Horizontal to Vertical Upward FlowL.E. Gomez, SPE, Ovadia Shoham, SPE, and Zelimir Schmidt,* SPE, U. of Tulsa;
R.N. Chokshi,** SPE, Zenith ETX Co.; and Tor Northug, Statoil
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SummaryA unified steady-state two-phase flow mechanistic model forprediction of flow pattern, liquid holdup and pressure drop is psented that is applicable to the range of inclination angles frhorizontal(0°) to upwardvertical flow (90°). The model is baseon two-phase flow physical phenomena, incorporating recentvelopments in this area. It consists of a unified flow pattern pdiction model and unified individual models for stratified, slububble, annular and dispersed bubble flow. The model canapplied to vertical, directional and horizontal wells, ahorizontal-near horizontal pipelines. The proposed model impments new criteria for eliminating discontinuity problems, proving smooth transitions between the different flow patterns.
The new model has been initially validated against existivarious, elaborated, laboratory and field databases. Followingvalidation, the model is tested against a new set of field data, fthe North Sea and Prudhoe Bay, Alaska, which includes 86 caThe proposed model is also compared with six commonly umodels and correlations. The model showed outstanding pemance for the pressure drop prediction, with a21.3% averageerror, a 5.5% absolute average error and 6.2 standard deviaThe proposed model provides an accurate two-phase flow menistic model for research and design for the industry.
IntroductionEarly predictive means for two-phase flow were based onempirical approach. This was due to both the complex naturtwo-phase flow and the need for design methods for industry.most commonly used correlations have been the Dukleret al.1
and Beggs and Brill2 correlations for flow in pipelines, and thHagedorn and Brown3 and Ros4/Duns and Ros5 correlations forflow in wellbores. This approach was successful for solving twphase flow problems for more than 40 years, with an updaperformance of630% error. However, the empirical approachas never addressed the ‘‘why’’ and ‘‘how’’ problems for twophase flow phenomena. Also, it is believed that no furtherbetter accuracy can be achieved through this approach.
A new approach emerged in the early 1980’s, namely,mechanistic modeling approach. This approach attempts tomore light on the physical phenomena. The flow mechaniscausing two-phase flow to occur are determined and modmathematically. A fundamental postulate in this method isexistence of various flow configurations or flow patterns, incluing stratified flow, slug flow, annular flow, bubble flow, chuflow and dispersed bubble flow. These flow patterns are shschematically inFig. 1. The first objective of this approach isthus, to predict the existing flow pattern for a given system. Tha separate model is developed for each flow pattern to prediccorresponding hydrodynamics and heat transfer. These modeexpected to be more reliable and general because they incorp
This paper (SPE 65705) was revised for publication from paper SPE 56520, presented atthe 1999 SPE Annual Technical Conference and Exhibition held in Houston, 5–8 October.Original manuscript received for review 20 October 1999. Revised manuscript received 20May 2000. Manuscript peer approved 9 June 2000.
SPE Journal5 ~3!, September 2000
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the mechanisms and the important parameters of the flow.current research is conducted through the modeling approach.plication of models in the field is now underway, showing tpotential of this method.
The mechanistic models developed over the past two dechave been formulated separately for pipelines and wellbores.lowing is a brief review of the literature for these two cases.
Pipeline Models.These models are applicable for horizontal anear horizontal flow conditions, namely,610°. The pioneeringand most durable model for flow pattern prediction in pipelinwas presented by Taitel and Dukler.6 Other studies have beecarried out for the prediction of specific transitions, such asonset of slug flow,7 or different flow conditions, such as higpressure.8 Separate models have been developed for stratiflow,6,9-11 slug flow,12-14 annular flow15,16 and dispersed bubbleflow ~the homogeneous no-slip model17!. A comprehensivemechanistic model, incorporating a flow pattern prediction moand separate models for the different flow patterns, was preseby Xiao et al.18 for pipeline design.
Wellbore Models. These models are applicable mainly for vercal flow but can be applied as an approximation for off-verticsharply inclined flow (60°<u<90°) also. A flow pattern predic-tion model was proposed by Taitelet al.19 for vertical flow, whichwas later extended to sharply inclined flow by Barneaet al.20
Specific models for the prediction of the flow behavior have bedeveloped for bubble flow21,22slug flow23-25and annular flow.26,27
Comprehensive mechanistic models for vertical flow have bpresented by Ozonet al.,28 by Hasan and Kabir,21 by Ansariet al.29 and by Chokshiet al.30
Unified Models. Attempts have been made in recent years tovelop unified models that are applicable for the range of inclition angles between horizontal(0°) and upwardvertical (90°)flow. These models are practical since they incorporate the innation angle. Thus, there is no need to apply different modelsthe different inclination angles encountered in horizontal, inclinand vertical pipes. A unified flow pattern prediction model wpresented by Barnea31 that is valid for the entire range of inclination angles (290°<u<90°). Felizola and Shoham32 presented aunified slug flow model applicable to the inclination angle ranfrom horizontal to upward vertical flow. A unified mechanistmodel applicable to horizontal, upward and downward flow coditions was presented by Petalas and Aziz,33 which was testedagainst a large number of laboratory and field data. RecenGomezet al.34 presented a unified correlation for the predictionthe liquid holdup in the slug body.
The above literature review reveals that separate compresive mechanistic models are available for pipeline flow and wbore flow. Only very few studies have been published on unifimodeling. The objective of this paper is to present a systemacomprehensive, unified model applicable for the range of inclition angles between horizontal(0°) andvertical (90°). This willprovide more efficient computing algorithms, because the mocan be applied conveniently for both pipelines and wellbor
1086-055X/2000/5~3!/339/12/$5.0010.50 339
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without the need to switch among different models. The propomodel will be evaluated against new field data, along with otpublished models and correlations.
Unified Model FormulationThe unified model consists of a unified flow pattern predictmodel and separate unified models for the different existing flpatterns. These are briefly described below.
Unified Flow Pattern Prediction Model. The Barnea31 model isapplicable for the entire range of inclination angles, namely, frupward vertical flow to downward vertical flow (290°<u<90°). Below is a summary of the applicable transition critefor this study, including the stratified to nonstratified, slug to dpersed bubble, annular to slug and bubble to slug flow.
Stratified to Nonstratified Transition.The criterion for thistransition is the same as the original one proposed by TaitelDukler,6 based on a simplified Kelvin–Helmholtz stability analsis given by
F2S 1
~12hL!2
vG2 dAL /dhL
AG
D >1, ~1!
where the superscript tilde symbol ‘‘;’’ represents a dimensionless parameter~length and area are normalized withd and d2,respectively, and the phase velocity is normalized with the cosponding superficial velocity!. F is a dimensionless group giveby
F5A rG
~rL2rG!
vSG
Adg cosu. ~2!
Slug to Dispersed Bubble Transition.The slug to dispersedbubble transition occurs at high liquid flow rates, where the tbulent forces overcome the interfacial tension forces, disperthe gas phase into small bubbles. The resulting maximum busize can be determined from
dmax5F4.15S vSG
vMD 0.5
10.725G S s
rLD 0.6S 2 f M vM
3
d D 20.4
. ~3!
Two critical bubble diameters are considered. The first iscritical diameter below which bubbles do not deform, avoidiagglomeration or coalescence, given by
Fig. 1–Flow patterns in pipelines and wellbores for horizontalto vertical flow patterns.
340 Gomezet al.: Unified Mechanistic Model for Steady-State Two-
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onow
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r-ingble
heg
dCD52S 0.4s
~rL2rG!gD 1/2
. ~4!
The other critical diameter is applicable to shallow inclination(610°) where, due to buoyancy, bubbles larger than this diammigrate to the upper part of the pipe causing ‘‘creaming’’ atransition to slug flow as follows:
dCB53
8
rL
~rL2rG!
f M vM2
g cosu. ~5!
Transition to dispersed bubble flow will occur when the maximupossible bubble diameter, given by Eq. 3, is less than both critdiameters given by Eqs. 4 or 5, namely,
dmax,dCD and dCB . ~6!
The transition boundary given by Eq. 6 is valid fora<0.52,which represents the maximum possible packing of bubbles fcubic lattice configuration. For larger values of void fraction, aglomeration of bubbles occurs, independent of the turbuleforces, resulting in a transition to slug flow. This criterion is givby
aNS5vSG
vSG1vSL50.52. ~7!
Annular to Slug Transition.Two mechanisms are responsible for this transition from annular flow to slug flow, causinblockage of the gas core by the liquid phase. The two mechaniare based on the characteristic film structure of annular flow:
1. Instability of the liquid film due to downward flow near thpipe wall. The criterion for the instability of the film is obtainefrom the simultaneous solution of the following two dimensioless equations:
Y51175HL
~12HL!2.5HL2
1
HL3 X2, ~8!
Y>22 ~3/2! HL
HL3~12 ~3/2! HL!
X2 ~9!
where X is the Lockhart and Martinelli parameter andY is adimensionless gravity group defined respectively by
X25
4CL
d S rL vSL d
mLD 2n rL vSL
2
2
4CG
d S rG vSG d
mGD 2m rG vSG
2
2
5
S dp
dL DSL
S dp
dL DSG
, ~10!
Y5~rL2rG!g sinu
S dp
dL DSG
. ~11!
Note that Eq. 8 yields the steady-state solution for the liqholdupHL , while Eq. 9 yields the value of the liquid holdup thasatisfies the condition of the film instability.
2. Wave growth on the interface due to large liquid supply frothe film. If sufficient liquid is provided, the wave will grow anbridge the pipe, resulting in slug flow. The condition for occurence of this mechanism is
HL>0.24. ~12!
Transition from annular to slug flow will occur whenever onof the two criteria is satisfied. A smooth change between themechanisms is obtained when the inclination angle varies overentire range of inclinations, or when a change occurs in theerationing conditions.
Phase Flow SPE Journal, Vol. 5, No. 3, September 2000
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Bubble to Slug Transition.The transition from bubble to slugflow occurs at relatively lower liquid flow rates compared to ttransition from slug to dispersed bubble flow. Under these contions the turbulent forces are negligible, and the transitioncaused by coalescence of bubbles at a critical gas void fractioa50.25, as follows:
vSL512a
avSG21.53~12a!0.5S g~rL2rG!s
rL2 D 1/4
sinu. ~13!
The bubble regime can exist at low liquid flow rates as givby Eq. 13, provided that the pipe diameter is larger thd.19@(rL2rG)s/rL
2g#0.5 and only for sharply inclined pipeswith inclination angles between approximately 60 and 90°.
Elimination of Transition Discontinuities. Mechanistic modelsfor the prediction of pressure traverses in multiphase flownotorious for creating discontinuities. This is the result of switcing from one flow pattern model to another as the transitboundary is crossed. Different models are used for different flpatterns to predict the liquid holdup and pressure drop, whmight result in a discontinuity. In order to avoid this problemthe proposed model, the following criteria were implementedsmooth the transitions between the different flow patterns.
Bubble to Slug and Slug to Dispersed Bubble TransitionNear the transition boundaries from slug to bubble or disperbubble flow, the liquid film/gas pocket region behind the slbody, namely,LF , becomes small. The short film/gas length cprevent the slug flow model from converging. Thus, to solve tproblem, when slug flow is predicted near these transition bouaries, the following constraints were developed:
if LF<1.2d and vSL<0.6 m/s ⇒bubble flow,
if LF<1.2d and vSL>0.6 m/s ⇒dispersed bubble flow.~14!
The valueLF /d51.2 is based on the mechanism that onceTaylor bubble length approaches the pipe diameter, it becounstable and might break into small bubbles. Under these cotions, for high superficial liquid velocities, due to turbulencetensity and bubble breakup and dispersion, the resulting flowtern will be dispersed bubble flow. However, for low superficliquid velocities, due to low turbulence intensity and coalesceof the small bubble to larger ones, the resulting flow pattern wbe bubble flow.
Slug to Annular Transition.A two-fold problem is associatedwith this transition boundary. First, a discontinuity in the pressgradient between slug flow and annular flow occurs. Also, if sflow is predicted near this transition boundary, due to the highrates, the film/gas zone becomes long, resulting in a very thinthickness, one approaching zero. This can prevent the slugmodel from converging. To alleviate the two problems, a trantion zone is created between slug flow and annular flow basethe superficial gas velocity. The transition zone is predicted bycritical velocity corresponding to the droplet model used by Taet al.19 as follows:
vSG,crit53.1S s g sinu~rL2rG!
rG2 D 0.25
. ~15!
Thus, for a given superficial liquid velocity, the transition regiis defined when the superficial gas velocity is greater thancritical gas velocity~given in Eq. 15! and less than the superficiagas velocity on the transition boundary to annular flow~predictedby the Barnea model31!. Hence, when slug flow is predicted in thtransition zone, the pressure gradient is averaged betweenpressure gradient under slug flow and annular flow conditioThe corresponding slug flow pressure gradient is calculated agiven superficial liquid velocity and the critical superficial gvelocity, given by Eq. 15. Similarly, the corresponding pressgradient under annular flow is calculated at the given superfi
Gomezet al.: Unified Mechanistic Model for Steady-State Two-Phase
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liquid velocity and the superficial gas velocity on the transitioboundary to annular flow, predicted by the Barnea model.31 Thisaveraging eliminates numerical problems and ensures a smpressure gradient across the slug to annular boundary.
Unified Stratified Flow Model. The physical model for stratifiedflow is given inFig. 2. A modified form of the Taitel and Dukler6
model is used here. Two modifications are introduced: the liqwall friction factor is determined by Ouyang and Aziz35 and theinterfacial friction factor is given by Bakeret al.36
Momentum Balances.The momentum~force! balances for theliquid and gas phases are given, respectively, by
2AL
dp
dL2tWLSL1t ISI2rLALg sinu50, ~16!
2AG
dp
dL2tWGSG2t ISI2rGAGg sinu50. ~17!
Eliminating the pressure gradient from Eqs. 16 and 17, the cobined momentum equation for the two phases is obtained aslows:
tWL
SL
AL2tWG
SG
AG2t ISI S 1
AL1
1
AGD1~rL2rG!g sinu50.
~18!
The combined momentum equation is an implicit equation forhL~or hL /d!, the liquid level in the pipe. Solution of the equationcarried out by a trial and error procedure, requires the determtion of the different geometrical, velocity and shear stress vaables. Under high gas and liquid flow rates, multiple solutions coccur. It can be shown that, in this case, the smallest of the thsolutions is the physical and stable solution.
Once the liquid levelhL /d is determined, the liquid holdupHL , can be calculated in a straightforward manner from geomecal relationships as follows:
HL5
p2cos21S 2hL
d21D1S 2
hL
d21DA12S 2
hL
d21D 2
p.
~19!
Once the liquid holdup is determined, the pressure gradient cadetermined from either Eq. 16 or 17. Either equation providesfrictional and the gravitational pressure losses, and neglectsaccelerational pressure losses.
Closure Relationships.The wall shear stresses correspondito each phase are determined based on single-phase analysisthe hydraulic diameter concept, as follows~Fanning friction factorformulation!:
tWL5 f L
rL vL2
2and tWG5 f G
rG vG2
2. ~20!
Fig. 2–Physical model for stratified flow.
Flow SPE Journal, Vol. 5, No. 3, September 2000 341
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The respective hydraulic diameters of the liquid and gas phaare given by
dL54AL
SLand dG5
4AG
SG1SI. ~21!
The Reynolds numbers of each of the phases are
NReL5
dL vL rL
mLand NReG
5dG vG rG
mG. ~22!
Taitel and Dukler6 proposed that both the liquid and gas wafriction factors, f L and f G , can be calculated using a standafriction factor chart. However, Ouyang and Aziz35 found this pro-cedure to be appropriate for the gas phase only. This is due tofact that the liquid wall friction factor can be affected significanby the interfacial shear stress, especially for low liquid holdconditions. Thus,f G is determined from a standard chart, whilef Lis determined by a new correlation developed by OuyangAziz35 that incorporates the gas and liquid flow rates, given a
f G516
NReG
for NReG<2,300, ~23!
f G50.001 375F11S 23104«
d1
106
NReGD 1/3G
for NReG.2,300,
f L51.6291
NReL
0.5161S vSG
vSLD 0.0926
. ~24!
The interfacial shear stress is given, by definition, as
t I5 f I
rG~vG2vL!uvG2vLu2
. ~25!
The interfacial friction factor for stratified smooth flow is takenthe friction factor between the gas phase and the wall. Howefor stratified wavy flow, as suggested by Xiaoet al.,18 the inter-facial friction factor is that given by Bakeret al.36
Unified Slug Flow Model. The unified and comprehensive analsis of slug flow, presented by Taitel and Barnea,37 is used in thepresent study with the following features: a uniform film along tliquid film/gas pocket zone; a global momentum balance on a sunit for pressure drop calculations, and a new correlation~Gomezet al.34! for the liquid holdup in the slug body. The original Taiteand Barnea37 model was extended to vertical flow by assumingsymmetric film around the Taylor bubble for inclination anglbetween 86 and 90°.
With the above characteristics, the original model is simplificonsiderably, as given below, avoiding the need for numerintegration along the liquid film region. The proposed simplifimodel is considered to be sufficiently accurate for practical apcations. Refer toFig. 3 for the physical model for slug flow.
Mass Balances.An overall liquid mass balance over a sluunit results in
vSL5vLLSHLLS
LS
LU1vLTBHLTB
LF
LU. ~26!
A mass balance can also be applied between two crsectional areas, namely, in the slug body and in the film regiona coordinate system moving with the translational velocity,vTB ,yielding
~vTB2vLLS!HLLS5~vTB2vLTB!HLTB . ~27!
A continuity balance on both liquid and gas phases resultsconstant volumetric flow rate through any cross section of the s
342 Gomezet al.: Unified Mechanistic Model for Steady-State Two-
ses
llrd
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nd
ser,
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elug
las
edcald
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g
ss-, in
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unit. Applying this balance on cross sections in the liquid slbody and in the liquid film region gives, respectively,
vM5vSL1vSG5vLLSHLLS1vGLS~12HLLS!, ~28!
vM5vLTBHLTB1vGTB~12HLTB!. ~29!
Eq. 28 can be used to determinevLLS , the liquid velocity in theslug body, since the other variables are given in the form of csure relationships. Then, the liquid film velocity,vLTB , can bedetermined from Eq. 27 for a given liquid holdup in this regioHLTB . Also, from Eq. 29 it is possible to determinevGTB , the gasvelocity in the gas pocket.
The average liquid holdup in a slug unit is defined as
HLU5HLLSLS1HLTBLF
LU. ~30!
Using Eqs. 26–28, the expression for the liquid holdup becom
HLU5vTBHLLS1vGLS~12HLLS!2vSG
vTB. ~31!
Eq. 31 shows an interesting result, namely, that the average liholdup in a slug unit is independent of the lengths of the differslug zones.
Hydrodynamics of the Liquid Film.Considering a uniformliquid film thickness, a combined momentum equation, similarthat in the case of stratified flow, can be obtained for the film/gpocket zone as follows:
tWFSF
AL2
tWGSG
AG2t ISIS 1
AF1
1
AGD1~rL2rG!g sinu50.
~32!
Solution of Eq. 32 yields the uniform~equilibrium! film thicknessor the liquid holdup in this region,HLTB . This value can be usedin a trial and error procedure, to determine the gas and liqvelocities in the slug and film/gas pocket regions, as discusbelow Eq. 29.
The liquid film length can be determined from
LF5LU2LS . ~33!
Fig. 3–Physical model for slug flow.
Phase Flow SPE Journal, Vol. 5, No. 3, September 2000
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The slug length,LS , is given as a closure relationship while thslug unit length,LU , can be determined from Eq. 26, as follow
LU5LS
vLLSHLLS2vLTBHLTB
vSL2vLTBHLTB. ~34!
Pressure Drop Calculations.The pressure drop for a sluunit can be calculated using a global force balance along aunit. Since the momentum fluxes in and out of the slug unit ctrol volume are identical, the pressure drop across this convolume for a uniform liquid film is
dp
dL5rU g sinu1
tSp d
A
LS
LU1
tWFSF1tWGSG
A
LF
LU, ~35!
whererU is the average density of the slug unit given by
rU5HLUrL1~12HLU!rG . ~36!
The first term on the right-hand side of Eq. 35 is the gravitatiopressure gradient, whereas the second and third terms reprthe frictional pressure gradient that results from the frictiolosses in the slug and in the film/gas pocket regions. No accetional pressure drop occurs in the slug unit control volume formlation.
Closure Relationships.The proposed model requires four closure relationships, namely, the liquid slug length,LS , the liquidholdup in the slug body,HLLS , the slug translational velocityvTB , and the gas velocity of the small bubbles entrained inliquid slug,vGLS. The closure relationships are given below.
A constant length ofLS530d and LS520d is used for fullydeveloped and stable slugs in horizontal and vertical pipes,spectively. For inclined flow, an average slug length is used baon inclination angle. However, for horizontal and near horizon(u561°) large diameter pipes (d.2 in.), the Scottet al.38 cor-relation is used, as given below
ln~LS!5225.4128.5@ ln~d!#0.1, ~37!
whered is expressed in inches andLS is in feet.The liquid holdup in the slug body,HLLS , is predicted using
where the slug superficial Reynolds number is calculated as
NReSL5
rLvMd
mL. ~39!
The slug translational velocity is determined from tBendiksen39 correlation, given by
vTB51.2vM1~0.542Agd cosu10.351Agd sinu!. ~40!
The gas velocity of the small bubbles entrained in the liqslug,vGLS, can be determined in the manner suggested by Haand Kabir,21 given in a later section by Eqs. 57 and 58. Note thfor this case the liquid holdup in the slug body,HLLS , should beused.
Unified Annular Flow Model. The model of Alveset al.27 de-veloped originally for vertical and sharply inclined flow has beextended in the present study to the entire range of inclinaangles from 0 to 90°, as given below. The physical modelannular flow is given inFig. 4.
The annular flow model equations are similar to the stratififlow model ones, since both patterns are separated flow. Theferences between the two models are the different geometricaclosure relationships, and the fact that the gas core in annularincludes liquid entrainment.
Momentum Balances.The linear momentum~force! balancesfor the liquid and gas core phases are given, respectively, by
Gomezet al.: Unified Mechanistic Model for Steady-State Two-Phase
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2tWF
SF
AF1t I
SI
AF2S dp
dL DF
2rLg sinu50, ~41!
2t I
SI
AC2S dp
dL DC
2rC g sinu50. ~42!
Eliminating the pressure gradients from the equations resultthe combined momentum equation for annular flow, namely,
tWF
SF
AF2t ISI S 1
AF1
1
ACD1~rL2rC!g sinu50. ~43!
Eq. 43 is an implicit equation for the film thicknessd ~or d/d! thatcan be solved by trial and error, provided the proper geometrivelocity and closure relationships are provided. These arescribed below.
Mass Balances.The velocities of the liquid film and the gacore can be determined from simple mass balance calculatyielding, respectively,
vF5vSL
~12E!d2
4d~d2d!, ~44!
vC5~vSG1vSLE!d2
~d22d!2 . ~45!
The gas void fraction in the core and the core average denand viscosity are given, respectively, by
aC5vSG
vSG1vSLE, ~46!
rC5rGaC1rL~12aC!, ~47!
mC5mGaC1mL~12aC!. ~48!
Fig. 4–Physical model for annular flow.
Flow SPE Journal, Vol. 5, No. 3, September 2000 343
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Closure Relationships.The liquid wall shear stress is detemined from single-phase flow calculations based on the hydradiameter concept.
The most difficult task in modeling annular flow is the detemination of the interfacial shear stress,t I , and the entrainmenfraction, E. By all means this is an unresolved problem evenvertical or horizontal flow conditions.
The definition of the interfacial shear stress for annular flow
t I5 f IrC
~vC2vF!uvC2vFu2
. ~49!
As suggested by Alveset al.,27 the interfacial friction factor canbe expressed by
f I5 f SC I , ~50!
where f SC is the friction factor that would be obtained if only thcore~gas phase and entrainment! flows in the pipe. Calculation off SC should be based on the core superficial velocity (vSC5vSG1EvSL) and the core average density and viscosity given, resptively, by Eqs. 47 and 48. The interfacial correction parameterI isused to take into account the roughness of the interface. Diffeexpressions forI are given by Alveset al.27 for vertical flow only.In the present study, the parameterI is an average betweenhorizontal factor and a vertical factor, based on the inclinatangle,u, as follows:
I u5I H cos2 u1I V sin2 u. ~51!
The horizontal correction parameter is given by HenstockHanratty41 as
I H511850FA , ~52!
where
FA5@~0.707NReSL
0.5 !2.51~0.0379NReSL
0.9 !2.5#0.4
NReSG
0.9 S vL
vGD S rL
rGD 0.5
~53!
and NReSLand NReSG
are the liquid and gas superficial Reynolnumbers, respectively. The vertical correction parameter is gby Wallis17 as
I V511300d
d. ~54!
The entrainment fraction,E, is calculated by the Wallis17 corre-lation, given by
E512e2[0.125(f21.5)], ~55!
where
f5104vSGmG
s S rG
rLD 1/2
. ~56!
Unified Bubble Flow Model. Extension of the Hasan and Kabir21
bubble flow model for the entire range of wellbore inclinatio
344 Gomezet al.: Unified Mechanistic Model for Steady-State Two-
-ulic
r-
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rent
on
nd
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angles was carried out by taking the component of the bubblevelocity in the direction of the flow, as given below~seeFig. 5 forthe bubble flow physical model!.
The gas velocity is given by
vG5C0vM1v0` sinu HL0.5, ~57!
where vM is the mixture velocity,C0 is a velocity distributioncoefficient,v0` is the bubble rise velocity andHL
0.5 is a correctionfor bubble swarm. In the present study, the velocity distributcoefficient C051.15, as suggested by Chokshiet al.,30 and thebubble rise velocity is given by Harmathy41 ~in SI units! as fol-lows:
v0`51.53S gs~rL2rG!
rL2 D 0.25
. ~58!
Substituting for the gas velocity in terms of the superficial veloity yields
vSG
12HL5C0vM1v0` sinu HL
0.5. ~59!
Fig. 5–Physical model for bubble flow.
TABLE 1– DATABASE FOR INDIVIDUAL FLOW PATTERN MODELS VALIDATION
Eq. 59 must be solved numerically to determine the liquid holdHL . Once the liquid holdup is computed, the gravitational africtional pressure gradients are determined in a straightforwmanner.
For dispersed bubble flow, the homogeneous no-slip model17 isused. Details of this simple model are omitted here for brevity
Results and DiscussionThis section includes the validation of the developed unifimodel with published laboratory and field data, and the perfmance of the model with new field data.
Unified Model Validation. Initially, the individual flow patternmodels for slug flow, stratified flow, bubble flow and annular flowere validated against several sets of available laboratorylimited field data.Tables 1 and 2present the range of data anthe validation results, respectively.
Unified Slug Model.Validation of the proposed slug flowmodel 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 rangupward inclination angles between 10 and 90° at 10° increme
2. the Nulandet al.42 data for 10, 20, 45, 60, and 80° includinliquid 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 tGomezet al.34 slug body liquid holdup correlation with publisheexperimental data~including additional data other than the abovmentioned three sets!. As can be seen, the correlation follows thtrend of decreasing slug liquid holdup as the inclination anincreases.
Comparisons between the predictions of the unified slug moand the experimental data were carried out for both the liq
Fig. 6–Comparison between predicted and measured slug liq-uid holdup „Ref. 34….
Gomezet al.: Unified Mechanistic Model for Steady-State Two-Phase
p,ndard
.
edor-
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holdup (HLU) and the pressure gradient, averaged over a sunit. The results for the different data sources are given in Ta2.
Unified Bubble Model.The data of Caetanoet al.22 were usedto test the model for bubble flow. Note that the Caetanoet al.datawere acquired in an annulus configuration with a 3-in. casinner diameter~ID! and 1.66-in. tubing outer diameter~OD!. Forthis reason the comparison was carried out only for the liqholdup. An equivalent diameter was used that provides the scross-sectional area and superficial velocities that occur inannulus. The results show excellent agreement with an aveerror and an average absolute error of22.3 and 2.7%, respectively.
Unified Stratified Model.The stratified flow model was testeagainst the liquid holdup data of Minami.44 The data were col-lected for air-water and air-kerosene. The model systematicunderpredicted the data, with an average error and average alute error of220.8 and 33.5%, respectively, as shown in TableNote that, as reported by Minami,44 the original Taitel andDukler6 model performed poorly against his data. Modificationboth the liquid wall friction factor and the interfacial friction factor, implemented in the present study model, improves the pretions of the stratified model considerably.
Unified Annular Model. As shown in Table 1, Alveset al.27
provided 2 new field data points, in addition to the 75 data poitaken from the Tulsa U. Fluid Flow Projects~TUFFP! database, inwhich the wells are under annular flow. The model of Alveset al.shows excellent agreement with the data: For the 2 data pointsaverage error and average absolute errors are 1.5%. For thdatabase points the average error is20.9% and the average absolute error is 9.8%.
Entire Unified Model Validation.Following validation of theindividual flow pattern models, the entire unified model wevaluated against the TUFFP wellbore databank, as reporteAnsari et al.29 The databank includes a total of 1,723 laboratoand field data, for both vertical and deviated wells. The data coa wide range of flow conditions: pipe diameter of 1 to 8 in.; orate of 0 to 27,000 B/D; gas rate of 0 to 110,000 scf/D andgravity of 8.3 to 112°API. Additionally, six commonly used corelations and models have been evaluated against the dataThey are those of Ansariet al.,29 Chokshiet al.,30 Duns and Ros,5
Beggs and Brill,2 Hasan and Kabir,21 and the modified Hagedornand Brown.3 The modifications of the Hagedorn and Brown corelation are the Griffith and Wallis45 correlation for bubble flowand the use of no-slip liquid holdup if greater than the calculaliquid holdup. Note that, except for the Beggs and Brill2 correla-tion, the other five methods were developed for vertical upwflow only. These methods are adopted in this study for deviawell conditions by incorporating the inclination angle in the gratational pressure gradient calculations. The proposed unimodel is the only mechanistic model applicable to all of the incnation angle range, from horizontal to vertical.
The overall performance of the unified model showed an avage error of23.8% and an absolute average error of 12.6%. THagedorn and Brown3 correlation showed a minimum averag
Flow SPE Journal, Vol. 5, No. 3, September 2000 345
TA
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346 Gomezet al.: Unified Mechanistic Model for Steady-State Two-Phase Flow SPE Journal, Vol. 5, No. 3, September 2000
c-
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t
o
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e
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error and absolute average error of 1.2 and 9.3%, respectivHowever, the databank includes about 400 data points colleby Hagedorn and Brown3 to develop their correlation. An objective comparison should exclude these data points from the dbank.
Unified Model Performance and Results.The ultimate goal ofany model is to predict the flow behavior under field conditioThe performance of the proposed unified model under field cditions was evaluated by comparison between its predictionsdirectional well field data provided by British Petroleum and Stoil. Two sets of data were provided. The first data set includesdata points while the second data set includes 65 cases. Theinclude wells with different flow conditions: pipe diameter27
8 to 7 in.; inclination angles of 0 to 90°; oil rate of 79 to 2,65B/D; gas rate of 42 to 23,045 Mscf/D, and water-cut of 0 to 80Of the total cases, 59 wells were producing naturally andremaining 27 were on artificial lift. Each data point included,addition to the geometrical and operational variables, the wellhpressure, the wellhead and bottomhole temperatures and thepressure drop.
Physical Properties.The pressure/volume/temperature~PVT!properties used were summarized by Brill and Beggs.46 The Glasocorrelation was used for the prediction of the solution gas/oiltio, oil formation volume factor and oil viscosity. The Standingzfactor was used in the calculations of the gas phase properThe Leeet al. correlation was used for the gas viscosity. Tgas/oil surface tension was predicted by the Baker and Swerdcorrelation. The liquid phase~oil and water! properties, namely,density, viscosity and surface tension, are calculated based ovolume fraction of the oil and water in the liquid phase. Tvolume fractions were calculated based on the in-situ flow raassuming no-slip between the oil and water.
For the gas lift wells, the gas properties are calculated aslows. Up to the point of gas injection, the calculations are pformed using the flow rate and specific gravity of the formatigas. At the point of gas injection, the formation gas flow ratecombined with the injection gas rate to give the total gas flow rawith a weighted average specific gravity based on the two flrates at standard conditions. From the point of injection tosurface, the PVT properties, including the solution gas oil ra~and hence free gas quantity!, are determined based on the combined total gas specific gravity. No tuning of the PVT data wdone.
Results and Discussion.Table 3 reports the pressure drop prediction performance of the unified model, along with thatChokshiet al.,30 Hagedorn and Brown3 and Ansariet al.,29 vs. thefirst data set~21 data points!. Note that Table 3. includes, inaddition to the pressure drop, the gas/liquid ratio and the wcut. The comparison shows good agreement, with an averageof 25.2% @and a corresponding standard deviation~s.d.! of 14.7#and an average absolute error of 13.1%~with a s.d. of 8.1! for theunified model. Corresponding errors for the other methods arfollows: 210.5% ~s.d. 12.2! and 12.3%~s.d. 10.3! for Chokshiet al.,30 211.7% ~s.d. 12.1! and 14.5%~s.d. 8.3! for Hagedornand Brown,3 and216.1% ~s.d. 14.0! and 17.5%~s.d. 12! for theAnsari et al.29 model.
Fig. 7 shows a comparison between the predicted results ofunified model and measured pressure drops for the 65 cases osecond data set. The predictions of the proposed unified mshow excellent agreement vs. this data set, with an averageof 0% ~s.d. 3.9!, as compared to 4.5%~s.d. 4.5! for the Chokshiet al.30 model. The average absolute error for the unified moand the Chokshiet al.30 model are 3.0%~s.d. 2.5! and 5.5%~s.d.3.2!, respectively.
The overall performance of the model was evaluated vs.combined two data sets, including all 86 well cases. The reswere compared with the predictions of only the Chokshiet al.30
model. In addition, a sensitivity analysis was carried out basedthe maximum deviation angle of the well, production meth~natural or artificial lift! and tubing diameter. All the results arsummarized inTable 4.
Gomezet al.: Unified Mechanistic Model for Steady-State Two-Phase
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For the combined data sets the unified model shows exceperformance, with an average error of21.3% ~s.d. 8.2! and ab-solute error of 5.5%~s.d. 6.2!. These results are also showgraphically inFig. 8. The Chokshiet al.30 model shows an average error and absolute error of 0.9%~s.d. 9.6! and 7.1%~s.d. 6.4!,respectively. As can be seen from Table 4, except for the thsmall diameter well cases, the unified model shows better permance than the Chokshiet al.model, especially for large diametetubing and deviated wells. It is believed that the unified slug flmodel is the main reason for this behavior, since it is more sable for directional flow. Both models perform equally well fothe entire range of water cuts.
ConclusionsA unified steady-state two-phase flow mechanistic model forprediction of flow pattern, liquid holdup and pressure drop wpresented that is applicable to the range of inclination angles fhorizontal(0°) to upwardvertical flow (90°). The model consistof a unified flow pattern prediction model and five individual unfied models for the stratified, slug, bubble, annular and disperbubble flow patterns.
The proposed unified model was evaluated and compared toother six most commonly used models or correlations. This wcarried out by running the unified model and the other methagainst the TUFFP wellbore databank.29 The databank includes atotal of 1,723 laboratory and field data for both vertical and deated wells. The overall performance of the unified model showan average error of23.8% and an absolute average error12.6%.
The performance of the unified model and of other modelscorrelations was evaluated against 86 new directional well fidata cases provided by British Petroleum and Statoil. The pretions of the unified model show excellent agreement with dawith an average error of21.3% and an absolute average error5.5%, with respective standard deviations of 8.2 and 6.2. A ssitivity analysis of the model performance was conducted wrespect to tubing diameter, method of lift and maximum wellboinclination angle. The unified model showed superior performaexcept for a limited number of small diameter wells.
The predictions of the unified model were carried out withoany tuning of either the model or the PVT data. It providesaccurate two-phase flow mechanistic model for research andsign for the industry.
Fig. 7–Comparison between unified model predictions anddata set No. 2 „65 cases ….
Flow SPE Journal, Vol. 5, No. 3, September 2000 347
TA
BLE
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348 Gomezet al.: Unified Mechanistic Model for Steady-State Two-Phase Flow SPE Journal, Vol. 5, No. 3, September 2000
.
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Nomenclature
A 5 area, L2, ft2
C 5 Blasius equation coefficientC0 5 flow distribution coefficient
d 5 diameter, L, ftE 5 entrainment fractionF 5 dimensionless group
FA 5 annular flow parameterf 5 Fanning friction factorg 5 acceleration due to gravity, L/t2, ft/sec2
h 5 liquid level height, L, ftH 5 liquid holdupI 5 interfacial annular parameterL 5 length, L, ft
NRe 5 Reynolds numberp 5 pressure, M/Lt2, lbf/ft2
S 5 perimeter, L, ftv 5 velocity, L/t, ft/sec
v0` 5 single bubble rise velocity, L/t, ft/secX 5 Lockhart and Martinelli parameterY 5 dimensionless group
Greek Letters
a 5 void fractiond 5 film thicknessm 5 viscosity, M/Lt, lbm/ft-secp 5 3.141 5926f 5 annular entrainment parameteru 5 inclination angle measured from horizontalr 5 density, M/L3, lbm/ft3
; 5 dimensionlessm, n 5 Blasius equation exponents
AcknowledgmentThis paper is dedicated to the memory of Dr. Zelimir Schmidt
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SI Metric Conversion Factorsbbl 3 1.589 873 E201 5 m3
Luis E. Gomez a member of Sigma Xi, is a PhD-degree candi-date at the U. of Tulsa in Tulsa, Oklahoma. e-mail: [email protected]. He previously taught in the MechanicalEngineering Dept. of the U. de Los Andes. Gomez holds a BSdegree in mechanical engineering from the U. de Los Andesand an MS degree in petroleum engineering from the U. ofTulsa. Ovadia Shoham is Professor of Petroleum Engineeringand Director of the Separation Technology Projects at Tulsa U.in Tulsa, Oklahoma. e-mail: [email protected]. He teaches andconducts research in modeling of two-phase flow in pipes andits applications in oil and gas production, transportation, andseparation. Shoham holds BS and MS degrees in chemical en-gineering from the Technion, Israel, and the U. of Houston, re-spectively, and a PhD degree in mechanical engineeringfrom Tel Aviv U., Israel. He served as a 1989–92 and 1998–2000member of the Production Operations Technical Committeeand as a 1991–92 member of the Forum Series in NorthAmerica Steering Committee. Zelimir Schmidt, deceased,was Professor of petroleum engineering and Director of Artifi-cial Lift Projects at the U. of Tulsa, Oklahoma. He spent 10 yearsas a production engineer with INA-Naftaplin in the former Yu-goslavia and served as a consultant to various companies be-fore joining the U. of Tulsa faculty. Schmidt held an engineer-ing degree from the U. of Zagreb and MS and PhD degrees inpetroleum engineering form the U. of Tulsa. He served as a1987–88 Distinguished Lecturer and was a 1994–95 Forum Se-ries in South America and Caribbean Steering Committeemember, a 1991–95 Editorial Review Committee member, and1981–82 and 1994–96 U. of Tulsa Student Chapter FacultySponsor. Rajan N. Chokshi is a program project managerwith TanData Corp. in Tulsa, Oklahoma. e-mail:[email protected]. His current interests are change man-agement, enterprise software architecture, and emergingtechnologies in computing. He has more than 15 years’ expe-rience in research and design of fluid-flow and artificial-liftproblems. He has developed software for and taught profes-sional courses in these areas and managed consultingprojects in the U.S., Canada, Venezuela, and India. Chokshiholds BS and MS degrees in chemical engineering from Gu-jarat U., India, and Indian Ints. of Technology, Kanpur, respec-tively, 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]. Hisresearch interests include multiphase flow, fluid mechanics,leak detection, and gas leakage/subsea blowouts. He previ-ously worked for Technical U. of Trondheim, Sintef Hydrody-namic Laboratory, and Reinertsen Engineering Co. Northugholds a BS degree in civil engineering from Technical U. ofTrondheim and an MS degree in fluid mechanics from Norwe-gian U. of Science and Technology.
Phase Flow SPE Journal, Vol. 5, No. 3, September 2000
