells.41 dyneslcm. This results in a trial-and-error procedure to determine E . Sonic conditions are reached when the Mach number becomes 1. 1. . I..016 cp.1)(11. If this region is predicted. This will require a caiculation of pressure gradients with both slug-flow and mistflow correlations. = 1 1. 4.97 cp. known as the Ros field method. From Eq. Transition Region. 4. Determine the flow pattern: From Fig. . From Fig.Unfortunately. where p . Example 4.2. across which the pressure gradient is infinite.0 + (1. The acceleration component of the pressure gradient can be approximated by Beggs and Brill" provided a derivation of Eq. From Eq. Ek.37 for bubble flow. and e = 0.. L1 = 1. (4. In this study.00006 ft.T. .o.. 4. .2. .60) . . neglecting kinetic energy effects: From Eqs. d = ft. 2. > AgBIS.. Slug Now.... This modification was meant to extend the Griffith and Wallis work to include the high-velocity-flow range. 2. The liquid holdup determined from Eq.2 or Eq. . 2.. 4.. In Sec. Consequently. 0rkiszewski8 tested several published correlations with field data and concluded that none was sufficiently accurate for all flow patterns.17 X lo-' N ~ be : fi = .65) MULTIPHASE FLOW IN WEL1. and & is constrained algebraically to be 0. The and slip density is calculated from where and v. .11 and 4. . 4. Orkiszewski proposed the last term in Eq.. A similar Griffith and Wallis development neglected the presence of a liquid film around the Taylor bubble and the possibility of liquid droplets being entrained in the Taylor bubble. . N.. . . For the boundary between bubble flow and slug flow..sThe = ~ 1 ~ 2 @ * ..208 psf/ft..13..63 to account for the distribution of liquid in these regions. he chose these criteria established by Griffith and Wallis. . which is equivalent to Eq. .. Determine pressure gradient.2 we will show that v. Fig.< NpSln. 4.. The slug-flow correlation was developed with the Hagedorn and Brown4 data.. I . (4.B.63 by performing mass and volume balances on a typical slug unit consisting of a Taylor bubbleI5 and a liquid slug. 4. 4.1 . .17.80 + 29.. . vb.. .42.2) as a function of relative roughness and Reynolds number for the liquid phase.32b and 4 . is a function of the gas and liquid densities and surface tension.12 as functions N R e b and NRCL..64) of where C1 and C2 are expressed in Figs.11--Griffith and Wallis13 C...and From Fig. (4.40. From Fig. ' ~ for bubble flow and the Duns and Ros7 method for mist flow. 4. which in turn is used to calculate the elevation component of the pressure gradient... = Orkiszewski developed Eq.31. . The friction pressure-gradient component for bubble flow is given by = 0. . From Eq.22. 3 2 ~ defined these. Bubble/slug transition: The acceleration pressure-gradient component for bubble flow was considered negligible. . 4. = ftlsec. . = 1 . . . 4. .. = 1...01 psf/ft = 0. . Slug flow exists if I. Bubbleflowexistsif 1.. 4. correlation. . NReb pLvbd = H 36 . .. and 4.22. 4. including the transition region between them.39. .. ..60 then is used to calculate slip density with Eq. From Eq.. Orkiszewski used the Duns and Ros flow-pattern transitions for the boundaries between slug flow and mist flow. The friction factor is obtained from a Moody diagram (Fig.21 = 30. Eqs.S . vsg = ftlsec.. 3.8 ft/sec is a good approximation of an average v. Orkiszewski adopted the Griffith14 suggestion that 0.8. 4. Flow-Pattern Prediction.. by the relationship vb BubbleFlow.. 4. Orkiszewski method selected the Griffith and W a ~ l i s ~ ~ . (4. . .36 for the Duns and Ros correlation. Orkiszewski Method.0175. 3. He then selected what he considered to be the most accurate correlations for bubble and mist flow and proposed a new correlation for slug flow. ... liquid holdup for bubble flow is determined from 5 I. Griffith and Wallis correlated the bubble-rise velocity. f l =0. (4.569logd + X. . 4.. these constraints are supposed to eliminate pressure discontinuities between equations for .12-Griffith and Wallis13 C2 comlatlon..0.70.000. . the value of r is calculated from one of these expressions. . . (ftlsec) Equation for l4.64 or Eqs..PpL). is in feet per second. ..013 l0gpL .38 + 0. 4.0. .68) where . . 4. X= . < 10. . . Fig. 4. .log v. . . . . Orkiszewski did not define criteria to establish which liquid phase is the continuous phase when both oil and water are present.01 d1.70. . . . ..70.... 4. This procedure will converge in one step if the new value of NReb does not result in a change among Eqs...Fig. 4. (4.0..681 d1. . rr and.. ...75 c10 >10 Orkiszewski used the data of Hagedom and Brown to calculate and correlate the liquid distribution coefficient. (4.. . . . . . . Estimate a value of vb.12.67 through 4. . Oil Oil and c10 >I0 r= 0. . .. . .. . l. .0. determination of v requires b an iterative procedure when using Figs. .78) Because the equation pairs do not necessarily meet at v = 10 ftfse~.73 4.0274 dl 371 + 1) + 0. .. 4. use the value in Step 3 as the next guess and go to Step 1.571 + 1 ) + 0. .13 shows the discontinuities can be sig' I nificant. . . (4. . . The procedure follows..045 logp. and v.167 log v.72 4... .. vb can be calculated from this set of equations. When NReb5 3. .0 .232 log v..000.1 1 and 4. ~o-~N. If they are not sufficiently close. . d is in feet.67 through 4.01271og(pL + 1) . . . . 0.. (4. .63 log d 1 where (4.. (4. . . . When C2 cannot be read from Fig. ..161 + 0. .66) Fig.73) + 0.)&. TABLE ~.. . . .709 . 4. .284 dl 41s and When NReb2 8.75) When 3. .000. .~ 2.. Compare the values of vb obtained in Steps 1 and 3.162 log v...12 was extrapolated so that vb could be evaluated at higher liquid Reynolds numbers. A method to eliinate them is described in the next section. . . . b 3. . .428 log d. .. . . Calculate vb using Eq.67 through 4.35 + 8. 37 .888 log d. 3. A good first guess is and p~ is in centipoise. . .12 or Eqs.= o. . (4. . 0 6 5 ~ . . .1 identifies liquid distribution coefficient equations. .. If v. . . ... . . .I--ORKISZEWSKI' LIQUID DISTRIBUTION COEFFICIENT EQUATIONS Continuous Liquid Phase Water Water Value of v.74 4. Continue until convergence is achieved.70) Because vb and NRebare interrelated.72) . Calculate NReb using the value of v from Step 1. . . . . ( . .397 + 0.. 4.0. .. . . .. . . .799 . . . . .77) . .0.. . Table 4.74 X log(pL r = 0. 4. . . . r.. The value of r is ccnstrained by these limits. Depending on the continuous liquid phase and the value of the mixture velocity. Fig.3 can be used for this purpose.. .. .. 0. 1. v b = (0. v b 1 . MULTIPHASE-FLOW PRESSURE-GRADIENT PREDICTION rr -2 S . = 8.000 < NReb< 8. . . .76 be modified Guess v b = 0. and E = 0. From Eq. with p. and v. ~ r i ~ ~ i a * suggested that coefficients in Eqs. . = l(0. .43)(0.7. were From Eq.2) by use of a Reynolds number given by NRe = 'Ad..0.4. A..78 are ineffective in eliminating the pressure discontinuities that result from the use of either pair of equations.63 can become less than the no-slip mixture density.159. . Because N.32. (4.84) = 476. 4..flow pattern is not bubble flow. 4.. 4.A. . Brill16 showed that the constraints given by : F g 4.83)(0. r = .. Is .75..3 N R= ~ ~ 0... pg =0..16 Because .66..61)(7. When this occurs. Petrobras S.41 dyneslcm.. Npvn = 50 + (36)(11. .54 < 476. 4.Example 4.. A logical replacement is From Eq. described in Sec. .97 continuous liquid phase.97) = Yes.38 . Given: po =0. .000.. Eq.O .ntsec Fig.00006 ft.0 ft/sec. . (4. @ = (0.. The problem results (1.. The friction pressure-gradient component for slug flow is :..43 ft/sec is correct. and between Eqs. .0.. problem in the computing algorithm. Determine flow pattern: From Eqs. 05 = 0..79 for the friction pressure-gradient component.488)(47.. where f is obtained from the Moody diagram (Fig.2. affect the accuracy of results.5) 0... when this occurs.428 log d .4-Using the Orkiszewski Method.). 2. 1... A second modification in the Orkiszewskicorrelation is to replace p... 4..66. Therefore... .73 for water as the N ~ e L= 0. by use of the 2.73..13--Liquiddistribution coeff icient discontinuities. SUPERFICIALMIXTURE VELOCITY..35) + (8..488)(47. . + C(l.5) = 7.. set A = 0.13.86 from discontinuities between Eqs. .68. = 11.61)(11....74 and 4. .... .97 cp. The liquid distribution coefficient also can become too large a negative number for high flow rates corresponding to large values of v. v. ..397 + 0. pn 'Personal communcation w ~ t h Tnggia. . ... d1.74 X 10-6)(2.. and 4.43 ft/sec. .17 - 105. 4.5) J so that the slopes of these curves are retained but the discontinuities are eliminated.287 .. 3. . 104..159 r .013 logp. for calculating pressure vs.5) = 2.84) .2. (4. = 11.97 From Eq. . . It also becomes necessary to replace Eq. ..97 . 4. Flow pattern is slug flow...65.0.72 and 4.86 X lo5)] -4 = 11. = . The Orkiszewski method can cause a convergence From Eq.59. Orkiszewski recommended the Duns and Ros method if either mist flow or the transition region between mist flow and slug flow is predicted. 4.065)(7.016 cp.63 log d . 4.61)(2. Calculate the Vertical.. . .4. 2..0.58 and 4.77. 4. .0.logv. The acceleration pressure-gradient component for slug flow was considered negligible. .518? 38 MULTIPHASE now IN WELLS . respectively. The resulting equations for water and oil.4.13. (1..159. . depth traverses in wells. .0)(0.. .. Multiphase-Flow Pressure Gradient for Example 3. 4..i.75 for oil as the continuous phase.2) Reynolds number given by Eq. Rio de Janeiro (1984).77 and4.0. Determine slip density: ModiJications.. .80) (1. .0.5 . C= 0.. 4. 4.81) where From Eq.1 < 0.. a.(0..488)(47. + 1) NReb = + 0.65. From Eq.2.. Because NReb > 8. where f 1s obtained from the Moody diagram (Fig. =8. .162 log v.26. 4.. Eqs.74.12cannot be used. This solves the convergence problem but also may = 2.01 log(j4. 4. attern map used by Aziz et a. 4. 4.2 or Eq.5 for Ny z 4.From Eq. Determine friction factor: NReL= 2.63.2 or Eq. Flow-Pattern Prediction. The liquid holdup for slug flow also is calculated from Eqs.63.22 andf is obtained from Fig.17.85 and 4. Bubble Flow. 4. Determine pressure gradient: From Eqs. f=0. however.85 and where VSL is in feet per second. the coordinates are the superficial velocities of each phase. These equations represent the flow-pattern transitions in Fig. and 4. 4. 4. can be predicted from Aziz et aL M e w The Aziz et method uses many of the fundamental mechanisms that form the basis of modem mechanistic models. Fig.17 for a Reynolds number given by The acceleration component of the pressure gradient was considered to be negligible for bubble flow.4.. is the rise velocity of a continuous swarm of small bubbles in a static liquid column.86 are attempts to validate the flow-pattern map for fluids other than air and water. is in pounds per cubic feet.) MULTIPHASE-FLOW PRESSURE-GRADIENT PREDICTION . is determined from Eq.0158. This velocity can be predicted from where the first term is the approximate velocity of the fluid mixture. (Reproduced with permission of the Petroleum Soc.79. X 10' and From Fig 2.90 and 4. and vb. < 26.3 1. Liquid holdup for . The friction component of the pressure gradient is determined from and where p. 2. and UL is in dynes per centimeter. Bubble flow exists if N < N1. 4.15 Aziz et al.14 shows the flow. vs is in feet per second. accounting for the nonuniform velocity and bubble concentration profiles across the cross section. The bracketed terms in Eqs. p. pL is in pounds per cubic feet. 3. 4. vb. For an air1 water system at atmospheric conditions. < N2 for Ny<4 or N 1 < N. 4. the bubble-rise velocity in a static liquid column is based on a Taylor bubble. Slug flow exists if N1< N.85 and 4. state that where C was given by alli is'^ as and Fig. 4.86 define the coordinates. bubble flow is calculated from where vbfis the rise velocity of small gas bubbles in a flowing liquid. 2. 3.1 4--Aziz etaL9flow-pattern map. For slug flow. 2.14. Slug Flow.96. and fiat presented by Govier et aLR Eqs. When the transition region is predicted. From Eq. as well as on data of Poettmann and Carpenter1 and Orkisz e ~ s k iThe wells from Iraq were flowing primarily through a cas.96.5. Their conclusion was based on a comparison of the predicted and measured pressure drops for 80 tests on 15 flowing wells in Iraq.A ) ( ) m i s . Example 4. : rn ..15-Chierici et a l l 0 & correlation.345) and (32.Orkiszewski8 - Griffith and Wallis13 - The friction factor is obtained from a Moody diagram (Fig. Al-Najjar and ~ l . 4. 4. a. Determine liquid holdup: From Eq.85 and 4. the pressure gradients must be calculated with both the slug-flow and mist-flow equations. 4. The transition region exists when N2 < Nx < N3 for Ny < 4. where Modifications..35)]~"~= 1.41 dyneslcm. Nl = 0.83) From Eq. (4.14.5. Mist Flow. .s o o f showed that improved l~ results could be obtained with the Aziz et al.-.5. From Fig. Thus.016 cp.. 1.. Method.. 2. method if the flow-pattern map in Fig. similar to the procedure described in the Duns and Ros method.90. and E = 0. 4. S I U ~ + (1 .(g) dZ = Fig. From Eq.00006 ft.. 4. pg =0... . linear interpolation is performed.97 cp.88) From Eq.86. recommended the Duns and Ros mist-flow method be used to calculate pressure gradient for this flow pattern. 4. + 1... = 8.2) as a function of relative roughness and a Reynolds number given by The acceleration pressure-gradient component was considered negligible for slug flow. Verification. > 26.95.~ ingltubing annulus. To obtain the pressure gradient. = 10.61 47.55 < 26.14 was replaced with the Duns and Ros map in Fig. Multiphase-Flow Pressure Gradient for Example 3.98.295 = 10. Transition Region. From Eq. slug flow exists. In Fig.2. Mist flow exists when Nx > N3for Ny < 4 or N.25 < 26.2)(7. Aziz et al.51[(100)(6. 4...5 for Ny > 4. note that the transition region does not exist for Ny > 4.61 .14. 4.174)(0.55 for Ny 2 4 . 1.. v. Determine flow pattern: From Eqs. Given: po =0.97. " .91. MULTIPHASE n o w rN WELLS vbf = (1. = (0.69 ft/sec. 4.5-Using the Aziz et al.. 4. Calculate the Vertical.5)(47..101) 2.and rn is determined from - The friction pressure-gradient component for slug flow is determined from . . 4. 4. 90 ft long. For each pipe size. Liquid holdup is determined from = 0. From Fig 2. and = .000. 4. Determine flow pattern: Orkiszewski predicted slug flow. liquid and gas rates were varied so that.105 for extrapolation also are plotted in Fig. E = 0. and the acceleration component is neglected for slug flow.103. Determine pressure gradient: 3.0 16 cp. Beggs and Brill proposed the following pressure-gradient equation for inclined pipe. 4. 4. Curves based on Eq. 4. and f 90".105 gives where Ek is given by Eq. f lo0. 4. a = 8. only slug-flow treatment is presented. For this range. 105 and E = d 0.92 + 32. slug-flow treatment is almost identical to that of Aziz et al.55 = 33.ug 0.86 (0. 4. 4. and mist-flow regions is identical to that of Orkiszewski. 0. Given: p.70 can result in discontinuities. all flow patterns were observed.21 Fig. Multiphase-Flow Pressure Gradient for Example 3. = 0. Fig. including directional wells. Determine friction factor: N R= ~ (1. f75".83)(0. 4. where C1 is obtained from Fig. Their original flow-pattern map has been modified slightly to include a transition zone between the segregated.92 + 32.00012. proposed that for liquid Reynolds numbers greater than 6. f=0.00006 ft. f35". f =0. Calculate the Vertical. The pressure-gradient friction component is identical to that of Aziz et al.00006 = 0.20 Chierici et al.17.4 1 dyneslcm.12 used by Orkiszewski. the inclination of the pipe was varied through the range of angles so that the effect of angle on holdup and pressure gradient could be observed. 4. Beggs and Brill Method. f 15".15 that the extrapolation equations proposed by Orkiszewski in Eqs.3. f20°. 4.6-Using the Chierici et al.233 psilft. 4.15 for liquid Reynolds numbers less than 6.2. given by . The only difference is in the slug-flow region. The Beggs and ~ r i l l l method was the l first one to predict flow behavior at all inclination angles.2 or Eq. 2.59 psflft = 0. Determine liquid holdup: From Eq. The correlations were developed from 584 measured tests.232 psilft.2 or Eq. 0.00012.17. 4.5 = 0.35 for all cases. Combining Eqs. From earlier work by Nicklin et a1. Fig. method to predict flow pattern and to calculate liquid holdup and pressure gradient in the bubble-.67 through 4. Method. Method. Liquid holdup and pressure gradient were measured at angles from the horizontal of 0°.and 1. 4. Chierici et al. transition-. Example 4. 4. 2. The Chierici et a/. MI 'I TIPHAYE-FI .00006 = 0.67 = 33. Values of C2 can be obtained from Fig. sections of acrylic pipe.5) = 2.5 From Eq.15 is identical to Fig.11 as a function of the bubble Reynolds number and is essentially 0. when the pipe was horizontal.97 cp. Determine pressure gradient: Chierici et al. Their test facility was 1. f5'.5-in. TheChierici etal. 1.61)(7.47 psflft E = d 0. Thus. Beggs and Brill chose to correlate flow-pattern transition boundaries with no-slip liquid holdup and mixture Froude number.and intennittent-flow patterns.488)(47.0158..16 illustrates the horizontal-flow patterns considered by Beggs and Brill. After a particular set of flow rates was established. On the basis of observed flow patterns for horizontal flow only. Slug Flow. 4. From Fig 2.. The pipe could be inclined at any angle from the horizontal.0158.104 and 4. Determine friction factor: and = 0. . 4. show in Fig.000.106.15 and show no discontinuities.97) 2. f 55". The fluids were air and water.17 shows both the original and the modified (dashed lines) flow-pattern maps. they prepared an empirical map to predict flow pattern.OW PRESSI IRE-GRADIENT PREDICTION Flow-Pattern Prediction.53 and The resulting prediction of liquid holdup essentially is identical to that of Aziz et al. .18 shows the variation of liquid holdup with pipe inclination for three of their tests.. . .11 1) L.110) L. . . . . . . . . .112) + . The liquid holdup that would exist ifthe pipe tverr horizontal is first calculated and then corrected for the actual pipe-inclination angle. . = 31&2!302. . . = 0.Segregated Stratified Input Liqurd Content. . (4. . For high flow rates corresponding to what now is called dispersed-bubble flow. . . . 4. . The liquid holdup that would exist if the pipe were horizontal is calculated first from The equations for the modified flow-pattern uansition boundaries are L. < 0. . .01 and NF. . . .18-Effect Distributed of inclination angle on liquid holdup. 2 0. . iL Fig. . < L. . . unless the pipe I S horizontal. .2468. . = 0 . 1. Annular Intermittent Plug Angle of Pipe From Horizontal. . . . .113) The following inequalities are used to determine the flow pattern that would exist if the pipe were horizontal. . . .4 and N.(4. . 1 . ~ . . .50".4 andNFr 2 Ll . .16-Beggs and Brilll1 horizontal-flow patterns.. . However. . . . 1 < 0. . . . . . . . . 4. Fig. . . .. .4 and L3 Distributed. The same equations are used to calculate liquid holdup for all flow patterns. 4.(4. This flow pattern is a correlat~ng parameter and. . > L. . . Intermittent. . L. . . . . gives no informat~onabout the actual flow pattern. liquid holdup essentially was independent of inclination angle. . .01 or I1. . . . . . . . . 4. . . . . . the empirical coefficients are different for each flow pattern. 1 r 0.(4. . . Different correlations for liquid holdup were developed for three hori:onral-flow patterns. ... < NFr 5 L. = 0.. .'452. Transition.000925A. . .01 and NFr < L. . . 5 1 ~ ~ ' ~. . . . Bubble . . . 0.ll or 1 r 0. . . 5 L.17-Beggs and Brill21 horizontal-flow-pattern map. . The liquid holdup was a maximum at about 50" from horizontal and a minimum at approximately . < 0. . Fig. 42 .4andL3 < N. . . . . Segregated. . . . and . . Liquid-Holdup Prediction. . .10A. degrees Fig. . ..016 cp. Payne et al. and c are determined from Table 4.TABLE 4. The coefficients e .3 7680 3 5390 h . Payne etal.. and From Fig 4. . and . Determine flow pattern: From Eq.... .&. When the flow pattern falls in the transition region. + (1 . Determine liquid holdup: From Eq. " M .ug =0.. 4..6140 0. . and pn is obtained from Eq.845 1.m.0173 0..41 dyneslcm. . Payne et al. Given: . .17 for an appropriate value of relative roughness.. is determined from the smooth pipe curve on the Moody diagram (Fig.2) or from Eq. Beggs and Brill introduced this correlation where 8 is the actual angle of the pipe from horizontal and C is defined by with the restriction that C r 0.17 forlL = 0. 4. On the basis of their limited data..0978 0.3-BEGGS AND BRILLll EMPIRICAL COEFFICIENTS FOR C Flow Pattern Segregated uphill Intermittent uphill c 0. 4. .IL. the liquid holdup must be interpolated between the segregated and intermittent liquid-holdup values as = AH..7-Using the Beggs and Brill Method..507. The ratio of the two-phase friction factor to the normalizing friction factor was correlated with the Beggs and Brill experimental data. . 2..22experimented with a 2-in.5056 The factor to correct liquid holdup for the effect of pipe inclination is given by to single-phase liquid flow.12 or 2.A)H. 3.1244 -0.. It is unlikely that the discontinuity at the 2. recommended these constant correction factors to improve liquid-holdup values. b. where Example 4. each with inclination angles of about f 10' from horizontal. found that the Beggs and Brill method unde*redicted friction factors...0. The original Beggs and Brill method has been found to overpredict pressure drops in producing wells....0609 e 00 1 1 2 960 0.63 x smaller value of y would ever be encountered. Liquid holdup was measured in all uphill and downhill sections by use of quick-closing ball valves. . The coefficients a. 2.2. where Modifications.109. Consequently.1 15 gives the liquid holdup corrected for the effect of inclination angle. Payne et al.2) or from Eqs. the horizontal-flow pattern is intermittent.700 . = 1 C VJ 4. Payne et al..4473 with the restriction that HL(o) )AL.2 1.0 to ensure that the correlation degenerates MULTIPHASE-FLOW PRESSURE-GRADIENT PREDICTION 2.. Eq.57 cp. Carefully metered mixtures of natural gas and water were flowed through the pipe at pressures between 400 and 500 psia.00006 ft..3 for the appropriate horizontal-flow pattern. resulting in However. Calculate the Vertical. .0868 0. . Eq.. This change improved the pressure-drop predictions for the Beggs and Brill method for rough pipes. E = 0. . For s that is used when l < y < 1. the resulting liquid holdup for 8 > 0' should not be less than . be obtained from the Moody diagram (Fig. improved results should be obtained if the Payne er al.2.. . . Because the Beggs and Brill method was based on data obtained in smooth pipe.016.2 for the appropriate horizontal-flow pattern. 1. . and h are determined from Table 4.. 2. However.123 contains discontinuities for y values of about and 1.4846 0. it is necessary that s = 0 for y = 1.. = P nvmd - "" . .1.-diameter flow loop constructed with Schedule 40 line pipe that was more than 500 ft long and contained three hills.065 b - TABLE 4.5824 f 9 .2-BEGGS AND BRILLll EMPIRICAL COEFFICIENTS FOR HORIZONTAL LIQUID HOLDUP Flow Pattern Segregated Intermittent Distributed a 0. . Friction-Factor Prediction.1 14. The two-phase friction factor is calculated from The normalizing friction factor.3050 . . recommended that the normalizing friction factor&.3692 0. Distributed uphill All patterns downhill No correct~on: = 0.980 0.5351 0. 4. a = 8. 2.. Multiphase-Flow Pressure Gradient for Example 3. liquid-holdup correction factor is applied for wells.( . also found that the Beggs and Brill method overpredicted liquid holdup in both uphill and downhill flow. g.0.13 by use of a Reynolds number defined as N.uo=O. f . Two modifications to the Beggs and Brill correlation frequently are used. nominal ID steel pipe... Fig.0. 4. the bubble/slug transition is described by NgvB.128..131) where .. f. Apply Payne er al. f= (1. Increased liquid viscosity accelerates the transition from slug to annular/mist flow.393952 < 0. The closed end of the U-shaped pipe could be raised or lowered to pennit flow at any angle from 0" to f90" from the horizontal.19 represent the bubble/slug... + (0.. For each oil phase tested.530. In downflow and horizontal flow. and 4. 1. 2. The slug/(annular/inist) transition was identical for horizontal and all upflow and downflow angles.. this transition becomes a function of viscosity only and is a vertical straight line. From 0" to . C. liquid viscosity had a significant effect on this transition. Each leg of the U was 56-ft long with 22-ft entrance lengths and 32-ft test sections on both uphill and downhill sides.124.574) = 0.330282 4. bubble flow did not occur for the range of flow parameters considered. Eq.473.429(log sin 0 + 1. From Eq. = Hu.. Approximately 1. Eq. 3.. The subscripts BS.079951 0. C = 0. The transition curves were drawn on each map.551627 15 519214 0.21..4.96(0.97)(0. C = (0.. Fluids were air and kerosene or lube oil.924)(0.. MULTIPHASE n o w IN WELLS .288657 C5 C..789805 0. C.. 4.0155) = 0.. f= fn e(03873) = 1.507) 2.119788 Therefore. Determine pressure gradient: .475686 0..473) (0. where Mukherjeeand Brill Method. S.3873.8081 39 4. 4.516644 0. = (0. and HL(..130. 3. (4 2 .493) = 0 .0155. bubble-/stratified-flow transition occurs below inclination angles of .574. C ..19 gives a flow chart to predict flow patterns by use of these flow-pattern transition equations.048 TABLE 4. Flow-Pattern Prediction. 4. = 0.121.129875 -0. (4.1. The bubblelslug transition was found to be linear at 45" with the axes and was best fit by Eq.0228. = 0..504887 -0..1 17.132 sin 0.133 gives the downflow and horizontal stratified-flow boundary.940 + 0. 4.13 1. From Fig.M. In horizontal flow. urements and more than 1.130 describes this transition....30" at lower rates.380113 0..0. 4... Determine friction factor: From Eq.. Their test facility consisted of an inverted U-shaped. respectively.. correction factor: HLo. bubble-/slug-flow transition occurs as gas velocities increase at a fixed liquid velocity. 5 0 ~ ~ . = 0.. However. At lower liquid rates. 4.00012.... p n = (0.. and ST in Eqs.4. and the stratified transitions. Two transitions were fitted for upflow.493) ln[2.0.. A transparent section in each leg allowed for flow-pattern observations and permitted the use of capacitance sensors to measure liquid holdup. Y = 1..507)~~~(11.30".262268 0.171584 56.343227 0. 4. From Eq.. 4. flow-pattern maps were drawn on log-log scales with dimensionless gas and liquid velocity numbers as the coordinates.87) 1004473(3.5-in.074 sin 0 From Eq.133 and in Fig.From Eq. These curves then were fitted with nonlinear regression equations. 4.123...8 1)Oom8] = .4-MUKHERJEE AND BRILLT2EMPIRICAL COEFFICIENTS FOR LIQUID HOLDUP Uphlll Downhill Strat~f~ed Downhill Other ..000 pressure-drop meas44 In downflow at higher liquid rates.. 4. slug/(annular/mist). 4. .132) This transition generates a family of curves for different inclination angles and liquid viscosities.2 for &Id= 0.. however. Therefore. From Eq.016)(0. (4. The Mukherjee and Brill12method was initiated in an attempt to overcome some of the limitations of the Beggs and Brill method and to take advantage of new instrumentation developed to measure liquid holdup. where x = log Ngv + 0. = loy.122.371771 0.500 liquid-holdup measurements were obtained for a broad range of gas and liquid flow rates.128. -0. Fig. 4.19-Flow chart to predict Mukherjee and Brilli2 flow-pattern transitions. Liquid-Holdup Prediction. The liquid-holdup data were correlated with an equation of the form HL = exp Fig. 4.20--Control volume for stratified flow.'* [(c,+ C2sinB + c,sin2B + c , N : ) ( A $ ; / ~ : ) ] . TABLE 4.5-MUKHERJEE AND BRILL1* ANNULAR-FLOW FRICTION-FACTOR RATIOS fR Three different sets of coefficients were developed for Eq. 4.135. One set is for uphill and horizontal flow: one set is for downhill stratified flow; and the third set is for other downhill-flow patterns. Table 4.4 gives the coefficients. 1.oo I HR 0.01 0.98 1.20 1.25 1.30 1 25 1.oo 1.oo 0.20 0.30 0.40 0.50 0.70 1.oo Bubble and Slug Flow. The pressure gradient for bubble flow and slug flow is determined from dP -, dL where f+ G p j g sin , 1 - E, ...................... (4.136) 10.00 Ek = PsVrn"~~ p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.137) or Eq. 2.17 for an appropriate relative roughness and a Reynolds number given by Eq. 4.138. Then, The friction factor is obtained from a Moody diagram (Fig. 2.2) or Eq. 2.17 for an appropriate relative roughness and a Reynolds number given by where p, is obtained from Eq. 3.21. Annular Flow. The pressure gradient for annular flow is determined from Eq. 4.137 and Stratified Flow. It is likely that stratified flow will occur only in highly deviated or horizontal wells. Mukherjee and Brill chose a separated-flow or two-fluid approach to develop a pressure-gradient-prediction method. Fig. 4.20 gives a control volume that defines all variables for this approach. A steady-state momentum balance on the gas and liquid phases yields Eqs. 4.142 and 4.143, respectively. and Mukherjee and Brill developed an empirical expression for the friction factor that depends on liquid holdup. Liquid holdup first is calculated from Eq. 4.135 and appropriate coefficients from Table 4.4. A ratio of holdup values then is calculated as dp A -= d~ - (swLPL t , W , ) - p L A d sine. ........ + (4.143) Either equation can be used to calculate pressure gradient. Mukhe j e e and Brill chose Eq. 4.142. To eliminate the effect of interfacial forces in large-diameter pipes, they suggested the addition of Eqs. 4.142 and 4.143 to obtain Eq. 4.144. and a friction-factor ratio,&, is interpolated from Table 4.5. A noslip friction factor,$,, is obtained from a Moody diagram (Fig. 2.2) MULTIPHASE-FLOW PRESSURE-GRADIENT PREDICTION A* = - (r,~, dL + T , , . ~ P- ) LA, + ~ d , ) sine. ~ g 45 - Mukherjee and Brill concluded that, for most cases involving stratified flow, l~quid holdup is sufficiently small. W, is then small compared with P . Consequently, Eq. 4.142 becomes , 5. Solve Eqs. 4.152 and 4.153 for r,,.~and twx. 6. Calculate the pressure gradient from Eq. 4.144 for large-diameter pipes or Eq. 4.145 for smaller-diameter pipes. Example 4.8-Using the Mukherjee and Brill Method, Calculate the Vertical, Multiphase-Flow Pressure Gradient for Example 3.2. Given: p, = 0.97 cp, p, = 0.016 cp, a = 8.41 dynes/cm, and , E = 0.00006 ft. 1. Determine flow pattern: From Eq. 4.130, I- From simple geometrical considerations basedon Fig. 4.20, it can be shown that 6, AL,Ag, and Pg are all related to hL and d. The PL. relations are II N, , and = 11.54 < 350.8. Therefore. flow pattern is not annularlmist. Angle > 0". Therefore, from Eqs. 4.128 and 4.129. I L where 6 is in radians. Hydraulic diameters for the gas and liquid phases also can be defined by and NLv= 11.87 < 18.4. Therefore, flow pattern is slug flow. 1 - Govier and ~ z i z suggested that the wall shear stresses can be ~' evaluated approximately by assuming single-phase flow to occur in the cross section occupied by a given phase. With these assumptions, these relations result. 2. Determine liquid holdup: From Eq. 4.135 for uphill coefficients, and 3. Determine friction factor: From Eq. 4.138, where f~and& = Moody friction factors based on Reynolds numbers defined by f From Fig. 2.2 for ~ld=0.00012, = 0.0155. b 4. Determine pressure gradient: From Eq. 4.136, and I L I Use the following steps to obtain the pressure gradient for stratified flow. 1. Calculate HL on the basis of Eq. 4.135. 2. Solve Eq. 4.147 ~teratively ford. A value of 0.001 is agood initial guess for 6. AL also is calculated from Eq. 4.147. 3. Solve Eq. 4.146 for h ~ l dCalculate dhg and dhL from Eqs. 4.150 . and 4.151. 4. Knowing 6 and P, Pg and PL are calculated from Eqs. 4.148 and 4.149. 4.2.2 Mechanistic Models. Most mechanistic models that predict two-phase-flow behavior in pipes are for an isolated mechanism, such as flow pattern, film thickness, or rise velocity of gas bubbles in liquid columns. Although several methods predict flow behavior for a single flow pattern, in this monograph we discuss only those that predict flow behavior for all flow patterns. Specifically, we describe the methods of Ansari et and Hasan and ~ a b i r . * ~ - ~ ' MULTIPHASE now IN WELLS Dispersed Bubble C Transition Superficial Gas Velocity, mls I BUBBLE FLOW SLUG FLOW CHURN FLOW ANNULAR FLOW Fig. 4.22-Typical flow-pattern map for wellbores.24 Fig. 4.21-Flow patterns in upward two-phase flow.3o(Reproduced with permission of the American Inst. of Chemical Engineers.) of about 0.25. With 0.25, the transition can be expressed in terms of superficial and slip velocities as vSS = 0 . 2 5 ~ ~0 . 3 3 3 ~ ~ .~ . .. The models first predict the existing flow pattern and then calculate the flow variables by taking into account the actual mechanisms of the predicted flow pattern. Two mechanistic model^,^^.^^ currently in the review process for possible publication in journals, are not described in this monograph. In addition, OLGA and TACITE are commercial computer programs that include proprietary steady-state mechanistic models. Because of their proprietary nature, details of the models are not included in this monograph. Answi et al. Method. Ansari et formulated a comprehensive mechanistic model for upward, vertical two-phase flow. This study was conducted as part of the Tulsa U. Fluid Flow Projects (TUFFP). Although the pressure-gradient equations imply that the model can be used for deviated wells, no attempt was made to account for the effect of inclination angle on the mechanisms considered. Because of the model's complexity and because it was developed with SI units, the development and examples also are presented in SI units. + .. .. . . . .... . . .... . (4.159) as where v, =slip or bubble-rise velocity given by H a r m a t h ~ ~ ~ This is shown as Transition B in Fig. 4.22. At high liquid rates, turbulent forces break down large gas bubbles into small ones, even at void fractions greater than 0.25. This yields the Barnea3* transition to dispersed-bubble flow as Flow-Pattern Prediction. Taitel et presented the basic work on mechanistic modeling of flow-pattern transitions for upward two-phase flow. They identified four distinct flow patterns and then formulated and evaluated the transition boundaries among them. Fig. 4.21 shows the four flow patterns: bubble flow, slug flow, chum flow, and annular flow. Barnea et modified these transitions to extend the applicability of the Taitel e t a / . model to inclined flows. B a n - ~ e a ~ ~ combined flow-pattern-prediction models applicable to different inclinationangle ranges into one unified model. On the basis of these studies, flow patterns can be predicted by defining transition boundaries among bubble, slug, and annular flows. where f is obtained from the Moody diagram (Fig. 2.2) for a no-slip Reynolds number. This is shown as Transition A in Fig. 4.22. At high gas velocities this transition is governed by the maximum a ~ packing of bubbles to give coalescence. Scott and ~ o u b con- ~ cluded that this occurs at a void fraction of 0.76, giving the transition for no-slip disperszd-bubble flow as This is shown as Transition C in Fig. 4.22. Transition to Annular Flow. The transition criterion for annular flow is based in part on the gas-phase velocity required to prevent fall back of the entrained liquid droplets in the gas stream. This gives the transition as BubbleISlug Transition. Taitel et a / . gave the minimum diameter for which bubble flow occurs as For larger pipe sizes, the basic transition mechanism for bubbly to slug flow is coalescence of small gas bubbles into large Taylor bubbles.15 Experimentally, this was found to occur at avoid fraction MULTIPHASE-FLOW PRESSURE-GRADIENT PREDICTION and is shown as the left Transition D in Fig. 4.22. Ban-~ea~~ proposed additional transition criteria that considered the effects of film thickness. One effect is the bridging of the gas the second criterion for annular flow is satisfied. slug flow.2) for a Reynolds number defined by D is a result of applying the two Barnea film thickness criteria..25. .169 must be solved implicitly for 4. The film instability mechanism also can be expressed in terms of the modified Lockhart and ~ a r t i n e l lpa.. 4. Eq.168) and B = (1 . The bubble-flow model is based primarily on the work by Caetano3' for flow in an annulus. dispersed-bubble flow and bubbly flow. by use of a NewtonRaphson approach.5 for n' gave the best results. the right Transition where fTp is obtained from a Moody diagram (Fig. and annular flow. Because of the uniform distribution of gas bubbles in the liquid and the lack of slippage between the two phases.. Combining Eqs." = 0. the slipthe modified Lockhart and M a ~ t i n e l l parameters. from geometric considerations.. 2.164 was modified as where.22 1 or 4.. Thus. 4. If 6 . = g sine@. 6 = d / d .PC)... with the rising bubble concentrated more at the center than along the wall of the pipe.. The two bubbleflow patterns.. turbulent velocity profile for the mixture. is greater than 4..160 gave H a ~ m a t h ~expression for bubble-rise velocity. Chum flow was not modeled because of its complexity.. 4.177 yields fF/fa Annular flow exists if the two Barnea criteria are satisfied and if vsS is greater than that at the transition given by Eq. 4. Eq.169 is first expressed as This is an implicit equation for the actual holdup for bubbly flow.. the two-phase parameters can be expressed as HLFis the fraction of the pipe cross section occupied by the liquid film.. then Eq. Bubble-Flow Model. Eq.. 's~~ To account for the effect of bubble swarm.~ h ~ dified the expression as (4.36 found that a value of 0. slippage is considered by taking into account the bubble-rise velocity relative to the mixture velocity. i~ rameters..)'( Other parameters are defined in the annular-flow model. Two-phase-flow parameters then can be calculated from and and The two-phase pressure gradient for bubbly and dispersedbubble flow is composed of two components. lated by solving a dimensionless combined-momentum equation. Zuber and ~ e n c mo. dispersed-bubble flow was approximated as a pseudo-single phase.169 can be solved for &... . HLF can be expressed in terms of the dimensionless film thickness... Another effect is the instability of the liquid film. With this simplification. Eq.36 are i~~ page velocity can be expressed as Y.163.. but was treated as a part of slug flow. XM and YM.core by a thick liquid film at high liquid rates. as H = 46_(1-6_). In Fig..176 and 4....222.22. If annular flow is predicted. First.. Flow-Behavior Prediction. The bridging mechanism is governed by the minimum liquid holdup required to form a liquid slug. Ansari etal. Because bubble flow is dominated by a relatively incompressible liquid phase.F.. The dimensionless film thickness can be calcu.165 is used to determine the first criterion for annular flow. 4. 4. 4. 4. This resulted in separate models for bubble flow. The elevation pressure gradient is The minimum dimensionless film thickness then is determined iteratively from The friction component is A good initial guess is 4. To account for the effect of liquid entrainment in the gas core.. there is no significant change in the density of the flow- . 4. Eq.. Ansari et aLZ4 presented physical models to predict flow behavior in each flow pattern.~ where the value of n' varies from one study to another.. and For bubbly flow. which causes a downward flow of the film at low liquid rates.. (~PI~L). Assuming a where Z is determined from either Eq. were considered separately in developing the model. assuming no entrainment in the core. 4. resulting in essentially no pressure drop from acceleration.191.193. 4. .23-Slug-flow schematic. are (v.4..as shown in Fig'4'23. . . (4. the root is unique. respsctively.24 ing fluids. v.197 with respect to HLTByields Mass balances for steady-state liquid and gas transfer between the liquid slug and the Taylor bubble.HUB) 0. . 4. .192) where dL is the constant film thickness for fully developed slug flow. .. V!TB.vH . Eqs. if a root exists in the interval of Vo and ~ h o h a m (0. respectively. . 4.195 as F(HLTB). . (4. . ) = [vTB.7gdL.1). as defined in Eq. 4.194. . veloc~tyof the gas bubbles In the l ~ q u ~ d 1s the dug DEVELOPED where the second term on the right side represents the bubble-rise velocity. For a 'lug unit. .vgu)(1 .197 and 4. 4.42 Eqs. . dL by use of the lrotz4' expression.( ..vgrB)(l . . 4. overall gas and liquid mass balances.. ~ and Schmidt.the root of Eq. .. HLTB. .188) + HLTB . . v g ~v.H m ) = (vrB .V K ~ ) + (4. . From geometrical considerations. . .. Eq. . Sylveste@ developed a simplified version using a correlation for slug void fraction. the acceleration pressure gradient is negligible compared with the other pressure-gradient components. also showed that. Thus.198) . = J196. 4. . . . VLTB can be expressed in terms of the Taylor-bubble void fraction to give o o O O 0 0 LIQUID SLUG The liquid-slug void fraction can be obtained from the correlation developed by ~ ~ l v e s t ewho~used data from Fernandes et r.189) The Taylor-bubble-rise velocity is equal to the centerline velocity plus the Taylor-bubble-rise velocity in a stagnant liquid column. VLTB. - (4. .. . . .197.. (4. . .HLTB).188 through 1. .194canbe solved iteratively toobtain the following eight unknowns that define the fully developed slug-flow model: B. 4. MULTIPHASE-FLOWPRESSURE-GRADIENT PREDICTION 49 .Jl .5 HLTB where The derivative of Eq.. and Vo and Shohamj3 showed that these e ~ g hequat tions can be combined algebraically to give Fig.177. .vLTB)]HLTB .B)vgLT(l HU) .185) . 4. can be determined readily from Eq.. and (vTB ..I S~mllarly. .185. .HgLS. 4. are given by Vsx = PvgTB(l.195 can be solved for HLTa by use of an iterative-solution method. and4. .190and 4. . (4. . . respectively.. .-loth studles assumed a fully developed slug flow. .HLTB) ( I + and where ' = H K ~ ( ~ T ~ . .198 suggest that the Newton-Raphson approach can be incorporated easily to determine HLTB. . .9 16 t@)( 1 . The velocity VLTB of the falling film around the Taylor bubble can be correlated with the film thickness.4.. Slug-Flow Model.. . Defining the left side of Eq.196.4.196) With vra and HgLSgiven by Eqs. . Fernandes er presented the first complete physical model for slug flow. The fluid velocity is nearly constant. F(HLTB) = (9. Therefore.186. . .. u . .. . where To determine all slug-flow variables. A 2...200 assumes that the liquid film around the Taylor bubble does not contribute to the elevation component. 1. 4. . Solve Eq.. 7. .. the friction component of the pressure gradient is (4. Annular-Flow Model. Wallis18 determined it empirically as where The shear stress in the film can be expressed as where Prs = pLHU where fF is obtained from a Moody diagram (Fig. However. .ENTRAINED For fully developed slug flow. . Using Eqs. presented a detailed description of the complex equations to predict flow behavior for developing slug flow. Solve Eq.19 1. .... The film thickness varies throughout the Taylor-bubble region. .. . .201) N ~ e . 4... 4... and The core is considered a homogeneous mixture of gas and entrained liquid droplets flowing at the same velocity.. . . = 5. Fig.. Conservation of momentum applied separately to the core and the film yields.194.a d . . Calculate V B . .. calculate kuand LTBfrom the definition of p.~ a ~ gave ia~ l o f detailed analysis of the mechanisms involved in annular flow. .VSL(~ FE) AF 4611 -a .. . Consequently. This gives 50 MULTIPHASE FLOW IN WELLS . the elevation component of the pressure gradient occurring across a slug unit is given by FE is the fraction of the total liquid entrained in the core.HLTB.2) for a Reynolds number defined by P L p L ~ H F . .24 shows a fully developed annular flow. . .. good initial guess is HLTB 0.. For fully developed slug flow. and 4. = 4. . .211) VF . Solve Eq. = 3... .. ... In developing slug flow.. use this step-by-step procedure.4. . .202) where f s is obtained from a Moody diagram (Fig. . 4.. McQuillan and W l ~ a l l introduced the concept of developing e~~~ slug flow during their study of flow-pattern transitions.HLrr). alli is'^ discussed the hydrodynamics of annular flow and presented the classic correlations for entrainment and interfacial friction as functions of film thickness. Therefore.. . determine HLTB. .185 or 4.(4. .15. where VF = qL(1 .188 for v ~ g Note that HLLS 1 -He= .199.212) and d.. 4. 4. . (4.. . Hewitt and ~ a l l . 4. Solve Eq.= Eq.186 for /?. . ~ a r n e contends a~~ that failure to consider the developing slug region can result in significant pressure-drop-prediction errors. Assuming that LLS = 30d. . the length of the cap at the top of the Taylor bubble is a significant portion of the total length of the Taylor bubble. 8.. Ansari et al...FE) . .190.191 for vgm 6.. 2. and HgLsfrom Eqs.. this monograph does not address the developing slug region. . Friction losses were assumed to occur only across the liquid slug and are neglected along the Taylor bubble. = @(I . respectively. Thus. . . . 4. .. .. . (4. .. . . for the core and the film. 2.196 through 4. ..189 for V ~ T B . Ansari et assumed that the acceleration component of the pressure gradient can be neglected...193 for VLTB Note that HgTB 1 . pipe in producing wells is so long that only minor errors should result if the entire slug-flow region is treated as fully developed flow. All subsequent models were based on this approach. . vgu. Solve either Eq.2) for a Reynolds number defi~~ed by + p g ( l . 4. Eq. 4.214 reduces to 7 where the superficial liquid-friction-pressure gradient is given by The basic unknown in Eqs. 4.228 and 4.229 is the dimensionless film thickness, 6. The pressure gradients in the core and film must be the same. ~ n - i r n ~ l i c i t equation ford can be obtained by equating Eqs. 4.228 and 4.229, giving wherefSL =the friction factor based on superficial liquid velocity and can be obtained from a Moody diagram (Fig. 2.2) for a Reynolds number defined by For the shear stress at the interface, where To simplify this equation, Ansari et al. used the dimensionless a p proach developed by Alves et al.36This approach defines the following dimensionless groups in addition to using the previously defined modified Lockhart and Martinelli parameters, XM and YM. and and f, = fscZ, ................................. (4.220) where Z = a correlating factor relating interfacial friction to the film thickness. On the basis of the performance of the model, Ansari et al. found that the Wallis expression for Z works well for thin films or high entrainments, whereas the Whalley and Hewite7 expression works well for thick films or low entrainments. Thus, Z = I and Eq. 4.230 then reduces to + 3006_forF, > 0.9 .................... (4.221) Eq. 4.233 can be solved iteratively by use of the Newton-Raphson approach to obtaind. If Eq. 4.233 is F(6), the derivative of Eq. 4.233 with respect to 6 yGlds - Combining Eqs. 4.218 through 4.220 yields The superficial friction-pressure gradient in the core is given by wherefsC is obtained from a Moody diagram (Fig. 2.2) for a Reynolds number defined by v,, = FEvsL + v, , and . . . . . . . . . . . . . . . . . . . . . . . . . . (4.226) Then, The pressure gradient for annular flow can be calculated by substituting the above equations into Eqs. 4.204 and 4.205. Thus, Once 6 is known, the dimensionlessgroups #F and #c can be obtained from the following forms of Eqs. 4.228 and 4.229, respectively. + pcg sin 8. . . . . . . . . . and and MULTIPHASE-FLOW PRESSURE-GRADIENT PREDICTION 51 The core density and viscosity are obtained from Eqs. 4.206 and 4.227, respectively. as and Because the pressure gradient in the film and core must be the same, the total pressure gradient can be obtained from either Eq. 4.23 1 or 4.232. Thus, = 0.36 x lo-' Pa.s. The superficial core velocity from Eq. 4.226 is vsc = (0.547)(1.208) + 1.173 = 1.834 mls. From Eq. 4.225, Ansari et al. neglected the acceleration pressure gradient for anl~ nular flow. Lopes and ~ u k l e f found that, except for a limited range of high liquid flow rates, the accelerationcomponent resulting from the exchange of liquid droplets between the core and the film is negligible. and relative roughness is Then, the flow is turbulent. From Fig. 2.2, Example 4.9-Using the Ansari et al. Mechanistic Model, Calculate the Vertical, Multiphase-Flow Pressure Gradient for Example 3.2. Similarly, from Eq. 4.217 the film Reynolds number is Given: E = 0.00006 ft = 1.83 x 10 m, d = 6.0 in. = 0.1524 m, yj~=3.97ft/sec= 1.208m/~,v~~=3.86ft/sec=1.173m/s, p~=47.61 lbm/ft3= 761.7 kg/m3, pg = 5.88 lbm/ft3= 94.1 kg/m3, DL = 8.41 dynes/cm= 8.41 x kg/s2, pL =0.97 cp=0.97 x Pa.s, and the friction factor from Fig 2.2 for a relative roughness of pg=0.016 cp=O.O16~ 10-3 Pa.s. 1.2008 X l o d 4 is Flow-Pattern Determination. fsr = 0.01708. 1. Check dispersed-bubble transition boundary (Eq. 4.161): The superficial gas-core and liquid-film pressure gradients are determined from Eqs. 4.224 and 4.216 as and Then, flow pattern is not dispersed bubble. 2. Check annular-flow transition boundary: From Eq. 4.163, Assuming that fFgsL I, the modified Lockhart and Martinelli pa= rameters can be estimated from Eqs. 4.167 and 4.168 as and Although vsg = 1.173 > 0.87 gives the annular-flow pattern, also check the Barnea transition criteria. From Eq. 4.209, Because FE =0.547, using Eq. 4.222 yields The entrainment fraction is calculated from Eq. 4.208. FE = 1 - exp[- 0.125(7.84 - IS)] = 0.547, Substituting XM, YM, Z, and HLF=46(1 -d) into Eq. 4.166 gives From Eq. 4.207, MITI TIPH49F FI OW IV WFI I C Solving the above equation for 8, b 3 0.1265. Therefore. b = (0.1265) : x (0.1524)= 0.0193 m and HZf; (4)(0.1265) (1 -0.1265) = 0.442. With the criterion from Eq. 4.165. Thus, the liquid film holdup is sufficient to bridge the pipe, and annular flow does not exist. 3. Check bubbly flow existence: With Eq. 4.158. the minimum pipe diameter at which bubbly flow can exist is and HLTE= 0.15 0.1658 - -= 0.13. 8.4383 = 8.4383, With the calculated HLTB as the new guess for HLTB. Then, bubbly flow can exist. 4. Check bubble/slug transition: From Eq. 4.160, From Eq. 4.159, the superficial liquid velocity at the transition is v,, and v,, = 3.4 m/s = 3[1.173 - (0.25)(0.151)(sin90")1 = 3.4 m/s > v , = 1.208 m/s. The flow pattern is then slug flow. SlugFlow Modeling. The step-by-step procedure described in the discussion of the slug-flow model is followed to determine all slug-flow variables. 1. Calculate 4.194 as = 8.031, m, H and HLTB = 0.13 0 00132 - - 0.1298. = 8.03 1 g ~ and v g u from Eqs. 4.190, 4.191, and , = 3.258 d s . The difference between the guessed and the calculated values of HLTB is very small. Therefore, HLTB3 0.13. 3. Solve Eq. 4.193 for VLTB: Note that HgTB= 1 - HLTB. HgTE = 1 - 0.13 = 0.87 and 05 and v,, = 1.2(2.381) v , = 9.916 (9.81)(0.1524)(1 - + (0.151)(0.826)~.~ 2.994 m/s. = [ f i ) ] 3.144 mls. = 2. Using Eqs. 4.196 through 4.199, determine HLTB. A good initial guess is HLTB = 0.15. 4. Solve Eq. 4.188 for VLLS: Note that HLLs = I - H g a . H, = 1 - 0.174 = 0.826, = (3.258 (3.258 - v,)0.826 and v, = 2.25 d s . + 3.1445)0.13, 5. Solve Eq. 4.189 for V (3.258 MULTIPHASE-FLOW PRESSURE-GRADIENT PREDICTION g ~ ~ : - 2.994)(1 - 0.826) = (3.258 - vgTE)(l- 0.13) 53 . 1.0 6.0Palm.12 m or if vSL > 0.13) /? = 0. ( ) el = [(I . The transition criterion given by Eq. When the gas void fraction exceeds 0. 4. v.97 x 10-')(0. generally follow a zigzag path when rising through a liquid. 4.1524) = 4. (0. and ultimately cause a transition to slug flow.202.0. as given by Eq.163 suggests that this transition criterion is independent of the liquid flow rate. Friction component of pressure gradient: p..0.287)(94..243.205 mls. Annular-Flow Transition.26 9.. Turbulent flow exists. Slug/Chum Transition.200. 4. is used for the slip or bubble-rise velocity.. and L. 4. the rise velocity of a Taylor bubble..160.826) = 8.779.. However. By use of a drift-flux concept. is dependent on the pipe diameter. = 142.. When y~> v. slug flow. chum.240) C is the flow coefficient given by Eq...205)(1 . BubbleISlug Transition.185 for g : 1. 4.287.. 7. Hasan and Kabir Method.57) = (1 .. 4. when y~< v.218 psilft. 4. 4. and annular flow.30 Hasan and ~ a b i r ~identified the same four flow patterns shown in ~v*~ Fig..25.240.287)(LTB). Eq. Total pressure gradient: = 4. increase its size..52. The terminal rise velocity of small bubbles given by Eq.994)(1 . the nose of the Taylor bubble sweeps the smaller bubbles ahead of it. coalesce with it.. 2.173 = ( 1 .. 4.2 ifd < 0. sin vs8 = 4(C.20 1. Hasan and Kabir adapted an approach very similar to that of Taitel et ~1. + (0.921 kPdm = 0.. Transition from bubble flow (the condition of small bubbles dispersed throughout the flow cross section) to slug flow (when the bubble becomes large enough to fill the entire cross section) requires agglomeration or coalescence. This results in collisions among bubbles.826) From Eq..02 m/s 2. 4. = 30(0. the rising smaller bubbles approach the back of the Taylor bubble. Hasan and ~ a b i r ~also developed ~-~' a mechanistic model to predict pressure gradients in wellbores.81) sin 90' = 4. To model flow-pattern transitions. . + vs).287)(4.25. When large bubbles are dispersed into small bubbles at high liquid rates... Elevation component of pressure gradient: From Eq. < 0. Dispersed-Bubble Transition.203.826) and + 8(3. other than very small ones.0. bubble coalescence cannot be prevented and transition to either slug.. with the consequent bubble agglomeration and formation of larger bubbles.12 m and ifv.241. MI'I TlPHAqF Fl OW IN WFI I S 10. Solve Eq. However. = 1. even though the gas void fraction exceeds 0. this transition boundary is a locus of constant mixture velocity values.0ifd > 0. Bubbles. They argued that the transition from slug to churn flow occurs when the liquid slug trailing a Taylor bubble attains the maximum possible void fraction of 0. .0.8)(2.163 also is used in the Hasan and Kabir mechanistic model to predict the transition to annular flow. chum flow. 4. which increases with an increase in the gas flow rate.2. The H a ~ m a t h y ~ ~ expression. = 3. transition to slug flow is inhibited. Transition to slug flow takes place at superficial gas velocities greater than that given by Eq.52.57 m .. From Fig.and v . which is possible in smaller-diameter pipes. 4.~ dicts the slug/churn transition. For the transition to dispersed-bubble flow.1)](9.240. 8. the transition then can be expressed by Eq.921.. 4.s.160 is dependent on fluid properties but independent of pipe diameter. Flow-Pattern Prediction. Assuming L a = 30d.2 Palm = 4.21: bubble flow. The Barnea and B r a ~ n e r ~model' pre~. where the turbulent intensity is maintained at the same level as that in dispersed-bubble flow.287M654.04 x 10e4Pa.2 Palm. Hasan et reported that a transition to slug flow is expected at a void fraction of 0..0.v. 4. Taitel er proposed From Eq.84 m. Eq.02 m/s From Eq. = (0..4) + (0. (4. or annular flow must occur. calculate Lsu and LTBfrom the definition of g : L . In other words.016 x 10-~)(1. 86 ftfsec = 1. where C and v are given by Eqs. fine liquid droplets flow in the gas core with a velocity the same as the gas phase while a thin liquid film creeps up the pipe wall.. .. The friction component can be computed by treating the multiphase mixture as a homogeneous fluid.97 ftlsec = 1.97 x 10-j Pa. section. is small and can be neglected.240): = 0. Calculate the Vertical.41 dynes/cm=8.97 cp=0. respectively. d=6. Steen and wallis5? suggested that.. Flow-Pattern Determination. the total pressure gradient can be obtained by Eq.241 and 4. xg. is defined in terms of the gas mass fraction. Thus.151] = 0..83 x m. > vsgBls.0.. ~ p~= 47. Multiphase-Flow Pressure Gradient for Example 3.246. the Lockhart and ~ a r t i n e l l i ~ ~ parameter.245 suggests that an accurate estimation of the liquid holdup is essential when computing the elevation component.208 m/s. 1.f. 2. but with different values for C. Acceleration.Flow-Behavior Prediction.. 4. Wallisis suggested that the wall shear stress around the vapor bubble be ignored. = and 1. because of the chaotic flow nature that tends to make the gas concentration profile flat.209. the friction pressure drop pertains to the gas interacting with the wavy liquid film. With this assumption..I [(1.1524 m. Because vs.. As in bubble flow.15 be used for C rather than a value of 1.00006 ft= 1. 4. for low-pressure systems. . Check bubblelslug transition (Eq. (4. indicating the similarity in evaluating the friction terms in slug and bubble flow. Bubble and Dispersed-Bubble Flow. ~ < 4 ~~ FE = 0.20 if v. Because of the different hydrodynamics in each flow pattern.v .857 log. 4.41 X l o w 3 kg/s2. applied in slug flow. 0 0 5 5 if v. vsg = 3. The drift-flux model of Eq.2) for a Reynolds number defined as The liquid holdup in the core can be given by this simple equation from WallisIs where X.88 lbm/ft3 = 94. and v given by C = 1. 1 = .248. however. is where v.pg=0. .s. However.. can be written as where and .10-Using the Hasan and Kabir Mechanistic Model.. Thus.246 also was . is defined by Eq..244 by use of Eqs. can be determined from the Moody diagram (Fig. 4.2 and . and fluid properties as because pL/pL This was recommended by Govier and ~ z i z * ~ would not be too different from p.bubble flow does not exist. is suggested that a value of 1.O16x Pa.160. 4.207. Slug and Churn Flow.f.2.151 d s ./p.2) + 0. pg = 5. components. in-situ mixture density. This component accounts for most of the pressure drop occurring in the bubble. Annular Flow. v s = 3. The product pLHLis very nearly equal to p. The total pressure gradient in twophase flow can be written as the sum of the gravitation or hydrostatic head (dp/dL). and the contribution of the friction component to the total pressure gradient is very small.2)(1.0 in. In bubble and dispersedbubble flow the expression for holdup. In annular flow. friction (dp/dL)f and acceleration (dp/dL). 4.246 and 4. is given by Eq.016 cp=O. 4.2. which is a large component of the total pressure gradient for annular flow. Hence. The estimation of the friction component presents some difficulty because some of the liquid flows downward in a film around the Taylor bubble while most of the liquid flows upward in the liquid slug. 4. > 4 . it is appropriate to estimate the liquid holdup in the central core rather than for the entire pipe cross c.1 kg/m3 OL =8.. ILc. HL.and slug-flow patterns.7 kg/m3.244 can be used with the mixture density calculated from the liquid holdup estimated with Eq.s. the gas void fraction is for the channel volume not occupied by the liquid film. the friction pressure gradient becomes Example 4. In annular flow. 4.572. To estimate total pressure gradient. the acceleration component can be neglected during all but the annular-flow pattern. p. The friction pressure gradient.61 lbm/ft3 = 761.. Eq. The contribution of the friction component is no longer negligible but is still small (typically 10% of the total gradient). 4.. the gas mass fraction calculation should include the entrained liquid droplets.250) where In general. =0.2 used for the slug-flow pattern..208) (4 . The liquid holdup for the core. and friction factor. when the liquid film is fully turbulent.pL =0. it .. It was recommended that the equations developed for slug flow also be used for the chum-flow pattern. Eq.173 d s . are made separately. HL.. Friction factor... estimations of holdup. the liquid entrainment can be obtained as a unique function of the critical vapor velocity from FE = 0 . . Given: E =0.1. In the case of annular flow. . 700 6 to 27.115 Palm. bubblepoint pressures calculated from different black-oil models can have errors in excess of 50%.3 to 46 24 to 86 'Includes dala from Poettrnann and Carpenter.5 to 70.64) + (94." M e s ~ u l a mC~ ~ a ~ h oand field data from several oil companies. 3.245): HL=land 1.3 to 69 600 to 23.247.38) + 0.7)(0. 3. ~ such programming inadequacies. = (761.2.249. and that adequate information is available to calculate fluid temperatures as a function of depth.6-RANGE OF DATA IN TUFFP WELL DATA BANK^^ Nominal Diameter Source Old TUFFP Databank' Govier and FogarasiS4 ~sheim~ Chierici et a/. The databank contained 1. 3. Calculate slip velocity (Eq. Depending on the depths of wells. an evaluation of six commonly used correlations and two mechanistic models was performed by Ansari et n1.401 p. Calculate holdup (Eq.000 0.173 = 0.Flo w Modeling. With this introduction. = (521.5 Pdrn. Friction component of pressure gradient: From Eq. that sufficient information is available to predict pressure/volurneltemperature (PVT) relationships. . Evaluation of the cor= 150. this can lead to serious errors in pressure-drop caIcuIations.233 psilft.600 720 to 27.l0 Prudhoe Bay (in. MULTIPHASE FLOW IN WELLS . 4. 4.36) = 521. 4. 4. 4." Espanol. Therefore. relations and models with the databank was based on these statistical parameters. C = 1. such as ensuring that the calculated liquid holdup always exceeds the no-slip holdup for upward flow. that a pressure-gradient-prediction method is specified.1)(0. neither dispersed-bubble nor chum flow exist.712 well cases with a wide range of data.TABLE 4.243): 5.567 114to27.3 Evaluation of Wellbore Pressure-Gradient-Prediction Methods 2.6.7. as shown in Table 4. The pressure drop in a well can be calculated with the computing algorithm described in Sec. The flow pattern is slug. Each requirement is apotential source of error when calculating pressure drop.3. Slug.24753The pressure-drop predictions from applying each correlation and model by use of the computing algorithm described in Sec. From Fig.266 kPdm or V. From Eq. am .' Fancher and Brown.400 740 to 55. When using empirical correlations or mechanistic models to predict pressure gradient.8 Hagedorn.244.DB =4.5 to 7 Oil Rate (STBID) 0 to10. = 0.2 Orkiszewsk1.3 Baxendell and Thomas. Many times the profile is unknown and vertical flow is incorrectly assumed. another common source of errors is the frequent absence of reality checks.248): 4.~ 2. Black-oil-model PVT calculations.265. This in turn can lead to very contradictory results when evaluating pressure-gradient-prediction methods with different PVT correlations. 4.'2 and Aziz et ~ 1may be vulnerable to .401 > 2. 4. such as Duns and ~ 0 sMukherjee and Brill.000 Gas Rate (MscflD) 1. .2)(2.914 200 to 110.150 8 to 1.5 1010.5 17to112 35 to 86 8. Correlations that have not been modified.246) and slip density (Eq. Elevation component of pressure gradient: From Eq. For slug flow. Correlations that have been modified normally include reality checks. ~ .4 kg/m3.) 1to8 2 to 4 Z7l8 to 6 2'18 to 5 5.7 were compared with measured data in a well databank developed at TUFFP. PVT property correlations often incorrectly predict the part of a well in which single-phase flow occurs. This requires that the profile of the well is known.81) sin90° = 5. Turbulent flow exists.1 Criteria for Comparison With Data.2.64 (1.000 Oil Gravity ("API) 9.4)(9.381. can be highly inaccurate under many real applications. 4. which include the prediction of mass transfer and fluid physical properties of the gas and liquid phases.5 P d m = 5. 4. A common error in deviated wells is to not use accurate well profiles in the marching algorithm. 1. 2. Total pressure gradient: = 5. For example. Check dispersed-bubble transition boundary (Eq. The excellent performance of the Hagedorn and Brown correlation can be explained only by the extensive data that were used in its development 'Personal wmmuncation with A.3 Overall Evaluation.052 1.378 0. model for annular flow is significantly better than all other methods. E2 indicates an average of how large the errors are. Aziz et al.883 1. DunsRos= Duns and Ros correlation HasKa = Hasan and Kablr model BegBr = Beggs and Br~ll correlat~onOrk ~ = Orkiszewski correlation and MukBr = Mukherjee and Br~ll s correlation evaluation also involved the use of a relative performance factor FT.029 0.226 4.'^ Aziz et Ansari er and Hasan and ~ a b i r .085 1.128 5.262 1.852 2. 11 gives results for those cases in the total databank that were predicted to be in annular flow for 100% of the well length. where E4 indicates the overall trend independent of the measured pressure drop. this can be attributed to the use of flow mechanisms. Duns and Ros. The correlations and models used for the comparison were a modified Hagedorn and ~ r o w n ?Duns and ROS? Orkiszewski8 with the Triggia* correction.The~ . Col.774 1.143 2.295 0.463 1.7-RELATIVE PERFORMANCE FACTORSz4 I 745 0. 2 ) and for deviated well cases only (Col.296 1.678 2.486 2.601 29 0. 7 through 10 give results for cases predicted to have slug flow over 100% of the well length.. The cases used for Cols.803 1.140 Ansari HagBr Aziz DunsRos HasKa Orkis MukBr EDB = entlre data bank. 7 and 8 were selected from the entire databank.838 3.128 0.764 2.Ansari= Ansan eta1 model. Ansari er al. Petrobrhs S.919 2.VSNH ..792 1. For the latter three methods..314 1. E6 indicates the scattering of the errors about their average error.683 70 0.3.000 755 0.. 4.546 0.712 0. 1 of Table4.000 0. AB =all well cases wlth 75% bubble flow AS =all well cases w~th100% slug flow VS =vertical well cases w~lh 100% slug flow SNH = all well cases wlth 100% slug flow wrthout Hagedornand Brown data VSNH = allvertical well cases wnh 100% slug flow w~thout Hagedorn and Brown data AAN = all well cases with 1Wooannular flow. An overall evaluation first was performed by use of the entire databank.4 Evaluation of Individual Flow-Pattern Models.062 1.678 2. 9 and 10 were selected from the reduced databank. To make the evaluation unbiased with respect to the methods..A. The .647 626 1.005 5.719 1. Rio deJaneim (1984). Col.7.318 5.121 0.3.798 2.7 gives an evaluation of the methods in terms of Fv with the best value for each column being boldfaced.575 3. which was defined by where E l indicates the overall trend of the performance relative to the measured pressure drop.n 1. Performance of individual flow-pattern models is based on sets of data that are dominant in one particular flow pattern. E3 indicates the degree to which the errors are scattered about their average percent error. whereas the cases used for Cols.043 6. HagBr = Hagedorn and Brown correlat~on.711 1.780 4.000 1. The performance also was checked for vertical well cases only (Col. concluded that the overall performance of their model was better than all other methods.000 0.403 4. indicating the best and worst performances. The performance of the Ansari er a / .213 1.086 1.940 4. the overall performances of the Hagedorn and Brown. 4 shows the results for all vertical well cases.273 4. Es also is independent of the measured pressure drop and indicates the magnitude of the average error. DW = dev~ated well cases. Beggs and Brill1' with the Payne et correction.Table 4. VW =vertical well cases. ANH = all well cases mthout Hagedornand Brown data..056 2.836 5.2 Comparison Method.284 4.977 387 0. and Hasan and Kabir methods are comparable.112 0.876 0. VNH = vertical cases w~thout Hagedornand Brown data.461 0. 5 shows the results for combined vertical and deviated well cases.VNH .201 2. Cols.998 4. Mukhe j e e and rill. ~ ~ was ~ evaluation accomplished by a comparison of the statistical ParXneterS. Tnggla. a second database was created that excluded 33 1 sets of data from the Hagedorn and Brown study..TABLE 4. as predicted by the transitions described by Ansari et ~ 1 . These results are shown in Col.081 0.551 4.688 4. resulting in Col.343 654 1.516 ED0 VW OW ANH AB AS VS SNH AAN . However.485 1.312 1.457 1..322 6.381 0.028 2.009 1.044 3. 4. To 4 ~ have an adequate number of cases for the bubble-flow-pattern database. Aziz = Azlz et a1 correlation. The minimum and maximum possible values for FT are 0 and 6. 6. respect~vely.214 1.386 1.108 1.700 0.600 1.142 0939 1. cases with bubble flow predicted over 75% of the well length were considered.3. For this reduced databank.585 1. 3)..909 1. 4.590 3. Col. These conclusions were derived from deviated wells with larger tubing sizes. .Angel and ~ e l c h o n and ~ i n k l e pres-~ . Flow-Pattern-Transition Prediction. the Ansari et al.267 and 4. Bubble-/Slug-Flow Transition. . . all are flow-pattern independent. = d.. .66-67conducted a combined experimental and theoretical study of upward. . Z4 for circular pipes. ~ O el was modified for the bubblelslug transition in an annulus by use of values of the gas void fraction measured at this transition. . determination of flow-pattern-transition boundaries. . 4. The new mechanistic models give reasonable results in both oil and gas wells.. (4. the Ansari et al. The transition from either bubble or slug flow to dispersed-bubble flow then becomes where the hydraulic diameter is defined by d. . . transition to slug flow was found experimentally to occur at average gas void-fraction values of 0. None of these data was used in the development of the pressure-gradient-prediction methods. . . 2. Each is described in following sections. For annular flow. They concluded that the empirical correlations by Orkiszewski and Duns and Ros appeared to be less efficient than mechanistic models. . The use of mechanistic models to predict multiphase-flow behavior in annuli is a recent trend. .270) M1'1T I P H I Y F F1 OW IN WFI 1 9 . . The ~ n i & et al. Although the Hagedorn and Brown correlation performed better than the others for deviated wells. .. The predictions either have applied correlations originally developed for flow in pipes by use of the hydraulic diameter concept or have applied correlations developed from data obtained from two-phase flow in an annulus. . . we strongly recommend that users exercise careful judgment based on evaluation of actual measured data to select a particular method. ~ ~ . Traditional methods to predict pressure drops. Caetano et a1. Pucknell et compared predicted pressure drops with measured pressure drops for 246 data sets collected from eight fields.4. . The theoretical part included development of flow-pattern-prediction models and the formulation of models for each flow pattern to predict average volumetric liquid holdup and pressure gradient. 11). . slug-flow model. development of flow-pattern maps. two-phase flow in vertical concentric and fully eccentric annuli. evaluation of pressure-gradient-prediction methods to calculate pressure drop in wells is a difficult task. 3. no single method gives accurate predictions of bottomhole flowing pressures in all fields. Most correlations make predictions without taking into account the existing flow pattern.. . such as that of Duns and Ros. Thus. . Despite the development of new mechanistic models. Salim and Stanislava compared methods that describe the flow of gaslliquid mixture in wells with more than 189 data sets collected from five different sources.. Except for the Winkler correlation. . The experimental study included the establishment of flow-pattern definitions.. In this context. . . . 62% of the pressure drops were predicted with errors of less than f 6%.67 and Hasan and Kabifi8 presented mechanistic models to predict flow behavior when gas andliquid si58 Eqs. model performed best (Cols. . BaxenGaither et ~ l . The Ansari er al. 4. . . the Taitel e t a ~ . . .266) and 4 and d. . . model performed best when Hagedorn and Brown dataare excluded for both all well cases and all vertical well cases (Cols. Reliable convergence is desirable in any multiphase-flow i model. when the data without Hagedorn and Brown are considered. In gas wells. respectively. . = d. These correlations were developed empirically from experimental data. proposed a method to predict both liquid holdup and pressure gradient on the basis of dimensional analysis and experimental data. 7 and 8). At high superficial liquid a~~ velocities. . are the inside casing diameter and the outside tubing diameter. their method exceeds the performance of the Ansari et al. but the Hasan and Kabir model performed somewhat better.20 for flow through a concentric annulus and 0. The Ansari et al. used Fanning friction factors throughout the study. 3). correlations were the most common procedure to predict liquid holdup and pressure gradient for multiphase flow in an annulus.. Caetano etal. 9 and 10). Most of their field data were taken from wells that exhibited the annularlmist-flow pattern. They reached four specific conclusions.. . Only 29 cases satisfied the bubble-flow criteria. In oil wells. typical of North Sea wells. . .~ r~ ented empirical correlations for the prediction of presssure gradients in annuli.268 constitute the criteria for the bubblelslug transition boundary at low liquid flow rates in concentric and fully eccentric annuli. The resulting equations for the transition then can be expressed by and ! i In the past.1 Caetano et al. . the ~ a r n e criterion for transition to dispersed-bubble flow was modified with the hydraulic-diameter concept.4 Pressure-Gradient Prediction in Annuli multaneously flow upward in an annulus. None of these empirical correlations can be considered general or comprehensive.66. and measurements of average volumetric liquid holdup and average pressure gradient for several tests in each flow pattern. . and Hasan and Kabir models behave no better than many of the traditional methods in predicting erroneous discontinuities. . mechanistic model gives the best results of all the methods evaluated. . respectively. . . ...15 for modflow through a fully eccentric annulus. The models proposed to predict flow-pattern transitions in an annulus are similar to those r proposed by ~ n s a r e al. The experimental part of the study involved flowing air with either water or kerosene up a vertical annulus. the Ansari et al. (4. . nor do they capture the complex flow behavior that occurs when gas and liquid flow simultaneously in an annulus. I. including a gas and gas-condensate field. none gave satisfactory results (Col. + dl . 68% of the pressure drops were calculated within f 15%.. When the Hagedorn and Brown data are included (Cols. give good results in oil wells. similar to their use in predicting regular pipe flow. . but can give very poor results in gas wells.. 4. 4. . For equiperiphery diameters larger than this. Method. In fact.. Overall. gashquid. Transition to Dispersed-Bubble Flow. They are preserved here to retain consistency with the original work.and modifications that have been made to the correlation. model is significantly better than all other methods (Col. model performed only as well as existing methods. which primarily is for slug-flow conditions. . The minimum equiperiphery diameter for which bubble flow occurs is where dEp is the equiperiphery diameter defined by d. Caetano et a1. The Hasan and Kabir model gave the best predictions for these cases. 4 and 5).dl . .. Clearly. HL. Bubble Flow. . The Reynolds number for bubble flow is defined by ~ + vsR . The basic concept for modeling the bubble-flow pattern is the slippage that takes place between the gas and the liquid phases. the total pressure gradient for bubble flow is Eq.272 and 4. . HL. . However. For pipe flow. 4. as given by Eq. 4. The model developed for the case of a constant film thickness constitutes a simplified form of the model developed by Fernandes et for upward vertical pipe flow.f is the Fanning friction factor evaluated for the homogenous mixture flowing in either the concentric or eccentric annulus.5. Higher values of void fraction will cause transition to slug flow. These components are referred to as elevation. 4. pipes encountered in producing wells are so long that to treat the entire slug-flow region as fully developed flow should introduce only minor errors. Applying this criterion to slip flow yields this transition boundary to dispersed-bubble flow for gas void fractions of 0. is determined by the method presented in Chap. . . The Fanning friction factor.283 also can be used for dispersed-bubble flow by replacing liquid holdup in Eq. found that the rise velocity of a Taylor bubble can be predicted by using the equiperipheral-diameter concept. . The model for dispersed-bubble flow is based on the no-slip. . . . 2.56 and 2. as given by Eqs. Therefore. Thus. Therefore. . The second configuration is for a developing Taylor bubble. For the case of a cap bubble with varying film thickness. LL.163. developed a hydrodynamic model for slug flow in annuli that considers two possible configurations.275. . 4. the translational velocity of the Taylor bubble is which is an implicit equation for the liquid holdup. . as discussed previously. For a uniform bubble-size distribution and a cubic lattice packing. 4. Successful modeling of two-phase-flow pressure gradient requires analyzing each pressure-gradient component as a function of the existing flow pattern.277.278. 4. . respectively. Chap. The bubble-swarm index. p ~ p is pTp = p L I L + p g ( l . homogeneous nature of this flow pattern. The figure shows a slug unit. respectively.57.. The friction-pressure-gradient component for annuli configurations is given by where the mixture velocity. consisting of a Taylor-bubble and a film region.52. . which can be used as the average thickness for the entire film zone. the elevation component is where and the in-situ liquid holdup. the model follows ~ . . The total pressure gradient for steady-state flow consists of three components. . ~ ~ the approach of McQuillan and ~ h a l l ewho studied the transition from bubble to slug flow.276 with no-slip liquid holdup. based on the in-situ liquid holdup. friction. The first configuration is for fully developed Taylor-bubble flow and occurs when the bubble cap length is negligible compared with the total liquid film length. . The transition is given by Eq. . Taitel et ~ 1 sug. The elevation component is evaluated rigorously through the slip density. . Mechanistic models were developed to predict flow behavior for all flow patterns in annuli configurations. Thus. 4. agrees with the value used by Ansari et Combining Eqs. Flow-Behavior Prediction.52 is the maximum allowable gas void fraction under dispersedbubble conditions. and holdup values corresponding to the different regions of the slug unit. . . and convective acceleration. . . The required minimum gas velocity is determined from a balance between gravity and drag forces acting on the largest stable droplet. Caetano et al.274. . f'. 3 gave a descriptive representation of slug flow in an annulus. 4. . . . Also given are the lengths. is given by Combining Eqs. . dispersed bubble. Each model is presented. Under this condition. given by ~ v. velocities. . Thus. gested that the mechanism causing transition to annular flow is related to the minimum gas velocity necessary to transport upward the largest liquid droplet entrained in the gas core.273..281) where the no-slip liquid holdup. is Transition to Annular Flow. This includes bubble. (4. 4. the film thickness reaches a constant terminal value. Slug Flow. . is given by Eq. .282. For this case. . . which consists of only a cap bubble. AL. . . Taylor-Bubble Translational Velocity. determined experimentally and used in the model. the film thickness varies continuously along the film zone and cannot be assumed constant. . slug. .177 and rearranging yields Dispersed-Bubble Flow. 0.177 gave the rise velocity of a single gas bubble in a bubbleswarm medium.279) . v. Caetano et al. p.13. . The mixture viscosity. and 4. 2 for annuli configurations. a m ~ d e for the developing Taylor-bubble l region is not included in this monograph. . . . The acceleration pressure gradient in bubble flow is normally negligible. n' = 0. . . followed by a liquid slug. Fig. . . = v. and annular flow. extended the model using characteristics observed for slug flow in annuli. 4.4. . Caetano et al. . Eq. .(4.25 shows the physical model for fully developed slug flow. . Similarly.177. a mass balance for the gas phase yields Liquid-Film Zone. Fernandes etal. . . . Slippage takes place in the liquid-slug zone as the bubbles rise through the liquid phase....290 and 4. .. kU. . The liquid film flowing downward and surrounding the Taylor bubble can be considered as a free-falling film. The free-falling film is assumed to have a thickness and flow-rate relationship at the bottom of the Taylor bubble identical to that for a falling film on the surface of a vertical plane or cylinder. for a concentric annulus the liquid holdup in the liquid-slug zone is HLrs = 0.. For a fully developed Taylor bubble. . 8.. The liquid and gas phases in the liquid-slug zone are assumed to behave like bubble flow. .. the in-situ liquid velocity in the liquid-slug zone becomes Overall Mass Balances..291) = superficial liquid velocity vsg = superficial gas velocity V L L ~ = in-situ liquid velocity in liquid slug v g ~ s= in-situ gas velocity in liquid slug VLTB = in-situ liquid velocity in liquid film V g = in-situ gas velocity in Taylor bubble ~ ~ VTB = Taylor-bubble transitional velocity HLLS = liquid holdup in liquid slug LLs = length of liquid slug LLF = length of liquid film Lsu = length of slug unit VSL Combining Eqs.. mass and volume balances are equivalent.288) Combining Eqs. An overall volume balance on the liquid phase yields where the slug-unit length. a mass balance for the liquid-slug region yields where the flow area for an annulus is A = T ( d c . For turbulent flow.. The flow of gas and liquid within a slug unit was assumed previously to be incompressible.. . They assumed that the liquid holdup at the bubblelslug-transition boundary was constant.. for a fully eccentric annulus.. 4.9086 and 113.. .. . The relationship between the film thickness and the film velocity for this case is given by The indices Cx and CM depend on the flow regime in the film zone. this yields Fig.000. The slip velocity can be approximated by the bubble-rise velocity in a bubble swarm as previously given by Eq. Thus.. were analytically determined by wallis18 to be 0.287.. .. n 2 .000. . respectively.. the holdup in the liquid slug also is constant and has the same value as at the transition conditions.80. .85. . the in-situ liquid velocity in the film surrounding the Taylor bubble.. NReLTB 1. 4. (4. (4. . . .39 recommended > ~' using CK = 0. Similarly. the liquid holdup in the film zone can be determined from the flow geometry and the film thickness. = L. This situation is expected to be valid when the entry region used to develop the velocity profile is less than the length of the film. 60 < For laminar flow. .25--Fully developed slug Mass Balance in Slug and Film Zone. is considered positive for downward flow. . . .. the indices CK and C . NReLTB 1.. . . .177 and 4. Therefore.25 yields where VLTB. where the film reaches a terminal constant thickness. Thus.df)... 4.Because the total volumetric flow rate is constant at any section in the liquid slug. .0682 and C . MULTIPHASE FLOW IN WELLS . Because gas in the liquid slug and in the Taylor bubble are both flowing upward. H m =0.291 yields A similar procedure can be applied for the gas phase. proposed that the liquid holdup in the slug . = 213.. 4. + L.. . HLTB is the liquid holdup at a cross section that contains the terminal film thickness. Following Barnea and ~ r a u n e rCaetano et al. . .. 4. ~ r o t z determined these values experimentally. . is L. The Reynolds number associated with the liquid-film flow is Liquid-Slug Zone.. A mass balance on the liquid phase between Planes A-A and B-B of Fig.~~ zone can be determined from the liquid holdup occuning at the transition boundary between bubble and slug flow. . is replaced by p.292 or Eq. respectively. An iterative procedure on the film thickness. but with different values. . . and Sc.Average Liquid Holdup. Thus. The gas and liquid droplets flowing in the annulus core are assumed to flow as a homogeneous mixture with the same velocity.301 to obtain liquid perimeters on the casing wall and liquid film core-mixture interface.. is the total area of the casing liquid film. f'. . Linear Momentum Equations.303 through 4. 4. The pressure-gradient components are determined for the liquid-slug zone and then averaged over the entire slug unit length. . a linear momentum balance for the inner (tubing) film yields where all the terms have the same meaning as given for similar terms in Eq.293. assumed the Taylor bubble to be a constant pressure region. The elevation pressure-gradient component for the slug unit is calculated through the slip density as Gas and Liquid where the slip density for the gaslliquid mixture in the liquid slug is Wail Wetted Perimeter Casing Film. HLrBand VLTB are the average liquid holdup and film velocity in the entire film zone. In th~s equation. t and rc.303..-(UTE LSU +~ ~ ( v L T B~ + m j .Fig.26 gives a schematic description of annular flow in a concentric annulus. . . The model is based on equilibrium fully developed flow. The gaslliquid interfaces are considered stable. p c is the density of the mixture in the core. 4 The acceleration pressure-gradient component is related to the amount of energy required to accelerate the liquid film. where p . to the existing upward in-situ liquid velocity in the liquid slug. 61 . The two liquid films are assumed to have uniform thickness. . S.26--Annular flow in concentric annulus.. . liquid film. 4. St For a fully developed Taylor bubble..307) . (2) acc Section A-A' HLTB = p . respectively. which is initially flowing downward. 4.pc)g = 0 . is determined by the method presented in Chap. and& is the area of the core occupied by the mixture. this equilibrium condition exists. The friction pressure gradient is obtained from Fig. 2. The total pressure gradient for the slug-flow pattern then can be expressed by combining Eqs. The phases are assumed to be incompressible. 6.299. (dp/dLS is the total pressure gradient for the casing . . the pressure gradient that occurs in the Taylor bubble and film zone is neglected. and HLTB can be determined from Eq. 4. 4. are the shear stresses at the casing wall and liquid film core-mlxture interface. all the pressure gradient for an entire slug unit is assumed to occur in the liquid-slug zone. S. The slug-unit average liquid holdup..296 for a fully developed Taylor bubble.(pL. A. 4. are the wetted MULTIPHASE-FLOW PRESSURE-GRADIENT PREDICTION S + r "2 . two combined-momentum equations can be written. 4. 4.298 through 4.305. HLSu. 4. . can be determined as where LLslku is determined from Eq. Because Caetano et al.67 where the Fanning friction factor.280. Interface GarlLiquid Perimeter Tubing Film Wall Wetted Perimete Tubing Fllrn. With this equilibrium condition and the linear momentum equations given by Eqs. respectively. A linear momentum balance for the gasldroplets mixture flowing in the core of the annulus yields Annular Flow. Similarly. is required.. The corresponding Reynolds number for the slug body is determined by Eq. be GasAiauid ~omogeneous Mixtun niform Tubing-Llquldilm Thickness. Ac (4. The conservation of linear momentum for the outer (casing) liquid film yields where (dp/dLk is the total pressure gradient for the mixture in the core. Pressure Gradient. 4.. Interface GulLiquid Perimeter Caring Film F li m \ Uniform Casing-LiquidThkkness.. hence. . the slug-body slip density given by Eq. . . .325. .... and SIi = n(d. . . . .@. . . .3 18. .(6. . = n(d. . respectively. .46. . . With the shear stress and geometrical relationships defined in Eqs.325) S . . . (4. . . ..309) A Ac' . which is evaluated by a Blasius-type expression given by Similarly. f: and = cc . =f. mixture density. . . ( d . S. . .. . . . . .(d.. . glven by Eqs.-L Ad S + r . respectively. . given by Eq. . tubing-liquid film. . .. The perimeters associated with the casing-liquid film and tubingliquid film. . 4. . .314) f = f.. .(d. . . .322) and V c = "sc- . . . . . (4.. and f ' =Fanning friction factor associated with the interface. 4. . = x d .Gr(dr + dl)] The wall shear-stress terms are given by X 4(dt [df + 26. . . .332 and Eq. =casing-liquid-film thickness and tubing-liquidfilm thickness. .. . . 2. 4. p is the density for the phase which wets the wall. . . . .) . 4. .26. . .320 for a Reynolds number based on the superficial mixture velocity. and A =annulus cross section For flow in an annulus. . . . .. . .321) s. .2. . respectively. . Ad = xd. . .df -@.) . . .310) and A . . . =in-situ mixture velocity in the core. ... and f is the associated Fanning friction factor.(4. .. . .6..315) where f j c =friction factor obtained with Eq. is evaluated by taking into account the annulusconfiguration geometry described in Chap. . . . .. . . . = mixture denslty in the core. . .. . . . . .. . the combined linear momentum equation for the casing-liquid film and the mixture in the core. .t. . . .HLc) . .). . (4. . . . .df .. . . .338. = nd. .3. . . .323) + 6.. . .(l + . (4. . (4.. . .X [d. 4. . . v is the in-situ average phase velocity. . .. 4. . . . .0. . . . .and S .220 and 4. the hydraulic diameters associated with each region can be written as -f dc l ' p ~ ~ ~ .. where T. . . . .. modified the Wallis interfacial-frictionfactor correlation for thln films or hlgh entrainment fractions in annular vertical flow.d: . . . .).. 'espectively. C. 4. .308) On the basis of the annulus geometry given in Fig.26 and because the films are considered uniform in thickness. . . . as given by Eq. 4. . . . . . . + sc. . . . The casing-liquidfilm area. (4. . . ..@. . (4.309 through 4. . . The resulting casing-film-interface and tubing-film-interface friction factors are. . =interfacial shear stress. .. 4. . .3 11. . .d. . .313) . and the core area for the mixture are. .+ . . . .'pcT. . . can be rewritten as By applying the hydraulic-diameter concept to the casing-liquid film.1 for laminar flow and n = . vi [ - CC (1 + 3m2)pc4] - .. . v. . . . . are S. . 4. . can be rewritten as d. 4. . + dl)] where r is the respective wall shear stress. .. ) .308. . . and core and with Eqs. . (4. . . . . . . . . .) f = CN". . . . . . the combined linear momentum equation for the tubingliquid film and the mixture in the core.. .6. . geometrical relationships for the relevant dimensions can be written. . . .. . . . . . . (4. . ..@. . . . . .25 for turbulent flow..(d. .Adl + r . . . . . ' . . .320) where n = .. given in Eq.317) and + 4(dc . . .. .315. . . (4. ...(d. . .). . (4. p. . . .(dl + dl)].f t ' p L ~ d . . . .. . . . . .221. .). . - bL- . . . . .. (4. . ACf = nd. . . The mixture density and velocity were defined as p c = pLHLc p. . . . .. .. . A c is given in Eq. .) [df .320 uses either 4fi d@. . . . .324) . where 4 and d. . . . the ~nterfaclal shear stress is approxlrnated by r. ... . . [fsc (I + 3oo$)pc$] . .312) . . . . The Reynolds number in Eq. . = 0- vg . .6 . . . . . . .(d. . . . . . . the tubing-liquid-film area. . . . . + 6. . are given by Eq. ..307. . . . (4. . . . . . . and hydraulic diameter in the core. . . - Neglecting the liquid film velocities in comparison with the core velocity. for both the wall and interface sides. . . . . . + 26. . . .) . .. . . . ~ v2 f . . depending on the wall shear or stress being evaluated. . .. . . . . . . .309 through 4..) (dl + 263 dl@. . . . . 4. The coefficient. Caetano et al. . (4. .) .3 1 1) where HLc and v. + 6. . .. .. . . . .. . .. . .' .(dl . . . . However.d. During the numerical determination of the average angle of view for an eccentric annulus. which flows in the form of two films wetting the bounding walls and as droplets entrained in the gas core. the integrand expression in Eq. as given by Eq. This core-mixture velocity is given by The in-situ liquid holdup in the core.) . 4. a reduces to 0. Using Eqs. in terms of the annulus pipe-diameter ratio and degree of eccentricity. 4. 4.4(d.) + [df . yields <w> = --l6 (df . <W.209 gave the Wallis18correlation. 4. Phase Continuity Equations. At equilibrium conditions. Consequently. For a fully eccentric annulus.)] I Eqs. the liquid droplets deposition and entrainment rates are equal for each liquid film. Summary of Governing Equations: Dimensionless Form. 4. is where K is the annulus pipe-diameter ratio.309 and 4. Similarly. The normalizing variable selected for length is the hydraulic diameter for the annulus configuration. given by Eq. and the integral result is the one given by Eq. 4. 4.288. if e # 0 .336 for concentric and eccentric annuli. Fig. and 4. is the total in-situ liquid holdup in the annulus. which flows in the core bounded by the two liquid films.309 through 4.4dc(dc .27 shows the results found for the tubinglcasing-liquidfilm-thickness ratio..) \ sin-(2)rdr and where vgcis the in-situ gas velocity in the core and HL.270. 4. The continuity for the liquid phase. 4.3 11. The liquid holdup for the casing-liquid film is defined as the ratio between the area for this film and the total annulus area.332 yields MULTIPHASE-FLOW PRESSURE-GRADIENT PREDICTION . A prediction for the ratio between the thicknesses of the tubing film and the casing film was developed by assuming equilibrium between the entrainment and deposition rates of liquid droplets.5.4d.df . resulting in this relationship between the tubing. 4. The governing equations developed previously for the annular flow model can be presented in dimensionless form.336. no analytical solution is found for Eq. + 26.336 becomes independent of the integration variable.334. where the gasAiquid mixture is assumed to be homogeneous. which predicted the liquid fraction entrained in the gas core of long pipes under annular two-phase flow. these singularities also appeared for extremely low values of pipe-diameter ratio. singularities were fwnd when the pipe-diameter ratio equaled zero. yields where <Wi > is the cross-sectional average of the angle of view associated with the tubing deposition area as seen by the entrained droplets. Hkf = in-situ liquid holdup in the casing film. and FE =fraction of liquid entrained in the gas core.4. The normalizing variables selected for velocity are the superficial liquid velocity for the in-situ velocities in the liquid films and the core-mixture velocity for the in-situ velocity in this region. the in-situ liquid holdup for the tubing-liquid film is given by and e is the degree of annulus eccentricity. TubinglCasing-Liquid-Film-ThicknessRatio. = n ( l > 1 . where H L f = in-situ liquid holdup in the tubing film. FE. The integration can be carried out numerically by applying a technique like Simpson's rule.335.and casing-liquid-film thicknesses. Consequently. respectively. + d.(d.335 and 4. is given by The total liquid holdup in the annulus is the sum of the liquid holdups in the core and in the two films. 4.27 should not be used for liquid-film ratio determination for an eccentric annulus in these singularity regions. the curves in Fig. The continuity for the gas phase. 414. > can be expressed in terms of geometrical variables by Eqs. a. Note that when e = 0 (concentric annulus).208 and 4. 8.KZ) where the parameter.d. Using Eqs.330 through 4.288 yields <w. . . .27-Liquid-film-thickness ratio in annuli.a]. (4.339 and 4.. and the dimensionless group. (4..340 by use of an iterative procedure to determine the dimensionless casing-liquid-film thickness. . The experimental part of their study involved flowing air through a stagnant water column in a vertical annulus....... ... G1....(I . . .. . .. are )~~ In Eqs.. (4... For a given set of phase flow rates.353) In Eqs... Fig. . .K)] . . ..339 and 4.j 3 . .345) Annulus Pipe Diameter Ratio. . The resulting elevation pressure drop was used to calculate the void fraction.H [ l . 4.. . . and (4.. . .a] ..347) . and 4.341 and 4. . and configuration geometry.. 4... YM. ( d ~ l d Land (dpldLlsc = superficial pressure gradients of the liquid phase and the core mixture. the dimensionless groups.. Eq. .. Hasan and KabiP8 investigated two-phase flow in vertical and inclined annuli. ..342.348) where FE =liquid fraction entrained in the core.. 4.. ...343) G2 = 'iK &(I .' J Similarly.344) 6. .... ..G2.Also.. the total pressure gradient can be determined by use of any of the linear momentum equations given by Eqs. X M .. . (4.. .. .. . .2 Hasan and Kabir Method.. 4. K = ddd. . .&(I .. . 4. . ... .. ... 6 .67 where "=f. .. fluid physical property values. An estimated friction pressure drop was subtracted from the measured total pressure drop.340.4..303 through 4. . .. G3.. .. 4... provide additional details on the nondimensionalizCaetano et ing process. .... .. 4. .. . .K ) ] . -&. a solution is obtained for Eqs.(4.. + GI = y .336 in nondimensional form is - 6 .. the modified Lockhart and Martinelli parameter. The theoretical part included development of a model by use of a drift-flux approach for the slip between phases... ... .and G4are given by G....... = &[l ... The void fraction in the test section was determined from pressure-drop measurements.. Finally.. . respectively. =-L j d... Eq...305.-c T K ( ~ a C 7 J ... .. . 4. The in-situ velocities and liquid holdups then can be calculated. .&[I 4 . ...327 can be written in nondimensional form as df -c = 4dC[1 -&(I . This also was assumed to be the case for flow in annuli. They concluded that the terminal rise velocity. causing an increase in the rise velocity. based on the total slug-unit volume. the four major flow patterns were recognized: bubble.e. Bubble Flow. However. Hasan and Kabi~-'~ found that models that predict the sluglchum flow transition on the basis of the "flooding" phenomenon do not appear to be applicable at high pressures. The approach used for bubble flow to determine the effect of annulus dimension and pipe-inclination angle on the flow parameter can be used for slug flow only if Eq.4 m/s and is equal to 0. Eq. The transition from chum (or slug) to annular flow was not investigated experimentally. because the mixture velocity is much higher than the bubble-rise veloc~ty during churn flow. Although imprecise. In Eq. . Rather. The presence of an inner pipe tends to make the nose of the Taylor Hasan bubble sharper. Hasan and ~ a b i ranived . Application of the drift-flux model for slug flow is more complicated than for bubble flow because of the different drift velocity of the small bubbles in the liquid slug compared with that of the Taylor bubbles. Weisman and Kang69 identified discrepancies between data and empirical correlations from a number of sources. Bubble/Slug Transition.27 1 to predict the onset of dispersed-bubble flow in vertical and inclined annuli. Hasan and Kabii did not investigate the churn-flow pattern because of its chaotic nature.28-Schematic of slug unit for slug flow in an annuius. a slight error in estimating bubble-rise velocity does not affect void-fraction estimation significantly. and ~ a b i fstated that their Taylor-bubble-rise data showed a linear i~ relationship with the diameter ratio. Expressions for void fraction in bubble. and chum flow were developed from relationships between the phase velocities. slug.355.28 shows a slug unit of length Lsu. and annular. 4. Therefore. Although the bubble shape is quite different from the classic Taylor bubble.. Rather. Chum Flow. consisting of a Taylor bubble of length LTB and a liquid slug of length LU. The annular-flow pattern was not investigated. (LLs/bu)-is approximately equal to 0. 4. In addition.240 also is applicable for annuli geometry. The circular channel data of Akagawa and Sakaguchi70 showed that the average volume fraction of gas in a liquid (1 . v.~ Flow-Behavior Prediction. the criteri~ 4. chum.2 was proposed for C1. 4. Hasan and Kabir suggested that the bubblylslug-transition criterion given by Eq. the bubble-rise velocity during chum flow is probably not much different from that for slug flow. no attempt was made to delineate this transition for annuli. As was the case with flow in circular pipes.Flow-Pattern-Transition Prediction. even for flow in annuli. The influence of pipe diameter becomes significant only when the diameter of the bubble exceeds 20% of the channel diameter.25 vsg for lower superficial gas velocities. once again. Slug/Churn Transition. vsgl(l . The average holdup for the slug unit can be expressed as where C1is the flow parameter analogous to Co in bubble flow.at ~ ~ expression for liquid holdup using for the rise velocity of a Taylor bubble in an annulus. 4. Trunsition to Annular Flow.-i.1 when vsg > 0.H m ) slug..) Fig.m (Reprinted with permission from Elsevier Science. Eq. was not affected significantly by either the inner pipe or the pipe-deviation angle from the vertical.163 is suggested for flow in annuli.355 alone is used to estimate the gas void fraction. d&. Assuming that this approximation also applies to flow in an annulus. They recommended the use of Eqs. they suggested that the analyses presented for bubble and slug flow can be applicable for the churn-flow pattern.269 through 4. slug.HLTB)varies linearly with v.357 can be rewritten as if vsg > 0. with a slope of C I . 4. 4.4 d s and as if vsg < 0. suggesting this expression for the Taylor-bubble-rise velocity for vertical annuli Hasan and ~ a b i @ ~ a simple approach to estimate the liquidused slug void fraction.. Assuming that the liquid slugs do not contain ~ ~ this any gas bubbles. it was suggested that the model proposed by Brauner and Bamea5' for circular channels be used for the transition to chum flow. Hasan and Kabir suggested that Eq. Fig. 4. on given by Taitel e t ~ 1in.4 d s . Slug Flow. a constant value of 1. Hasan and ~ a b i r demonstrated that a pipe deviation of up to 32' from the ver'~ tical did not seem to affect the rise velocity of small bubbles. Moreover. this approach indicates that the flow parameter is not influenced significantly by either inner pipe diameter or pipe-inclination angle.246 also can be used in bubbly and dispersed-bubble flow. 4. This criterion suggests that the transition to chum flow occurs when the gas void fraction in the liquid slug following the Taylor bubble exceeds 52%. Thus. 356 for circular pipes. The agreement between experimental results and the annularflow model's predictions is less satisfactory than for models developed for other flow patterns. The agreement is weaker for the fully eccentric annulus case.8 shows that the model's prediction of liquid holdup is very good. The model's performance was obtained by comparing the predicted results with measured data.-ID outer pipe and 1.-OD inner pipe at a pressure of about 45 psi. The quick-closing ball valves used by caetano3' to measure liquid holdup have a lower accuracy in annular flow than for all other flow patterns. with average percent error and standard deviation values slightly higher than 109c. Using Eq. The agreement between the model and experimental results for pressure gradient also is good. The standard deviations ranged from 4 to lo%. Table 4. - L 4. - 4. 4. As in the liquid-holdup prediction. Hasan7'~" analyzed s~~ the void-fraction data gathered by Ney7) and F ~ e n t e for circular pipes in the chum-flow pattern. the model performs very well. C a e t a n ~ ~ ~ gathered liquidholdup.This was caused by the very low superficial phase velocities used when acquiring the data in this configuration. The model performs equally well for either fluid pair or annulus configuration. The degree of scatter about the mean is also good.8-EVALUATION OF CAETANO et a/.15 was appropriate for the parameter CI. The model performance is only fair with airkerosene. Although lower values of pressure gradients wereencountered in the fully eccentric annulus because of the lower friction losses. an accurate estimation of Cl is very important in the prediction of void fraction. the 66 flow was sometimes unstable because of a heading phenomenon.5 Evaluation of Annulus Liquid-Holdup and Pressure-Gradient-PredictionMethods Published liquid-holdup and pressure-drop data for two-phase flow through annular geometries are scarce.0 61 AlrMlater 4 75 4 32 8 69 11 69 2 84 1 11 5 46 2 43 7 72 4 03 3 32 6 86 8 86 97 1 15 94 9 33 AlrlKerosene 52 1 -0 46 -2 77 3 44 Concentr~c $/dl Eccentric $/dl Dispersed bubble Concentr~c HL Eccentric HL Concentric $/& Eccentr~c dp/dL Slug Concentric HL Eccentric HL 7 33 12 03 1 67 112 -3 84 2 93 .259. with a standard deviation of about 5. Taking into consideration this variation might improve the performance of the model.8 reports this performance in terms of the average percent error.8 shows that the model's prediction of liquid holdup is very good. The model performed slightly better for the airlwater mixture. the model performed slightly better for the airlwater mixture. LIQUID-HOLDUP AND PRESSURE-GRADIENT PREDICTIONS FOR TWO-PHASE FLOW IN CONCENTRIC AND FULLY ECCENTRIC ANNULIS' €1 E3 AlrMlater Bubble Concentric 4 Eccentric HL A~rIKerosene -2 17 . In general. independent of either physical properties or annulus configuration. the agreement between the model and experimental results is again very good.TABLE 4. The bubble concentration profile in chum flow may be dissimilar to that for slug flow because of the characteristic churning motion of this flow pattern. Caetano et and Hasan and KabiP8 used these data to evaluate their respective models. suggesting that the model adequately accounts for eccentricity. with vm given by Eq. 4. this value might decrease with an increase in the gas flow rate.8 shows that the model's prediction of Equid holdup and pressure gradient is adequate for aidwater flow in either annulus type.The same value was proposed for use in annuli. with a slight overprediction for both airlkerosene and aidwater mixtures.355. pressure-drop. Table 4.0%. the accuracy of experimental measurements obtained for the annular-flow pattern must be considered. No significant differences were found in model performance for the two annuli. The actual reported holdup values MULTIPHASE FLOW IN WELLS . E l .8 shows that the model's prediction of liquid holdup is good.0%. For slug flow.3. and the percent standard deviation. For bubble flow. Table 4.2. Results are provided for different flow patterns and for both concentric and fully eccentric annuli. For annular flow. showing low values of both average percent error and standard deviation. 4. with a slight overprediction tendency for airlwater flow and a slight underprediction tendency for airkerosene flow.89 1 39 -2 46 .66-in. The agreement between the model and experimental results for pressure gradient is also good.74 3 94 1 75 35 1 .1 CaetanoetaL Method. confirming reasonable scatter about the mean. with a slight underprediction tendency of less than 3. The standard deviation about the mean is below 3. This is because of the low overall holdup values characteristic of annular flow. Hasan concluded that a value of 1. and flow-pattern data for airlwater and air1 kerosene two-phase flow through an annulus with 3-in. For dispersed-bubble flow.0%. The slight overprediction of both the average liquid holdup and pressure gradient might result from treating the liquid holdup in the liquid slug as a constant. However. Table 4. Table 4. which can cause a lower degree of measurement accuracy. However. E3. Under these conditions.5.07 Concentr~c $/dl Eccentric $/a -2 3 1 - 23 17 - Annular Concentric HL Eccentr~cHL Concentr~c $/dL Eccentr~c$/dL 66 00 25 22 - - However. as given by Eqs.1 06 10 60 15 00 17 11 -1403 5 80 9 97 8 00 4.256 and 4. 04% and the airkerosene data by 9.84% and 2. Good agreement alsoexisted between the Sadatomi eral.3 and 4. often based on experience not documented in the literature.e. Others. a deposition-rate expression was adopted.16%. the airlwater liquid-holdup data are overestimated on the average by 5.047 0.034 0. No existing comprehensive model properly accounts for all the effects of inclination angle on flow behavior. Among these.0214 for the gas void fraction in all flow patterns in this data set. therefore. for the Caetano er al.026 AirMlater 0. Caetano et al.success and is accepted in the petroleum industry. The model predicted an average error of 0.200 Airwater 0 022 0. to refine the model. The general overestimation by Hasan and Kabir suggests that a lower value of C. by use of one investigator's flow-pattern map with another's pressure-gradient equationnone has any technical merit and all should be avoided. 4. it was not possible for Caetano to acquire data covering a broad region in this flow pattern.057 0. Two mode l ~are~ ~ . Hasan and Kabir also compared their model predictions with the airlwater data of Sadatomi et who used a 30-mm-OD.75 applied the isotropy concept.049 0. Caetano's airlwater bubble-flow data were predicted with an average percentage error of 2.. Finally. for the homogeneous model. many of the variables used to predict flow behavior in mechanistic models - . Film-thickness ratio is the third major aspect to influence the performance.0 for C.60% and a standard deviation of 2. The film-thickness ratio became dependent on the adopted isotropy of scattering and independent of flow interference on the liquid droplets entrained in the gas core.400 1 900 5.005 AirIKerosene 0.0%. These higher percentage-error values reflect the generally lower values of liquid holdup rather than any inaccuracy in predicting the absolute values. no doubt will be published. Any improvements in accuracy are purely accidental.. they also are much more complicated and difficult to understand.75%. Hasan and KabiP8 reported statistical comparisons between predicted liquid-holdup values and measured values from C a e t a n ~ for flow in a concentric annulus.070 5. The percentage .041 .100 AirIKerosene 3.TABLE 4. Which correlation or model should be used? Many companies have their favorites.246. Average percentage error and percentage standard deviation were not reported because liquid-holdup values in this flow pattern were very low. the modified Hagedorn and Brown4 and Beggs and Brillll methods appear to be best. Thus. Table 4. data with more detailed characteristics for this flow pattern would be required.1 and 7. The Hasan and Kabir model predicts these data with an average percent error of 0. the Hasan and Kabir model overestimates the air/ water slug-flow data by 7.OW PRESS1 IRE-GRADIENT PRFDlCTlON 8.6 General Observations Engineers who perform multiphase-flow design calculations for wellbores are clearly faced with an immediate dilemma.023 and a standard deviation of 0. is needed in Eq.17% and 5. although agreement is better for the airkerosene system.3. should perform better than the Hagedorn and Brown method in deviated wells. respectively. with El = 3. the pressure measuremknt system used by Caetano may not be adequate for the cross-sectional-dependent and axial-dependent annular-flow pattern. the rigorous Caetano et al. 4. data and Hasan and Knbir predictions for slugand churn-flow patterns. slug-flow model predicted the liquid-holdup data for both systems better. 4. then a small overall liquid-holdup value will be predicted. model. several conclusions can be drawn from published evaluations and experience. could not use these insights.67% with the homogeneous model proposed by Caetano er al.01 for the airlwater data and by 0. Considerations of different drift fluxes in the Taylor bubble and the liquid slug reduced the percent standard deviation by only 0. The Caetano et annular-flow model also showed this. Nevertheless. compared with values of -2.2 Hasan and Kabir Model.) El A~rMlater AirIKerosene 2. The evaluations presented in Secs.94% and E3 = 7.013 0.0. resulting in percentage errors that exaggerated the inaccuracy of the method.4%. or a lower value of v.040 0.9 also presents the liquid-holdup predictions of the Hasan and Kabir model vs.93%.!&EVALUATION OF HASAN AND KABIR LIQUID-HOLDUP PREDICTIONS FOR TWO-PHASE FLOW IN (Reprinted with permission from Elsevier Science.043 were sometimes less than the accuracy possible with the ball-valve approach. decisions often are made without awareness of the serious limitations or the availability of more accurate methods.020 8. Future models that improve significantly on Ansari et al. Hutchinson et ~1. The Beggs and Brill method attempts to account for the effect of inclination angle on liquid holdup and. Mechanistic models offer the most accurate methods to predict pressure gradients in wellbores. respectively.980 2. respectively. )~ No comparisons were made on pressure drop.5 are inconclusive and based on a very limited set of data. compared with an underestimation of 1.5. Among the empirical correlations presented.2%. Caetano's chum-flow data.0.063 0. It is a mass-transfer-process type that does not take into account the particle size and the inherent type of deposition mechanism. A second major aspect in the annular-flow-model performance is its strong dependence on the entrainment liquid fraction. However.93%. compared with values of . If a large entrainment fraction is predicted because of a high superficial gas velocity.90% and a percentage standard deviation of 3. because of gas compressor limitations.i%omprehensive model has enjoyed grea. general overestimation of the liquid holdup is evident for both fluid systems. Unfortunately. the Ansari et al. For bubble flow. However. respectively.9 presents the statistical results.97% for the airkerosene data.043 0. Hasan and Kabir also proposed a simplified approach for slug flow in which all the gas is assumed to have the same drift flux as the Taylor bubble.160 7. FE.98% for the Caetano airkerosene bubble-flow data. The Hasan and Kabir model predicted an average percentage error of 1.600 7 040 9 930 E3 E 4 E6 Bubble Dispersed Bubble Slug Churn AirMlater 4. are proprietary.07%.8% and E3 = 9. Although several attempts have been made to create hybrid correlations-i.018 0. ~ peer review.033 AirIKerosene 0. while percentage standard deviations for these two sets of data are MI 'I TIPH4SE-FI. As a result.000 0. However. Although excellent models are available to predict flow patterns at all inclination angles. Also. Unfortunately. For slug flow. alli is'^ reported that viscous liquids normally exhibit a smaller entrainment fraction than predicted by his correlation. Caetano's airkerosene data appear to support this contention. 15-mm-ID annulus. G ~ d n e reported that the r~~ rate of deposition was dependent on mechanisms that account for the existing flow field and are determined by liquid droplet size. The apparent low value of the average error supports the use of a smaller value for the parameter C1.46% and 4.004 for the airkerosene data. when most of the bubbles still flow through the channel core. 4. 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G.: "Pressure Drop Correlations for Inclined Two-Phase Flow. Alves." SPEPE (November 1988) 474.28 1.H. Jr." AlChE J. Wilkes. R. Griffith. 19.39. New York City (1969). Trans.34. ." paper SPE 20628 presented at the 1990 SPE Annual Technical Conference and Exhibition.43.: "Prediction of Pressure Gradients for Multiphase Flow in Tubing.: "SluglChum Transition in Upward GasLiquid Flow. Chem. Prac. Hewitt. Steen. Oklahoma (199 I).: "Experimental Study of Pressure Gradients Occurring During Continuous Two-Phase Flow in Small-Diameter Vertical Conduits. Eng. Trans. Chokshi. G. 23. Energy Res. Vo.: "A Mechanistic Model for Two-Phase Vertical Slug Flow in Pipes. 10.: "Evaluation of Inclined-Pipe Two-Phase Liquid Holdup and Pressure-Loss Correlations Using Experimental Data. AIME. Massachusetts (1962).741. Tech. mechanistic models will be developed that are sufficiently accurate for a broad range of gas.E.: "Flow Pattern Transition for Vertical Downward Two-Phase Flow. Oxford." Drill.F.D. Davies. J.206.470.R. Harmathy. Nicklin. AIME. and Dunn. U. Energy Res. G. 40. Brill.B.S. Payne." JPT(May 1973) 607. Hasan. M. K. 49. D. 285. Y.58. Caetano. Successful modeling must begin with two-phase oiUwater flow before the more complicated multiphase-flow behavior can b e addressed." Trans. D. 257. and water flow rates. NYO-31142-2 (1964).." Intl. 39. Tech. Prog. Hawell.: "The Correlation of Liquid Entrainment Fraction and Entrainment Rate in Annular Two-Phase Flow.. Shoham. G. 297..: "Velocity of Large Drops and Bubbles in Media of Infinite or Restricted Extent. I.B. Beggs." J. Inc. J. E. Brill. ofTulsa." PhD dissertation.. McQuillan.. both in annular flow and in the Taylor-bubble region of slug flow. References I. Trans.M. K. Weisman. Energy Res. (March 1992) 114.: "Pressure Drop in Wells Producing Gas and Condensate. U. Slug. (1964) 100." Bulletin. U. Baxendell. Scr.P. Tech.. Multiphase Flow (1992) 18.53.F.625-Inch Fully Eccentric Plain Annular Configuration. 62. (1971) 26.641. Sato. and Dukler. Espanol.R. Communications (1988) 66. Multiphase Flow ( 1975) 2.W. J. I. and Fogarasi. Trans.. G. and Chum Flow in Vertical Two-Phase Up-Flow. (March 1992) 114. Hasan.B. E.F. Y.N. Camacho. (January/February 1994) 33. and Sakaguchi.: "Comparison of Correlations for Predicting Pressure Losses in High Gas-Liquid Ratio Vertical Wells. Tech. E. P. Oklahoma (1970).: "Flow Pattern Transitions in Vertical and Upwardly Inclined Lines.P.: "Comparison of Three methods for Calculating a Pressure Traverse in Vertical Multi-Phase Flow.: "Experimental Study of Pressure Gradients Occurring during Continuous Two-Phase Flow in Small Diameter Vertical Conduits.H. Austin. Part 11: Modeling Bubble.." J. R." MS thesis. 68. Oklahoma (1970). A.E. U. Shoham." JPT(August 1958) 59. and Pressure Drop in Bubbly. 75. Richardson. C..G. H." J P T (March 1963) 309. Slug and Annular Flow. .J. AIME. 2 2 9 4 9 . Brill.F. Slug. and Vervest. J." paper SPE 26682 presented at the 1993 Offshore European Conference. C.and Two-Phase Vertical Flow Through 0. Pucknell. of Texas. Tech.. Prac. Oklahoma (1968). U. 65. J.. Tulsa. Angel. AIME. of Tulsa.: "Supplement to paper SPE 20630. 57. 222.Mason.Y. Cdn.: "Fluctuation of Void Ratio in TwoPhase Flow.." PhD dissertation. of Tulsa. Fuentes. Ros. S.279. 14. H.A. 213." Chem. Oklahoma (1968). and Chum Flow. Aberdeen. H. M." Drill. 73. of Tulsa.: "Upward Vertical Two-Phase and Flow Through an Annulus.: "Void Frdction in Bubbly.: "Inclined Two-Phase Flow: Flow Pattern. J. and Kirkpatrick... 0. 58.: "A Study of Multi-Phase Flow Phenomena in Directional Wells.: "Producing Wells on Casing Flow-An Analysis of Flowing Pressure Gradients." J.: "Two-Phase Flow in Vertical Noncircular Channels." MS thesis. & Prod.: "Upward Vertical Two-Phase Row Through an Annulus.271. G. N. Messulam. Texas (1964) 56.V. Ansari. and Kang.: "Simultaneous Flow of Gas and Liquid as Encountered in Well Tubing. 55." MS thesis. 63. Caetano. S. U. 101.G. 7-10 September." MS thesis. 228. E. Caetano. Govier. Trans. A.R. S..R. Texas (May 1994). Salim.D.and Brill. J.W. Tech. (October-December 1975) 14."J.: "Deposition of Liquid or Solid Dispersions from Turbulent Gas Streams: A Stochastic Model. Hagedorn. J. T. of Texas.A. 76.H.: "Single.996 X 0. U. Cdn. Eng. 72. Winkler." PhD dissertation.R.S.: "Deposition of Particles from a Gas Flowing Parallel to a Surface. Hasan. A." Intl. Gaither. 69. Part I: Single-Phase Friction Factor. Gardner. Multiphasr Flow (1982) 8.: "AnEvaluationofRecent Mechanistic Models of Multiphase Flow for Predicting Pressure Drops in Oil and Gas Wells.R. 67.. Void Fraction." Chem. 74." Intl. A.K. 64. Hasan. 60.." paper SPE 28671 available from SPE. of Tulsa. Oklahoma (1968).: "Single. Pet.C. Akagawa. Multiphase Flow (1981) 7. Pet.M.58. C. Ney. JSME (1966) 9. Taylor Bubble-Rise Velocity and Flow-Pattern Prediction.: "Comparison of Correlations for Predicting Multiphase Flowing Pressure Losses in Vertical Pipes.K. J. Sadatorni.J." JPT (October 1961 ) 1037. Tulsa. 61..2 13.J.F. A. 59." Intl. J. 77.: "Low-Ratio Gas-Lift Correlation for Casing-Tubing Annuli and Large-Diameter Tubing." J.: "Evaluation of Methods Describing the Flow of Gas-Liquid Mixture in Wells. Trans. of Tulsa. Texas (1968).E.28. A Comprehensive Mechanistic Model for Upward Two-Phase Flow in Wellbores. 71. Energy Rrs. Tulsa. Winkler. M. 54. et al. 1. A.. 104." Intl." Particulate Phenomena and Multiphase Transport. P. New York City (1988) 1 . J. U. Tulsa. 66.C. Hewitt. G. C. and Welchon. and Saruwatari. and Kabir. Eng.419.: "A Study of the Directional Well. Tulsa. A. Shoham. Hutchinson. O. AIME. Austin. and Stanislav..and TwoPhase Fluid Flow in Small Vertical Conduits Including AnnularConfigurations. Hemisphere Publishing Co.: '"Two-Phase Flow in Vertical and Inclined Annuli.W. 70." MS thesis. 0. P. J.. . = where q is in BID and 4h the choke diameter in inches. pressure ratio for different choke sizes. 5. orifice. When the pressure ratio reaches a critical value. Eq. However.e... or choke).= c CDTSC .0 are possible for methane at pressures between 2. venturi.000 psia and 150°F..25 and 1.. in Eqs... As the pressure ratio declines. 33.6 Fig..9 are often used. a gas with k = 1. the pressure ratio y =p21p1 is replaced by the critical-pressure ratio... Because sonic velocities are high for si~gle-phase liquids.9.1. Single-phase liquid flow seldom occurs in wellhead chokes because wellhead pressures are almost always below the bubblepoint pressure of the produced fluids.. Values of k for hydrocarbon gases at lower pressures are typically between 1.611 0.3.~)... 5.3' AND 5.16 K 3. The resulting Eq. Prediction of flow behavior for multiphase flow through chokes relies heavily on a knowledge of single-phase flow through restrictions.. flow rate increases.- fIXL0 C L ( W I e ROTATW CERAMIC CALl8RATION MnL Fig. This is much more difficult than for single-phase gas flow. Fig.3 and 5. such as friction.. Gh P T psia "R 27. 5.4 depend on the shape of the opening to the restriction.. Fig. To predict the flow behavior for multiphase flow through chokes requires that one first predict the boundary between critical and subcritical flow.7 shows a typical graph of gas flow rate vs. and for an upstream pressure and temperature of 1. The flow coefficient.3.6 shows the flowcoefficient behavior for a nozzle. The correct ratio of specific heats for hydrocarbon gases varies with pressure and temperature.8 shows that values as high as k = 2.25 c s " x 'i - .696 psia 519. the ratio of the restriction diameter to the pipe diameter. k is approximately 1. the flow rate is constant for a given choke size.. The proper choice depends on whether a calculation is made of the critical-pressure ratio below which totalmass flow rate is constant.000 and 4.. 5. 5. 5. For all pressure ratios below y. 5.1. C can be determined experimentally and depends primarily on the type of restriction (i.2 Single-Phase Gas Flow.. and the Reynolds number.. For gases. Choke diameter is frequently called "bean" size and is measured in 6 4 t h ~ of an inch.865 14..1--CONSTANTS AND UNITS FOR EQS.2-Multiple orifice valve wellhead choke design (after Willis Oil Tool CO. .56 for this gas.53.1--Choke schematic.. flow behavior is always subcritical. 5. 5. Eq.4 Symbol 9sc Table 5.. Fig.3' is applicable for both critical and subcritical flow. Discharge-coefficient values in Eq. or if the sonic velocity of a multiphase mixture is estimated. In oil field units. nozzle. the flow rate through the choke becomes constant.For practical reasons. 5.4 for both customary and SI units. All irreversible losses are accounted for through a discharge coefficient.865 101.2 accounts for all irreversibilities...1 gives values for the constants in Eqs.7915 For air and other diatomic gases.. Although a value of CD = 0. s .. this equation becomes Fig. critical flow for gases is often estimated to occur at a pressure ratio of 0.68 "R 844. for a compressible fluid. describes single-phase flow of an incompressible liquid through a choke. for critical flow.3. the length of the restriction..82 and 0. 5. 5. Bernoulli's equation can be combined with an isentropic (adiabatic-frictionless) equation of state. 5. 5. where C. y.3 Multiphase Flow. which can be developed from a combination of Bernoulli's equation4 and conservation of mass. values between 0.000 psia and a temperature of 50°F. For this reason.325 kPa 273. we describe single-phase liquid and gas flow before introducing multiphase flow. 5. respectively. p ... and the Reynolds number.1 and 5.4 and the critical-pressure ratio from Eq..3-Cerarnic choke disk operation (after Wiliis Oil Tool Co.6259 0.865 is given in Table 5.5 is 0. The critical-pressure ratio for a gas with a ratio of specific heats k = CplC.1 Single-Phase Liquid Flow. This can have a significant effect on the predicted critical-pressure ratio and resulting critical-flow rate...5 predicts a critical-pressure ratio of 0.57 SI Metric m31d mm kPa K 1. C. 5. is given by Customary MscflD ~n.5. TABLE 5. for lower values of R1. This permits a calculation of yc.6 assumes the derivative of the flow rate with respect to pressure ratio is zero at the critical boundary.6. Several authors have developed methods to predict the critical-flow boundary in multiphase flow.9. Perkins developed the expression . A direct substitution method. They assumed isentropic flow through the restriction. Ashford and Pierce8 developed an expression for total-mass flow rate of a multiphase mixture. the parameter n and the upstream in-situ gas mass fraction or quality. 5. D = -+ n + k . In-situ gasniquid ratio at upstream conditions.I0 performed a combined experimental and theoretical study that resulted in these equations to determine y.). are determined from Eqs.1 1 . and and A Eq.f~. 5. Perkins. an incompressible liquid. Determining yc from Eq. R1.6 and shows that at values of R1 above about 10.l 2 In Eqs. 9 L .6 requires an iterative procedure to determine values of y. A value of yc is first assumed in Eq. 0 7 @ ~ . A value of 0.7 is dimensionless and any set of consistent units can be used.l. x. Fig. is adequate.. respectively. as a function of in-situ gadliquid ratio for different values of k. Sachdeva et al. agree to within a predetermined tolerance. Ashford and Pierce. 5. However.1800 1.significantly lower values of yc are predicted.He combined equations for conservation of mass and ~ isentropic expansion of a homogeneous multiphase mixture to anive at an expression for total mass flow rate.9 gives a plot of Eq.6 0. Eq.5 is a recommended first guess.m 16/e4 in. 5. and where where ws. 5. 5.8 1 1 . no liquid flashing in the choke. in which the calculated value of yc is used for the next guess until the calculated and guessed values ofy.10 and 5. Pressure ratio. where Fig.8. 5. 5...OW Critical Flow + Subcritical 4 w 2W 0 d m 09 04 0. 5.4-Internal flow paths for velocity-controlled subsurface safety valves (after Brill and Beggs3). Using the same assumptions as Ashford and Pierce in Eq.400 EQURLIZlNG W C > g 12w B n 0 g 1. the criticalflow boundary is similar to single-phase gas.8 and 5. Eq. 5.can be calculated easily as the ratio of the superficial gas and liquid velocities determined at conditions immediately upstream of the choke.7 requires an iterative procedure. Sachdeva et al. and Critical-Flow Boundary. 5. Perkins6 derived an equation to predict the critical-pressure ratio that follows very closely the approach of Ashford and ~ i e r c e .= 0 . LOCKING $ 1. ( ~ . y Fig. and a homogeneous mixture.5-Dependence of choke flow rate on y. 5. where Fig. is greater than 32.I3 studied sonic velocity in two-phase systems as a function of flow pattern.~). NRe to define the boundary between critical and subcritical flow. and In the Perkins expression.17 is the effective sonic velocity for the gas phase and shows that the effective sonic velocity is governed primarily by the sonic velocity of the gas phase because the second term in the denominator is small. The curves in Fig. a composite sonic velocity does not exist because each phase is continuous in the axial direction. 5. For example. the disturbance propagates with different velocities in both phases in the axial airection.6-Flow coefficients for liquid flow through a nozzle (after Crane CO.12 was defined as As for the case of Ashford and Pierce and Sachdeva et al.8 ftfsec and the mixture Froude number is greater than 600. 5. (ft-1bf)l (Ibm-OF). which can be used The parallel expression for the effect~ve sonic velocity in the liquid phase is . Wallis said the sonic velocity of a homogeneous mixture passes through a minimum at a no-slip void fraction of 0. He assumed a homogeneous mixture and contended that the assumption was valid as long as v. Nguyen et al.0012. Wallis Sonic Velocity. If a pressure pulse is imposed on the liquid and gas simultaneously.10. Fortunati Fortunatill presented an empirical method that can be used to calculate both critical and subcritical multiphase flow through chokes. a density ratio of 0. Nguyen et al. 5. x = weight fraction of a given phase in the flowing stream and G =heat capacity at constant volume. Reynolds Number based on 6 . 5. Wallis presented an expression to calculate the sonic or compressibility-wave velocity of a homogeneous mixture. C presumably can be evaluated at either upstream or downstream conditions.. In Eq. The polytropic expansion exponent for mixtures used in Eq.9 psia. 5. y Fig. an airlwater mixture at atmospheric pressure will have an air sonic velocity of 1. an iterative procedure is required to determine y. and a minimum mixture sonic velocity of only 75 ftlsec. Sonic Velocities.12 where The sonic velocity of the homogeneous mixture does not necessarily lie between the sonic velocities of each phase and in some circumstances may be far less than either. Eq.10 were based on a downstream pressure of 19.100 Wsec.7--Gas flow performance for different choke sizes. An effective sonic velocity exists in each phase that is influenced by the other phase.. Eq. 5.14 should be used to calculate the Fortunati mixture velocity from the actual mixture velocity and downstream pressure.9 compares the Fortunati critical-flow boundary with boundaries predicted by Ashford and Pierce. 5. For stratified flow. Using experimental data..5..16. Fortunati developed Fig. Fig. 5. 5.Pressure Ratlo. I' presented an empirical correlation to predict critical multiphase flow through Thornhill-Craver-type chokes (see Fig.. Subcritical-Flow Behavior. Critical flow exists if y c y. combined expressions for sonic velocities of each phase flowing within an elastic boundary with aconcept that the wave front sequentially passes through zones of liquid and gas within the homogeneous mixture. Critical-Flow Behavior. Through the application of a least-squares regression-analysis procedure to a series of high-pressure natural gaslwater tests. a simple homogeneous Bernoulli-type equation was adopted. OmaAa er al. 5..1 1. .0 10 100 1. a composite sonic velocity expression was developed for an idealized slug unit. 5.01 0. 4 h = 6 4 t h ~ of an inch. In contrast to stratified flow. e m 3 Fig. .000 10.. Gilbert Type.9--Comparison of critical-flow boundary correlations for multiphasechokes (after Gouldg). Nguyen et al. this empirical correlation was developed.8-Ratio of specific heats for methane (after Perkins6). Eq. ~ i n d stated that a generalized expression with l ~ some simplifying assumptions for the flow of gas and oil through a knife-edged choke is given by where This expression has been the basis for several modifications by use of experimental and field data.= v*.000 100.. Omafia et a1. Fortunatill s~ecifies = 19. and vi2 in the second term of the denominator in Eq. these oilfield units should be used: qL. Eq.18 are quite small. psia Fig.1).20 gives the resulting expression.2 : 10 I 1 1 1 1 I111 1 I 1 111111 1 1 1 1 1 1 100 1.19 gives the result.. ..74 X 10-2pl - . A dimensional analysis of the problem of multiphase flow through chokes yielded pertinent dimensionless groups. or if v. and Sachdeva et al. = 1. In the next five equations.000 Pressure. gadliquid ratio.. . Experimental and field tests confirm that accurate prediction of subcritical flow rates through restrictions is very difficult. Sachdeva et aL1° presented the following equation to calculate flow rate through a choke.000 10.. and pl = psia. Thus.u~= dyneslcrn.001 0.. 5. 5. N. R. 0.WO p. Following extensive tests of subcritical twophase flow through velocity-controlled subsurface safety valves.. This generalized equation can be used.. 5.- and For homogeneous flow.6 ' Subcritical Flow Critical Flow 1.< = STBLlD. TUFFP Model..9 h psia as a basis for correlation. p = 1bm/ft3. giving it a larger influence on the effective sonic velocity in the liquid phase. 5...0. 2 gives values of a..3 summarizes the results.54x 10-3 0.TABLE 5. 5.=is in STBL/D. 400. = 0. Sachdeva et a .588 0. Assume that the wellhead pressure and temperature are 1.2. R.6 with k = 1. ° found that. In Example 3. 5.12~10-3 1.500 4.446. 5.~ 0. Table 5. 1.118 0.6 = 0.0633 ft = 0.r rn.88 b c 0.200 where pl is in psi.89 1.76 in. For cases of critical flow. 600. = . r r . .25x 10-3 3 . Table 5. Determine the choke diameters required to maintain the well flow rates when the separator pressures downstream of the choke are 200.10-Mixture velocity for critical and subcritical flow (after Fortunatill).e.?=y.\ .235 0.pl.22 with y = y. Determine the flow status of the given pressure conditions for pl = 1.700 psia and 180°F. 800.546 0.22 should be evaluated with p.93 1.200 psia.7. a choke is located immediately downstream of the wellhead.962.993 in. if the flow is deterl1 mined to be critical from Eq. and c proposed by different investigators. r r AYIT TI t n n r r r r r n r r T n r c r r n x . &lpl. Rp is in scf/STBL.470 0. and The Eq.000.700 psia.c r -7- nrnr\. 2.706 Status Critical Critical Critical Su bcritical Subcritical Subcr~tical 1.2--CHOKE CONSTANTS Investigator Ros16 Gilbert1' Baxendelll Achonglg a 2.3.in. and 4h is in inches. y < y.9 psia Fig.546 3. 1. Gilbert: and 63.650 TABLE 5.000 1.00 1.3--SUMMARY OF RESULTS m (psi4 200 400 600 800 Y 0.1-Determine Choke Diameters Required To Maintain Well Flow Rates. Sachdeva et al. 5.. respectively. critical-flow correlations. the flow rate should be calculated from Eq. Example 5.6 trial-and-error solution yields y. i.353 0. with &= 19. determine the critical-pressure ratio. and 1. 5. yc Pressure Ratio.b. determine 4husing the Gilbert and the Omafia et al. Using Eq. n n ~ ~ ~ . = 0. qL. and that all fluid properties in Eq. 5.: then dCh= 0. 8 6 ~ 1 0 . 64 Omaiia et al. .000 psia.33 for the equivalent length of pipe.4-SUMMARY OF RESULTS Pr (Psi4 G h 64. .() I d . . The Darcy-Weisbach equation (Eq. and d. there is generally a higher degree of turbulence in the component than in a straight pipe at the same Reynolds number. seat reduced = 300-lbm steel venturi ball-cage gate valves = 125-lbm iron-body Y-pattern globe valves = 125-lbm brass angle valves.005 in. . .000 1. . and at p2 = 1. each one is replaced by an equivalent length of pipe that would yield the same pressure drop as the component.200 psia. . .4 Flow T h r o u g h Piping C o m p o n e n t s When a single-phase gas or llquid flows through apipe fittingorpiping component. 2.(144) PnlVmB1 and C. = 0. For multiple components. 5. = 0. .6 63. 2. d. . TABLE 5.757ft3/sec. . . dd= 2 = 90" plpe bends. the Gilbert and Omaiia et al. One way to determine the pressure drop across a component is to add an "equivalent length to the straight pipe. 30 diameters bng (K=30/) 4 d =600-lbm steel wedge gate valves.200 63.33) When solving Eq. .16 be used to account for the increased turbulence in the com~onent?For multi~le components. it is recommended that the fully turbulent friction factor &en b * . For cases of subcritical flow. At p2 = 800 psia. composition disk = 125-lbm brass globe valves.4 68.= 90" plpe bends.6 63. C2.) Status Critical Critical Critical Subcr~tical Subcr~tical Subcritical 200 400 600 800 1.50. When several components occur in apiping system. 5. . Clearly.50. 64 at p2 = 1. . use the TUFFP correlation with a CD of 0.~). . or d = K. all the equivalent lengths can be added to the actual pipe length before performing the pressure-drop calculation. .6 Table 5. composition disk 0.11-Variation of reslstance coefficient Kwith size and type of selected components (after Crane CO. correlations give significantly different results for critical flow. . . . .0838 ft = 1. p.6 64. 2 Ap = 2g. MULTIPHASE FLOW IN WELLS .. . . it is appropriate to calculate the frictional pressure drop or head loss caused by these components. = -in.= 125-lbm iron-body wedge gate valves =600-lbm steel wedge gate valves -0 = 90" pipe bends.. . .04 Ibm/ft3.. = 64 (64th~ in. dd= 1 6 # Fig. . = @& in.778 ft3/sec and q. . . q.6 in. = 0. 5. 64 74. (5.- 0 =Schedule 40 pipe. dd= 3 -0. = 0. Note that this CD was selected to force the Gilbert equation and the TUFFP model to yield the same choke diameter at the critical-flow boundary.6 74. = 27. . . ..4 shows a summary of the results.9) can be written as 3. . d. .4 .. . . 64 Ibf/ft2 = 0.D. y. Energy Res. New York City (1969). and Pipe. 16. Texas (1946)." SPEDC (December 1993) 27 1. for many piping components have been determined e ~ ~ e r i m e n t a l lFig. Brill. R. "Flow of Fluids Through Valves.: One-Dimensional Two-Phuse Flow. R. & Prod.: 'Two-Phase Flow in Piping Components. Oklahoma (1984). They also discovered that.E.E. J.00006 ft. and Beggs. Thus.E..R. K.15 X d = 6. Cook. F. T. I." JPT(September 1975) 1145. R. 13. J. Ashford. 5-8 October. Texas. Shell Oil Co. 6." paper SPE 2682 presented at the 1969 SPE Annual Fall Meeting. 12." Appl.W..P. New Orleans.: "An Analysis of Critical Simultaneous Gas-Liquid Flow Through a Restriction and Its Application to Flowmetering.11 gives some of them. Z. 10.: Principles of Oil Well Production. K = 0. and Greiner. New York City (1964). Instructions and Parts List.: 'Two-Phase Flow Through Chokes. . 2. 5. H.-diameter. Nguyen. (September 1986) 108. Darby.L. acceptable results are obtained by use of single-phase. Winter. 3. of Tulsa. P. E. five 90" elbows and a gate valve exist next to each other immediately downstream of the well." Technical Paper 410. Crane Co. Amsterdam. In Example 3. T.2-Estimate t h e Pressure D r o p for Multiphase Flow T h r o u g h Piping Components." paperSPE 3742 presented at the 1972 SPE European Spring Meeting." paper SPE 15657 presented atthe 1986 SPE Annual Technical Conference and Exhibition.32 is modified for multiphase flow conditions." JPT (August 1974) 849. 11.. 7. Estimate the pressure drop across these components. Shell Oil Co. P. F. P. N. H. 14. et al. U . 5. D. when piping components are located very close to each other. 17.. 5.2. G. 1969.E.L..: ''Two-Phase Flow Through Wellhead Chokes. Fittings. and E = 0. Sookprasong.: "Critical and Subcritical Flow of Multiphase Mixtures Through Chokes.: Gas Production Operations. Gould. Colorado 28 September-1 October. Catalog 4230A. Houston. Houston (October 1961). Example 5. McGraw-Hill Book Co. 4. 5. and Pierce. when Eq. frictional losses are higher than for individual components and undetermined longer equivalent lengths are necessary.: Two-Phase Flow in Pipes. 19. H. Wallis. Achong. = 3.K. Prac.11. Tech.15 K = 1.: "sonic velocity inTwoPhase Systems.D. OGCl Publications.: "Determining Multiphase Pressure Drops and Flow Capacities in Downhole Safety Valves.: "Multiphase Flow Through Chokes. Ornaiia. equivalent-length values.: "Discussion of An Evaluation of Crit~cal Multiphase Flow Performance Through Wellhead Chokes." Internal Company Report. 9' elbow and 0 = 65. 8.46 psi. T. Long Beach.8 for a gate valve.. 16-18 May.J.B..: "Flowing and Gas-Lift Well Performance. Beggs. Gilbert. Ros. Multiphase Flow (198 1) 7. Inc. 9.0 in.. California (1971). Nind. New York City (1996). 197..The resistance coefficients. Res. for multiphase flow.Tulsa. They concluded that. and Dottenveich. Tulsa.: Chemical Engineering Fluid Mechanics." I. Houston (October 1967). 18. 15. Fortunati. (1954) 126. Marcel Dekker Inc. Baxendell..: Reponon the Calibration ofPosifive Flow Beans as Manufactured by Thornhill-Craver Company. ( 1960) 9. Inc. and Schmidt. J. Kingsville. 20. F.: "Bean Performance-Lake Maracaibo Wells." Intl. New York City (198 1).. From Fig." internal company report. Sachdeva. Denver. L n o w THROUGH RESTRICTIONSAND PIPING COMPONENTS . Brill. N.C.: "Revised Bean Performance Formula for Lake Maracaibo Wells. Sci.B.Texas College of Ans and Industries. Willis Oil Tool Co. Oklahoma (1991).~ Sookprasong er aL20 investigated multiphase flow through several piping components.H. W. McGraw-Hill Book Co.L.. Jr. for a 6-in.. Perkins. References 1.F. et al." Drill.3 1 1. M.P. . . (%. Flow rates calculated in the unstable flow region indicate intermittent production. WELL DESIGN APPLICATIONS .3 presents a typical tubingintake curve for producing wells. For injection. can be set to zero in Eq. BWPD Fig. Fig.Apt. heading.600 6. or loading up of wells. h h . and friction loss in the flowline.2 is im~ortant ~roduction iniection cases. 6.Apch)f= friction pressure losses through the flowline. nodes. psep. 6. Note that in the case of injection. is called the tubing-intake curve. Consequently. choke.5 shows. In Eq.i = Prep + A P ~ (AP~. the plot of hf q (assuming a fixed wellheadlseparator pressure) vs.000 4. 6. starting at a fixed wellhead or separator pressure.2 also can be used to generate gradient curves. Note that. the tubing performance curve (TPC) is used for both injection and production. 6. 6.. Pressure losses resulting from friction and acceleration always occur in the direction of flow.000 8. Fig.6). 6.. during production the tubing intake represents tubing performance. Gradient curves are -0 Rate. tubing performance could be called tubing outflow because the fluid is pumped from the tubing into the reservoir. & =total hydrostatic pressure loss. 6. and the friction and acceleration terms should be negative. =pressure loss because of acceleration. for or 6. q Fig.Fig. The separator or the wellhead pressure normally is known and the methods to determine the pressure losses in each segment were discussed in previous chapters. or flowing pressure vs.+ Apt + A P ~ + Ap. in the absence of flowlines. as the injection rate increases. tubing-intakecurve for producing wells. 6.1-Possible a flowing w e t 6 pressure losses in the producing system of of various nodes (from Mach et al.4-Typical systems graph for injection wells..2-Location Fig.4 shows a set of typical tubing performance curves for an injection well. It is important to note that the flow condition represented by this curve will be stable only if dp/dq r 0 at any given point. Eq. 6. whereas hydrostatic pressure loss occurs in the direction of elevation gain. in Eq. 6.= bottomhole flowing pressure at Node 6. + ) ~ .2 should be replaced by pump discharge pressure. The algebraic signconvention to determine pressure losses in Eq.psep can be set equal to the wellhead pressure.3-Typical ' L 2. depth curves. the separator pressure. tubing.. However. In the rest of this chapter. This should be avoided through proper design and will be discussed in the example problems.2. the bottomhole injection pressure decreases.2. 6. the friction and acceleration terms should be positive for production. In this chapter. or restrictions. This relationship can be expressed mathematically as P.0b0 Injection Rate. psep =separator pressure. and hp. Am. where p.. As Fig. is in darcies. ft = A / n .500 4 .. psig 1. AOFP is the maximum flow rate.000.1 gives the input data. The flow rate corresponding to atmospheric bottomhole flowing pressure is defined as the absolute open flow potential (AOFP) of the well. representing a shut-in condition.. For single-phase oil or liquids. very useful in the design of artificial-lift systems.500 3. In oil wells.75 + s.. whereas the flow rate is always zero when the bottomhole pressure is the average reservoir pressure. (see Appendix B).. rnd. 6..500 Rate. Input Data.6 is stated by Darcy's lawlO. a. p . (6. this term is insignificant. p* = 1 cp. Productivity index. 6. A 90% drawdown is likely when pumps are located just above the top perforations and operated at optimum condition with minimum fluid level above the suction. Darcy's .GLR = 800 GLR = 100 -a.this simple equation often is used to estimate flow rates from oil wells quickly. If fluid level has not risen.... r. 1. q. psia.08 x 1 0 . h. increasing the&.. IPR is defined in the pressure range between the average reservoir pressure and atmospheric pressure.. 6. =effective permeability of the fonnation to oil...583 ft. ft... Fig. Check actual viscosity under reservoir conditions. k. cp.3 Inflow Performance Inflow performance relationship (IPR)is defined as the functional relationship between the production rate from the reservoir and the t~ bottomhole flowing pressure. TABLE 6.. Normally..5-Typical set of gradient curves for different GLR's. . pr =average reservoir pressure. The actual flowing bottomhole pressure also depends on the separator pressure and the pressure loss in the flow conduits up to the depth of midperforation.1-INPUT 10. + Dq.0..000 2. STBUD Fig. .' DATA FOR EXAMPLE 6. Estimate production rate assuming a 90% drawdown. optimized wells flowing naturally or with gas lift should produce with 50 to 80% drawdown.. J . ~ $ 0 (6. Fig. such as gas lift or bottomhole pumps.6-Typical IPR curve.... 6.7 shows that an IPR based on Darcy's law for oil or liquid is a straight line.~ k . ft 500 1.466 ft.500 2.000 3.J l W f ) . If viscosity is not the problem.000 Water cut 0 Oil viscosity. is the absolute valMULTIPHASE FLOW IN WELLS i 1 IL For re = 1. 6. B.5.4) . po =viscosity of oil. and Dqo =pseudoskin caused by turbulence.... the problem may be caused by skin. r. bbUSTBO.GLR = 400 -x. law simplifies to go = 80 koh -(& . the pump is the problem.6 presents a typical IPR based on Darcy's law for single-phase liquid flow... A 30% drawdown is common when sand production is suspected. p+f . the IPR shown in Fig. = wellbore radius. po. Example 6.. 6. cp Unknown Formation volume factor.2 Productivity Index. = 0. 6. =radius of drainage. A 60% drawdown can be assumed without any knowledge of the system.= ttomhole flowing pressure.1 Single-Phase Liquid Flow..3.+GLR = 50 -&- -+ = 200 GLR -e. md k Pay thickness. B. check fluid level in annulus. psig b Fig.1-Diagnosing Pumping-Well Problem. 6. and P" 2.. c. Average reservoir pressure. or reducing drawdown.. If fluid level has risen. Assuming reasonable reservoir drawdowns.. (p. ft2.. Table 6. b.1 6 15 Permeab~lity. and no turbulence...3) where qo =oil flow rate into the well.GLR = 1. I . at which the well can produce with atmospheric pressure at the bottomhole.. q . s Pumped off (negligible fluid column above pump section) where k.. s = 0.. h = net thickness of the formation. d. diagcose possible problems.. as illustrated in Sec. =formation volume factor of oil. h ( p ~ P w f ) 40 = Solution.000 1. ~ i l b e r first proposed well analysis using this relationship. where A is area of circular drainage. STBOID. ft. (see Appendix B).llfor radial flow as 7. sf=total skin. d O '"Ow Pressure. For a well producing 20 STBOD. . especially for low-permeability reservoirs. psia. 63.. bbl/STBO Unknown 0 Skin. r. Determine productivity index. 7.5 1 ft and apply Darcy's law for radial flow.22 STBOlpsi-D AOFP = 3. % =effective permeability to gas.. cp.2 3. T. Successful buildup tests also help determine average pressure p.7. cp p. Special well tests. J - = 1. . Using Fig. md 16.. 3. p s ~ g Dr~lledhole slze (open hole).3.2 30 08 40 1. MscfD. On the basis of Darcy's law. (6.3 Single-Phase Gas Flow. . Determine AOFP. . acres (43. q . including 1.22- Assume r. Input Data. where qg =gas flow rate.75 + st] - q.p b ) . STBOlD Fig. p. Thus. are used to determine the drainage shape and re. Transient well-test i n t e r p r e t a t i ~ n . several information sources are available.2-Oilwell IPR Problem. 6. . q.P q ) ' STBO/D psi ' The productivity-index concept is not applicable for gas wells because the IPR for a gas well is nonlinear.example. A.h '= p s . Format~on volume factor. In [ 2o 0. I .03 x 10-$kKh(p: . DFSlGN APPl TCATTONF I1 . In injection wells. pr.75 + s. B. 6. the buildup test is called a falloff test and the drawdown equivalent test is called an injectivity test.. bbllSTBO Average reservoir pressure. md. ~ n .4 Sources of Information. STBO/D psi ' 6.672 STBOlD Rate. 1.000 12. Darcy's law for single-phase gas is . If properly conducted and interpreted.3-FACTORS FOR DIFFERENT SHAPES AND WELL POSITIONS IN A DRAINAGE AREAt4 S Sk~n. and skin. Z= gas compressibility factor determined at average temperature and average pressure.25 200 160 0 - Permeab~l~ty.490 ft. (pr . and jig = gas viscosity calculated at average pressure and average tempemture. well-test ~nterpretation methods quite often yield the most representative values of reservoir parameters. plot the IPR curve. These values normally are the volumetric average values In the radius of in- W .TABLE 6.2-INPUT DATA FOR EXAMPLE 6. d.. = 0. . re = 2. Example 6. . Table 6. 2.2 provides the input data. Solution. + Dq. fraction. (no turbulence) s ue of the slope of the IPR straight line and is a constant for a singlephase liquid reservoir. Well logs and cores also are used to determine k and h.7-Productivity-index graph.560 ft /acre) TABLE 6. . .0. "F Well draina e area. 01 1 wscos~ty. Pay th~ckness. . among others. ff h. Calculate drainage radius. such as extended drawdown or reservoir limit tests.6) I J= 1. Tr =average reservoir temperature. buildup ~~ ~~ and drawdown tests are used to determine khlp and st. . Reservo~r temperature.. When performing IPR calculations. 7.08 x 10-'k. = 1. such as (khlp). 6.3.For . OR. 4 gives the input data. r. the pr and AOFP uniquely define it. Solution. the drainage area is rectangular or of some other noncircular shape.365 Skin. whereas logs and cores determine the value of k at discrete points around the wellbore.3.3.9) ~ # " [ l n ( x ) .500 20 -5.. AOFP. acres Wellbore radius. ~ ~ r [ l n . Input Datu.~ k d ( p :. 5 . . Gas compressibil~ty factor. STEP 1 TABLE 6. E. . acres Wellbore rad~us. "F md Gas permeability. For a gas well. T. The drainage radius. p.03 x 1 0 .[ln(x) . psig Pay thrckness. jig. rnd 16. ft cp Average gas viscosity.8) 9g=-p . h. 1. .5 gives AOFP values for different skin factors.7 gives the IPR data used in this figure.. is dimensionless.p i f ) . . 3. .365 1 ~$.. In (re/r. Pay thickness. cp Formation volume factor. Table 6. tabulate and draw the IPR curve.. s 200 1 200 0.5--EXAMPLE OF SKIN FACTORS AND AOFP'S Permeabil~ty.. From Darcy's law (neglecting turbulence). Darcy's law can be modified for a bounded drainage area of different shapes as 7. .1 1. is Fig. Solve Part 1 for a square drainage area and a skin of zero. Because IPR is linear. draw IPR's and tabulate skin vs.75 s. . 6. . Table 6. ] ( 0.3-IPR Problems for Oil and Gas Wells. Applications of equations based on radial geometry to a noncircular drainage area could lead to substantial error.) . Most reservoir engineering calculations assume radial-flow geometry. . . 6.bbWSTBO Spacing.. G. pslg Spac~ng. (6. l . 'Sensltlv~ty parameter 5 2.~ ) + s. AOFP is Solution. .3. Radial geometry implies that the drainage area is circular and the well is located at the center of the drainage circle. . . Table 6. Table 6.4-INPUT DATA FOR EXAMPLE 6. . . . Calculate the AOFP..8 shows IPR lines for each skin factor. . h.] for oil and 7..5 Boundary Effects. . For an oil well. .. ft Sk~n.O. In many cases.3. . + ~ q . .'s Oil viscosity.9 gives this IPR curve..08 x 10-'k&(p.75 + s.500 80 0.6 gives the input data. 6. . 90 = Reservoir temperature.75 for gas.) -0. p.3forvariousdrainage shapesand well locations...7) ..08 x 10-'kOh . for an oil well. Average reservoir pressure.2 80 0.0. The productivity index for oil then is J = = ~ L = 2. . . re. ft Skln.75 = 7..6-INPUT vestigation. Pr .0. .Pbf 7. . s -5 AOFP (STBOID) 604 21 6 186 163 110 78 23 DATA FOR EXAMPLE 6. 10.r ' I TABLE 6. (6. .019 1.] + wherexisgiven in Table6. (6. Example 6. . Solution. Z Average reservoir pressure. . qomax Input Data. Fig. . 4 . From Table 6. From Part 1 .50 1.22. -1. ft r. Note that A is the drainage area and h / r . STEP 2 -1 0 1 5 10 50 TABLE 6.I 3.Pw. 3. during depletion. the bottomhole flowing pressure falls below the bubblepoint pressure.500 2.15 Fig.9 Gas Rate.950 14.6 -0-Phase Flow.000 2. 83 . 6.&IPR lines for an oil well in Example. as in gas wells. 6. Though quite simple to apply. twophase. a pressure/volume/temperature (PVT) analysis generates a phase envelope in a pressure-temperature diagram. 750 Fig. p. quite often is supported by field data and is more universally used for both oil and gas wells. and the corresponding temperatures are plotted. the Vogel IPR is not commonly supported by field data from multirate oilwell tests.94PR curve for a gas well in Example 6. Before discussing composite IPR.3 (Part 1). the IPR in oil wells is of the form A = Black Oil B =Composite C = Wet Gas D = Dry Gas A B E = Kolut~on Gas x =pr Temperature. Such information helps determine the type of IPR equation to be used. 6. Vogel IPR. A Vogel IPR curve can be generated if either the AOFP. 6.10. On the basis of fluid samples taken at reservoir conditions.500 1. 6. qg.3. Multipoint or backpressure testing of gas wells is a common procedure to establish gas-well performance curves or deliverability. For the case of an oil reservoir. pr.000 1. TABLE 6.030 7. such as Case E in Fig. WELL DESIGN APPLICATIONS Ferkovich IPR.10 presents a typical black-oil pressuretemperature diagram showing the physical state of the fluids. When the average reservoir pressure. Phase Behavior of Hydrocarbon Fluids. single-phase flow occurs when the bottomhole flowing pressure is above the bubblepoint pressure of the reservoir fluid at the reservoir temperature. For either case.0 250 500 Rate.and two-phase flow in the reservoir. on the other hand. Through simulation of saturated oil reservoirs. During reservoir depletion.880 15.190 Flow rate normalization with qo. The pressure profile in the drainage flow path determines the areas of single.17.10-Typical phase diagram for black oil.500 3. as shown in Eqs. Darcy's law is used above the bubblepoint pressure and a Vogel or a Fetkovich IPR is used below the bubblepoint. resulting in a combination of single. 6.10 where Fr is below the bubblepoint pressure.. either a vogel16 or a FetkovichI7 IPR is recommended. hydrocarbon phase behavior is reviewed briefly. 6. or a combination. Darcy's law is applicable only for singlephase flow.000 500 0 6.400 13. q STBOID . Vogel developed this IPR. MscflD Fig. 6. 6. DATA USED IN FIG. and a flow rate and the corresponding bottomhole flowing pressure are known. one can identify the type of reservoir fluid. Consequently. 6. such as single-phase.230 12.seemed to make this IPR work in a multiphase system.15 through 6.are known or the reservoir pressure. This IPR can be modified easily for a composite reservoir system.3 (Part 2). and the reservoir pressure. The ~etkovich" IPR. . the bottomhole and wellhead flowing pressures. O R Fig. The general conclusion from these backpressure tests is that. the pressure continues to drop unless maintained by fluid injection or flooding.7-IPR (pslg) w (~31~) 0 4.10. Fetkovich applied these tests to oil wells with reservoir pressures above and below the bubblepoint pressure. The IPR of such systems is called a composite IPR.and two-phase flow on the basis of the black-oil phase diagram shown in Fig.440 10. go-. a buildup test for pr and a flow test with a bottomhole gauge are needed. For the case of two-phase flow. For composite reservoirs shown in Fig. The dimensionless variables are defined as follows. .584 Table 6. . Average reservoir pressure. hours 'Sens~tw~ly parameter TABLE G.584 hours (316 days). 5 ]-3. WFl I DFSlGN APPI ICATlnN9 Fig.Tr dimensionless time. Dimensionless wellbore pressure drop. where w = fracture width and xf= fracture half-length. . producing at a constant rate.10 presents IPR's for five different times to the start of pseudosteady-state flow.20 equation is valid this for the transient radial-flow period. R Format~on volume factor.P&)] 141. CincoL. . defined in Eq. @ 0 VlSCOSlty. Table 6..crxfZ and . AS this equation states. when pseudosteady-state flow begins.2qy$. ko t ~= q and 0. 100. .13 . = 948 0. ... where @Pocrrt t. (6. f i Permeabll~ty. . .13) Also known as the transient IPR equat~on. 6. = 948 -hours.' t. For gas flow. 40 = k0h(i7. .x. .23 @Poctr. . are k0h[p. . psig Net pay thckness. E. qo. 6.7. . For homogeneous reservoirs. . . re.7 psia or 0 psig) in Eq.000 7. is given by. Calculae the duration of infinite-acting radial flow and tabulate the AOW.87(1. . 6. 10.TABLE 6.21)]+ )~ log . . .12246logr' + Note that for a single-phase oil reservoir. ~ a r l o u ~ h e r ' ~ and Aganval et al. . Solution. Duration of infinite-acting radial flow. r ~ ~for . Fig. respectively.14.2 1 05 IPR'S FOR EXAMPLE 6. Consequently.is C/o = kjw/kr/.23+0. .Z ' fracture Cfo. . . 6.oil and gas.000 1 2.13 presents a few of these types of curves. . The fluid is considered slightly compressible with constant viscosity. 6 p . r. at late times or after depletion starts. It is important to note that the early-time pressure behavior depends on Cfo However. pD. ..13-Constant rate type curve for finite-conductivity fracture in a closed-square Q5 . .000264kOr t'rj = 1 6 2 .~ ( 0 .5 (s?B"~"/"D) 79 48 40 35 (hours) 0. s. . c. .000. . .6[{10g[ 0. s Total compressibility. respectively.psi Wellbore storage coetflclent.9--INPUT DATA FOR EXAMPLE 6. t p S . for oil and gas. transient IPR curves can be generated for fractured reservoirs by use of curves from Fig. . 000)' 1 = 7. f ) Homogeneous Reservoir With Induced Vertical Fracture. As in the homogeneous reservoir case. h. the real-gas pseudopressure function23 is the used and gas properties are evaluated~at initial reservoir pressure. the longer the transient period. + 0. transient production period before the onset of the pseudo-steady state flow. . md 16. many low-permeability gas reservoirs produce a major part of the recoverable reserve in this transient period. are f 0. B 0 [ [ l o g ( ~ ) . the pressure response is influenced by the shape and size of the drainage area.2 x 10 -Q. .3.14) Example 6. and Infinire Homogeneous Reservoir. t. 6. ' D = '' = 1.I9 presented information on transient IPR. .. . .000264kgt @b*cO. making transient IPR's useful in the prediction of production decline. Input Data.p. . 1. .5-mansienl IPR Pmblem. . bbl/STBO Skin. AOFP (p..9 gives the input data. ft Poros~ty.1 10 100 1. et ~ 1 .q. . .4. ~ ' Meng and Brownz2 modified them for wells with an induced fracture at the center of a reservoir with different closed rectangular drainage areas. B.. bbllpsi Time. the lower the permeability of the rock. 424q.~ .13. Wellbore radius./.cp 1 1 I.l&TRANSIENT 2.87s] + log(t)] @p. average reservoir pressure and the AOFP at any time uniquely define the IPR for that time.5 Dralnage radlus.2 x 1 0 . (6. 14.developed curves for finite-conductivity vertical fractures. . It is characterized by flowing times less than tp. .584 t 31 1 121 10-5 0 001 0 1. the well known infinite-acting semilog approximation for a well with a skin.= 162.000 20 0. .. . The exponent n ranges between 0. Input Data. Consequently.5 and 1. respectively. ft Calculations (from Appendix 8) 6 6 800 0.3. Table 6. L. . 6. this coefficient increases with increasing values of k and decreases with an increasing skin factor.. re. Darcy's law suggests that the productivity of a vertical well is directly proportional to the net thickness or the productive length of the well. in horizontal wells. These IPR's are used to generate analytical production-decline curves.f Darcy's law (liquid) TABLE 6.4 Hor~zontal permeab~l~ty.4-Horizontal-Well Problem. In Italy's Rospo Mare field. Average reservoir pressure. m.9reh and where a is one-half the major axis of a drainage ellipse in a horizontal plane..8 gives the input data. a horizontal well was reported to produce 10 times greater than its vertical neighbors. % . However. the productivity is not directly proportional to the length.9 0 0123 2 i g 9C P i Rate.4. these wells can be drilled perpendicular to the natural fracture planes and substantially improve productivity. Golan and Whitson9 presented a detailed discussion of this IPR. 6. h. is - - 6. psia Wellbore radius. ft Horizontal-well length. ft R e s e ~ o l temperature. md k+.12-Horizontal-well drainage model (from Joshile).. Fig. 6. L.11-Different forms of IPR's. Fig. The Fetkovich IPR is customized for a well and is obtained by multipoint backpressure testing. C changes as a result of depletion.21 80 0. as presented in Example 6. 9 permeability anisotropy = Subscripts h and v refer to horizontal and vertical. h. low-permeability reservoirs with high vertical permeability. horizontal wells can be very long. It also is important to note that for gas wells.8--INPUT DATA FOR EXAMPLE 6. such as flow after flow or isochronal testing. The productive length of a horizontal well can be considerably greater.12 presents a schematic of the drainage region for a horizontal well. According to Joshi. Horizontal Wells. rd and IOni. r. This equation is called the oil or gas deliverability equation. Calculate the flow rate through a horizontal gas well at a bottomhole flowing pressure of 400 psia. For heterogeneous. md 4.770 1.0 often is attributed to non-Darcy flow effects. For pseudosteady-state production without rate-dependent skin. All variables are in oilfield units and can be converted easily for a gas-well IPR. The coefficient Crepresents the reservoir productivity index and is a strong function of phase mobility. Fig. this equation is applicable at all stages of depletion.65 16 2. Vert~cal permeab~l~ty. O F r Gas speclflc grav~ty.. n less than 1. Assuming pseudosteady-state flow. T.490 0. Unless limited economically.7 Transient IPR.. naturally fractured reservoirs. neither C nor n changes. located at the middle of the pay thickness. Example 6. p.0 for both oil and gas wells. ft Dramage radius. yg Net pay thickness. The decline curves are only valid in the < for L > hIOniand (U2) 0. for U 2 ~ 0 .. Such increases also are possible in very thin. Solution.18an IPR for a horizontal well of length. acres IPR Perforat~on ~ntewal. For 011 wells below the bubblepoint pressure. psla Resewo~r temperature. c. Sk~n. use the Vogel IPR.pb = bubblepoint pressure and %b =oil production rate at the bubblepoint pressure. given 9. Calculate pdvs. permeability. O F Permeab~l~ty. and q. Mathematically. This technique is used widely in the design. Wellhead temperature. ~n r.. @. and skin among others. The system sensitivity study is a major engineering application of production-systems analysis. ft h. may mitigate these problems. "API Gas spec~flc gravlty.14 elucidates this point graphically before actual problems and sensitivity studies are discussed. in. k. pslg Perforat~on ft top. 3.. and the like) are accounted for by combining these losses with the appropriate total-system losses. A step-by-step procedure to calculate transient IPR is presented. Once the parameter is optimized. p. Calculate the fracture penetration ratio xf/x. through perforations. r.). p. determine the dimensionless pressure PD('D@ C/D? TABLE 6. If these curves do not intersect.. shotslft Tunnel diameter. md k. any artificial-lift method will lower this intake curve by either reducing the total pressure loss in the tubing through aeration of the fluid column or by adding a net positive pressure change. 7.. @. In the input-data . From the type curves. Do the following..O 20 12and . A system sensitivity is defined as the functional relationship of the production or injection rate with any system parameter. 6. allowing the determination of surface production rate through the whole production system..8 shows how to use transient IPR to generate production decline curves for homogeneous reservoirs. Water cut. Wellhead pressure. Dralnage area.g. However. yg GOR. % 011 gravlty. Calculate dimensionless time r ~ for any assumed time r and ~ f for known parameters k.. Calculate %. L. such as tubing internal diameter. such coupling allows the reservoir to produce fluids into the wellbore and enables the piping system to lift these fluids to the separator at the surface. Ap.w. 5.4 Production-Systems Analysis Production-systems analysis is a simple engineering tool to couple IPR with the tubing intake..' In. Pay th~ckness. Calculate the dimensionless fracture conductivity Cm 2.ly. Steps 1 through 5 are followed in the range of pressures considered to generate IPR curves. Fig.4 3. Additional pressure losses in the production system (e. T. ft Perforation shot dens~ty. is assumed for a closed reservoir. 7. The intersection of these curves is the solution point or natural flowing point and determines the producing rate and the pressure. such as gas lift or subsurface pumps.h..35 7 x~IY. Dp. go below the bubblepoint pressure using Vogel's equation. In 4. y. Intake Curve C .14-Typical qo qpuq systems graph. The graphical presentation of the coupled IPR with the tubing-intake curves often is called the systems graph. 6. wellhead pressure. 6.11-INPUT DATA FOR EXAMPLE 6. indicating this well's inability to produce to the surface. . other parameters are considered.'s Wellbore rad~us. Intake Curve B is the natural flowing curve with production rate of go.5 160 darcy 20 4 0. 'Sensttn~ty parameter 0. xf.Pb' where. to the fluid column at the bottom of the tubing string by use of a subsurface pump.r/ among others 4. and . where. scNSTBO Water speclflc gravity. induced fractures.I d I \ L for single-phase oil or gas. Intake Curve A does not intersect the IPR curve. the well probably is loading up and artificial-lift methods. In Tublng ID. 6.Vogel.6 Resewolr pressure. Sequential optimization in single-variant steps is a convenient way to learn the procedure and troubleshoot. such as those in the tubing or reservoir system. chokes. Example 6.865 70 Hagedorn and Brown 5.I (A) Rate Fig.. Calculate gob and J at the bubblepoint pressure using kb and J gob =Pt . econornic evaluation. A.012 0 824 and 1 5 For gas wells. multivariant optim i ~ a t i o n ~ ~ one to determine the most profitable well design enables configuration while considering a number of system parameters simultaneously. The as production rate improves to qpUmp a result of the new pumped well. This is accomplished by repeating IPR or intake curves in the system graph for different values of the sensitivity parameter. A drainage geometry x. For gas.500 180 1. 1. m(p) is used instead of pressure.8 0 35 0.. gasloil ratio (GOR). with and without induced hydraulic fractures. This also is called sequential optimization. O F Tublng correlation Caslng ID. This figure shows a typical IPR curve with three hypothetical intake curves. Tunnel length. However.65 500 1 07 100 6. and troubleshooting of oil and gas wells. A few example systems analyses with a computer program called PERFORM^^ will show their importance in optimized production-systems design and troubleshooting. Wellhead temperature. y.11 gives the input data. To study the rate decline. in. Example 6. combination of 1. shotslft Tunnel diameter. This unstable flow condition is very apparent from the location of the point of intersection between the IPR and the intake curves.65 300 1 03 50 8. Additionally.012 2 75 Fig. O F Permeability. 10. psig Perforation top.200 160 1. Table 6. r. 6. As mentioned earlier.824-in. psia Reservoir temperature. % 011 gravity. A successful fracture stimulation treatment can reduce the skin to . f l Perforat~on shot density. resulting in a more costeffective treatment. a matrix acid treatment or a fracture treatment can be used to remove skin.6. This is a typical low-permeability gas reservoir with very little liquid production. However. Tunnel length.5 20 -4.7-Production-Systems-Analysis Problem on Effect of Skin Improvement by Acidizing a n Oil Well. ft Skin. tubing ID.15 shows that the damaged well with a high skin of 12 produces about 22 STBOD with both tubing sizes. the unstable flow condition continues until production reaches about 60 STBOD.12-INPUT DATA FOR EXAMPLE 6. Note that this stimulation effect improves the inflow performance to an extent that the potential for surging with any size tubing is eliminated. The rate difference is so marginal that it may need further verification with another method of calculation. if the total skin is reduced to 5 or less through hydraulic fracturing. GOR.TABLE 6.8-Production-Systems-Analysis Problem on Effect of Fracturing a Gas Well. p. Solution. 6. The pump suction pressure is 50 psig. this well offers excellent opportunity for stimulation. &. Oil Rate. k. Dp. use of a very efficient matrix acid treatment to remove skin resulting from damage yields a skin of 11 with a very marginal increase in the production rate. % Water cut. Note that hydraulic fractures can remove damage and then impose some negative skin. T.13 provides the input data. IPR Perforation interval. the well production doubles to about 16 STBOID. Table 6. For the 1.5-in.5 10 0 25 0. Solution. although the well surges with the larger 1. the well would surge if the calculated rate was 18 STBOD or less. L. scflSTBO gravity. Example 6. production will range from 89 to 107 STBOD. even for the smaller 0.16-0il rate sensitivity to total skin. 3. Water specif~c Wellhead pressure. Tubing ID. "API yg Gas specific grav~ty.6-Production-Systems-Analysis Problem on Effects of Skin and 'lhbing ID. Porosity. and 17 3. 0. tables for the following examples. Input Data.' s Wellbore radius. in.15-Effectsof ance. tubing ID and a suitable stimulation optimizes the production system design. tubing ID. h. -2. md Pay thickness. This vertical well produces with a beam pump installed 100 ft above the top perforation. vertical-fractured reservoir. STBOID tubing ID and skin on production perform- kb . time must be the sensitivity variable. T OF Tubing correlation Casing ID.. If the source of damage is properly diagnosed. the sensitivity parameters are indicated by an asterisk.16. Example 6.. 6.840 70 Hagedorn and Brown 5. QT . in. However.4 or less. Fig.7 Reservoir pressure. in.17 gives rate-decline curves with transient IPR's for a homogeneous reservoir and a finite-conductivity. The high reservoir pressure of 6. Table 6.5 160 Darcy 20 4 0 35 12 0.5-in. Thus. The system graph presented in Fig. Input Dau. ft .800 psia makes it a perfect candidate for fracture stimulation. a nonfractured gas well. The WELL DESIGN APPLICATIONS Input Data. Thus. optimum production refers to the maximum economic production. i\ 8 BOPD A -1 0 5 Skin 10 15 20 'Sensitwtty parameter Fig. On the basis of the sensitivity analysis presented in Fig.5-in. Give a critical review of skin sensitivity analysis and assume the skin caused by damage is 6 out of a total skin of 17. This spread in the flow rate is enough to question the accuracy of the pressure-loss correlation. depending upon the tubing size. q'~. This problem shows important effects of tubing ID and skin damage on production performance of an oil well.12 provides the input data Solution. The larger tubing size not only increases the production by a factor of five but also completely stabilizes the production. ~ h acres . tubing ID. 6. Drainage area. This problem compares the transient rate declines over a year for a fracture-stimulated well vs. in. L. With systems analysis for the well in this example. % 01 1 gravity.10-Production-Systems-~nal~sis problem on ~ f fect of Water Cut. Water cut. the Mukherjee and Brill. 6.494 1. The bottomhole flowing pressure.995 Time. Input Data.13--INPUT DATA FOR EXAMPLE 6. in.891 psia. Chap. &. 6. varies between 2. This problem is a typical water-cut-sensitivity analysis in an oil well with three-phase oil. The lower value was calculated by the Orkiszewski method and the higher value by the Aziz er al.0 3. ~ - Example 6. ~h~ ~~~~~ieral. a judgment based on experience and real data analysis is often the best guide to the selection of the proper or most accurate pressure-gradient method. and the production rates calculated from them can vary substantially. and gas production. In. Fig. ..16-Gradient curves with different pressureloss methods. pslg Perforat~on ft top. nonfractured g a s well. To design the fracture.5 160 Translent 50 4 kcb Poros~ty. Errors in production-rate predictions can be caused by other sources. an additional fracture-length-sensitivity analysis for optimization should be performed.05 50 2.8 Reservo~r pressure.- ~~~- . The Orkiszewski and Hagedom and Brown pressure-loss-calculation elingbf - - - - ~ . h.9-Production-Systems-Analysis Problem on Effect of Pressure-Gradient-Prediction Method. yg Liqu~d y~eld. 0. T. p s ~ a Reservoir temperature. T O F Tublng correlat~on Casing ID. and Beggs and Brill methods predict conservative production rates in that order. 6.TABLE 6.. such as the fluid properties calculated from empirical models.800 220 0. pi.65 1 1.17-Comparison of production rate decline for fractured vs. Dp. md Pay thckness. The gradient curves in Fig. This example illustrates the discrepancies in pressure-drop calculations for a typical oil well with associated water and gas production. Consequently. in. Wellhead pressure.18. 4 discusses many methods. Understanding this problem should clarify any other sensitivity analysis where the same sensitivity parameter is repeated. The well in this problem shows almost a twofold increase in production for the whole year with placement of an infinite-conductivity vertical fracture of 500 ft half-length. r.14 provides the input data Solution.. Dralnage area. R Skln. Tublng ID. y. These four methods underpredict the rate compared with the Orkiszewski and Hagedom and Brown methods. A dilemma often facing engineers who must solve multiphase-flow problems is how to decide the most suitable pressure-gradient-prediction method to calculate pressure losses in the tubulars. hi^ is a discrepancy of nearlv 30%. days Fig. s Wellbore radius.. a transient IPR curve as opposed to a Darcy-type steady-state relationship is used. % 4. bbl/MMscf Water speclflc gravity. 6. reh. Wellhead temperature. acres IPR Perforation interval. methods predict lighter pressure gradients in the production tubing. Tunnel length. "API Gas spec~fic grav~ty.07 600 8. Aziz et al. 6. k. shotslft Tunnel d~ameter. "F Permeability. I8 show the pressure profiles in the 9. Table 6. which is based on an empirical approach.245 and 2. Example 6. ln. method is based on mechanistic modvertical upward flow and comes within 5 % of the Beggs and Brill prediction. and Mukherjee and Brill methods. water.35 7 0. in.810-ft-deep oil well producing water and associated gas. Ansari eral. R Perforat~on shot dens~ty.840 70 Cullender and Smlth 4.4 10 0 55 0. The pressure gradient or the pressure profile in the tubular is calculated with the different methods and presented in Fig. calculated by different methods. resulting in higher fluid deliverability when compared with other methods used in this example. pslg Perforat~ontop.196-Water-cut sensitivity. 6.11-Production-Systems-Analysis Problem on Effect of Vertical Permeability in Horizontal Well. ff Perforation shot density. ~n Tunnel length.19A is the systems graph with tubing-intake curves plotted for each water-cut value considered. T O F Caslng ID. T.07 100 8. although it affects the rate from a horizontal well quite dramatically. in. The vertical permeability increases over a range of 0. In 5. In addition to these problems.01.07 200 100 9. y. 6. Fig. As the sensitivity plot in Fig.15 gives the input data. 6.. the oil production rate also falls because of the increased water cut. STBUD Water cut.000 0J . Tublng ID.19B shows. Pay th~ckness. Wellhead pressure. . ~n Dralnage area. ft .20A shows dramatic Improvement in productivity in the horizontal well over the vertical well. yg GLR. scfISTBO Water speclflc gravlty. "API Gas speclflc gravlty.TABLE 6.000 3. Water cut.000 6.. It is important to know that vertical permeability does not affect the production rate from a vertical well when Darcy-type reservoir-flow equations are used. 13.. This example uses an openhole gas well to compare the performance of vertical vs.65 300 1. T. % Fig.0 2. L.15--INPUT DATA FOR EXAMPLE 6.35 6 05 10 0.65 360 1. shotsift Tunnel diameter. the Dressure gradients in the tubing increase because of the density difference between oil and water and also because of the reduction of dissolved gas in the fluid system. In.. T.m 4. y. The intersections of the tubing-intake curves with the IPR curve give the production rates for each water cut. Table 6. "API Gas speclflc gravlty. horizontal wells with vertical permeability as the sensitivity variable.650 70 Hagedorn and Brown 5 921 2 441 kh Porosity.16 provides the input data.03.ma 2. Dp. s Wellbore radius.12-Production-Systems-Analysis Problem on Effect of Choke ID. Wellbore correlat~ons* 500 20 32 0. formation of oillwater emulsions in the production tubing at higher water cuts may cause increased viscosity and affect the oil production rate more drastically.14--INPUT DATA FOR EXAMPLE 6.9 Rate.psla ReSe~Olr temperature.441 Ansari et a1 Z6 Hagedorn and Brown*' Ok1szewsk12~ Beggs and BrillZg Mukherjee and Br1113~ AZIZ a1.acres IPR Perforation interval. O/O 01 1gravlty. Example 6. "F Tublng correlation Casing ID. 6. In. @. WELL DESIGN APPLICATIONS . Fig. Solution. Tubing ID. "F Permeab~l~ty. ft h. l md. This is an example of surface-choke-ID sensitivity with the Ashford and Pierce3* correlation. Table 6. hh. 6. Fig. 0 o I \ 1 I 5. STBOID Water Cut. Solution. This is a primary reason for economic successes of horizontal wells in naturally fractured reservoirs with high vertical permeabilities. Reservoir temperature.. 6.001 to 0. -- Example 6.000 200 36 0 44 0 35 160 DarcyNogel 44 4 0. yg GOR..07. md k. p.' % 01 1gravlty. pslg Perforat~ontop.810 70 5. ft Wellhead temperature.m 0 10 20 30 40 50 60 70 80 90 100 I 01 1 Rate. &.3' et TABLE 6. Skln. As the water cut increases. % 'Sensltlv~iyparameter Input Data.10 Reservoir pressure. Wellhead temperature. In. scfISTBL Water speclfic gravlty.09 32 0..05. The systems graph presented in Fig. 'Senslt~ltyparameter Input Data. Appropriate use of artificial-lift methods often may mitigate this problem.19A-Systems graph for water-cut sensitivity.20B shows that the horizontal-well performance improves substantially as the vertical permeability increases. O F Wellhead pressure. r. MscflD 0 0. ft Skln. As Fig. This often is called the gradient reversal point and happens at about 700 scftSTBL in this example.07 0 1. "API Gas speclflc gravrty.. s Wellbore rad~us.22B shows.17 provides the input data.O.22A is a production-systems graph for GLR sensitivity. md Pay thrckness.5 160 Darcy/Josh~ 10 2.500 F Reservoir pressure. 'Sensltlv~fyparameter Permeab~lrty (radral).' in. 0. GLR sensi- oI Gas Rate.in. yS Llqurd yreld. Solutiorz. psrg Vertrcal well Separator temperature..8 6. increasing the mixture velocity and the friction losses in the tubing. k.03.08 0.11 2. Wellhead pressure. In.02 004 0..20A-Systems graphs comparing horizontal vs. Choke correlatron 'Sensll~v~fy parameter Input Data. Poros~ty. kv.000 0 54 TABLE 6. r. Drarnage area. in.208-Horizontal-well cal permeability. this reversal phenomenon causes the dome shape of the liquid rate vs. 5.0.21B shows how the choke sizes control the production rates. This example demonstrates this sensitivity and provides an introduction to the application of multiphase-flow theory to artificial-lift design.'k. O F 180 Horrzontal permeabrlrty. h.06 0. s Wellbore radius.9 0.16-INPUT DATA FOR EXAMPLE 6. p.. L. The systems graph in Fig. reh.Table 6.0. h. Example 6..65 2. 6. DATA FOR EXAMPLE 6. producing well is the algebraic s i m of the reduced hydrostatic gradient and the increased friction gradient.ft Water cut. 6. psla Reservoir temperature. Solution. in.OOl 600 5. 90 Fig. Tunnel diameter. psra R~S~NOI~ temperature..O F Tubing correlation Casrng ID. pi. The larger the choke size. in. y. fi. Fig. % 01 1 gravrty.1 0. md . T O F Tublng correlation Casrng ID. L.psrg Perforatron top. 30 20 1. there is a threshold GLR above which thereduction in hydrostatic gradient is less than the increase in friction gradient.0.000 70 Ansari et a/. rn.75 Ashford and Prerce 44 Perforatron top. Horizontal Flowlrne Flowlrne length.5 160 DarcyNogel 30 30 0. As the pressure declines in the direction of flow. 6. Drarnage area.826 2. Choke ID. L.TABLE 6. T O . vertical well performance with vertical permeability variation in the horizontal well. T. ft Flowlrne correlatron Flowlrne ID. R Separator pressure. Perforation shot denstty. Fig. acres IPR Porosity. ps. ft Skn. in Tublng ID. Dp.500 140 10 20 0 2.18 gives the input data. Thus. Table 6. the higher the production rate. acres IPR Water cut.001. shotstft Perforation interval. A . the total effect of increasing GLR on the pressure losses in.12 J Vertical Permeability. R Tunnel length. Wellhead temperature.375..012 1. yg GLR. GLR uniquely affects oilwell production rates. T. +. % 01 1 gravrty.13-Production-Systems-Analysis Problem on Effect of Gasbiquid Ratio (GLR). With increasing GLR.65 800 1 07 4 20 7 0 35 10 0. "API Gas speclfrc gravrty. k+. md Pay thckness. the compressible gas phase expands and 3.867 400 Dukler 2. 6. O 3.441 Vertrcal permeab~lrty. y. in. reh. % Horizontal-well length.17-INPUT Reservoir pressure. performance sensitivity with vertiMI11 TIPHASF FI OW 1 WFI 1 T U .865 1. r. 6. in. 6. % 4. md Fig.500 70 Duns and Ros 3. bbltMMscf Water speclfrc gravrty. Increased GLR reduces the hydrostatic gradient and the mixture density of the fluids in the tubing.000 gains velocity.5.12 4. Tubrng ID. Input Data. scf1STBL Water spectftc gravrty. particularly gas lift.0 1.21A shows the change in tubing-intake curves with differing choke sizes. wellhead pressure.O 35 160 DarcyNogel 30 30 0.5 d=0. Artificial-lift methods help manipulate the tubing-gradient curves for a predesigned boaomhole oil or liquid rate.000 1. Dra~nage area. % Oil gravity. 6.4 0.5. DATA FOR EXAMPLE 6.23 presents a typical pressure traverse for a downhole-pumped well. If gas is injected into the production string at the lowest possible entry point. the GLR should be increased to 700 scf/STBL. 6. 6.000 1. For optimized production. psia Reservoir temperature. In.7 0. fl h.3 0. and the corresponding bottomhole flowing pressure.22A-Production-systems graph for GLR sensitivity.8 1 Choke Diameter. these wells cannot produce to the surface without artificial lift. O Permeab~llty (rad~al). Llquld Rate. shotslft Tunnel diameter..1 Subsurface Pump. @.5 0.700.13 4.75 0. 6. Without the pump this well is dead.000 GLR. Skin. T. % Wellhead pressure. 6. Tubing ID. are identified from the IPR curve. Opbmum GLR = 700 scfISTBL 0. q. &.h.500 Fig..23 and 6. this well produces with a GLR of 175 scf/STBL. with the fluid column at Point A in the tubing.22B--GLR sensitivity for gas-lift design and optimization. Perforatron interval.21 B-Choke-ID sensitivity. In this case. In. IPR Water cut. ft 180 30 20 1. "API Gas specifrc gravlty. yg GLR. d=O d=0.. 3. With the same dead-oil gradient. psig Perforat~on fl top. as Figs. rn.35 7 08 10 100 6.867 Fig. ~ hacres . y..07 20 4 0. scfISTBL 2.6 0. STBUD Fig. in. 6. Tublng correlation Casrng ID. in. 0 500 1. A designed rate. Wellhead temperature.375 d = IPR 01 0. L.000 I 2. r.200. Perforation shot dens~ty. In. p.2 0.500 F TABLE 6. Tunnel length.' scWSTBL Water speclfic gravity. 6. p.000 3. Pay th~ckness.5m Fig. Fig..1.500 2. This is also the basic principle of gas lift.865 70 Duns and Ros 5. kb Porosity.500 3. 6. s Wellbore radius. the oil level in the annulus should be at Point B with the pump suction 91 - WELL DESIGN APPLICATIONS . the maximum liquid production rate is 983 STBL/D and occurs at the optimum GLR of 700 scf/STBL.18-INPUT Reservoir pressure. m \ . go.. the GLR can be increased. Before any artificial-lift installation. 'Sens~t~vlty parameter O F tivity curve.21A-Systems graph for surface-choke-ID sensitivity. with a resultant increase in the liquid production rate to 983 STBLiD. with a designed flowing 1 200 .012 1. md k.. muf.5 Artificial Lift The previous section introduced artificial-lift diagnostics based on systems analysis principles. T. 1. This section presents the application of multiphase-flow principles in the design of artificial-lift methods. 0 0 500 1 . In many cases..24 show.500 2. Dp.65 175. Artificial-lift methods can be used to mitigate the problem. ~ ' al. &. The pressure difference. 6. as presented in Example 6. or even interstitial water.presented the best known procedure to calculate the minimum stable-gas velocity.228). showed that to unload a gas well. The gas is injected through a port in the gas-lift valve with a designed differential pressure. Ap. 6. at Point C.6. For simplicity.13. Any dynamic gas-gradient method can be used. 6. is then fixed. Historically. the bottomhole flowing pressure is obtained from the IPR curve. From the GLR sensitivity study in Example 6. Gas wells often produce a liquid phase. in low-permeability gas wells. However. particularly if the proper PVT relationships given in Appendix B are used. This permits complete unloading of liquids from the wellbore. causing severe well-loading problems. such as oil. and designed wellhead pressure.24-Pressure Pr traverse for gas lift. even completely stopping it with time.53 Gas Lift.9. This design generally results in reduction of the production tubing diameter to achieve sufficient gas velocity. psig I +I < 2 Fig. &h. 6. When the pump discharges the fluid into the tubing at a rate. 6. at the port. the wellbore liquids are produced to the surface and the well does not accumulate any liquids. allowing the liquid to flow to the surface with a predesigned flowing wellhead pressire. is calculated for the optimum GLR. the gas velocity may not be sufficient to lift the liquid phase to the surface unless the tubing diameter is reduced to attain the liquid unloading velocity. a constant fluid gradient is calculated below the point of gas injection. Lower velocities result in accumulation of liquid in the well. wellhead pressure.23-Pressure traverse for bottomhoie pumps. where the optimum GLR was 700 scfISTBL and the corresponding production rate was 983 STBUD with a wellhead pressure of 100 psig. Normally. pressure difference is between the pump intake and the pump discharge. This problem can be particularly severe in condensate or retrograde condensate reservoirs.23 shows. If the oil is produced through the annulus. the largest droplet in the gas stream must be carried to the surface. the tubing forms the gas-injection string and needs to be blocked at the bottom. we highly recommend the use of compositional models presented in Appendix C. This gas-well-loading problem can be mitigated by increasing the gas velocity in the wellbore or by using some form of artificial lift.24 presents an example gas-lift pressure traverse in the production tubing. Gas-well-loading problems often can be diagnosed from production-systems-analysis plots. To identify any problem with retrograde-condensation-induced loading. with the discharge pressure at the depth shown.L :-I r Pwb Pwh Pressure. such as plunger lift or gas lift. At that optimum rate. Gas-lift design is optimized based on a GLR sensitivity study. ~ In j ~ wells with high liquid velocity this may not be a problem. Continuous lift of a constant-quality fluid can be achieved only by proper tubing string design. the prediction of the conditions conducive to this problem has . the effective permeability to gas can be severely reduced if condensation occurs near the wellbore. p. as in Example 6.ur r 3~ been very u n r e l i a b ~ e~ ~ ~n eer ~ 1 . Surface production rates and phase characteristics often can be quite deceptive in predicting gas-well-loading problems.oil non is called gas-well l ~ a d i n g . With the available surface casing injection pressure.24. qo. Note that the tubing is packed above the pay zone. the optimum rate.33 Fig. go. gas lift aerates a liquid column with injected gas. this well also is assumed dead before the installation of gas lift. at the point of injection. The port size of the gas-lift valve then can be determined with the Thornhill-Craver equation8 or any type of choke equation for the passage of the designed injection gas rate with the differential pressure. condensates. Fig.h. the dead-oil gradient is used for simplicity.. as discussed in Example 6.13 (Fig. 6. between the casing and the tubing. Normally. 6. Theoretically. all the multiphase pressure-loss methods revert to the single-phase-flow case based on superficial gas velocity. which is a precondition to a gas-lift installation that ensures injection into the tubing only through the valve port. Unlike with Turner et the continuous film model presented earlier by ~ u k l e r . as shown in Fig. Dukler proposed a method to determine the minimum gas velocity required to move the liquid .3. The intersection of the IPR curve with the tubing-intake curve in the unstable zone is agood diagnostic of such problems. qo. The upstream injection pressure is determined from the calculated casing gradient. If the velocity of the producing gas is sufficiently high. This tubing-gradient curve is generated by use of one of the two-phase-flow models with the designed rate. The higher the volume and pressure. the injection gas rate is calculated by subtracting the produced gas rate from the optimized gas rate that was calculated from the sensitivity study for the optimum liquid rate and GLR. This may drastically reduce the gas velocity in the wellbore. particularly with low reservoir pressures. In gas-condensate reservoirs. the greater the plunger-lift efficiency. Plunger-lift efficiency also is very sensitive to the volume and pressure of gas in the annulus. To calculate the fluid pressure at the point of injection (Point B). After determining the flowing fluid pressure in the production string. intermittent gas lift and plunger lift (which is intermittent in nature) are a second choice because of higher cost and associated operational problems. the pump must be designed to proThis vide a pressure boost of Appump. Ap. In Fig. it is even conceivable that a well producing dry gas may have liquid in the wellbore. the fluids flow to the surface at the specified wellhead pressure. and the calculated gas injection rate through the annulus. As Fig. This should allow lifting the well at rate q to the surface at a predesigned . h h . the injection gradient or casing gradient is calculated. Ap. This liquid accumulation eventually can create enough hydrostatic pressure in the wellbore to curtail gas production severely. This phenomej . 6. Depending on the phase behavior of the gas.6 Gas-Well Loading 6. particularly in the presence of a large retrograde envelope.. This is particularly important for unloading wells after completion or workover. T O F Tubing correlat~on Casing ID. and p. g = gravitational acceleration.. 1bm/ft3. In. yg GOR. =terminal velocity.25-Gas-well-loading erosion envelopes. Water spec~flc Wellhead pressure. film on the tubing wall continuously upward.. 1. In Fig.14-Effect of ntbing ID on Gas-Well Unloading and Erosion. Gaswell loading does not occur if annular flow is maintained throughout the wellbore. p. the more efficient the unloading. Table 6. =gas density. drop-terminal-velocity equation can be written as Tublng Olameter. In this example.39. 3 0 F To unload a well. For a spherical droplet. Velocity depends on permeability and pressure.990 80 Gray 4 408 0. and Z= gas compressibility factor. p. the gas rates for all the tubing ID'S considered in the 1-md permeability where v. rego or^^^ reported an error in the coefiicient of this equation in the original Turner et paper. A =area of cross-section of flow.25 with the tubing-ID sensitivity plots for two different permeabilities.19--INPUT DATA FOR EXAMPLE 6. is in dyneslcm. scfISTBO gravity. 2. acres IPR Completion Water cut. 6. O Permeab~l~ty.^.163.' in. the critical gas flow rate. Velocity is the key to liquid unloading. However. The maximum stable droplet diameter can be determined from the relation35 where o = surface tension.is 0. p s ~ g . Pay th~ckness. Coleman et a1. 6. ~ i r n eet 01.. % 01 1 gravity. From Eq. and p . two gas wells are studied: one with low permeability and a suspected loading problem and one with higher permeability and a suspected erosion problem.. the equation presented here has been corrected and does not include the Turner et u1. psia Reservo~r temperature. The higher the velocity is.-suggestedthat wellhead conditions controlled the liqr uid load up in most cases.2.. CD.' md k. q. which can be substituted into Eq. the terminal velocity is given by the relation developed by Foust et aL3* Example 6. Turner et al. can be calculated as where q. The similarity of Eq.0.. ft h. This was substantiated by Coleman er It is important to note that the critical unloading rate determined from the liquid droplet model may not be applicable to slug flow. 6. 4. 32.. As long as the actual predicted production rates are higher than the corresponding unloading rate. Skin. p. showed that the drag coefficient for the droplet. Tublng ID.Wconcluded from field tests that this 16% upward correction in terminal velocity is not needed to calculate the minimum gas flow rate or critical gas rate required to keep low-pressure gas wells unloaded. The best way to mitigate any flow-related problem is to identify the potential problem before it manifests and design the system around it. do =droplet diameter. Substituting this value of CD and converting the units of surface tension. A modified Turner unloading rate is used. Solution. p. a. These authors suggested that the minimum gas velocity in the well had to equal the terminal velocity of the largest liquid droplet in the flowstream.03 200 5. is in ft/sec. 'Sens~tlvityparameter 4.19. T. 1 6. The gas-well-unloading and erosion-rate envelopes are superimposed in Fig. 6. Wellhead temperature. and CD =drag coefficient for the droplet. suggests that gas-well loading will not occur if annular flow is maintained throughout the wellbore.19 gives the input data. WELL DESIGN APPLICATIONS Fig. a tubing ID sensitivity study must be performed to optimize tubing size. ~ . gas-well loading from downward flow of a liquid film on the tubing wall also can be considered.174 ft/sec2.39 With the annular transition criteria described in Sec. the wells are unloaded. Based on the terminal-velocity equation presented here. T = temperature. ftlsec. ft . y. 3 0 20 0 30 160 Darcy Open hole 0 54 0 65 150 1. the Turner er al. s Wellbore rad~us. 4. rsh. to dyneslcm. Drainage area. are in 1bmlft3.0.18 to obtain Assuming spherical droplets. which are controlled by tubing ID. = critical gas-flow rate. which predicts the transition to annular flow. arbitrary 16% upward correction to this velocity. in.2 for the Ansari e t ~ l .000 160 1. "API Gas speclflc gravity. The Turner et al. in. r.TABLE 6. 91 .= liquid density.25.14 Resewoir pressure. This example helps identify a gas-well-loading problem and provides a method of solution.0. O R . ft2.44. Consequently. lbm/ft3. ~ ~ mechanistic model.2 1 to Eq. the gas velocity has to exceed this terminal velocity at every point in the tubular until the liquid droplet reaches the surface.p = pressure. MMscf/D. psia. ~ b m l s e c . Input Data.. 6. model is accepted as a more reliable method of liquid unloading.824. A tubing ID sensitivity analysis is performed. ft. 6. Perforation top. where v. which was repeated in other references to this equation. This equation is bows will be more susceptible to erosion because more solids will based on extensive empirical information gathered by Shirazi et impinge on the projected area of the incoming pipe. pipe size. For the low permeability of 0.has . This means that for solid-laden gases. flow-stream velocity. In fact.~ a~b.^*-^^ Unlike the previous equations. this one is more mated50. 6. This type of plot should be used as a template for such studies.Abm. and fluid density. pipe material.55 x lo5 psi and an allowable penetration rate of h = 10 miVyear. the American Petroleum Inst. The model to compute the maximum penetration rate in elbows for carbon-steel material is . =parcause of solid particle impingement is directly proportional to the ticle density. more rigorous mechanistic model that accounts for a number of variables. Salama and Venkatesh4j developed this model to predict the penetration rate for an elbow. For the higher permeability of 1 md. 6. where the particle-impact velocity is lower than the flowstream velocity. 6. Example 6.23 to mitigate pipe erosion. allowing for some acceptable penetration rates resulting from erosion. = sand sharpness factor (empirical). Consequently.eter. the API study recommends C factors of 100 for continuous flow and 150 for intermittent flow. erosional velocity can be calculated from this empirical equation. F. this percentage pipe wall and not the flow-stream velocity.25. It is often more economic to produce wells at optimum rates. this ve- This equation assumes hardness T = 1. This adjustment is necessary because with high fluid density and viscosity most of the sand grains may pass through the core of the elbow without impinging on the wall. 6. J-!xf Stagnation =Om v L Fig.23 and 6. q. psi: and d = elbow diameter. this results in curtailment of production and. must be adjusted for the number of particles actually impinging on the pipe wall. ftlsec. the gas rate does not improve while the gas velocity in the tubing goes down. where h =erosion penetration rate.24 are independent of penetration rate.d = sand production . This equation seems to be more reasonable for limiting the designed fluid velocity in tubing to control any erosional damage to the pipe. 6. is very conservative.7 Erosional Velocity Zone Tee EquiMlent StagnationLength Partlde Initial Position Unear flJd V. they cannot be used for such calculations. with normal production decline. Assuming particle impact velocity equals flow stream velocity in Eq. This assumption is not valid for liquid flow. including flow geometry. total weight of impinging solids. affects the economics. will decrease to 30% for liquids because most of the fluids will be canied in the center of the flow stream without impacting the wall of the elbow. 6. This problem uses a value of C = 100. = partlcle impact velocity.51that for low-density gas systems. in. in the case of pipe flow. 6. Ibm lft3. Continuous excessive erosion often leads to mechanical failure. Salama and Venkatesh suggested that the applicability of Eq. ( A P I )recommends limiting the maximum velocity in the flow string to a critical value called erosional velocity. pipe d~amshown that the volume of ductile metal eroded from the surface be. the only parameter considered in these equations is density. the erosional velocity will be lower than for liquids canying solids. This flow-regime dependence of erosion calls for consideration of fluid properties. leakage. Elbows are a common piping component and are affected the most by erosion damage. tubing.9 125 for carbon steel). Allowing for a 10 miVyr erosion rate. Salama and Venkatesh4j reviewed the API equation for erosional velocity and concluded that the form can be justified from a Bernoulli-type relationship. p is the density of the flowing fluid in 1bm/ft3. 6. rate.3 md (as the tubing ID is increased from 1 in. such as the density. is the erosional velocity in ftlsec. Because Eqs. In Fig. viscosity. q.26 be limited to gas flow only. the problem should not exist.where A = empirical constant (0. fluid viscosity. The mitigation procedure used in this section should be used as guidelines for proper design. and p. millyr. they recommended the following relationship to calculate the erosional velocity caused by liquid impingement on a steel surface.to 5. noting that the particle-impact velocity in gas flows (with low density and viscosity) nearly equals the flow-stream velocity.). The prediction of erosional velocity is based on penetration rates in an elbow geometry. As the density increases.25. But this subject is very controversial and often not emphasized in design. thus reducing erosion. Sand in gases with lower densities causes more erosion than sand in liquids with higher densities. elThe other variables were defined previously. In this case. However. API recommends an unspecified C value of less than 100. Shirazi e t c 1 1 . i~ ~ w i c z . For sand-laden fluids. thus. sand-particle sharpness. To ensure a predesigned life of flow strings.000-psi maximum pressure-drop range in the flow conduit for clean twophase fluids. ~ ~ presented a new. B = Brinell ~ no ~~ ~ age caused by abrasive wear m e ~ h a n i s m . assuming a 3. suggesting no imminent chance of a loading problem. or both. For multiphase mixtures.14 illustrates the use of Eq. suggesting that the erosional velocity can be increased by reducing the density of the flowing fluid or decreased by increasing the density of the flowing fluid. in. Thus.23. Fp = penetration factor for steel based on 1-in. 100% of these solids general because vp is the true characteristic impact velocity on the will impinge on the pipe. and compositior?. the predicted v. It is estiand other^. sand density. Erosion limits the life of flow strings.45 where v. ~~ To control erosion. For sandfree flow.case are substantially higher than the corresponding unloading rates. ~hardness. the erosional velocity is compared with the mixture velocity in a pipe and the no-slip mixture density is used in Eq.45 The presence of sand in the flowing fluid results in erosion dam. the sand flow rate. and C is an empirical constant. Erosion is caused by cavitation or bubble collapse and by the impingement of liquid or solid particles on the pipe wall. causing the well to load in 3-in. erosion problems are predicted to occur for tubing ID'S of less than 2 in. as long as the producing gas rates are lower than the erosion envelope. However.d. However. This does not agree with experimental observations for sand-laden fluids.Iocity C Erosion is the physical removal of pipe material in contact with a flowing fluid.000. particle size. In producing wells. T = elbow metal hardness. ft3m: v.26-Concept of equivalent stagnation length. the problem may arise in the future. Salama and Venkatesh obtained this expression for erosional velocity in ftlsec. whereas the geothermal gradient below the base of permafrost was found by those authors and Lachenbruch et aLS8to be in the range of 1. The distance of penetration in the stagnation zone. they will not close if the actual operating or closing pressure of the dome is lower than the test-rack setting of this pressure. . This problem could be mitigated by use of a fluid valve that is insensitive to temperature. 2. sand size and density. scbsurface pumps.27-Dimensionless impact velocity nomograph. Fp.20 and a penetration factor. v.TABLE 6. ans son^^ presents a case study.4OF1100 ft. marine environment./if. repeat Steps 1 through 6 for different assumed velocities and plot h vs. L (in.. Dimensionless impact velocity. These problems are related to the low temperatures in the permafrost or marine environment. the tubulars cool down to the low ambient temperature. p. prior production through the well immediately before unloading to establish a flowing temperature gradient also could solve this problem. vp. On the basis of this equivalent stagnation length and assuming the sand particle is traveling in a one-dimensional flow field. F. 5. or both. Shirazi et presented this procedure in detail. 4. The difference in temperature gradients under flowing and static conditions is particularly instrumental in most of the reported production problems. Calculate the equivalent stagnation length. As a result. from Table 6. Fig. L DESlGN APPl TCATlOW .29 and 6.1 Permafrost/Marine Environment. = 1. Permafrost is the frozen or semifrozen alluvial formation found in arctic climates. pipe insulation. during cold startup. NRe. This leads to solidification of fluids if the temperature in the affected tubulars falls below the pour point and wax-appearance temperature in waxy The problem may be particularly severe if there is a gas-lift casing or pressure valve. the nitrogen pressure in the dome was substantiallv lower than the set valve closure pressure. and fluid properties. The hydrate-forming temperature also is marked to locate the potential zone or depth interval of hydrate formation. Hanson mentions that the valve was set to close at the flowing temperature at valve d e ~ t hwhich was 50 to 60°F higher than the actual static temperature . Before a shutdown. based on the flow-stream or the fluid velocity. from Eq. from Fig. thus preventing the transfer of gas to the next lower valve. 7.27.8. and hydrothermal grar y 'Fig.). also called the equivalent stagnation length.28 illustrates the unique dients found in oermafrost.~ ~ 6. the sand particles must penetrate the fluid layer. ~ c ~ a u presented a graphical relationship to calculate the characteristic impact velocity.26. 6. This is an iterative calculation and can be programmed easily on the and McLaury er also basis of this procedure. 6. of tion problems have been r e p o ~ t e d . fluid velocity. Thus. spherical glass beads flow-stream velocity. Compute the maximum penetration rate. There is also a significant difference in the flowing and static temperature gradients in these zones. 2. these problems can be avoided with such proper precautions as inhibitor injection. Because the pressure of nitrogen in the dome actuates these valves. However. Calculate NR. a ratio of characteristic impact velocity to the flow-stream velocity (vplv). and normal earth's crust.~ 3. QT .30. where a negative hydrothermal gradient60 in sea water follows a positive geothermal gradient below the seabed.1°F/100 ft. for the actual allowed h value from this plot. subsurface safety valve. flow regime.8 Special Problems This section 1s intended to alert the reader to some of the potential pipe-flow-related problems in permafrost and marine environments. 3.Abm for steel elbows and tees). from Ref. v (ftlsec). Depending on the duration of shut-in in such environments.68 in. Determine the characteristic impact velocity. q.8 to 3.27.4 to 5. To that impinge on the target wall. 6. is in lbm~ft. is in centipoise and fluid density. To calculate the erosional velocity for an allowed yearly penetration rate. 6. rounded corners Rounded. Depending on the geographical area. permafrost thickness may exceed 2. and @ from Eqs.) where fluid viscosity. and is determined from this relationship. and the particle diameter. Similar problems60-62caused by abnormal temperature gradients also have been observed in marine pipelines.28 in. 1. on the basis of a study of 34 wells by Godbole and Ehlig-Economidess7). vp/v.000 f t . L. 6. dp (in. ~ e cause of permafrost on Alaska's North Slope. d (in.45 locity depends on such factors as pipe geometry and size. is a function of the elbow internal diameter. Determine the dimensionlessimpact velocity. @. depending on the characteristics of the produced fluid. cooling tubulars during startup. v.28.27 presents this graphical relationship with three dimensionless groups. from Eq.26SAND-SHARPNESS FACTORS6 Description Sharp corners. 6. 6. 6. angular Semirounded. These pressure valves are very sensitive to the temperature of the nitrogen-charged dome. The characteristic impact velocity is calculated based on a simple mode154~55 defines a stagnation zone presented in Fig. proportional to the ratio of the mass of fluid being displaced by the particle to the mass of the impinging particle The following procedure should be used to calculate the maximum penetration rate caused by erosion damage in elbows for a given sand-flow rate. h. an accurate knowledge of temperature gradient is absolutely necessary for multiphase-flow-related design calculations. Select the W 1 . Particle Reynolds number.d (ft3/D). or other restrictive (choke) devices within or near the low-temperature zone. Fig. in a marine environment with subsea wells. A dimensionless parameter. 52 (or use 3. 6. v over the range of allowed h values. 1. Because the phase behavior of hydrocarbon fluids is highly temperature-dependent. wax deposition and hydrate problems62 may be expected. or the initial flow period after a shutdown. for the known flow-stream velocity. where L.).. if possible. some unusual produc~ ~ .Many ~ ~ them are related to the abnormal geothermal gradients in the area. Select particle sharpness factor. The permafrost zone is abnormally cold with low geothermal gradients (0. 6. where the gas-lift pressure valve just below the permafrost will not close during unloading. 6. particularly in a high-pressure and low-temperature operation.61 Gas hydrates are crystalline solid compounds resembling dirty ice in appearance.0801 MULTIPHASE n o w IN WELLS I co2 . They are commonly composed of water and a combination of methane. In marine pipelines. Inhibition/Dissociation. where the gas contains moisture or free water or load-water produced during post-fracture cleanup. Hydrates also concentrate hydrocarbons.0677 0. especially near the permafrost zone. also known as thermodynamic inhibition. To mitigate these problems.0460 0. Economics permitting. carbon dioxide.*-64 Gas hydrates (also called clathrates for encaging) are crystalline compounds that result when water forms acagelike molecular structure around very specific smaller guest molecules.29-Gas-hydrate envelope in a pressurehemperature phase diagram. propane. and hydrogen sulfide. Heating the system above the hydrate-formation temperature at system pressure. 6. In arctic wells. Adding or injecting an inhibitor. can identify pipe segments ideal for hydrate formation during cold startup.29. Fig. 6. The injection of inhibitors moves the hydrate stability conditions toward higher pressures and lower temperatures. One cubic foot of hydrate may contain as much as 180 scf of gas.65. 4. and high-pressure oil and gas wells.2 Gas Hydrates. 2. Determine the geothermal and hydrothermal gradients before the pipeline design. normal butane. 3. 2.0171 0.Hydrate Formation Temperature Temperature. riser pipes. If a hydrate-free system cannot be designed. workover. to the hydrate system to reduce its stability at the system pressure and temperature. nitrogen.65New inhibiTABLE 6. ethane. hydrates often form during the startup phase. During cold startup of producing wells after shut-ins for well testing. are particularly susceptible to hydrate blockage even during normal operation because of Joule-Thompson cooling. When designing gas or multiphase-flow pipelines in gashydrate-prone systems.. 1. if possible. such as methanol or glycol. the oil and gas industry uses properly insulated pipeline segments or hot oil pumping when the problem occurs. hydrate-related blockage in the pipeline should be considered while troubleshooting flow or production problems. The common method of hydrate inhibition is by inducing thermodynamic instability to the hydrate phase. as shown in Fig.0077 0. Displace the pipe segments identified in Step 4 with chemical inhibitors. generation of a gas-hydrate envelope in a pressureltemperature phase diagram. Unlike ice. Ideal situations for gas-hydrate formation include 1. 3. Gas-hydrate phase behavior and kinetics are well understood. Sloan6] presented a very comprehensive review of these processes. during any shutdown. consider the phase behavior of gas hydrates and.21-EXAMPLE Component Methane Ethane Propane Iso-butane mbutane Iso-pentane n-pentane Hexane Heptane Nitrogen OF GAS COMPOSITION Mole Fraction 0.0842 0. they are found anywhere in the world. 5. Formation of solid gas hydrates in oil and natural gas wells or transmission pipelines63 causes physical blockages to flow and is a major nuisance. hydrates form anywhere water and the guest hydrocarbon molecules come in contact at temperatures above or below 32OF and at elevated pressures dictated by the composition-dependent phase behavior. In high-pressure oil and gas wells. This is accomplished by the following method.28-Geothermal and hydrothermal gradient in different environments. permanent chemical injection lines also are used for mitigation. 4. Determine the phase behavior of the flowing fluids in the operating minimum and maximum pressure and temperature range experienced between the reservoir and first stage of separator.0093 0. Reducing the system pressure below hydrate stability at system temperature. however. The formation of gas hydrates is another very common problem. iso-butane. Subsurface restrictions. hydrates are a nuisance in flowing wells or pipelines. Changing composition or removing one of the components of hydrate. etc. O F Fig. such as chokes or safety valves. 5.0024 0. Being solid. 6. especially after shutdowns or during cold startups. such as diesel. At or near subsurface safety valves or other choke devices where Joule-Thompson cooling is present. After combining the information in the first three steps.0051 0. To avoid or mitigate these problems. Similar phase envelopes can be drawn for wax deposition. or routine surface-facility maintenance. especially in high-pressure and marine environments.0063 0. Determine the flowing temperature gradient in the well or pipeline before the pipeline design.Temperature - Thermal G a i n rdet ----. 6.8. avoid the formation of gas hydrates or hydrate plugging. where they can totally or partially block fluid passage. such as the hydrocarbon or the water.6741 0. High-pressure gas wells or flowlines. methanol. Although the very low temperature of a permafrost environment provides the perfect condition for gas-hydrate formation. AIME. Trans.: "Determining Multiphase Pressure Drops and Flow Capacities in Down-Hole Safety Valves.G." SPEJ (December 1983) 885.P. and Domingues-A. J.: "Sluny Pipeline Design and Operation Pitfalls to Avoid. Mexico City.: "Apply NODAL^^ Analysis to Production Systems. Coleman. Table 6.M... requiring higher pressure and/or lower temperature to form hydrates. T." SPEPF (May 1994) 143.D. Trans. Tulsa. Gilbert.. Oklahoma (1984) Chap. Per. A.M.J. Mach. 21. Texas (1982).C.. and Venkatesh. et 01.21 gives the gas composition. H.R. New York City (1964). Jr." Dwight's Energydata Inc." Technical Note No. 80 wt% methanol in water moves the hydrate equilibrium line to the left. 6. Eng. RecornmendedPractice forDesign and Installation of Offshore Production Plarfom Piping System. 234. Trans." JPT (June 1967) 829. Al-Hussainy. Carroll. Mukherjee. Al-Hussainy.: "Pressure Drop in Wells Producing Oil and Gas. 19-23 March. RP14E. SPE. inhibiting the agglomeration of smaller hydrate nuclei t o larger masses." Well Servicing (Januaryffebmary 198 1) 38.. AIME. 21-24 September.S. Proano. Washington.H. 285.E. M. Dallas (1995). 12. and Treatment Scheduling in the Optimum Hydraulic Fracture Des~gn. Inc.A.H.: "Expanding the Range for Predicting Critical How Rates of Gas Wells Producing From Normally Pressured Waterdrive Reservoirs... E.. Energy Res. S. 237. 5.: "Avoid Premature Liquid Loading in Tight Gas Wells Using Prefrac and Postfrac Test Data. Prentice-Hall Book Co. 11. 3. Houston." JPT (May 1966) 637.: "Determination of Average Reservoir Pressure From Build-Up Surveys. 20. Note that the hydrate equilibrium line (Line A) in the middle of the gaslliquid enve~obe determines the thermodynamic stability conditions of pressure and temperature. Prog. This shows a little shrinkage of the hydrate-phase envelope with the addition of methanol. Hubbard." J. 249. Versluys. Thus. Davalath and BarkePo presented a detailed review of design considerations for gas-hydrate prevention in deepwater gas-well completions... 7." JPT (November 1969) 1475." JPT (March 1991) 334.M. 3 1. Gulf Publishing Co. and Fogarasi.. WFI I DFSIGN APPl ICATTOYS 17.. and Ramey.: Advances in Well Test Analysis.. 28. Monograph Series.: "The Isochronal Testing of Oil Wells." paper 3.. API.: "Systems Analysis Use for the Design and Evaluation of High-Rate Gas Wells. 30 September-3 October. A.. and Whitson.N. AIME.. S. F.: "A New Concept in Continuous-Flow Gas-Lift Design. 2. Erosional Velocity Limitations for Offshore Wells. . PennWell Publishing Co." Chem. 23. 234. New Jersey (1991). K. E... Nevada.: "Systems Analysis as Applied to Producing Wells.E. K.J. Mach. 10. New York City (1980). R. Muskat..M.E. C. 14. AIME... Fetkovich. Brown. P. D. SPE. : "A New Look at Predicting Gas-Well Loadup. Houston. Trans.J. Dietz. 192. AIME.: "Transient Vertical FracPressure Behavior Tor a Well With a F~nite-Conductivity ture. H. T h e following example illustrates the hydrate-phase envelope with respect to a typical North Sea gas-phase envelope. Turner. (1960) 56. Inhibition of hydrates by pumping hot oil into the hydrate system is commonly practiced to mitigate any existing problems during cold startup.: "A Comprehensive Mechanistic Model forTwo-Phase Flow in Wellbores. 32. and Brown. 27.: "Experimental Study of Pressure Gradients Occurring During Continuous Two-Phase Flow in Small-Diameter Vertical Conduits. (20 September 1982) 123. ASME (1976). Texas (1967) 1.: "Augmentation of Well Productivity With Slant and Horizontal Well\. and Brown. Richardson. 30. R. Monograph Series. M.. R.: Pressure Buildup and Flow Tests in Wells. Aziz.. McGraw-Hill Book Co. T. 291. P. Trans. Vogel. and Russell.. Denver. 240.G. and Pierce. M. E." J." SPEJ (September 1970) 279. Trans.S." JPT (August 1995) 693.M. Richardson. Darcj. copyrighted by Intl." Technology of Artificial Liff Methods. 6. Samaniego-V. R. 4. chemical inhibitor injection with pipe insulation (if necessary) is a more c o m m o n method to prevent hydrate formation. Trans.Z.S." SPEJ (August 1978) 253. reprinted by SPE. called kinetic inhibitors. Nind. G. third edition.: "Predicting Two-Phase Pressure Drops in Vertical Pipes." Oil & Gas J. M. McGraw-Hill Book Co. Trans.P.: "Fluid Mechanics and Heat Transfer in Vert~cal Falling Film Systems. 22. S. 293.E. References 1. 270. Salama. K. Beggs. Joshi. Earlougher.: "Pressure Drop Correlations for Inclined Two-Phase Flow." SPEJ (August 1977) 271.: "A Study of Two-Phase Flow in Inclined Pipes.E.: "Understanding Gas-Well Load-Up Behavior." JPT (August 1965) 955: Trans. Neotechnology Consultants Ltd. DC (1981). 18. "'PERFORM' Well Performance Analysis Program. Analytical Treatment. H. 15. 43. Inc. (December 1985) 107.A.A.: "Generalization of the APl RPI4E Guideline for Erosive Services. Jr.: Hydrocarbon Phase Behavior. AIME. 291. Matthews. AIME." paper SPE 9424 presented at the 1980 SPE Annual Technical Conference and Exhibition. N. AIME. AIME." SPEPE (August 1990) 212. 9. Oudeman. AIME. AIME (1930) 86.A.G. Lea. Ansari. Calgary (December 1989).." JPT(July 1992) 782." paper SPE 1253 1 presented at the 1983 Offshore Technology Conference. Upchurch. Cinco-L. 246. as shown by Line B. 24. Texas (1977) 5. Golan. are being developed that allow the system to exist in the hydrate thermodynamic stability region. If the pressure and temperature in this g a s well lie o n the left side of this envelope during cold startup. H." JPT (May 1973) 607. Hagedorn..: "Comments on the Prediction of Minimum Unloading Velocities for Wet Gas Wells. Las Vegas. Tech. Trans. AIME.549. Meng.E." paper presented at the 1981 ASME Energy Sources Technical Conference and Exhibition. AIME. et 01. 111 and Home. et al.: "Flowing and Gas-Lift Well Performance." JPT (April 1965) 475. A. 44. R. 289. Odeh. J. J. Ashford. Tech.G. 4.. Proano." Drill.: Well Performance. Dukler. et al. 13. 297.. Agarwal. 18-20 May. J. AIME. Jr. J: "Mathematical Development of the Theory of Flowing Oil Wells. 42. and Pack. Crouch. and Aude. M.. an understanding of such hydrate-phase behavior forms the basis of its thermodynamic inhibition and is highly recommended for troubleshooting and solving hydrate-related problems. (July-September 1972) 11. However.: "Production Optimization of Oil and Gas Wells by NODAL^^ Systems Analysis.B.R. and Brill. Trans. J. F..E." JPT (January 1968) 83. and Brown. 14.E. Trans. K. 2-5 May. e t a l . G. Thompson.L.. Mach.E. 19.W. J. 41. Trans. Inc." JPT ( S e p tember 1975) 1145. S. 8. 26. Gregory. 275. H. second edition.: "An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: I." paper SPE 4529 presented at the 1973 SPE Annual Meeting.M. D. (1954) 126.A. Cdn. R. Injection of inhibitor moves this curve t o the left. Govier. J. 45. 37. and Brill. and Dukler. C. 29.: "Multivariate Optimization of Production Systems. Shirazi. Foust..: Principles of Unit Operations.: "Inflow Performance Relationships for Solution-Gas Drive Wells.: "Analysis and Prediction of Minimum Flow Rate for the Continuous Removal of Liquids From Gas Wells. Dallas. M.: Les fontaines publiques de la ville d e Dijon (1 856).S.. Houston (1989).: "Evaluation of API RPIIE. With the same gaslwater stream.F. 25. AIME.: The Flow of Homogeneous Fluids Through Porous Media.V.E.J. Englewood Cliffs.N.: "Improved Prediction of Wet-Gas-Well Performance. Coleman. Mach. AIME. et al.C. John Wiley & Sons.. et al. E. hydrates may cause blockage to the line.tors. "Pseudosteady-State Flow Capacity of Oil Wells With Limited Entry and an Altered Zone Around the Wellbore. 36. 46. J..38. T h e area to the left of this curve provides the ideal stable-gas-hydrate envelope. Boston (1982). 39." SPEPE (August 1989) 321.: "Coupling of Production Forecasting.D..000 Ibmm o l k r dry gas and 2 Ibm-mollhr water was generated with the PengRobinson equation of state from Appendix C and is presented in Fig." JPT (March 1991) 329.W." Trans. H. K. New York City (1937). J.. Trans. 35. The extent of control of the hydrate-phase envelope depends o n the composition of the producing hydrocarbons and the type and amount of hydrate inhibitor added.: "Application of Production System Analysis to Determine Completion Sensitivity on Gas Well Production.29... Fracture Geometry Requirements. injection of 3. Ahmed."paper SPE 16435 presented at the 1987 SPELow Permeability Reservoirs Symposium. Colorado. A.: Principles of Oil Well Production. Prac.: "Application of Real Gas Flow Theory to Well Testing and Deliverability Forecasting. 33. W. 16." Trans." JPT(June 1988) 729. 38. T. 18-22 January. and Ramey. T. A. 34. J. Human Resources Development Corp. A phase envelope for this g a s in a feed stream with 1.C. H.. and Brown. AIME. 40. 255. AIME. Richardson. A. & Prod. 243... Orkiszewski. Truns.1 presented at the 1979 Congreso Panamericano de Ingenieria del Petrdeo. Trans.B. E. Trans.607 Ibm-mol/hr.. 287. K.E. K. : "Erosion of a Two-Dimensional Channel Bend by a Solidniquid Stream. 291. Gas Processors Suppliers Assn. C.A. College Station...D. 59. Hanley. 87. Fifth Intl. E. Patil. B. V. 145. New York City (1993) 259.H.47. Ethane or Methane. Shirazi." Liquid-Solid Flows. P. 64. 23-26 September. Lachenbruch. No. (1980) 18. Jr. Tulsa. er al.: "Natural Gas Hydrates in Alaska: Quantification and Economic Evaluation.: "Factors Modifying the Erosive Wear Equations for Metals. Tech." paper SPE 20770 presented at the 1990 SPE Annual Technical Conference and Exhibition. 5 1. Benchaita. 27-19 March. E. Project 825-85. D.. Trans." J. CSME (1985) 9. Eng. McLaury. and R0binson." API Offshore Safety and Anti Pollution Research (OSAPR) Project No. (1979).: "Hydrate Inhibition Design for Deepwater Completions.." J.. E. Bakersfield. ASME. S.B. and Tolle. 65.Chen.D. Ind. M." J. G." paper SPE 13593 presented at the 1985 SPE California Regional Meeting.TTPHASE FLOW IN WFI I $ ." Trans." APl OSAPR Project No... 62. ASME. E. J. Tolle. No.. 3. 213.J.: "A Procedure to Predict Solid Particle Erosion in Elbows and Tees. AIME." Proc. (1984) 106.: "A Model to Predict Solid Particle Erosion in Oilfield Geometries.: "Natural Gas Hydrates. 58. B11. Texas (May 1977).R. of Tulsa. et al. Oklahoma (1993). OkIahoma (1985).T.: "Study of Hydrate Dissociation by Methanol and Glycol Injection. 159.:"Permafrost. Ind.: "HydrateFormation and Equilibrium Phase Compositions in the Presence of Methanol: Selected Systems Containing Hydrogen Sulfide." Israel J. Geo. and Rabinowicz. McLaury. New York City (1990).T. College Station.D. Fluid Engineering Div.. 3. and Jadid. and Rabinowicz. Godbole. Pressure Vessels and Piping Div.: "Use of Laboratory and Field Testing To Identify Potential Production Problems in the Troll Field. G. U. Conference on Erosion by Liquid and Solid Impact. M. M. E." MS thesis. Bakersfield. U. Sloan. ASME.S. Texas A&M Research Foundation. (1983) 105.: "Design of Fittings to Reduce Wear Caused by Sand Erosion.: "The Wear Equation for Erosion of Metals by Abrasive Particles." paper SPE 13635 presented at the 1985 SPE California Regional Meeting.. Benchaita.: "Erosion of a Pipe Bend by Solid Particles Entrained in Water. and Barker. 57. 2. R." RR-87. 6. Sloan.: "A Particle Tracking Method to Predict Sand Erosion Threshold Velocities in Elbows and Tees.98. 48. Eng. E.. Texas A&M Research Foundation. 54. Blanchard.. Davalath. MUI. S.K. Griffith.P.J. 55. and Kamath..L. P. 56.. 52. J. Cambridge. Res. Texas (March 1976). Alaska.H. New York City (1994) 189. J. and Greenwood. and Ehlig-Economides. E.S. California.. 193.D. 49. Heat Flow and Geothermal Regime at Prudhoe Bay. 63. Rabinowicz. 2. Hanson.C. Carbon Dioxide. J. California. Marcel Dekker f Inc... Tulsa.: "Detection and Prevention of Sand Erosion of Production Equipment. Trans. B.W. 60..A. Griftith.9301. 61. 50.215. Rabinowicz. Weiner. ASME.B. No. D. Jr. P. 53." SPEPE (February 1989) 34.: "ErosionofaMetallic Plate by Solid Particles Entrained in a Liquid. Ng.: "Pilot Gas-Lift System at Prudhoe Bay. Sira." Codes and Standards in Global Environment. et al. A." SPEDC (June 1995) 115. New Orleans.: Clathrate Hydrates o Natural Gases. S." JPT (December 1991) 1414: Trans.S. 27-29 March. drag coefficient CFD = dimensionless fracture conductivity CK. = formation volume factor of oil. L~ pipe dlameter ratio for annulus. = specific heat at constant volume.263 f= Moody or Darcy-Weisbach friction factor f= no-slip volume fraction. U m relative performance factor in Eq. u t 2 gravitational conversion factor geothermal temperature gradient.33 equilibrium constant. 4. L3/L3 penetration factor.256 E2 = absolute average percent error in Eq. mol/mol moles of liquid. mUt3T ratio of specific heats.7 and 5.4. L 3 5 3 & = formation volume factor of water. 5. m parameter in Eq.259 E4 = average error in Eq..Appendix A Nomenclature and SI Metric Conversion Factors a = abscissa A = Helmholtz energy. 5. 4. mL2/t2 slip gaslliquid volume fraction (liquid holdup).262 Eg= standard deviation in Eq. L enthalpy. 2. L2/t2T d = pipe diameter. L ~ I L ~ holdup ratio in Eq. 4.2 C = liquid-holdup parameter in Eq. 4. Cp/C.140 injection rate. L e = eccentricity of annulus.29 p i p length. ~ ~ / m c = wellbore storage coefficient. L2/t2 mass.9 D = turbulence coefficient DBc= distance between pipe centers. L3/t permeability anisotropy mechanical equivalent of heat thermal conductivity.CM film indices in Eq. L31L3 llquid holdup from liquid film. L3/L3 c = compressibility.294 = CL = specific heat of liquid. L2/t2T C. Ln e = intrinsic specific energy.264 sand sharpness factor acceleration of gravity. 4.= total formation volume factor. 4. L liquid mole fraction.1 17 C = spec~fic heat. 4. = relative error Ek = dimensionless kinetic-energy pressure gradient in Eqs. m volumetric heat capacity. 4. mol formation resistivity factor liquid entrainment fraction.241 Cp = specific heat at constant pressure. m/Lt2T number of moles NnMFYCf A T 1 ' R F . L3/L3 f = Fanning friction factor f= fugacity.196 Bg = formation volume factor of gas. effective permeability. L2 A = well drainage area. 4. Ut liquid interface height.260 Eg = absolute average error in Eq.59 latent heat of vaporization.53 El = average percent error in Eq. L L piplng component factor in Eq. L2 A = parameter defined in Eq. g~ = h= h= h= h= hL = H= HL = HLF = HR = i= Iani = J= k= k= k= K= K= K= Kw = K' = L= L= L= L= LB= 4= m= rn = m@) = M= M= n= friction geometry parameter for annulus moles of feed mixture. 4. 4.96 real-gas pseudopressure function molecular weight. r n ~ ~ / t ~ A = pipe cross-sectional area.240 and 4. = g= g = . L2/t2 e = error e. = flow coefficient in Eqs. L3/L3 B. 4. T/L convective film coefficient. L erosion penetration rate. L2/t2T CD = discharge coefficient. 5. L parameter in Eq. L2/t2 net reservoir thickness. 4. L D = dimensionless parameter in Eqs. m/t3T specific enthalpy.258 E3 = percent standard deviation in Eq.9 and 4. L 2 / t 2 ~ C.1 and 5. 4. L3/L3 B. mlLt2 L F= F= F= FE = Fp = Frp= F. L4t2/m C = correction factor C = flow coefficient in Eqs. moUmol Watson characterization factor power-law consistency index in Eq. mol horizontal well length. 4. 5 13 xf= fracture half length.5 y = ratio of downstream to upstream pressure ND = d~mens~onless d~arneter numbers In y.123.116 S = entropy. 4. L3/t 4. L co = outside casing crit = critical x= mass fraction C = calculated. 4. and 4. IIT Npr=Prandtl number in Eq. = volumetric flow rate at standard conditions.13 and 5. = specific volume. 13/L3 p = density. m/t2 Rp = producing gasloll ratio. L~ Y = liquid-holdup inclination angle correction in Eq.25 b = parameter in Eqs. 5. 4.29 y = mole fraction of component in vapor phase n' = bubble-swam parameter In Eq. 2. L 1 = thermal conductivity. l/t q = heat flow rate.4 = parameter in Eq. apparent U = overall heat-transfer coefficient..129 MULTIPHASE Flow IN WEL1. Wt b = bubble. @ = parameter in Eqs. = absolute-open-flow potential. 2. = d~mensionlessvelocity number In Eq.122.336.168 NFr= Froude number in Eq.2 interfacial friction parameter in Eqs. L NE = dimensionless number in Eq.177 y = parameter in Eq. S = saturation 4.103 N. OAPI q = volumetric flow rate. 4.14. circular. Ut E = absolute pipe roughness. m/13 R = in-s~tu gashiquid volume ratlo. 4. L Np = liquid viscosity number in Eq.3 4.12 & = minimum d~mensionlessfilm thickness . atmospheric. m/Lt2 s = parameter In Eqs. = Nusselt number In Example 2. mL2/t3 I= liquid d~stnbution ' coefficient in Eqs.134 NGr= Grashof number in Eq. 4. Ut b = bubble in flowing liquid f v.14. = critical pressure ratio Eqs. 4. L n = number of components . bulk v*= sonic velocity. 4. L t ~ l m ~ Q = heat flux. corrected. Ln 8 = inclination angle from horizontal R = gas constant. L3L3 NL = I~quid-viscositynumber in Eq.94 ratio of Taylor bubble to slug unit length NqL= dimens~onlessliqu~d flow rate number in Eq. 4. 5. 4. 4. L2/t2 a = air. a t 2 u = specific internal energy. L n = polytropic expansion exponent In Eq.27 y. L3/L3 r= radius. T 1 = upstream conditions 2 = downstream conditions T = metal hardness. 4. = well radius. Wt v E = effective sonic velocity. boiling point. = drainage boundary width. m L 2 / t 2 ~ €J = reduced boiling point R = gasloil ratio. m/t ch= choke w = fracture width. and 4. 5. L p = absolute viscosity.S . ~ ~ / t r] = Joule-Thompson coeffic~ent. L2/t NN. 4. moWmol c = casing. mL2/t*T t = time. 5. m ~ / t ~ rd = drainage radius. = dimens~onless density ratio in Eq. L ci = inside casing W. m/t3T acc = acceleration an = annulus v = velocity. U t .. L3/m bs = bubble in static liquid B/S = bubblelslug boundary V = vapor mole fraction.26 coefficient of thermal expansion.98 . 4.221 and 4. rad U: = interface chord length. 4. L v = kinematic velocity.85 and Fig. critical V = volume. 4. A = difference N.132 Nd = pipe-diameter number in Eq. t Subscripts T = temperature. 4.4 void fraction Np = dimensionless pressure number in Eq. L3/L3 a = surface tension.48 E = convergence tolerance N = dimensionless number In Eq. L NLC= corrected I~quid-vicosity number in E~gs.93 gas compressibility factor.125 w = Pitzer acentric factor S = slip velocity number qj = parameter in Fig.86 and Fig. = angle of view average in Eqs..4 S = surface area. p = pressure. gas core x = mole fraction of component in liquid phase.63.8 y = parameter in Eq. m/t3 1 = no-shp gadliquid volume fraction.97 YM = Lockhart and Martinelli parameter in Eq.53 and 2. 5. = oil gravity. 4. moVmol CA = concentric annulus x = parameter In Eq. mol cem= cement w = mass flow rate. 4. 4.9 and 5.6 vertical distance.124 N = d~mensionless parameter in Eqs. L y = specific gravity Pch = parachor YAP. L3/t y = shear rate. 4. 4. l3 calc = calculated V = moles of vapor. 4. L2/t R = annulus concentncity parameter in Eq.109 mole fraction of component in feed mixture.n = parameter in Eqs. XM = Lockhart and Martinelli parameter in Eq. m/lt2 @ = poroslty p~ = d~mensionless wellbore pressure drop @ = fugacity coefficient P = penmeter. 4. 2. 2. Nwe = Weber number in Eq.8 1 q.47 dL = liquid film th~ckness.222 NLv= liquid-veloc~tynumber in Eq. 2.167 n ' = power-law flow behavior index in Eq.7 and 5.3 thermal diffusivity.10 = drainage boundary length. 4. 4.232 c . moYmol Ngv gas-velocity number in Eq. 4.335 and 4.72 through q.28 subtended angle by interface chord NRe= Reynolds number dimensionless film thickness N. 4. = flow-pattern coordinate in Eq. L N = flow-pattern coordinate in Eq. L ~ L ~ a = percent standard deviation s = skin factor r = shear stress. m/Lt r.23 1 and 4. 02 = std m3/d scf/D x 2. friction.75 i = inlet. erosional.536 240 E + 0 0 =kJ/K cp x 1. reservoir. s rni . differential. terminal tf= wellhead flowing ti= ins~de tubing to = outslde tubing TB = Taylor bubble TP = two-phase Tr/M = transitionlmist boundary U= upstream = vertical vp = vapor pressure V= vapor w = water. dissolved. formation.02 = m3/kpa bbVpsl x 2. 2.02 = m3/kg ft311bm x 6. downstream D = droplet Dw = dimensionless at wellbore e = environment. restriction s = smooth pipe. fracture fc = forced convection fl= flowline F = film.242 796 E . sandface sc = standard conditions. minerals max= maximum meas = measured mf = midpoint flowing min = minimum M-R = Metzner and Reed n = no-slip. transition.154 591 E .589 873 E-01 =m3 E . dead.152 125 E-01 =kg Ibm x 4.03 = kW/m2 ' E .069 E+01 = k l / m 3 ~ E-W=kW Btulhr X 2.83 1 685 E .930 71 1 Btu/hr-ft2 X 3. tubing. deposits. 5 4 * E + 00 = m/kg in.d = dewpoint.2.~/lbf 8 057 652 X ft x 3. static. fluid. steam. s E + 0 0 =mN/m dynelcm x 1.urn2 md x 9.O* 'F (OF-32)ll.8 =K "Flft x 1. producing. dry.5/(131.730 735 x BtuAbm x 2.03 = kw/m2/K Btuhr-ft2-OF x 5.02 = K/kPa O~-in. Fortunati F C = friction-concentric annulus FE= friction-eccentric annulus g = gas h = hydraulic.67)/1. interface.83 1 685 E .290 304 E-02 =m3 ft3 x 2.259 979 E -04 =.326 E+OO=W/kg BtuJlbm-OF x 4.186* E + 00 = kJ/kg-K BtulOR x 7. ~ 2 .5 + "API) =g/cm3 bbl x 1.0* E-03 = P a . slip. solution.02 = m3/s ft3/sec x 2.894 757 E . wellbore. particle pc = pseudocritical p r = pseudoreduced pss = pseudosteady state q = at Inclination Angle q.60 1 846 E + 0 4 =kgls Ibmhr x 1. earth. wall wf = bottomhole flowing wh = wellhead o = solid Superscripts A = assumed C = calculated L = liquid V= vapor O -= ideal = average S Metric Conversion Factors I "API 141.535 924 E-01 =kgmol lbm mol x 4.74 and 2. hydrostatic./lbm x 1. flash. normalizing NC= noncircular o = oil p = pipe. horizontal HT= heat transfer in Eqs. reduced.048* E-01 = m ft/sec x 3.0 1 = std m3/std m3 scf/STBO x 1. initial.54* E+OI = m d a E+OO =kPa psi X 6.048* E-01 = d s ft/sec2 x 3.869 233 miYyr x 2.305 916 Btu x 1. inner I= interfacial k = kinetic L= liquid LF= liquid film LT= liquid slug m = mixture. quartz r = rough pipe.g = "c " F (OF + 459.535 924 E + 0 1 =kg/m3 Ibm/ft3 x 1.801 175 'Conversion factor 1 exact.822 689 E+OO=Wm E .83 1 685 E+Ol =mm in. free. emulsion el = elevation est = estimated E = effective E = eccentric annulus A EP = equiperipheral f= film. surface choke s d = sand sep = separator sv = safety valve SC= supeficial core Sg = superticial gas SL = superficial liquid S/M = sluglmist boundary S/Tr = slug/transition boundary SU = slug unit r = total.055 056 E+00=kJ ~ t u l f t ~ x " ~ 942 .048* E-01 = d s 2 E-02 =m2 ft2 x 9.678 263 E + 00 = W/mK ~tu/(hr-ft2-"~lft)1. psia. API gravity of the liquid phase and specific gravity of gas. These are often called PVT properties. If the mixture composition is known. each with its own properties. These assumptions imply that the composition of the oil and gas do not change with pressure and temperature. Such flow and fluid property changes can be predicted with either compositional or black-oil models. 2 Hydrocarbon Physical Properties Crude oils and natural gases contain several pure organic and inorganic chemical components. Clearly. the correlations should be used with proper caution. these quantities may be estimated from and where n. in some cases verbatim or with minor changes. The assumptions in the blackoil model are that. both weights being corrected for the buoyancy of air. "The specific gravity of a petroleum oil and of mixtures of petroleum products with other substances is the ratio of the weight of a given volume of the material at a temperature of 60°F. Gas solubility is defined as the volume of gas dissolved per unit volume of liquid at a fixed temperature and pressure. These correlations are very useful in determining the fluid and rock physical properties required in the application of the multiphase flow theories presented in this monograph. and to provide supplementary information when necessary. B-1provides a way to estimate these quantities for undersaturated oil at reservoir pressure. =number of components in the mixture. at any fixed temperature. Sometimes the weight of the MULTIPHASE FLOW IN WELLS . The compositional models presented in Appendix C can represent the mass-transfer phenomena more accurately and are applicable universally. If the system composition is not known.i =critical pressure of Component j. densities. Some of these PVT properties are developed from reservoir fluid samples belonging to particular geographical areas and may not be applicable in other areas where reservoir fluid composition is very different. boiling points. the use of laboratory data measured on representative samples taken from the reservoir is strongly recommend whenever available. phase densities. Consequently.. Most black-oil-model-related fluid physical properties1-28can be determined by use of pressure/volume/temperature (PVT) cells. the phy sical properties of reservoir fluids are pressure.1~5 However. The critical pressure and temperature of a multicomponent mixture are called pseudocritical pressure. Figs. Table B-1 presents selected physical properties of several of these individual components. and. Although PVT property correlations are very widely used in the petroleum industry world102 wide. Formation volume factor determines the change in the volume of a phase at different pressures and temperatures. This appendix presents a detailed discussion of these black-oil parameters.. . This constant composition assumption may be valid for the oil phase. Table B-1shows a very wide range of molecular weights. viscosities. In . very active mass transfer occurs between the liquid and the gas phases. prissure. and p. Tpc. Gj =critical temperature of Component j. gas evolves from solution in the oil. in some cases. 1 Introduction Integration of the pressure-gradient equation requires determination of individual phase velocities. An important limitation of the black-oil model is its inability to predict retrograde condensation phenomena. Most of the fluid and rock properties included in this appendix have been reported previously. In dynamic conditions of multiphase flow in pipes. but can cause errors in predicting the physical properties of the gas phase.respectively. & and pseudocritical temperature. 6 and 7 present more complete data. the pressure and temperature of the fluids change continuously. B . this appendix. Most of these correlations are empirical in nature and are based on a limited quantity of representative samples of data."9 This often is referred to as yo at 60°F/600F. This appendix presents the most widely used and reliable fluid and rock physical property correlations. yj = mole fraction of Component j . OR. these pseudocritical properties are used as correlating parameters in many empirical correlations. As the pressure decreases in the direction of flow below the bubblepoint. the liquid phase has a fixed gas solubility and formation volume factor. the oil specific gravity corrected to 60°F (the value normally reported) is used. Refs. surface tension at different pressures and temperatures. we decided to incorporate some of these materials.to the weight of an equal volume of distilled water at the same temperature.and temperature-dependent. This appendix methods to determine fluid physical properties required in the black-oil model. B-1 through B-3 can be used to estimate Tpc and&. However. Fig. increasing the gas velocity and the oil density and viscosity. Compositional models are recommended for these cases. The black-oil model should be avoided for very volatile or light crude oils or gas condensates. and critical temperatures and pressures among the components constituting reservoir fluids (crude oils or natural gases).Appendix B Fluid and Rock Properties B. 3 238.08977 - 51. normally are used to estimate the pseudoreduced temperature. CgHZ0 mdecane. B. up to the bubblepoint pressure of the oil.6 .5 805. The physical properties of dead oil are a function of the API gravity of the oil. In Eq...7 332.4 1.02 2.. oil is not determined at 60°F.3" 62.. CsHI4 n-heptane.1160 0. B-I.37 - 0. CH4 Ethane. C5Hl2 mhexane.04 30.07 44.26 142. gas solubility determines the mass transfer between the liquid and gas phases.258.365.15 72.. B 4 ) and crude oil is often called undersaturated.0 10.1531 0.35 41.76.306 3..4 9 3.. it can be corrected to 60°F by use of the technique described in Ref.5 763.4 139.7 734.79 - - 0.297.7 416.1 671. CO 'Apparent denslty In lhqu~d phase.3.I 1 490.1531 Critical Temperature (OR) 343. (In Ref. B-3.. oil physical properties also depend on gas solubility. N2 Air (02+ N2) Carbon dioxide. CloHZ2 Nitrogen.12" 36. B.04228 0.112 227.005313 0. API gravity of oil is defined as with yo at 60°F.1 through B-4.9 278.423.6 240 Critical Pressure (psi4 668 708 616 529 551 490 489 437 397 361 332 304 493 547 1..127..2 258.8 96.ft3) 19' 22.1162 0.6 615. yo =specific gravity of stock tank or dead oil at 60°F/600F. YAPI..9 209. Fig..2 7 7. 9.07630 0.O . CBHIB mnonane.08 18. where retrograde condensation = T ..08432 0. is called dead oil.208 181 737 507 Constituent Methane. Table 5 is used for hydrometer measurements at other than 60°F. C4H10 Isopentane.3 146.10 58.3 -317.07380 0. Tpc and g.8 82. C7H16 n-octane. B.6 212.01 34.1 Liquid Density (Ibm. 0 2 Carbon monoxide.6 142.8 665.071 1.. In black-oil models. B-1-Approximate correlation of liquid pseudocritical pressure and temperature with specific gravity (after Trubes).4 383.1 Gas Solubility. Table 7 allows the correction of volume at a given temperature to volume at 60°F.9 1 303.. in addition to the pressure. H2 Oxygen.. Fig. pressure. In Eqs. the gas gravity must be known.07380 - O 1972 Gas Processors Suppliers Assn.9 1.01 Normal Boiling Point ("R) (OF) 201 .070. and p = pressure of interest.22" 31. and API gravity of oil. and temperature..4 1 972.1 541. 1 atm (Ibm/ft3) 0. The pressure at which the liquids are weighed is not specified. O R .. 103 . Gas solubility in oil increases as the pressure increases.07924 0.3 828.TABLE B-1-PHYSICAL PROPERTIES OF HYDROCARBONS AND SELECTED COMPOUNDS6 Gas Density at 60°F.66" 35.. T Pr With gas in solution.23 128. T~ (B-3) '-2 I and where T = temperature of interest.. in the absence of gas in solution.) The API gravity of water is 10.20 114. and pressure.. the specific gravity at reservoir pressure corrected to 60°F is the ratio of the oil density at reservoir pressure and temperature corrected to 60°F to the density of distilled water at 60°F and 1 atm.8 845.96 39. If the oil gravity is reported at other than 60°F.12 72. B-5.109.1 350. FLUID AND ROCK PROPERTIES SpecificGravity of Undersaturated Reservoir Liquid at Reservoir Pressure Corrected to 60°F Fig.3 59.96 44.01 45.7 765.3 .4 1. Molecular Weight 16. CsH8 Isobutane...4 -313.02 32. B-2 applies to bubblepoint liquids by use of the specific gravity corrected to 60°F..44" 38. "Density at saturation pressure. C4Hlo mbutane. In Fig.5 1.92 44.7 668. To use Fig..9 556.29 28...023. For light oils or condensates.43. C2H6 Propane.0 .. B-3 applies to condensate well fluids and natural gases.01 28.4 155. Tp... This parameter fails to predict any retrograde condensation effects.6 3 . in which case the specific gravity is either corrected to 60°F to give yo at 60°F/600F or is reported as yo at the given temperature.5" 49. Above the bubblepoint pressure.2 345.2 ..08 45.18 100. temperature.7 162.320.12 58.2 .6 672.. gas solubility stays constant (Fig..5 . psia.9 470..41 42..00 28.0 . Gas solubility is defined as the volume of gas dissolved in one stock tank barrel of oil at a fixed pressure and temperature. 9...7 36.15 86. temperature and pressure must be absolute.3 Oil Physical P r o p e r t y C o r r e l a t i o n s Oil. H2S Water Hydrogen.4 547. C02 Hydrogen sulfide. g. C5Hl2 mpentane.1 549. TABLE B-2--COEFFICIENTS FOR VAZQUEZ AND BEGGS13 CORRELATION Coefficient CI c . psia. T= temperature.0178 1 .000 O R 1. They assume a flash-vaporization process. leading to faster flow of gas. which can be written as ~ s Standing Correlation. Tp. and TeP. The mathematical form is where yo =specific gravity of oil. Fig. This is because of the higher transmissibility of gas.0). The gas specific gravity. Gas Gravity.= actual separagas tor temperature. In the flash-vaporization process.19 which is considered to be more representative of the separation process experienced in the reservoir flow system. R. B-7. Consequently.7245 0.I870 & 23. OF. =effective molecular weight of tank oil obtained from Fig.0937 25. effects are suspected.0362 1. If the spec~fied gravity. and YAPI= gravity of oil. the differential-vaporization process is defined as the separation of the solution gas that is liberated from an oil sample as a result of a change in pressure andlor temperature. yg =gas gravity at the actual separator conditions of psep and Tep..100 Pseudocritical Temperature. Several empirical correlations to determine gas solubility are presented... Glas0 Correlation. Fig. and leaving behind the oil from which it was liberated. the amount and composition of each phase should be predicted by flash-vaporization calculations presented in Appendix C. psia. IS at any pressure other than 100 psi.931 Vazquezand Beggs Correlation. OF. refers to a separator pressure of 100 psig and can be calculated from where yg =specific gravity of gas (air = 1.psep = actual separator pressure. "API. Lasater Correlation. . B-4 compares gas solubility with these two types of vaporization processes. it should be adjusted to yg 100 with Eq. yg (air = 1) Fig. B-5. This is quite different from the differential-vaporization process. 700 800 900 1. a correlating parameter obtained from Fig. Vazquez and ~ e presented ~ an improved empirical correlation to estimate gas solubility. 8-3-Correlation of pseudocritical properties of condensate well fluids and miscellaneous natural gases with gas gravity (after Brown et al. In this case. M. use of this parameter may lead to substantial error in calculations. 2 YAPI 5 30 YAPI ' 3 0 0. It is often convenient to calculate the gas solubility from the nomograph presented in Fig. B-2--Correlation of liquid pseudocriticaltemperature with specific gravity and bubblepoint (after Trube8). and yg =mole fraction of gas. B-9. yg. The Standing correlation2I states where the values for the coefficients are presented in Table B-2. ~ a s a t e r l * presented the empirical relationship to calculate the solution gas/oil ratio or gas solubility as where yg 100 =gas gravity at the reference separator pressure of 100 psig.lo). This ~ o r r e l a t i o n ~ ~ is based on North Sea crude-oil data. yg 100. the gas liberated from the oil in a PVT cell during pressure decline remains in contact with the oil from which it was liberated. B-6. with continuous removal from contact with the source oil before establishing equilibrium with the liquid phase. p = system pressure. can be calculated by solving Eq.a = 185. B-5. B.1437.. use For YAPI > 30. .32657...843208. On the basis of their large data base. replacing R. Eq.1 by setting R.was 101. = volume of oil at standard conditions..3. OF. OF. = 0. OR.. comparison of flash and differential gas sol- The accuracy of this correlation is believed to decrease for solu'~ tion gasloil ratios exceeding 1. It can be expressed as Sunding Correlation. and temperature. This correlation16 is based on a wide range of data covering Southeast Asia (including Indonesia). in Eq. p.IS). bbl.O. The oil formation volume factor is measured in reservoir barrels per stock tank barrel. Above the bubblepoint pressure or for undersaturated crudes.000 measured data values for the formation volume factor of oil below the bubblepoint pressure. and the Middle East. with Rp... Al-Marhoun Correlation. or very ~ s l ~ close to the Vazquez and ~ e ~reference pressure of 100 psig. To calculate the formation volume factor of oil. The oil formation volume factor increases continuously with pressure as more gas dissolves in the oil until the pressure reaches the bubblepoint or the saturation pressure. . pSep. c. including the Vazquez and Beggs correlation from Eq. where T is in OF. This observation led to the adoption of 100 psig as their reference pressure for specific gravity of gas. As the pressure increases above the bubblepoint pressure. it also is imperative to determine the bubblepoint pressure before the oil formation volume factor is determined. equal to the producing gastoil ratio.vol. this is very convenient. They used it to correlate data into two groups on the basis of API gravity. the oil stops dissolving more gas and the formation volume factor decreases because of the compressibility of the liquid.voir to surface conditions.39844 1. and solving for the corresponding pressure. The bubblepoint pressure. bbVSTBO. Thus. and C3.~ a r h o u npresented this correlal~ tion based on Middle East crude oils. pb. and (V. STB. =~ ume of oil at pressure.~' was developed primarily from California crude oils. A I ... Kartoatmodjo and Schmidt also proposed an empirical equation to correct specific gravity referred at any other separator pressure and temperature. A few of the more commonly used correlations are presented here.. y. yg = specific gravity of the solution gas. the formation volume factor is always equal to or greater than 1 . B-15 should be obtained from Eq. s where T= temperature. YAPI. scftSTB0.285 psi. ( b ) p . B-19. B-4-Idealized ubilities.3.Analysis of the expression for p i shows that the correlation cannot be used for pressures in excess of 19. B-6 for pressure after replacing R. B-6 or Fig. B. The gas solubility. Table B-3 gives values for the coefficients C1. B-15 in graphical form as Fig. R. Fig. C2. T= temperature. . and R. B-16 can be used to determine the formation volume factor where . Bob can be calculated from the Standing correlation (Fig. b = 1. It is important to note that the oil formation volume factors are determined by use of different equations above and below the bubblepoint pressure.1.. Rp. Based on empirical correlation of 6. oil. scfISTB0.400 s c f / S T ~ 0 . B. B-9 or Eq.. as presented by Vazquez and Beggs Correlation. in Eq.2 Oil Formation Volume Factor and Bubblepoint Pressure.. Kartoatmodjo and Schmidt observed that the mean separator pressure. North and South America.. c = . where B = oil formation volume factor.6. RD. Vazquez and Beggs proposed this mathematical expression. above the bubblepoint pressure. B-8 describes this behavior. (B-IS) where yg =gas specific gravity... B-9. Bob. d = . B-16 can be calculated with one of several methods. B. stand& also presented Eq. B. =gas solubility.. For noncomputer determination of the formation volume factor. A mathematical definition is Differential b E - 3 0 V) Pressure Fig.877840.. yglOO. T = temperature. The volume in barrels occupied by one stock tank barrel of oil with the dissolved gas at any specified pressure and temperature is defined as the formation volume factor of oil. This is the oldest and most commonly It used empirical ~orrelation.00012 . y =specific gravity of the stock-tank . and yg loo = gas specific gravity at 100 psig. There are different correlations to calculate the oil formation volume factor and the bubblepoint pressure. = stock-tank oil specific gravity. (B-18) where R. Kartoatmodjo and Schmidt Correlation. and e = 1. with the produced gasloil ratio. =solution gas/ oil ratio. The Standing correlation can be expressed in this mathematical form below the bubblepoint pressure. pb. The isothermal compressibility.. The bubblepoint pressures are calculated from the solution gastoil ratio correlations given in Sec. Consequently.). For y ~ p l 30. Kartoatmodjo and Schmidt evaluated all the correlations presented in this section to determine gas solubility and found the Vazquez and Beggs correlation to be the most accurate. T..9759 + 0.. This is a measure of the volumetric shrinkage of oil from the reser- ..3. . . B-9.can be calculated from Eq.337~ 27. . . B-5--Properties of natural mixtures of hydrocarbon gas and liquids at bubblepoint pressure (after Standing2').C3 yApIIT Table B-4 tabulates the values of coefficients C ] . . p is the pressure at which Formation volume factor is to be calculated. . and C3 in Eq.~ the isothermal compressibility of oil. and Bib is a "correlating number" defined by where a = . C2.914328 11. propriate corrections for nonhydrocarbon content and paraffinicity.18 0. B-8 to solve for bubblepoint pressure by replacing R. c.393 G 2 c 3 lo-s & G I -1.7447 log p i ..302 18(log pi) . with Rp.84246 10. The Glass correlations were developed and validated against North Sea crude oils. . . .172 56.811 x 1 0 . TABLE B-3--COEFFICIENTS FOR THE VAZQUEZ AND BEGGS CORRELATION13 TABLE B-4--COEFFICIENT VALUES FROM VAZQUEZ AND BEGGS Coefficient YAPI 30 YAP1 '30 Coefficient c l YAPI 5 30 YAPI 730 cl 4.677~ 1. Glaso Correlation.751x 4. . . .7669 + 1. B-20.Example Fig. The Standing1' PVT correlations are widely used in the oil industry. . 2 .0. yg 100. (B-22a) . G l a s modified the Standing correlations to make them independent of oil type and suggested ap- Glaso also presented a bubblepoint-pressure correlation and suggested a procedure to account for the presence of nonhydrocarbon content of the crude oil system.62 0. although they are based primarily on California crude oils.100~ 1.670 x 1. The proposed bubblepoint-pressure correlation19 can be expressed as log pb = 1. can be determined from this Vazquez and Beggs empirical correlation. . Vazquez and Beggs modified Eq. . Claso's oil formation volume factor correlation can be expressed mathematically as where Gas specific gravity. These correlations do not correct for other oil ~ ~ ~ types or nonhydrocarbon content. and presented this mathematical expression. . . . . =gas solubility. for N2.:y:. OF. = 2.0 .. scftSTB0.. and a. d = 3. and 0. For nonhydrocarbon content.8 16. Bd--Bubblepoint pressure factor vs. psia.4. and a. b = 0. pb. gas mole fraction (after Lasaterl2).b.Fig. Al-Marhoun15 developed his formation volume factor correlation on the basis of crude oils from the Middle East and proposed this empirical relationship.. where F = R". C. CCO2= 1..593. a "correlating number" defined by I - where R. 172. where CH2S H2S correction factor and yH2S mole fraction of H2S = = in total surface gases. I L AI-Marhoun Correlation. yo = stock-tank-oil gravity. with p. a.1437. (B-24b) where y o =mole fraction of C02 in total surface gases. Correction factors.742390.. and a1 through as = coefficients of the correlation with the following values: Gas Mole Fraction Fig..8ycO2T-' 553.. and c = 1.699.32657. C02. yg = average specific gravity of the total surface gases.130.323294.. b = 0. L where T = temperature. O R .38088 X b = 0. Al-Marhoun's bubblepoint-pressure correlation can be expressed as I I . = 0.c = coefficients with respective values of 0.87784. B-6-Molecular weight vs.. For volatile oils.715082. and H2S are calculated by i- where yN2=molefraction of nitrogen in total surface gases. and e = 1. = .366. a = 5. where C=correction factor...989.027.O. Kartoatmodjo and SchmidtI6 proposed a formation volume factor correlation at and FLUID AND ROCK PROPERTIES . c = . tankoil gravity (after Lasaterl2). T= temperature..202040.1. Glasa suggested these corrections to the bubblepoint pressure. a.. i Kartoatmodjo and Schmidt Correlation. and ( P ~=corrected bubblepoint ) ~ pressure.. where the coefficients are a=0. B-&Variation of formation volume factor isotherm with pressure at constant temperature. B-27 and the isothermal oil compressibility.9153 .16. . This formation volume factor is recommended for the flash-vaporization process normally encountered in pipe flow. . but solve forp with R. . C" = 6.3 Oil Density.. . ~ ~ MUI-TIPHASE o w IN WELLS n . . . . is calculated from this empirical relationship. = 0. .3. The formation volume factor above the bubblepoint pressure can be calculated with Eq. The bubblepoint-pressure correlation is presented in two ranges of API gravity oil.50. pb = bubblepoint pressure.and differential-vaporization formation volume factors. . . and YO=specific gravity of stock-tank oil. These correlations were developed with nonlinear regression analysis. -0. Rp. (B-29b) B. where F = R ~ ~ ~ Y ~ + 0. . . . Pressure.0001F'. . where 'T=temperature. .is recommended. In this case. below the bubblepoint pressure that can be expressed mathematically as B. B-90 gives an empirical conversion factor between flash. B-12a and B-12b. . . such as in inflow performance relationship (IPR). . .45T. For y ~ 5p ~ 30. For application to reservoir flow. psia Fig. . B. The density for saturated crude oils below the bubblepoint pressure can be calculated at a given presssure and tem- Fig. Eq. (B-27) OF. equal to the produced gasloil ratio.98496 + 0. Bob is estimated from Eq. B-29a and B-29b are identical to Eqs. ? & Y ~ ~ ~ ~ yg loo= specific gravity of gas at 100 psig.35505 2 7 ylW ' where p is in psia.8257 x P ~ t 5 0 0 y ~ 6 1 3 0 . . . c. B--hart to determine oil formation volume factor with the Standing ~ o r r e l a t i o n . . I 0. a differential-vaporization formation volume factor Example Note that Eqs. In gasloil.56. . gas gravity. Then. pressure. . . given the oil compressibility. . oil density can be calculated by first determining the density at the bubblepoint pressure.56.1 w 80 40 20 z 0 10 8 . pod.. Eq. 8-37 can be used for density when p >pb. (After Katz et Reproduced with permission of the McGraw-Hill Cos. in O R . B-11 to determine dead-oil viscosity if the API gravity of the crude oil and the temperature are known. should be calculated based on the free-gas gravity.6 0. Methane is the first hydrocarbon component to evolve into the free-gas phase at or below the bubblepoint pressure. is calculated as the gas solubility at or above the bubblepoint pressure.. y d . both the free and the dissolved-gas gravity must be greater than 0.3 0 6 4 2 2 4 0. 10 .56. i FLUID AND ROCK PROPERTIES 109 . ~ e a l ~ O presented a graphical correlation in Fig.5 Oil Viscosity.. . yg.oo . . standingZ1presented this mathematical equation to represent Beal's correlation for dead-oil viscosity. the formation volume factor.. . : 15 20 25 30 35 40 45 50 55 W 65 Crude Oil Gravity. .4 0.2 01 . where yo = stock-tank oil specific gravity and ygd = dissolved-gas gravity described in Sec. with specific gravity of 0. B-10-Dissolved-gas gravity correlation. and the specific gravity of the dissolved gas. The dead oil is defined at atmospheric pressure and at any fixed system temperature without dissolved gas. This section presents empirical correlations to calculate the oil viscosity based on frequently available hydrocarbon-system parameters. R. the composition of each phase changes. Rp. In thermodynamically stable mixtures of oil and gas.1 Dead-Oil Viscosity Correlations. These limits can be expressed mathematically as B. Solubility. as the equilibrium is disturbed with change in pressure under isothermal conditions. the free-gas gravity can be calculated by perature if the solution gasloil ratio. . The viscosity of crude oil with dissolved gas is an important parameter in pressure-loss calculations for flow in pipes or in porous media. as functions of pressure and temperature. For empirical correlations. such as density and viscosity. and gas solubility. .WO 2 2w 5 g m . the oil viscosity should be determined in the laboratory for the required pressure and temperature ranges. . . . . . However. . B. . where produced gasloil ratio. two-phase-flow calculations. B. B-10 to predict the dissolved-gas gravity. . For saturated oils above the bubblepoint.3. such as temperature. the physical properties of the free gas. This dead-oil viscosity then is corrected for the system pressure condition. Because methane is the lightest component in natural gas.000 E? 1 . Katz er used Fig. are known.3. . while the upper limit of the free-gas gravity is the average specific gravity of the total separated gas. Most commercial computer programs fail to account for the changing composition of the free and dissolved gas as pressure and temperature change. . this also forms the lower limit of the dissolved-gas gravity. . as a function of the API gravity of the crude oil and the gas solubility of the oil phase. "API Fig. Beal Correlation. . .3. ft3/bbl residual oil and y.) On the basis of a simple material balance.2% 2. Such sequential vaporization increases the specific gravity of both the free and the dissolved gas. r yd 2 0. Whenever possible. ygd. expressed at standard conditions. T.5. (B-34) Fig. B. . . oil gravity.4 Specific Gravity of Free and Dissolved Gas. the dead-oil viscosity is determined first. which changes the gas solubility and the specific gravity of the dissolved gas. at 1-atm pressure and temperature.3. Normally dead-oil viscosity is determined in the laboratory whenever PVT analysis is done. B-11--Gas-freecrude viscosity as a function o temperature f and oil gravity at 60°F and atmospheric pressure (after Bea120). B. ygd. . .4. Heavier hydrocarbon components vaporize with further lowering of pressure. . .5rnxocro1. Glas014 presented an empirical correlation based on North Sea data. . . It is based on 460 dead-oil viscosity measurements and can be expressed mathematically as pd = l o x .75261og(T)-269718 where and MUL~PHASE now IN WELLS .) TI I63 and T is in OF. . . where Glas~ Correlation. . Sutton and Farshadla found this correlation to be the most accurate dead-oil viscosity correlation among the three described here.1 .000 psia Fig. . T= system temperature. oil gravity = 16 to 58OAP1. B-38 was developed for data in the following ranges of temperature and API gravity.5~) ~. . 1. and This correlation was developed from these ranges of data: pressure = 132 to 5. (B-38) -~ .) 5. . .-c-- 0 1 0 0 1m 3000 4 .p. up to the bubblepoint pressure. Beggs and R o b i n ~ o n ~ ~ proposed a different empirical correlation to determine the dead-oil viscosity. Oil viscosity decreases with rising pressure as the solution gas increases. . . There are few empirical correlations to determine the viscosity of saturated or undersaturated crude oil systems. B-39 to determine the live-oil viscosity. . where X . Eq. In its empirical form. psta 000 ' ' ' .------L 1 -h 0 2 -020 1 -0 1 m 1. .. . p= p. .ooo1. . This correlation16 recommends the following correction of the dead-oil viscosity presented in Eq.5.2 Saturated Crude Oil Viscosity. . . Kartoatmodjo and Schmidt Correlation. temperature = 70 to 295OF.0 02023yAP. . .3 13 log T . .447.1 5 ~ ~ ~ ~ 5 4 8 .. . . . respectively: 50 5 T 5 300 and 20. Beggs and Robinson Correlation.oor.0 x 108)~-Z8177(logyAp.moa.ooJ~. and gas solubility = 20 to 2. where a = 10. = (16.070 scfISTBO. 1 . This correlation can be expressed as p. .soJ 3 . Pressure. w 5 000 " lo 9 m 0 0 I . The reservoir oil viscosity depends on the solution-gas content.265 psia. . This c o r r e ~ a t i o n ~ ~ is based on 2. .3. .pb. .(B-37) B. . 4 1 2 I : Pressure. . . OAPI. . The empirical form of this equation is = 10(3 0324 . Kartoatmodjo and Schmidt Correlation. B-12-Variation of oil viscosity with pressure. psi . . m UndersaturatedPressure.20 Beggs and Robinson Correlation.141 x ~ O ' ' ) T 444(logYAPI)a. . this correlationI6 is a combination of all three previous ones and can be expressed as p. YAP1 =gravity of oil system. = (3. . .36.0 . OF. .073 saturated oil viscosity measurements. In multiphase pipe flow.28 To account for the effect of water salinity..=density of water at standard conditions.35 1. . viscosity = 0. methane is more soluble than ethane.5 15 psia.).4 Ibm/ft3.3 to 59S0API. B-13-Surface tension of crude oils at atmospheric pressure (after Baker and Swerd~off*~).0 . .. The surface tension of live oil with dissolved gas.. .0753 . ... =gas solubility in pure water. . B .. . Specific gravity or density of water is also an important property and is normally readily available from an oilfield water analysis. . where A=2. =specific gravity of water at standard conditions.. B-44.199 scfISTB0. .. . where p. . . . + (R. The solubility of methane in water can be used to estimate the solubility of natural gas in pure water with an accuracy of 5% or better.3 to 2. FLUID AND ROCK PROPERTIES R. B. The density of pure water at standard conditions is 62. weight percent of NaC1. The effect of increasing pressure is to increase gas solubility.~ ) Tand. . the physical properties of water play an important role in multiphase flow calculations. Saturation Pressure. the surface tension of dead oil is correlated vs. the density of water can be calculated from This correlation is based on data in the following ranges: pressure = 141 to 9. .1 M o r r e c t i o n to dead-oil surface tension.1 Water Density.9 x T-(1. .. The solubility of hydrocarbon gas components is inversely proportional to their molecular weights.2 Gas Solubility in Water. . CS = salinity correction factor = 1.. . p.0107 -(5.. Above the bubblepoint pressure.4 Water Physical Properties where Water is often a very important liquid component in the oil and gas production system. The effect of pressure on surface tension of oil with dissolved gas can be determined from Eq. . = R. Methane is the most soluble component of natural gas in water.. . B. .. ethane is more soluble i n ~ the than propane. . psia Fig.~ ) TB=0. = viscosity of undersaturated oil. .45 X 1 0 . R. where p.23 B. ... .C...75 x 10-7) ~ ~ (3. C = -(8. .. . . Fig. temperature and the API gravity of the oil..3. viscosity. ..3 Undersaturated Crude Oil Viscosity. Ref. gas specific gravity = 0. Because the precise effects of temperature on dead-oil surface tension is unknown. Fig.~ ~ of the im~ measure balance in the interfacial molecular forces for two phases in contact.. ~ c ~ a states that ~ solubility of each component is two to three times greater than that of the next heavier paraffinic component. The gas solubility is corrected for the effect of water salinity.Oil Gravity at 6O0F. 19 is the source of this gas-solubility correlation. . surface tension. gas solubility = 90. . (B-47) where T is in O F and (Rsw)b =gas solubility in brine. Vazquez and Beggs Correlation..( 3 . .. Dodson and Standing3' suggested the corrections presented in Fig. . Vazquez and BeggsI3 proposed this correction to the saturated crude oil viscosity at the bubblepoint pressure for pressures above the bubblepoint pressure. . .~~ The empirical work of Baker and S ~ e r d l o f fpresented in Fig. = aJ. commonly is used to estimate the surface tension of crude oil at atmospheric pressure (dead oil).5. B. . B. . gas solubility.5 1 1 to 1. Ibml ft3. . B-16..4. gashiquid and liquidliquid surface or interfacial tension values are used to determine flow patterns and liquid holdup.12+(3. scf1STBW. . . 4~8o . B-13.. Fig. B-14.3. . and so forth. & =formation volume factor of water. a..[0. 5 9 ~ o ."API Fig. and oil gravity = 15.6 Surface Tension. . (B-44) where x = percent correction from Fig. .. . pressure relationship. .02 x 10-")T2.000173T]S. IbmIft3.1 17 to 148 cp. can be obtained with an appropriate correction to sod for the dissolved gas. ..4. extrapolation beyond the data presented in Fig. This correlation16 allows correction of the saturated crude oil viscosity at the bubblepoint. Neglecting gas solubility in water. a. . In this figure. cp. = A + Bp + C p 2 . . and compressibility are of particular interest.26 x 1 0 . (B-46) ~. as a function of pressure. . Kartoatmodjo and Schmidt Correlation. . Thus. B-15 can be used to estimate methane solubility in pure water. sod. .13 is not recommended. and B. and y. Among the physical properties of water.0.r/ loo). . rising pressure increases the viscosity of oil because of its compressibility. based on Eq... . . Surface t e n s i ~ nis a ~ .29Consequently. formation volume factor. B-12 presents this viscosity vs.~ ) ~ + ( 1 ..3 ) ~ . = density of water at any pressure and temperature.. . pwr. . where S = salinity of water. . B-4 1 for undersaturated pressure. B-14 presents this correction to the surface tension of the dead oil as a percentage reduction of the dead-oil surface tension. thus decreasing surface tension. pob.. ~ h m e d recommends the use of this gas-solubility correlation '~ for water. . psia. T = system temperature.. ft3hbl Fig.. Because gas solubility is very low in water compared to that in oil. water formation volume factor is assumed to be 1. 0 ~ ~ 5 2 ( 4 ...ical polynomial relation to determine the formation volume factor of water. X $ "st Temperature.=water formation volume factor above bubblepoint pressure.and p = system pressure @ >pb). psia.l.8 X 10. B= ..l Correction for Gas in Solution &Y 0 5 1 0 1 5 20 25 Gas~Water Ratio.. bbVSTBW. 0 E la-. In actual field situations.... B-15-Solubility of methane in pure water (after Culberson and M~Ketta*~). psi.. presented an empirthus. Water formation volume factor above the bubblepoint pressure changes as a result of the compressibility. Assuming negligible thermal expansion. = T.ll.)= isothermal compressibility coefficient of the gas-free makr.rl. Temperature. where 8. O F f : Fig. B-1GEffect of salinity on gas solubility: TDS=total dissolved solids (after Dodson and Standing31). A = 3. this empirical relationship is proposed.0 . and T=tem- 4 B. For gas-free water. cw =water compressibility.0. psi . where (c...60.OW IN WFI 1 9 . B-17-Compmsibility of pure water.'' The top graph yields cw for pure water while the bottom graph is used to determine a correction factor for cw to correct for the effects of gas solubility. Bwb = water formation volume factor at the bubblepoint pressure.8536-0.=formation volume factor of water. water shows gas solubility and. The compressibility of water with natural gas in solution can be estimated from Fig. and p = system pressure. MlJI TTPHASE R. B-17.9267 x . p =pressure. ~ o u l d ~ O . pb.3 Water Formation Volume Factor.(8..000134p. it is common practice to neglect compressibility and gas solubility of water. including effects of gas in solution (after Dodson and Standing31).4.1. shrinks under pressure reduction.'qp. ~ e e h a presented an empirical correlation to determine the n ~ ~ isothermal compressibility coefficient of gas-free and gas-saturated water. O F .. '. ~ ~ + C= (3. OF. psia. where &..T. Fig. . No data are available on the effects of gas solubility. ..45 Surface Tension of Water. The other parameters in this equation have been defined pre- . Numbere et proposed this mathematical expression for the salinity correction factor. this adjustment is suggested. Gas density is defined as the mass per unit volume of gas.. ~ o c o t and Hough er ~ 1 . from the work of Hough et al. and R = universal gas constant. and Cs = salinity correction factor.4. psi . B. . B-21). . ft3. This equation of state is called the real gas law. in-~ t~~ 3 vestigated the surface tension of waterlgas systems. ~ ~ . These intermolecular collisions are perfectly elastic. .I . His empirical F correlation relating water viscosity with temperature in O is presented as This equation is called the ideal gas law. . . is an empirically determined constant.. =gas solubility in water. B .. temperature. where p = absolute pressure. These molecules are assumed to have no attractive or repulsive forces between them. According to the kinetic theory of gases. psia. Although these r e s u ~ t s . scfISTBW. . v ..)~= isothermal compressibility coefficient of the gas-saturated water. van wingen3j reported the effect of temperature on water viscosity. . . B. T ~. Data on the viscosity of oilfield water are very scarce. It is important to note that the production of oil and water mixtures often causes emulsions to form in the well. Based on empirical studies.Temperature. and V = volume of gas. the equations of state are modified for real gases to include the gas deviation factor or compressibility factor.4 Water Viscosity. the viscosity of an oiVwater emulsion can be many times greater than the viscosity of either constituent liqFLUID AND ROCK PROPERTIES where the compressibility factor. Nevertheless.. M = molecular weight of gas. . Fig B-19 shows the kinematic viscosity. volume. one can linearly interpolate between the curves for 74°F and 280°F and obtain acceptable estimates of the surface tension of water. salinity (after Frick35). .5. . Thus. Z. and T = temperature. .. T=absolute temperature. These are mixtures of multicomponent gases and are called real gases. weight percent of NaCI. .. psi-'. Hydrocarbons or natural gases are not pure or single-component gases. and R. TDS. A volume fraction weighted mixture viscosity calculation procedure does not work in this case. Gas occupies the volume of its container without regard to shape or size. ~ ~ . . B. . . where S=salinity of the water. B-20 to estimate water viscosity as a function of salinity and temperature.1 Gas Density. Ibm. apply rick's^^ correction chart in Fig. From the real gas law.pM8 - . is the =density of gas. Matthews and Russell' presented Fig. OF. their approach is highly questionable in its prediction of water surface tension as a function of such variables as water salinity and gas composition. where (c.. where c = isothermal compressibility coefficient of the brine. To account for the salinity of water. . 3. rn = mass of gas. Meehan also proposed a correlation to determine the isothermal compressibility of gas-saturated water. rn = weight of gas. . the basic assumptions of negligible intermolecular forces and insignificant volume of molecules compared with the volume occupied by the gas are not valid. (B-56) where p. "F Fig. of water at saturated condition vs. B-19-Kinematic viscosity of steam and water as a function of temperature (after Prats5). . . An equation of state is a mathematical relationship between pressure. B-1&Ratio of brine viscosity to pure water viscosity vs. This figure also provides a method to correct for pressure. actual laboratory viscosity data should be used in this case. For real gases. For large dissolved-solids content in water. Katz et combined their works into a single diagram (Fig. often called the Z factor. B-18 to estimate brine viscosity. Water viscosity increases with pressure and dissolved solids and decreases with increased gas solubility. ps = V m . ~ ' all experiments have suggest that been valid. and they collide at random. gas is composed of a large number of molecules of insignificant volume compared to the total volume of the container. OR. and temperature for a fixed mass of gas and is expressed mathematically for n moles of gas as . As discussed in Chap. n =number of moles of gas. V = volume of gas (gas container). 5 Gas Physical Properties A gas is defined as a homogeneous fluid of low density and viscosity without a definite volume. However. . u~d phase. % Fig. . l l - Temperature.. B.. . If the specific Gavity of the natural gas exceeds 0. is defined as the ratio of the gas density to that of air. .. .S.96 and its specific gravity is 1. . . . psia ft3/lbm-mol O R .. . . all hydrocarbon gases have the same gas deviation factor. . Z P . p g = 2 .. expressed mathematically as Thus.O F Fig. The real-gas deviation factor is a very Important vanable used to calculate gas density and gas formation volume factor. Eqs. .. .. . . . .000 8.. the specific gravity of gas is Yg = (z)sc z. . .ough et a\ 37 methane I 2. . yg.o . Ma. The specific gravity of gas. . Brown et al. . . l l . .. . .. . 6-20-Water viscosities at various salinities and temperatures (after Matthews and Russell1). .75 a correction for high molecularweight gas should be applied. O R . B-21-Effect of Pressure and temPerature on Surface tension of water (after Katz et aL26). The specific gravity of free gas is a function of pressure and temperature because it depends on the composition and quantity of gas transferred between the liquid and the free-gas phase. T = absolute temperature of gas. B-59 should be replaced by y ~ as defined in Eq.. psia. . This law states that at the same reduced pressure and temperature. . . .. . Thus. .. 114 B. ..000 10oOOC Pressure.73. p.. .. Note that the term "pseudo" is used MULTIPHASE n o w IN WELLS . To determine this parameter. and R = 10. . . .96.. . . viously. ... . yg. at standard conditions.M8 - yg = 28... in Eq. f ugh et a/ 37 methane 20 10 1 0 . . .2 Real-Gas Deviation Factor. o l O ~ ~ l l l l l l l l . .. . B-3j.000 6.. .(B-57) The molecular weight of air.. l o correlated the pseudocritical pressure and temperature with the specific gravity oigas (Fig.. ... (B-59) I where yg = specific gravity of free gas (air = 1).. . p = pressure of gas.000 4. (B-58) Mg Gas density in Ibm/ft3 can be determined easily by combining the real gas law with the definition of specific gravity of gas. . . ... . . . B-35. ..19 Stewart er proposed adjustments to the pseudocritical pressure and temperature for highmolecular-weight natural gases. psla Fig.is 28.l through B-4 define the critical and reduced pressures and the critical and reduced temperatures. Z= real-gas deviation factor. Standing and ~ a t z l l used the law of corresponding states. .. . . 7s~ T . .7 1 . . Several empirical correlations are available. such as hydrocarbon or natural gases... (B-62) . For ease of calculation with computer programs. 7 ~ ~I l . . ..5 1 . psia. . - where Tpc = pseudocritical temperature. . . and After determining the pseudocritical pressure and temperature. . .@. . Then.)'ex~(Gr p. .." to represent the properties of pure gas mixtures. . .. Case 1: Natural Gas Systems. ppc = pseudocritical pressure. B-3. .Pseudoreduced Pressure Pseudoreduced Pressure Fig. 5 ~ : . B-22). and yg =specific gravity of the gas mixture (air = 1). B-22. FI 1ill3 AND ROCK PROPFRTTF9 . = 187 and + 330y.. .) + 1. O R .0. . . T. B-22. it is an 1 I-constant empirical equation used to fit Z-factor curves in Fig. 2 p2 + A..p.(] + ~. Eqs. . . . . .. the gas deviation factor can be obtained from Fig. . The is ~ Dranchuk and Abu-Kassem c o r r e ~ a t i o n ~presented for the sake of completeness. = 706 . . . . . Standing2' presented a set of empirical equations to determine the pseudocritical pressure and temperature to approximate the curves in Fig. . l y i . B-3 and B-4 can be used to calculate the reduced pressure and temperature. B-22-Real-gas deviation factor for natural gases as a function of pseudoreduced pressure and temperature. L A. these mathematically expressed correlations are convenient for Z-factor determination. . . representing the Standing and Kat7I1 Z-factor nomogram (Fig. . . and can be expressed as Case 2: Gas Condensate Systems. . MULTIPHASE FLOW IN WELLS . . g. B. ft3/scf. . y ~ + = mole fraction of hydrogen sulfide in the gas mixture. B-3. cp.= pseudocritical pressure. require correction. The viscosity of the gas mixture at atmospheric pressure is estimated from and A. ) ~ ~ . that ratio is applied topgu.4 Gas Formation Volume Factor. such as carbon dioxide. B-70 or Fig. . The commonly used oilfield unit for viscosity is centipoise. such as the Newton-Raphson iteration technique. OR.. OR. which cannot be used for sour gases. . =viscosity of Component j at the desired temperature and atmospheric pressure. . and hydrogen sulfide. OR. B-23. .E and wherepg. Molecular weight is related to gas gravity by The gas viscosity at reservoir pressure is estimated by determining the ratiop.4 . = Tpc . the Standing and Katz Z-factor chart can be extrapolated beyond that reduced pressure because of the linear relationship of Z vs. The pseudoreduced temperatures and pressures for use in Fig. .2 X. B-23. where 1 poise is defined as a viscosity of 1 dyne-s/cm2. 0. defined mathematically by pk B. nitrogen.lpgu at the appropriate temperature and pressure from Fig. Natural gases often contain nonhydrocarbon components. + where Tp. . and hydrogen sulfide. presented a widely used method to estimate natural Carr et gas viscosity. . . Isothermal gas compressibility can be defined as the change in volume per unit volume of gas for a unit change in pressure. where Bg =gas formation volume factor.0 5 Tpr 53. Eq. Td. n =number of components in the gas. B-24. The viscosity of a fluid is defined as the ratio of shear stress to shear rate. thus. B-3 and B-4 and Fig. B-23 can be used with the gas molecular weight or gas gravity to estimate the gas viscosity at reservoir temperature and atmospheric pressure. Fig. Standing and Katz developed their Z-factor correlation based on mixtures of hydrocarbon gases with molecular weights less than 40. In the presence of nonhydrogen gases. ~ / +219Mg+ T).obtained from either Eq. Then. OR.= corrected pseudocritical temperature. .42presented a semiempirical equation to calculate gas viscosity. This method suggests the following adjustments to the pseudocritical properties used to determine the Z factor from the Standing and Katz chart. & = mole fraction of Component j. . . for pp. The fonnation volume factor of gas can be calculated from the real gas law. .0. and T= absolute temperature. B. . This correlation represents the Standing and Katz Z-factor chart within 0. This equation. p. psia.A 1 throughA 11 were determined from nonlinear regression on 1. . . .6 Gas Compressibility. At nonhydrocarbon gas-content values below 5%. The 11 c0nstants.7210. ..2 I p p r < 30 and 1.T = reservoir temperature. T. and E =pseudocritical temperature adjustment factor. If the gas composition is not known. at constant Tp. B-24 are estimated from Eqs.500 data points from the Standing and Katzll Z-factor chart.5. Lee et al. 0 2 ~ .p. =gas density at reservoir pressure and temperature. Although presented for ppr up to 15. and Mj = molecular weight of Component j from Table B-1.0. lbrn/ft3.=reduced gas density and is defined by B. Y = 2. can be presented as .13 expressed mathematically as where the CoefficientA is the sum of the mole fractions of hydrogen sulfide and carbon dioxide in the gas mixture or With the real gas law to replace volume in the previous equations and after proper differentiation. .585% average absolute error and is applicable over the ranges. there is negligible effect on the Zfactor. . . psia. X = 3. .3 Correction for Nonhydrocarbons. p = pressure. Wichen z ~ ~ and ~ z i presented a simple gas compressibility correction procedure to compensate for the presence of hydrogen sulfide and carbon dioxide in natural gases. psia. . = pseudocritical temperature. B-64 is implicit in Z and must be solved by some iterative method. It requires knowledge of the gas composition and the viscosity of each component at atmospheric pressure and in-situ temperature. and Mg =apparent molecular weight of the gas mixture.5.(B-66) ( 09 where K = (9.4 0 .5 Gas Viscosity. =viscosity of the gas mixture at the desired temperature and atmospheric pressure.5 + (986/7') +O. . = 0. . . Higher concentrations of nonhydrocarbon gases can cause substantial error when calculating the compressibility factor and. .5. .5. . the gas viscosity at atmospheric pressure and desired temperature must be corrected by use of the inserts in Fig. > 15.OIMg. pp. = corrected pseudocritical pressure. such as nitrogen. carbon dioxide.where p. They present pseudoreduced compressibility at two different ranges of compressibility values. T. 33O.75 at pJep= 14. From Eq. cp.067 = 1. To estimate the gas compressibility. to the pseudocritical pressure.88. B-25 and B-26). in psia and absolute temperature. B-60. Solution Gas/Oil Ratio (from Vazquez and Beggs). Also given are yR = 0. From Eq. B-18. p.7)~ 669 psla. e ~ ~ He defined isothermal gas compressibility. as the ratio of pseudoreduced compressibility.(37. Trube presented correlations to estimate the pseudoreduced compressibility as a function of pseudoreduced pressure and temperature (Figs. Determine the black-oil-model mass-transfer parameters and the physical properties of the gas and oil phases at a location in the well where the pressure is 1. Note that these two correlations are similar. T r ~ b presented a correlation to estimate gas compressibility.000 scfISTB0. as = 1. B-5.10. From Eq. ygd = 0. From Eq. cR.5)(0. = 168 + (325)(0.7 psia and & = 60°F. B-9. in O R .5)(0.7) . From Eq.61 1bm/ft3. B-35. and Rp = 1. Ppc. T.197 bbVSTBO. Note that for an ideal gas. = 677 = + (15)(0.700psia and the temperature is 180°F. B-61. From Eq.7) .41). = 47. From Eq. B-30.(12.131 + 0. B-23-Viscosity of natural gases at 1 atm (after Carr et a1. p.7)~ = 389"R I I' . Example B. A crude oil and natural gas mixture is flowing through a wellbore.Molecular We~ght Fig. Z = I ..l-Calculate Black-Oil-Model Mass-Transfer Parameters and Physical Properties. From Eq.. Specific Gravity of Free Gas. Oil Density. B..0 + 0. y ~ p l = . Pseudocritical Pressure and Temperature Cfrom Standing). From Fig. where Z = gas deviation factor at absolute pressure. Specific Gravity of Dissolved Gas Cfrom Katz). and Oil Formation Volume Factor (from Vazquez and Beggs). B-8 for YAPI > 30°. 853. Gas Compressibility Factor Cfrom Standing and Katz). B-4. B-3.(0.2)(5.88)1351] = 0. From Fig. From Fig.~' p = [10. 4 . B-7 1 . Ter From Eq. From Eq. From Eq.35 1 . B-72.243) = 1. From Eq. Y = 2. From E q . . p. = 29 dyneslcm.016 cp. + 100) -0. Fig. 7 ) e r ~ (5. From Eq.Pseudoreduced Pressure. Gas Formation Volume Factor. B-24-Effect of temperature and pressure on gas visco~ity.7. B-13.715(281 . Gas Viicosity (from Lee et al. Surface Tension Viom Baker and Swerdlofn. B-22. = (28. = 1 =2. 118 MULTIPHASE FLOW IN WELLS .44(281 + 150)-O'~~ 0. a. B-14.56~~.7) = 20.56)'. B-40. Oil Viscosity (from Beggs and Robinson).97)(0. M. pod = PSEUDOREDUCED TEMPERATURE.).243)(5. [ Gas Density. = ( 1 0 ) . Fig.~ ( 1 2 8 . From Eq. From Eq. B-25--Correlation of pseudoreduced compressibility for natural gases (after T r ~ b e ~ ~ ) . %=29. B-59. and Z = 0.' = 0. From Fig. and b = 5.97 cp.4 . B-37. B-69.28 1brnAbm-mol.515 ](2. cf= formation or rock compressibility. From Eq. scf/STBO.9 x l ~ .2 Total Compressibility. This empirical c o r r e ~ a t i o n ~deterto ~ mine the total formation volume factor can be expressed mathematically as B.. In all transient fluidflow calculations ia porous media. This correlation can be expressed mathematically as log B. . different methods to determine individual fluid compressibilities already have been presented.I. Al-Marhoun Correlation. B-27--Correlation of pseudoreducedcompressibilityfor an undersaturated oil (after Trubes). pW 5 6 7 8 9 K ) I5 Fig.6. 4 .ppe. Total Formation Volume Factor. where C=2. B-44.aol 3 4 Pseudoreduced Pressure. the isothermal compressibility coefficient of Phase i is FLUID AND ROCK PROPERTIES . psi.must be calculated.T = temperature. So =average oil saturation: c = isothermal compressibilitv coefficient of the oil phase. = a correlating number and is defined as where c. It is often convenient to express the formation volume factor below the bubblepoint pressure in terms of total formation volume factor.I. sure. O F . Fig. where B. scfISTB0.b =gas solubility at bubblepoint. and a=0. d = 2. =formation volume factor of oil. psi . In this appendix. where B.0. = total system compressibility. Total formation volume factor is defined as the ratio of the total volume of hydrocarbons per unit stock-tank barrel of oil at the prevailing pressure and temperature.0062 1. c. psi . B. c =isothermal compressibility coefficient of the water phase. This can be expressed mathematically as where F = Ryyb. R.6.644516. Glas@I4 presented an empirical correlation to calculate the total formation volume factor. and So +&. B.080135 The unit of compressibility is reciprocal of pressure (Ilpsi). and Bg =gas formation volume factor.=water saturation. psi.1. and e = .~ . S.+Sg= l . is defined as the change in volume per unit volume for unit change in pressure at constant temperature. independent of the number of phases present. Sg =gas saturation.I. cg = isothermal compressibility coefficient of the gas phase.l.Compressibility. and y = phase specific gravity. R.I.. b = c = 0. B-26-Correlation of pseudoreduced compressibility for natural gases (after T r ~ b e ~ ~ ) . bbVscf. denoted as c. which also is known as the two-phase formation volume factor. Pseudoreduced Pressure. = gas solubility at system conditions.6 Composite Fluid Properties Individual phase properties for formation volume factor and compressibility can be combined into composite properties. bbVSTBO. p = system pres~ ~ ~ ~ s . psia. . = 0.47257 log B. the total isothermal compressibility.761910.079340. psi. The total rocufluid system compressibility can be expressed mathematically as + 0.y'.1.724874. . By definition. =water formation volume factor. Formation Compressibility. the effect of gas solubility can be ignored.. the apparent compressibility of oil. ~ e e h a presented a correlation that includes the effects of gas solubility (Eqs.I t O S D 1 . or from the Trube correlation8 shown in Fig.below the bubblepoint pressure can be calculated by Water Compressibility. from the correlations presented earlier in this section. Oil compressibility is always positive because the volume of an undersaturated liquid decreases as the pressure increases. coa. and R. 9 1 I N ~ * eOROSlTY AT ZERO NET PRESSURE. bbVSTBW.. B-Wore-volume compressibilityat 75% lithostatic pressure vs.. Because the gas solubility in water is very low. the compressibility of oilfield water can be defined as where R. bbYSTBO. initial sample porosity for friable sandstones (after N e ~ r n a n ~ ~ ) ... Dodson and Standing3' presented a correlation to estimate the compressibility of water (Fig. Fig. B. Rock or formation compressibility under isothermal conditions can be defined as -L(%). . Accordingly. initial sample porosity for limestones (after Newman45). $1 L Fig. c. respectively. B-50 and B-51a). Figs. bbYscf. Oil com- There are different correlations for rock compressibility. = formation volume factor of water. Eq. Oil Compressibility. The isothermal compressibility of an undersaturated oil (above the bubblepoint pressure) can be defined as co = where B. Tpr The oil compressibility then can be estimated from where B.= pseudocritical pressure from Fig. . B-27. and cg in Eq. =gas solubilities in oil and water. B O ap where Vp = pore volume. For an isothermal condition.= . c.). B-81 can be replaced by the previous expression to yield pressibility can be determined from laboratory experiments. Howevn ~ ~ er. and Bg =gas formation volume factor. cpr. Trube correlated pseudoreduced compressibility. B-28 through B-31 present the rock compressibility correlation after N e ~ m a n .. B-17). B-1 and Tpc = pseudocritical temperature from Fig.ap =p o a p ~(dp"). B-83 can be modified slightly to include the effects of solution gas on the change in liquid phase volumes and presented as where g. Taking into account the gas in solution. pp. =oil formation volume factor. and pseudoreduced temperature. B-2&Poravolume compressibility at 75% lithostatic pressure vs. where Bi =formation volume factor of Phase i.1 -( as. = = INITIAL POROSIN AT ZERO NET PRESSURE. each for a fixed type of rock. ~ is strongly recomIt ~ MULTIPHASE FLOW IN WELLS . B.1. 0 .with pseudoreduced pressure. v. initial sample porosity for consolidated sandstones laf. gas evolves from the liquid. K.16 by use of oil compressibility.... (B-91) This correction. separator gas gravity. B. B.. O R . The differential-liberation process more accurately represents gas liberation from liquid during flow through a porous rock or reservoir.. B-25. it is clear that these correlations are questionable... Kartoatmodjo and Schmidt16 presented an empirical method to convert flash-liberation oil formation volume factors to differentialliberation oil formation volume factors based on tank oil gravity. initial porositv for unconsolidated sandstones (after mended that laboratory data for this parameter be used wherever possible. Fc. B-15. Kesler and Lee47 and othe r ~ ~ have investigated~and reported on the enthalpy of crude . Moses46 has shown that the error resulting from the use of flash data for a differential-liberation case.8 Enthalpy Enthalpy is the property of a fluid that determines the heat required to raise or lower its temperature.16 Fluid physical properties based on flash liberation are more appropriate for use in tubing and flowline calculations. ~hvsical oroverties . although this often was not reported. B-30-Pore-volume compressibility at 75% lithostatic pressure vs. Most of the fluid physical property correlations presented in this appendix were developed with flash data.. ~ ~ . ter N e ~ m a n ~ ~ ) . separator pressure. ~ oil fractions for liquids and vapors. Differential Liberation o f Gas Accurate determination of rock and fluid physical properties is very important for any pipe flow calculations used in oil and gas production engineering. and Bod = differential formation volume factor. calculated from Eq.8. B-28 through B-3 1. However... However. for well-performance calis culations..A h . or differential liberation for a flash-data case. is defined as where Tb = average cubed boiling point and yo =oil specific gravity. B@= flash formation volume factor from Eq. Sec. Rg. The Kesler and Lee correlations involve obtaining base values of the enthalpy at zero pressure (denoted by hO) and then correcting these for pressure effects according to the relation h = hO . The conversion factor... it often is assumed that c. However. the fluid ... B-32 through B-39 give the base enthalpies and require information on the API gravity and the Watson characterization factor of the crude (or crude fraction).Fig. it is important to note that the gas compressibility is at least an order of magnitude higher than the rock or liquid compressibility.. The pressure correction Ah is determined as FL(fTDAND ROCK PROPERTIFS . . can be calculated by equations of state (see Appendix C). such as flash or differential liberation. From Figs. B-21a... c. B-31-Pore-volume compressibility at 75% lithostatic pressure vs. ~ .. K. or B-27. also called the enthalpy correction factor.. B-81. Formation volume factors above the bubblepoint pressure then can be obdned from Eq.. can be up to 20%... at best. B-18. =cg. increasing the volumetric flow rate of the gas andcontinuously changing the liquid and gas physical properties. . B-90 also can be used to convert formation volume factor at the bubblepoint pressure obtained from Eq.7 Flash vs.6 presented a detailed discussion of gas compressibility.. As the fluid produces from the reservoir through the well and flowlines. B-27. Eq. separator temperature. is given as where T = temperature. Depending on the type of gailiberation. needed in Eq.. rock compressibility.. B.. While the gas compressibility is on the order of the liquid or rock compressibilities are typically on the order of 10-5 or In gas reservoirs. the differential-solution gas/ oil ratio cannot be converted by application of this method... vary.. B. An empirical method for correction of base enthalpies for pressure follows..1 Crude Oils and Fractions. and reservoir temperature. 9. B-19 or 8-28. Figs. B. ..5. when the API gravity and the pseudocritical temperature or pressure of the crude are known. .00045T . . O F Fig. "F Fig. B-32-Enthalpy of petroleum fractions (&= 10. B46 gives heat capacities of saturated liq1 0 0 ~ J g 0 4 0 0 5 m e a l uids.2 discusses the liquid and vapor enthalpies of water. after Kesler and Lee47). B-3 and B-4. B-34--Enthalpy of petroleum fractions (& = 11. w. and the quantities Tpc. Fig. CpO. . ature and of the latent heat of condensation and evaporation of the of crude fractions.)(') found in ) is Figs. B-8. The corrections for pressure given in Figs. defined by Eqs. B-41 and B-42. . . the liquid and vapor phases can be estimated by taking the chord secant of the enthalpies over the desired temperature range. . . Enthalpies calculated by this procedure take with temperinto account changes in the isobaric specific heat. 8-40 through B-42 depend on the reduced temperature and pressure. . B-43. 0 Fig. . and acentric factor. . where ( A ~ I R T ~ ~ ) ( Ois found in Fig. B-45 corrects for pressure. If they are not available from measurements or other sources. CpO. L . its heat capacity may be estimated from the GambillSOrelationship. . B-43. Sec. . and are independent of the Watson characterization factor. . B-33-Enthalpy of petroleum fractions (6 10. the values of Kw and w may be estimated from Fig. When only the density or API gravity of the crude is known. T=O to 600"F.200"F. T= 600 to = 1. (AWRT~. . . . I 0. The base enthalpies. . . Temperature. cpo 0. The second term of Eq.Temperature.are referred to a value h0 = 0 at T = . Average specific heats. T=O to 600°F. 122 Fig. B 4 gives heat capacities of selected hydrocarbon and nonhydrocarbon gases and vapors at atmospheric pressure. Fig.ppc. hO.200°F. B-40. . w. . Kesler and Lee4' provide additional details.8. and RTp. O F where T is less than 300°F. . after Kesler and Lee47). after Kesler and Lee4').IM are found in Fig.388 +r= (B-95) iY0 Temperature. B-93 is omitted when the reduced temperature is less than 0. Kw. 1 3.194.33 420 431.167.9 910.h .91 441.8 732. .5 656.8 1. .0 1. B-35-Enthalpy of petroleum fractions (&= 11.198.9 763.201.07 218.3 h. B-49.3 995.2 1. is found by multiplying given in Fig.21 500 520 540 544.(T .76 Absolute Pressure (psi4 1 Latent Heat Liquid Vapor & b 69.9 871.2 802.83 350 358.163.1 588.43 381. .8.708.000 1. T 2 T. B-47 and B-48 give the enthalpies of water and steam at saturation pressures and temperatures.01 100 134. after Kesler and Lee4').179.4 809.13 161.5 1. ature above the saturation value. .71 440 460 467.9 1. . O F Fig. .8 1.130.8 1.7 982.4 1. .3 910. .3 1. . .85 396. B-49 is somewhat lower 123 AN^ ancu ~ o n ~ c o r r c ~ .150.TABLE B-SENTHALPY OF WATER AND STEAM AT SATURATED CONDITIONS53 Enthalpy (Btu/lbm) Temperature (OF) 101.7 1.500 1.696 29.4 1. .0 1. The differencebetween these two quantities represents the heat required to convert water from liquid to its vapor (steam) and is called the latent heat of evaporation. .203.203.40 374.9 843. Table B-5 presents the enthalpy of water and steam at the saturation pressure and temperature. (B-97) L.3 806.133.59 466. At temperatures higher than its boiling point.2 825. .186.37 471.3 826.97 500 566.000 3. = 11.70 418.4 1.200"F.2 1.2 1.78 640 660 680 695..3 1.9 536.5 671.21 200 212 250 300 327.105.80 812.97 417.20 600 620 635. .203.204.000 1.25 193.204.198.3 548.716 5 10 11525 14.067.205.2 Water and Steam. T=600 to 1. .7 785.7 1.2* 800 900 1.165.035.00 480 486.3 2.787.0 216.64 330.1 888.40 464. . T= 0 to B.25 250 300 308. . .8 1. steam is said to be superheated. . . .34' ' Critical properties. cs. . .62 150 200 247.72 117.7 678.177.17 167.0 714.8 1.187.5 1. . 1.143. .7 649.204. .1 1.206.5 739.2 1.060.7 1.4 1.T.204. = c. . .04 393.4 3.130.2 1.37 700 705. .0 390. .1 556.366.007.3 1.4 562. .48 269.9 611.3 (Btullbrn) 1.5 452.7 1.7 463.2 1.8.5 624.12 600 680.82 67. .80 1. .O 863. .7 171.43 321. . Figs.202. after Kesler and Lee47).7 714.6 823.68 962.9 1.019. above its saturation value as given in Table B-5. .3 945. B-36-Enthalpy of petroleum fractions (& 600"F. .8 970.125. by the excess temperthe mean specific heat.145.7 0 150 162. = h . . .543. .lW 1pw 400.97 376.2 2. Ah.78 409. . .193.192.4 1.7 1. .4 503.000."F Fig.3 1.6 755.56 560 580 596.8 646.135.000 2.59 487.094.60 298.6 1.5 310.0 2.87 130.104. . This product is called superheat. Temperature.0 1.82 350 381. .6 542.201.80 511.1 977. . FT rlin .82 400 600 700 3.2 1.6 1. .4 757. T. .. .42 449.326.150.99 180.).2 686.4 1.6 910.2 1. .7 794. . (B-96) Note that the mean specific heat of steam at atmospheric pressure over a temperature range indicated in Fig.53 355.9 1. .191.2 588. Temperature.2 1. The additional enthalpy of superheated steam.4 616. . 5. For fluid-saturated rocks.after Kesler and Lee47). MULTIPHASE FLOW IN WELLS . T = 600 to 1 . SomertonSJ provided data in Figs.5. O F than the actual specific heat of superheated steam at a given temperature. 6-37-Enthalpy of petroleum fractions (& = 11. R Fig.20OoF. B-46 shows the specific heat of saturated water. B-44.=12. O F Fig. 124 Fig. This is an example of the F i g B-51 shows lhe heat [hat the heal of a is ihe mass average of *a corresponding to the porosities of Samples 1 and 8 of Table B-7. i B.200 F.8. after Kesler and Lee47). the heat capacity may be estimated from where Mf=effective volumetric heat capacity of the rock in terms of the rock and fluid properties. Those calculated heat capacities are based on reported heat capaci L of its constituents. B-50 and B-51 and Table B-6 on the heat capacity of dry ties of pure quartz and pure calcite. after Kester and Lee4'). L I L Temperature.Temperature. "F Reduced Pressure. 600°F.3 Reservoir Rocks and Minerals.8. B-W-Enthalpy of petroleum fractions (G 12. Prats prescribed this as - - Temperature. 6-4GEnthalpy departure from zero-pressure simple-fluid function (after Kesler and Lee4'). shown in Fig. Fig. ~ i g B3&Enthalpy of petroleum fractions (&. They agree rather well with the experimental values of the two samples. t=600 to = 1. T = o to . includingelements. "F Fig..S. of several minerals.. . .w and RtPcNasfunctions of oil gravity and Watson's characterization factor. . and clays in ca~mol-oK. water and gas). anhydrite. ppc. . . oil. B-44-Heat et capacity of gases at 1 atm (after Touloukian FLUID AND ROCK PROPERTIES 125 . = isobaric volumetric heat capacity of oil.f = volume fraction of noncondensable gases in the vapor phase. = steam density. p. B-98 is the volumetric average of the densities of its constituents.P. . I(. 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 Oil Gravity. . M. including calcite. . ! CW= isobaric heat water Per unit mass.. sulfides. is the volumetric avenge density of the solid constituents of the rock m a t r i ~ . and p. = isobaric volumetric heat capacity of water. . . N~~~ that the water is in the liquid state only at the highest pressure. ~ e l g e s o provides coefficients to calculate isobaric heat capan~~ cities. Fig. . respectively.60 provide both tables and graphs of isobaric heat capscities in caVg-"K for nonmetallic solids. . L Fig.Reduced Pressure. pa. ~ ' .. water. S = saturation of fluid phases (oil. sulfates. (8-100) Thermal conductivity is a property of substances that controls the rate of heat transfer by a conductive process. .9 Thermal Conductivity + s. a Reduced Pressure. 4 = isobaric volumetric heat capacity of solid. oxides. B-42-Enthalpy departure from zero-pressure-deviation function low-temperature part (after Kesler and ~ ~ e ~ ~ ) . "API Oil Gravltv. p. = latent heat of vaporization of water. ~ B. and gas phases. carbonates. TO"loukian a1. B-43-Tw. when those samples b e saturated with methane or water at 62OOF md several pressures between 11. Fig. P.W0 psis. . ~ ~ ~8b l themophysic~propertiesof selected materials.7 and 3. of the fluid-saturated rock in Eq.). p. 'APl Temperature. + where p... (B-99) where) = formation or rock porosity. The solid density. . (after Kesler and Lee47). p.P. dolomite. and other oxygen compounds.01 + ~ J ( s ~ P . and AT= ternperature change under The average pf. B-41-Enthalpy departure from zero-pressure-deviation function high-temperature part (from Kesler and Lee47).~ ~ Table B-6 gives the calculated heat capacities for the first eight samples in Table B-7. as a function of temperature. micas. M. = P a ( l . =densities of the solid (rock). Mg =isobaric voiumetric heat capacity of gas. 5 33. Touloukian et reviewed methods to estimate the thermal conductivity of mixtures of gases. except at low values of reduced pressure and high values of reduced temperature. B-53.0 46. The thermal conductivity of saturated water.6% with a maximum deviation of 4.500 psia 3. poorly sorted Medium hard.Water Methane Water Methane Water Methane Water . B-54 and B-55.5 34.5 35.6 35. shown in Fig.9 33.0 35. to estimate the thermal conductivity in BtuID-ft-OF. and suggest that the following relation be used "because of its simplicity and moderate reliability" for "approximate engineering calculations. temperature at several values of pressure and for the saturated vapor. B-52.196 0.. B-52 are values for some pure organic liquids.199 0.6 36. mplp.5 33. Lenoir and and Eckert and Drake* presented thermal conductivities of superheated steam vs.78 <yo <0.5 36.7 33. values for organic liquids are not very sensitive to changes in pressure.7 33.7 35. Comparing measured values with predicted values.6 34.4 TABLE B-7-DESCRIPTION OF TEST SAMPLESs4 Principal Minerals .3 49. Thermal conductivities of gases at atmospheric pressure are plotted vs.3 47.6 33..1 2 3 4 5 6 7 8 9 10 Sandstone Sandstone Silty sand Silty sand Siltstone Siltstone Shale Limestone Sand Sand Description Well consolidated. 1.7 40.6 33. fine-grained Unconsolidated.273 0. is several times larger than that of pure hydrocarbons and exhibits a maximum at about 260°F.225 0.0 34.2 Gases. based on Lenoir's work62 shown in Fig. laminated Granular. medium-fine grain Poorly consolidated.186 0.4 35.. revealed an arithmetic deviation of 1.1 33.1 Liquids.7 33.TABLE B-&CALCULATED HEAT CAPACITIES OF FLUID-SATURATED ROCKS54 (heat capacity at 620°F and pressure in BtUft3-OF) Sample 1 2 3 4 5 6 7 8 Rock Sandstone Sandstone Silty sand Silty sand Siltstone Siltstone Shale Limestone 14.2 33.6 45.1 34.2%.6 39.1 32.0 39. coarse-grained Porosity 0.6 35.1 34.0 34.6 39.0 33.0 34.9 39.7 34.Quartz Sample Rock ..6 32.3 34.6 40. Eq.95 and 32OF< T < 392OF. As can be seen in that figure.38 0. 126 MULTIPHASE FLOW IN WELLS . A.1 35.0 35.6 39. uniform texture Unconsolidated.7 36.7 47.9 34.5 35. Thermal conductivities of saturated organic liquids decrease with increasing temperature. B-53.34 (%) 80 40 20 20 20 25 40 100 100 Clay Mineral Trace kaolinite Illite(?) Kaolinite type Kaolinite type Kaolinite type lllite Illite/kaolinite Other Trace pyrite Feldspars Feldspars Feldspars Feldspars Feldspars Calcium carbonate B. poorly sorted Medium hard.0 34. temperature in Figs.0 32. For 0.3 32.2 36.6 33.7psia 500 psia 1.000psia Dry Methane .34. Cragoe6] proposes the relationship where fi =mole fraction of the jth gas component.5 33.9 48. from those of constituents.9 34.0 32.9 33. For petroleum fractions and hydrocarbon mixtures." REDUCED PRESSURE p.1 32.071 0. broken Hard Hard.6 39. B-45-Pressure correction to the molar heat capacity of gases (after Chemical Engineers' Handbooksz).5 33.9.9 32.9.207 0.0 33.8 35.0 32. medium-coarse grain Poorly consolidated. Fig." ~ e n o i fgives the ratio of two liquid thermal conductivities at the i~ same temperature but different pressures as where E is given in Fig.4 35.3 48.9 32.. B-101 gave "average and maximum errors of 12 and 39%.296 0.3 34.7 36. B.5 32. Plotted in Fig..6 39. . . and the effect of pressure. 30% Zn) Copper.50~.45fq + 1.11 0..5 0. B-47-Xnthalpies of water and steam a s a function of the saturation pressure (reference temperature is 32"F). . no distinction is made about the fluids other than the brine. In these expressions. =bulk density in grams per cubic centimeter.0. .000 720 820 480 1.10 0.1. . .031 0. and pressure.260 4. pure Brass (70% Cu. .37 and fluids and minerals .1065 0.. ~ . ..200 890 600 240 820 500 940 550 2.0. f is the fractional volume of quartz in the sand.. .500 5.900 Thermal D~ffus~vity a (ft2/D) At 68°F 87. psla Fig. For unconsolidated quartzitic sands saturated with water and oil.1. . . . The data points 4 D l O O ~ l l O 4 m 5 m I O D 7 m Temperature. . (B-107) where k = permeability in millidarcies. . . the thermal conductivity at 125°F is given by corresponding to those in Kern River oil sands. .65(1 . .125)(~. . .73 104.. which is generally minor.104 and B. .. = 0. . . 3 + 0. (B-106) where4 is given by Eq..2 21.113 0.).2 10. .37k0'0 + + 0.f. . 6-&Heat kian et capacity of saturated pure liquids (after Toulou- Fig. B-103. . . Eqs. .96 31.9%) Impure (99. .4%) Wrought (C < 0.. . . .105 are correlations based on samples having porosities between 0. F = formation resistivity factor.860 1. temperature. (B-104) where the thermal conductivity may be estimated from = 4.11 0.800 5.q5 5.735 -1 .65 18.93 Reservoir Rocks and Other Solids.200 960 600 230 890 790 463 1. . pure 01961 Prentre-Hall Inc Thermal Conductivity 1 (BtuID-ft-"F) At 68°F At 212°F At 1.12 B.12~).28 x l O V 3 (.092 0 0915 0.350 11 to 19 1.1 no 460 310 16..2%) Steel. .~%.112"F 2. .% . . The effect of temperature is given by Af(T) = 1 .2 22. 1% C Stainless steel (18 Cr. . including the effects of fluid and mineral content.. "F Saturation Pressure. particle size. 8 Ni) 5 n . pure Nickel Pure (99.106 0. pure Concrete Iron Pure Cast (C . .82). . . is estimated from . . Somerton et and Anand er have reported extensively on the thermal conductivity of reservoir rocks.830 1. for quartzitic sands unless the mineral thermal conductivity can be estimated from its composition.5%) Lead. . .28 and 0.75@ . .8 4. mild. . . . Values of M f are added to the value of at the base pressure.. .21 0. .TABLE B-&THERMOPHYSICAL PROPERTIES OF SELECTED MATERIALS5 Density (I brn/@) At 68°F 169 532 559 119 to 144 493 454 490 710 556 556 487 488 456 P lsobar~c Heat Capac~ty (Btullbm-OF) At 68°F 0. (B-105) Mf = Ap x 10-~(0.054 CP Metals and Alloys Aluminum. B. .214 0.0 15. and p. . ~ Af A..390. .45 to 0.8 16.108 0. . .13 36. . . B.^^ for the thermal conductivities of sandstones at 68°F fully saturated with one liquid.160 0 100 200 300 400 Siltstone Siltstone EShale .30.108.f. The effect of temperature on the thermal conductivity of liquidsaturated sandstones is given by .-800 . . "F Temperature. .220 I 0. . . - 500 F 600 900 1. "F Fig.d refers to dry reservoir rock. -700 -.240 . and m is Archie's cementation factor."68 b 0.000 Temperature. The thermal conductivity of the dry consolidated sandstone at 68OF may be estimated from in 15% for 85% of the values when the ratio ALIAjd within the is range of 1. In this equation. The agreement between values of thermal conductivity reported in the literature and those calculated by Eq.ij 0.180 0. 9-S-Heat capacities of some reservoir rocks (after S~merton~~). B-56 show the typical effects of fluid content and porosity on the thermal conductivity of unconsolidated oil sands. . MULTlPHASE FLOW IN WELLS . 128 Fig.Saturation Temperature. (B-109) . and T is in O R .000psi increase in effective stress.200 0. 1 refers to liquid. . Anand et give the relation Superheat. the subscriptjl refers to the reservoir rock at full liquid saturation. O Fig. .320 0. . a refers to air. . B-48-Enthalpies of water and steam as a function of the saturation temperatures (reference temperature is 32"F).53 in Fig. . Heat added (Btullbm)= mean specific heat x superheat OF). . . For consolidated sands. .300 where AJl is given by Eq. . O F Fig. .20 to 2.280 e d $ 0. B-51--Comparison of calculated heat capacities of reservoir rocks and principal constituents (after SomertonS4).260 & 0. B-108 is said to be with0. B-4SMean specific heat of superheated steam. The effect of stress on the thermal conductivity of consolidated sandstones is to increase the thermal conductivity value by "1 to 2% per 17. Fig.4 Thermal Diffusivity. A similar volumetrically weighted geometric mean may be used to approximate any average value when no better procedure is available.30 9. B-57 shows the measured thermal conductivity of Berea sandstone vs. B-58 shows the effect of temperature on the thermal conductivities. serlc~te.6 15. Such averages must consider the relative volumes of the fluids present in the reservoir rock.665 0.035"F TABLE B-l&DESCRIPTION OF SAMPLES USED IN TABLE 8-1170 Prlnc~palMlnerals Feldspar ("/.0 78.9 8. b~ot~te Hal~te 10 60 Clay. B-58.2 78. or from the values obtained with the equations given previously. For condensing steam. medlum gralned Small vugs.7 16. laminated.833 0.l. 864°F a -p 0.2.660 0.698 1. Repeat values were generally lower because of reactions that occurred in the samples as they were heated to temperatures of 1.8 118. very fine grained Well consol~dated.8 126. the film coefficient of heat transfer.045OF.895 0.5 21. it appears that Eq. the temperature history of a sample may have a significant effect on thermal properties.2 8.200 0 205 0 265 0.950 1.000 ~ t u / f t ~ . very flne grained Crystall~ne Well consolidated.100..0 116. clay a -B quartz 1. sencite Ca C03 50 Clay. oil. large to very fine grained Porosity 0. measured by use of two different techniques. iron oxides. B.816 0.694 0.280 Fig. .134.0 115.16 21. clay CaC03-CaO +COp. .581 0. . med~um-coarse gralned Hard. B-59 shows the effect of temperature on the thermal diffusivity of dry consolidated rocks.53OoF a -B 128.5 26. gralned fine Well consolidated.96 0. B.9 Thermal Reactions a -B quartz 1. is large and generally adequate.497 Thermal Diffusivity Steady State (ft2/~) 90°F 0.624 0.020°F. Thermal diffusivities are related to the thermal conductivity and volumetric heat capacity by the relationship Values calculated in this manner are quite adequate for engineering applications.780 0. andlo and. and gas when the thermal conductivity of the brine-saturated sample is available.8 13. (B. clay 40 35 Clay. calcite Quartz Sample Bandera sandstone Berea sandstone Boise sandstone L~mestone Shale Rock salt Tuffaceous sandstone Description Well consol~dated. Errors in this approach should be recognized as being potentially significant.. Thus.0 107.936 2.the*conductivity of the wetting phase has the dominant effect..2 128. such as Fig. Ac~ cording to Anand et a ~ .8 19. Repeat measurements were always lower because of reactions that occurred as the samples were heated to temperatures of 1.186 0 170 0.5 137. B-107 should be biased accordingly. calculated values actually may be better than directly measured values. B-108 could be used. Table B-9 presents information on the thermal diffusivities of several dry consolidated rock samples at 200°F. where the value oflJ. The liquid conductivity used in Eq. . together with thermal conductivities of dry saturated rocks given in Table B-9 and Fig. Table B-9 presents data on the thermal conductivity of several dry consolidated rock samples. Table B-8 reported the thermal conductivities of some metallic and nonmetallic materials.800°F. .010 0.643 Sample Bandera sandstone Berea sandstone Boise sandstone Limestone CaO (after reaction) Shale Rock salt Tuffaceous sandstone Thermal Conductivity Unsteady State (Btu/D-ft-"Fat 200°F) 275°F Initial Repeat . temperature when the sample is saturated with four different fluids. Tables B-9 and B-10 give the initial sample properties. Thermal conductivities of reservoir rocks containing several fluids are treated differently.660 0. -"~. In the absence of specific data.TABLE B-&THERMAL CHARACTERISTICS OF TEST SAMPLES70 Thermal Diffusivity Bulk Density Unsteady State (Ibmlft3) (ft2/D at 200°F) Initial Repeat Initial Repeat .504 - quartz 1. Fig.7 11. Table B-11 gives thermal conductivities of selected minerals found in reservoir rocks. Because of difficulties in the measurement techniques.2 134.55 13.. B-108 or consolidated samples. B-57. $.7 0. clay a -B quartz 1.9 140.5 Coefficients of Heat Transfer.9.924 0. clay - Dehydration.780 0.1 135. respectively. may be estimated from Eq.444 0.516 0.5 0.~when NRe > 2. Average values may be estimated from figures.30 9.35 25 Calc~te. clay 65 10 Calc~te. pumice lap~lli.113) .69 hJ = 48.& are the thermal conductivities of the oil and gas phases. Fluid flow increases the apparent thermal conductivity of porous rocks.2 0. The following approximate averaging procedure is useful to estimate the value of the thermal conductivity of a porous medium containing water.9.3 131. .015OF.80°F.821 0. calculated from thermal diffusivities and densities measured at 20°F.04OoF. Tables B-9 and B-10 give initial sample properties. to estimate thermal conductivities of saturated consolidated rocks other than sandstones.55 quafiz 1.948 0.492 0.I (%) Other . turbulence prevails (NR.4 266.8 76.4 Ibm/ft3). O F Fig..97 17.7 106... where the Reynolds number.60). and r. at "Value used by Somerton et a166 MULTIPHASE FLOW IN WELLS . is greater than 9.114.4 84. In the temperature range of 350 to 6W°F.5 185.= density of water at standard conditions (62. B-SMeneralized chart for liquid thermal conductivities and effect of pressure (after LenoiPz).0 18 cp.1 178. the film coefficient of heat transfer is found from70 when NR. NR.8 186. B-52-Thermal conductivity of saturated pure liquids (after Touloukian et a1. =steam-equivalent injection rate as condensed water in barrels per day. L.9 Thermal Conductivity (Bt~/ft"F-D) 49.9 178. the value of factors representing the water properties [(1. N R ~is defined as .3 181. =inner radius of the conduit in feet.85ri. the approximate film coefficient is derived as Mineral Calcite Dolomite Siderite Aragonite Anhydrite Gypsum Halite Quartz Chert Pyrite Hornblende Albite Microcline Orthoclase Kaolinite lllite Montrnorillonite Chlorite Biotite Muscovite 165.7 161. where i. R Fig.] does not vary by more than f 10%. For hot water.100) whenever i.2 Temperature.8 159.O 1 65.2 38./pt8)~.8 71. B. all data are from Horaip5and were obtained by use of naturally Occurring m~nerals room temperatures and pressures.Reduced Pressure.4 237.7 34. Because the viscosity of saturated steam over typical temperature ranges is approximately 0.1 'Except tor kaolinlie. 0 0 1 0 0 ~ 3 0 0 4 W s O O ~ 7 0 0 TABLE B-11-THERMAL CONDUCTIVITY OF ROCK-FORMING NATURAL MINERALS'S Density (Ibm/fP) 169..36 28. = viscosity of steam at the injection temperature in centipoise. =viscosity of the hot water in centipoise. Accordingly.6 171. The dimensionless Prandtl number is where &=specific heat of water at constant pressure in Btu/ Ibm-OF andp. > 2.2 159.100.6 51.p. also is given by Eq.35 41. p.0 32.7 3 .9 29.8 198.2 32. thermal conductivity of water in Btulft-D-OF. exceeds 2.49 11 2.3 176. is the .1 38" 30.7 163.8 306.9 176. "F Fig.0165 - 050 Porosity 4 Berea Sandstone Temperature.033 in. B-56-Thermal conductivity of unconsolidated oil sands (after Somerton et Fig. .:. B-57-Thermal conductivity of Berea sanddone-edfects of temperature and fluid saturant (after Anand et W8 N .75 . f 0.Temperature.so). it generally is found that PA4 0. . . . = [n(0. . B-60shows aplot of the results calculated from Eq. - when 350°F<T<6000F. . "F 1 .072 conductivities of gases at 1 atm (after Tou- 3 L . . n - Air Satur ted i f= 0. "F Fig.023 to 0. For gases. .. Fig. . and the subscript sc refers to standard conditions.375 Temperature.00 Brine Saturated I f =0. . is given by n rrrn bwn Accordingly. (B-118) . 1. C.\ _I d x 0. B-115 for gas properties.20t ------A -5.. . the film coefficient is found by evaluating Eq.92 f 0. B-55-Thermal loukian et al. including the con= version factor associated with the oilfield units for gas.01096)/2]-2 P8'-i . . the film coefficient is given by wncu D P ~ D F W T ~ C Q . . Soivenf Saturated I f =0. For hot gases. B-117.07 and the Reynolds number. Fig. > r.. . B-54--Thermal conductivities of gases at 1 atm (after Touloukian et L K r u p i c ~ k1 ~ ~ a = 2. Limestone ..Shale Rocksalt Temperature. IbmID-cp-ft Fig. 6-61--Correlation to estimate film coefficientof heat transfer for gas flow at high ~elocities. Fig. Ibmlft . "F Temperature. WawnOxlde 6 7 8 . thus possibly plugging the injection wellbore or other parts of the flow system. ~ c ~ d a r n s ~ ~ gave the approximate film coefficient as hf..5 132 wglpsl. .. andp. milelhr. ft. scale deposit on the inside wall of the A pipe also would be an asset if there were no danger of the deposit for 1. Often values are specific to the project and seldom available unless measurements are made. Values of the coefficient of heat transfer caused by deposits of scale and dirt.. a significant scale deposit between pipe and insulation is an asset. Mscf/D. cp.Boise ---. and r.p6 . where& = thermal conductivity ofair.~ MULTIPHASE FLOW IN WELLS . Values of hd.LIn-mrtone 5. 8-----Tuff being dislodged. =density of air. where low heat losses are desired.. B-61 shows the values of 9 for hot-gas injection obtained from Eq. of about 200°F. cp.- Banderasmdstone Berea Sandstone ---Shale -.. 'F Fig.viscosity. Bt~lft-D-OF 8.. 4 -.-Tun 2 3 -. Boise Sandstone .000 Btu/ft2-D-OF. < 50. 103 104 BID-A Fig. p =density. Btulft-D-OF..: = Inner radius of the conduit. McAd a d 9 gave the coefficient of heat transfer caused by forced convection (air currents) at the outer surface of a conduit. Ag =thermal conductivity.1 . as d when wg > 192pgri and w q = i c 10' lbrn/D.-Roclc 0. Values of hf. For surface injection lines exposed to winds. In injection and production lines. because it results in a relatively high resistance to heat flow. hd.000. B-59-Thermal diff usivity relative to that at 200°F-samples subjected t6 increasing temperatures (after Somerton and Boozer70).-Berea Sandstone Sandstone . re =external radius of the conduit exposed to air.... fl kSC8 350°F < T c 600°F BID I 1 1 1 1 1 1 1 1 Ly..2 - 1 2 3 4 67---- -Bandera Sandstone -. B-GO-Correlation to estimate film coefficientof heat trane fer for water.. i=injection rate.vw = wind velocity normal to the pipe.. pa. 364)1(2rLvwpaJc/pa). seldom are known with any degree of confidence. 1bm/ft3.000 < NR. hfc. M c ~ d a m used~ typis~ a cal value of 48. Stulft-D-OF.. For the surface temperatures at r. B-119. B-58-Thermal conductivity relative to that at 200°F-samples sub ected to increasingtemperatures (after Somerton and Boozer7 ). = viscosity of air. This value is recommended in the absence of better information..--. NRe= [!h(4. Fig. ft. and Russell. McGraw-Hill Book Co. E. Beal." Trans. Prats. SPE.: "A Comparison of Enthalpy Prediction Methods. 14.C.A. R. New York City (1964)..L.F. 24. L. Engineering Data Book. 3. 37.: "Interfacial Tension Between Water and Oil Under Reservoir Conditions. and Keyes. 25. Texas (1967) 1.: Recovery of Oil in the United Stares.P. Jr.B.: Applied Hydrocarbon Thermodynamics. Gould. H. 23.R. Natural Gas. (1944) 173. R." Trans. Culberson.H.: "Natural Gasoline and the Volatile Hydrocarbons. D.. R. 57. and Danner. McCain." H~drocarbon Processing (March 1976) 153. AIME (1958) 213.K. Properries of Saturated and Superheated Steam. 32." JPT (September 1975) 1140. New York City (1962) 1. Numbere. they generally are applied irrespective of wind direction because there is n o easy way to correct for differences in the wind direction.M. Lee. 17. Crude Oils and Its Associated Gases at Oil Field Temperatures and Pressures. J. McGraw-Hill Book Co.0. 55. W. Richardson. I I." Hydrocarbon Processing (May 1972) 119. Trans.: "A Pressure-Volume-TemperatureCorrelation for Mixtures of California Oils and Gases. M.: Volumetricand Phase Behavior of Oil Field Hydrocarbon Systems.. Richardson. Richardson.F.: Hydrocarbotl Phase Behavior." Trans. Chem.: "Improve Prediction of Enthalpy of Fractions.: "The Viscosity of Natural Gases. D. and Schmidt. California (November 1977) 17. N.R. Texas (1981).J. O. API. K.: "Calculation of T s for Sour Gases." paper SPE 23556 available from SPE.375.P. Rzasa.: J.: "Viscosity of Hydrocarbon Gases Under Pressure.: "Evaluation of Empirically Derived PVT Properties for Gulf of Mexico Crude Oils. Friable. AIME(1957) 210. Touloukian. Vazquez. G.94. Prausnitz. Pet. 12.C.C.. 30. S. third edition. C. Beggs. New York City (1950) 127.L. K. SPE. Katz. Texas (1971) 3. et al. A. B. P. et al. 52.: Oil Recovery by Steam Injection. 49. et al. Y. Hankinson. and Aziz. 19. et 01.C.B.. New York City (1970) Chap. J. Monograph Series. Carr. Wichert.... Monograph Series.R.L.119 and B. and Testing Crude Oil. 27. Frick. Gambill.: "Prediction of Pseudo-Critical Parameters for Mixtures. and Farshad. Intl.D. C. 6. Richardson. Richardson.. J. 27-30 September. Jr." AIChE J. Inc. Jr.H. Ahmed.409. AIME (1951) 192. IFVPlenum. R. T." Trans. 341. 16." J.33.: "Specific Heat-Nonmetallic Solids. E. Oklahoma (1948). Brigham.L.: "The Viscosity of Air.J.W. 16. Moses. Thomas. Keenan. Petroleum Publishing Co. Chemical Engineers' Handbook. Al-Marhoun. T. Prac. P. M.: "Calculation of Z-Factors for Natural Gases Using Equations-of-State. 38. Gas Processors Suppliers Assn. Trans. W.S. API. R.. reproduced in Petroleum Production Handbook. ninth edition.. IFVPlenum. Apparatus and the Water-Methane System. B." paper presented at the 1959 AlChE Meeting. B..000 psia. Edminster. J.D. Connecticut (1940)..N. C. Meehan. 47.). Stewart. M.: "Bubblepoint Pressure Correlation. Tarakad.R." Thermophysical Properties of Matter. Chap. Kobayashi. 26. ninth edition. W.. Tulsa. 28.: "Pressure-Volume-Temperature and Solubility Relations for Natural Gas Water Mixtures. B. W." Chem.S..B.: "Pore-Volume Compressibility of Consolidated.H. Trube. New Orleans. 40.. Baker. SPE. and Katz.S.: "Thermodynamics of Hydrothermal Systems at Elevated Temperatures and Pressures. F.." JPT(June 1980) 968. Reid. Tech. 0..: "Finding SurfaceTension of Hydrocarbon Liquids.H. Dodson. S. M c ~ d a m sWillbite.: "Correlations for Fluid Physical Property Prediction.379. SPE. Houston (I 989).. Chilton (eds.~~ exposed pipes carrying high-temperature fluids. Burkhard. and Standing. Eng.. Tulsa. 3 1. Y." and Allen and ~ o b e r t s ' have provided additional data o n ~ radiation effects.D. R.120 are valid only for wind velocities that are normal t o the pipes. AIME. C. Science (1967) 267. 41. and Poling. Inc. 6." Trans.: The Reservoir Engineering Aspects of Waterflooding.A.C. Texas ( 1962) 11. Richardson. Dallas. D." Hydrocarbon Processing (April 1969) 106. 33." Ind. 35.. 15. 48. Gulf Publishing Co. Producers Publishing Co. Sutton. AIME (1946) 165..N. D.H. Standing. Texas (June 1991). Perry and C. 18. Hocott. van Wingen." Trans. Prac. 13. Although strictly speaking Eqs.: Thermodynamic Properties of Steam. M." Keynote Address presented at the 1998 SPE Annual Technical Conference and Exhibition. 16. AIME (1957) 210. Inc.: "Some Thermal Characteristics of Porous Rocks. New York City (1987). W. AIME ( 1939) 132.57. M. M.I.W.A.H. Tulsa. Z.B. 44.: Thermal Recover): Monograph Series. R. Inc. AS. Farouq Ali. 9." Natural Gasoline Assn. 184.: Handbook of Natural Gas Engineering. and Standing. McGraw-Hill Book Co. fourth edition.: "PVT Correlation for Middle East Crude Oils.H. 2.: "Engineering Applications of Phase Behavior of Crude Oil and Condensate System." Trans. M. AIME (195 1 ) 192. Dranchuk. W. Brown.: "Predict Natural Gas Properties. (1957) 275. and Katz. M. 46.. Earlougher. 269.355.: Correlationsforfhysical Properries of Petroleum Reservoir Brines. Missouri. (2 January 1956) 125. AIME (!942) 146.L. New York City (1959). Cdn..: "Phase Equilibria in Hydrocarbon-water Systems: 111-Solubility of Methane in Water at Pressures to 10." JPT (August 1974) 833." Ame. F. H." SPERE (February 1990) 79. C. 39. 33.E. Kesler. Stundard2500.when the given units are used.H.: "New Correlations for Crude Oil Physical Properties. and Beggs. Inc.. Combustion Engineering Inc. and Wood. Trube. D.. Bradford.. 53." Pet. 45. D. T.: "A Correlation for Water Compressibility. AIME.. G. Standing. R. 269." JPT (July 1986) 715. Gulf Publishing Co. Texas (1982) 7.: "You Can Predict Heat Capacities.223. and McKetta. Kartoatmodjo. 237.: "Surface Tens~on Methane-Propane of Mixtures. & Prod.K. of America. FLUID AND ROCK PROPERTIES 29.: "Vertical Two Phase Steam-Water Flow in Geothermal Wells. Technical Data Book. New York City (1970) Chap. 10. D.L. Glas0." Oil & Gas J.729. (July-September 1975) 14.: Pressure Buildup and Flow Tests in Wells. and Lee." JPT (August 1966) 997: Trans. M. Pennsylvania (1970)." Drill. and Eakin. et al. Water. fifth edition. Standing. and Burrows.. (November 1980) 125. "Compressibility of Natural Gases. 5.: Advances in Well Test Analysis. AIME. and Robinson.: "Interfacial Tensions at Reservoir Pressures and Temperatures.:'The Impact of Hurricanes on Technical Conferences and Exhibitions. 54.35. and Phillips. Weinaug.L." JPT (May 1980) 785. 51.: "Generalized Pressure-Volume-Temperature Correlations.: The Properties of Petroleum Fluids. Windsor..: "Density of Natural Gases. Lasater.G.." Trans.B. Touloukian. SPE.." Thermophysical Properries of Matter. Oklahoma ( 1972) Sec. Sampling. Stanford U. Kansas City.264." JPT (February 1973) 129. (1943) 25. F. 36. 42.B. and Unconsolidated Reservoir Rocks Under Hydrostatic Loading. B.. Measuring. T. Steam Tables. Monograph Series. Eng. M. 50. 20.S. Hough. D. H. N. New York City (1973). 5. John Wiley & Sons Inc. 22. Matthews.M.F. Eng... and Voo. Houston (1961). Radiation effects m a y be important for uninsulated . Gonzalez. D..: The Properties of Gases and Liquids. Richardson. 285." Drill..R. McGraw-Hill Book Co. & Prod.E..G. Texas (1977) 5. .G. 4. New York City (1936).: Petroleum Production Handbook..B.: "Estimating the Viscosity ofcrude Oil Systems. (1976) 22. 34... API. Craig. Palo Alto. J. Somerton." Trans. 21. J. AIME (1953) 201. AIME. References I. 7. SPE. (June 1957). M. 56. 8. Helgeson.. Jr. W." Trans. Adamson. Oklahoma (1973)." JPT (May 1988) 650. A. Truns.L. Petroleum Research Inst.and Swerdloff. AIME (1958) 213. Newman.: "Specific Heat-Nonmetallic Liquids and Gases. 130.: "Compressibility of Undersaturated Hydrocarbon Reservoir Fluids. and Abu-Kassem. 3. Jr. 60. A. U. Inc.L. E.S. Colorado (1966) 97. E. Lenoir.H. (1951) 47. C. Tulsa.: "Thermal Conductivity-Nonmetallic Liquids and Gases. W.: "Effect of Pressure on Thermal Conductivity of Liquids. T.: "Predicting Thermal Conductivities of Formations From Other Known Properties. E.: Heat and Mass Transfer. IFJPlenum." J. J. Eng.-I. of America." Thermophysicul Properties of Matter.S." JPT (May 1967) 607.: 'Thermal Characteristics of Porous Rocks at Elevated Temperatures.M. Lenoir. 97. (1967) 7. New York City (1970) Chap. 5. K.H. Krupiczka.58." Chem. Chem." Intl. 8. Grim. Keese. AlME (1960) 219. 63. Willhite.: 'Thermal Behavior of Unconsolidated Oil Sands. Clark Jr." Perroleurn Refiner (1956) 35. Somenon.O.. New York City (1959). Handbook of Physical Constants. Somerton. 162.. 62.: "Thermal Conductivity of Gases.: 'Thermal Conductivity of Rock-Forming Minerals. 66. and Boozer.F. Prog.A. 67.H. MULTIPHASE FLOW IN WELLS . Somerton. J.H. New York City (1968). Measurements at High Pressure. Inc. No. and Roberts... lnc.P. S.: Clay Mineralog?: McGraw-Hill Book Co. J. G. 64." Trans... R. 1278. McGrawHill Book Co. S. Oklahoma (1978) 1. 122.. R. 72.: Production Operations. McAdams. 61. 70. and Comings. Horai.418. No. New York City (1954). 71.P.223. Cragoe. andChu. 69." SPEJ (October 1974) 513. McCraw-Hill Book Co. J. Touloukian. thirdedition. DC (1929). Inc.S.P.R. Geological Soc. G.G. Allen.: "Overall Heat Transfer Coefficients in Steam and Hot Water Injection Wells. 68. 65.M. Oil & Gas Consultants Intl. revised edition. W. (ed. Anand. Geophysical Research (1971) 76..). Washington." SPEJ (October 1973) 267.: Heat Transmission. Eckert.M. et al.D. and Gomaa. Y. and Drake. Bureau of Standards. W.: "Analysis of Thermal Conductivity in Granular Materials. R. 59..: Miscellaneous Publication No. Eng. Boulder. W.W. . . C-8 can be simplified as Once convergence is obtained for V/F.5 for (V/F).. and replacing y. . Thus. . . ... K... Thus..(C-7) v + g ( K . V=O and 4. The previous procedure requires values for K. . C-7 and either Eq. This imposes a second level of implicitness in Eq. .. several methods exist to determine K values.4 Bubblepoint and Dewpoint Curves C3. ..... . . Convergence is achieved when C. Similarly. C-1 and C . this equation to determine the mole fraction of the ith component in the liquid phase can be written as XI f ' ( g )J V = - 1 z. . .. a bubblepoint pressure is the minimum pressure at which a liquid will not vaporize. . . . . .. . assuming K values can be calculated at each iteration step for V/E A second-order Newton-Raphson convergence scheme4 can be used to solve the Rachford and Rice equation for V/F with a reasonable convergence to~erance. . % x i ... C-10 is computation intensive. require the determination of the V/F value. . . .Ud0ruturat..3 Solution Procedure.. . . . . .. f'(V/F). . (c-5) This equation is often called the Rachford and Rice equation. n .. (C-11) (C-6) where the derivative. It is important to note that the solutions for xi and y. .2 Vapor Phase.. . for instance 0. The recommended way to determine K values. .. .5 .. .. Therefore.1 Liquid Phase.. . At the bubblepoint pressure. .. C-7 and C-9. To solve it. . . is from EOS's.1l2 . . . Given a first guess. .F = x . so are the dependent variables for which a solution is needed. ... .. . based on Eqs.3 ) Individual component balances are z.. For temperatures less than the critical temperature. .2 The overall material balance is F = L + V.. . . C .. .. . Mole fraction of each component in the feed mixture. results in ' 'I . . C-1 or C-9....~ procedure requires that The these variables be known: Number of components. C-3 may be rewritten as A= 1 F . the composition of each phase can be determined from Eq. and U F with Eqs. Equilibrium constant for each component. . . .. . It is implicit in V/F. solving Eq. To solve for V/E Eqs. . the bubblepoint curve separates a single-phase liquid from the two-phase region.9 gives this procedure. . .. Solving Eq. . combining Eqs. . (V/RJ... . Because phase compositions are unknown in flash calculations. . . K values must be known. . = q K. On further simplification.. C. . . C-7 and C-9 are combined by use of this definition of mole fractions. . L + y . . . . (C-4) Eq. As mentioned earlier. . .. . respectively..1) 2.. V . . C-1-Typical phase diagram. a. is obtained by differentiating Eq. . . .. . respectively. Sec. Eq. .... . . Therefore. C-2. C-7 and C-9 gives Temperature Fig. .(K.. such as those given in Fig. C-9 simplifies to y. . .3. . .. C-4 for x. . . .where the subscript j refers to the iteration step count. an improved value of (V/F) can be estimated from C3... A first guess for the mole ratio. temperature. . .. solution of Eq.. .( c . C. ...rcroir Or Therefore. . and composition of each phase. .d O i l t. at the pressure. C-1 and C-5 gives Eq.. . however. . . there is no vapor and all feed is in the saturated-liquid phase. . v . This section presents a solution scheme for the Rachford and Rice equation. including use of charts. Note that K values are themselves phase-composition-dependent. (C-12) = 1 .. . .g .. .4 foryi and using Eqs. . C-10. C-10 with respect to V/F.. .and i 136 MULTIPHASE n o w IN WELLS . = ' 3 K g . .. psia Ethane Conversion Pressure. psia. .) K-value chart. . .p= system pressure. . T.000 psia Fig. Note that. Therefore. . . . depending on whether a bubblepoint or a dewpoint pressure is to be calculated. V = F. . At a dewpoint pressure. OR. Eq. C-2-Example Hill Cos. . two dewpoint pressures can exist. L = 0. at the bubblepoint or wherep. . . . . C. OR. . temperature. . . The wilsons correlation can be used to estimate K values initially. . . (C-15a) and dewpoint pressure. and : = y. Wilson proposed this simplified thermodynamic expression to estimate K values. . C-14 or C-15. . This procedure is iterative in pressure at any temperature below the critical point. . . . .T = system =acentric factor of Component i. . =critical temperature of Component i. .. . At a given temperature. . and o.16) Bubblepoint and dewpoint pressures also are calculated with iterative methods. . 2. (C. . The problem is to determine iteratively the pressure and the K values of each component in the feed at that calculated pressure to satisfy Eq. The known feed composition is also the phase composition. .I separates the single-phase gas and two-phase regions. . psia Pressure. . the hydrocarbon mixture exists in asingle phase. . (After Katz etaL3 Reproduced with permission of the McGraw- The dewpoint curve in Fig. .Pressure. = cntlcal pressure of Component i. . C-7 simplifies to x. psia. and the composition of both phases.1.14 is satisfied within a reasonable tolerance. For methane it is very small.I .7 Equilibrium Constants From EOS's . These K values are substituted into Eq.5 Bubblepoint Pressure For most applications in the multiphase flow of hydrocarbon mixtures in pipes. then the iteration is stopped and the last assumed bubblepoint pressure is accepted as the solution. the fugacity coefficient of any component is defined for vapors as then the assumed bubblepoint pressure is low and the next guess must be a higher pressure. .169374 86 and 8 = 15. the ideal assumptions used in deriving Eq. T. the fugacity coefficient of Component i in each phase is a function of the system pressure. the system pressure is denoted by pb. Such equilibrium depends modynamics of vapor~liquid on the fugacity of each component in each of the phases. C-20. and n = number of components in the system. For monatomic gases. To determine w.15. psia.5. in atmospheres. w is essentially zero. i. p. It is evident from this definition that fugacity has units of pressure. A good starting value for pb can be obtained from Starting with a first guess of bubblepoint pressure from Eq. K values are calculated with EOS's.e. Tc With this initial guess for dewpoint pressure. equal fugacities of a component in each phase implies zero net mass transfer of that component between the phases. the fugacity of a pure component is defined as then the assumed bubblepoint pressure is high and the next guess must be a lower pressure. w. is calculated from Eq. = critical pressure. The important parameters controlling the phase behavior are pressure. given the pressure and temperature at the computation node and the overall mixture composition. a. and Tb =normal boiling point at 1 atm. Otherwise. the iteration is stopped. C.hydrocarbons. = reduced vapor pressure ( =fip/pc) and f i p = vapor pressure.25 18 . In mixtures. the acentric factors also can be estimated from the Edmister8 correlation as logp. If + then the assumed dewpoint pressure is lower than the actual dewpoint pressure. C-16) for 4 . or K values. The ratio of the fugacity to the system pressure is and called the fugacity coefficient. a new bubblepoint pressure is assumed based on these criteria: During the iterative process. Kvalues are determined by use of EOS's.6 Dewpoint Pressure The first guess of dewpoint pressure can be obtained by combining Eqs. the reduced vapor pressure at a reduced temperature T. C-2 are unrealistic.The acentric factor. w represents the acentricity or nonsphericity of a molecule. where 8 = T b 4 = reduced boiling point. the criteria for thermodynamic vaporniquid equilibrium is that the fugacities of every component in each phase must be equal. L = liquid phase. C-15 and C-16 as MULTIPHASE FLOW IN WELLS .In p. for a pure component is defined as6 where pvp. p = system pressure. C. A hydrocarbodliquid mixture at any point in the pipe is in equilibrium with a vapor mixture at the local pressure and temperature.4721 In O + 0. For hypothetical or pseudocomponents. phase composition. These K values then are substituted into Eq. Physically. The acentric factor also can be determined from the Lee and Kesler7 vapor-pressure relationships as (C. a new dewpoint pressure is assumed based on the following criteria.28862 In 8 . O R .167 . psia. if These equilibrium constants can be calculated based on the therequilibria. C. resulting in thermodynamic equilibrium of phases. Otherwise.972 14 + 6.( = Tfl) of 0. For a hydrocarbon/liquid/gas mixture in equilibrium.0. Thus. For higher molecularweight ."R. If where f = fugacity. Acentric factors for pure components can be determined experimentally. If then the assumed dewpoint pressure is higher than the actual dewpoint pressure. If Eq. such as those used to characterize the heavy ends (C7+) in a multicomponent hydrocarbon mixture. or where f = fugacity. thus depend on all these parameters.. w increases.09648 8-I 1. temperature. =critical temperature. and Z = compressibility factor.18) where a = . At the bubblepoint. The iteration is continued until convergence within the pre-set tolerance is achieved.43577 06. Bubblepoint-pressure calculations can be initialized with the Wilson correlation (Eq. V = vapor phase. Fugacity also can be considered as vapor pressure modified to represent the escaping tendency of molecules from one phase into the other. C-26 for pure components.7 is required.6875 8 . An important objective in flash calculations is to determine the individual phase compositions. C-14. The equilibrium constants.13. and temperature.Tb C. Mathematically. C-15 to check if this equation is satisfied within a reasonable tolerance. If it is. and Peng-Robinson (PR). van Redlich-Kwong (RK). and V = molar volume. OR. Thus.8.8. 10. R = gas constant. where A* = ap/R2T2. in terms of compressibility factor. Combining this EOS with the definition of fugacity coefficient presented earlier.TABLE C-1--CONSTANTS FOR FOUR COMMON EOS's (After Reid eta/. and T.8 Cubic EOS's An EOS is an analytical expression relating the pressure to the temperature and molar volume of fluids.I2 These modifications allow the use of SRK and PR EOS's for mixtures of hydrocarbons. psia. and .176w2 Peng-Roblnson 2 -1 0.) Equat~on van der Waals Redl~ch-Kwong Soave L 0 1 w 0 0 b RTc ~ P C 0 08664RTc PC 0.Parameters a and b characterize the molecular properties of the individual components. Secs. where ku = 0. as are the constants for the four EOS's. is the reduced temperature. replacing molar volume. C . the equilibrium constant may be found with The fugacity coefficients for each component in each phase are defined by use of EOS's. ft31rnol. such as volatile oils and gas condensates. This results in the following mathematical criteria at the critical point. Alternative forms of these EOS's.or vapor-phase compressibility factors.)] . Z. from the SRK EOS.1 SRK EOS. are used often to calculate phase compressibility factors and fugacity coefficients. C-32a and C-32b give the alternatives for liquid and vapor phases.42748p pc [ l + b(1 - V)] 8 where ku = 0.T. C-29 and C-30 are integers and are presented in Table C-1. Ideal gas laws assume that the volume of molecules is insignificant compared to the volume of the container. w is the acentric factor of the pure component. the two most widely used in terms of the phase compressibility factors.' Reproduced with permission of the McGraw-Hill Cos.48 + 1. C. The term cubic EOS implies that the volume terms in it can be cubic. Soave-Redlich-Kwong (SRK). respectively.42748RPT2.574~10. T. It is clear from the definition of mixture fugacity coefficient that. Parameters u and w in Eqs. and An alternative from in terms of the compressibility factor. In Table C-1.37464 + 1. Eqs.8.73 psi-ft3Abmmol-OR.07780RTc PC 0.45724@2 PC (1 2 + ku(1 . and that there are no attractive or repulsive forces between the molecules.~ ~ : der Waals. Parameter b is the covolume and reflects the volume of molecules.54226~ 0. There are four well-known cubic E O S ' s ~ . respectively.1 and C. B* = bp/RT: p = system pressure.2 present the SRK and PR EOS's. because at thermodynamic vaporfliquid equilibrium fiv = fiL . Parameters a and b are calculated based on van der ~ a a l sobservations that the critical isotherm has a zero slope and '~ an inflection point at the critical point. T = system temperature. andp.5 PC Jj 2 1 0 0.26992w2 and for liquids as and where @ = fugacity coefficient of Component i in the vapor phase : and @f-= fugacity coefficient of Component i in the liquid phase. Cubic EOS's have been developed that do not use these restrictive assumptions. from the real gas law is v where ZL and Zv= liquid-phase and gas. respectively. Both have been modified with suggested mixing rules recommended by ~ o a v e "and Peng and Robinson. C. A generalized form for a cubic EOS is Soave" and Peng and ~ o b i n s o n used these criteria'to deterl~ mine Parameter a and then modified the resulting expression empirically by use of the acentric factor to reproduce pure hydrocarbon vapor pressures. Parameter a is a measure of the intermolecular attractive forces between the molecules.08664RTc PC 64 a 27 Rq -PC 0. Soavel1 derived these relationships to calculate the fugacity coefficients for each component in the liquid and gas phases. are the critical temperature and critical pressure of the pure component. For these EOS's. The ideal and real gas laws presented in Appendix B are two EOS's. it is possible to apply these equations to both vapor and liquid phases. respectively.. (C-42) dent. fugacity coefficients of each component in each phase. temperature. 4. With appropriate EOS's.. phase compositions required in determining phase physical properthe appropriate EOS can be solved either by direct or iterative meth. C-16 and the known values. y. C-28.45724. C-9 4. ods. The fugacity coeflicients are calculated with Eqs.. C-33a and C-33b.2 PR EOS.constants are highly iterative. T. and fioi =0. 7.. . ' ~ . cf)r.' ~ 2. based on the PR E O S ' s ..5422% .. and the overall system composition.9 Solution Algorithm nents.8. If the convergence criterion in Step 6 is satisfied for all compoC. Using fugacity coefficient ratios from Eq.0..07780.and a 5.. C-43a and C-43b give cubic equations for the each component. C-16 tnal-and-errorprocedure uslng EqS C-10 through C-13 I x. the smallest root for the i= l liquid cubic equations corresponds to that of the liquid phase. n = number of components in the system... K. C. = 0. the values of the equilibrium constants are used to calculare To determine the compressibility factor. = 0. C-28. 4.~ ) and librium constants. w~th C-38 Eq (C-38) Prlnt (C-39) >o 0001 Fig. Eq. The equilibrium constants.... the compositions of the liquid and gas phases obtained from flash calculations can be used to determine the and : . 6. + 140 MULTIPHASE FLOW IN WELLS . 7.. The equi. respectively.v wfth Eqs. The input data required for this calculation are the system pressure. Compare the equilibrium constants in Step 2 with the calculated values in Step 5 using the following convergence criteria: where and & = liquid-phase and gas-phase compressibility factors.. and M) and p. The function f can be cal. liquid and vapor phases. these calculated values are used as the new guesses. the solution procedures for mixture equilibrium q where C. C-3--Algorithm to perform vaporlliquid equilibrium calculations. p."with Eqs.08664. The largest root for the vapor cubic equations corresponds to the comand pressibility factor of the vapor phase.4 values for each component. with Eq. C-33 . C-40 and C-41 Dererm~ne and Zv from 4 approprfateroots of EOS (C-37) Calculate the following #\ wfth Eq C-7 @ i w ~ tEq 6-9 h K. are calculated with Eq.The next nine equations define variables used in Eqs.. C-28. On the basis of Eq. The following is a step-by-step algoculated on the basis of the equation for Soave presented in Table rithm* to calculate the equilibrium constants. Only the C Cj and& values are different.37464 1. Consequently.26992w f . can be calculated from Eq.. I Est~rnate @ values.ties. through C-42. as previously discussed for the SRK EOS..where preassigned convergence tolerance ( 1 1 0 . Cb=0. a. Zv. estimate or assume K.w .. I propenles of each component &. and It is important to note that the compressibility factors used in the fugacity coefficient for each phase are phase-composition-depen. C-10 to perform flash calculations. 3. C-38 and C-39 &. Otherwise.. C-1.16..42748 and Cb = 0. in the liquid or gas phase. Eqs. @. 2. T.. 1.. On the basis of assumed K! values from Eq. the equilibrium constants also are composition-depena i= [1 + fol(l dent. T . For the PR EOS. These equations are cubic equations that yield a single real root and Steps 3 through 6 are repeated until convergence is achieved. w~th C-7 Eq. Thus. in the single-phase region and three real roots in the two-phase region. calculate the equilibrium constants or K c values for each component. these are C.. for C. use Eq. . can be calculated from n ML = I= X . .4.. and + 0.6564296 0.0.TABLE C-2-PARAMETERS FOR V'CORRELATION (After Hankinson and Thomson. independent of any EOS. Hankinson and Thomson proposed this empirical model to calculate molal liquid volume.and phase-behavior predictions.65 Fig. this correction can greatly improve the volumetric. . . W S R K is the acentric factor derived from the SRK EOS.. by methods proposed by Lee and Kesler7 and presented in Sec. .15 Soave" and Graboski and ~ a u b e r tsuggested that these corrections are not necesl~ sary for pure hydrocarbon systems. of Chemical Engineers.T.. . C-46 are usually not sufficiently accurate..346916 .)~'~.05527757 0.43 1 .82 0.1753972 5.1050570 P/o) .25 < T.2717636 0. This correlation was developed for the saturated-liquid densities of pure compounds and their mixtures.3053426 0. (C-55) VAPOR-/LIQUID-PHASE EQUILIBRIUM 141 . The modifications were based on an empirically determined correction factor called the binary interaction coefficient. where. C-49 are calculated with the following mixing rules. .95.. .98 0. ...407302 -0. C-47 is the molal liquid volume. such as H2S and C02....0. In the presence of associated nonhydrocarbons. C.. A better approach is to use the correlation proposed by Hankinson and Thomson.1901563 0. .. Soave" suggested some modifications to calculations of the A and B parameters. the vapor and liquid densities can be determined from and where 0.) where 0. and w are calculated following expressions~ Vs in Eq." It can be calculated. .18 Reproduced with permission of American Inst. M .81446(1 . ...0.2834693 7.2.. Thus. . (C-48) I with If nitrogen andlor hydrogen are present.442388 0. Hankinson and Thomson Correlation.O. and T.0. Liquid densities calculated from Eq.T..89 1. .58 1. .lO. (C-50a) Once the compressibility factors of each phase are determined. .. O I ~ R Kcan be replaced in Eq. . .. .0.02276965 0. ..01379173 0.2828447 b c 0. < 1.. = Ta. 2 T~ = '=I n xd ~ ~ c i .05759377 -0.. . . C.06379110 ..2960998 0.08057958 .2851686 0. C-49 by w from a more general source. . C-3 is a computer flow chart to perform a flash calculation with the algorithm.l. From Eq. .85 3.0. ..190454(1 .0.2368581 -3. C-46. .11 83987 1. .. 10 Densities where V* is the characteristic molal volume of the liquid and and v are two correlating parameters given by : " v:' .OO 0..307061 9 0. These parameters consider molecular interactions. .. .5218098 0.18 The method can be applied to a variety of liquids with sufficient accuracy and reliability. ML. < 0. . The relevant independent variables in Eq.. . the liquid density can be defined as and where where the apparent molecular weight of the liquid.05468500 .) Average Absolute Error Paraffins Olefins and diolefins Cycloparaffins Aromatics All hydrocarbons Sulfur compounds Fluorocarbons Cryogenic liquids Condensable gases a 0.18 C.2905331 0. Mi=molecular weight of the ith component. T. . . Hankinson and Thomson18 provided the values of V * and WSRK for a list of organic and inorganic pure components relevant to hydrocarbon fluids. .391715 .25 < T..1703247 . Alani and ~ennedyl' also developed an equation similar to the van der Waals equation to determine the molal liquid volume. .. .23 1. .5 189. (C-58) C. .. given by where Ci=reduced temperature of Component i... .. . & = pseudocritical pressure of the crude.. . V.and Zipperer-type equations to determine the viscosity of the live oil. ... System pressure and temperature also must be known.. where Mc7+=molecular weight of C7+and yc. PC4 i-C5 PC5 mC6 *C7 PC8 V * = V. . a2 = 0. .5 271 ..1023. V'.. used Herning. > 1.. Surface tension is a measure of imbalance in the interfacial molecular forces for two phases in contact. V * . If observed viscosity data are available. i. a0 a4 = 0.e.~ ( T .. defmed by The viscosity of C7+ used in this correlation can be calculated by any of the dead-oil-viscosity correlations presented in Appendix B.I9 or Lee et There are few methods to determine liquid viscosity based on compositional models. The last resort in determining the characteristic molal volume.?' proposed this equation to calculate VC7+ To determine the viscosity of hydrocarbon liquids. Like any other intensive property. C. ..12 Surface Tension where p. TF = pseudocritical temperature of the crude. ft3/lbm-mol.. The surface tension of a liquid in equilibrium with its own vapor decreases as the temperature of the system decreases.058533.. . lbrn/ft3... and p . =oil density at the prevailing system condition.. .11 Viscosity 351 5 Based on compositional models. . Lohrenz er al.. 4 1. . and n = number of components in the mixture. ft3/lbm. .023364.5 co2 N2 l~ Hankinson and ~ h o m s o nalso presented a correlation to calculate the characteristic volume.... ... viscosity is a function of pressure.0 231. from Eq.. a3 = .. together with the molecular weight and specific gravity of the heptane-plus component. . ~ h m e observed that the Lohrenz et d~ correlation is very sensitive to the density of the oil and the critical molar volume of the C7+ fraction.) Component Parachor 78. . Lohrenz er developed this empirical correlation to determine the viscosity of saturated or live oil from its composition.9 225. a1 =0. =viscosity of Component i at atmospheric pressure and system temperature. For T .23 Reproduced with permission of the American Chemical Soc. and Clark Correlation.0 108 0 150. OR...and TABLE C-3--PARACHORS FOR PURE SUBSTANCES (After Weinaug and Katz.. b. the critical molar volume of the C7+fraction should be adjusted to validate this correlation. the surface tension becomes zero as the phase interface vanishes.1 Lohrenz. wherep. Tffci and pot =live oil viscosity at system temperature and atmospheric press ~cp. This relationship for liquids can be mathematically expressed as Lohrenz et al. . At the critical point. defined mathematiis cally as ti=mixture viscosity parameter of Component i. gas viscosity can be calculated with described in Appendix B.if data are not readily available. Bray. .5.040758. This viscosity 1s calculated from these equations. hydrogen sulfide. ) O 94 6. xC7+ mole fraction of C7+. . . . and c are determined from Table C-2. In multiphase pipe flow. gasfliquid and liquid/liquid surface. and composition of the hydrocarbon fluid. 11. is to equate it to the critical volume. .. . C48. psia. (C-65) and for T... Ps = 34( 10 . . C...or interfacial-tension values are used to determine flow regimes and liquid holdup. where the mixture viscosity parameter. temperature.0 4 0 1 77.3 181. xi =mole fraction of Component i.0093724. . expressed in terms of methane through heptane-plus components.. V. .+ =specific gravity of C7+.0. .. ... There are a few empirical correlations to predict surface tenMULTIPHASE FLOW IN WELLS . nitrogen. = critical volume of Component i. =reduced oil density. VC7+ =critical molar volume of C7+.5. Ma =apparent molecular weight of the liquid mixture. the methods of Cam et al. Mi=molecular weight of Com= ponent i. c1 c 2 c 3 kc4 where a.0 312.. one must know the composition of the liquid. t. = 0. and carbon dioxide. enthalpy at (pl. For example. and temperature. For complex hydrocarbon mixtures. In Eq. p L and pV.3 The surface tension is calculated in dyneslcm. H. p. These variables are very useful thermodynamic properties1 and often can be related to well-operating variables.K). with AO = Helmholtz energy at any reference state or an ideal gas state.sion of pure liquids and their mixtures. and enthalpy for a pure material or for a mixture of constant composition can be written as and . The terms and (@ .is defined as where (PC. They relate any thermodynamic property (enthalpy. C. T = system temperature. =density of the gas phase. T ) . 1bm/ft3.+=parachor of the plus fraction or the hydrocarbon mixture. derive the departure function for Helmholtz energy as (@ . The departure function for enthalpy. y.H p 2 k 2 are called the departure functions (from ideality).h = a temperature-independent parameter called the parachor. TI) and ( p 2 . This relationship can be expressed mathematically as where a = surface tension and P. TI = @ . The parachor of hydrocarbon mixtures can be calculated from z ~ ~ this Weinaug and ~ a t empirical relationship. For pure liquids in equilibrium with their own vapor.AH. T. the enthalpy of a fluid system. xi =mole fraction of Component i in the oil phase.767396(103). and = reference volume. S ~ ~ d suggested~ use of Macleod's e n ~ the relation1 between the surface tension of liquid and the phase densities. the liquid. The entropy departure function.4004065(10-4). S. (Jlmol . V= system volume.MaL =apparent molecular weight of the oil phase.4. T. For example. c. p = system pressure.are expressed in g/cm.13 Thermodynamic Properties1 where 9 =entropy at reference state. in this case) at any state represented by p. where AHis enthalpy departure from the ideal condition calculated from the EOS. The variation of thermodynamic properties between two equilibrium states is independent of the path chosen from one state to the other. The superscript 0 indicates the fluid is an ideal gas. is presented as Note that the SRK EoSIO can be presented as In this section. can be calculated as and where p. With the SRK EOS. in this case) at the same temperature. =density of the oil phase. (Wc+ =molecular weight of the plus fraction. Weinaug and Katz23 presented parachor values of a selected set of pure components.6148734.1. H.and vapor-phase densities. C-68. and a1 through a4 =coefficients with the following values: a1 = . Reid et al.558855. shown in Table C-3. a3 = 3.Hp ]) v where A = Helmholtz energy at any specified state. and other such parameters) in the system change. entropy. Thus.=apparent molecular weight of the gas phase. as H = f ( p . can be represented at any pressure. Katz et aL3 used the Macleod-Sugden equations to calculate the surface tension of liquid. it can be shown that the departure functions for the Helmholtz free energy. T2) the enthalpy change can be represented as where the values of Parameters a and b are presented in Table C. temperature. and n = total number of components in the system. T to a reference state (p = 0.).T1)can be found from Hp 1. a 2 = 2. =mole fraction of Component i in the gas phase. (Jlmol). p. All departure functions can be calculated from an EOS. we present a method based on EOS's to estimate the enthalpy and entropy departure from ideality. Between two states at ( p l . as long as the pressure and temperature at each of the states are known. It is important to estimate thermodynamic-property variations as the other independent variables (pressure. and a4= 3. A parachor is a constant. where the constant-pressure heat capacity. such as temperature change in the fluid. Ibm/ft3. Mu. The Netherlands (1 873). Kobayashi.: Hydrocarbon Phase Behavior. Eng. No.G. et.H.E. D. 2. D. Then." Cdn. G.B.N.L.p). fourth edition. (A . Houston (1989).G.D. W. Leiden." J. and Thomson. Chem. Washington.S.. Inc.. Proc. J. Eng. No.. C.L. 9." JPT (August 1966) 997. D. G.. The Variation of Surface Tension with Temperature and Some Related Functions.: "Volume of Liquid Hydrocarbons at High Temperatures and Pressures. Fugacities of Gaseous Solutions.: 'The Variation of Surface Tension. VI. B. Fund.H.. (1943) 25.B.B. and Robinson.H.: "Applied Hydrocarbon Thermodynamics. 3.: "A New Two Constant Equation of State. Trans. 20.653. New York City (1959).: "A Modified Soave Equation of State for Phase Equilibrium Calculations 1 : Hydrocarbon System.233." J. DC (1980) No. 13. Lee.L.: 'Two and Three Phase Equilibrium Calculations for Coal Gasification and Related Processes.. Peng. H. 6. D. 23." Ind. Gulf Publishing Co. McGraw-Hill Book Co. Slot-Petersen.C. Chem. N. M." AIChE 3." paper 15C presented at the 1968 AIChE Annual Meeting. 14.: "Calculating Viscosities of Reservoir Fluids From Their Compositions.: "Equilibrium Constants From a Modified Redlich-Kwong Equation of State.. AIME. Soave. (1978) 17. An Equation of State.264. References 1. (1972) 27. J. New York City (1969). (1979) 25. Chem. and Robinson.W. Bray. Dew. 4.: The Propcrfies ofGases and Liquids. 8.: "Viscosity of Hydrocarbon Gases Under Pressure. and Kesler. Trans. 11. Lohrenz.I. Peng. A. and Kennedy.S.E. 19.595.35. T h e reference-state molar volume can b e calculated from where the reference-state p r e s s ~ r e .C. Des. 10. and Poling."Ind. Reid. 133.: "A Generalized Thermodynam~cs tion Based on Three-Parameter Corresponding States. Am Chem. Carnahan. Chem.: "The Viscosity of Natural Gases." Petroleum Refiner (1958) 37.. M.A O ) and (S.: "On the Continuity of the Liquid and Gaseous State. Inc. AIME (1960) 219.288." Trans. New York City (1987).59. D.." Chemical Reviews (1949) 44. G. Luther. Symposium Series. Chem. D. 231.~ Z i s the compressibility factor.510.A. Ohio..: "A New Correlation for Saturated Densities of Liquids and Their Mixtures. B. Katz.. 1197. R.It is important to note that both of the departure functions. v/I. (1924) 125. Lee. Sci. Cleveland. J." AIChE J. 22. S. J. 4. Soc. G." PhD dissertation.. (1975) 21. 12..H. 13. R.: "Surface Tension of Methane-Propane Mixtures. Eng. D. Gonzalez. al. B.443. and Burrows. and Robinson...:Handbook ofNatural Gas Engineering. K. Wilson. Alani. van der Waals. Carr. (1955) 77.: "On the Thermodynamics of Solutions." Ind. and Kwong. (1976) 15. Sugden. 237. D. Redlich. C. Graboski..can b e set to unity at any cho~~.T. 0 . J. and Clark. 15. AIME (1954) 201.L. Eng. and Katz. McGraw-Hill Book Co. T. D. M. Sigthoff..: "The Volumetric and Thermodynamic Roperties of Fluids. J. R. Part 4: Compressibility Factors and Equations of State." JPT (October 1964) 1171. sen unit system and R is chosen consistently. Pitzer." Trans. 17. 144 MULTIPHASE FLOW IN WELLS . C.: "A Systematic and Consistent Approach To Determine Binary Interaction Coefficients for the Peng-Robinson Equation of State. B. (1976) 53. John Wiley & Sons. T h e reference state also = can be chosen as the system pressure ofp0 =p.E. American Chemical Soc. AIME. Ahmed." Thermodymmics of Aqueous Systems with Industrial Applications.R. Prausnitz. 4-7 May.32. and Eakin.B.." SPERE (November 1989) 488. and Wilkes. Weinaug.O. B. 5.: 'Two and Three Phase Equilibrium Calculations for Systems Containing Water. 173. 16. Correla7. T. depend o n o r the reference-state molar volume. H." Chem.3427. Peng. Hankinson. Eng. Edmister.: "A Modified Redlich-Kwong EOS Application to General Physical Data Calculations. and Daubert.: AppliedNumericalMethods. 21. 18. Soc. 286 1.40 8.60 5.668 3.286 1.960 198.570 159.00 4.580 7.60 H40 J55 C75 N8O H40 J55 C75 N80 H40 H40 J55 J55 C75 N80 H40 H40 J55 J55 C75 N8O H40 555 C75 N80 H40 H40 J55 J55 C75 C75 C75 N8O N80 N8O PI05 PI05 H40 J55 0.133 0.995 1.460 23/8 2.340 102.25 3.900 % 1.070 7.900 1.020 10.375 2.875 2.30 240 2.660 1660 1 660 1.290 144.90 2.441 2.286 1.20 1.810 TBC NonUpset (Ibf) 6.500 2.668 3.660 1.890 7.450 71.920 5.660 1.160 37.) (In.25 3.750 10.260 9.270 10.300 14.500 3.380 1.910 2.200 2 200 2 200 2.170 41.180 5.90 1% 1.190 0.) T8C Non. The information is taken from Production Operations by T.125 0.145 0.640 7.610 1.30 2.90 2.460 3.254 0.910 2.190 0.800 38.286 1.700 9.76 2.063 2.156 0.75 2.76 3.824 0.955 0955 0.70 5.375 2.375 2 875 2.180 177.901 1 947 1.70 1.500 3.063 3.140 0 125 0.375 2.230 5.286 2. Appendix D i TABLE D-1-TUBING MINIMUM PERFORMANCE PROPERTIES Integral Jotnt Jo~nt Yield Strength Box OD (in.770 7.780 35.30 18.668 3.70 6 50 8.900 1.560 14.40 2.060 10.516 1 516 1.140 0.110 2. Allen and A.313 1313 1.955 1.668 3.920 10.870 19.590 7.460 20.980 43.460 3.960 29.190 8.70 4.500 0.640 14.875 2.WO 1.70 120 1.70 1. Inc.75 2.33 2.970 149.660 1660 1.640 6.995 1.350 11.640 14.900 186.76 2.660 1 660 1660 1.040 39.410 136.217 0.70 3.) 1.330 9.900 1.95 6.730 0.330 7.375 2.P.254 0.217 0.290 24.050 1.550 1.95 4.960 52.751 1..156 0.165 2.259 2.100 72.90 2.063 3.049 1.180 22.516 1.516 1.40 1% 1.070 66.910 2.041 1.500 3.70 1.995 2.710 190.500 3.947 1.600 44.090 7.375 2.50 8.550 5. Oklahoma (1978).060 20.133 0.955 0.40 6.760 27.054 2 054 2 054 2054 2.70 5.910 T8C Upset (Ibf) 13.120 11.410 1.810 4.041 1.780 72.960 Z1/t6 4.516 1.995 2.286 1.340 6.33 2.900 2.) (in.375 2.40 8.10 2.960 15.610 19.901 1.970 14.940 22.657 1 550 1.145 0.167 0.133 0.910 2.050 1.25 3.880 1.890 36.480 99.315 1.867 2.516 1.60 4.049 1.00 4.375 2.110 2.441 1.910 2.00 4.060 5.70 6.680 5.875 2.520 11.260 71.660 1.270 5.070 11.901 1. & Oil Gas Consultants Intl.80 4. ~obkrts.500 2.910 3.500 2 500 19.668 2.875 2.125 0.113 0.750 11.160 15.145 0.875 2.875 2.313 1.460 3.867 2.960 30.500 3.480 11.200 15.900 1.180 30.375 2.441 2.063 3.060 26.730 0.875 2.10 2.063 3063 3 063 3.970 21.657 1 657 1.910 2.76 2.730 56.286 1.660 1.824 1.120 31.890 26.360 14.516 1.370 Integral Joint (Ibf) Threaded and Coupled (T8C) Tubing Size Nomlnal Weight Coupled Outs~de Dlameter Nominal OD (In.910 14.60 5.900 7.140 53.145 0.T 8 C Integral Wall lns~de Dr~fl Non.286 1.441 2.310 138.990 135.000 13.217 0.500 11.880 1.80 7.325 2 325 2.60 6.70 5.040 9.315 1.900 1 900 1.60 4.875 2.130 35.740 11.113 0 113 0.) (tn ) ( ~) n (in ) (in.063 2.875 2 875 2.40 8 60 6.441 2.773 1 901 1 773 2.550 1 550 1.690 10.516 1.570 15.460 .880 1 880 2.875 3.200 14.40 2.090 26.600 6.875 2.930 103.940 60.165 3 500 3.875 C75 C75 N8O N80 PI05 PI05 2.14 1. Tulsa.347 2.970 50.510 15.690 49.347 2.590 4.133 0.430 49.955 0.730 0 730 0955 0.347 2.500 67.140 0.80 6 40 6.313 1.110 2.410 15.867 1.770 50.72 2.910 71.360 29.675 28 1 ' 2.900 1.110 2.875 2 875 2.516 2200 2.280 9.910 2.610 5.970 59.20 120 180 1.500 3.580 12.660 135.049 1.315 1.430 97.80 180 1.360 4.680 10.657 1.480 2. Diameter (Ibflfl) (lbfm) (Ibflfl) Grade (In.780 15.560 209.540 21.and D-2 provide the standard tubing and casing informa1 tion required for pipe-flow calculations.230 10.955 0.308 0 217 0 308 0217 0 308 0824 0.680 10.125 0.33 2.740 36.50 6.200 2.920 12.347 2.500 14.580 98.700 19.530 10.70 4.200 2.947 1.80 1.370 26.063 2.347 2.875 1 2.063 2.259 2.6M) 1.190 0.14 1.I i Tubing and Casing Properties Tables D.200 2.500 41.610 1.00 4.100 9.156 0.140 0.180 31.360 8.490 135.751 1.60 5.460 3.940 31.14 1.840 11.710 52.875 2875 2.156 0.460 3.280 15.190 0.350 10.041 1.14 1.260 260.190 0.960 63.530 21.) (on.167 0.410 1380 1.668 3 460 3.773 1.470 14.010 20.40 2.050 1315 1.25 4.110 2.254 0.50 6.375 2.30 2.400 94.650 1 610 1610 1.380 1.940 26.000 13.041 1.Upset Upset Dr~fl Upset Upset Jo~nt Thickness Diameter D~ameter Upset Regular Spec.824 0.370 0.650 1.900 1.160 5.040 14.30 2.980 11.286 1.668 3.947 1.290 7.120 15.) Internal Collapse Yield Reststance Pressure (PSI) (PSI) 7.920 5.259 2.660 1.360 105.751 2.40 2.820 96.560 126.500 30.660 8.375 2.80 4.710 10.110 2.375 2.516 1.167 0.049 1.063 3.167 0.113 0.165 2.000 13.650 5.50 8.970 36.668 3.420 53.901 1.751 1.280 5.95 4.050 1.75 2.250 35.O.75 2.72 172 172 1.270 10.286 1.325 2325 7.570 6.250 8.380 1.955 1.290 7.880 1.33 2.995 1.063 3. 710 8.950 2.210 1.656 7.844 6.980 234 284 334 313 367 442 519 693 597 797 672 897 746 996 814 1.625 9.460 4.410 14.360 4.470 2.040 12.059 5.000 5.545 5.060 4.470 7.510 6.253 0296 0.892 7.040 4.950 8.400 0.210 7.370 2.335 M I 'ITIPHASF FI OW T WFI I S Y .250 0290 0.430 372 434 417 486 688 788 1.825 4.560 8.00 38.690 9.767 4.390 15.980 7 2.025 6.640 6.415 0 288 0 352 0 417 0.040 2.920 7.350 7.625 2.00 35.340 6.283 4.050 6.550 7.453 0 498 0.110 172 202 229 217 247 348 428 502 445 548 643 65/8 2.796 7.715 7.511 Minimum Collapse Pressure (psi) Coupling OD (in.250 8.440 10.390 7.120 4.186 997 1.) 5.460 9.004 5.270 4.921 5.00 17.160 11.110 2.140 13.653 4.) Nominal We~ght (Ibflft) 9.620 T5I8 315 346 490 575 769 674 901 798 1.400 4.563 5563 5.625 9 625 9 625 9.140 6.780 10.413 6.500 0.231 0.151 6.00 20.500 8.560 4 494 4 408 4.750 6.00 49.050 6. Dimensions ID (In.892 4 778 4 670 6.450 1.810 11.060 12.060 2.040 13.400 8.50 17.380 6.540 10.972 7.310 4.310 2.362 0 244 0.720 2.760 10.500 8.160 14.600 7.352 0.160 9.337 0.328 0.70 33.520 3.270 6.180 13.276 6 184 6.200 245 314 266 340 481 586 677 641 781 904 1.00 44.049 5.275 0.080 5.870 5.825 7.910 4.810 12.540 0.625 9.304 0.00 26 00 29.656 7.087 2.530 3.361 0.550 6.930 4.720 3.580 184 7.362 0.350 NBO Pl10 H40 77 J55 101 154 N80 O P II J55 N ~ O PITO 4% 6.940 130 7.557 Drift (in.430 0.969 5 879 5.00 28.00 24.60 13.520 Mlnlmum Internal Yleld Pressure (PSI) Shon Thread Long Thread OD Sue (an.120 10.320 13.190 12.450 0.791 5 675 6.00 20.304 0.790 5.140 10.020 12.890 3.830 5.924 5.110 4.408 0.240 12.810 8.830 10.656 7.055 887 1.320 7.300 0.500 8.700 7.) 0.270 3.500 0.600 10.264 0.410 6.860 9.826 4.00 1500 18 00 14 00 15.560 5.800 122 176 6.50 15 10 11.969 6.875 6.830 11.510 8.450 3.050 6.00 32.656 7.70 39 00 24.656 7.880 13.740 10.050 6.640 9.640 9.625 7.700 9.500 8.00 32.625 9.010 11.500 7.000 5 000 5.00 40.970 4.094 6.420 162 223 270 279 338 406 5 3.320 3.170 10.280 7.375 0.240 12.460 244 233 279 6.700 8.320 11.50 13.000 5.770 J55 3.00 28.) 3 965 3.TABLE D-1-TUBING MINIMUM PERFORMANCE PROPERTIES (continued) Threaded and Coupled (TBC) Coupled Outs~de D~ameter Integral Jo~nt Jolnt Yleld Strength Wall Grade (~) n lns~de (~) n Dr~ft (~n ) Thickness D~arneterD~arneter TABLE D-2--CASING MINIMUM PERFORMANCE PROPERTIES Jolnt Yield Strength (1.300 10.630 5% 3.276 5.066 85/8 1.625 8 097 8 017 7.960 N80 PI10 H40 3.00 20.700 212 133 169 207 182 223 396 311 495 388 2.220 0.500 9.666 5.00 23.760 6.538 6.205 0.020 6.386 H40 2.040 8.000 3 920 3.960 8.456 6.340 7.) 4 090 4.00 24.450 4.272 0.560 7.921 7.00 36.190 J55 4.640 2.796 5.563 5 563 6.656 7.151 4.570 10.00 26.900 6.230 8.50 11.120 11.860 3.900 10.656 8.050 7.390 7 390 7.390 7.012 4 950 4.331 6 241 6.180 12.140 5.366 6.850 8.750 4.740 4.040 4.100 5.020 8.00 Wall (in.490 13.380 5.875 3 795 3 701 4435 4 369 4.000 IbO .180 5.00 23.656 7.240 9.765 6.795 6.490 7.240 4.00 32.500 7.887 4.400 10.290 11.890 9.475 0.317 0.670 14.220 9.40 29. 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