A Mechanistic Model for Gas/Liquid Flow in Upward Vertical Annuli

Transcription

1 A Mechanistic Model for Gas/Liquid Flow in Upward Vertical Annuli.. Yu, H.-Q. Zhang, and M.X. Li, PE; and. arica, PE, the University of ulsa ummary In this study, a mechanistic model is developed to predict flow patterns, pressure gradient, and liquid holdup for gas/liquid flow in upward vertical annuli. he flow-pattern-transition model consists of a modified Zhang et al. (3a) unified model for dispersedbubble flow and annular-flow-pattern transitions, a aetano (986) model for bubbly-flow transition, and a modified Kaya et al. () model for slug- to churn-flow transition. he hydrodynamic models are developed on the basis of the dynamics of slug flow, and the film zone is used as the control volume. he two liquid films are taken into account in the -slug-flow and annular-flow model developments. he churn-flow model is developed on the basis of the Zhang et al. (3a) unified model for pipe flow by using a much shorter slug length. Introduction As shown in Fig., an is formed by a pipe being located inside a larger pipe. Fluid flows through the area bounded by the outer-pipe (casing) inner wall and the inner-pipe (tubing) outer wall. here are two parameters to define the configuration: pipediameter ratio and the degree of eccentricity. he pipe-diameter ratio is given by K = d d, () where d is the outer diameter of the tubing and d is the inner diameter of the casing. he degree of eccentricity accounts for the displacement of the tubing center from the casing center and is expressed by e = DB ( d d ) () In the petroleum industry, multiphase flow occurs during the production and transportation of oil and gas through horizontal, inclined, or vertical pipes and wells. Gas/liquid two-phase flow in an can be found in a variety of practical situations. In high-rate oil and gas productions, it may be beneficial to flow fluids vertically through the configuration between well tubing and casing. Gas kick is an important phenomenon that is caused by the influx of gas into the wellbore of an oil well. In this situation, gas/liquid two-phase flow occurs in an and may cause a blowout, which could damage the production system. It is necessary to study the two-phase-flow behavior in an to prevent this from happening. Because of the importance of multiphase flow in an, many researchers have conducted studies to model the characteristics of flow. he modeling methodology can be divided into two categories: empirical correlations and mechanistic models. he mechanistic models for flow are flow-pattern dependent, and the models take the fluid physical properties and geometry into account on the basis of the conservation equations. everal flow-pattern-transition models have been developed in recent years [e.g., Kelessidis (986) for upward opyright ociety of Petroleum Engineers his paper (PE 48) was accepted for presentation at the PE Annual echnical onference and Exhibition, ew Orleans, Louisiana, UA, 4 7 October 9, and revised for publication. Original manuscript received 4 July 9. Revised manuscript received 5 January. Paper peer approved 8 March. concentric and eccentric flows based on the aitel et al. (98) model, and Hasan and Kabir (99) for inclined flows based on the drift flux approach]. Very few hydrodynamic models predict liquid holdup and pressure gradient. aetano et al. (99a; 99b) developed a mechanistic model for gas/liquid twophase flows in concentric and fully eccentric annuli. In comparison, many advances have been made in modeling multiphase flow in pipes. ome of the approaches may be applied to flow. he Zhang et al. (3a) unified model first integrates flow-pattern transition and individual hydrodynamic predictions by using the same basic governing equations. Most researchers treated the as pipes, with the hydraulic diameter dh = d d (3) Metin and Ozbayoglu (7) introduced a representative diameter for a fully eccentric : d = d d (4) R From experiments it has been found that the flow patterns in annuli are different from pipe-flow patterns (Fig. ). It is known that flow patterns have a significant effect on the momentum, heat, and mass transfers. herefore, the configuration must be considered in the development of hydrodynamic models for flow. he objective of this study is to theoretically study gas/liquid two-phase flow in upward vertical concentric and fully eccentric annuli. he flow-pattern-transition models and hydrodynamic models for individual flow patterns will be developed to predict liquid holdup and pressure gradient. he model prediction results will be compared with the ulsa University Fluid Flow Projects (UFFP) -flow data to evaluate the model performance. Hydrodynamic Models he new model is developed on the basis of the dynamics of slug flow, and there are five flow patterns considered during the model development: slug flow, annular flow, churn flow, bubbly flow, and dispersed-bubble flow. In this section, the individual hydrodynamic models for these flow patterns are presented. Modeling of lug Flow. In the present study, the slug-flow model is developed on the basis of the approach of the Zhang et al. (3a) unified model. he configuration is considered in the model development. he entire liquid-film zone of the slug unit is used as the control volume that includes casing and tubing liquid films. A schematic of an slug flow is shown in Fig. 3. Mass-onservation Equations. Assuming incompressible flow and no liquid entrainment in the gas core, the input liquid-massflow rate at the lower boundary of the film zone can be written as: L AH LF ( v vf ) L AH ( v vf ), and the output liquid-massflow rate at the upper boundary in the film zone is L AH L ( v v ). Here, H LF and H are the liquid holdups in the casing and tubing films, respectively; H L is the liquid holdup in the slug body and v F and v F are the velocities of casing and tubing films, respectively. he fluid velocity in the slug body is v = vl vg. For steady-state slug flow, the input mass-flow rate is equal to the output mass-flow rate across the liquid films. herefore, the massbalance equations for the liquid and gas phases can be obtained, respectively, as August PE Production & Operations 85

2 d d DB DB oncentric DB= e= Partially Eccentric DB=(d -d )/4 e=.5 Fully Eccentric DB=(d -d )/4 e= Fig. Annular-flow configuration. H ( v v ) = H ( v v ) H ( v v ) (5) L LF F F ( HL )( v v ) = ( HLF H )( v v ), (6) where v is the gas core velocity in the film region. Overall Mass Balances. onsidering the gas and liquid flow in the slug unit and the incompressibility of gas and liquid, the mass-balance equations for liquid and gas can be written, respectively, as l v = l H v l ( H v H v ) (7) U L L F LF F F l v = l ( H ) v l ( H H ) v, (8) U G L F LF where l U is the slug unit length, l is the slug length, and l F is the film length, and lu = l. Momentum Equations. he forces acting on the liquid films include momentum exchange between slug body and liquid films, frictional forces acting at the wall, static pressure difference between lower and upper boundaries, frictional forces acting at the interface, and gravitational force. he momentum equations for casing and tubing films are derived separately. he liquid flowing from the slug body to the aylor-bubble region splits into two parts: casing film and tubing film. For z Bubbly flow Dispersed bubble flow Front Back lug flow hurn flow Annular flow Fig. Flow patterns in upward vertical-concentric- flow (aetano 986). casing film, the momentum input at the lower boundary is L AH LF ( v F v ) v F. he momentum output at the upper boundary is L AH L ( v v ) v. H L is the liquid holdup corresponding to the casing film. For fully developed slug flow, the input massflow rate is equal to the output mass-flow rate and both can be expressed as L AH LF ( v F v ). herefore, the momentum output can be expressed as L AH LF ( v F v ) v. hus, the momentum exchange between casing film and slug body can be obtained as L AH LF ( v F v )( v F v ). he frictional force acting on casing film at the wall is F F l F. he frictional force acting on the interface between casing film and gas pocket is I I l F. All forces acting on the casing film and tubing film should be in balance, and hence the momentum equation for the casing and tubing films can be obtained, respectively, as ( P P) L( vf v )( vf v ) II FF = Lgsin HLF A HLF A (9) ( P P) L( vf v )( vf v ) II FF = Lgsin l l H A H A. F F () imilarly, the momentum equation for the gas pocket can be written as l F H LF v F v v F v H LF v F asing liquid film ( P P) ( v v)( vs v) = II II gsin =. ( H H ) A LF () H L H H L ontrol volume Eqs. 9 and and Eqs. and yield the combined momentum equations for the casing- and the tubing-film flows, respectively: ubing liquid form ( v v )( v v ) L v v v v F F ( )( ) FF II HLF A ( HLF H ) A II ( ) sin H A ( H H ) A g = L LF LF Fig. 3 ontrol volume used in slug-flow model () 86 August PE Production & Operations

3 ( v v )( v v ) L v v v v F F ( )( ) FF II H A ( HLF H ) A II ( ) sin. H A ( H H ) A g = L LF (3) ranslational Velocity of aylor Bubble. Hasan and Kabir (99) proposed an equation for aylor-bubble drift velocity in upward annuli: v =. 345(. K) gd ( )/ (4) D L G L he translational velocity can be expressed as v = v vd (5) he coefficient is considered to be the ratio of the maximum to the mean velocity of a fully developed velocity profile, and it varies with flow state. According to icklin (96), Bendiksen (984), and Zhang et al. (3a), is. in laminar flow and.3 in turbulent flow. In the transition region (, < Re < 4,) between laminar and turbulent flows, is given by =.. 7 (, )/, (6) R e lug Length. It has been proposed that slug length is related to pipe diameter, but the closure relationship for slug length varies with different models or correlations. According to aitel et al. (98) and Barnea and Brauner (985), the slug length for vertical pipes can be estimated by applying the representative-diameter concept in slug flow: l = 6 d (7) R lug Liquid Holdup. Zhang et al. (3b) developed a mechanistic model to predict slug liquid holdup based on the balance between turbulent kinetic energy of the liquid phase and the surface free energy of dispersed gas bubbles in the slug body. his model can be used in slug flow by using the representative diameter. hear tresses. he shear stresses in the combined momentum equations are calculated as = f v F F L F / (8) = f ( v v ) v v / (9) I I F F Friction Factors. he Andritsos and Hanratty (987) correlation modified by Zhang et al. (3a) is used to calculate the interfacial friction factors for casing and tubing films. he wallfriction factors are calculated with f = n Re, () where = 6, n =, for Re,; and =.46, n =., for Re 3,. he discontinuity of the friction factor in the transition region between laminar flow and turbulent flow is addressed by interpolation. he Reynolds numbers for the casing and tubing films and the gas core are defined, respectively, as Re = 4A v F F FL /( F I )/ L, () Re = 4A v F F FL /( F I )/ L, () Re = 4A v /( )/ (3) G I I L Pressure Gradient. he average pressure gradient is calculated across the entire slug unit: dp dp dp = LF L l l U (4) he pressure gradients for the liquid-film zone and the liquidslug zone include gravitational losses and frictional losses. acceleration loss is considered. he frictional pressure gradient includes three parts: the casing and tubing liquid-film friction terms and the gas-core friction term: dp fflvf Lg F = A LF F f v A F L F (5) dp f g vg vl = ( ) (6) d L R Modeling of Annular Flow. For annular flow, the momentum equations for casing and tubing flows can be obtained by removing the momentum-exchange terms from Eqs. and 3: I I II ( HLF H ) A H A ( HLF H FF ( L) gsin = H A ) A (7) I I II ( HLF H ) A HLF A ( HLF H ) A FF ( L) gsin =. H A LF (8) he relationships between superficial velocities and the local fluid velocities are vl = HLFvF H vf HLv (9) v = ( H H H ) v, (3) G LF L where H L is the liquid holdup in the gas core. he continuity equation in the gas core can be expressed as v ( H ) = v F v (3) LF G E L Wall and Interfacial Perimeters. On the basis of the geometry and the assumption of uniform film thickness, the perimeters of the casing and tubing films are given, respectively, as = ( d ) (3) I = ( d ) (33) I he wetted-wall perimeter for the casing and tubing films, respectively, is F F = d (34) = d (35) he hydraulic diameter for the casing and tubing films and for the gas core are given, respectively, as ( ) d = 4 A, (36) F F F I ( ) d = 4 A, (37) F F F I August PE Production & Operations 87

4 ( ) d = 4 A (38) I I Wall Friction Factor. he wall friction factor for annular flow is calculated by the aetano (986) correlation, which takes the configuration into account. Interfacial Friction Factor for Annular Flow. everal correlations of interfacial friction factor have been tried in the present study, and the selection is the Asali (984) equation improved by Ambrosini et al. (99). asing/ubing Liquid-Holdup Ratio. he casing/tubing liquidfilm-holdup ratio is derived on the basis of the aetano (986) liquid-film-holdup equation and liquid-film-thickness ratio: H H LF = K (39) Liquid-Entrainment Fraction. he Wallis (969) correlation is used to calculate the liquid entrainment in the gas core of annular flow. Pressure Gradient. he pressure gradient of annular flow includes frictional pressure gradient and gravitational pressure gradient. acceleration term is considered. he frictional and gravitational pressure gradients are given, respectively, as dp fflvf f v F F = A A F F L F (4) dp LHLF HLF g = ( ( )) (4) G Modeling of hurn Flow. Annulus churn flow is similar to slug flow, but more chaotic, frothy, and disordered. he liquid bridging the pipe is also much less compared to slug flow. his is because of the higher gas void fraction in the liquid slug that breaks the continuity of the liquid between successive aylor bubbles. As this happens, the slug collapses, falls back, and interacts with the following slug. here is no available mechanistic model to predict the hydrodynamic behaviors of churn flow in pipes. Most researchers apply the slug-flow model in churn flow without any modifications. In this study, a hydrodynamic model for churn flow is proposed on the basis of the slug-flow model in the Zhang et al. (3a) unified model with necessary modifications. Only one liquid film is considered by using the representative diameter. Different slug liquid-holdup equations and slug lengths are used in the churn-flow model. he combined momentum equation for liquid film and gas pocket is given by L( vf v)( vf v) ( v v)( v v) FF HLF A ( HLF ) A II H A ( H ) A ( L ) gsin =. LF LF (4) aitel et al. (98) treated churn flow as the entry region of slug flow. hurn flow was considered to have short liquid slugs and aylor bubbles. In this study, the slug length for churn flow in an upward is set as l s = 4. d (43) R lug Liquid Holdup. everal slug-liquid-holdup equations have been tried in the present study. he Gregory et al. (978) slug liquid holdup gives the best prediction, and it is used in the churn-flow model. lug ranslational Velocity. he slug translational velocity of the liquid slugs in churn flow can be expressed in the form proposed by icklin (96) and Bendiksen (984). Modeling of Bubble Flow. Bubble flow includes both dispersedbubble flow and bubbly flow. For dispersed-bubble flow, the gas and liquid phases are assumed to be homogeneously mixed. Liquid holdup and pressure gradient are calculated on the basis of this assumption. For bubbly flow, slippage between liquid and gas phase is considered, and the aitel et al. (98) model (i.e., transition from bubble to slug flow) constitutes a rudimentary model for bubble flow. Flow-Pattern-ransition Models ransition to Annular flow. he annular-flow-pattern transition model is based on the Zhang et al. (3a) unified model. he transition to annular flow occurs when the film length becomes infinitely long, which makes the momentum-exchange term in the slug-flow momentum equation zero. Given the superficial liquid velocity v L, the superficial gas velocity v G can be calculated through an iteration process. Guessing v G and the casing-film velocity v F, and on the basis of the known variables, the film liquid holdup H LF and the gas-core velocity v can be calculated. hese parameters can be used to solve the combined momentum equations for annular flow to calculate new casing- and tubing-liquidfilm velocities. he two liquid-film velocities are used to calculate the new gas-core velocity and the new gas superficial velocity: H LF ( HL ( v v ) vl)( vg vlfe ) = vv LF / vv (44) G E ( ) (45) v = v v H v H H L F LF F L v = v (. H ) v F (46) G LF L E everal iterations are required to reach an accurate value of v G, and the curve of v G vs. v L is used as the boundary between churn flow and annular flow. ransition to Dispersed-Bubble Flow. At high gas-flow rates, the Zhang et al. (3b) mechanistic model for transition between slug flow and dispersed-bubble flow is used. At lower gas-flow rates, the Barnea (987) model is used as the transition boundary. lug Flow to Bubbly Flow. he bubbly- to slug-flow-pattern transition model is developed on the basis of the aetano (986) model. According to the experiments of aetano (986), the average value of the void fraction is. at the bubbly- to slug-flow-pattern transition for flow in a concentric. In a fully eccentric, the critical void-fraction value becomes.5. he lower value of the void fraction in a fully eccentric is because of the migration of gas bubbles into the wide region of the cross-section area that creates a higher local void fraction. he bubble shape might be another reason for a different critical void fraction. he bubbles in concentric- and fully eccentric- flows are not the same, and this affects the aylor-bubble rise velocity. On the basis of the critical void fraction, the flow-pattern-transition criteria are given as relations between superficial gas and liquid velocities for a concentric and a fully eccentric : 5. vg =. 4vL. 36 ( L G) g L (47) 5. vg =. 8vL. 36 ( L G) g L (48) lug Flow to hurn Flow. he Kaya et al. () model is used for the slug- to churn-flow-pattern transition that was developed on the basis of the drift flux approach. his approach was first proposed by engesdal et al. (999) and then modified by Kaya et al. (). he global void fraction in a slug unit can be expressed as = v G. vm ( 35. sin / (49) 54. cos ) ( g( L g) d / L) 88 August PE Production & Operations

5 v L, m/s. d v L, m/s v G, m/s Dispersed bubble lug Annular Bubble hurn ew model prediction Fig. 4 Flow-pattern map for air/water flow in upward vertical concentric. Different void fractions have been used as the slug- to churnflow-pattern transition criteria. hokshi (994) found that the transition from slug to churn flow occurred when the void fraction was.8, while Garber and Varanasi (997) introduced another value of.73. Kaya et al. () used the value of.78 as the transition gas void fraction. In the present study,.64 is used as the void fraction for the transition from slug flow to churn flow in gas/liquid flows. Flow-Pattern Map. he experimental flow-pattern maps obtained for air/water and air/kerosene flows in a concentric are given in Figs. 4 and 5, respectively. Fig. 6 shows the flow-pattern map for air/water flow in a fully eccentric. Fig. 4 also shows that the predictions of flow-pattern-transition models generally agree with the experimental results. However, there is a discrepancy for the slug- to churn-flow-pattern transition at high superficial liquid velocities. It appears that the transition to annular flow in experiments happens at lower v G than model prediction. v L, m/s..... v G, m/s Dispersed bubble lug hurn Bubble Annular ew model prediction Fig. 6 Flow-pattern map for air/water flow in upward vertical fully-eccentric.... v G, m/s Dispersed bubble Bubble lug hurn Annular ew model prediction Fig. 5 Flow-pattern map for air/kerosene flow in upward vertical concentric. he model predicts higher superficial liquid velocity for dispersed- to bubble-flow-pattern transition in concentric- flows. Fig. 6 shows that the flow-pattern-transition model performs well for air/water flow in a fully eccentric. olution Procedure Fig. 7 shows the overall solution flow chart for the present model. Flow pattern is first determined on the basis of the input variables, and then the hydrodynamic parameters are calculated using Input: d, d, d, v L, v G, ε, θ, ρ L, ρ g, μ L, μ g, σ ingle phase gas or liquid? Dispersed bubble flow? Bubbly flow? Annular flow? hurn flow? lug flow calculation ingle phase calculation Dispersed bubble flow calculation Bubbly flow calculation Annular flow calculation hurn flow calculation Output: Flow pattern, dp/, H L, Fig. 7 Overall flow chart for present model. August PE Production & Operations 89

6 Input variables to calculate v, l, H L, and guess a value of l F Input variables to calculate F E, guess a value for H LF alculate v, H L, H LF, and H alculate v F, H LF, H LF, H and v alculate v F from Eq. (5). heck whether v F and v F satisfy Eq. () and doiteration Guess a value for v F and calculate v F from Eq. (5). heck whether v F and v F satisfy Eq. (7) and do iteration. ew value of l F calculated from Eq. (3) ew H LF calculated from Eqs. (8) l Fnew l Fnew.? l Fnew Output the respective hydrodynamic models. Fig. 8 shows the solution flow chart for slug flow. he slug-flow characteristics and liquid holdup pressure gradient are calculated through this program. he unknown variables in the slug-flow model include v F, v F, H LF, H, H LF, and v. Fig. 9 shows the flow chart for annular flow. he annular-flow liquid holdup and pressure gradient and geometrical parameters are calculated through this program. he flow charts for churn flow, bubbly flow, and dispersed-bubble flow are not provided in the paper. Model Performance he experimental data of aetano (986) are the only data available to be used in the model evaluations. he test section was 6 m long with a 3-in. casing inner diameter (d ) and a.66-in. tubing outer diameter (d ). he fluids were air/water and air/kerosene. According to the eccentricity of annuli and fluid combinations used in the experiments, the experimental data are grouped as air/water in concentric, air/water in the fully eccentric, and air/kerosene in the concentric. Each set of experimental data is compared with the new-model calculation corresponding to the same flow condition. omparison riteria. tatistical parameters are used to examine the model performance against the aetano (986) experimental data on the basis of relative error and actual error, ei = ( V i V i ) V i cal exp / exp (5) e j = ( Vcalj Vexp j) (5) V exp represents the experimental measurement, and V cal represents the model prediction. ix statistical parameters are defined in the following. Average relative error, ε = ( e i ) (5) i= Absolute average relative error, Fig. 8 Flow chart for slug-flow calculation. ε = ( e i ) (53) i= tandard deviation about the average relative error, ε3 = ε ( ei ) /( ) (54) Average actual error, i ε 4 = ( e j ) (55) j= Absolute average actual error, ε 5 = ( e j ) (56) j= tandard deviation about the average actual error: ε6 = ( ej ε4) j H LFnewH LFold.? H LFold Output refers to the total number of data points. ( ) (57) Overall Model Evaluation Results. he overall evaluation results of the present model against the aetano (986) data for liquid holdup and pressure gradients are shown in able, in which wc, we, and kc refer to the flow conditions of air/water flow in a concentric, air/water flow in fully eccentric, and air/ kerosene flow in concentric, respectively. aetano s model performance is also shown in able. Because of the convergence problem in aetano s computer code, only part of the experimental data is compared (7 out of 67 data for air/water flows in a concentric, 64 out of 77 data points for air/kerosene flows in a concentric, and 47 out of 99 data points for air/water flows in an eccentric ). he present model gives much smaller values than the aetano model for all the six statistical parameters for liquid holdup and pressure gradient. Fig. 9 Flow chart for annular-flow calculation. 9 August PE Production & Operations

7 ABLE PREE MODEL AD AEAO OVERALL EVALUAIO WIH LIQUID- HOLDUP AD PREURE-GRADIE MEAUREME omparisons tatistical Parameters H L ε ε ε 3 ε 4 ( ) ε 5 ( ) ε6 ( ) H Lnew_wc H Lcatano_wc H Lnew_kc H Lcaetano_kc H Lnew_we H Lcaetano_we dp/ ε ε ε 3 ε 4 (Pa/m) ε 5 (Pa/m) ε6 (Pa/m) dp/ new_wc dp/ caetano_wc dp/ new_kc dp/ caetano_kc dp/ new_we dp/ caetano_we ,53.3,.8 Figs. through 5 show the comparisons of the aetano (986) experimental measurement of liquid holdups and pressure gradient with the values predicted by the present model and by aetano s model. For all three flow conditions, the present model gives better predictions. aetano s model gives more-scattered predictions for liquid holdup and reasonable predictions for pressure gradient based on limited converged calculations. he big error points predicted by aetano s model are mostly in churn flow. Performance of Individual Hydrodynamic Models. he present hydrodynamic models for slug flow, churn flow, annular flow, dispersed-bubble flow, and bubble flow are evaluated separately against the aetano (986) experimental data. ables through 5 show the evaluation results of liquid holdup and pressure gradients for slug-flow, churn-flow, bubbly-flow, and dispersed-bubble-flow models, respectively. hese individual hydrodynamic models give good predictions of liquid holdup and.8 oncentric Annulus % oncentric Annulus % HLcal.6.4 % dp/ cal, Pa/m % dp/ exp,pa/m Fig. Present-model predictions vs. measured liquid holdups and pressure gradients for air/water flow in concentric..8 oncentric Annulus % oncentric Annulus % H Lcal.6.4 % dp/ cal, Pa/m % dp/ exp, Pa/m Fig. aetano-model predictions vs. measured liquid holdups and pressure gradients for air/water flow in concentric. August PE Production & Operations 9

8 .8 Air-Kerosene oncentric Annulus % 8 Air-Kerosene oncentric Annulus % H Lcal.6.4 % dp/ cal, Pa/m 6 4 % dp/ exp, Pa/m Fig. Present-model predictions vs. measured liquid holdups and pressure gradients for air/kerosene flow in concentric. HLcal Air-Kerosene Air-Kerosene oncentric Annulus oncentric Annulus % 8 % % dp/ cal, Pa/m 6 4 % dp/ exp, Pa/m Fig. 3 aetano-model predictions vs. measured liquid holdups and pressure gradients for air/kerosene flow in concentric. pressure gradient reflected by the small values of the statistical parameters. he errors are bigger in the bubbly-flow pressure gradient of air/water flow in a fully eccentric. his might be caused by the experimental errors mentioned by aetano (986). Part of the data was obtained at very low superficial velocities, and the flow was unstable under this condition because of the heading phenomenon. able 6 shows the evaluation results for annular flow. he annular-flow model does not perform as well as the models for other flow patterns. his may be related to the uncertainties in the experimental measurements of annular flow and the strong dependence of the model on the closure relationships, such as interfacial-friction factor and liquid-entrainment fraction, which need to be improved. onclusions A mechanistic model has been developed for gas/liquid flow in upward vertical annuli on the basis of the dynamics of slug flow. he model considers five flow patterns: slug flow, annular flow, churn flow, bubbly flow, and dispersed-bubble flow. he mechanistic H Lcal Fully Eccentric Annulus % % dp/ cal, Pa/m Fully Eccentric Annulus % % dp/ exp, Pa/m Fig. 4 Present-model predictions vs. measured liquid holdups and pressure gradients for air/water flow in fully eccentric. 9 August PE Production & Operations

9 H Lcal Fully Eccentric Annulus % % dp/ cal, Pa/m Fully Eccentric Annulus % % dp/ exp, Pa/m Fig. 5 aetano-model predictions vs. measured liquid holdups and pressure gradients for air/water flow in fully eccentric. ABLE LUG-FLOW-MODEL EVALUAIO WIH LIQUID-HOLDUP AD PREURE- GRADIE MEAUREME omparisons tatistical Parameters H L ε ε ε 3 ε 4 ( ) ε 5 ( ) ε6 ( ) H Lnew_wc H Lnew_kc H Lnew_we dp/ ε ε ε ε 4 (Pa/m) ε5 (Pa/m) ε 6 (Pa/m) 3 dp/ new_wc dp/ new_kc dp/ new_we ABLE 3 HUR-FLOW-MODEL EVALUAIO WIH LIQUID-HOLDUP AD PREURE- GRADIE MEAUREME omparisons tatistical Parameters H L ε ε ε 3 ε 4 ( ) ε 5( ) ε6( ) H Lnew_wc H Lnew_kc H Lnew_we dp/ ε ε ε 3 ε 4 (Pa/m) ε5 (Pa/m) ε 6 (Pa/m) dp/ new_wc dp/ new_kc dp/ new_we ABLE 4 BUBBLY-FLOW-MODEL EVALUAIO WIH LIQUID-HOLDUP AD PREURE- GRADIE MEAUREME omparisons H L tatistical Parameters ε ε ε 3 ε 4 ( ) ε 5( ) ε6( ) H Lnew_wc E-8..7 H Lnew_kc H Lnew_we dp/ ε ε ε 3 ε 4 (Pa/m) ε5 (Pa/m) ε 6 (Pa/m) dp/ new_wc dp/ new_kc dp/ new_we ,6.3,7. August PE Production & Operations 93

10 ABLE 5 DIPERED-BUBBLE-FLOW-MODEL EVALUAIO WIH LIQUID-HOLDUP AD PREURE-GRADIE MEAUREME omparisons tatistical Parameters H L ε ε ε 3 ε 4 ( ) ε 5( ) ε6( ) H Lnew_wc H Lnew_kc H Lnew_we dp/ ε ε ε 3 ε 4 (Pa/m) ε5 (Pa/m) ε 6 (Pa/m) dp/ new_wc dp/ new_kc dp/ new_we 7.3E ABLE 6 AULAR-FLOW-MODEL EVALUAIO WIH LIQUID-HOLDUP AD PREURE- GRADIE MEAUREME omparisons H L tatistical Parameters ε ε ε 3 ε 4 ( ) ε 5( ) ε6( ) H Lnew_wc H Lnew_kc H Lnew_we dp/ ε ε ε 3 ε 4 (Pa/m) ε5 (Pa/m) ε 6 (Pa/m) dp/ new_wc dp/ new_kc dp/ new_we model includes flow-pattern-transition models and hydrodynamic models for individual flow patterns. he hydrodynamic models for slug flow and annular flow are developed by considering two liquid films and by taking configuration into account. he new churn-flow model is developed on the basis of the Zhang et al. (3a) unified modeling approach for slug flow. he flow-pattern-transition models and hydrodynamic models are evaluated with the aetano (986) experimental data. he flow-pattern-transition model performs well for air/water flow in concentric and fully eccentric annuli. he individual hydrodynamic models for slug flow, churn flow, and dispersed-bubble flow give good predictions of liquid holdup and pressure gradient. he annular-flow model gives reasonable predictions, but it does not perform as well as the other flow-pattern models. he annular-flow model needs to be improved further with respect to the closure relationships and needs to be evaluated with additional experimental results. Eccentricity is not used as an input parameter in the present model. However, when compared with experimental results of air/ water flows in both concentric and fully eccentric annuli, similar performances are observed for flow-pattern, liquid-holdup, and pressure-gradient predictions. his indicates that the eccentricity effect is not significant on the multiphase-flow hydrodynamics. In comparison with the aetano (986) model, the overall performance and stability of the present model are much better, especially for liquid-holdup prediction. menclature A = cross-section area = constant D = pipe diameter DB = distance between centers of casing and tubing in an dd = eccentricity f = Fanning friction factor F E = liquid entrainment g = acceleration of gravity H L = liquid holdup K = pipe-diameter ratio l = length Re = Reynolds number p/ = pressure gradient P = pressure = perimeter v = velocity Greek Letters = gas void fraction = viscosity = pipe-inclination angle = density = shear stress ε = statistical parameter ubscripts = casing, gas core, critical A = concentric aetano_wc = aetano model for air/water flow in concentric aetano_we = aetano model for air-water flow in fully eccentric aetano_kc = aetano model for air-kerosene flow in concentric cal = calculation or prediction D = drift exp = experimental F = film F = casing film F = tubing film G = gas G = gas casing G = gas tubing H = hydraulic I = interfacial 94 August PE Production & Operations

11 I = casing interfacial I = tubing interfacial L = liquid L = liquid holdup in gas core LF = liquid film LF = casing-liquid-film holdup = tubing-liquid-film holdup L = slug liquid holdup L = slug liquid holdup corresponding to casing liquid film M = mixture new_wc = present model for air/water flow in concentric new_we = present model for air/water flow in fully eccentric new_kc = present model for air/kerosene flow in concentric R = representative diameter = slug, surface L = superficial liquid G = superficial gas = tubing, translational, turbulent U = slug unit WG = gas wall Acknowledgment he authors wish to thank the UFFP member companies for supporting this research project. References Ambrosini, W., Andreussi, P., and Azzopardi, B.J. 99. A physically based correlation for drop size in annular flow. Int. J. Multiphase Flow 7 (4): doi:.6/3-93(9) Andritsos,. and Hanratty,.J Influence of interfacial waves in stratified gas-liquid flows. AIhE J. 33 (3): doi:./ aic Asali, J Entrainment in Vertical Gas-Liquid Annular Flow. PhD dissertation, University of Illinois, Urbana-hampaign, Illinois. Barnea, D A unified model for predicting flow-pattern transitions for the whole range of pipe inclinations. Int. J. Multiphase Flow 3 ():. doi:.6/3-93(87)9-4. Barnea, D. and Brauner, Holdup of the liquid slug in two phase intermittent flow. Int. J. Multiphase Flow (): doi:.6/ 3-93(85)94-7. Bendiksen, K.H An experimental investigation of the motion of long bubbles in inclined tubes. Int. J. Multiphase Flow (4): doi:.6/3-93(84)957-. aetano, E.F Upward vertical two-phase flow through an. PhD dissertation, University of ulsa, ulsa, Oklahoma. aetano, E.F., hoham, O., and Brill, J.P. 99a. Upward Vertical wo- Phase Flow hrough an Annulus Part I: ingle-phase Friction Factor, aylor Bubble Rise Velocity, and Flow Pattern Prediction. J. Energy Resour. echnol. 4 (): 3. doi:.5/ aetano, E.F., hoham, O., and Brill, J.P. 99b. Upward Vertical wo- Phase Flow hrough an Annulus Part II: Modeling Bubble, lug, and Annular Flow. J. Energy Resour. echnol. 4 (): 4 3. doi:.5/ hokshi, R Prediction of Pressure Drop and Liquid Holdup in Vertical wo-phase Flow hrough Large Diameter ubing. PhD dissertation, University of ulsa, ulsa, Oklahoma (August 994). Garber, J.D. and Varanasi,.R Modeling of n-annular Flow in Gas ondensate Wells. Paper 9765 presented at the AE International ORROIO 97 5nd Annual onference and Exposition, ew Orleans, 4 March. Gregory, G.A., icholson, M.K., and Aziz, K orrelation of the liquid volume fraction in the slug for horizontal gas-liquid slug flow. Int. J. Multiphase Flow 4 (): doi:.6/3-93(78)93-x. Hasan, A.R. and Kabir, wo-phase flow in vertical and inclined annuli. Int. J. Multiphase Flow 8 (): doi:.6/3-93(9)989-y. Kaya, A.., arica,., and Brill, J.P.. Mechanistic Modeling of wo- Phase Flow in Deviated Wells. PE J. 6 (3): PE-7998-PA. doi:.8/7998-pa. Kelessidis, V Vertical upward gas-liquid flow in concentric and eccentric annuli. PhD dissertation, University of Houston, Houston, exas. Metin,.O. and Ozbayoglu, M.E. 7. Analysis of two-phase fluid flow through fully eccentric horizontal annuli. Presented at the 3th International onference on Multiphase Production echnology 7, Edinburgh, UK, 3 5 June. icklin, D.J. 96. wo-phase bubble flow. hemical Engineering cience 7 (9): doi:.6/9-59(6)857-. Oliemans, R.V.A., Pots, B.F.M., and rompé, Modelling of annular dispersed two-phase flow in vertical pipes. Int. J. Multiphase Flow (5): doi:.6/3-93(86) aitel, Y., Barnea, D., and Duckler, A.E. 98. Modelling flow pattern transitions for steady upward gas-liquid flow in vertical tubes. AIhE Journal 6 (3): doi:./aic engesdal, J.Ø., Kaya, A.., and arica, Flow-Pattern ransition and Hydrodynamic Modeling of hurn Flow. PE J. 4 (4): PE PA. doi:.8/57756-pa. Wallis, G.B One Dimensional wo-phase Flow. ew York: McGraw-Hill. Zhang, H.-Q., Wang, Q., arica,., and Brill, J.P. 3a. Unified Model for Gas-Liquid Pipe Flow via lug Dynamics Part : Model Development. J. Energy Resour. echnol. 5 (4): doi:.5/ Zhang, H.-Q., Wang, Q., arica,., and Brill, J.P. 3b. A unified mechanistic model for slug liquid holdup and transition between slug and dispersed bubble flows. Int. J. Multiphase Flow 9 (): doi:.6/3-93()-8. ingting Yu is a consultant in P Group researching multiphase flow in pipes and wells. he holds a B degree from hina University of Petroleum and an M degree in petroleum engineering from University of ulsa. Hong-Quan Zhang is an associate professor of petroleum engineering at the University of ulsa, the principal investigator of UHOP, and an associate director of UFFP. hong-quan-zhang@utulsa.edu. From 998 to 3, he was a senior research associate of UFFP. Before joining the University of ulsa in 998, he was an associate professor and professor at ianjin University. In 993 and 994 as an Alexander von Humboldt Research Fellow, he conducted researches at the Max Planck Institution of Fluid Mechanics and the German Aerospace Research Establishment in Göttingen, Germany. Zhang holds B and M degrees from Xian Jiaotong University and a PhD degree from ianjin University. He is an associate editor for PE Journal. Mingxiu (Michelle) Li is working on high viscosity oil water emulsion rheology. he holds a PhD in mechanical engineering from the University of Edinburgh, a B degree in energy and power engineering, and an M degree in engineering thermophysics are from Xi an Jiaotong University, hina. he is an PE member. em arica is a professor of petroleum engineering and the director of ulsa University Fluid Flow Projects (UFFP) and ulsa University Paraffin Deposition Projects (UPDP) at he University of ulsa. cem-sarica@utulsa.edu. His research interests are multiphase flow in pipes and flow assurance. arica holds B and M degrees in petroleum engineering from Istanbul echnical University and a PhD degree in petroleum engineering from he University of ulsa. He has previously served as a member of PE Production Operations and Books ommittees. He currently serves as a member of PE Projects, Facilities and onstruction Advisory ommittee. He served as a member of PE Journal Editorial Board between 999 and 7. August PE Production & Operations 95