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A Comprehensive Mechanistic Model forUpward Two-Phase Flow in WellboresA.M. Anssri, Pakistan Petroleum Ltd.; N .D. Sylvester, U. of Akrow and C. Sarioa, O. Shoham, andJ.P. Brill, U. of Tulsa

s % 40630Summary. A comprehemive model is formulated to predict the flow behavior for upward two-phase flow. This model is composed ofa model for flow-pattern prediction and a set of independent mechanistic models for predicting such flow characteristics as holdup andpressure dmp in bubble, slug, and annul= flow. fbe comprehensive model is evaluated by using a well data bank made up of 1,712 wellcases covering a wide variety of field data. Model peifortnance is also compared with six commonly used empirical correlations and theHasan-Kabr mechanistic model. Overall model performance is in good agreement with the data. In comparison with other methods, thecomprehensive model performed the best.

IntroductionTwo-ph.a.se flow is commonly encountered in the PeVO1eum, chefi-cal, and nuclear indushies. This frequent occurrence presents thechallenge of, understanding, analyzing, and designing two-phasesystems.

Because of the complex nature of two-phase flow, the problemwas first approached through empirical methods. The trend hasshifted recently to the modeling approach, Tbe fundamental postu-late of the modeling approach is the existence of flow patterns orflow configurations. Vwious theories have been developed to pre-dict flow patterns. Separate mcdels were developed for each flowpattern topredictflow characteristics like holdup and pressure drop.By considering basic fluid mechanics, the resulting models can beapplied with more confidence to flow condkions other than those.used for their development.

Only Ozon et al.] and Hasan and Kab@ published studies oncomprehensive mechanistic modeling of two-phase flow in verticalpipes. More work is needed to develop models that describe thephysical phenomena more rigorously.

The purpose of this study is to formulate a detailed comprehen-sive mechanistic model for upw%d two-phase flow. The compre-hensive model fmt predicts the existing flow patmn and then calcu-lates the flow vtiables by taking into account the actu.atmechanisms of the predicted flow pattern. The model is evaluatedagainst a wide range of experhhental attd field data available in theupdated Tulsa U, Fluid Ftow Projects (TUFFP) weU data bank. Theperformance of the mcdel is also compared with six empirical cor-relations and one mechanistic model used in the field.

FlowPattern PredictionTaitd et al? presented the basic work on mechanistic modeling offlow-pattern Wmsitiom forupwardtwo-phme flow. Theyidentiledfour distinct flow patterns (bubble, slug, chum, and annular flow)and formulated and evaluated tie transition boundaries among themWig: 1).,Bamea et al.4 later modified the transitions to extend theapphcabdity of the model to inclined flows. Bamea5 then combinedflow-pattern prediction models applicable to different inclinationangle ranges into one unified model. Based on these different works,flow pattern can be predicted by detining transition boundaxksamong bubble, slug, and annular flows.

Bubble/Slug Emsition.Taitel et a[.3 gave the minimum diameterat which bubble flow occurs as.

[ 1tin= 19.01 - . . . . . . . . . . . . . . . . . . . . . . . . (1)For pipes larger than this, tie basic transition mechanism for bubbleto slug flow is coalescence of small gas bubbles into large Taylorbubbles. This was found experimentally to occur at a void fraction

C -2pyriaht 1994 SC&W of Petr&a.m Engineers

SPE Production& Facilities, May 1994

of about 0.25. Using this vatuc of void fraction, we can express tbetransition in terms of superficial and slip velocities

VSg=0.25V, +0.333USL, . . . . . (2)

where v. is the slip or bubble-rise velocity given by

[1%

v, = 1.53 gc7L(pL-pG) . . .---r

. (3)

This is shown as Transition A in Fig. 2.

Dispersed Bubble ltansition. Athigh liquid rates, turbulent forcesbreakkwge gas bubbles down into small ones, even at void fractionsexceeding 0.25. This yields the transition to dispersed bubble flows:

() 0,5vs.= 0.725 + 4.15 .% + SL . . . . . . . . . . . . . . . . . . . ..(4)Thii is shown as Transition B in Fig. 2.

At high gas velocities, this transition is governed by the maxi-mum packing of bubbles to give mdcscence. Scott and Kouba7concluded that this occurs at a void fraction of 0.76, giving the tramsition for no-slip d%persed bubble flow as

%E=3J7%L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

This is shown as Tramitio C in Fig. 2.

Transition to Anmdar FIow. The transition criterion for a.nnukmflow is based on the gas-phase velocity required to prevent the en-trained tiquid droplets from. falling back into tbe gas stream. Thisgives the transition as

[1 %guL(pLpG)!J~g = 3.1 P: (6)shown as Transition D in Fig. 2.

Bamea5 modified the same transition by considering the effectsof film thickness on the transition. One effect is that a thick liquidfdm bridged the gas core at high liquid rates. The other effect is in-stability of the liquid film, which causes downw%d flow of the filmat low liquid rates. The bridging mechanism is governed by theminimum liquid holdup required to forma liquid slug

HW> 0 .12, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(7)where HLF is the fraction of pipe cress section occupied by the liq-uid film, assuming no entrainment in the core. The mechanism offilm instability can be expressed in terms of the modified Lockhartand Martinelii parameters, Q and YM,

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ttB:::&E _sLUG FLOW

t

tCHURNFLOW

I

D............................................. . .,.. . . . . .. . . .1ANNULARFLow

To account for the effect of the liquid enmainment in the gas core,Eq. 7 is modified here as

()zfw+aLc* >0.12 . . . . . . . . . . . . . . . . . . . . . . . . .. (12)Annular flow exists if vsg is greater than that at tbe tmnsitio giv-

en by Eq. 6 and if the two Bamea criteria am satisfied. To satisfy theBamea criteria, Eq. 8 must first be solved implicitly for ~ti. Hmi s then calculated from Eq. 11; ifEq. 12 is not satisfied, annular flowexists. Eq. 8 cm usually be solved for afin by using a se.mnd-ordmNewton-Rapbso approach. Tbus, EqT8 can be expressed as

F&) = YM 2-15H~ FM . . . . . . . . . . . . . . . . . . (13)WJI1.5HH)and

1 .5HLWMIV&) = wJ1-1.5ffu)

+ (2-1.5HM)VwH~(~5.5HW)Wm(l-1.5HW)2 . .

The minimum dimensionless film thickness is then determined it-eratively from

F(laimj )!lfij+, dtij-m . . . . .

J(15)

. . . . . . . (14)

Fig. lFlow patterns in upward fwo-phase flow.

0.01 0.1 1 10 100SUPERFICIAL GAS VELOC3TY (M/S)

Fig. 2~pical flow-patiern map for wellbores.

YM = 2-15H~ x~, . . . . . . . . . . . . . . . . . . . . . . . . .wJ1-1.5ffm) (8)

.[J 3SLwhere XH = ~ . . . . . . . . . . . . . ...(9)

Sc

~w = g sin O(p=-pd

()

,. . . . . . . . . . . . . . . .dp

(10)z ~c

and B=(lFE)2(fr&SL). Fem geometric considerations, ffLF can beexpressed in term of minimum dimensionless film thickness, ~ti,as

Hm=@tin(l+tin). . . . . . . . . . . . . . . . . . . . . . . . . .. (11)

144

A good initial guess is C& =0.25.

FIow-Sehavior PredictionAfter tbe flow patterns me predicted, the next step is to developphysical models for the flow behavior in each flow pattern. This stepresulted in separate models for bubble, slug, and annular flow.Chum flow bas not yet been modeled bemuse of its complexity andis treated as part of slug flow. The models developed for other flowpatterns ze discussed below.

Bubble FlowModef.The bubble flow model is had on Caetanos8work for flow in an anmdus. The bubble flow and dispersed bubbleflow regimes are considered separately in developing the model forthe bubble flow pattern.

Because of the uniform distribution of gas bubbles in the liquidand no slippage between the two phases, dispersed bubble flow canbe approximated as a pseudmingle phase. WLtb this simptificmim,the two-phase pammetm can be expressed as

PIP=PLh +P&aiL....... . . . . . . . . . . . . . . . . . . .(l6)/%=#LaL+##-&). . . . . . . . . . . . . . . . . . . . . . . . . . (17)

andvrP=v~v~L+v~g, . . . . . . . . . . . . . . . . . . . . . . . . . . .. (18)

where ).L=vJvm. . . . . . . . . . . . . . . . . . .. (19)

For bubble flow, the slippage is considered by taking into accountthe bubble-rise velocity relative to the mixture velocity. By assum-ing a turbulent velocity profile fortbe mixture with the rising bubbleconcentrated more at the center than along the pipe waif, we can ex-press the slippage velocity as

%=vg-L2vm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (20)Hannathy6 gave an expression for bubble-rise velocity @q. 3).

To account for tie effect of bubble swarm, Zuber and Hench9 mod~.tied &is expression

[1 % H;, . . . . . . . . . . . . . . . . . . . . .~~ = 153 WAPL-%) fP; (21)

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(a) DEVELOPED SLUG UNIT (b) DEVELOPING SLUG UNITFig. S-Schematic of slug flow.

where the value of n varies from one study to another. b the presentstudy, ?/=0.5 was used to give the best results. Thus, Eq. 20 yields

[1 %g%(!%k ) @5 = k _ ,2yM, . . . . . . . . . . . . ( 221.53 P; L I-HLThis gives an implicit equation for the actual holdup for bubble

flow. The two-phase flow Patzmeters can now be cafctdated tlom

P=P=P.H. +P,(l-HJ . . . . . . . . . . . . . . . . . . . . . . . . . . (23)md#=p=pLH+j@-ffJ. . . . . . . . . . . . . . . . . . . . . . .. (24)

The two-phase pressure gm.dient is made up of three components.TbUs,

(a ($). (J (d.~ + ~ . . . . . . . . . . . . . . . . .. (25)()pz, =ppgsiutl. . . . . . . . . . . . . . . . . . . . . . . . . . .. (26)() =fTPPrP%, . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (27)&f 2d

where~p is obtained from a Moody diagram for a Reynolds numberdefined by

N&r*. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (28)

Because bubble flow is dominated by a relatively incompressibleliquid phase, there is no significant change in tbe density of the flow-ing fluids. This keeps the fluid velocity neady constant, resulting ines2entiaJly no pressure drop owing to acceleration. Therefore, theacceleration pressure drop is safely neglected, compared with theother pressure drop components.

Slug Flow ModeI. Fermndes et al.lo developed the fmt tbomughphysical model for slug flow. Sylvesterl 1 presented a simplified ver-

SPE Prodnctio & Facilides, May 1994

sion of this model. Tbe basic simplification was the use of a correla-tion for slug void fraction. These models used an impo~t 8ssump-tion of fully developed slug flow. McQui13an and Whalley12introduced the concept of developing flow during their study offlow-pattern transitions. Because of the basic difference in flow ge-ometty, the model keats fully developed and developing flow sepa-rately.

For a fuly developed slug unit (Fig. 3a), the overall gas andfiqtddmass balances give

S$ &%rB(l-HLr?) + k%s(%ts) . . . . (29)and v~L = (l-/Y)vmHW -~VL#LrB, . . . . . . . (30)

respectively, where

B=.%JLsw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(?1)Mass balances forliquid and gas from liquid slug to Taylor bubble

give(v,8cc-v~)Hm =[vw(-VL,J]HL,J . . . . . . . . . . . . . (32)

and (vwv8J(l-ffm) = (VTB-V8T.J(l-HLT,J. . . (33)The Taylor bubble-rise velocity is equal to the centerline velocity

plus the Taylor bubble-rise velocily in a stagnaut liquid colutmu i.e.,

[1 ,AUm1.2vm + 0.35 = . . . . . . . . . . . . . . . . . (34)pLSimilarly, the veloci~ of the gas hubbies in the liquid slug is

1/.

[1gu= 1.2,,+ 1.53 -. PA, . . . . . . . . . .. (35)

P?where the s=ond term on tfte right side represents the bubble-risevelocity defined in Eq.21.

fhevelocityofthe falliigfimcanb+ correlatedwitbthe filmthickness with the Brotz13 expression,

VLm==, . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (36)where fiL, the constant film thickness for developed flow, can be ex-pressed in terms of Taylor bubble void fraction to give

vm=9.916[gd(l-&) ]W. . . . . . . . . . . . . . . . . . . . .. (37)

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Tbeliquid slug void fraction can be obtained by Sylvestersll cor-relation and fmm Femandes et al. slo and Schmidts 14 data,

VssSm = 0.425 + 2.65vm . . . . . . . . . . . . . . . . . . . . . . . .. (38)

Eqs. 29 or 30,31 through 35,37, &d 38 can be solved iterativelyto obtain the following eight &mnvns that Min. the slug flowmodck & HLm, H81.s, V8TB, VLTB, VgU, VI,LS, and vrB. Vo and Sho-ham15 showed that these eight equations can be combined algebra-ically to give

(9.916 @)(l-_~H,,HVTB(l-H,TB) +x= 0,

. . . . . . . . . . . . . . . . . . . . .. (39)

.[vm.HgH[l.53[-~(1-H,u)O}]... . . . . . . . . . . . . ....- . . .. (40)

With VB and Hgf-s given by Eqs. 34 and 38, respectively, ~ canbe readily determined from Eq. 40. Eq. 39 is then used to fmd HLmwith an iterative solution method. De fting the left side of Eq. 39 asF(ffLT,s), then

F(HU,) = (9.916 @( I--)05HLrurB(I-HLm) +.l.

. . . . . . . . . (41)Taking the derivative of Eq. 41 with respect to HLTB yields

F(ffm) = VTB + (9.916@

[x (1--0+ HLTB 1J~

.. . . . . . . . . . . . . . . . . . . . .. (42)HLTB, the root of Eq. 39, is then determined iteratively flom

F(HL@H ErBj+L = LT8j-q . . . . . . . . . . . . . . (43)

The step-by-step pmcedme for determining all slug flow vari-ablesis as follows.

1. Catculate WB and HgLS from Eqs. 34 and 38.Z. Using Eqs.40tbrough 43, determine H~TB. A good initial

guess is HLTB=O.15.3. Solve Q. 37 for VLTB. Note that HgTB=I-HLTB.4, Solve !3q. 32 for v~. Note that HI,U=l-H8M.S. Solve Eq. 35 for Vg=.6. Solve Eq. 33 for vgTB.7. Solve ~. 29 or 30 forff.,8. Assuming that& =30d, calculate ,&u and LTB from the defini-

tion of~.To model developing slug flow, as in Fig. 3b, we must determine

the existence of such flow, This requires calculating and comparingthe cap length with the total length of a developed Taylor bubble.The expression for the cap length is12

[2

LC=&II~a+_ I-H 1NLm(L,B)-&..........(44)where WgTB and HNUB me cahlated at the terminal film thickness(called NusseIt fhn thickness) given by

[ 1113~N= ;dVNL.B#L(l-ffNLIB)8(PL-P,) . . . . . . . . . . . . . . . . . . . (45)146

TIE geommy of the film flow gives HNUB in terms of 8N as

()

2HNH. =l- 1-$ . . . . . . . . . . . . . . . . . . . . . . . . . . .. (46)

To determine vNgTB, the net flowrate of & can be used to obtain

Ngm=vrB_ (T8_gM)-. . . . . . . . . . . . . . . . . . (47)

Tbe length of the liquid slug can be calculated empirically fromLU=Cd, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (48)

where C was found16to vary from 16t045. Weuse C=30 in thisstudy. This gives the Taylor bubble length as

Lm=[LH/(l#)I& . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (49)From the comparison of& and LTB, if & ? LTB, the flow i s devel -

oping slug flow. This requires new values for L~, f&, andV&B calcula~d emherfor developed flow.

For L&., Tayl or bubble volume can be used

iB

% =~

A~(L)dL, . . . . . . . . . . . . . . . . . . . . . . . . . . .. (50).

where A& can be expressed in tennsof local holdup hLTB(L),which i turn can be expressed in terms of velocities by using Eq.32. This gives

~JL)=[l-(vr=i51)The volume w be expressed in terms of flow gmmetry as

()L% + Lu

Grz= YsgAp ~ -vgDAP(lHuJ:. (52)

Substitution of Eqs.51 md52into Eq.50 gives

()

L~ + LU LBVS* -V@(lHIM) ~TB

%1[ ~_v,,- vu)HmJzz 1do..................... (53)F.q. 53 can be integrated and then simplified to give

~~+(yP~+5=054)where u=-v~g~, . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. (55)

~ = %-YSU(2-HJ% LLS, . . . . . . . . . . . . . . . . . . . . . . . .. (56)

~nd ~ = %-vuIfw. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (57)&

Aft er ca lcul ati ng L$8, the other local parameters can be calcu-lated from

V:rB(O= ~-VIZ . . . . . . . . . . . . . . . . . . . . . . . . . . .. (58)

(vTrvU)HU

d : L; Jz. . . . (59)

In calculating pressure gradiems, we consider the effect of vary-ing film thickness and neglect the effect of friction along the Taylorbubble.

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For developed flow, the elevation component occurring across aslug unit is given by

() =[(l-i$pts +~pglgsinO, . . . . . . . . . . . . . . . . . . (60)dLewhere pU=pLHm +pJ1-Ifu). . . . . . . . . . . . . . . . . . .. (61)

The elevation component for developing slug flow is given by

() [(l~*)pU +B*pm.lgsinO, . . . . . . . . . . .. (62)dLewhere prm is based on average void fraction in the Taylor bubblesection witi varying film Wlckness. It is given by

P,B. =p,HLm+P,(l-Hma), . . . . . . . . . . . . . . . . . .. (63)

where HLTLM is obtained by integrating Eq. 59 and dividing by L*m,giving

2(uTrYw)HmHL7BA =

&G%-. . . . . . . . . . . . . . . . (64)

The friction component is the same for boththedeveloped and de-veloping slug flows because it occurs only across the liquid slug.This is given as

(.)dp =f_(l~), , . . . . . . . . . . . . . . . . . . . . . . . .. (65)m,where ~should bereplaced by,O*for developing flow. f~canbecalculated by using

R,U =pUvmd/pB. . . . . . . . . . . . . . . . . . . . . . . . . . . .. (66)

For the pressure gradient due to acceleration, he velocity in the@lm must be considered. The liquid in the slug experiences decel-eration as its upward velocity of VLLS changes to a downward veloc-ity of VLTB. The sm. Iicpid atso experiences acceleration when itexits from the film with a velocity VLTB into an upward moving liq-uid slug of velocity VT-I-S. ff the two changes in the liquid velocityoccur within the same slug unit, then no net pressure drop due to ac-celeration exists over that slug tmh.f hishappens when tbe slugflow is stable. The correlation used for slug length is based on itsstable length, so the possibility of a net pressure drop due to accel-eration does not exist. Therefore, no acceleration component ofpressure gradient is considwed over a slug unit.

Annular Flow ModeL A discussion on the hydrodynamics of annu-h flow was presented by Wallis. 17 Along with this, WafIis alsopresentd tie classic correlations for entrainment and interracialfriction as a function of film thickness. Later, Hewitt and Hall-Tay-lorlg gave a detailed analysis of the mechanisms involved in an an-nular flow. AI1 the models that followed later are based on this ap-proach.

A fully developed annular flow is shown in Fig 4. The conserva-tion of momentum applied separately to the core .amd the fdm yields

().% -zj~t-p.A.gsin6=0 . . . . . . . . . . . . . . . . . . . . (67)c()nd AP ~ +riSirpSF-pLAFg sin6= O. . . . . . (6g)F

The core density, PO is a no-slip density because the core is con-sidered a homogeneous mixture of gas and en fmined fiquid dropletsflowing at the same velocity. Thus,

PC= PL&C+P&aLC), . . . . . . . . . . . . . . . . . . . . . . . . . (69)FEv~L

where ,IK = .SX + EVSL

. . . . . . . . . . . . . . . . . . . . . . . . .. (70)

SPE production& Facilities, May t994

GAS CORE

LIQUID FILM

ENTRAINEDLIQUID DROPLET-

VF

ST1. .. I.. ..-. ... . .. ... . . .. . ,~ .

.

F

rF

i

-e8

Dc -

i

Fig. 4-gchematic of annular flow.

FE is the fraction of the total liquid enmined in the core, given bywallis as

FE = l-sxP[4.125(u.,+~ 1.5)1, . . . .; . (71)

where uCtir=lO, OOO~(~)fi. . . . . . . . . . . . . . . . . . . . . (72)

The shear stress in the film can be expressed as$

rF=f#L~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (73)

where f~ is obtained from a Moody diagram for a Reynolds numberdefined bypLvFdHF, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (74)~t, = ~L

where vF=_=w . . . . . . . . . . . . . . . . . ..(75)

~ddHF=@(14Jd .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (76)This gives

[- 12f_ZF=. @-FJP~ &4J . . . . . . . . . . . . . . (77)147

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Eq. 77 reducks to

()F _ d (l-FE)z f~ d~4 [4C!(ldJ]2f~L dL ~L , . . . . . . . . . . . . . . . . . . . . . (78)where the superfmid liquid friction pressure grad~ent is given by

()dp _ .fsLPLv&, .,, ,,, ,,, ,,, ..,,,,, ,,, .,,,,, ,,,z SL 2d (79)fiL is the fdiO. f?iCtOr for Supefi.ial liquid velocity and can be

obtained from a Moody diagram for a Reynolds number defined by

N ~$L=pLv,Ld/pP . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (80)For the shear stress at the interface,

Ti=~jPc!J~/8, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (81)

where vC=v~c/(1-2c3) z . . . . . . . . . . . . . . . . . . . . . . . . . . .. (82)

and~=fScZ, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (83)where Z is a correlating factor for in ferfacial friction aid the fibnthickness. Based on the performance of the model, tie Wallis ex-p~ssion for Zworks well for thin films or high entminments, where-as the WhMey and Hewitt19 expression is good for thick tilnts orlow entittments. fbu.s,

Z=l+300~ for FE> 0.9 . . . . . . . . . . . . . . . . . . . . . . . . . (84)

mdZ=l+24($13~ for FE

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TABLE lRANGE OF WELL DATANominal Di ameter Oil Rate

SourceGas Rate O il Grav it y

(in.) (STBID) (MSCFID) (API)Old TUFFP Data I108 01010,150Bank

1,5 to 10,567 9.5 to 70,5

Govier and 2t04 8 to 1,600 114 tO 27,400 17t0 112Fogaras%Asheim23 2718 tO 6 720 to 27,000 740 to 55,700 35 to S6Chierici et aI,Z4 27L3 to 5 0.3 to 69 6 to 27,914 8.3 to 46Prudhoe Bay 5% to 7 600 to 23,000 200 to 110,000 24 to 66.Includes dat a f rom Poenmmn and Catwnter,= Fmcher and Brown,% Ha9edom,27 Baxendall and 7homas,28 0rkiszews!4 ,29ESPmOl,m MMSU18.M,3* canacho,~ and field dti from seveml 03 cnmPanias.

ing from the exchange of liquid droplets between the core and the E3 indicates the degree of scattering of the errors shout their averagefilm is negligible. value.

Average ermc

EvahtationThe evaluation of the comprehensive model is cank?d out hycompming tie pressure drop from the mcdel with the measured datain the updated TUFFP well data bank that comprises 1,712 wellcases with a wide range of data, as given in Table 1. The perfor-mance of the model is also compared with that of six correlationsand another mechanistic model tlmt am commonty used in the petro-leum indus~.

Criteria for Comparison with DataThe evaluation of the model using the databank is hazed on the fol-lowing statistical parameters

Average percent error:

(104)().E,= +~eri x100, . . . . . . . . . . . . . . . . . . . . . . . .r. ,APM-APi. . .where ed = . . . . . . . . . . . . . . . . . . . . . . . . (105)Apiw

EI indicates the overatt trend of the perfommce, relative to themeasured pressure drop.

Absolute average percentage error

() E,= ~~le.1 x1OO. . . . . . . . . . . . . . . . .i.1 . . . . . . . . (lo6)Ez ind ic at es how large the errors are on the average.

Percent standard deviation

()E,= }~ej , . . . . . . . . . . . . . . . . . . . . . . . . . . . . .i= 1 (108)where e; = ApjCOIC-Api~,.r . . . . . . . . . . . . . . . . . . . . . . .. (109)

E4 indicates the overall trend independent of the measured pressuredzop.

Absolute average errcx

-() .Es= ;~lefl . . . . . . . . . . . . . . .i = 1 (1 10)Es is ASO independent of the measured pressure drop and indicatesthe magnitude of the average error.

Standard deviationr,= j (e,;_J2.. . . . . . . . (111),= ,E6 indicates the scattering of the results, independent of the mea-sured pressure drop.

Criteria for Comparison With OtherCorrelations and ModelsThe ccmelations and models used for the comparison area modifiedHagedorn and Bmwn,z7 Duns andROS,330rkiszewski29 with Trig-gia correction,~ Beggs and B2i1135 with Palmer correction,36 Muk-heriee and Brill.37 Aziz et al.. 38 ad RISLUI and Kabir.239 The com-pmison is accomplished bycomp.ming the statistical parameters.The comparison involves the use of a relative performance factordefined by

. J__eri -El)z IE,I-IE,J E2-E2E3. ~ . . . . . . . . . . .n-1 (107) F. = 1?21 I-IE1 .1 +-j= I m nunMO\ELHAGBRAzlz

DUNRSHASKABEGBRORKISMUKSR

EDB17120.7000.6651.3121.7191.9402,9824,2844.883

WV10861,1210.6001.1081,6782.0052.9065.2734.647

TASLE 2-RELATIVE PERFORMANCE FACTORSDW VNH__@~ ANH AB6261.3780.9192.0851,6782,2013.4452.3226.000

7550.0810.B760.8031.7111.8363,3215,8363,909

13810.0000,7741,0621.7921,7803,4144.6884.601

290.1432.0290.2821.1260.0092.82s1.2264.463

10521.2950.3861.7982.0562,5752,S833.12S5,342

6541.4610.4851.7642.0282.5902.5953.3165.140

SNH7450.1120.4571.3141.8522,0443.2613.5514.977

VSNH AAN387 700.142 0.0000.939 0,5461.486 0,2142.298 1.2131,996 1.0433,262 1.9724.403 8..::4.683

!BD.e.tim databank VW..erma! well caseq Dw=detia!ed well cases VNH.VB,liC4 wll ea$e$ without H.g8dom and Brown dam ANH41 well cases Wi!hout Hagedorn and3mw data A&d wll cams with 75% bubble fluw AS41 well cams with 100% slug 110!+ VS=vetical well cams with 100% slug flow SNH=dl w.U cases wilh100% s[ug flowWithout Hagtiorn and Brow. dal% VSNH=vec7ica wII cases with 100% slug flow Wm.! Hagedorn and Bmn dalx ,&AN4] well oases w%h 100% annular flow HAGBR.I.gedom and Brown corrwti.n; A217.=A2iz.! .1. cormlatlm DUNRS=Dun3 and R.. com!atiow HASKA=H.Sa. and Kabir mechmlstic modet BEG8R=Bww and .3rl! corml%x)RKIS.OrWzews.ki cerrela!iox MUKBR=Mukherjee and 8till correlation.

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E,-E, IE41-IE4 I- + m+ E3mm-E3dn IE4JIE, Inun

ETE5mh E=E6Mn+ E5m-E5ti + E6muE6& . . . . . . . . . . . . . . . . . . ,. (112)

The minimum and maximum possible vafues for Fw are O and 6,intilcating the best and worst performances, respectively.

The evacuation of the model in terms of Fw is given in TabIe 2,with the best value for each column being boldfaced.

OveraU Evahmtion. The ov?rall evaluation involves the entirecomprehensive model so as to study the combined petionnance ofall the independent flow pattern behaviomnodels together The eval-uation is first performed by using the entire &ta bank, resulting inCOL 1 of Table 2. Model performance is also checked for verticalwell cases only, resulting in Col. 2 of Table 2, and for deviated weUcases only, resulting in CoL 3 of Table 2. To make the comparisonunbiased with respect to fbe correlations, a second datsba.w wascreated that excluded 331 sets of data from the Hagedorn and Brownstudy. For this reduced data bank, the results for all vertical wellcases are shown in Col. 4 of Table 2, and the results for combinedveticaf and deviated weil cases are shown in COL 5 of Table 2.

Evaluation of Individual Ftow Pattern ModeLs. ~e perfmmanceof individual flow pattern models is based on sets of data that mdominant in one particular flow pattern, as predicted by the bansi-tions described earlier. For the bubble flow model, well cases wifhbubble flow existing for more than 75% of the welf length we con.sidered in order to have an adequate rmmber of cases. These resultsare shown in COL 6 of Table 2. Cols. 7 through 10 of Rable 2 giveresults for well cases predicted to have slug flow exist for 100% oftbe well fength. The cases used for COL 7.and 8 were selected fromthe entire data bmk, whereas the cases used foc Cols. 9 and 10 and11 were selected from the reduced data bank fhat eliminated theHagedorn and Bmum data, which is one-third of all the vertical wellcases. Finally, Col. 11 of Table 2 gives results for those cases in the10M data bank that were pmdkxed m be in a.mmlar flow for 100%of the well length.

Complete performance results of each model or correlationagainst idvidual statistical parameters (El, E6) are give in thesupplement to this paper.m

ConclusionsFrom Cofs. 1 through 11 of Table 2, the performance of the mcdeland other empiricaf correlations indicates the fcdSowing,

1. The overall perfomumce of the comprehensive mcdel is spe-rior to afl other me fhods considered. However, the ovemfl perfofm-a.nces of the Hage.dmn and Brown, Aziz et aL, Duns and Rm, andHasan and Kabir models are comparable to that of the model. Forthe last three, this can be attributed to the use of flow mechanismsin these methods. The excellent performance of the Hagedorn andBrown correlation can be explained cmly by tie extmsive data usedin ifs development and modifications made to the correlation. Infact, when the data without Hagedorn and Brown wetl cases are comsidered, the model pmfonned the best (Cols. 4 and 5).

2. Although the Hagedorn and Brown correlation perfmmed bet-ter than the other correlations and models for deviated wells, noneof the me fbods gave satisfacto~ results (Cccl. 3),

3. Only 29 well cases were found with over75% of the well lengthpredicted to be i bubble flow. The model performed second best tothe Efasan and Kabir mechanistic model for bubble flow (Col. 6).

4. The petiotma.nce of the slug flow model is exceeded by fheHagedorn and Bmvm correlation when the Hagedorn and Brown&ta are imluded i the data bank (Cols. 7 and 8). The model per-formed best when Hagedorn and Brow data are not included for allwell cases and all vertical well cases (Ccds. 9 and 10).

5. The performance of the annufar flow models is significantlybetter than all other methods (Col. 11).

6. Several variables in the mechanistic modeI, such as bubble risevelocities and film thickness, we dependent cm pipe inclinationangle. Modifications to include inclination angle effects on thesevariables should fmther improve model performance.

AcknowledgmentsWe thank the TUFFP member companies whose membership feeswere used to fund part of fbis reseacb projecf, and Pakistan Petro-Ieum Ltd. for the fnancial support provided A.M. Ansari.

ReferencesL Ozon, P.M., Ferschmider, G., and Chwekoff, A.: A New M.kiphme

Flow Model Pdicm Resmre and Temperature Profiles? paper SPE16535 presented al the 1987 SPE Offshore Emope Ccmference, Aber-deen, Sept. S1 1, 19S7.

2. Hamn, A.R. and Kabir, C, S,:A Smdy of Mulfiphase flow Behavior inVertica f VMt s, SPEPE (M&y 1988) 263.

3. Taitel. Y.. Bamea. D.. and Dnkter. A.E.: Moddline Flow Paftem Tmn-Sitios for Steady Upward Gas-Liquid Flow in VeI%l Tubes, AIChEJ. [ 1 9$4) > 26.345.. . . . . . . .,-.. .

4. Barnea, D., Shoham,O..amlTahel, Y: F1.aw PattemTnusitionfor VeI-dcat Dowward Two-Phase t%w, Chem E.g. Sci. (1982) 37, 74I.

5. Bamea, D.: A Unified Model for Predicting Flow-Pattern Tmnsiti.mfor tie Whole Raoge of Pipe inclinations; Intl. J. Mulliphase Flow(1987) 13,1.

6. Hammthy, T.2 Velocity of Lacge Drops and Bubbles i Media of Inti-nite m Remitted Extent, AJChEJ. (1960) 6,281.

7. SCOR, S.L. and Kouba, G.E.: Advances i Slug Flow Characterizationfor Hmimrdat and Slightly Imfimd P@efines? P2P,, SPE 20628 pr-esented at the 1990 SPE Annw.t Technical Cmference and Exbibitim,New Odans, Sept. 23-26.

8. Caetano, E.F.: Upward Vertical llvcHimse ROW T&o.gh an .&mu-lus~ PhO dissmta[ ion, U. of Tulsa, T!m, OK (1985).

9. Zuber, N. and Hench, L: Sfeady Stale and Transient Void Fraction ofBubbtiE Systems and 2fmir operating Limits. par! 1: Steady Mate Op.erafion, Genemt Electric Report 62GL1OO (1962).

10. Femandes, R.C., Setnaih T., and Dukfer, A.S.: Hydmdynandc Mcdelfor Gas-Liquid Slug Flow in Vemicaf Tubes, A{ChEJ. (1986) 29,981.

IL Sylvester, N,D.: A Mec hanistic Model for fWo-Phase Verdcaf SlugFlow in Pipes; ASMEJ, Energy Rcwumes Tech. (1987) 109,206.

12. McQnilIan, K.W. and Whalley, P.B.: Flow Patterns in Vecricd Tvm-Phase Flow 1/1, J. Mdtiphas, Flow (1985) 32, 161.

13. Brotz, W.: VJber die Vmausberechmmg da Abmptiomgesch-windiS-keit von Gasen in Stmrnemkm Ftumiekeitsscbicbkn. Chem. Ire -. Tech,( 195 4) 2 6, 470 .

14. Schnd&, Z.; Expenmnml Study of Gas-L@td Flow inaPipeline-Riser.fY.fm MS fJmis. U. of TM.% Tds. (1976).

15. Vo, D.T. and Shoham, 0.: *A Note on the Existence of a Solution forRvo-Fbase S1ug Ftow i VertimJ Fipes? ASME J. Energy RexxocesTech (19S9) 111,64.

16. Dukler,A,K, MaroD, D. M., and Branmr, N.; C

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26. F?acher, G.H,, and Brown, K.!2 TrvXctio. of pressure Gradiem$ forMultiphase F30W in T.bing, Tram.. AfME (1963) 2.%, 59.

27. Hagedorn, A.R.: E.qmimental SmdyofRessure Gradients &curringduring Continuous fW*Phase Flow in Small Diameter Verticaf Con.dubs, PhD dissertation, U. of Texas, Austin (1964).

28. Baxendell, P.B.: The Calculation of Pressure Gradients in High RateFlov?ig Wells; JPT (Oct. 1961) 1023.

29. Grkiszewski, J.: %edicdng Two-Phase ~SSm DrOPS in VtiCalPip.% JPF(Jme 1967) 829.

30. f?.spanol, H.J.W ~cmnparisonof Three Methods fore?.lculating aFms-surehveme iVerticd Mulfi-Phase Flow, MS thesi s. U. of2Msa.Tul-sa, OK (1968).

31. Messufam, S.A.G.: Comparison of Correlations for Redicting Muki-phase 3%wimgprewre f.nsms in Verdcal PiF@s. MS t hesi s. U. oPDd-S+ Ttdm, OK (1970)

32. Camach.a, C.A.: comparison of Correlations for PTedcdng RessureLosws inHQh Gas-Liquid Ratio Vedicrd WelisYMS fhesii. U. ofTulsa,fuka, OK (1970).

33. Duns, H. lr. and Ros, N.C.3.: Vertical Ftow of Gas and Liquid Mixturesin Wells. Pmt.. 6fb World Pet. Con% (1963) 451.

34. Brill, J.P.: Discontimilies in the Orfi=wsid Correlation for predict-ingpressure-enfs in Wells, J. E.eqyRes. Tech, (M.xch, 1989)41,34.

3S. Beggs, H.D. and Bdll, J.P.: A Sfudy of Two-Phase Flow in fnctiiedPipes, - JPT(?vfay 1973) 6+77.

36. Palmer, C.M.: .Evalua60n of fnclimd Pipe Twc-Phase Liquid HoldIIpCorrelations Using Expetimmcal Data, MS thesis, U. oflldsa. Tu@OK ( 1975 ).

37. Muklmjee, H. and Brill, J. P.: Ressure Drop Correlations for fnctinedIVc-Phase Flow, J. Energy Res. Tech. fDec. 1985).

38. tiIz, K, Govier, G.W., and Fogarasi. M.: pressure Drop in Wells Re-ducing Gil and Gas, J. Gin. P.L Tech. (July-gept. 1972) 38.

39. Kabir, C.P., and Hasan, AR.: performance of a TWo-Phase GadLiquidFlow Mcdel i Vmical Wells, JPSE (1990) 4,273.

40. .4mari, A.M. et d,: ,Supplement to paper SPE 20630, A Comprehm-sive Mechanist ic Mcdd for UpwardTWc-Pba.$e Flow in WeUbores,pa-per SPE 28671 available at SPE headquarters, Richardson, TX.

Nomenc la tu rea = coefficient defined in Eq. 55A = cross-sectional area of pipe, L, m2b = coefficient defined in Eq. 56c-= cceffkient defined in Eq. 57C = constant factor relating friction factor to Reynolds

number for smoofh pipesC= coefficient defined in Eq. 48d. pipe diameter, L, me = error function

El= average percentage error, %E2 = absolute average percentage error, %Es= standard deviation, %E~ = average error, miLt2, psiES= absolute average error, mlL12, psiEs= standard deviation, mlLt2, psif= friction factor

FE . fraction of liquid entrained in gas coreFT = relative performance f=tor, defined in Eq. 112g = gravity acceleration, m.fs2h = local holdup fractionH. average holdup &actionL. Iengfb along the pips, mn = number of well casesz= exponent to account for the swarm effect on bubble

rise velociw

NW= Reynokk number~ = pressufe, m/Lt2, psiq = flowrate, L3{t m3fss. wetted perimeter, L, mv = velocity, Ut mlsV= volume, L3, m3X= Lockhart and Martielli parameterY= Lockhart and hfardnelli parameterZ = empirical factor defining interracial friction~ = Iengfb ratio, defined in F.q. 31C5 = fdm thickness, L, mC3 = ratio of fti thickness to diameterX = differences = absolute pipe m@mss, L, m@ = angle from horizontal, md or &gA = no-slip holdup fractionP = dynamic viscosity, kg/ins. kghr$v = kinematic viscosity, L21t, m21sqp = density, mJL3, kg/m3~ = Solace Ensio, mltz, dynelcmr = shear stiess, @Lt2, Nlm3@ = dimensionless groups defined in Eqs. 94 and 95

Subscriptsa = amelera,tionA = averagec = Taylor bubble cap, core

crit = criticale = elevationf. frictionF= filmg.ga3H = hydraulici = ith elementI= interracialL = liquid

IS= liquid slugm . mixtureM. modified

mu= maximummill . Inil-humN= Nimseltp = piper = relatives = slipS = superficial

SU = slug unitt. total

3B = Taylor bubbleTP. two-phase

Supermript* = developing slug flow

S1 Metric Conversion Factorin. X2,54* E+OO = cm

T.mvmhn factor is exact SPRPF

OIIOi.4 SPE ma.u!ctipt mceiv@d for review Sept. 2, 1930. Revb%d mm.szrht -bedSept. 2s, 1993. Paw? acmpted lor Lmbl!cation DWC 6, 1993. Fawr (SPE 2Ce30) fimt Pr6-~#:2~ 1990 SPE Annual Tecim@d Cnnfemnm 4 Ei+M+io held i New Orbs.,,

1

SPE production.9 Facitides. May 19P4 151

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A CONL)REHENSIVE MECHANISTIC MODEL FOR UPWARD10 TWO Pm SE FLOW I N WELL130RES J wuu !UQ.

High.Water.Cut Gas Wellst SPE Prod, Eng, J, (Aug.1987), 165-177,26Chlerlol, G. L,, Culeol, Q, M, ,and Soloccl, G,: Two.PhaseVertical Flow In 011 Wells . . Predlotlon of PressureDrop, SPE J. Pet, Tech. (Aug. 1974), 927-938.26 Poet? mann, F, H., and Car penter, P. G,: TheMultlphase Flow of Gas, 011 and WaterTht ough Vertical Fiow Str Ings withApplication to the Design of Gas. LiftInstallations, API Drllllng and ProduotlonPrlictlaes, 257- 317 (1962).27 Fanoher, G. H., and Brown, K. E,: Pr edlc4 Ionof Pressure Gradients for Multi phase Flew InTubing, Trans. AM4E(1 963), u, 59-69.28 Hagedorn, A, R,: ~~

~lRa.d~, Ph, D.Dissertation, The University of Texas atAustin (1964),

28 Baxendell, P, B,: The Caloulatlon of Pr es$ureGradients in High Rate Flowing Well), SPEJ. Pet, Tech, (@S, 1961), 1023.300rklszew sKI, ..t,: Predicting Two-PhasePressure Drops in Vertical Pipes, SPE J.

Pet, Tech. (June 1967), 829=838.31 Espanoi, H, J. H.: Q21tlfJJf~ff~~

lRMLWU4ulikp~ZSe FI ou , M, S. ThesisThe University of Tuisa (1968).32 Mes*uiam, S, A, Q,: ~ o~

~=ELf@lQ~~*M*S. Thesi8, The University of Tuisa (1970),

33 Camaeho, C, A,: @mg6111~~&L&fit&~ M.S* Th@~is, TheUniversity of Tulsa (1970),

34 Duns, H,, Jr, and Ros, N, C, J,: VertiaalFIGW of Gas and Liquid Mixtures in Welis, ~Worid pet, Congr es% 451,(1063),

35 Briii, J. P,: Dieoontinuitieo in theQkiczewski Oorreiat ion for Predicting180

Pressure Gradients in Wells-, ASME JERT,111, 34 . 36, (March, 1989).36 Beggs, H. D, and Briil, J. P.: A Study of

Two- Phase Fiow ir~ lnolined Pipes, Jilts~., 607.617, (May, 1973),

37paimer , C, M,: Evacuation of inc!!ned PipeTwo- Ph#jse Liquid Hoidup Correlations UsingExperimental Data, M, S, Thesis, TheUniversity of Tuisa (1975).

38 Mukherjee, H. and Brill, J. P.: PressureDrop Correlatior?s for inclined Two- PhaseFlow, Trans. ASM5 JERT (Dee., 1985).3gAziz, Y., Govier, G. W, and Fogar asi, M.:Pressure Drop in Welis Produoing Oil and

@AS, ~= 48, (JuiY .September, 1972),

fWMENCiATURElWGr.@h L

aAbcccdDeEl132B3E4EsE6L8hIfLnnNPQReRPFs

coefficient defined in Bq. 46cross-sectional area of pipe, mzcoefficient defined in Hq. 47coefficient defined in Eq. 48constan: factor reldtg friction factor to Reynoldsnumber for smooth pipmcoefficient defined in Eq. 3$differential change in a variablepipe diameter, merror functionavarago percentage error, %absolute average porcontage error, %standard deviation, %averago error, psiabsoluto average error, psistandard deviation, psifriction factorfraction of liquid entrained in gas corogrswity aoceleratlon, m/s2local holdup fracttonaverago holdup fractionlongti~ along the pipe, mexponent relating friction factor to Reynoldsnumbu for smooth pipmexpentmt to account for the swarm effect on bubbler i so volooitynumber of well cases successfully traversedpressure, psi [ N/m2 1flow rate, m9/sReynolds numberRolat{vo Porfortnmtao Factor, defined in Iiq. 92wetted porimotor, m

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velocity, mlsvolume, mLockhart and Martinelli parameterLockhart and Martinelli parameterempirical factor dcfinitig interfaclal friction

Icmgth ratio, defined in I@ 27film thickness, mratio of fihn thickness to diwnetcrdlfferenoeaiwolute pipe roughness, mdimensionless groups, defined in Eqs, 79 and f?f)no-slip holdup fractiondynamic viscosity, kg/m+ikinematic viscosity, m2/sangle from hortizontnl, rad or degdensity, kg/m3surface tcttsion, dyntdcrnshear stress, N/m9

accelerationA avoragoc Taylor bubble cap, comcrit critical9 elevationf frictionP filmo ga8H hydrauIiei ith elementI interfacingL liquidLS liquid slugM mixturemn min{mumN Nussoltr relatives slipS sttporfioialw slug unitt totalm Taylor bubbleTp two.phase

* dtwelopin~ slug flow

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TA r KE 1M XGE O FWE LL Di SVi

TABLE 3STAT1871CALRESUEE5 US ING AL L VS KI IC AL WE LL C !MES

E2w Es{%11 5 . 119-219.3

[&i)-75

-17.7-18-623252-0509?8.0

{Ril [Ri]9 5 9 1 7 3 .981 .3 144 .998 .4 M2.5

102 .0 1 7 6 . 3121 .7 Ha s1 5 4 .9 2 9 8 .81 4 72 2 1 1 .0

RFP(-15.3806S877.5428.38212.91315-37417X2.2

mm . m QLEak2 Gad@& 0 1 Gr at & zOIL ] Smom @W/Dj A@n i3AGBR 10 . sWIODEL 14 .5Ku z 34 .G1-8 0-10150 1-5-10567 9-5-70.5

21.924 8-?600 1 1 4 -2 74 0 0 1 7 -1 122:-8 72 0-2 700 0 74 0-5 570 0 35-86 16 .7

m um 21 .1MUiiR 20-9

23 .038.522-0Re& 3 c k e &~2 4 2:-7 0 -3 -5 3 47 - 4 48 -44 9 60 :

&5 02-68 6-27914 8 .3465&-7 60 0 -2 3 00 0 20 0 -1 MIOOO 24 -8 6

Uiitrmecal=T AB LE 4

!STAT ISTZCAL RE-SULX?5X7 *S INGALL V8E? l TCALW E LL CA SE S W THOU T H M3 EDOR W AN= BR OWN= O AT AE2{%]

MODEL 10.ZHAGRR 12 .2A212 12 8DI)NROS 1 5.0

1 8 2 ?MUR%R 20-6ORKIS 27 .4

ES~OAJ14 .817 .018 .022 .823 .3!XL646 .8

~~i) [Ri)-6 .4 !3 7 -8

-122 x3(L6-21.s 1 2 6 . 333 .1 1 3 5 281 S 16 7 292 .5 181-97 7 .5 2 2 3 .4

g)1 7 2 2207 .22 1 6 . 12 0 9 . 22 3 5 62 3 9 . 5362-4

(-)5 . 0 0 06 . 8 0 18 . 4 5 9

10 .81419-1662 1 s 0 122 .400

%ch ad cs d a?a & om F mm ma ml a nd &qxo*&r2 6 . F~&a ~d ~2 7 .Hage dm n a md Brm m ~- BaxcndeU arid 3bri9as29. OrEszewsl@O.E 5p afi o1 3 u M eS Sa am =. an d ~cbc@ !Md da t a &om sm era ? oii

STA7 7 S7 7 CAL RES1 3 Z 3 S USRIGN.Lm w VEm ?c iLm m3 . cAsEsEs(%) [Ei) ls j) (Ri)12.3 -3.0 109.0 164 .41 4 -8 1 3 .1 1 2 2 .1 166 .22 7 .1 -6 .4 1 6 5 .8 216-714 .7 -90 9 15 4 .6 2 8 0 . 519 .8 1 1 0 .3 1 7 6 .5 191 .325 .7 1 5 2 .6 2 1 5S) 1 9 3 . 07 1 .9 2 95 .6 4 5 3 -5 538 .1

E2:% ]MODEL 8 .6HWBR 10 .6DWWROS 1 8.1J lz1 2 10 2M3KSR la 2

24 .5oRm s 60 -7

(-- )5 .0008 3 3 49 2 6 1

35-68543 .14058 .808

1 1 8 . 5 1 5

MODEI . 1 2 1 1 7 .1 9 3 IOL3 163 -2 5 3 7 3fm z X2 2 16 8 -20-8 1 16 .S 19 0 -4 7 3 4 9Iz&Gss 9 2 1 3 .6 -28 5 1 02 4 3 3 7 s .4 7 -101m ? NiROs 1 2 2 1 8 5 3 3 -4 1 1 0 9 177 .7 8 .4700R81S 16 .1 3+7 2 12 2 XW3 2733 8X53

4.4 2 0 2 41 -3 1 3 4 3 2 0 7 s 10 .1021 7 s 2 0 .2 78 .7 1 5 8 s 217-2 14 .751

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~ 6Sl&k~ ~Z2 S USKG ALL WELL CASESWn H Cm ER 75% 3 K? ? FLOW

E2 l%[%] ~) (Ril (Ril [p s i] nIKQDEL 3 -2 3 .7 -*k x-8 6 7 -0 7 6 -9 5 .0 0 0Az iz 3 2 a7 -3 0 -3 6 8S) 7 9 -1 5 2 8 6omaS 3-3 42 -269 69-4 90.6 5.493D?mRos 3.6 40 -47-8 77.5 8.2 6.374HAG8Ra8 4s -44-9 7 8 .7 8 0 -1 6 .5 1 18EG3R 3 -8 4 .S 4 6 -6 79 2 1 0 2 -6 6 .8 4 2MEKBR 7 -3 3 -8 -154 .0 155.6 83-3 12.852

TA%LE8SBmSKXL R8S?iEXS t.SIXG ALL VEK7iCALUELLCA6ESWTIH mLY%sLz??Flnwwr7HoliTHAGEDmN A??=BRtwn.= Dfsm

E2 Es E4 E5 RPF~o~) [%) Q-) rp s ) & (-)BIODEL 162 20S -7 s 1 0 .7 1 9 8 .7 5 .s3 1A212 1 9 .1 2 4 .1 59 126.7 226-3 5.896

17-0 21.1 14.4 140-5 252.6 7.118mmRos 2 4 2 2 8 -3 M3Lxo 16 9 4 2 4 1 -9 2 2 .6 9 4ORKXS 2 8 .6 4 3 -5 1 0 1 s 1 9 9 s 3 2 1 -2 24.619

24-7 262 118-9 177.3 ~~~ 25.873KGK3 R 332 2 4 2 1 5 2 .3 2U5 .4 2 5 3 .3 3 2 .3 1 9

E2[%]Az32 1 4 . 8MODEL 16 2HAGBR 10 -1Oms 14 .6

15 .5IXm ROS 15 -1. 21 3

T=IS 7Sr A=i X AI . . RESETS USING A i l .W EL L CA SE S W IT H K )(E % S LU G F LOW

:9 .8 5 .6 1 0 2 3 173-82 0 .4 1 3 .0 1 0 1 2 1 6 0 .81 4 .8 -19 -7 80 .4 1 7 6 . 82 6 .3 1 7 .4 1 1 6 .3 2 1 2 .921 -3 43 .7 1 1 4 . 8 1 8 4 . 921 .4 56 .6 m 82 17 0 .72 1 3 9 9 -1 1 5 3 .2 1 9 7 2

ii

-MBLE 9SmnSTici lL RESULTS ALL WELL. CaSESWITH ml% Amn.rmR FiQw

Z2f%) %) gi) [Ri) gil

MODEL 9 .7 1 2 .4 -2 1 .8 8 0 .7 1 3 2 -9J !2 J z 1 2 .4 1 6 5 2 2 2 1 0 8 .1 1 Q5 .4EL%G8R 1?5.1 1 6 .4 7 0 .6 1 2 8 .7 1 4 8 .2DUNR OS 2 0-0 2 4 .8 -7 9 -0 1 7 4 .9 2 2 3 -1MUKBR 2 s -5 1 9 . 9 2 0 2 .1 2 1 9 .9 1 9 6 .7SEE 322 18.0 250.7 261-9 180Sor?ms 7 8 .7 6 8 2 5 0 4 .0 5 4 4 .9 4 0 7 . 9

RPF(-16 . 0 1 67 . 4 1 37.CW58 . 8 2 0

1 3 . 1 8 11 5 2 7 62 4 . 1 4 6

RPF(-)5.0005-8968 . 6 5 2

1 1 2 8 31 7 . 4 0 92 0 . 5 1 54 5 . 8 1 0

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!000000000:.:0:Q:?33.%7~o00

t

}SLUGFLOW

t

Doo

0

t

t~..$.. . . .. . .$... .,,,..0,

.$.* .. *

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