-
Predicting Two-.Phase Pressure Drops in Vertical Pipe
J. ORKISZEWSKI * ESSO PRODUCTION RESEARCH CO.HOUSTON, TEX.
FIG. I-GEOMETRICAL CONFIGURATIONS IN VERTICAL FLOW.
ANNULAR - SLUG ANNULAR - MISTTRANSIJION
C 0B
SLUG
A
BUBBLE
the free-gas phase is small. The gas is present as smallbubbles,
randomly distributed, whose diameters also varyrandomly. The
bubbles move at different velocities depend-ing upon their
respective diameters. The liquid moves upthe pipe at a fajrly
uniform velocity and, except for itsdensity, the gas phase has
little effect on the pressure gra-dient.SLUG FLOW (FIG. IB)
In this regime, the gas phase is more -pronounced. AI~though the
liquid phase is still continuous, the gas bubblescoalesce and form
stable bubbles of approximately thesame size and shape which are
nearly the diameter of thepipe. They are separated by slugs of
liquid. The bubblevelocity is greater than that of the liquid and
can be pre-dicted in relation to the velocity of the liquid slug.~2
Thereis a film of liquid around the gas bubble. The liquidve-lodty
is not constant-whereas the liquid slug alwaysmoves upward (in the
direction of bulk flow); the liquidin the film may move upward but
possibly at a lower ve-locity, or it may move downward. These
varying liquidvelocities will result not only in varying wall
friction losses,but also in a "liquid holdup" which will influence
flowingdensity. At higher flow velocities, -liquid can even be
en-trained in the gas bubbles. Both the gas and liquid phaseshave
significant effects on the pressure gradient.TRANSITION FLOW (FIG.
IC)
The change from a continuous liquid phase to a con-tinuous gas
phase occurs in this region. The liquid slugbetween the bubbles
virtually disappears, and a significantamount of liquid becomes
entrained in the gas phase. Al-though the effects of the Hquid are
significant, the gasphase is more predominant.
A method is presented which can accurately predict,with a
precision of about 10 percent, two-phase pressuredrops in flowing
and gas-lift production wells over a widerange of well conditions.
The method is an extension ofthe work done by Griffith and Wallisll
and was found tobe superior to five other published 11'lethods. The
precisionof the method was verified when its predicted values
werecompared against 148 measured pressure drops. The
uniquefeatures of this method over most others are that
liquidholdup is derived fr01n observed physical phen;mena,
thepressure gradient is related to the geometrical distributionof
the liquid and gas phase (flow regimes), and the methodprovides a
good analogy of what happens inside the pipe.It takes less than a
second to obtain a predl1ction on theIBM 7044 computer.
ABSTRACT
INTRODUCTION
Original manuscript received in Society of Petroleum Engineers
officeAugust 8, 1966. Revised manuscript of SPE 1546 received March
1, 1967.Paper was presented at 41st Annual Fall Meeting held in
Dallas, Tex.,Oct. 2-5, 1966. Copyright 1967 American Institute of
Mining, Metal-lurgical, and Petroleum Engineers, Inc.
':'Presently with International Petroleum Co. Ltd., Talara,
Peru.llReferences given at end of paper.':":'All four regimes could
conceivably exist in the same well. An exam-
ple would be a deep well producing light oil from a reservoir
which isnear its bubble -point. At the bottom of the hole, with
little free gaspresent, flow would be in the bubble regime. As the
fluid moves up thewell, the other regimes would be encountered
because gas continuallycomes out of solution, and the pressure
continually decreases. Normally,however, flow is in the slug regime
and rarely in mist, except for con-densate reservoirs or
steam-stimulated wells.
The problem of accurately predicting pressure drops incflowing
or gas-lift wells has given rise to many specializedsolutions for
limited conditions, but not to any generallyaccepted one for broad
conditions. The reason for thesemany solutions is that the
two-phase flow is complex and-difficult to analyze even for the
limited conditions studied.Under some conditions, the gas moves at
a much highervelocity than the liquid. As a result, the down-hole
flowingdensity of the gas-liquid mixture is greater than the
cor-responding density, corrected for down-hole temperatureand
pressure, that would be calculated from the producedgas-liquid
ratio. Also, the liquid's velocity along the pipewall can vary
appreciably over a short distance and resultin a variable friction
loss. Under other conditions, the li-quid is almost completely
entrained in the gas and hasvery little effect on the wall friction
loss. The difference invelocity and the geometry of the two phases
strongly in-fluence pressure drop. These factors provide the basis
forcategorizing two-phase flow. The generally accepted cate-gories
(flow regimes) of two-phase flow are bubble, slug,(slug-annular)
transition and annular-mist. ** They areideally depicted in Fig. 1
and briefly described as follows.BUBBLE FLOW (FIG. IA)
The pipe is almost completely filled with the liquid and
JUNE, 1967 8119
-
ANNULAR-MIST FLOW (FIG. 1D)The gas phase is continuous. The bulk
of the liquid is
entrained and carried in the gas phase. A film of liquidwets the
pipe wall, but its effects are secondary. The gasphase is the
controlHng factor.
To cope with the complex problem, the many publishedmethods were
analyzed to determine whether anyonemethod was broad enough, or had
the ingredients to bebroad enough, to accurately predict pressure
drops over awide range of well conditions. The methods were first
cate-gorized. Certain methods were selected from each cate-gory to
predict pressure drops for two selected well caseswhose flow
conditions were significantly different fromthose originally used
in developing the various methods.Finally, the predicted pressure
drops using the more prom-ising methods were compared against known
values takenfrom 148 cases having widely varying conditions of
rate,GOR, tubing size, water cut and fluid properties.
BASIS FOR SELECTING METHODS STUDIEDBased upon similarity in
theoretical concepts, the pub-
lished methods were first divided into three categories.From
each category certain methods were selected, basedon whether they
were original or unique, and were devel-oped from a broad base of
data. The discriminating fea-tures of the three categories are
shown.CATEGORY I (REFS. 1,3-6, 9)
Liquid holdup is not considered in the computation ofthe
density. The density is simply the composite density ofthe produced
(top-hole) fluids corrected for down-hole tem-perature and
pressure. The liquid holdup and the wallfriction losses are
expressed by means of an empiricallycorrelated friction factor. No
distinctions qre made amongflow regimes.CATEGORY 2 (REFS. 7, 8,
10)
Liquid holdup is considered in .the computation of thedensity.
The liquid holdup is. either correlated separatelyor combined in
some form with the wall friction losses.The friction losses are
based on the composite propertiesof the liquid and gas. No
distinctions are made amongflow regimes.CATEGORY 3 (REFS. 2,
11-13)
The calculated density term considers liquid holdup. Li-
qUi.d holdup .is determined from some concept of slip Ve-lOCIty
(the dIfference between the gas and liquid veloci-ties). The wall
friction losses are determined from thefluid properties of the
continuous phase. Four distinct flowregimes are considered.
Of the 13 methods categorized, two from each categorywere
selected for further study. The methods of Poettmannand Carpenter/
and Tek4 were picked from Category 1.Most of the methods in this
category are extensions of thePoettmarin-Carpenter work. In the
second category, theHughmark and Pressburg methodS was selected;
the Hage-dorn and Brown methodlO was not available at the timeof
the initial screening, but it was included in the final de-tailed
evaluation. There are really only two methods inCategory 3. The
Griffitht2 and the Griffith and Wallisl1methods are synonymous; the
Nicklin, Wilkes, and David-son methodl3 is for special conditions
and parallels thework of Griffith-Wallis. The other method is that
of Dunsand Ros.2
RESULTS OF THE COMPARISONThe five methods initially selected,
whose results were
hand calculated, were compared by determining the devi-ation
between predicted and measured pressure drops forthe first two well
conditions listed in Table 1. Fig. 2 com-pares the predictions for
Well 1. The results were similarfor Well 2. The most accurate
methods (Duns-Ros andGr,iffith-Wallis) were then programmed for
machine com-putation and further tested against 148 well
conditions.*
Neither method proved accurate over the entire rangeof
conditions used. Although the Griffith-Wallis methodwas reliable in
the lower flow-rate range of slug flow, itwas not accurate in the
higher range. The Duns-Ros meth-od exhibited the same behavior
except that it was also in-accurate for the high-viscosity oils in
the low flow-raterange. The Griffith-Wallis method appeared to
provide thebetter foundation for an improved general solution
al-though its predicted values were in greater error (21.9percent)
than Duns and Ros (2.4 percent). The heart ofthis method,
prediction of slip velocity, is derived fromphysical observation.
However, since friction drop was
':'The data in Table 1 are from 22 Venezuelan heavy-oil wells.
In addi-tion to the data presented in Table 1, the data used are
from the pub-lications of Poettmann and Carpenter,l Baxendell and
Thomas,3 Fan-cher and Brown,6 and Hagedorn and Brown.9 These
represent 126 addi-tional pieces ot data.
TABLE I-PHYSICAL CONDITIONS AND FLOW RATES OF HEAVY-OIL WELLS
STUDIEDOil Measured Wellhead Flow String
Well Oil Rate GOR Water Gravity Depth Pressure Diameter
MeasuredNo. (B/D) (scf/bbl) Cut (%) CAPI) (ft) (psig) (in.) Llp
(psi)-1- 320 4020 30 10.3 4360 250 8.76 810
2 175 6450 17 9.5 4360 300 8.76 9253 1065 765 15.1 3825 550
2.992 6504 1300 252 14.6 3940 150 8505 3166 1430 14.4 3800 700 5506
1965 232 14.4 3720 300 9007 1165 957 15.6 4240 700 8508 1965 1500
13.5 4570 850 6509 2700 267 15.6 4175 300 1200
10 855 185 12.9 4355 250 145011 2320 1565 13.6 4670 910 74012
2480 858 18.6 4575 650 90013 1040 472 18.6 4400 400 95014 1490 341
13.0 4065 500 105015 1310 335 13.6 3705 500 95016 1350 185 12.9
4160 150 135017 788 222 16.0 4210 350 140018 1905 962 14.1 4487 580
72019 967 193 13.3 4766 250 130020 1040 385 12.5 4505 250 110021
1585 865 12.9 4692 400 fI 75022 1850 575 18.7 3924 700 800
8l,lO TOllRN AT 0);' l.)];'TnnT t:"TTlIlA" nrtT:'l"
T'W''l'\.Tf''lt, T ..... ......, ,,.
-
+ 8.712.7
+ 0.724.2
+ 5.410.8
+ 1.210.3
+16.441.4
3000
+ 2.320.0
-16.936.6
+ 1.732.1
+ 2.427.0
+22.718.7
2.111.1
This Duns HagedornMethod and Ros and Brown
+ 0.311.8
- 0.810.8
I-I--
I I
- +___~ L _
10% BAND I (114 DATA POINTS)-~'---t-------t-----------
: ~
400 600, 1000MEASURED 6P - PSI
-~. --_..j------,----+_._-_.- -------- ..__._.~ .._--I i DATA
INSIDE 10%
i BANtf!\iOTPfbTTEDi I
200
400
300
3000,..----rD"..,-A-rT-rA-S-O-U....-R-C-rE-r'--,.----------,
-- 0 BAXENDElL at al.o FANCHER at al.Do HAGEDORN at al. POHlMANN
at al. THIS PAPER
I !
2000
~ 1000D..
~ 600 ------1----+--=:--,Wl-VCwI:Xc..
FIG. 3-THIS WORK (MODIFIED GRIFFITH AND WALLIS PREDICTION).
TABLE 2-SUMMARY OF DEVIATIONS BETWEENMEASURED AND PREDICTED
PRESSURE DROPS
Prediction Method
Table 1 - Heavy-Oil Wells(22 Wells, low to medium
velodtie,s, 10 to 20 API oils)Avg. error, percent - 1.2Std.
deviation, percent 10.4
Baxendell-Thomas3(1 Well, 25 rates mostly high
velocities, 34 0 API oil)Avg. error, percentStd. deviation,
percent
Fancher-Brown6(1 Well, 20 rates medium to
high velocities, 95 percentwater cut)Avg. error, percentStd.
deviation, percent
Hagedorn-Brown9(1 Well, medium to high
velocities, 16 water runs,16 oil runs of 10 to 100 cp oil)Avg.
error, percent + 0.1Std. deviation, percent 8.2
Poettmann-Carpenter1(49 Wells, low to medium
velocities, 15 wells highwater cut, rest36 to 54 API oils)Avg.
error, percent - 1.0 + 5.8 -13.0Std. deviation, percent 12.0 12.4
22.2
sufficiently evaluated (e.g., flow in the casing annulus andin
the mist-flow regime). The method's precision might befurther
improved if the liquid phase distribution could bemore rigorously
analyzed.
This method is accurate over a broader range thanprevious
correlations. For a prediction method to be gen-eral, it must be
expressed in terms of flow regime andliquid distribution. The other
methods, which were not de-
Over-all Results(148 Well conditions)
Avg. error, percentStd. deviation, percent
Results from GroupedData Sources
4000
10 0 API OIL320 BOPD
/------i--- 4020 GOR -
1000 2000 3000WEll DEPTH - FEET
FIG. 2-COMPARISON OF PRESSURE PROFILES CALCULATED BYVARIOUS
METHODS FOR WELL 1 (TABLE 1).
200 THPo
1000
ViQ.
The results of the study, summarized in Table 2, arepresented as
the deviations between predicted and meas-ured values for the
modified Griffith-Wallis, the Duns-Ros and the then recently
published Hagedorn-Brown1V)V)wet.::a.. 600f--+--....
....
~
CONCLUSIONSFor general engineering work, the modified
Griffith-Wal-
lis method will predict pressure drops with sufficient ac-curacy
and precision over a wide range of well conditions.I recommend its
use. However, the method should be usedwith discretion for those
well conditions which were not
negligible in the work, the method for predicting frictionlosses
is an approximation and therefore open to Limprove-ment. On the
other hand, the Duns and Ros work in thisrange (which they termed
plug flow) is presented as a com-plex set of interrelated
parameters and equations, and istherefore difficult to relate to
what physically occurs in-side the pipe.
The Griffith-Wallis work was extended to include
thehigh-velocity flow range. In modifying the method, aparameter
was developed to account for (1) the liquid dis-tribution among the
liquid slug, the liquid film and en-trained liquid in the gas
bubble and (2) the liquid holdupat the higher flow velocities. This
parameter served tobetter approximate wall friction losses and
flowing den-sity, and was principally correlated from the earlier
pub-lished data of Hagedorn and Brown.9 The data from Table1 were
also used to determine the effects of pipe diameteron the
parameter. The details of the parameter evaluationare given in
Appendix C and a brief description of themodified Griffith-Wallis
method is outlined in AppendixA.
JUNE, 1967 881
-
veloped in this manner, are only useful in the range
ofconditions from which they were developed.
NOMENCLATURE
A p = flow area of pipe, sq ftB o = oil formation volume factor,
bbl/STB
C, C 2 = parameters used to calculate bubble rise velocitiesfrom
Eq. C-5, dimensionless, to be evaluatedfrom Figs. 8 and 9
dh = hydraulic pipe diameter (4 X Ap/wetted perime-ter), ft
D = depth from wellhead, ft/::iD = increment of depth,ft
f =..Moody friction factor, dimensionless, to be eval-uated from
Fig. 6
Fg = flowing gas fraction, dimensionlessg = accelerafon of
gravity, ftl sec2
g c = gravitational constant, ft-lb(mass)/lb(force)-sec2(Lh =
bubble-slug boundary, dimensionless(L)111 = transition-mist
boundary, dimensionless(L)s = slug-transition boundary,
dimensionless
N b = 1,488 vbd" pLI,ILL, bubble Reynolds number,
dimen-sionless
NEe = 1,488 V D h P/IL, Reynolds number, dimensionlessp =
pressure, psia
/::ip = pressure drop, psip = average pressure, psia
Ppc = pseudo-critical pressure, psiaP" = reduced pressure,
dimensionlessP = pressure, lb/sq ftq = volumetric flow rate, cu
ft/sec
qo = oll rate, B/DR = produced GOR, scf/STB
R 8 = solution gas, scf/STBT pa = pseudo-critical temperature,
oR
T r =reduced temperature, dimensionlessT = average temperature,
ofv = fluid velocity, ft/sec
Vb = bubble rise velocity (velocity of rising gas bubblerelative
to preceding liquid slug), ft/sec
Vbz = base bubble rise velocity for Eq. C-9, ft/secV 8 = slip
velocity (difference between average gas and
liquid velocities), ft/secVgD = qg eVpdga)/A p , dimensionless
gas velocity
z = gas compressibility factor, dimensionlessy = fluid
specificgra"Vity, dimensionless
r = liquid distribution coefficient, to be evaluated fromEqs.
C-ll through C-16, dimensionless
M = viscosity, cp~ID = Moody pipe relative roughness factor
(Fig. 7) and
Duns-Ros mist flow factor (Eqs. C-21 and C-22),
dimensionless
p = density, lbl cu ftp= average flowing density, lb/cu ft
7f = friction-loss gradient, lb/sq ft/ft(J' = surface tension,
Ib/sec2
SUBSCRIPTSg = gasL = liquid0= oilt = total
ACKNOWLEDGMENT
The author wishes to thank the Creole Petroleum Corp.for
supplying data in Table 1 from 22 Venezuelan heavy-oil wells.
REFERENCES
1. Poettmann, F. H. and Carpenter, P. G.: "The Multiphase Flowof
Gas, on, and Water Through Vertical Flow Strings", Drill.and Prod.
Prac., API (1952) 257.
2. Duns, H., Jr., and Ros, N. C. J.: "Vertical Flow of Gas
andLiquid Mixtures from Boreholes", Proc., Sixth World Pet.
Con-gress, Frankfort (Jnne 19-26, 1963) Section II, Paper
22-PD6.
3. BaxendeIl, P. B. and Thomas, R.: "The Calculation of
Pres-sure Gradients in High-Rate Flowing Wells", J. Pet.
Tech.(Oct., 1961) 1023-1028.
4. Tek, M. R.: "Multiphase Flow of Water, Oil, and Natural
GasThrough. Vertical Flow Strings", J. Pet. Tech. (Oct.,
1961)1029-1036.
5. Yocum, B. T.: "Two-Phase Flow in Well Flowlines", Pet.
Eng.(Nov., 1959) B-40.
6. Fancher, G. H., Jr., and Brown, K. E.: "Prediction of
PressureGradients for Multiphase Flow in Tubing", Soc. Pet. Eng.
J.(March, 1963) 59-69.
7. Baker, W. J. and Keep, K. R.: "The Flow of Oil and
GasMixtures in Wells and Pipelines: Some Useful Correlations",J.
lns~. of Pet. (May, 1961) 47, No. 449, 162-169.
8. Hughmark, G. A. and Pressburg, B. S.: "Hold-Up and Pres-sure
Drop with Gas-Liquid Flow in a Vertical Pipe", AIChE J.(Dec., 1961)
7, No.4, 677-682.
9. Hagedorn, A. R. and Brown, K. E.: "The Effect of Liquid
Vis-
2000 3000400 600 1000MEASURED 6P -PSI
200100 I-.L-~_-L.-._~--'-_~ -'-_-'-~
100
3000 DATA SOURCEo BAXENDELL et al.
. 2000 0 FANCHER et al.A HAGEDORN et al. POETTMANN et al. THIS
PAPER
[ v; 1000J:l.
J:l. ..
-
where A p = pipe area, sq ft,
':'This method is a composite of the following:Method Flow
Regime
APPENDIX ADESCRIPTION OF MODIFIED GRIFITH AND
WALLIS METHOD*
(A-3)
at in-
APPENDIX B
DETERMINATION OF FLOW REGIME
W t = total mass flow rate, Ib/ sec,**qy = gas volumetric flow
rate, cu ftlsec.**
With the abO've conditions and Eq. A-2, Eq. A-I maythen be
expressed in a more convenient form. ***
Griffith and Wallis have defined the boundary betweenbubble and
slug flow,l1 and Duns and Ros have definedthe boundaries for the
remaining three regimes.2 The flowregime may be determined by
testing whether the variablesqui q t or vyD, or both, fall within
the limits prescribed.
':":'All volumetric (q) and mass (w) flow rates are those of the
pro-duced fluids that are corrected f0'r temperature, pressure and
gas solu-bility.
':":":'L",Z is taken as positive downward. The pressure should
be madediscontinuous with depth should the denominator approach
zero, or be-come negative. to estabish the shock front that
characterizes sonic ve-locities.
t1Pk = [_1_ P+ T f 2 -1 t1Dk,144 1 - Wt qy/4,637 A p p "
where for average temperature-pressure conditionscrement k,
p = average fluid density, Ib/cu ft,b..p = pressure drop,
psi,
p = average pressure, psia.Since temperature is related to
depth, Eq. A-3 may be in-cremented by either fixing .6.D and
solving for .6.p, or viceversa. However, since pressure usually has
a greater in-fluence on the average fluid properties than
temperature,6.p should be fixed because the change in average
fluidproperties would then be more gradual in going from
oneincrement to another. The value of ,6.p should be around10
percent of the absolute pressure, which is known forone point in
the increment, but should not be greaterthan 100 psi for that
increment.
Pressure drops can be calculated, using Eq. A-3 in thefollowing
manner.
L Pick a point in the flow string (e.g., wellhead or
bot-tom-hole) where the flow rates,fluidproperties, tempera-ture
and pressure are known.
2. Estimate the temperature gradient of the well.3. Fix the 16.P
at about 10 percent of the measured or
previously calculated pressure, which may be at either thetop or
bottom of the increment. Find average pressure ofincrement.
4. Assume a depth increment ,6.D and find average depthof
increment.
5. From the temperature gradient, determine averagetemperature
of increment.
6. Correct fluid properties for temperature and pressure.7.
Determine the type of flow regime from Appendix B.8. Based on Step
7, determine the average density (p)
and the friction loss gradient (Tf) from Appendix C.9. Calculate
,6.D from Eq. A-3.10. Iterate, if necessary, starting with Step 4
until as-
sumed ,6.D equals calculated t1D.11. Determine values of p and D
for that increment.12. Repeat procedure from Step 3 until the sum
of the
6.D's equals the total length of the flow string.A detailed
example of the above calculated procedure is
given ,in Appendix D.
(A-2)
(A-I)
BubbleSlug (density term)Slug (friction gradient
term)TransitionAnnular-mist
Griffith l2Griffith and Wallisl1This workDuns and Ros2Duns and
Ros2
-dP = TfdD + (gp/gc)dD + (pv/gc)dv ,where P = pressure, Ib/sq
ft,
Tf = friction-loss gradient, Ib/sq ft/ft,D = depth, ft,g =
acceleration of gravity,ft/sec2,
gc = gravitational consant, ft-lb(mass)/Ib(force)-sec2,
p = fluid density, Ib/ cu ft,v = fluid velocity, ftl sec.
The procedure was credited to Griffith and Wallis becauseslug
flow occurred in 95 percent of the cases studied. Al-though the
mist-flow regime could not be evaluated, theDuns-Ros method was
used because it appeared to bemore accurate and logical than the
Martinelli method rec-ommended by Griffith. In two-phase flow, both
Tf and pare influenced by the flow regime type, and all three
termsare functions of temperature and pressure. Therefore, touse
Eq. A-I, (1) the flow string must be incremented sothe fluid
properties do not change markedly within anyof the increments, (2)
the flow regime type and correspond-ing variables of p and Tf must
be determined for each in-crement and (3) each increment must be
evaluated by aniterative procedure.
The kinetic energy term is significant only in the mist-flow
regime.2 In mist flow VL
-
The above variables are defined as
VgD = quC4y pdg u)/Ap(Lh = 1.071 - (0.2218 v//dh , with the
limit
(L) ~ 0.13(L)s = 50 + 36 vgD qdqg(L)1JI = 75 + 84 (vg qdqgt75
,
where vgD = dimensionless gas velocity,Vt = total fluid velocity
(qt/Ap), ft/sec,PL = liquid density, lb/cu ft,u = liquid surface
tension, Ib/sec2
Limitsqg/qt < (Lhqg/qt> (L)B, V gD < (L)s(L)M > Vg
> (L)sVgD > (L)M
Flow RegimeBubbleSlugTransitionMist
(B-1)
(B-2)(B-3)(B-4)
where r is a coefficient correlated from oilfield data.Griffith
and Wallis correlated the bubble rise velocity Vbby the
relationship
(C-5)where C1 is expressed in Fig. 8 as a function of
bubbleReynolds number (Nb = 1,488 Vb dhPd,fLL) , and C2 is
ex-pressed in Fig. 9 as a function of both N b and liquid Rey-nolds
number.
N Re = 1,488 pLdllvt/{LL (C-6)where Vt equals total velocity of
liquid and gas (qt/A p),ft/sec.
Fig. 9 was extrapoiated* * so that Vb could be evaluatedat the
higher Reynolds numbers. When C2 cannot be read
':"~The parallel work of Nicklen, Wilkes and Davidson13 provided
thebasis for the extrapolation. It showed that bubble rise velocity
was in-dependent of Nb in the Reynolds number range of 9 X 103 to 1
X 105The correlation of bubble rise velocity was found comparable
to Eq. 0-5when Nb was around 8 X 103 The results were incorporated
into theabove extrapolation.
BUBBLE FLOW (REF. 12)The void fraction of gas (Fg ) in bubble
flow can be ex-
pressed as
APPENDIX CEVALUATION OF AVERAGE DENSITY AND
FRICTION LOSS GRADIENT
where V s = slip velocity in ft/sec. Griffith suggested thata
good approximation of an average V s is 0.8 ft/sec.* Thus,with Eq.
C-1, the average flowing density can be computedas
0.0500.04.......
0.03 "v0.02
VIVI
0.015 Z0.01 ::t:0.008 Cl0.006 ::>0.004 0
lll:0.002
>0.0010.0008. i=0,0006 ::30.0004
w0.0002 lll:0.00010.00005
2 3 4 6 10 20 3040 60 100
1=l-
f-' ..
I- ~~ -,-, ;"
,~~",
""
I",~~ ASP HALTED CAST IRON _" ~I~'&"'1 ~ CAST IRON
0:-'
-
FIG. 8-GRIFFITH AND WALLIS' Cl VS BUBBLE REYNOLDS NUMBER.
(C-19)
(C-20)
(C-21)(C-22)
TRANSITION FLOWDuns and Ros approximated pandT! for transition
flow.
The method is first to calculate these terms for both slugand
mist flow, and then linearly weight each term withrespect to VgD
and the limits of the transition zone (L)8and (L)]I' The terms vgD'
(L)M and (L)8 are defined inAppendix B. The average density term
would be
Duns and Ros express the friction-loss gradient as
1000 2000 3000 4000 5000 6000
(1488 q d P )
REYNOLDS NUMBER - NRe Ap~L h L
FIG. 9-GRIFFITH AND WALLIS' C2 VS BUBBLE REYNOLDSAND REYNOLDS
NUMBERS.
- - (L)M - vgD [_] + vgn - (L)8 [_jP - (L)M - (L)8 P slug (L)M -
(L)8 P mist
. . . . . . . . . . (C-17)The friction gradient term would be
weighted similarly. Amore accurate friction-loss prediction is
claimed if the gasvolumetric flow rate for mist flow is taken
as
and when Vt > 10,
r > - voA p (1 - p). . . . . (C-16)- qt + vbAp PL
The above constraints eliminate pressure discontinuitiesbetween
flow regimes.
N 1.41----+_-uI-
~uB: 1.2 I-----..q----+-~==----+--ou
qg = Ap(L)]I (pL/gat Y4 . . . . . . (C-18)MIST FLOW
The average flowing density for mist flow is given in Eq.C-2.
Since there is virtually no slip between the phases,Fg is
where V g is the superficial gas velocity, and f is again
ob-tained from Fig. 6 as a function. of gas Reynolds number(Nne =
1,488 pgdhvg/,fLg) and a correlated form of theMoody relative
roughness factor ~/D that was developedby Duns and Ros. In their
correlation, they limit ~/D tobeing no smaller than 10-3 but no
greater than 0.5. Be-tween these limits, ~/D is determined from Eq.
C-21 if N 10is less than 0.005 and from Eq. C-22 if N,e is greater
than0.005.
~/D = 34 a/(pgv/dh ) ~/D = 174.8 a (N w )o.302/(pgv/dn) ,
where N 10 = 4.52 X 10-7 (vgfLda)2 Pg/PL'DEVELOPMENT OF T!FOR
SLUG FLOW
A new method was -developed to correlate the friction-loss
gradient for slug flow because neither the Griffith andWallis
method, nor the Stanlei4 method (an outgrowthof the Griffith-Wallis
work) proved accurate for the wellconditions studied. (The
Griffith-Wallis data were taken
Use EquationC-lIC-12C-13C-14
. . . . . . . (C-15)
V t
1010
~ j..--IVI~
WaterWaterOilOil
ContinuousLiquidPhase
0.30
0.40
0.10
r 2 -0.065 Vt
UI-Zw
~ 0.20u..l.L.Wou
from Fig. 9, the extrapolated values of Vo may be calcu-lated
from the following set of equations.
When No ~ 3,000,Vo = (0.546 + 8.74 X 10-6 Nne) Vgdn (C-7)
When No 2 8,000,Vb = (0.35 + 8.74 X 10-6 Nne) Vgdh (C-8)
When 3,000 < No < 8,000,Voi = (0.251 + 8.74 X 10-6 Nne) V
gdn ,
1 J 2 13.59,fLL (C 9)Vo ~ T Vbi + 1 Vbi + PL Vd
n -
The wall friction-loss term, which has been
independentlyderived, is expressed as
T! = fPLVt[qL + VbAp + rj . . . . . (C-I0)2gcdn qt + VbAp
The friction factor .is obtained from Fig. 6 and is a func-tion
of the Reynolds number given by Eq. C-6 and the ~/D
_ obtained from Fig. 7. The liquid distribution coefficient rmay
be determined by the equation which meets the fol-lowing
conditions.
r = [(0.013 log ,fLL)/dnUS] - 0.681 + 0.232 log V t- 0.428 log
dn (C-lI)
r = [(0.045 log fLL)/dno.799] - 0.709 - 0.162 log Vt- 0.888 log
dh (C-12)
r = [0.0127 log (fLL +1)/dnl.415] - 0.284+ 0.167 log V t + 0.113
log dn (C-13)
r = [0.0274 log (fLL + l)/dnl.371] + 0.161+ 0.569 log dn - log
Vt {[O.Ollog (fLL + 1)/dn1.571]+ 0.397 + 0.63 log dn} , (C-14)
but .is constrained by the limits
00 10 20 30 40 50
(1488 V d p )
BUBBLE REYNOLDS NUMBER - Nb fL: h L
JUNE, 1967 835
-
FIG. lO-EFFECT OF VELOCITY ON WATER DISTRIBUTIONCOEFFICIENT.
FIG. ll-OIL DISTRIBUTION COEFFICIENT AFFECTED BY BOTHVELOCITY
AND VISCOSITY.
0.942
Dead Oil Viscosity:at 100F 89 cpat 210F 8.8 cp
126F150F
1000L..----1-0.L.00----2-0-'-0-0----30-'-0-0---4-:-:-'000
DEPTH - FT
The average temperature (T), read from Fig. 12 is 127.5F.
160r-----,-------r----~---__,
':'These are conveniently found in Frick's Petroleum Production
Hand-booTe, Vol. II (Ref. 16).
APPENDIX D
EXAMPLE OF TWO-PHASE PRESSUREDROP CALCULATION
FIG. l2-TEMPERATURE VS DEPTH - WELL 22.
u..o
I
~
1401-----t-----'--=:;;;l:;;;;;;o-.......,=----j--------j::>l-e:(D:::~
1201-----t------t-----j--------j~wJ-
TABLE' 3 - FLOW RATES AND PHYSICAL CONDITIONSOF HEAVY-OIL WELL
22
1,850 Oil SpecificSTB/D Gr~vity (Yo)
Produced GOR (R) 575 scf/ Gas' SpecificSTB Gravity (Yg) 0.75
3,890 ft Wellhead Pressure 670 psia0.249 ft Tubing Area Ap
0.0488
l:iq ft
An example calculation of the modified Griffith-Wallismethod is
presented to illustrate the details of the proce-dure outlined in
Appendix A. In this example we will pre-dict the pressure drop for
heavy-oil Well 22 (Table 1). Theinput well data required for the
calculation are given inTable 3. In addition, we will need the
following correla-tions* that correct fluid properties for pressure
and tem-perature:
Gas pseudo-critical properties. (Katz et al.) T pc , ppc.Gas
compressibility (Brown et al.) z.Live oil viscosity (Chew and
Connally) p,.Oil formation volume factor (Standing) B oSolution gas
(Lasater) R .For calculational convenience,the
temperature-viscosity-
depth data contained in Table 3 should be plotted.
Thetemperature-depth plot is shown in Fig. 12, and log
vis-cosity-log temperature plot is shown in Fig. 13.
The detailed procedure for the calculation of the pre~sure drop
for the first increment (k == 1) is as follows.
1. Based on the 670-psia wellhead pressure, fix Ap at 100psi.
Assume I~D to be 540 ft. The average pressure
-
FIG. 13-DEAD OIL VISCOSITY VS TEMPERATURE - WELL 22.
~'Parentheses indicate the page number in Frick's booklG, where
thevarious correlations are found.
':":'Live oil viscosity. Dead oil viscosity, a parameter in the
correlation,is read from Fig. 13.
2. With the condition determined in Step 1, the fluidproperties
are corrected for temperature and pressure.
From FricklG the following values are obtained.
R. = 115 scf/bbl (page 19-9).*B a = 1.073 bbl/STB (page
19-25).
pp@ = 665 psia (page 17-6).T po = 415R (page 17-6).
p" = 18 cp** (page 19-40).The gas compressibility c is
determined as
Tr
= T + 460 = 587.5 = 1.42T po 415
P 720pr = Ppc = 665 = 1.08 ,
and from Frick (page 17-15),Z = 0.875.
(B-2)
- 0.097 ]
Evaluate Tl with Eq. C-lO:_ 0.034 (55.8) (6.72)2 [ 0.129 + 1.74
(0.0488)
Tj - 64.4 (0.249) 0.328 + 1.74 (0.0488)= 2.26 lb/ sq ftl ft.
5. The depth increment from Eq. A-3 is
therefore, r = - 0.097.The ~/D value from Fig. 7 is 0.0006. With
this value andthe calculated N Re of 7,720, a friction factor of
0.034 isread from Fig. 6.
Evaluate p with Eq. C-4:7.77 + 55.8 (1.74) (0.0488)
0.328 + 1.74 (0.0488)+ (-0.097) (55.8) = 24.91b/cu ft.
(Lh = -22. Since (L)n has the limit of 0.13,... (L)B = 0.13.
(L)8 = 50 + 36 (9.53) (0.129)/0.199 = 272. . (B-3)Because qy/qt
> (L)B and VUD < (L)8' the fluids are in
slug flow.4. The equations given in the Slug Flow section of
Appendix C are used to calculate pand TI'Determine Reynolds
number, bubble Reynolds number
and slip velocity (Vb)'Nne = 1,488 (55.8) (0.249) (6.72)/18 =
7,720 . (C-6)
Since the bubble rise velocity is a nonlinear
correlation.iteration is necessary. Therefore, assuming Vb = 1.75,
bub-ble Reynolds number is
N b = 1,488 (55.8) (0.249) (1.75)/18 = 2,010.C2 cannot be read
from Fig. 9. Thus the extrapolationequation (Eq. C-7) is used since
N b < 3,000.
----
Vb = [0.546 + 8.74 X1O-G(7,720)] Y32.2 (0.249)== 1.74
ft/sec.
Determine liquid distribution coefficient r and frictionfactor
f. Eq. C-13 is used to evaluate r since Vt < 10:
= [0.0127lc g (18 + 0] _0.284 + 0.16710 6.72r (0.249y"4l5 g
+ 0.113 log 0.249 = - 0.097,
Test limiting r with Eq. C-15:- 0.097 2 - 0.065 (6.72)
2 - 0.436;
The corrected densities arePL = WL/qL = 7.20/0.129 = 55.81b/cu
ftPu - wyqr! = 0.565/0.199 = 2.84 Ib/ cu ft.
3. The variables described in Appendix B are calculatedand then
are tested against the boundary limits to deter-mine the flow
regime.
Test Variables:V t = qtlA p = 0.328/0.0488 = 6.72 ft/sec.
qu/qt = 0.199/0.328 = 0.607.
VUD = 0.199 [\/0.534 (55.8)] /0.0488 = 9.53 . (B-1)
Boundary Limits:
(L) = 1 071 _ ~.2218 (6.72)2B' 0.249
30080 100 200TEMPERATURE _ OF
\i\.'\'\'\\\\\
I\.'"\.
I ' I
10860
Q., 60u
The corrected volumetric flow rates areqL = 6.49 X lO-5 qoBa =
6.49 X lO-5 (1,850) (1.073)
= 0.129 cu ft/sec-7 (f + 460).qy = 3.27 X lO z qo (R - R s )
--_-
P= 3.27 X lO-7 (0.875) (1,850) (575-115)
(587.5)/720= 0.199 cu ftl sec
qt = 0.128 + 0.199 = 0.328 cu ft/sec.The corrected mass flow
rates are
WL = qo (4.05 X lO;-3 Yo + 8.85 X lO-7 yyR8 )= 1,850 [4.05 X
lO-3 (0.942) + 8.85 X 10-7
(0.75) (115)]= 7.201b/sec
W y = 8.85 X lO-7 qoyy{R - R 8 )= 8.85 X lO-7 (1,850) (0.75)
(575 - 115)= 0.565 Ib/ sec
Wt = 7.20 + 0.57 = 7.771b/sec.
10080
JUNE, 1967 837
-
FIG. 14-CALCULATED VS MEASURED PRESSURE DROP - WELL 22.
\~ o MEASURED\ - CALCULATED\
'"\
~\
~\
~
o
500
3500
1000
4000600 800 1000 1200 1400 1600
PRESSURE - PSI A
3000
.... 1500u..
P +- Tf 1
7.77 (0.199) )]4,637 (0.0488)2 (720) _.24.9 +- 2.3 1- 529
ft.
The true value of .D.D1 is near 529 ft. The calculation
willconverge very closely to this value even when the assumed!:::.Z
is off by 1O percent of the assumed value (540 ft)because, under
these well conditions, the pressure gradientis primarily controlled
by the relatively temperature-insen-sitive density head. However,
under those circumstanceswhere the friction ,gradient, which is
temperature sensitive,is significant, iteration would be necessary
should the cal-culated value of l::,.D differ from the assumed
value by+- 10 percent.
6. The top of the next increment is fixed at 529 ft and770 psi,
and Steps 1 through 6 are repeated for the newconditions.
7. The procedure is continued until ~ .!:::.D is equal to
thetotal depth. The calculated pressure profile is comparedagainst
the measured profile in Fig. 14. ***
SlS8 JOURNAL OF PETROLEUM TECHNOLOGY
829830831832833834835836837838
