Gomez et al Unified Mechanistic Model for Steady-State.pdf

Unified Mechanistic Model for Steady-State Two-Phase Flow: Horizontal to Vertical Upward Flow L.E. Gomez, SPE, Ovadia Shoham, SPE, and Zelimir Schmidt,* SPE, U. of Tulsa; R.N. Chokshi,** SPE, Zenith ETX Co.; and Tor Northug, Statoil Summary A unified steady-state two-phase flow mechanistic model for the prediction of flow pattern, liquid holdup and pressure drop is presented that is applicable to the range of inclination angles from horizontal (0°) to upward vertical flow (90°). The model is based on two-phase flow physical phenomena, incorporating recent developments in this area. It consists of a unified flow pattern prediction model and unified individual models for stratified, slug, bubble, annular and dispersed bubble flow. The model can be applied to vertical, directional and horizontal wells, and horizontal-near horizontal pipelines. The proposed model implements new criteria for eliminating discontinuity problems, providing smooth transitions between the different flow patterns. The new model has been initially validated against existing, various, elaborated, laboratory and field databases. Following the validation, the model is tested against a new set of field data, from the North Sea and Prudhoe Bay, Alaska, which includes 86 cases. The proposed model is also compared with six commonly used models and correlations. The model showed outstanding performance for the pressure drop prediction, with a ⫺1.3% average error, a 5.5% absolute average error and 6.2 standard deviation. The proposed model provides an accurate two-phase flow mechanistic model for research and design for the industry.

Introduction Early predictive means for two-phase flow were based on the empirical approach. This was due to both the complex nature of two-phase flow and the need for design methods for industry. The most commonly used correlations have been the Dukler et al.1 and Beggs and Brill2 correlations for flow in pipelines, and the Hagedorn and Brown3 and Ros4/Duns and Ros5 correlations for flow in wellbores. This approach was successful for solving twophase flow problems for more than 40 years, with an updated performance of ⫾30% error. However, the empirical approach has never addressed the ‘‘why’’ and ‘‘how’’ problems for twophase flow phenomena. Also, it is believed that no further or better accuracy can be achieved through this approach. A new approach emerged in the early 1980’s, namely, the mechanistic modeling approach. This approach attempts to shed more light on the physical phenomena. The flow mechanisms causing two-phase flow to occur are determined and modeled mathematically. A fundamental postulate in this method is the existence of various flow configurations or flow patterns, including stratified flow, slug flow, annular flow, bubble flow, churn flow and dispersed bubble flow. These flow patterns are shown schematically in Fig. 1. The first objective of this approach is, thus, to predict the existing flow pattern for a given system. Then a separate model is developed for each flow pattern to predict the corresponding hydrodynamics and heat transfer. These models are expected to be more reliable and general because they incorporate *Deceased. **Now with TanData Corp. Copyright © 2000 Society of Petroleum Engineers This paper (SPE 65705) was revised for publication from paper SPE 56520, presented at the 1999 SPE Annual Technical Conference and Exhibition held in Houston, 5–8 October. Original manuscript received for review 20 October 1999. Revised manuscript received 20 May 2000. Manuscript peer approved 9 June 2000.

SPE Journal 5 共3兲, September 2000

the mechanisms and the important parameters of the flow. All current research is conducted through the modeling approach. Application of models in the field is now underway, showing the potential of this method. The mechanistic models developed over the past two decades have been formulated separately for pipelines and wellbores. Following is a brief review of the literature for these two cases. Pipeline Models. These models are applicable for horizontal and near horizontal flow conditions, namely, ⫾10°. The pioneering and most durable model for flow pattern prediction in pipelines was presented by Taitel and Dukler.6 Other studies have been carried out for the prediction of specific transitions, such as the onset of slug flow,7 or different flow conditions, such as high pressure.8 Separate models have been developed for stratified flow,6,9-11 slug flow,12-14 annular flow15,16 and dispersed bubble flow 共the homogeneous no-slip model17兲. A comprehensive mechanistic model, incorporating a flow pattern prediction model and separate models for the different flow patterns, was presented by Xiao et al.18 for pipeline design. Wellbore Models. These models are applicable mainly for vertical flow but can be applied as an approximation for off-vertical sharply inclined flow (60°⭐ ␪ ⭐90°) also. A flow pattern prediction model was proposed by Taitel et al.19 for vertical flow, which was later extended to sharply inclined flow by Barnea et al.20 Specific models for the prediction of the flow behavior have been developed for bubble flow21,22 slug flow23-25 and annular flow.26,27 Comprehensive mechanistic models for vertical flow have been presented by Ozon et al.,28 by Hasan and Kabir,21 by Ansari et al.29 and by Chokshi et al.30 Unified Models. Attempts have been made in recent years to develop unified models that are applicable for the range of inclination angles between horizontal (0°) and upward vertical (90°) flow. These models are practical since they incorporate the inclination angle. Thus, there is no need to apply different models for the different inclination angles encountered in horizontal, inclined and vertical pipes. A unified flow pattern prediction model was presented by Barnea31 that is valid for the entire range of inclination angles (⫺90°⭐ ␪ ⭐90°). Felizola and Shoham32 presented a unified slug flow model applicable to the inclination angle range from horizontal to upward vertical flow. A unified mechanistic model applicable to horizontal, upward and downward flow conditions was presented by Petalas and Aziz,33 which was tested against a large number of laboratory and field data. Recently, Gomez et al.34 presented a unified correlation for the prediction of the liquid holdup in the slug body. The above literature review reveals that separate comprehensive mechanistic models are available for pipeline flow and wellbore flow. Only very few studies have been published on unified modeling. The objective of this paper is to present a systematic, comprehensive, unified model applicable for the range of inclination angles between horizontal (0°) and vertical (90°). This will provide more efficient computing algorithms, because the model can be applied conveniently for both pipelines and wellbores, 1086-055X/2000/5共3兲/339/12/$5.00⫹0.50

339

d CD ⫽2

冉

0.4␴ 共 ␳ L⫺ ␳ G 兲g

冊

1/2

共4兲

.

The other critical diameter is applicable to shallow inclinations (⫾10°) where, due to buoyancy, bubbles larger than this diameter migrate to the upper part of the pipe causing ‘‘creaming’’ and transition to slug flow as follows: d CB ⫽

f M v 2M ␳L 3 . 8 共 ␳ L ⫺ ␳ G 兲 g cos ␪

共5兲

Transition to dispersed bubble flow will occur when the maximum possible bubble diameter, given by Eq. 3, is less than both critical diameters given by Eqs. 4 or 5, namely, d max⬍d CD and d CB .

Fig. 1–Flow patterns in pipelines and wellbores for horizontal to vertical flow patterns.

The transition boundary given by Eq. 6 is valid for ␣ ⭐0.52, which represents the maximum possible packing of bubbles for a cubic lattice configuration. For larger values of void fraction, agglomeration of bubbles occurs, independent of the turbulence forces, resulting in a transition to slug flow. This criterion is given by

␣ NS ⫽ without the need to switch among different models. The proposed model will be evaluated against new field data, along with other published models and correlations. Unified Model Formulation The unified model consists of a unified flow pattern prediction model and separate unified models for the different existing flow patterns. These are briefly described below. Unified Flow Pattern Prediction Model. The Barnea31 model is applicable for the entire range of inclination angles, namely, from upward vertical flow to downward vertical flow (⫺90°⭐ ␪ ⭐90°). Below is a summary of the applicable transition criteria for this study, including the stratified to nonstratified, slug to dispersed bubble, annular to slug and bubble to slug flow. Stratified to Nonstratified Transition. The criterion for this transition is the same as the original one proposed by Taitel and Dukler,6 based on a simplified Kelvin–Helmholtz stability analysis given by F

2

冉

1

2 ˜ L /dh ˜L ˜v G dA

˜ L兲2 共 1⫺h

˜A G

冊

⭓1,

共1兲

where the superscript tilde symbol ‘‘⬃’’ represents a dimensionless parameter 共length and area are normalized with d and d 2 , respectively, and the phase velocity is normalized with the corresponding superficial velocity兲. F is a dimensionless group given by F⫽

冑

␳G v SG . ␳ ⫺ ␳ 共 L G 兲 冑dg cos ␪

共2兲

Slug to Dispersed Bubble Transition. The slug to dispersed bubble transition occurs at high liquid flow rates, where the turbulent forces overcome the interfacial tension forces, dispersing the gas phase into small bubbles. The resulting maximum bubble size can be determined from

冋 冉 冊

v SG d max⫽ 4.15 vM

0.5

册冉 冊 冉

␴ ⫹0.725 ␳L

0.6

2 f M v 3M d

冊

.

共3兲

Two critical bubble diameters are considered. The first is the critical diameter below which bubbles do not deform, avoiding agglomeration or coalescence, given by 340

v SG ⫽0.52. v SG ⫹ v SL

共7兲

Annular to Slug Transition. Two mechanisms are responsible for this transition from annular flow to slug flow, causing blockage of the gas core by the liquid phase. The two mechanisms are based on the characteristic film structure of annular flow: 1. Instability of the liquid film due to downward flow near the pipe wall. The criterion for the instability of the film is obtained from the simultaneous solution of the following two dimensionless equations: Y⫽ Y⭓

1⫹75H L 共 1⫺H L 兲 2.5H L

⫺

1 H L3

2⫺ 共 3/2兲 H L H L3 共 1⫺

共 3/2兲 H L 兲

共8兲

X 2,

共9兲

X2

where X is the Lockhart and Martinelli parameter and Y is a dimensionless gravity group defined respectively by

X 2⫽

Y⫽

冉 冉

4C G ␳ G v SG d ␮G

冊 冊

共 ␳ L ⫺ ␳ G 兲 g sin ␪ . dp dL SG

冉 冊

⫺n

2 ␳ L v SL 2 2 ⫽ d ⫺m ␳ G v SG 2

4C L ␳ L v SL d d ␮L

冉 冊 冉 冊 dp dL

dp dL

SL

,

共10兲

SG

共11兲

Note that Eq. 8 yields the steady-state solution for the liquid holdup H L , while Eq. 9 yields the value of the liquid holdup that satisfies the condition of the film instability. 2. Wave growth on the interface due to large liquid supply from the film. If sufficient liquid is provided, the wave will grow and bridge the pipe, resulting in slug flow. The condition for occurrence of this mechanism is H L ⭓0.24.

⫺0.4

共6兲

共12兲

Transition from annular to slug flow will occur whenever one of the two criteria is satisfied. A smooth change between the two mechanisms is obtained when the inclination angle varies over the entire range of inclinations, or when a change occurs in the operationing conditions.

Gomez et al.: Unified Mechanistic Model for Steady-State Two-Phase Flow

SPE Journal, Vol. 5, No. 3, September 2000

Bubble to Slug Transition. The transition from bubble to slug flow occurs at relatively lower liquid flow rates compared to the transition from slug to dispersed bubble flow. Under these conditions the turbulent forces are negligible, and the transition is caused by coalescence of bubbles at a critical gas void fraction of ␣ ⫽0.25, as follows: v SL ⫽

冉

1⫺ ␣ g共 ␳ L⫺ ␳ G 兲␴ v SG ⫺1.53共 1⫺ ␣ 兲 0.5 ␣ ␳ L2

冊

1/4

sin ␪ .

共13兲

The bubble regime can exist at low liquid flow rates as given by Eq. 13, provided that the pipe diameter is larger than d⬎19关 ( ␳ L ⫺ ␳ G ) ␴ / ␳ L2 g 兴 0.5 and only for sharply inclined pipes with inclination angles between approximately 60 and 90°. Elimination of Transition Discontinuities. Mechanistic models for the prediction of pressure traverses in multiphase flow are notorious for creating discontinuities. This is the result of switching from one flow pattern model to another as the transition boundary is crossed. Different models are used for different flow patterns to predict the liquid holdup and pressure drop, which might result in a discontinuity. In order to avoid this problem in the proposed model, the following criteria were implemented to smooth the transitions between the different flow patterns. Bubble to Slug and Slug to Dispersed Bubble Transitions. Near the transition boundaries from slug to bubble or dispersed bubble flow, the liquid film/gas pocket region behind the slug body, namely, L F , becomes small. The short film/gas length can prevent the slug flow model from converging. Thus, to solve this problem, when slug flow is predicted near these transition boundaries, the following constraints were developed: if L F ⭐1.2d and v SL ⭐0.6 m/s ⇒bubble flow, if L F ⭐1.2d and v SL ⭓0.6 m/s ⇒dispersed bubble flow. 共14兲 The value L F /d⫽1.2 is based on the mechanism that once the Taylor bubble length approaches the pipe diameter, it becomes unstable and might break into small bubbles. Under these conditions, for high superficial liquid velocities, due to turbulence intensity and bubble breakup and dispersion, the resulting flow pattern will be dispersed bubble flow. However, for low superficial liquid velocities, due to low turbulence intensity and coalescence of the small bubble to larger ones, the resulting flow pattern will be bubble flow. Slug to Annular Transition. A two-fold problem is associated with this transition boundary. First, a discontinuity in the pressure gradient between slug flow and annular flow occurs. Also, if slug flow is predicted near this transition boundary, due to the high gas rates, the film/gas zone becomes long, resulting in a very thin film thickness, one approaching zero. This can prevent the slug flow model from converging. To alleviate the two problems, a transition zone is created between slug flow and annular flow based on the superficial gas velocity. The transition zone is predicted by the critical velocity corresponding to the droplet model used by Taitel et al.19 as follows:

冉

␴ g sin ␪ 共 ␳ L ⫺ ␳ G 兲 v SG,crit ⫽3.1 ␳ G2

冊

Fig. 2–Physical model for stratified flow.

liquid velocity and the superficial gas velocity on the transition boundary to annular flow, predicted by the Barnea model.31 This averaging eliminates numerical problems and ensures a smooth pressure gradient across the slug to annular boundary. Unified Stratified Flow Model. The physical model for stratified flow is given in Fig. 2. A modified form of the Taitel and Dukler6 model is used here. Two modifications are introduced: the liquid wall friction factor is determined by Ouyang and Aziz35 and the interfacial friction factor is given by Baker et al.36 Momentum Balances. The momentum 共force兲 balances for the liquid and gas phases are given, respectively, by ⫺A L

dp ⫺ ␶ WL S L ⫹ ␶ I S I ⫺ ␳ L A L g sin ␪ ⫽0, dL

共16兲

⫺A G

dp ⫺ ␶ WG S G ⫺ ␶ I S I ⫺ ␳ G A G g sin ␪ ⫽0. dL

共17兲

Eliminating the pressure gradient from Eqs. 16 and 17, the combined momentum equation for the two phases is obtained as follows:

␶ WL

冉

.

冉

␲ ⫺cos⫺1 2

共15兲

Thus, for a given superficial liquid velocity, the transition region is defined when the superficial gas velocity is greater than the critical gas velocity 共given in Eq. 15兲 and less than the superficial gas velocity on the transition boundary to annular flow 共predicted by the Barnea model31兲. Hence, when slug flow is predicted in the transition zone, the pressure gradient is averaged between the pressure gradient under slug flow and annular flow conditions. The corresponding slug flow pressure gradient is calculated at the given superficial liquid velocity and the critical superficial gas velocity, given by Eq. 15. Similarly, the corresponding pressure gradient under annular flow is calculated at the given superficial Gomez et al.: Unified Mechanistic Model for Steady-State Two-Phase Flow

共18兲

The combined momentum equation is an implicit equation for h L 共or h L /d兲, the liquid level in the pipe. Solution of the equation, carried out by a trial and error procedure, requires the determination of the different geometrical, velocity and shear stress variables. Under high gas and liquid flow rates, multiple solutions can occur. It can be shown that, in this case, the smallest of the three solutions is the physical and stable solution. Once the liquid level h L /d is determined, the liquid holdup, H L , can be calculated in a straightforward manner from geometrical relationships as follows:

H L⫽

0.25

冊

1 SL SG 1 ⫺ ␶ WG ⫺ ␶ IS I ⫹ ⫹ 共 ␳ L ⫺ ␳ G 兲 g sin ␪ ⫽0. AL AG AL AG

冊冉

hL hL ⫺1 ⫹ 2 ⫺1 d d ␲

冊冑 冉

1⫺ 2

hL ⫺1 d

冊

2

. 共19兲

Once the liquid holdup is determined, the pressure gradient can be determined from either Eq. 16 or 17. Either equation provides the frictional and the gravitational pressure losses, and neglects the accelerational pressure losses. Closure Relationships. The wall shear stresses corresponding to each phase are determined based on single-phase analysis using the hydraulic diameter concept, as follows 共Fanning friction factor formulation兲:

␶ WL ⫽ f L

␳ L v L2 2

␳ G v G2 . 2

共20兲

SPE Journal, Vol. 5, No. 3, September 2000

341

and

␶ WG ⫽ f G

The respective hydraulic diameters of the liquid and gas phases are given by d L⫽

4A L SL

and

d G⫽

4A G . S G ⫹S I

共21兲

The Reynolds numbers of each of the phases are N ReL ⫽

dL vL ␳L ␮L

N ReG ⫽

and

dG vG ␳G . ␮G

共22兲

Taitel and Dukler6 proposed that both the liquid and gas wall friction factors, f L and f G , can be calculated using a standard friction factor chart. However, Ouyang and Aziz35 found this procedure to be appropriate for the gas phase only. This is due to the fact that the liquid wall friction factor can be affected significantly by the interfacial shear stress, especially for low liquid holdup conditions. Thus, f G is determined from a standard chart, while f L is determined by a new correlation developed by Ouyang and Aziz35 that incorporates the gas and liquid flow rates, given as f G⫽

16 N ReG

for

N ReG ⭐2,300,

冋 冉

共23兲

␧ 10 f G ⫽0.001 375 1⫹ 2⫻104 ⫹ d N ReG

Fig. 3–Physical model for slug flow.

冊册 1/3

unit. Applying this balance on cross sections in the liquid slug body and in the liquid film region gives, respectively,

N ReG ⬎2,300,

for

f L⫽

6

冉 冊

1.6291 v SG 0.5161 v SL N Re L

0.0926

.

共24兲

The interfacial shear stress is given, by definition, as

␳ G共 v G⫺ v L 兲兩 v G⫺ v L兩 ␶ I⫽ f I . 2

共25兲

The interfacial friction factor for stratified smooth flow is taken as the friction factor between the gas phase and the wall. However, for stratified wavy flow, as suggested by Xiao et al.,18 the interfacial friction factor is that given by Baker et al.36 Unified Slug Flow Model. The unified and comprehensive analysis of slug flow, presented by Taitel and Barnea,37 is used in the present study with the following features: a uniform film along the liquid film/gas pocket zone; a global momentum balance on a slug unit for pressure drop calculations, and a new correlation 共Gomez et al.34兲 for the liquid holdup in the slug body. The original Taitel and Barnea37 model was extended to vertical flow by assuming a symmetric film around the Taylor bubble for inclination angles between 86 and 90°. With the above characteristics, the original model is simplified considerably, as given below, avoiding the need for numerical integration along the liquid film region. The proposed simplified model is considered to be sufficiently accurate for practical applications. Refer to Fig. 3 for the physical model for slug flow. Mass Balances. An overall liquid mass balance over a slug unit results in LS LF ⫹ v LTB H LTB . v SL ⫽ v LLS H LLS LU LU

共26兲

A mass balance can also be applied between two crosssectional areas, namely, in the slug body and in the film region, in a coordinate system moving with the translational velocity, v TB , yielding 共 v TB ⫺ v LLS 兲 H LLS ⫽ 共 v TB ⫺ v LTB 兲 H LTB .

共27兲

A continuity balance on both liquid and gas phases results in a constant volumetric flow rate through any cross section of the slug 342

v M ⫽ v SL ⫹ v SG ⫽ v LLS H LLS ⫹ v GLS 共 1⫺H LLS 兲 ,

共28兲

v M ⫽ v LTB H LTB ⫹ v GTB 共 1⫺H LTB 兲 .

共29兲

Eq. 28 can be used to determine v LLS , the liquid velocity in the slug body, since the other variables are given in the form of closure relationships. Then, the liquid film velocity, v LTB , can be determined from Eq. 27 for a given liquid holdup in this region, H LTB . Also, from Eq. 29 it is possible to determine v GTB , the gas velocity in the gas pocket. The average liquid holdup in a slug unit is defined as H LU ⫽

H LLS L S ⫹H LTB L F . LU

共30兲

Using Eqs. 26–28, the expression for the liquid holdup becomes H LU ⫽

v TB H LLS ⫹ v GLS 共 1⫺H LLS 兲 ⫺ v SG . v TB

共31兲

Eq. 31 shows an interesting result, namely, that the average liquid holdup in a slug unit is independent of the lengths of the different slug zones. Hydrodynamics of the Liquid Film. Considering a uniform liquid film thickness, a combined momentum equation, similar to that in the case of stratified flow, can be obtained for the film/gas pocket zone as follows:

冉

冊

1 ␶ WF S F ␶ WG S G 1 ⫺ ⫺ ␶ IS I ⫹ ⫹ 共 ␳ L ⫺ ␳ G 兲 g sin ␪ ⫽0. AL AG AF AG

共32兲

Solution of Eq. 32 yields the uniform 共equilibrium兲 film thickness or the liquid holdup in this region, H LTB . This value can be used, in a trial and error procedure, to determine the gas and liquid velocities in the slug and film/gas pocket regions, as discussed below Eq. 29. The liquid film length can be determined from L F ⫽L U ⫺L S .

Gomez et al.: Unified Mechanistic Model for Steady-State Two-Phase Flow

共33兲 SPE Journal, Vol. 5, No. 3, September 2000

The slug length, L S , is given as a closure relationship while the slug unit length, L U , can be determined from Eq. 26, as follows: L U ⫽L S

v LLS H LLS ⫺ v LTB H LTB . v SL ⫺ v LTB H LTB

共34兲

Pressure Drop Calculations. The pressure drop for a slug unit can be calculated using a global force balance along a slug unit. Since the momentum fluxes in and out of the slug unit control volume are identical, the pressure drop across this control volume for a uniform liquid film is dp ␶ S ␲ d L S ␶ WF S F ⫹ ␶ WG S G L F ⫽ ␳ U g sin ␪ ⫹ ⫹ , dL A LU A LU

共35兲

where ␳ U is the average density of the slug unit given by

␳ U ⫽H LU ␳ L ⫹ 共 1⫺H LU 兲 ␳ G .

共36兲

The first term on the right-hand side of Eq. 35 is the gravitational pressure gradient, whereas the second and third terms represent the frictional pressure gradient that results from the frictional losses in the slug and in the film/gas pocket regions. No accelerational pressure drop occurs in the slug unit control volume formulation. Closure Relationships. The proposed model requires four closure relationships, namely, the liquid slug length, L S , the liquid holdup in the slug body, H LLS , the slug translational velocity, v TB , and the gas velocity of the small bubbles entrained in the liquid slug, v GLS . The closure relationships are given below. A constant length of L S ⫽30d and L S ⫽20d is used for fully developed and stable slugs in horizontal and vertical pipes, respectively. For inclined flow, an average slug length is used based on inclination angle. However, for horizontal and near horizontal ( ␪ ⫽⫾1°) large diameter pipes (d⬎2 in.), the Scott et al.38 correlation is used, as given below ln共 L S 兲 ⫽⫺25.4⫹28.5关 ln共 d 兲兴 0.1,

共37兲

where d is expressed in inches and L S is in feet. The liquid holdup in the slug body, H LLS , is predicted using the Gomez et al.34 unified correlation, given by H LLS ⫽1.0e ⫺(7.85⫻10

⫺3 ␪ ⫹2.48⫻10⫺6 N

ReSL )

,

0⭐ ␪ ⭐900 ,

共38兲

where the slug superficial Reynolds number is calculated as N ReSL ⫽

␳ Lv M d . ␮L

共39兲

The slug translational velocity is determined from the Bendiksen39 correlation, given by v TB ⫽1.2v M ⫹ 共 0.542冑gd cos ␪ ⫹0.351冑gd sin ␪ 兲 .

共40兲

The gas velocity of the small bubbles entrained in the liquid slug, v GLS , can be determined in the manner suggested by Hasan and Kabir,21 given in a later section by Eqs. 57 and 58. Note that for this case the liquid holdup in the slug body, H LLS , should be used. 27

Unified Annular Flow Model. The model of Alves et al. developed originally for vertical and sharply inclined flow has been extended in the present study to the entire range of inclination angles from 0 to 90°, as given below. The physical model for annular flow is given in Fig. 4. The annular flow model equations are similar to the stratified flow model ones, since both patterns are separated flow. The differences between the two models are the different geometrical and closure relationships, and the fact that the gas core in annular flow includes liquid entrainment. Momentum Balances. The linear momentum 共force兲 balances for the liquid and gas core phases are given, respectively, by Gomez et al.: Unified Mechanistic Model for Steady-State Two-Phase Flow

Fig. 4–Physical model for annular flow.

⫺ ␶ WF

⫺␶I

冉 冊

SF SI dp ⫹␶I ⫺ AF AF dL

冉 冊

SI dp ⫺ AC dL

⫺ ␳ L g sin ␪ ⫽0,

共41兲

F

⫺ ␳ C g sin ␪ ⫽0.

共42兲

C

Eliminating the pressure gradients from the equations results in the combined momentum equation for annular flow, namely,

␶ WF

冉

冊

SF 1 1 ⫺ ␶ IS I ⫹ ⫹ 共 ␳ L ⫺ ␳ C 兲 g sin ␪ ⫽0. AF AF AC

共43兲

Eq. 43 is an implicit equation for the film thickness ␦ 共or ␦ /d兲 that can be solved by trial and error, provided the proper geometrical, velocity and closure relationships are provided. These are described below. Mass Balances. The velocities of the liquid film and the gas core can be determined from simple mass balance calculations yielding, respectively, v F ⫽ v SL

v C⫽

共 1⫺E 兲 d 2 , 4 ␦ 共 d⫺ ␦ 兲

共 v SG ⫹ v SL E 兲 d 2 . 共 d⫺2 ␦ 兲 2

共44兲

共45兲

The gas void fraction in the core and the core average density and viscosity are given, respectively, by

␣ C⫽

v SG , v SG ⫹ v SL E

共46兲

␳ C ⫽ ␳ G ␣ C ⫹ ␳ L 共 1⫺ ␣ C 兲 ,

共47兲

␮ C ⫽ ␮ G ␣ C ⫹ ␮ L 共 1⫺ ␣ C 兲 .

共48兲

SPE Journal, Vol. 5, No. 3, September 2000

343

Closure Relationships. The liquid wall shear stress is determined from single-phase flow calculations based on the hydraulic diameter concept. The most difficult task in modeling annular flow is the determination of the interfacial shear stress, ␶ I , and the entrainment fraction, E. By all means this is an unresolved problem even for vertical or horizontal flow conditions. The definition of the interfacial shear stress for annular flow is

␶ I⫽ f I␳ C

共 v C⫺ v F 兲兩 v C⫺ v F兩 . 2

共49兲

As suggested by Alves et al.,27 the interfacial friction factor can be expressed by f I ⫽ f SC I,

共50兲

where f SC is the friction factor that would be obtained if only the core 共gas phase and entrainment兲 flows in the pipe. Calculation of f SC should be based on the core superficial velocity ( v SC ⫽ v SG ⫹E v SL ) and the core average density and viscosity given, respectively, by Eqs. 47 and 48. The interfacial correction parameter I is used to take into account the roughness of the interface. Different expressions for I are given by Alves et al.27 for vertical flow only. In the present study, the parameter I is an average between a horizontal factor and a vertical factor, based on the inclination angle, ␪, as follows: I ␪ ⫽I H cos2 ␪ ⫹I V sin2 ␪ .

共51兲

The horizontal correction parameter is given by Henstock and Hanratty41 as I H ⫽1⫹850F A ,

Fig. 5–Physical model for bubble flow.

共52兲

where

冉 冊冉 冊

0.5 2.5 0.9 2.5 0.4 兲 ⫹ 共 0.0379N Re 兲 兴 关共 0.707N Re vL SL SL F A⫽ 0.9 vG N Re SG

␳L ␳G

0.5

共53兲

and N ReSL and N ReSG are the liquid and gas superficial Reynolds numbers, respectively. The vertical correction parameter is given by Wallis17 as

␦

I V ⫽1⫹300 . d

共54兲

The entrainment fraction, E, is calculated by the Wallis17 correlation, given by E⫽1⫺e ⫺[0.125( ␾ ⫺1.5)] ,

angles was carried out by taking the component of the bubble rise velocity in the direction of the flow, as given below 共see Fig. 5 for the bubble flow physical model兲. The gas velocity is given by v G ⫽C 0 v M ⫹ v 0⬁ sin ␪ H L0.5 ,

where v M is the mixture velocity, C 0 is a velocity distribution coefficient, v 0⬁ is the bubble rise velocity and H L0.5 is a correction for bubble swarm. In the present study, the velocity distribution coefficient C 0 ⫽1.15, as suggested by Chokshi et al.,30 and the bubble rise velocity is given by Harmathy41 共in SI units兲 as follows:

共55兲

v 0⬁ ⫽1.53

where

␾ ⫽104

冉 冊

v SG ␮ G ␳ G ␴ ␳L

1/2

共56兲

.

共57兲

冉

g ␴共 ␳ L⫺ ␳ G 兲

␳ L2

冊

0.25

共58兲

.

Substituting for the gas velocity in terms of the superficial velocity yields

Unified Bubble Flow Model. Extension of the Hasan and Kabir21 bubble flow model for the entire range of wellbore inclination

v SG ⫽C 0 v M ⫹ v 0⬁ sin ␪ H L0.5 . 1⫺H L

共59兲

TABLE 1– DATABASE FOR INDIVIDUAL FLOW PATTERN MODELS VALIDATION Data Source Minami (Ref. 44) Nuland et al. (Ref. 42) Felizola and Shoham (Ref. 32) Schmidt (Ref. 43) Caetano et al. (Ref. 22) Alves et al. (Ref. 27)

Flow Pattern Inclination Stratified Slug Slug Slug Bubble Annular

␪ ⫽0° 10°⬍ ␪ ⬍60° 0°⬍ ␪ ⬍90° ␪ ⫽90° ␪ ⫽90° ␪ ⫽90°

Pipe Diameter (in.)

Fluids

3 4 2 2 Annulus 2.5

Air-kerosene/water Dense gas (SF6)-oil Air-kerosene Air-kerosene Air-kerosene/water Natural gas-Crude

Liquid Density Pressure (psia) Data Points (lbm/ft3) 50/62.4 51 50 50 50/62.4 27

50 145 250 225 45 1750

100 52 72 15 19 2 (75) Total⫽260

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TABLE 2– INDIVIDUAL FLOW PATTERN MODELS VALIDATION RESULTS Pressure Gradient Data Source Minami (Ref. 44) Nuland et al. (Ref. 42) Felizola and Shoham (Ref. 32) Schmidt (Ref. 43) Caetano et al. (Ref. 22) Alves et al. (Ref. 27)

Liquid Holdup

Flow Pattern

Inclination

Average Error (%)

Abs. Average Error (%)

Average Error (%)

Abs. Average Error (%)

Stratified Slug Slug Slug Bubble Annular (2 points) Annular (75 points)

␪ ⫽0° 10°⬍ ␪ ⬍60° 0°⬍ ␪ ⬍90° ␪ ⫽90° ␪ ⫽90° ␪ ⫽90° ␪ ⫽90°

¯ 7.5 20.6 ¯ ¯ 1.5 ⫺0.9

¯ 10.2 25.0 ¯ ¯ 1.5 9.8

⫺20.8 ⫺6.7 0.6 ⫺9.3 ⫺2.3 ¯ ¯

33.5 9.6 13.2 15.0 2.7 ¯ ¯

Eq. 59 must be solved numerically to determine the liquid holdup, H L . Once the liquid holdup is computed, the gravitational and frictional pressure gradients are determined in a straightforward manner. For dispersed bubble flow, the homogeneous no-slip model17 is used. Details of this simple model are omitted here for brevity. Results and Discussion This section includes the validation of the developed unified model with published laboratory and field data, and the performance of the model with new field data. Unified Model Validation. Initially, the individual flow pattern models for slug flow, stratified flow, bubble flow and annular flow were validated against several sets of available laboratory and limited field data. Tables 1 and 2 present the range of data and the validation results, respectively. Unified Slug Model. Validation of the proposed slug flow model was carried out using the following sets of data: 1. the Felizola and Shoham32 data provide detailed slug characteristics, liquid holdup and pressure drop, for the entire range of upward inclination angles between 10 and 90° at 10° increments; 2. the Nuland et al.42 data for 10, 20, 45, 60, and 80° including liquid holdup and pressure drop; 3. the Schmidt43 data for vertical flow with liquid holdup only. Fig. 6 presents a typical comparison of the predictions of the Gomez et al.34 slug body liquid holdup correlation with published experimental data 共including additional data other than the above mentioned three sets兲. As can be seen, the correlation follows the trend of decreasing slug liquid holdup as the inclination angle increases. Comparisons between the predictions of the unified slug model and the experimental data were carried out for both the liquid

Fig. 6–Comparison between predicted and measured slug liquid holdup „Ref. 34…. Gomez et al.: Unified Mechanistic Model for Steady-State Two-Phase Flow

holdup (H LU ) and the pressure gradient, averaged over a slug unit. The results for the different data sources are given in Table 2. Unified Bubble Model. The data of Caetano et al.22 were used to test the model for bubble flow. Note that the Caetano et al. data were acquired in an annulus configuration with a 3-in. casing inner diameter 共ID兲 and 1.66-in. tubing outer diameter 共OD兲. For this reason the comparison was carried out only for the liquid holdup. An equivalent diameter was used that provides the same cross-sectional area and superficial velocities that occur in the annulus. The results show excellent agreement with an average error and an average absolute error of ⫺2.3 and 2.7%, respectively. Unified Stratified Model. The stratified flow model was tested against the liquid holdup data of Minami.44 The data were collected for air-water and air-kerosene. The model systematically underpredicted the data, with an average error and average absolute error of ⫺20.8 and 33.5%, respectively, as shown in Table 2. Note that, as reported by Minami,44 the original Taitel and Dukler6 model performed poorly against his data. Modification of both the liquid wall friction factor and the interfacial friction factor, implemented in the present study model, improves the predictions of the stratified model considerably. Unified Annular Model. As shown in Table 1, Alves et al.27 provided 2 new field data points, in addition to the 75 data points taken from the Tulsa U. Fluid Flow Projects 共TUFFP兲 database, in which the wells are under annular flow. The model of Alves et al. shows excellent agreement with the data: For the 2 data points the average error and average absolute errors are 1.5%. For the 75 database points the average error is ⫺0.9% and the average absolute error is 9.8%. Entire Unified Model Validation. Following validation of the individual flow pattern models, the entire unified model was evaluated against the TUFFP wellbore databank, as reported by Ansari et al.29 The databank includes a total of 1,723 laboratory and field data, for both vertical and deviated wells. The data cover a wide range of flow conditions: pipe diameter of 1 to 8 in.; oil rate of 0 to 27,000 B/D; gas rate of 0 to 110,000 scf/D and oil gravity of 8.3 to 112°API. Additionally, six commonly used correlations and models have been evaluated against the databank. They are those of Ansari et al.,29 Chokshi et al.,30 Duns and Ros,5 Beggs and Brill,2 Hasan and Kabir,21 and the modified Hagedorn and Brown.3 The modifications of the Hagedorn and Brown correlation are the Griffith and Wallis45 correlation for bubble flow and the use of no-slip liquid holdup if greater than the calculated liquid holdup. Note that, except for the Beggs and Brill2 correlation, the other five methods were developed for vertical upward flow only. These methods are adopted in this study for deviated well conditions by incorporating the inclination angle in the gravitational pressure gradient calculations. The proposed unified model is the only mechanistic model applicable to all of the inclination angle range, from horizontal to vertical. The overall performance of the unified model showed an average error of ⫺3.8% and an absolute average error of 12.6%. The Hagedorn and Brown3 correlation showed a minimum average SPE Journal, Vol. 5, No. 3, September 2000

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

336 1,747 537 511 1,044 527 1,841 1,135 1,196 894 344 1,490 1,758 1,898 1,494 1,382 5,716 872 2,118 1,498 2,847

68.0 20.2 87.0 5.0 45.0 60.0 0.0 36.9 59.2 62.3 80.0 0.2 55.6 0.3 27.0 70.2 33.1 66.7 26.7 0.3 9.0

Water Cut (%)

Average error „%… Std. dev. avg. error Average absolute error „%… Std. dev. abs. avg. error

Case

Gas/Oil Ratio (scf/stbl) 1,851 1,282 1,990 2,220 1,518 2,588 1,540 1,371 1,386 1,817 2,998 1,160 1,638 840 1,016 1,199 802 2,006 767 1,221 680

⌬ P Measured 1,743 1,325 1,842 2,308 1,650 2,146 1,094 1,319 1,268 1,496 2,539 908 1,240 722 1,136 1,267 716 1,664 973 1,249 813

⌬ P Calculated 1,826 1,333 1,885 2,479 1,405 2,444 1,105 1,383 1,100 1,413 2,783 791 1,301 743 1,041 1,076 512 1,883 678 1,153 632

⫺5.8 3.4 ⫺7.4 4.0 8.7 ⫺17.1 ⫺29.0 ⫺3.8 ⫺8.5 ⫺17.7 ⫺15.3 ⫺21.7 ⫺24.3 ⫺14.0 11.8 5.7 ⫺10.7 ⫺17.0 26.9 2.3 19.6 À5.2 14.7 13.1 8.1

⌬ P Calculated

À10.5 12.2 12.3 10.3

⫺1.4 4.0 ⫺5.3 11.7 ⫺7.4 ⫺5.6 ⫺28.2 0.9 ⫺20.6 ⫺22.2 ⫺7.2 ⫺31.8 ⫺20.6 ⫺11.5 2.5 ⫺10.3 ⫺36.2 ⫺6.1 ⫺11.6 ⫺5.6 ⫺7.1

Error (%)

Chokshi et al. (Ref. 30)

Error (%)

Present Study

1,684 1,161 1,804 2,261 1,349 2,265 1,189 1,338 1,120 1,289 2,579 815 1,145 787 876 1,078 659 1,751 821 1,036 820

⌬ P Calculated

À11.7 12.1 14.5 8.3

⫺9.0 ⫺9.4 ⫺9.3 1.8 ⫺11.1 ⫺12.5 ⫺22.8 ⫺2.4 ⫺19.2 ⫺29.1 ⫺14.0 ⫺29.7 ⫺30.1 ⫺6.3 ⫺13.8 ⫺10.1 ⫺17.8 ⫺12.7 7.0 ⫺15.2 20.6

Error (%)

Hagerdon and Brown (Ref. 3)

TABLE 3– PERFORMANCE OF UNIFIED MODEL AND OTHER METHODS FOR DATA SET No. 1 „Ref. 3…

1,819 1,219 1,837 2,554 1,260 2,448 1,012 1,223 948 1,284 2,881 703 1,290 676 966 960 469 1,822 592 1,090 562

⌬ P Calculated

À16.1 14.0 17.5 12.0

⫺1.7 ⫺4.9 ⫺7.7 15.0 ⫺17.0 ⫺5.4 ⫺34.3 ⫺10.8 ⫺31.6 ⫺29.3 ⫺3.9 ⫺39.4 ⫺21.2 ⫺19.5 ⫺4.9 ⫺19.9 ⫺41.5 ⫺9.2 ⫺22.8 ⫺10.7 ⫺17.4

Error (%)

Ansari et al. (Ref. 29)

error and absolute average error of 1.2 and 9.3%, respectively. However, the databank includes about 400 data points collected by Hagedorn and Brown3 to develop their correlation. An objective comparison should exclude these data points from the databank. Unified Model Performance and Results. The ultimate goal of any model is to predict the flow behavior under field conditions. The performance of the proposed unified model under field conditions was evaluated by comparison between its predictions and directional well field data provided by British Petroleum and Statoil. Two sets of data were provided. The first data set includes 21 data points while the second data set includes 65 cases. The data include wells with different flow conditions: pipe diameter of 7 2 8 to 7 in.; inclination angles of 0 to 90°; oil rate of 79 to 2,658 B/D; gas rate of 42 to 23,045 Mscf/D, and water-cut of 0 to 80%. Of the total cases, 59 wells were producing naturally and the remaining 27 were on artificial lift. Each data point included, in addition to the geometrical and operational variables, the wellhead pressure, the wellhead and bottomhole temperatures and the total pressure drop. Physical Properties. The pressure/volume/temperature 共PVT兲 properties used were summarized by Brill and Beggs.46 The Glaso correlation was used for the prediction of the solution gas/oil ratio, oil formation volume factor and oil viscosity. The Standing z factor was used in the calculations of the gas phase properties. The Lee et al. correlation was used for the gas viscosity. The gas/oil surface tension was predicted by the Baker and Swerdloff correlation. The liquid phase 共oil and water兲 properties, namely, density, viscosity and surface tension, are calculated based on the volume fraction of the oil and water in the liquid phase. The volume fractions were calculated based on the in-situ flow rates, assuming no-slip between the oil and water. For the gas lift wells, the gas properties are calculated as follows. Up to the point of gas injection, the calculations are performed using the flow rate and specific gravity of the formation gas. At the point of gas injection, the formation gas flow rate is combined with the injection gas rate to give the total gas flow rate, with a weighted average specific gravity based on the two flow rates at standard conditions. From the point of injection to the surface, the PVT properties, including the solution gas oil ratio 共and hence free gas quantity兲, are determined based on the combined total gas specific gravity. No tuning of the PVT data was done. Results and Discussion. Table 3 reports the pressure drop prediction performance of the unified model, along with that of Chokshi et al.,30 Hagedorn and Brown3 and Ansari et al.,29 vs. the first data set 共21 data points兲. Note that Table 3. includes, in addition to the pressure drop, the gas/liquid ratio and the water cut. The comparison shows good agreement, with an average error of ⫺5.2% 关and a corresponding standard deviation 共s.d.兲 of 14.7兴 and an average absolute error of 13.1% 共with a s.d. of 8.1兲 for the unified model. Corresponding errors for the other methods are as follows: ⫺10.5% 共s.d. 12.2兲 and 12.3% 共s.d. 10.3兲 for Chokshi et al.,30 ⫺11.7% 共s.d. 12.1兲 and 14.5% 共s.d. 8.3兲 for Hagedorn and Brown,3 and ⫺16.1% 共s.d. 14.0兲 and 17.5% 共s.d. 12兲 for the Ansari et al.29 model. Fig. 7 shows a comparison between the predicted results of the unified model and measured pressure drops for the 65 cases of the second data set. The predictions of the proposed unified model show excellent agreement vs. this data set, with an average error of 0% 共s.d. 3.9兲, as compared to 4.5% 共s.d. 4.5兲 for the Chokshi et al.30 model. The average absolute error for the unified model and the Chokshi et al.30 model are 3.0% 共s.d. 2.5兲 and 5.5% 共s.d. 3.2兲, respectively. The overall performance of the model was evaluated vs. the combined two data sets, including all 86 well cases. The results were compared with the predictions of only the Chokshi et al.30 model. In addition, a sensitivity analysis was carried out based on the maximum deviation angle of the well, production method 共natural or artificial lift兲 and tubing diameter. All the results are summarized in Table 4. Gomez et al.: Unified Mechanistic Model for Steady-State Two-Phase Flow

Fig. 7–Comparison between unified model predictions and data set No. 2 „65 cases….

For the combined data sets the unified model shows excellent performance, with an average error of ⫺1.3% 共s.d. 8.2兲 and absolute error of 5.5% 共s.d. 6.2兲. These results are also shown graphically in Fig. 8. The Chokshi et al.30 model shows an average error and absolute error of 0.9% 共s.d. 9.6兲 and 7.1% 共s.d. 6.4兲, respectively. As can be seen from Table 4, except for the three small diameter well cases, the unified model shows better performance than the Chokshi et al. model, especially for large diameter tubing and deviated wells. It is believed that the unified slug flow model is the main reason for this behavior, since it is more suitable for directional flow. Both models perform equally well for the entire range of water cuts.

Conclusions A unified steady-state two-phase flow mechanistic model for the prediction of flow pattern, liquid holdup and pressure drop was presented that is applicable to the range of inclination angles from horizontal (0°) to upward vertical flow (90°). The model consists of a unified flow pattern prediction model and five individual unified models for the stratified, slug, bubble, annular and dispersed bubble flow patterns. The proposed unified model was evaluated and compared to the other six most commonly used models or correlations. This was carried out by running the unified model and the other methods against the TUFFP wellbore databank.29 The databank includes a total of 1,723 laboratory and field data for both vertical and deviated wells. The overall performance of the unified model showed an average error of ⫺3.8% and an absolute average error of 12.6%. The performance of the unified model and of other models and correlations was evaluated against 86 new directional well field data cases provided by British Petroleum and Statoil. The predictions of the unified model show excellent agreement with data, with an average error of ⫺1.3% and an absolute average error of 5.5%, with respective standard deviations of 8.2 and 6.2. A sensitivity analysis of the model performance was conducted with respect to tubing diameter, method of lift and maximum wellbore inclination angle. The unified model showed superior performance except for a limited number of small diameter wells. The predictions of the unified model were carried out without any tuning of either the model or the PVT data. It provides an accurate two-phase flow mechanistic model for research and design for the industry. SPE Journal, Vol. 5, No. 3, September 2000

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9.1

⫺1.1

8.8 4.9

⫺3.3 ⫺0.1

0°⬍ ␪ ⬍90°

7 28

86 cases

to 7 in.

Inclination

Diameter

Database 79 to 2,658 B/D

Oil Rate

8.2

10.8

⫺0.1

À1.3

5.8

⫺3.3

4.1 13.0

5.1

⫺2.2

0.1 ⫺4.3

2.7

Standard Deviation

1.7

Average Error (%)

Entire Database

Tubing 7 d⫽2 8 in. Tubing 1 d⫽4 2 in. Tubing 1 d⫽5 2 in. Tubing d ⫽7 in.

Naturally flowing Gas Lifted

Vertical ␪ ⫽90° Horizontal to vertical 0°⬍ ␪ ⬍90° Deviated wells 45°⬍ ␪ ⬍90°

Classification

Overall 86

31

28

24

Diameter 3

Production 59 27

64

19

Inclination 3

No. of Wells

42 to 23,045 Mscf/D

Gas Rate

5.5

4.0

6.7

5.9

5.5

3.2 10.5

6.0

4.2

2.5

Absolute Average Error (%)

Present Study

0 to 80%

Water Cut

6.2

2.7

6.4

9.0

2.1

2.5 8.7

6.9

3.4

1.3

Standard Deviation

0.9

4.1

⫺3.3

1.7

⫺0.9

4.6 ⫺7.2

0.7

0.4

7.4

Average Error (%)

9.6

5.7

12.0

9.5

4.6

4.7 12.4

10.7

5.2

4.0

Standard Deviation

7.1

6.0

8.7

7.3

3.5

5.7 10.5

8.1

4.0

7.4

Absolute Average Error (%)

Chokshi et al. (Ref. 30)

TABLE 4– OVERALL PERFORMANCE OF UNIFIED MODEL AND SENSITIVITY ANALYSIS RESULTS

6.4

3.7

8.8

6.1

2.0

3.4 9.6

6.9

3.3

4.0

Standard Deviation

max NS R S SC SL SG TB U V W

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

maximum no-slip radians slug body superficial core superficial liquid superficial gas Taylor bubble total slug unit vertical wall

Superscripts ⬃ ⫽ dimensionless m, n ⫽ Blasius equation exponents Acknowledgment This paper is dedicated to the memory of Dr. Zelimir Schmidt. References Fig. 8–Overall performance of the unified model vs. the entire new database „86 cases….

Nomenclature A C C0 d E F FA f g h H I L N Re p S v v 0⬁ X Y

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

area, L2, ft2 Blasius equation coefficient flow distribution coefficient diameter, L, ft entrainment fraction dimensionless group annular flow parameter Fanning friction factor acceleration due to gravity, L/t2, ft/sec2 liquid level height, L, ft liquid holdup interfacial annular parameter length, L, ft Reynolds number pressure, M/Lt2, lbf/ft2 perimeter, L, ft velocity, L/t, ft/sec single bubble rise velocity, L/t, ft/sec Lockhart and Martinelli parameter dimensionless group

Greek Letters

␣ ␦ ␮ ␲ ␾ ␪ ␳ ␶ ␴

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

void fraction film thickness viscosity, M/Lt, lbm/ft-sec 3.141 5926 annular entrainment parameter inclination angle measured from horizontal density, M/L3, lbm/ft3 shear stress, M/Lt2, lbf/ft2 surface tension, M/t2, lbf/ft

Subscripts c crit CB CD F G H I LS L M

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

core critical critical buoyancy critical diameter film gas horizontal interface liquid slug liquid mixture

Gomez et al.: Unified Mechanistic Model for Steady-State Two-Phase Flow

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SI Metric Conversion Factors bbl ⫻ 1.589 873 ft ⫻ 3.048* ft2 ⫻ 9.290 304* ft3 ⫻ 2.831 684 in. ⫻ 2.54* lbf ⫻ 4.448 222 lbm ⫻ 4.535 924 psi ⫻ 6.894 757 *Conversion factors are exact.

E⫺01 E⫺01 E⫺02 E⫺02 E⫹00 E⫹00 E⫺01 E⫹00

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

m3 m m2 m3 cm N kg kPa SPEJ

Luis E. Gomez a member of Sigma Xi, is a PhD-degree candidate at the U. of Tulsa in Tulsa, Oklahoma. e-mail: [email protected]. He previously taught in the Mechanical Engineering Dept. of the U. de Los Andes. Gomez holds a BS degree in mechanical engineering from the U. de Los Andes and an MS degree in petroleum engineering from the U. of Tulsa. Ovadia Shoham is Professor of Petroleum Engineering and Director of the Separation Technology Projects at Tulsa U. in Tulsa, Oklahoma. e-mail: [email protected]. He teaches and conducts research in modeling of two-phase flow in pipes and its applications in oil and gas production, transportation, and separation. Shoham holds BS and MS degrees in chemical engineering from the Technion, Israel, and the U. of Houston, respectively, and a PhD degree in mechanical engineering from Tel Aviv U., Israel. He served as a 1989–92 and 1998–2000 member of the Production Operations Technical Committee and as a 1991–92 member of the Forum Series in North America Steering Committee. Zelimir Schmidt, deceased, was Professor of petroleum engineering and Director of Artificial Lift Projects at the U. of Tulsa, Oklahoma. He spent 10 years as a production engineer with INA-Naftaplin in the former Yugoslavia and served as a consultant to various companies before joining the U. of Tulsa faculty. Schmidt held an engineering degree from the U. of Zagreb and MS and PhD degrees in petroleum engineering form the U. of Tulsa. He served as a 1987–88 Distinguished Lecturer and was a 1994–95 Forum Series in South America and Caribbean Steering Committee member, a 1991–95 Editorial Review Committee member, and 1981–82 and 1994–96 U. of Tulsa Student Chapter Faculty Sponsor. Rajan N. Chokshi is a program project manager with TanData Corp. in Tulsa, Oklahoma. e-mail: [email protected]. His current interests are change management, enterprise software architecture, and emerging technologies in computing. He has more than 15 years’ experience in research and design of fluid-flow and artificial-lift problems. He has developed software for and taught professional courses in these areas and managed consulting projects in the U.S., Canada, Venezuela, and India. Chokshi holds BS and MS degrees in chemical engineering from Gujarat U., India, and Indian Ints. of Technology, Kanpur, respectively, and a PhD degree in petroleum engineering from the U. of Tulsa. Tor Northug is a principal engineer in the R&D Dept. of Statoil in Trondheim, Norway. e-mail: [email protected]. His research interests include multiphase flow, fluid mechanics, leak detection, and gas leakage/subsea blowouts. He previously worked for Technical U. of Trondheim, Sintef Hydrodynamic Laboratory, and Reinertsen Engineering Co. Northug holds a BS degree in civil engineering from Technical U. of Trondheim and an MS degree in fluid mechanics from Norwegian U. of Science and Technology.

Gomez et al.: Unified Mechanistic Model for Steady-State Two-Phase Flow

SPE Journal, Vol. 5, No. 3, September 2000