This document presents a comprehensive mechanistic model for predicting upward two-phase flow in wellbores. The model consists of a flow pattern prediction model and separate models for key flow characteristics like holdup and pressure drop for each predicted pattern of bubble, slug, and annular flow. The model is evaluated against a large well data set and shows good agreement. It performs better than other commonly used empirical correlations and an existing mechanistic model.

1. A Comprehensive Mechanistic Model for
Upward Two-Phase Flow in Wellbores
A.M. Anssri, Pakistan Petroleum Ltd.; N .D. Sylvester, U. of Akrow and C. Sarioa, O. Shoham, and
J.P. Brill, U. of Tulsa
s % 40630
Summary. A comprehemive model is formulated to predict the flow behavior for upward two-phase flow. This model is composed of
a model for flow-pattern prediction and a set of independent mechanistic models for predicting such flow characteristics as holdup and
pressure dmp in bubble, slug, and annul= flow. ‘fbe comprehensive model is evaluated by using a well data bank made up of 1,712 well
cases covering a wide variety of field data. Model pei’fortnance is also compared with six commonly used empirical correlations and the
Hasan-Kabr mechanistic model. Overall model performance is in good agreement with the data. In comparison with other methods, the
comprehensive model performed the best.
Introduction
Two-ph.a.se flow is commonly encountered in the PeVO1eum, chefi-
cal, and nuclear indushies. This frequent occurrence presents the
challenge of, understanding, analyzing, and designing two-phase
systems.
Because of the complex nature of two-phase flow, the problem
was first approached through empirical methods. The trend has
shifted recently to the modeling approach, Tbe fundamental postu-
late of the modeling approach is the existence of flow patterns or
flow configurations. Vwious theories have been developed to pre-
dict flow patterns. Separate mcdels were developed for each flow
pattern topredictflow characteristics like holdup and pressure drop.
By considering basic fluid mechanics, the resulting models can be
applied with more confidence to flow condkions other than those.
used for their development.
Only Ozon et al.] and Hasan and Kab@ published studies on
comprehensive mechanistic modeling of two-phase flow in vertical
pipes. More work is needed to develop models that describe the
physical 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 patm’n and then calcu-
lates the flow vtiables by taking into account the actu.at
mechanisms of the predicted flow pattern. The model is evaluated
against a wide range of experhhental attd field data available in the
updated Tulsa U, Fluid Ftow Projects (TUFFP) weU data bank. The
performance of the mcdel is also compared with six empirical cor-
relations and one mechanistic model used in the field.
FlowPattern Prediction
Taitd et al? presented the basic work on mechanistic modeling of
flow-pattern Wmsitiom forupwardtwo-phme flow. Theyidentiled
four distinct flow patterns (bubble, slug, chum, and annular flow)
and formulated and evaluated tie transition boundaries among them
Wig: 1).,Bamea et al.4 later modified the transitions to extend the
apphcabdity of the model to inclined flows. Bamea5 then combined
flow-pattern prediction models applicable to different inclination
angle ranges into one unified model. Based on these different works,
flow pattern can be predicted by detining transition boundaxks
among bubble, slug, and annular flows.
Bubble/Slug ‘Emsition.Taitel et a[.3 gave the minimum diameter
at which bubble flow occurs as.
[1dtin= 19.01 - ‘. . . . . . . . . . . . . . . . . . . . . . . . (1)
For pipes larger than this, tie basic transition mechanism for bubble
to slug flow is coalescence of small gas bubbles into large Taylor
bubbles. 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 tbe
transition 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 forces
breakkwge gas bubbles down into small ones, even at void fractions
exceeding 0.25. This yields the transition to dispersed bubble flows:
()
0,5
vs.
= 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 Kouba7
concluded that this occurs at a void fraction of 0.76, giving the tram
sition 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.nnukm
flow is based on the gas-phase velocity required to prevent the en-
trained tiquid droplets from. falling back into tbe gas stream. This
gives the transition as
[1
%guL(pL–pG)
!J~g = 3.1 —
P: ‘ ‘“””””’’’””””””’””’”””””’
(6)
shown as Transition D in Fig. 2.
Bamea5 modified the same transition by considering the effects
of film thickness on the transition. One effect is that a thick liquid
fdm 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 film
at low liquid rates. The bridging mechanism is governed by the
minimum 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 of
film instability can be expressed in terms of the modified Lockhart
and Martinelii parameters, Q and YM,
143

2. tt
B:::&E _sLUG
FLOW
t
t
CHURN
FLOW
I
D
......................................................,....“.
. . .
,.
. . . . . .
. . .
.
1
ANNULAR
FLow
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 the
Bamea criteria, Eq. 8 must first be solved implicitly for ~ti”. Hm
is then calculated from Eq. 11; ifEq. 12 is not satisfied, annular flow
exists. Eq. 8 cm usually be solved for afin by using a se.mnd-ordm
Newton-Rapbso” approach. Tbus, EqT8 can be expressed as
F&) = YM- 2-1”5H~ FM . . . . . . . . . . . . . . . . .. (13)
WJI–1.5HH)
and
1 .5HLWM
IV&”) =
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. l—Flow patterns in upward fwo-phase flow.
0.01 0.1 1 10 100
SUPERFICIAL GAS VELOC3TY (M/S)
Fig. 2—~pical flow-patiern map for wellbores.
YM = 2-1”5H~ x~, . . . . . . . . . . . . . . . . . . . . . . . . .
wJ1-1.5ffm)
(8)
.[
J3SLwhere XH = ~ . . . . . . . . . . . . . ...(9)
Sc
~w = g sin O(p=-pd
()
,. . . . . . . . . . . . . . . .
dp
(10)
z ~c
and B=(l–FE)2(fr&SL). Fem geometric considerations, ffLF can be
expressed 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 Prediction
After tbe flow patterns me predicted, the next step is to develop
physical models for the flow behavior in each flow pattern. This step
resulted in separate models for bubble, slug, and annular flow.
Chum flow bas not yet been modeled bemuse of its complexity and
is treated as part of slug flow. The models developed for other flow
patterns ze discussed below.
Bubble FlowModef.The bubble flow model is had on Caetano’s8
work for flow in an anmdus. The bubble flow and dispersed bubble
flow regimes are considered separately in developing the model for
the bubble flow pattern.
Because of the uniform distribution of gas bubbles in the liquid
and no slippage between the two phases, dispersed bubble flow can
be approximated as a pseudmingle phase. WLtb this simptificmim,
the two-phase pammetm can be expressed as
PI’P=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 account
the bubble-rise velocity relative to the mixture velocity. By assum-
ing a turbulent velocity profile fortbe mixture with the rising bubble
concentrated 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-%) f
P;
(21)

3. (a) DEVELOPED SLUG UNIT (b) DEVELOPING SLUG UNIT
Fig. S-Schematic of slug flow.
where the value of n’ varies from one study to another. b the present
study, ?/=0.5 was used to give the best results. Thus, Eq. 20 yields
[1
%g%(!%k) @5 = k_l,2yM, . . . . . . . . . . . . (22
1.53 —
P; L I-HL
This 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 ‘($). (J (d.+ ~ + ~ . . . . . . . . . . . . . . . . .. (25)
()
dp
z,
=ppgsiutl. . . . . . . . . . . . . . . . . . . . . . . . . . .. (26)
()
p =fTPPrP%
—, . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (27)
&f 2d
where~p is obtained from a Moody diagram for a Reynolds number
defined by
N&r*. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (28)
Because bubble flow is dominated by a relatively incompressible
liquid phase, there is no significant change in tbe density of the flow-
ing fluids. This keeps the fluid velocity neady constant, resulting in
es2entiaJly no pressure drop owing to acceleration. Therefore, the
acceleration pressure drop is safely neglected, compared with the
other pressure drop components.
Slug Flow ModeI. Fermndes et al.lo developed the fmt tbomugh
physical model for slug flow. Sylvesterl 1 presented a simplified ver-
SPE Prodnctio” & Facilides, May 1994
sion of this model. Tbe basic simplification was the use of a correla-
tion for slug void fraction. These models used an impo~t 8ssump-
tion of fully developed slug flow. McQui13an and Whalley12
introduced the concept of developing flow during their study of
flow-pattern transitions. Because of the basic difference in flow ge-
ometty, the model keats fully developed and developing flow sepa-
rately.
For a fuly developed slug unit (Fig. 3a), the overall gas andfiqtdd
mass balances give
‘S$ ‘&%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
,A
Um=1.2vm + 0.35 = . . . . . . . . . . . . . . . . . (34)
pL
Similarly, the veloci~ of the gas hubbies in the liquid slug is
1/.
[1
“gu= 1.2,”,+ 1.53 -. PA, . . . . . . . . . .. (35)
P?
where the s=ond term on tfte right side represents the bubble-rise
velocity defined in Eq.21.
‘fhevelocityofthe falliigfimcanb+ correlatedwitbthe film
thickness 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)
145

4. Tbeliquid slug void fraction can be obtained by Sylvester’sll cor-
relation and fmm Femandes et al. ‘slo and Schmidt’s 14 data,
Vss
‘Sm = 0.425 + 2.65vm”
. . . . . . . . . . . . . . . . . . . . . . . .. (38)
Eqs. 29 or 30,31 through 35,37, &d 38 can be solved iteratively
to obtain the following eight “&mnvns that Min. the slug flow
modck & 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, ~ can
be readily determined from Eq. 40. Eq. 39 is then used to fmd HLm
with an iterative solution method. De fting the left side of Eq. 39 as
F(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
14J~”
.. . . . . . . . . . . . . . . . . . . . .. (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 comparing
the cap length with the total length of a developed Taylor bubble.
The expression for the cap length is12
[
2
LC=&II~a+_ I-H
1~NLm(~L,B)-&, ..........(44)
where WgTB and HNUB me cahlated at the terminal film thickness
(called NusseIt fhn thickness) given by
[ 1
113
~N= ;dVNL.B#L(l-ffNLIB)
8(PL-P,)
. . . . . . . . . . . . . . . . . .. (45)
146
TIE geommy of the film flow gives HNUB in terms of 8N as
()
2
HNH. =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 from
LU=C’d, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (48)
where C’ was found16to vary from 16t045. Weuse C’=30 in this
study. This gives the Taylor bubble length as
Lm=[LH/(l#)I& . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (49)
From the comparison of& and LTB, if & ? LTB, the flow is devel-
oping slug flow. This requires new values for L~, f&, and
V&B calcula~d emherfor developed flow.
For L&., Taylor 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=i(51)
The volume w be expressed in terms of flow gmmetry as
()
L% + Lu
Grz= YsgAp ~ -vgDAP(l–HuJ:. (52)
Substitution of Eqs.51 md52into Eq.50 gives
()
L~ + LU LB
VS* — -V@(l–HIM) ~
‘TB
%
‘1[
~_(v,,- vu)Hm
Jzz
1
do.....................
.
(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 ~ = %-vu
—Ifw. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (57)
&
After calculating 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 Taylor
bubble.
SPE production&Facilities, May 1994

5. For developed flow, the elevation component occurring across a
slug unit is given by
()y =[(l-i$pts +~pglgsinO, . . . . . . . . . . . . . . . . .. (60)
dLe
where pU=pLHm +pJ1-Ifu). . . . . . . . . . . . . . . . . . .. (61)
The elevation component for developing slug flow is given by
()A ‘[(l~*)pU +B*pm.lgsinO, . . . . . . . . . . .. (62)
dLe
where prm is based on average void fraction in the Taylor bubble
section 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)Hm
HL7BA =
&G%-
. . . . . . . . . . . . . . . . (64)
The frictio’n 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~canbe
calculated 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 it
exits 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 velocity
occur within the same slug unit, then no net pressure drop due to ac-
celeration exists over that slug tmh.’f hishappens when tbe slug
flow is stable. The correlation used for slug length is based on its
stable length, so the possibility of a net pressure drop due to accel-
eration does not exist. Therefore, no acceleration component of
pressure 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 also
presentd tie classic correlations for entrainment and interracial
friction 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
()A.% -zj~t-p.A.gsin6’=0 . . . . . . . . . . . . . . . . . . .. (67)
c
()
*and AP ~ +riSi–rpSF-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 droplets
flowing 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—
ENTRAINED
LIQUID DROPLET-
VF
ST“1. .“.
I
-.. ..-. .. .. . ....
. . . .. . ,~ .“
.
‘F
rF
i
-e
8Dc -
i
Fig. 4-gchematic of annular flow.
FE is the fraction of the total liquid enmined in the core, given by
wallis 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 number
defined by
pLvFdHF
—, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (74)N~t, = ~L
where vF=_=w . . . . . . . . . . . . . . . . . ..(75)
~ddHF=@(14Jd .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (76)
This gives
[- 1
2
f_
ZF=. @-FJ’P~ &4J . . . . . . . . . . . . . . (77)
147

6. 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 fibn
thickness. 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 or
low entittments. ‘fbu.s,
Z=l+300~ for FE> 0.9 . . . . . . . . . . . . . . . . . . . . . . . . . (84)
mdZ=l+24($1’3~ for FE <0.9. . . . . . . . . . . . . .. (85)
Combining Eqs. 81 through 83 yields
()
~,_d Z dp
4(1-2@4 ~ SC’
. . . . . . . . . . . (86)
The supetilcial friction pressure gradient in the core is given by
(*)sc=f- (87)
wherefSc is obtained from a Moody diagram for a Reynolds number
defined by
N%
=pcv’#i//4,” ., . . . . . . . . . . . . . . . . . . . . . . . . . . .. (88)
vsc = FEW + VS8, . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (89)
and#c=pJU +p,(l-,lLc), . . . . . . . . . . . . . . . . . . . . . .. (90)
Tbe pressure gradient for annulw flow can be calculated by sub-
stituting the above equations into Eqs. 67 and 68. llns,
($JC=*(*),C +pwsin6’ . . . . . . . . . . . . . ..(9l)
()
~d @ _ (1-FJ2 f, &
dLF
()()*-3(1~~3 f,’ d ~
()
dp
+p&slne. . . . . . . . . . . . . . . .
-4&f&@3 m ,C
(92)
The basic unknown in the above equations is the dimensionless
film thickness, ~. An implicit equation for ~ can be obtained by
equating Eqs.91 and92. This gives
()
dp
4’XIJWW5 = ,C-(p’-pc)gsine
()
(l-FE)z fF dp
_—— —
64d3(ldJ3fSL W ~L
=0. . . . . . . . . . . . . . . .
—
. . . . . .. (93)
To simplify this equation, the dimensiodess approach developed
by Alves et aL20 is used. This approach defines the following d~.
mensionless groups in addition to previously &tined modified
Lockhart Mutinelli parameters, XM and l’M.
~;= (dp/dL)c-gpcsin8
(dp/dL)xc
(dp/dL)F -gPL sin 9
and& =
(dp/dL)~L “
. . . . . . . . . . . . . . . (94)
. . . . . . . . . . . . . . . . . . . . .. (95)
By using the moditied Lockftart MartineHi parameters, Eq. 93 re-
duces to
WM
y~-4d(l&.5(14J]25 —=0. . . . . . . .. (96)
+ [Q(14J]3
l%. above eqwtions can be solved iteratively to obtain &If Eq.
96 is F(&), then taking the derivative of Eq. 96 with respect to ~
yields
.zt4(l-21)]
“@ ‘[@(14J12[14_(14JI~5
-4a(14J[l%_(14J] 2.5
2.5Z[4( l-@]
3x;[4(~_@]
-@(l-@ [14@d)]35- [@(l+)]’
. . . . . . . . . . . (97)
& the root of Eq. ;6. 77ftus,
.: . . . . . . .. (98)dj+,=b.-~. ....................
-J F@j)
Once &is known, the dimensionless groups #F andcjc can be ob-
fained from the followhg form of Eqs. 91 and 92
@~=_Z.-
(1-tiJ5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(99)
{drz=l.
....................(100)
Alves20stated that Eq. 100canbe expresseda$
~; = %+-YM
~
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(101)
The total pressure gradient can then bc obtained from either F.q. 94
or 95 because the pressure gradient in the film and core must be the
same.’fhus,
Or(*)==($)F=@~(*)w+gpsin6(103)
Note that the above total pressure gradimt equations do not in-
clude accelerational pressure gfadient. ‘fhis is based on results
found by Lopes and Dukfer21 indicating that, except for a limited
range of high liquid flow rates, the accelerational component result-
SPE Production& Facilities, May 1994

7. TABLE l—RANGE OF WELL DATA
Nominal Diameter Oil Rate
Source
Gas Rate Oil Gravity
(in.) (STBID) (MSCFID) (“API)
Old TUFFP’ Data I108 01010,150
Bank
1,5 to 10,567 9.5 to 70,5
Govier and 2t04 8 to 1,600 114 tO 27,400 17t0 112
Fogaras%
Asheim23 2718 tO 6 720 to 27,000 740 to 55,700 35 to S6
Chierici et aI,Z4 27L3 to 5 0.3 to 69 6 to 27,914 8.3 to 46
Prudhoe Bay 5% to 7 600 to 23,000 200 to 110,000 24 to 66
.Includes data from Poenmmn and Catwnter,= Fmcher and Brown,% Ha9edom,27 Baxendall and 7homas,28 0rkiszews!4 ,29
ESPmOl,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 average
film is negligible. value.
Average ermc
Evahtation
The evaluation of the comprehensive model is can’k?d out hy
compming tie pressure drop from the mcdel with the measured data
in the updated TUFFP well data bank that comprises 1,712 well
cases with a wide range of data, as given in Table 1. The perfor-
mance of the model is also compared with that of six correlations
and another mechanistic model tlmt am commonty used in the petro-
leum indus~.
Criteria for Comparison with Data
The 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 the
measured pressure drop.
Absolute average percentage error
()
“
E,= ~~le.1 x1OO. . . . . . . . . . . . . . . . .
i.1
. . . . . . . . (lo6)
Ez indicates 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 pressure”
dzop.
Absolute average errcx
-()
.Es= ;~lefl . . . . . . . . . . . . . . .
i= 1
(1 10)
Es is ASO independent of the measured pressure drop and indicates
the magnitude of the average error.
Standard deviation
rE,= j (e,;_J2—.. . . . . . . . (111)
,= ,
E6 indicates the scattering of the results, independent of the mea-
sured pressure drop.
Criteria for Comparison With Other
Correlations and Models
The ccmelations and models used for the comparison area modified
Hagedorn 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 a“d RISLUI and Kabir.239 The com-
pmison is accomplished bycomp.ming the statistical parameters.
The comparison involves the use of a relative performance factor
defined by
.
J__
(eri -El)z IE,I-IE,J E2-E2
E3. ~ — . . . . . . . . . . .
n-1
(107) F. =
1?21 I-IE1 .1 +-
j= I m nun
MOEL
HAGBR
Azlz
DUNRS
HASKA
BEGBR
ORKIS
MUKSR
EDB
1712
0.700
0.665
1.312
1.719
1.940
2,982
4,284
4.883
WV
—
1086
1,121
0.600
1.108
1,678
2.005
2.906
5.273
4.647
TASLE 2-RELATIVE PERFORMANCE FACTORS
DW VNH__@~ ANH AB
626
1.378
0.919
2.085
1,678
2,201
3.445
2.322
6.000
755
0.081
0.B76
0.803
1.711
1.836
3,321
5,836
3,909
1381
0.000
0,774
1,062
1.792
1,780
3,414
4.688
4.601
29
0.143
2.029
0.282
1.126
0.009
2.82s
1.226
4.463
1052
1.295
0.386
1.798
2.056
2,575
2,S83
3.12S
5,342
654
1.461
0.485
1.764
2.028
2.590
2.595
3.316
5.140
SNH
745
0.112
0.457
1.314
1.852
2,044
3.261
3.551
4.977
VSNH AAN
—
387 70
0.142 0.000
0.939 0,546
1.486 0,214
2.298 1.213
1,996 ‘ 1.043
3,262 1.972
4.403 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 and
3mw” 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 flow
Without 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.

8. E,-E, IE41-IE4 I
- + m“
+ E3mm-E3dn IE4JIE, I
nun
ETE5mh E=E6Mn
+ E5m-E5ti + E6mu–E6&’
. . . . . . . . . . . . . . . . . . ,. (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 entire
comprehensive model so as to study the combined petionnance of
all the independent flow pattern behaviomnodels together The eval-
uation is first performed by using the entire &ta bank, resulting in
COL 1 of Table 2. Model performance is also checked for vertical
well cases only, resulting in Col. 2 of Table 2, and for deviated weU
cases only, resulting in CoL 3 of Table 2. To make the comparison
unbiased with respect to fbe correlations, a second datsba.w was
created that excluded 331 sets of data from the Hagedorn and Brown
study. For this reduced data bank, the results for all vertical well
cases are shown in Col. 4 of Table 2, and the results for combined
veticaf and deviated weil cases are shown in COL 5 of Table 2.
Evaluation of Individual Ftow Pattern ModeLs. ~e perfmmance
of individual flow pattern models is based on sets of data that m
dominant in one particular flow pattern, as predicted by the bansi-
tions described earlier. For the bubble flow model, well cases wifh
bubble flow existing for more than 75% of the welf length we con.
sidered in order to have an adequate rmmber of cases. These results
are shown in COL 6 of Table 2. Cols. 7 through 10 of ‘Rable 2 give
results for well cases predicted to have slug flow exist for 100% of
tbe well fength. The cases used for COL 7.and 8 were selected from
the entire data bmk, whereas the cases used foc Cols. 9 and 10 and
11 were selected from the reduced data bank fhat eliminated the
Hagedorn and Bmum data, which is one-third of all the vertical well
cases. Finally, Col. 11 of Table 2 gives results for those cases in the
10M data bank that were pmdkxed m be in a.mmlar flow for 100%
of the well length.
Complete performance results of each model or correlation
against idvidual statistical parameters (El, E6) are give” in the
supplement to this paper.m
Conclusions
From Cofs. 1 through 11 of Table 2, the performance of the mcdel
and other empiricaf correlations indicates the fcdSowing,
1. The overall perfomumce of the comprehensive mcdel is s“pe-
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, and
Hasan and Kabir models are comparable to that of the model. For
the last three, this can be attributed to the use of flow mechanisms
in these methods. The excellent performance of the Hagedorn and
Brown correlation can be explained cmly by tie extmsive data used
in ifs development and modifications made to the correlation. In
fact, when the data without Hagedorn and Brown wetl cases are com
sidered, 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, none
of the me fbods gave satisfacto~ results (Cccl. 3),
3. Only 29 well cases were found with over75% of the well length
predicted to be i“ bubble flow. The model performed second best to
the Efasan and Kabir mechanistic model for bubble flow (Col. 6).
4. The petiotma.nce of the slug flow model is exceeded by fhe
Hagedorn 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 all
well cases and all vertical well cases (Ccds. 9 and 10).
5. The performance of the annufar flow models is significantly
better than all other methods (Col. 11).
6. Several variables in the mechanistic modeI, such as bubble rise
velocities and film thickness, we dependent cm pipe inclination
angle. Modifications to include inclination angle effects on these
variables should fmther improve model performance.
Acknowledgments
We thank the TUFFP member companies whose membership fees
were used to fund part of fbis reseacb projecf, and Pakistan Petro-
Ieum Ltd. for the fnancial support provided A.M. Ansari.
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Nomenclature
a = coefficient defined in Eq. 55
A = cross-sectional area of pipe, L, m2
b = coefficient defined in Eq. 56
c-= cceffkient defined in Eq. 57
C = constant factor relating friction factor to Reynolds
number for smoofh pipes
C’= coefficient defined in Eq. 48
d. pipe diameter, L, m
e = error function
El= average percentage error, %
E2 = absolute average percentage error, %
Es= standard deviation, %
E~ = average error, miLt2, psi
ES= absolute average error, mlL12, psi
Es= standard deviation, mlLt2, psi
f= friction factor
FE . fraction of liquid entrained in gas core
FT = relative performance f=tor, defined in Eq. 112
g = gravity acceleration, m.fs2
h = local holdup fraction
H. average holdup &action
L. Iengfb along the pips, m
n = number of well cases
z’= exponent to account for the swarm effect on bubble
rise velociw
NW= Reynokk number
~ = pressufe, m/Lt2, psi
q = flowrate, L3{t m3fs
s. wetted perimeter, L, m
v = velocity, Ut mls
V= volume, L3, m3
X= Lockhart and Marti”elli parameter
Y= Lockhart and h’fardnelli parameter
Z = empirical factor defining interracial friction
~ = Iengfb ratio, defined in F.q. 31
C5 = fdm thickness, L, m
C3 = ratio of fti thickness to diameter
X = difference
s = absolute pipe m@mss, L, m
@ = angle from horizontal, md or &g
A = no-slip holdup fraction
P = dynamic viscosity, kg/ins. kghr$
v = kinematic viscosity, L21t, m21sq
p = density, mJL3, kg/m3
~ = Solace Ensio”, mltz, dynelcm
r = shear stiess, @Lt2, Nlm3
@ = dimensionless groups defined in Eqs. 94 and 95
Subscripts
a = amelera,tion
A = average
c = Taylor bubble cap, core
crit = critical
e = elevation
f. friction
F= film
g.ga3
H = hydraulic
i = ith element
I= interracial
L = liquid
IS= liquid slug
m . mixture
M. modified
mu= maximum
mill . Inil-hum
N= Nimselt
p = pipe
r = relative
s = slip
S = superficial
SU = slug unit
t. total
3“B = Taylor bubble
TP. two-phase
Supermript
* = developing slug flow
S1 Metric Conversion Factor
in. 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 -bed
Sept. 2s, 1993. Paw? acmpted lor Lmbl!cation DWC 6, 1993. F’awr (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

10. A CONL)REHENSIVE MECHANISTIC MODEL FOR UPWARD
10 TWO Pm SE FLOW I N WELL130RES. Jwuu!UQ.
High.Water.Cut Gas Wellst SPE Prod, Eng, J, (Aug.
1987), 165-177,
26Chlerlol, G. L,, Culeol, Q, M, ,and Soloccl, G,: ‘Two.Phase
Vertical Flow In 011 Wells .. Predlotlon of Pressure
Drop,” SPE J. Pet, Tech. (Aug. 1974), 927-938.
26 Poet? mann, F, H., and Car penter, P. G,: “The
Multlphase Flow of Gas, “011 and Water
Tht ough Vertical Fiow Str Ings with
Application to the Design of Gas. Lift
Installations, ” API Drllllng and Produotlon
Prlictlaes, 257- 317 (1962).
27 Fanoher, G. H., and Brown, K. E,: “Pr edlc4 Ion
of Pressure Gradients for Multi phase Flew In
Tubing, ” Trans. AM4E(1 963), u, 59-69.
28 Hagedorn, A, R,: ~~
~lRa.d
~, Ph, D.
Dissertation, The University of Texas at
Austin (1964),
28 Baxendell, P, B,: “The Caloulatlon of Pr es$ure
Gradients in High Rate Flowing Well),’ SPE
J. Pet, Tech, (@S, 1961), 1023.
300rklszew sKI, ..t,: “Predicting Two-Phase
Pressure Drops in Vertical Pipes,’ SPE J.
Pet, Tech. (June 1967), 829=838.
31 Espanoi, H, J. H.: Q21tlfJJf~ff
~~
lRMLWU4ulikp~ZSe FI ou , M, S. Thesis
The University of Tuisa (1968).
32 Mes*uiam, S, A, Q,: ~ o~
~=ELf@lQ~
~*
M*S. Thesi8, The University of Tuisa (1970),
33 Camaeho, C, A,: @mg6111~
~&L&fit&
~ M.S* Th@~is, The
University of Tulsa (1970),
34 Duns, H,, Jr, and Ros, N, C, J,: ‘ Vertiaal
FIGW of Gas and Liquid Mixtures in Welis, ”
~Worid pet, Congr es% 451,
(1063),
35 Briii, J. P,: “Dieoontinuitieo in the
Q’kiczewski Oorreiat ion for Predicting
180
Pressure Gradients in Wells-, ” ASME JERT,
111, 34 . 36, (March, 1989).
36 Beggs, H. D, and Briil, J. P.: ‘A Study of
Two- Phase Fiow ir~ lnolined Pipes, ” Jilts
~., 607.617, (May, 1973),
37paimer , C, M,: “ Evacuation of inc!!ned Pipe
Two- Ph#jse Liquid Hoidup Correlations Using
Experimental Data, ” M, S, Thesis, The’
University of Tuisa (1975).
38 Mukherjee, H. and Brill, J. P.: ‘Pressure
Drop Correlatior?s for inclined Two- Phase
Flow, ” Trans. ASM5 JERT (Dee., 1985).
3gAziz, Y., Govier, G. W, and Fogar asi, M.:
“Pressure Drop in Welis Produoing Oil and
@AS,” ~= 48, (JuiY .
September, 1972),
fWMENCiATURE
lWGr.@h L
a
A
b
c
c
c’
d
D
e
El
132
B3
E4
Es
E6
L
8
h
If
L
n
n’
N
P
Q
Re
RPF
s
coefficient defined in Bq. 46
cross-sectional area of pipe, mz
coefficient defined in Hq. 47
coefficient defined in Eq. 48
constan: factor reldtg friction factor to Reynolds
number for smooth pipm
coefficient defined in Eq. 3$
differential change in a variable
pipe diameter, m
error function
avarago percentage error, %
absolute average porcontage error, %
standard deviation, %
averago error, psi
absoluto average error, psi
standard deviation, psi
friction factor
fraction of liquid entrained in gas coro
grswity aoceleratlon, m/s2
local holdup fractton
averago holdup fraction
longti~ along the pipe, m
exponent relating friction factor to Reynolds
numbu for smooth pipm
expentmt to account for the swarm effect on bubble
riso volooity
number of well cases successfully traversed
pressure, psi [ N/m2 1
flow rate, m9/s
Reynolds number
Rolat{vo Porfortnmtao Factor, defined in Iiq. 92
wetted porimotor, m

11. v velocity, mls
v volume, m’
x Lockhart and Martinelli parameter
Y Lockhart and Martinelli parameter
z empirical factor dcfinitig interfaclal friction
Icmgth ratio, defined in I@ 27
film thickness, m
ratio of fihn thickness to diwnetcr
dlfferenoe
aiwolute pipe roughness, m
dimensionless groups, defined in Eqs, 79 and f?f)
no-slip holdup fraction
dynamic viscosity, kg/m+i
kinematic viscosity, m2/s
angle from hortizontnl, rad or deg
density, kg/m3
surface tcttsion, dyntdcrn
shear stress, N/m9
.“yw%m.ii
acceleration
A avorago
c Taylor bubble cap, com
crit critical
9 elevation
f friction
P film
o ga8
H hydrauIie
i ith element
I interfacing
L liquid
LS liquid slug
M mixture
mn min{mum
N Nussolt
r relative
s slip
S sttporfioial
w slug unit
t total
m Taylor bubble
Tp two.phase
* dtwelopin~ slug flow

12. TArKE 1
MXGE OFWELL DiSVi
TABLE 3
STAT1871CALRESUEE5 USING ALL VSKIICAL WELL C!MES
E2
w
Es
{%1
15.1
19-2
19.3
[&i)
-75
-17.7
-18-6
232
52-0
509
?8.0
{Ril [Ri]
959 173.9
81.3 144.9
98.4 M2.5
102.0 176.3
121.7 Has
154.9 298.8
1472 211.0
RFP
(-1
5.380
6S87
7.542
8.382
12.913
15-374
17X2.2
mm. m QLEak2 Gad@& 01 Grat&z
OIL] Smom @W/Dj A@n
i3AGBR 10.s
WIODEL 14.5
Kuz 34.G
1-8 0-10150 1-5-10567 9-5-70.5
21.9
24 8-?600 114-27400 17-112
2:-8 720-27000 740-55700 35-86
16.7
mum 21.1
MUiiR 20-9
23.0
38.5
22-0
Re&3cke &
~24 2:-7 0-3-5347”- 448-44960 :
&5 02-68 6-27914 8.346
5&-7 600-23000 200-1 MIOOO 24-86
U’iitrmecal=
TABLE 4
!STATISTZCAL RE-SULX?5X7*SINGALL V8E?lTCAL
WELL CASES WTHOUT HM3EDORW AN= BROWN= OATA
E2
{%]
MODEL 10.Z
HAGRR 12.2
A212 128
DI)NROS 15.0
182?
MUR%R 20-6
ORKIS 27.4
ES
~OAJ
14.8
17.0
18.0
22.8
23.3
!XL6
46.8
~~i) [Ri)
-6.4 !37-8
-122 x3(L6
-21.s 126.3
33.1 1352
81S 1672
92.5 181-9
77.5 223.4
g)
1722
207.2
216.1
209.2
2356
239.5
362-4
(-)
5.000
6.801
8.459
10.814
19-166
21s01
22.400
%chadcs da?a&om Fmmmaml and &qxo*&r26. F~&a ~d ~27.
Hagedmn amd Brmm~- BaxcndeU arid 3bri9as29. OrEszewsl@O.
E5pafio13u MeSSaam=. and ~cbc@ !Md data &om smera? oii
STA77S77CAL RES13Z3S USRIG
N.Lmw VEm?ciLmm3. cAsEs
Es
(%) [Ei) lsj) (Ri)
12.3 -3.0 109.0 164.4
14-8 13.1 122.1 166.2
27.1 -6.4 165.8 216-7
14.7 -909 154.6 280.5
19.8 110.3 176.5 191.3
25.7 152.6 215S) 193.0
71.9 295.6 453-5 538.1
E2
:%]
MODEL 8.6
HWBR 10.6
DWWROS 18.1
Jlz12 102
M3KSR la2
24.5
oRms 60-7
(-- )
5.000
8334
9261
35-685
43.140
58.808
118.515
MODEI. 121 17.1 93 IOL3 163-2 5373
fmz X22 168 -20-8 116.S 190-4 7349
Iz&Gss 92 13.6 -285 10243 37s.4 7-101
m?NiROs 122 185 33-4 1109 177.7 8.470
0R81S 16.1 3+72 122 XW3 2733 8X53 u
c14.4 202 41-3 1343 207s 10.102
17s 20.2 78.7 158s 217-2 14.751

13. ‘~ 6
Sl&k~ ~Z2’S USKG ALL WELL CASES
WnH CmER 75% 3K?? FLOW
E2 l%
[%] ~) (Ril (Ril [psi] n
IKQDEL 3-2 3.7 -*kx-8 67-0 76-9 5.000
Aziz 32 a7 -30-3 68S) 79-1 5286
omaS 3-3 42 -269 69-4 90.6 5.493
D?mRos 3.6 40 -47-8 77.5 8.2 6.374
HAG8Ra8 4s -44-9 78.7 80-1 6.511
8EG3R 3-8 4.S 46-6 792 102-6 6.842
M’EKBR 7-3 3-8 -154.0 155.6 83-3 12.852
TA%LE8
SBmSKXL R8S?iEXS t.SIXG ALL VEK7iCAL
UELLCA6ESWTIH mLY%sLz??Flnwwr7HoliT
HAGEDmN A??= BRtwn.= Dfsm
E2 Es E4 E5 RPF
~o~) [%) Q-) rps) & (-)
BIODEL 162 20S -7s 10.7 198.7 5.s31
A212 19.1 24.1 59 126.7 226-3 5.896
17-0 21.1 14.4 140-5 252.6 7.118
mmRos 242 28-3 M3Lxo 1694 241-9 22.694
ORKXS 28.6 43-5 101s 199s 321-2 24.619
24-7 262 118-9 177.3 ~~~ 25.873
KG’K3R 332 242 152.3 2U5.4 253.3 32.319
E2
[%]
Az32 14.8
MODEL 162
HAGBR 10-1
Oms 14.6
15.5
IXmROS 15-1
.
213
T=IS 7
SrA=iXAI.. RESETS USING Ail.
WELL CASES WITH K)(E% SLUG FLOW
:9.8 5.6 1023 173-8
20.4 13.0 1012 160.8
14.8 -19-7 80.4 176.8
26.3 17.4 116.3 212.9
21-3 43.7 114.8 184.9
21.4 56.6 m82 170.7
213 99-1 153.2 1972
ii
).
‘~
I
-MBLE 9
SmnSTicilL RESULTS ALL WELL. CaSES
WITH ml% Amn.rmR FiQw
Z2
f%) %) gi) [Ri) gil
MODEL 9.7 12.4 -21.8 80.7 132-9
J!2Jz 12.4 165 222 108.1 1Q5.4
EL%G8R 1?5.1 16.4 70.6 128.7 148.2
DUNROS 20-0 24.8 -79-0 174.9 223-1
MUKBR 2s-5 19.9 202.1 219.9 196.7
SEE 322 18.0 250.7 261-9 180S
or?ms 78.7 682 504.0 544.9 407.9
RPF
(-1
6.016
7.413
7.CW5
8.820
13.181
15276
24.146
RPF
(-)
5.000
5-896
8.652
11283
17.409
20.515
45.810
--
-:1
..
.-

14. !
0000
0000
0:.:0:
Q:?
33.%7~o
00
t
}
SLUG
FLOW
t
Do
o
0
t
t
~
..$.
. . . .
. . .
$.’.
. .,,
,..
0,
.$
.*
‘.
.*”
t
C1-llol;A;lW&AR
Fig. l-Flow patterns in upward two.phase flow.
201 I I I I [ I
i
(o) DEVELOPED SLUG UNIT
!2Q
3
z 0.1
A
a
6
ii
K
u.!
o. 0.0’
a
u)
0.00i
BUBBL’T / BARNEA
TRANSITION
I
I
/
I[ANNULAR
D
I
A SLUGOR CHURN
I
I
I
1
I
I
II
i
1“ I I I I
SUPERFICIAL GAS VELOCITY (m/S)
FIu. 2-Typlcel flow pattern map for wellborss.
q
t.k
,w
t: ‘
u ,
.n
{?=
I II I
02 0.1 1 10 100
~
,aso~
NT
4L
r ‘GTB
LyB
‘GTB 1
V~TB
DEVELOPING
J TAYLOR
BUBBLE -
-1.-0
‘v QL O?do
(b) DEVELOPINGSLUGUNIT
Fig, 3-Sohemetlo diagram of slug flow,
164

15. I ,’
“1
GAS CORE
,, . .
LIQUID FILM ~ _
. ,,
,,
ENTRAINED
.,
LIQUID DROPLET
~
1
,, . . IC . ‘
1:
I ~F
;::::,
F19.4-Sc,hemellc diagram o! annular flow, ~
,,
!.
,,
i, ,.
10‘ .
I‘?2(
+ CALCULATED PRESSURE
T
t
‘
9.0
ANNhLAFl
x MEASUREDPRJNWRE
J.
S1
:,:
,4
h
,,
,:
I
i
,
,,,
,., PRESSURE ( PSI ) :
1
Fig. 5.-Performanoe of tho co~praltan$lve model-typloal prenaure profile,
1,
,,, ,,.
I
I
:.
,
I
16s
,,,