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Pet. Sci. (2018) 15:135–145 https://doi.org/10.1007/s12182-017-0193-y ORIGINAL PAPER A mechanistic model of heat transfer for gas–liquid flow in vertical wellbore annuli 1 2 1 • • Bang-Tang Yin Xiang-Fang Li Gang Liu Received: 20 July 2016 / Published online: 27 October 2017 The Author(s) 2017. This article is an open access publication Abstract The most prominent aspect of multiphase flow is Keywords Gas–liquid flow Vertical annuli Heat the variation in the physical distribution of the phases in the transfer Tubing liquid film Casing liquid film flow conduit known as the flow pattern. Several different flow patterns can exist under different flow conditions which have significant effects on liquid holdup, pressure gradient 1 Introduction and heat transfer. Gas–liquid two-phase flow in an annulus can be found in a variety of practical situations. In high rate As oil and gas development moves from land or shallow oil and gas production, it may be beneficial to flow fluids water to deep and ultradeep waters, multiphase flow occurs vertically through the annulus configuration between well during production and transportation (Chen 2011). The tubing and casing. The flow patterns in annuli are different flow normally occurs in horizontal, inclined, or vertical from pipe flow. There are both casing and tubing liquid films pipes and wells. Gas–liquid two-phase flow in an annulus in slug flow and annular flow in the annulus. Multiphase can be found in a variety of practical situations. In high rate heat transfer depends on the hydrodynamic behavior of the oil and gas production, it may be beneficial to flow fluids flow. There are very limited research results that can be vertically through the annulus configuration between well found in the open literature for multiphase heat transfer in tubing and casing. For surface facilities, some large, under- wellbore annuli. A mechanistic model of multiphase heat utilized flow lines can be converted to dual-service by transfer is developed for different flow patterns of upward putting a second pipe through the large line (i.e., flowing gas–liquid flow in vertical annuli. The required local flow produced water in the inner line and gas in the annulus). parameters are predicted by use of the hydraulic model of During gas production, liquids may accumulate at the steady-state multiphase flow in wellbore annuli recently bottom of the gas wells during their later life. In order to developed by Yin et al. The modified heat-transfer model remove or ‘‘unload’’ the undesirable liquids, a siphon tube for single gas or liquid flow is verified by comparison with is often installed inside the tubing string, which would form Manabe’s experimental results. For different flow patterns, it a gas–liquid two-phase flow in the annulus. Flow-assurance is compared with modified unified Zhang et al. model based problems, such as hydrate blocking (Jamaluddin et al. on representative diameters. 1991; Li et al. 2013; Wang et al. 2014, 2016) and wax deposition (Zhang et al. 2013; Bryan 2016; Theyab and Diaz 2016), are strongly associated with both the hydraulic & Bang-Tang Yin and thermal behavior. For example, they are related to the yinbangtang@163.com fluid velocity, liquid fraction, slug characteristics, pressure gradient and convective-heat-transfer coefficients of dif- School of Petroleum Engineering, China University of Petroleum (East China), Qingdao 266580, Shandong, China ferent phase and flow patterns in multiphase flow. There- fore, multiphase hydrodynamics and heat transfer in an College of Petroleum Engineering, China University of Petroleum, Beijing 102249, China annulus need to be modeled properly to guide the design and operation of flow systems. Edited by Yan-Hua Sun 123 136 Pet. Sci. (2018) 15:135–145 Compared to theoretical studies of multiphase hydro- dynamics (Liu et al. 2007; Wang and Sun 2009), and multiphase heat transfer in pipe flow (Zheng et al. 2016; Gao et al. 2016; Karimi and Boostani 2016; Rushd and Sanders 2017), there are very limited research results in the open literature for multiphase heat transfer in wellbore annuli. Davis et al. (1979) obtained a model for calculating the local Nusselt numbers of stratified gas/liquid flow in turbulent liquid/turbulent gas conditions. The model was tested with heat-transfer experiments for air/water flow in a 63.5-mm inside diameter (ID) tube. Guo et al. (2017) established a mathematical model of heat transfer in a gas- Bubbly Dispersed- Front Back Churn Annular drilling system, considering the flowing gas, formation flow bubble flow Slug flow flow flow fluid influx, Joule–Thomson cooling and entrained cuttings in the annular space. However, the multiphase flow effect Fig. 1 Flow patterns for upward vertical flow in an annulus (Caetano 1986) on the heat transfer was not considered. Shoham, et al. (1982) undertook experiments on heat- model for gas–liquid flow that is consistent with the transfer for slug flow in a horizontal pipe. He found a substantial difference in heat-transfer coefficient existed hydrodynamic model in vertical wellbore annuli. between the top and bottom of the slug. Developing heat- transfer correlations of different flow patterns was the aim of most previous studies (Shah 1981). Twenty heat-transfer 2 Modeling correlations were compared in Kim’s study (Kim et al. 1997). He collected the experimental data from the open 2.1 Single-phase flow literature and recommended the correlations for different flow patterns. However, the errors with the experimental When fluids flow through an annulus, the surrounding temperature is cold, heat is lost from the fluids to the for- results by Matzain (1999) were large. Later, a compre- hensive mechanistic model about heat transfer in gas–liq- mation, resulting in a decline in temperature, as seen in uid pipe flow was obtained (Manabe 2001). It was compared with the experimental data, and the performance was better. However, there were some inconsistencies in h (l+dl ) annular and slug flow. It needed to be modified. Zhang et al. (2006) developed a unified model of multi- phase heat transfer for different flow patterns of gas–liquid pipe flow at all inclinations – 90 to ? 90 from the hori- zontal. The required local flow parameters were predicted by use of the unified hydrodynamic model for gas/liquid pipe flow developed by Zhang et al. (2003a, b). However, it is not 1 fit for the gas–liquid flow in an annulus, because the flow patterns in annuli are different from pipe flow patterns, as seen in Fig. 1 (Caetano 1986). A new heat-transfer model for gas–liquid flow in vertical annuli needs to be established. A hydraulic model was developed to predict flow pat- terns, liquid holdup and pressure gradients for steady-state gas–liquid flow in wellbore annuli (Yin et al. 2014). The major advantage of this model compared with previous mechanistic models is that it is developed based on the dynamics of slug flow, and the liquid-film zone is used as the control volume. The effects of the tubing liquid film, casing liquid film and the droplets in the gas core area on the mass and momentum transfers are considered. Multiphase h (dl ) heat transfer depends on the hydrodynamic behavior of the flow. The objective of this study is to develop a heat-transfer Fig. 2 Heat transfer from the annulus to the formation 123 Pet. Sci. (2018) 15:135–145 137 Fig. 2. In the steady state, there is no heat loss from the t ¼ annulus to the tube. For the fluids in the annulus, the conservation of mass is: where t is the flowing time, s; r is the wellbore radius, m. The heat transfer from the annulus to the well wall, q ðÞ q v ¼ 0 ð1Þ l l dl may be expressed in terms of an overall heat coefficient (Hansan and Kabir 1994), where q is the density of liquids, kg/cm ; v is the velocity l l of liquids in the annulus, m/s; dl is the length of the element. q ¼ 2pr UðÞ T T ð8Þ 1 co a a wb The conservation of momentum is: where r is the outside casing radius, m; T is the fluid co a d dp spd temperature in the annulus, C; U is the overall heat- q v ¼ q g sin h ð2Þ l l l dl dl A transfer coefficient of the annulus, W/(m C). where p is the unit pressure, MPa; g is the gravity accel- 1 r r lnðÞ r =r r lnðÞ r =r 2 co co co ci co w co eration, m/s ; h is the inclination angle, ; s is the friction; ¼ þ þ ð9Þ U r h k k a ci a cas cem d is the annulus diameter, m; A is the cross area of the annulus, m . where r is the inside casing radius, m; h is the convec- ci a The energy equation is: tive-heat-transfer coefficient of fluids in the annulus, W/ (m C); k is the conductivity of the casing wall, W/(m cas d 1 d q q v e þ v ¼ ðÞ pv q v g sin h ð3Þ l l l l l C); k is the conductivity of cement, W/(m C). cem dl 2 dl A Combining Eqs. (7) and (8) gives, where e is the internal energy of the unit, J/kg; q is the 2pkðÞ T T e wb e heat loss from the annulus to the formation, W. q ¼ ¼ 2pr UðÞ T T 1 co a a wb Using the mass balance, we can reduce Eqs. (2) and (3) w c l p further: q ¼ ðÞ T T ð10Þ 1 a e dp dv spd ¼q v q g sin h ð4Þ l l where c is the fluid heat capacity at the constant pressure dl dl A in the annulus, J/(kg C). d p dv q l 1 q v e þ ¼q v q v g sin h ð5Þ l l l l l l w c k þ r U T 1 p e co a D dl q dl A 0 A ¼ 2p r U k co a e Or where A is the local parameter defined in Eq. (10). dh dv q dv q l 1 l 1 ¼v g sin h ¼v g sin h Combining Eqs. (6), (8) and (10) yields, l l dl dl Aq v dl w l 1 dh dv c l p ð6Þ ¼v g sin h ðÞ T T ð11Þ l a e dl dl A where h is the enthalpy, J/kg; w is the mass flow rate of The enthalpy gradient can be written in terms of the liquids in the annulus, kg/s. temperature and pressure gradients: Using the definition of the dimensionless temperature dh dT dp T proposed by Hasan and Kabir (1991), we may write an D ¼ c g c ð12Þ p l p dl dl dl expression for heat transfer from the wellbore/formation interface to the formation as where g is the fluid Joule–Thomson coefficient in the annulus, C/MPa. 2pkðÞ T T e wb e q ¼ ð7Þ Combining Eqs. (11) and (12) gives dT 1 1 dv dp a l where T is the formation temperature, C; T is the þ ðÞ T T þ v þ g sin h g c ¼ 0 e wb a e l p dl A c dl dl wellbore temperature, C; k is the conductivity of forma- ð13Þ tion, W/(m C); the dimensionless temperature, defined by Hasan and Kabir (1991), T , can be easily estimated from Defining a dimensionless parameter U ,as the following models. dv dp dp pffiffiffiffiffi pffiffiffiffiffi 10 U ¼ q v þ q g sin h q g c a l p l l l l 1:1281 tðÞ 10:3 t ; 10 t 1:5 D D D dl dl dl T ¼ 0:6 ðÞ 0:4063þ0:5lnt 1þ ; t [1:5 D D We can write Eq. (13)as where t is the dimensionless flowing time, 123 138 Pet. Sci. (2018) 15:135–145 dT 1 1 dp k ¼ðÞ 1 H k þ H k m l g l l þ ðÞ T T þ U ¼ 0 ð14Þ a e a dl A q c dl where H is the liquid hold up; k is the conductivity of l l Equation (14) is the energy conservation equation of liquid, W/(m C); k is the conductivity of gas, W/(m C). N is the mixture Nusselt number (Zhang et al. 2006). fluids in the annulus. Nu We can write Eq. (14)as m m m N N l Re Pr m 2 l N ¼ qffiffiffiffi ð19Þ dT 1 1 1 dp Nu 2=3 f l m m þ T ¼ T U ð15Þ a e a 1:07 þ 12:7 N 1 0 0 Pr dl A A q c dl where f is the friction factor of the mixture; l is the Up to this point, only mathematical manipulations have m l viscosity of liquid, mPa s; l is the viscosity of the mix- been done to the enthalpy equation, and the analysis has m ture, mPa s; l is the viscosity of gas, mPa s, been carried out rigorously without simplification. Now if g Eq. (15) is used for an onshore production well, then l ¼ðÞ 1 H l þ H l l l m g l assuming the surrounding formation temperature is a linear m m N and N is the Reynolds number and Prandtl number function of depth, it can be expressed as Re Pr of the mixture. T ¼ T g L sin h ð16Þ e ei e q v d m R m m N ¼ where T is the temperature of formation at wellbore Re ei intake, C; g is the formation thermal gradient, C/100 m; c l pm m m L is the depth, m. N ¼ Pr If Eq. (15) is used for the riser of an offshore production well, then the surrounding sea temperature is not a linear where q is the density of the mixture, kg/m ; v is the m m function of depth. It will be calculated according to the velocity of the mixture, m/s; c is the specific heat of the pm actual environment (Wang and Sun 2009). mixture, J/(kg C), If, for a certain segment of the wellbore, U , c , g , g , h, a p l e c ¼ðÞ 1 H c þ H c pm l pg l pl v dv /dl and dp/dl can be approximately constants, com- l l bining Eqs. (15) and (16) and integrating, yields an explicit where c is the specific heat of liquid, J/(kg C); c is the pl pg equation for the temperature: specific heat of gas, J/(kg C). 0 0 So the associated parameters will be modified, T ¼ðÞ T g L sin hþðÞ T T expðÞ L=A þ g A sin h a ei e i ei e 1 r r lnðÞ r =r r lnðÞ r =r 1 dp U co co co ci co w co 0 0 ¼ þ þ ½ 1 expðÞ L=Aþ ½ 1 expðÞ L=A U r h k k qc dL A ab ci ab cas cem ð17Þ w c k þ r U T m pm e co ab D A ¼ 2p r U k co ab e 2.2 Bubbly flow and dispersed-bubble flow dv dp dp U ¼ q v þ q g sin h q g c ab m pm m m m m In bubbly flow and dispersed-bubble flow, the gas holdup is dl dl dl small and the gas superficial velocity is low, the gas phase where the subscript b represents in bubble flow or dis- is distributed as small discrete bubbles in a continuous persed-bubble flow; the subscript m represents the mixture liquid phase. So bubbly flow and dispersed-bubble flow can properties of gas and liquid, A is the local parameter be treated as pseudo-single-phase flow. The fluid physical defined in Eq. (20). properties are adjusted based on the liquid holdup. Zhang Equation (15) can be written as et al. correction (2006) for bubbly flow will be modified dT 1 1 1 dp based on ‘‘hydraulic diameter’’. a þ T ¼ T U ð20Þ a e ab 0 0 dl A A q c dl pm b b m d ¼ d d ð17Þ R ci tuo Then Eq. (20) can be used in the heat transfer in bubble where d is the hydraulic diameter, m; d is the inside R ci flow and dispersed-bubble flow based on the modified diameter of casing, m; d is the outside diameter of tubing, m. tuo convective-heat-transfer coefficient h . ab Then the convective-heat-transfer coefficient of Eq. (9) for bubbly or dispersed-bubbly flow is obtained from 2.3 Annular flow N k Nu h ¼ ð18Þ ab As shown in Fig. 3, there are tubing and casing films in annular flow in annuli, which is different from the annular where k is the conductivity of the mixture, W/(m C), 123 Pet. Sci. (2018) 15:135–145 139 H (l+dl ) H (l+dl ) where w is the mass flow rate of the gas core, kg/s; c is tuf gc gc pgc H (l+dl ) cf the specific heat of the gas core, J/(kg C); v is the gc velocity of the gas core, m/s; q is the density of the gas gc core, kg/m ; g is the Joule–Thomson coefficient of the gc tufc gas core, C/MPa; U is the overall heat-transfer coeffi- gc cient of the gas core, W/(m C), and defined as ccf 1 r d ci c U ðÞ r þ d h gc tuo tu gc where r is the outside radius of the tubing, m; d is the tuo tu cff thickness of the tubing film, m; d is the thickness of the casing film, m; h is the convective-heat-transfer coeffi- gc cient of the gas core, W/(m C). U is the overall heat-transfer coefficient of the tubing tuf Casing liquid film, W/(m C), and defined as film 1 r þ d ðÞ r þ d lnðÞ ðÞ r þ d =r tuo tu tuo tu tuo tu tuo ¼ þ U r h k tuf tuo tuf tu H (l ) cf H (l ) tuf where h is the convective-heat-transfer coefficient of the H (l ) gc tuf Tubing liquid film tubing film, W/(m C); k is the conductivity of the tubing tu wall, W/(m C). Fig. 3 Control volume and temperatures in annular flow (q is the tufc For the tubing film, the energy conservation equation is heat transfer from the tubing film to the gas core, W; q is the heat ccf as follows, transfer from the gas core to the casing film, W; q is the heat cff transfer from the casing film to the formation, W) dT w c 1 dp tuf gc pgc þ T T þ U ¼ 0 tuf gc tuf dl w c B q c dl tuf ptuf tuf ptuf gc flow in pipes. Assume there are no temperature and heat- ð22Þ transfer exchanges in the vertical direction. The tempera- ture and heat changes are only caused by the heat transfer where w is the mass flow rate of the liquid tubing film, tuf in the radial direction. Heat distribution in annuli includes kg/s; q is the density of the liquid tubing film, kg/m ; tuf three parts: in casing film, tubing film and gas core. The c is the specific heat of the liquid tubing film, J/(kg C); ptuf heat-transfer models in annuli are obtained by a method U is the local constant and defined as follows: tuf similar to that above. dv dp dp tuf For the gas core, the energy conservation equation is as U ¼ q v þ q g sin h q g c tuf tuf tuf tuf tuf tuf ptuf dl dl dl follows, dT 1 1 1 dp gc where v is the velocity of the liquid tubing film, m/s; g þ T T T T þ U gc cf tuf gc gc tuf tuf 0 0 dl A B q c dl pgc gc gc gc is the Joule–Thomson coefficient of the liquid tubing film, ¼ 0 C/MPa. ð21Þ For the casing film, the energy conservation equation is as follows, where T is the temperature of the gas core, C; T is the gc cf dT 1 w c cf gc pgc temperature of the casing film, C; T is the temperature tuf þ ðÞ T T T T cf e gc cf dl C w c A cf pcf of the tubing film, C; c is the specific heat of the gas cf gc pgc 0 0 1 dp core, J/(kg C); A , B , U are the local parameters gc gc gc þ U cf q c dl pcf cf defined as follows: ¼ 0 ð23Þ w c gc pgc A ¼ gc 2pðÞ r d U where w is the mass flow rate of liquid casing film, kg/s; ci c gc cf q is the density of liquid casing film, kg/m ; c is the w c cf pcf gc pgc B ¼ 0 gc specific heat of liquid casing film, J/(kg C); C is the local 2pðÞ r þ d U cf tuo tu tuf parameter defined in Eq. (23), dv dp dp gc U ¼ q v þ q g sin h q g c gc gc pgc gc gc gc gc w c k þ r U T cf pcf e co cf D dl dl dl C ¼ cf 2p r U k co cf e 123 140 Pet. Sci. (2018) 15:135–145 U is the overall heat-transfer coefficient of casing film, cf For fully developed laminar flows of the liquid film and W/(m C), and defined as gas core, the Nusselt number approaches a constant value. 1 r r lnðÞ r =r r lnðÞ r =r ci co co ci co w co According to Zhang et al. (2006), the Nusselt numbers for ¼ þ þ U ðÞ r d h k k fully developed laminar flows are calculated by, cf ci c cf cas cem where h is the convective-heat-transfer coefficient of 7:541 3:657 d cf i if N ¼ 3:657 þ 0:5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð27Þ Nu casing film, W/(m C). 2 0:5 2 r r tuo ci U is the local parameter defined in Eq. (23), cf gc N ¼ 3:657 ð28Þ Nu dv dp dp cf U ¼ q v þ q g sin h q g c cf cf pcf cf cf cf cf dl dl dl The associated hydraulic parameters are calculated by the Yin et al. model (Yin et al. 2014), such as the thickness of where v is the velocity of the liquid casing film, m/s; g is cf cf the liquid film, d and d , the hold up, liquid and gas c tu the Joule–Thomson coefficient of the liquid casing film, physical properties . C/MPa. Equations (21)–(23) are the heat-transfer models for 2.4 Slug flow gas–liquid annular flow in annuli. Then the fluid temper- ature in the wellbore is calculated by a weighted average 2.4.1 Film region method based on holdup. T ¼ H T þðÞ 1 H H T þ H T ð24Þ The flow characters of the film region are similar to the ta tuf tuf tuf cf gc cf cf annular flow. The difference is that the gas core with liquid where H is the holdup of the liquid tubing film; H is the tuf cf droplets changes into the Taylor bubble. So, the overall holdup of the liquid casing film. heat-transfer coefficients of Eqs. (21) and (23) should The convective-heat-transfer coefficients for the casing change from U to U . Then the heat transfer of the film gc T film, the tubing film and gas core are obtained by the fol- region can be calculated. lowing equations 1 1 ¼ ð29Þ gc tuf cf U h N k N k N k T T lf gc lf Nu Nu Nu h ¼ ; h ¼ ; h ¼ tuf gc cf d d d tufR gcR cfR where U is the overall heat-transfer coefficient of the Taylor bubble, W/(m C); h is the convective-heat- where k , k are the thermal conductivities of the liquid lf gc transfer coefficient of the Taylor bubble, W/(m C). film and gas core, W/(m C); d , d , d are the tufR cfR gcR ‘‘hydraulic diameter’’ of the tubing film, the casing film N k Nu h ¼ ð30Þ and gas core, m. TR d ¼ðÞ r þ d r ¼ d tufR tuo tu tuo tu where k is the thermal conductivities of the Taylor bubble, d ¼ r ðÞ r d ¼ d cfR ci ci c c W/(m C); d is the ‘‘representative diameter’’ of the TR Taylor bubble. d ¼ðÞ r d ðÞ r þ d gcR ci c tuo tu qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 The Nusselt numbers for the liquid film and gas core are d ¼ 2 ðÞ r d ðÞ r þ d TR ci c tuo tu calculated by using the correlations for single-phase con- 2.4.2 Slug region vective heat transfer. The Petukhov correlation (1970)is used for turbulent liquid-film flow: There are small discrete bubbles in a continuous liquid f 0:25 if if if N N if 2 Re Pr l phase. The flow characters are similar to bubbly flow. So, N ¼ qffiffiffi ð25Þ Nu 2=3 l if if the heat transfer can be calculated by the bubbly flow model. 1:07 þ 12:7 ðÞ N 1 2 Pr 2.4.3 Slug unit where i is tubing film or casing film, i = tu or c. f is the if friction factor at the wall in contact with the liquid film. The fluid temperature in the wellbore is calculated by the q v d c l if ifR pl if l if ifR N ¼ N ¼ weighted average method based on holdup. Re Pr l k ½ H T þðÞ 1 H H T þ H T l þ T l tuf tuf tuf cf T cf cf F ls s T ¼ ð31Þ The Dittus and Boelter correlation (1985) is used for turbulent gas core flow, where T is the temperature of the fluid in slug flow, C; T s ls gc gc 0:8 gc 0:33 N ¼ 0:023ðÞ N ðÞ N ð26Þ Nu Re Pr is the temperature of the fluid in the slug unit, C; T is the 123 Pet. Sci. (2018) 15:135–145 141 temperature of the Taylor bubble, C; l is the length of the fluids is shown in Tables 1 and 2. The simulated annulus liquid film, m; l is the length of the slug unit, m; l is the outer diameter is 76.2 mm, and the inner diameter is s U length of the whole slug, m. 42.2 mm. The basic parameters are shown in Table 3. Figures 6 and 7 show comparisons between the simu- lations of new models and experimental measurements 3 Solution procedure (Manabe 2001) of the convective-heat-transfer coefficients for single-phase gas and liquid flows, respectively. The Figure 4 shows the overall solution flowchart for the present good agreement between the simulations and experimental model. First, the parameters of fluid properties are provided. measurements for single-phase gas and liquid flows indi- Then the flow pattern is determined based on the input cates that the new models are reliable and the selected variables and then all the flow conditions, such as flow correlations are appropriate. pattern, liquid holdups, local fluid velocities of the liquid There are few experimental research results in the open film and gas core, and slug characteristics, are predicted by literature for multiphase heat transfer in annuli. The unified use of the hydraulic model of steady-state multiphase flow in Zhang et al. model (2006) is verified by comparison with wellbore annuli recently developed by Yin et al. (2014). Manabe’s experimental results for different flow patterns in Based on flow patterns, further calculations are performed a crude-oil/natural gas system, and good agreement has in different subroutines. If the flow pattern is single-phase been observed in the comparison. So, the unified Zhang flow, single-phase heat-transfer calculation will be performed; et al. model is modified to calculate the heat transfer of if the flow pattern is bubble or dispersed-bubble flow, the gas–liquid flow in annuli based on the ‘‘hydraulic diame- corresponding hydraulic model and heat-transfer model will ter’’ of the annuli, Eq. (17), and the results are compared be called for calculation. For annular flow, corresponding with the present mechanistic heat-transfer model for gas– hydraulic and heat-transfer calculations will be made. For liquid flow in annuli. slug flow, corresponding hydraulic and heat-transfer calcu- Figures 8, 9, and 10 are comparisons of convective- lations will be made, as seen in Fig. 4. Figure 5 is the heat-transfer coefficient for bubble flow, annular flow and flowchart for the present annular flow heat-transfer model. slug flow predicted by the present model and the modified unified Zhang et al. model (2006). For the bubble flow, the data points are located inside the 10% error band. The 4 Comparisons with the unified Zhang model agreement is good. It shows that the influence of annulus geometry is small for the low gas volume fraction. For the The heat transfer of single-phase gas or liquid flow can be annular and slug flow in annuli, most of the data points are calculated by the present model if the thermal conductivity located inside the 30% error band and all are overesti- of the tubing wall is infinite. The composition of simulated mated. It may because there is a tubing liquid film and a Input the fluid and flowing parameters Yes Single phase Single phase Single phase gas or liquid hydraulic calculation heat transfer No Yes Bubble flow Bubble flow Bubble flow hydraulic calculation heat transfer No Yes Annular flow Annular flow Annular flow hydraulic calculation heat transfer No Slug flow heat Slug flow calculation Output transfer Fig. 4 Overall flowchart for present model 123 142 Pet. Sci. (2018) 15:135–145 casing liquid film in the wall and the geometry is different Annular flow from the modified unified Zhang et al. model. It may cause hydraulic calculation the hydraulic parameters and fluid physical properties to change a lot, leading to larger convective-heat-transfer Hydraulic parameters coefficients. N , N , N Re Pr Nu 5 Conclusions and discussion h , h , h cf tuf gc A heat-transfer model for gas–liquid flow in vertical annuli is developed in conjunction with the mechanistic hydro- dynamic model of Yin et al. (2014), which can predict flow U , U , U cf tuf gc pattern transitions, liquid holdup, gas void fraction, pres- sure gradient, and slug characteristics in gas–liquid two- Local parameters phase flow in vertical annuli. The heat-transfer modeling is based on energy-balance equations and analyses of the temperature differences and variations in the tubing liquid T , T , T cf tuf gc film, casing liquid film, gas core, Taylor bubble and slug body. Fluid temperature The heat-transfer model for single gas or liquid flow is of annular flow verified by comparison with Manabe’s experimental results (2001). Good agreement has been observed in the com- Fig. 5 Flowchart for present annular flow heat-transfer model parison. For different flow patterns, it is compared with unified Zhang et al. model modified based on ‘‘hydraulic diameter’’. For bubble or dispersed-bubble flow, the error is lower than 10%. With the gas void fraction and gas flow Table 1 Composition of natural gas velocity increasing, the error will be larger but lower than Components Molar fraction, % 30%. In other words, the difference between the new model and modified unified Zhang et al. model will be small if the N 1.82 gas void fraction and velocity is small. The difference will CO 0.65 be large when it changes. The modified method based on C 93.83 ‘‘hydraulic diameter’’ is no longer applicable for slug and C 2.98 annular flow in vertical annuli when the gas void fraction C 0.59 increases. It may be 1.3 times larger than the new model. i-C 0.08 Experimental investigations of heat transfer in vertical are n-C 0.05 required to improve the model performance. Table 2 Composition of oil 123 Pet. Sci. (2018) 15:135–145 143 Table 3 Basic parameters for the simulation SG single-gas flow, SL single-liquid flow, BUB bubble flow, ANN annular flow, SLU slug flow +20% Present model Present model +20% Zhang et al. model Zhang et al. model -20% -20% 0 400 800 1200 0 400 800 1200 2 2 h , W/(m °C) h , W/(m °C) SGexp SLexp Fig. 6 Comparison of single-phase gas flow model simulations and Fig. 7 Comparison of single-phase liquid flow model simulations measured convective-heat-transfer coefficient and measured convective-heat-transfer coefficients h , W/(m °C) SGcal h , W/(m °C) SLcal 144 Pet. Sci. 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Petroleum Science – Springer Journals
Published: Oct 27, 2017
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