ABSTRACT

Mechanistic models for multiphase flow calculations canimprove our ability to predict pressure drop and holdup inpipes especially in situations that cannot easily be modeledin a laboratory and for which reliable empirical correlationsare not available. In this paper, a new mechanistic model,applicable to all pipe geometries and fluid properties is pre-sented. New empirical correlations are proposed for liq-uid/wall and liquid/gas interfacial friction in stratified flow,for the liquid fraction entrained and the interfacial friction inannular-mist flow, and for the distribution coefficient used inthe determination of holdup in intermittent flow.

INTRODUCTION

Empirical models often prove inadequate in that they arelimited by the range of data on which they were based and,generally, cannot be used with confidence in all types of flu-ids and conditions encountered in oil and gas fields. Fur-thermore, many such models exhibit large discontinuities1 atthe flow pattern transitions and this can lead to convergenceproblems when these models are used for the simultaneoussimulation of petroleum reservoirs and associated production

facilities. Mechanistic models, on the other hand, are basedon fundamental laws and thus can offer more accurate mod-eling of the geometric and fluid property variations.

All of the models presented in the literature are either in-complete2,3, in that they only consider flow pattern determi-nation, or are limited in their applicability to only some pipeinclinations4,5. A preliminary version6 of the model proposedhere that overcomes these limitations was presented in 1996.

For most of the flow patterns observed, one or more em-pirical closure relationships are required even when amechanistic approach is used. Where correlations availablein the literature are inadequate for use in such models, newcorrelations must be developed. In order to be able to achievethis, access to reliable experimental data is important.

A large amount of experimental data has been collectedthrough the use of a Multiphase Flow Database7 developed atStanford University. The database presently contains over20,000 laboratory measurements and approximately 1800measurements from actual wells. Based on subsets of thesedata, the previously proposed model6 included a detailedinvestigation of the annular-mist flow regime and new corre-lations for the liquid fraction entrained and for interfacialfriction. This model has since been refined based on addi-

THIS IS A PREPRINT - SUBJECT TO CORRECTION

A MECHANISTIC MODELFOR MULTIPHASE FLOW IN PIPES

Nicholas PetalasStanford University

Khalid AzizStanford University

PUBLICATION RIGHTS RESERVEDPAPER NO. 98-39. THIS PAPER IS TO BE PRESENTED AT THE 49TH ANNUAL TECHNICALMEETING OF THE PETROLEUM SOCIETY OF THE CANADIAN INSTITUTE OF MINING,METALLURGY AND PETROLEUM HELD IN CALGARY, ALBERTA, CANADA ON JUNE 8-10,1998.

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tional investigations of the stratified and intermittent flowregimes, and is the subject of this paper.

FLOW PATTERN DETERMINATION

The procedure for flow pattern determination begins withthe assumption that a particular flow pattern exists and isfollowed by an examination of various criteria that establishthe stability of the flow regime. If the regime is shown to beunstable, a new flow pattern is assumed and the procedure isrepeated. Figure 1 shows flow pattern transitions based onthe superficial velocities of the phases where the stabilitycriteria (transition boundaries S1, S2, etc.) considered in thismodel are sketched. The procedure for flow pattern determi-nation is illustrated in Figure 3, where it is seen that the ex-amination of the dispersed bubble flow regime is the first tobe considered.

Dispersed Bubble Flow

The dispersed bubble flow region is bounded by two crite-ria. The first is based on the transition to slug flow proposedby Barnea2 where a transition from intermittent flow occurswhen the liquid fraction in the slug is less than the value as-sociated with the maximum volumetric packing density ofthe dispersed bubbles (0.52):

48.0<LsE Eq. 1

The same mechanism is adopted in this model with theexception that the liquid volume fraction in the slug is notobtained from the correlation proposed by Barnea, but fromthe Gregory et al.8 correlation given below:

39.1

66.81

1

+

=m

LsV

EEq. 2

where mV is expressed in meters/sec. This transition isshown as line I1 in Figure 1.

A transition from dispersed bubble flow to froth flow canalso occur when the maximum volumetric packing density ofthe dispersed gas bubbles is exceeded (line D1 in Figure 1):

52.0>=m

SGG

V

VC Eq. 3

If the criteria given by Eq. 1 and Eq. 3 are not satisfied,dispersed bubble flow is not possible and the possibility ofstratified flow is examined next.

Stratified Flow

Determining the stability of the stratified flow regime re-quires the calculation of the liquid height, which can be ob-tained by writing the momentum balance equations for thegas and the liquid phases as was done by Taitel and Dukler3:

0sin =θρ−τ+τ−

−

cLLiiLwLL

g

gASS

dL

dpA Eq. 4

0sin =−−−

− θ

g

gAρSτSτ

dL

dpA

cGGiiGwGG Eq. 5

These can then be combined, eliminating the pressure gra-dient terms, and expressed in terms of the dimensionless liq-

uid height, Dhh LL =~, using the geometric relationships

outlined by Taitel and Dukler. The shear stresses are givenby the following relationships:

c

GGGwG

g

Vf

2

2ρ=τ Eq. 6

c

LLLwL

g

Vf

2

2ρ=τ Eq. 7

c

iiGii

g

VVf

2

ρ=τ Eq. 8

These definitions differ from those proposed by Taitel andDukler mainly in that pipe roughness is not ignored whendetermining the friction factors. In addition, the definition forthe gas/liquid interfacial shear does not require the assump-tion that the gas phase moves faster than the liquid phase(thereby assuming the shear stress to be based on the gasphase velocity alone). A new approach is also used whendetermining the liquid /wall interfacial friction factor, Lf , asdiscussed below.

The friction factor at the gas/wall interface (Eq. 6) is de-termined from an approach similar to that used in single-phase flow with the actual pipe roughness and the followingdefinition of Reynolds number:

G

GGGG

VD

µρ=Re Eq. 9

where GD is the hydraulic diameter of the gas phase.

For the liquid/wall interface (Eq. 7) it was found that theuse of an approach similar to single phase flow is not appro-priate and, instead, the following empirical relationship isused for calculating the wall/liquid interfacial friction factor:

731.0452.0 SLL ff = Eq. 10

The friction factor based on the superficial velocity, SLf , isobtained from standard methods using the pipe roughnessand the following definition of Reynolds number:

L

SLLSL

VD

µρ=Re Eq. 11

During downhill flow, it is possible for the dense phase toflow faster than the lighter phase. For this reason, the defini-tion of the gas/liquid interfacial shear (Eq. 8) is based on thequantity LGi VVV −= , which can become negative under

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certain conditions. The interfacial friction factor is calculatedfrom the empirical relationship:

ρρ×+= −

2335.16 Fr)Re105.0004.0(

GG

LLSLi

V

Dgf Eq. 12

The Froude number is defined asL

LL

gh

V=Fr .

Once the liquid height is known, the stability of the strati-fied flow pattern can be determined. The approach used byTaitel and Dukler, which uses an extension of the Kevin-Helmholtz wave stability theory, is also used in this model.This attempts to predict the gas velocity at which waves onthe liquid surface are large enough to bridge the pipe:

( )

L

LG

GGLLG

dh

dAgA

D

hV

ρ

θρ−ρ

−= cos1

Eq. 13

This transition is represented by line S1 in Figure 1. Thetransition criterion expressed by Eq. 13 does not apply when

°−=θ 90 . Experimental data, however, suggest that strati-fied-like annular flow can occur at such inclinations. In orderto account for this, when 02.0cos ≤θ , 02.0cos =θ is substi-tuted in Eq. 13.

At steep downward inclinations, Barnea proposes amechanism whereby stratified flow can change to annular,even at relatively low gas rates. This occurs when the liquidheight is small and the liquid velocity is high. Liquid dropletsare sheared off from the wavy interface and deposited on theupper pipe wall, eventually developing into an annular film.The condition for this type of transition to annular flow,shown as line S4 in Figure 1, is given as2:

L

LL

f

hgDV

θ−> cos)~

1(Eq. 14

It should be noted that Lf is calculated as per Eq. 10; not thedefinition proposed by Barnea.

At the higher upward pipe inclinations the predicted liquidheight has the tendency, given the transition criterion of Eq.13, to predict stratified flow where none is known to exist.For this reason, and to ensure continuity between flow pat-tern transitions, the present model limits stratified flow tohorizontal and downhill angles only. This approach is alsosupported by the fact that stratified flow is only observed forsmall upward angles in large-diameter pipes.

Thus, when 0≤θ , if the gas phase velocity is less thanthe transitional value given by Eq. 13 and the liquid phasevelocity is less than that of Eq. 14, the flow pattern is strati-fied. Although no distinction is made in this model betweenstratified smooth and stratified wavy flow for the purposes ofdetermining pressure drop and liquid volume fraction, thetransition between these two regimes is considered in flowpattern predictions. Taitel and Dukler propose that waves

will form on the liquid surface once the gas velocity is in-creased beyond (line S2 in Figure 1):

LGL

GLLG

Vs

gV

ρρθρ−ρµ≥ cos)(4

Eq. 15

The sheltering coefficient, s, is given as 0.01. In Xiao etal. 5 and in the present model, s is taken as 0.06, based on astudy by Andritsos9. This value is said to be more suitable,especially for gas flow with high viscosity liquids.

During downflow, waves can develop on the flowing liq-uid independent of interfacial shear from the gas flow. Thecriterion for the appearance of waves can be expressed interms of a critical Froude number which varies from 0.5 to2.2 depending on roughness and whether the flow is laminaror turbulent. Barnea2 recommends a limiting value of 1.5 forthe critical Froude number. When interfacial effects are con-sidered in the calculation of the liquid height, this limit canpredict smooth flow even at high liquid rates where the flowis known to be wavy. Reducing the limit to 1.4 appears toresolve this problem. Thus the transition from stratifiedsmooth to wavy flow based on this mechanism is (line S3 inFigure 1):

4.1Fr >=L

L

gh

VEq. 16

Annular-Mist Flow

The treatment of the annular-mist flow regime is similar tothe approach used for stratified flow and is based on thework of Taitel and Dukler3 and Oliemans et al.10. The modelis based on the assumption of a constant film thickness andaccounts for the entrainment of the liquid in the gas core.Slip between the liquid droplets in the gas core and the gasphase is not accounted for. Momentum balance on the liquidfilm and gas core with liquid droplets yields:

0sin =θρ−τ+τ−

−

cfLiiLwLf

g

gASS

dL

dpA Eq. 17

0sin =θρ−τ−

−

ccciic

g

gAS

dL

dpA Eq. 18

The geometric parameters can be expressed in terms of the

dimensionless liquid film thickness, DLL δ=δ~ , and theliquid fraction entrained, FE. The shear stresses are given by:

c

fLfwL

g

Vf

2

2ρ=τ Eq. 19

( )c

fcfcci

ig

VVVVf

2

−−ρ=τ Eq. 20

The friction factor for the liquid film is computed usingany of the standard correlations with the pipe roughness andthe film Reynolds number as expressed by:

4

L

fLff

VD

µρ

=Re Eq. 21

In order to solve Eq. 17 and Eq. 18, two additional quanti-ties need to be determined: the interfacial friction factor, if ,and the liquid fraction entrained, FE. These are determinedempirically and are given by:

2.0

074.0735.01

=

− SL

SGB

V

VN

FE

FEEq. 22

305.0

085.0

2Re24.0 f

cccc

i

DVf

f

ρσ= Eq. 23

Where the dimensionless number, BN , is defined as

L

GSGLB

VN

ρσρµ=

2

22

Eq. 24

Having determined the liquid film thickness, it is nowpossible to test for the presence of annular-mist flow. Ba-rnea2 presents a model for the transition from annular flowbased on two conditions. The same mechanisms are used inthe present model, although they are revised to account forthe differences in the modeling assumptions.

The first of the transitions proposed by Barnea is based onthe observation that the minimum interfacial shear stress isassociated with a change in the direction of the velocity pro-file in the film. When the velocity profile becomes negativestable annular flow cannot be maintained and the transitionto intermittent flow occurs. This transition mechanism isonly relevant during uphill flow. The minimum shear stress

condition may be determined by setting 0~ =δ∂τ∂

L

i .

( ) ( )f

ffSL

cL

Lf

E

EE

gD

FEVf

23

23322

2

1

sin

12

−−

=θ

−ρ−ρ

ρEq. 25

The liquid fraction in the film is given by:

( )LLf

f A

AE δ−δ== ~

1~

4 Eq. 26

Eq. 25 can be solved using an iterative procedure to obtainthe liquid film height at which the minimum shear stress oc-

curs,min

~Lδ (line A1 in Figure 1).

The second mechanism proposed by Barnea for annularflow instability occurs when the supply of liquid in the filmis sufficient to cause blockage of the gas core by bridging thepipe. This is said to take place when the in situ volume frac-tion of liquid exceeds one half of the value associated withthe maximum volumetric packing density of uniformly sizedgas bubbles (0.52). Hence, the transition from annular flowoccurs when (line A2 in Figure 1):

24.0or)52.01(21 ≥−≥ LL EE Eq. 27

Bubble Flow

When the liquid fraction in the slug (Eq. 2) is greater than0.48 and the stratified, annular and dispersed bubble flowregimes have been eliminated, the flow will either be inter-mittent, froth or bubble flow.

Bubble flow is encountered in steeply inclined pipes and ischaracterized by a continuous liquid phase containing a dis-persed phase of mostly spherical gas bubbles. It can exist ifboth of the following conditions are satisfied:

1. The Taylor bubble velocity exceeds the bubble veloc-ity. This is satisfied in large diameter pipes (Taitel etal.11) when

( ) 21

219

ρ

σρ−ρ>g

DL

GL Eq. 28

2. The angle of inclination is large enough to preventmigration of bubbles to the top wall of the pipe (Ba-rnea et al.12):

γ≤θb

bgd

CV

22

24

3cos l Eq. 29

The lift coefficient, lC , ranges from 0.4 to 1.2, the bubble

distortion (from spherical) coefficient, γ, ranges from 1.1 to1.5 and a bubble size, bd , between 4 and 10mm is recom-

mended. For this model, lC is taken as 0.8, γ as 1.3 and abubble diameter of 7 mm is used. The bubble swarm risevelocity in a stagnant liquid, bV , is given by13:

( )θ

ρ

σρ−ρ= sin41.14

1

2L

GLb

gV Eq. 30

When both of the above conditions are satisfied, bubbleflow is observed even at low liquid rates where turbulencedoes not cause bubble breakup. The transition to bubble flowfrom intermittent flow as suggested by Taitel et al.11 occurswhen the gas void fraction (during slug flow) drops belowthe critical value of 0.25 (line I3 in Figure 1). The calculationof the gas void fraction for slug flow is discussed below.

Intermittent Flow

The intermittent flow model used here includes the slugand elongated bubble flow patterns. It is characterized byalternating slugs of liquid trailed by long bubbles of gas. Theliquid slug may contain dispersed bubbles and the gas bub-bles have a liquid film below them.

As stated above, a transition from intermittent flow occurswhen the liquid fraction in the slug exceeds the value associ-ated with the maximum volumetric packing density of thedispersed bubbles (Eq. 1, line I1 in Figure 1). The samemechanism can occur at low liquid rates when sufficient liq-

5

uid is not available for slug formation. To account for thissituation, an additional transition criterion is imposed (line I4in Figure 1).

24.0≤LE Eq. 31

The liquid volume fraction calculated for slug flow is dis-cussed below. Although it is not treated as a separate flowpattern for the purposes of phase volume fractions and pres-sure drop determination, the elongated bubble flow regime isdefined here as the portion of intermittent flow for which theliquid slug contains no dispersed bubbles of gas. This condi-tion is arbitrarily represented in the model by the regionwhere 90.0≥

sLE (line I2 in Figure 1).

The liquid volume fraction may be determined by writingan overall liquid mass balance over a slug-bubble unit. As-suming that the flow is incompressible and a uniform depthfor the liquid film14:

( )t

SGLsGdbtLsL

V

VEVVEE

−−+= 1Eq. 32

GdbV represents the velocity of the dispersed bubbles, tV is

the translational velocity of the slug, and LsE is the volumefraction liquid in the slug body (Eq. 2). All of these quanti-ties need to be determined from empirical correlations.

The translational velocity of the elongated bubbles isgiven by Bendiksen15 as:

dmt VVCV += 0 Eq. 33

The parameter 0C is a distribution coefficient related to thevelocity and concentration profiles in dispersed systems andunder special conditions is related to the inverse of theBankoff K factor. Zuber and Findlay16 have confirmed em-pirically its application to other flow patterns, including slugand annular flow. Nicklin et al.17, in their study of the risevelocity of Taylor bubbles, have found that for liquid Rey-nolds numbers greater than 8,000, 0C =1.2, whereas at lower

Reynolds numbers 0C approached 2.0. It is generally takento be 1.2, although for this analysis it is determined from thefollowing empirically derived correlation:

( ) 031.00 Resin12.064.1 −θ+= mLC Eq. 34

The modified Reynolds number in Eq. 34 is based on themixture velocity and liquid properties:

L

mLmL

DV

µρ=Re Eq. 35

The elongated bubble drift velocity, dV , can be calculated

from the Zuboski18 correlation:

∞= dmd VfV Eq. 36

Where ∞= Re316.0mf for 1<mf , otherwise 1=mf , and

L

dL DV

µ

ρ= ∞

∞ 2Re Eq. 37

Bendiksen15 gives the elongated bubble drift velocity athigh Reynolds numbers as:

θ+θ=∞∞∞

sincos dvdhd VVV Eq. 38

The drift velocity of elongated bubbles in a horizontal sys-tem at high Reynolds numbers is given by Weber19 as:

( )L

GLdh

gDV

ρρ−ρ

−=

∞ 56.0Bo

76.154.0 Eq. 39

The Bond number,( ) 2Bo gDGL

σρ−ρ

=

The drift velocity of elongated bubbles in a vertical systemat high Reynolds numbers is obtained from a modified formof the Wallis20 correlation

( ) ( )L

GLdv

gDeV

ρρ−ρ−= β−

∞1345.0 Eq. 40

The coefficient, β, is given by:( )lnBo424.1278.3Bo −=β e Eq. 41

Finally, the volume fraction liquid (Eq. 32) can be calcu-lated once the velocity of the dispersed bubbles in the liquidslug is obtained from:

bmGdb VVCV += 0 Eq. 42

0C in Eq. 42 is determined from Eq. 34 and the rise ve-locity of the dispersed bubbles is calculated from21:

( ) θ

ρ

σρ−ρ= sin53.141

2L

GLb

gV Eq. 43

The empirical nature of the correlations used for determin-ing the liquid volume fraction (Eq. 32) requires that certainlimits be imposed on the calculated values. The first suchcondition affects Eq. 42 where it is possible, under certaindownflow conditions for the calculated value of GdbV to be-

come negative. In these situations, GdbV is set to zero. Inother situations, it is possible for Eq. 32 to yield values for

LE that are greater than 1.0. In these cases, LE is set equal to

LC .

When none of the transition criteria listed above are met,the flow pattern is designated as "Froth" implying a transi-tional state between the other flow regimes.

CALCULATION OF PRESSURE DROP AND LIQ-UID VOLUME FRACTION

Once the flow pattern has been determined, the calculationof the pressure drop and phase volume fractions can be de-termined as detailed below.

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Dispersed Bubble Flow

The calculation of the liquid volume fraction in dispersedbubble flow follows the procedure used for the dispersedbubbles in the slug in intermittent flow. Thus,

bmGdb VVCV += 0 Eq. 44

0C is determined from the empirical correlation given in

Eq. 34, and the rise velocity of the dispersed bubbles, bV , iscalculated from Eq. 43. The volume fraction is then obtainedfrom:

Gdb

SGL V

VE −= 1 Eq. 45

If 0≤GdbV , the volume fraction is then obtained from:

m

SGL VC

VE

0

1−= Eq. 46

In cases where the value of LE calculated by Eq. 45 or Eq.

46 is greater than 1.0, LE is set equal to LC .

Once the liquid volume fraction is known, the pressuregradient is determined from:

θρ+ρ=

− sin

2 2

cm

c

mmm

g

g

Dg

Vf

dL

dpEq. 47

The friction factor, mf , is obtained from standard methodsusing the pipe roughness and the following Reynolds num-ber:

m

mmm

VD

µρ=Re Eq. 48

The mixture density and viscosity are calculated in theusual way:

GGLLm EE ρ+ρ=ρ Eq. 49

GGLLm EE µ+µ=µ Eq. 50

Stratified Flow

The liquid volume fraction during stratified flow is sim-ply, from geometric considerations:

A

AE L

L = Eq. 51

The pressure gradient is obtained from either Eq. 4 or Eq.5.

Annular-Mist Flow

As is the case for stratified flow, the volume fraction liq-uid during annular-mist flow is determined from geometricconsiderations once the liquid film thickness is known:

SLSG

SGLL VFEV

VE )

~21(1 2

+δ−−= Eq. 52

The pressure gradient is obtained from either Eq. 17 or Eq.18.

Bubble Flow

The volumetric gas fraction during bubble flow is ob-tained from:

t

SGG V

VE = Eq. 53

The translational bubble velocity is defined as:

bmt VVCV += 0 Eq. 54

Zuber and Findlay16 have shown that the distribution pa-rameter, 0C , for dispersed systems can range from 1.0 to 1.5,the higher values being associated with high bubble concen-trations and high velocities at the center line (laminar flow).When the flow is turbulent and the velocity and concentra-tion profiles are flat 0C approaches 1.0. For the present

method, 0C is taken as 1.2. The bubble swarm rise velocity

in a stagnant liquid, bV , is given by Eq. 30. The value of

GE thus obtained, is limited to the range:

m

SGGG V

VCE =≤≤0 Eq. 55

The pressure gradient is given by:

θρ+ρ=

− sin

2 2

cm

c

mmmL

g

g

Dg

Vf

dL

dpEq. 56

The friction factor, mLf , is obtained from standard meth-ods using the pipe roughness and the following Reynoldsnumber:

L

mLmL

VD

µρ=Re Eq. 57

Intermittent Flow

The calculation of the volume fraction liquid for intermit-tent flow has already been described (Eq. 32). The pressuredrop may be obtained by writing the momentum balanceover a slug-bubble unit:

τ+τ+

πτ

+

θρ=

−

A

SSL

A

DL

L

g

g

dL

dp

GdbGdbLfLff

Lss

u

cm

1

sin

Eq. 58

Unfortunately, no reliable methods exist for the calcula-tion of the slug length, sL , nor for the length of the bubble

region, fL . Furthermore, although it is known that the fric-

7

tional pressure gradient in the gas bubble is normally smallcompared to that in the liquid slug, no reliable method isavailable for calculating it. Xiao et al.5 have modeled thebubble region by assuming it to be analogous to stratifiedflow. This treatment contradicts observations made in thelaboratory. In view of these uncertainties, the following sim-ple approach is selected:

( )AM

SL

fr

frcm

dL

dp

dL

dp

g

g

dL

dp

η−+

η+θρ=

−

1

sin

Eq. 59

Here, the quantity η is an empirically determined weightingfactor related to the ratio of the slug length to the total slug

unit length,u

s

L

Land is calculated from:

( )LELC −=η 75.0 Eq. 60

with the condition that 0.1≤η .

The frictional pressure gradient for the slug portion is ob-tained from:

gD

Vf

dL

dp mmmL

frSL

ρ=

2

2 Eq. 61

The friction factor, mLf , is calculated from standard methodsusing the pipe roughness and the Reynolds number given byEq. 57.

The term that still needs to be defined in Eq. 59 is the fric-tional pressure gradient calculated for annular-mist flow.This is obtained by using the liquid fraction given by Eq. 32to determine the liquid film height, assuming the flow patternto be annular-mist. Thus, from Eq. 52,

( ) ( )

+−−=δSG

SGSLLL V

VVFEE11

2

1~Eq. 62

The frictional pressure gradient based on annular-mistflow is then calculated from:

DdL

dp wL

frAM

τ=

4

Eq. 63

The shear stress wLτ is obtained from Eq. 19. Note that

Eq. 63 is obtained by adding Eq. 17 to Eq. 18 (to eliminatethe interfacial component) and removing the hydrostaticterm.

When the calculated film height (Eq. 62) is less than4101 −× , a simple homogeneous model with slip is used,

where:

Dg

Vf

dL

dp

c

mmm

frAM

ρ=

22

Eq. 64

The friction factor, mf , is obtained from standard methodsusing the pipe roughness and Reynolds number given in Eq.48. The mixture density and viscosity are obtained from Eq.49 and Eq. 50, respectively.

Froth Flow

Froth flow represents a transition zone between dispersedbubble flow and annular-mist flow and between slug flowand annular-mist. The approach used in this model is to in-terpolate between the appropriate boundary regimes in orderto determine the transition values of the in situ liquid volumefraction and pressure drop. This involves a number of itera-tive procedures in order to determine the superficial gas ve-locities at the dispersed bubble, annular-mist and slug transi-tions to froth. Once SGV at each transition is known, the vol-ume fraction and pressure drop values at the transitions arecalculated and a log-log interpolation between these values ismade for each quantity.

RESULTS

The model’s overall performance has been evaluated usingthe following approaches.

a) The behavior of the model was examined over a widerange of flow rates and fluid properties using three-dimensional surface plots. This was done over the com-plete range of upward and downward pipe inclinationsand both, pressure gradient and volumetric liquid frac-tion were analyzed.

b) Data were extracted from the Stanford Multiphase FlowDatabase for which pressure gradient, holdup and flowpattern observations were available. This resulted in atotal of 5,951 measurements consisting of variations influid properties, pipe diameters, and upward as well asdownward inclinations. The model was then comparedwith these experimental observations.

c) Finally, these same experimental data were analyzedusing a number of existing methods and the results com-pared to the new model.

Flow pattern maps and three dimensional surface plots areshown in Figures 4 to 27 for an air/water system at standardconditions and for an oil/gas system at reservoir conditions(see Table 1). The pipe inclinations shown include horizon-tal, 10° upward, vertical upflow (+90°), and 10° downward.Each plot covers a range of superficial gas velocity of 0.01ft/sec to 500 ft/sec and of superficial liquid velocity from0.01 ft/sec to 100 ft/sec.

The coloring of the three dimensional plots is consistentwith that of the flow pattern maps so as to show the locationof the flow pattern transitions. The superficial velocity axesappear on the X-Y plane with the appropriate parameter(pressure gradient or liquid volume fraction) plotted on the

8

vertical axis. Over one hundred different gas and liquid rateshave been calculated per plot, equating to over 10,000 calcu-lated points. This was done to insure that the model behavespredictably over the entire practical range of flow rates andpipe inclinations. The effect of fluid properties on flow pat-tern transitions can easily be seen in these plots. Althoughsome discontinuities in pressure gradient and liquid volumefraction are present at the transitions from stratified flow,overall, the model exhibits generally smooth behavior andconsistent trends between flow patterns.

The distribution of experimental data points according toangle of inclination is shown in Figure 28. The conventionused in expressing the range of inclinations in this figure isthat the lower number in the specified range is inclusive,whereas the higher number is not. Thus, “0° to 10°” impliesthe range where 0° ≤ θ < 10°. Although more than half ofthe data fall in the range of 0° to 10° inclination, it can beseen that there is also a fair sampling (1,164 points, or ~20%)of downhill data.

The model predictions for liquid volume fraction are plot-ted against the experimental measurements in Figure 29.Figure 30 shows a similar plot for the pressure gradient cal-culations. The model is able to predict the in situ liquid vol-ume fraction to within an accuracy of 15% in 3,663 of the5,951 cases (62%). This is shown in Figure 31, where thesenumbers are compared with other methods. The pressuregradient is predicted to the same accuracy for 2,567 cases(43%), as shown in Figure 31.

Some of the methods with which comparisons are beingmade are limited to specific ranges of pipe inclination. Tomake the comparisons of Figure 31 more meaningful, theyare shown grouped according to downward, near-horizontaland upward pipe inclination in Figure 32, for the volumefraction liquid, and in Figure 33, for the pressure gradient.

COMMENTS

The empirical correlations introduced in this paper,namely:• Eq. 10 and Eq. 12 for stratified flow,• Eq. 22 and Eq. 23 for annular-mist flow,• Eq. 34, Eq. 40 and Eq. 60 for intermittent flow,were developed in accordance with the following two objec-tives.

a) Accurately reflect the expected/observed behavior of thequantity being estimated.

b) Ensure that the correlation’s behavior results in smoothtransitions between adjacent flow patterns.

The application of both of these criteria involved relyingon statistical analysis of differences between predicted andexperimental values as well as the study of numerous surfaceplots (such as those shown in Figures 4 to 27). This did not

always result in the “statistically best” correlation beingadopted.

Solving the momentum balance equations in stratifiedflow (Eq. 4 and Eq. 5) and annular-mist flow (Eq. 17 and Eq.18) poses certain problems because of the presence of multi-ple roots. Primarily, it is necessary to determine which of theroots is the physical one. Xiao et al.5 assume that it is thelowest root. There does not however seem to exist a clearrationale for this assumption. Figure 2 shows the variation ofliquid height versus superficial gas velocity at a fixed super-ficial liquid rate for an air/water system at standard condi-tions (2.047” I.D. pipe, at 2° upward inclination). It can beseen that in the range where multiple roots occur ( SGV =57.144 to 112.67 ft/sec), the selection of the lowest root ver-sus the highest root results in significant changes to the pre-dicted liquid height. This large variation in liquid heightbased on different roots suggests that the selection of oneroot over the other simply affects the value of SGV at which atransition to another flow pattern occurs. In order to preventdiscontinuities, it is important to ensure that, whether thelowest or the highest root is used, the same root is used in allthe calculations. In the present model, the lowest root is se-lected for the ensuing calculations. Having to determine allof the roots presents a further complication as can be seen byexamining Figure 2 when SGV =58.0 ft/sec. It is seen thatthree roots exist under these conditions: 0.0586, 0.0646, and0.508. The lower two roots can only be detected by investi-

gating values of Lh~

within less than 0.005 of each other. Forthe range of validity of the solutions (0 to 1.0) this could re-quire over 200 iterations. For certain combinations of fluidrates and pipe inclination the required resolution of liquidheight or liquid film thickness is of the order of 0.001, whichcan require over 10,000 iterations.

A final note is warranted regarding discontinuities thatarise from friction factor calculations when the flow changesfrom laminar to turbulent. The traditional approach is to usethe laminar flow friction factor when the Reynolds number isless than 2,000. In the present model, the approach followedis to use the turbulent friction factor wherever it is greaterthan the laminar flow value. This results in a smoother corre-lation and since the application of single-phase friction factorcorrelations to multiphase flow situations is, at best, arbi-trary, the approach is believed to be reasonable.

CONCLUSIONS

A new mechanistic model has been presented which is ap-plicable to all conditions commonly encountered in the petro-leum industry. The model incorporates roughness effects aswell as liquid entrainment, both of which are not accountedfor by previous models. The model has undergone extensivetesting and has proven to be more robust than existing mod-els and is applicable over a more extensive range of condi-tions.

9

The empirical correlations that are necessary within themodel can only be improved with accurate and consistentdata over a wide range of conditions of commercial interest.To this end, efforts are in progress to obtain additional datain order to expand the Stanford Multiphase Flow Database.

Further testing of the new model will be undertaken whichwill include as much actual field data as can be obtained. It isexpected that results from these tests will more adequatelydemonstrate the ability of the model to predict reasonablyaccurate pressure drops and holdup under operating condi-tions.

ACKNOWLEDGEMENTS

The work described in this paper has been made possiblethrough the support of the Reservoir Simulation IndustrialAffiliates Program at Stanford University (SUPRI-B) and theStanford Project on the Productivity and Injectivity of Hori-zontal Wells (SUPRI-HW). Portions of this research havealso been supported by the U.S. Department of Energy con-tract DE-FG22-93BC14862. The development of the modelhas also greatly benefited through numerous discussions withMr. Liang-Biao Ouyang of Stanford University.

NOMENCLATUREA Cross-sectional area

0C Velocity distribution coefficientD Pipe internal diameterE In situ volume fractionFE Liquid fraction entrainedg Acceleration due to gravity

Lh Height of liquid (Stratified flow)L Lengthp PressureRe Reynolds numberS Contact perimeter

SGV Superficial gas velocity

SLV Superficial liquid velocity

Lδ Liquid film thickness (Annular-Mist)ε Pipe roughnessη Pressure gradient weighting factor (intermittent

flow)θ Angle of inclinationµ Viscosityρ Densityσ Interfacial (surface) tensionτ Shear stressx~ Dimensionless quantity, x

Subscriptsb relating to the gas bubblec relating to the gas coref relating to the liquid film

db relating to the dispersed bubblesG relating to the gas phasei relating to the gas/liquid interfaceL relating to the liquid phasem relating to the mixtureSG based on superficial gas velocitys relating to the liquid slugSL based on superficial liquid velocitywL relating to the wall-liquid interfacewG relating to the wall-gas interface

REFERENCES

1. Aziz, K. and Petalas, N., “New PC-Based Softwarefor Multiphase Flow Calculations,” SPE 28249, SPEPetroleum Computer Conference, Dallas, 31 July-3August, 1994.

2. Barnea, D. “A Unified Model for Predicting Flow-Pattern transitions for the Whole Range of Pipe Incli-nations,” Int. J. Multiphase Flow, 13, No. 1, 1-12(1987).

3. Taitel, Y., and Dukler, A. E. “A Model for predictingFlow Regime Transitions in Horizontal and NearHorizontal Gas-Liquid Flow,” AIChe Journal, 22, 47(1976).

4. Ansari, A. M., Sylvester, A. D., Sarica, C., Shoham,O., and Brill, J. P., “A Comprehensive MechanisticModel for Upward Two-Phase Flow in Wellbores,”SPE Prod. & Facilities, pp. 143-152, May 1994.

5. Xiao, J. J., Shoham, O., Brill, J. P., “A Comprehen-sive Mechanistic Model for Two-Phase Flow in Pipe-lines,” paper SPE 20631, 65th ATC&E of SPE, NewOrleans, September 23-26, 1990.

6. Petalas, N., and Aziz, K., “Development and Testingof a New Mechanistic Model for Multiphase Flow inPipes,” Proceedings of the ASME Fluids DivisionSummer Meeting, Volume 1, FED-Vol. 236, pp. 153-159, July, 1996.

7. Petalas, N., and Aziz, K., “Stanford Multiphase FlowDatabase - User’s Manual,” Version 0.2, PetroleumEngineering Dept., Stanford University, 1995.

8. Gregory, G. A., Nicholson, M.K. and Aziz, K., “Cor-relation of the Liquid Volume Fraction in the Slug forHorizontal Gas-Liquid Slug Flow,” Int. J. MultiphaseFlow, 4, 1, pp. 33-39 (1978).

9. Andritsos, N., “Effect of Pipe Diameter and LiquidViscosity on Horizontal Stratified Flow,” Ph.D. Dis-sertation, U. of Illinois at Champaign-Urbana (1986).

10. Oliemans, R. V. A., Pots, B. F., and Trope, N., “Mod-eling of Annular Dispersed Two-Phase Flow in Verti-

10

cal Pipes,” Int. J. Multiphase Flow, 12, No. 5, 711-732 (1986).

11. Taitel, Y., Barnea, D. and Dukler, A. E. “ModelingFlow Pattern Transitions for Steady Upward Gas-Liquid Flow in Vertical Tubes,” AlChe Journal, 26,pp. 345-354 (1980).

12. Barnea, D., Shoham, O.,Taitel, Y. and Dukler, A.E.,“Gas-Liquid Flow in Inclined Tubes: Flow PatternTransitions for Upward Flow,” Chem. Eng. Sci., 40, 1pp. 131-136 (1985).

13. Zuber, N., Staub, F.W., Bijwaard, G., and Kroeger,P.G., “Steady State and Transient Void Fraction inTwo-Phase Flow Systems,” 1, Report EURAEC-GEAP-5417, General Electric Co., San Jose, Califor-nia, January 1967.

14. Govier, G. W. and Aziz, K. “The Flow of ComplexMixtures in Pipes,” Van Nostrand, Reinhold (1972),reprinted by Robert E. Kriger Publishing Co., Hunt-ington, New York, 1977.

15. Bendiksen, K. H. “An Experimental Investigation ofthe Motion of Long Bubbles in Inclined Pipes,” Int. J.Multiphase Flow, 10, pp 1-12 (1984).

16. Zuber, N., and Findlay, J.A., “Average VolumetricConcentration in Two-Phase Flow Systems,” J. Heat.Transfer, Trans. ASME, Ser. C, 87, pp. 453-468(1965).

17. Nicklin, D. J., Wilkes, J. O., and Davidson, J. F.,“Two Phase Flow in Vertical Tubes,” Trans. Inst.Chem. Engrs., 40, pp. 61-68, (1962).

18. Zukoski, E.E., “Influence of Viscosity, Surface Ten-sion, and Inclination Angle on Motion of Long Bub-bles in Closed Tubes,” J. Fluid Mech., 25, pp. 821-837 (1966).

19. Weber, M. E., “Drift in Intermittent Two-Phase Flowin Horizontal Pipes,” Canadian J. Chem. Engg., 59,pp. 398-399, June 1981.

20. Wallis, G. B. “One-Dimensional Two-Phase Flow,”McGraw-Hill, New York, 1969.

21. Harmathy, T.Z., “Velocity of Large Drops and Bub-bles in Media of Infinite or Restricted Extent,” AIChEJ., 6, pg. 281 (1960).

Table 1. System Properties for Flow Pattern Maps

Air/WaterSystem

Oil/GasSystem

Pipe diameter 2.047 in 6.18 in.

Gas Density .08 lb/ft3 8.139 lb/ft3

Liquid Density 62.4 lb/ft3 52.53 lb/ft3

Gas Viscosity 0.01 cP 0.018 cP

Liquid Viscosity 1.0 cP 2.757 cP

Interfacial Tension 72.4 dyne/cm 20 dyne/cm(Absolute) PipeRoughness

0.00015 ft 0.01 ft†

† This high value of roughness is used to represent an open-hole well com-pletion.

S1S2

I1

I2

I3A1

A2

D1

I4

VSL

VSG

S3

S4

Figure 1. Transitions used in flow pattern determination

0.01

0.10

1.00

0.01 0.1 1 10 100 1000VSG

hL/D

Air/Water at StandardConditions

2.047" Pipe, +2° Inclination

VSL= 0.05 ft/sec

Figure 2. Stratified Flow: Multiple Roots in Liquid HeightCalculation for Increasing VSG (Air/Water System)

11

Yes

Flow isSTRATIFIED

Yes

No

STRATIFIEDSMOOTH

STRATIFIEDWAVY

No

Yes

Yes YesDISPERSED

BUBBLE

FROTH

FROTH

ANNULARMIST

BUBBLE

ELONGATEDBUBBLE

SLUG

24.0MA

≤−LE

max

~~LL δδ <

( )V

g

s VGL L G

L G L

≤−4µ ρ ρ θ

ρ ρcos

V

ghL

L

≤14.

Flow downwardor horizontal?

θ ≤ 0

( )V

gD h

fL

L

L

≤−1

~cosθ

FC A

VdA

dhG

G

L

L

2

2

211

~

~

~

≤

CG ≤ 0 52.

( )

cos

.

θγ

ρ ρ σ

ρ

≤

>−

>

3

4 2

19

0 75

2 2

2

0C

d

V

g

Dg

E

l

bub

L G

L

L slug

Begin Mechanistic Model

Calculate~hL

Calculate~δ L

48.0≥LsE ELs ≤ 09.

48.0<LsE

No

No

Yes

Yes

No

Yes

No

No

24.0SLUG

>LE

No

No

Yes

Yes

Figure 3. Flow pattern determination for mechanistic model

12

0.01 0.1 1 10 100

VsG (ft/sec)

0.01

0.1

1

10

100

VsL

(ft/

sec)

Dispersed Bubble

Slug

StratifiedWavy

Annular-Mist

Froth I

Stratified Smooth

Elongated Bubble

Figure 4. Flow pattern map for air/water system at 0°inclination (horizontal)

0.01

0.1

1

10

VsG(ft/sec)

0.01

0.1

1

10

VsL(ft/sec)

1E-006

0.0001

0.01

1

100

∆∆∆∆Pfr (psi/ft)1E-006

0.0001

0.01

1

100

∆∆∆∆Pfr (psi/ft)

Figure 5. Frictional pressure gradient for air/water system at0° inclination (horizontal)

0.010.1

1

10VsG (ft/sec)

0.01

0.1

1

10

VsL(ft/sec)

0

0.2

0.4

0.6

0.8

1

EL0

0.2

0.4

0.6

0.8

1

EL

Figure 6. Liquid volume fraction for air/water system at 0°inclination (horizontal)

0.01 0.1 1 10 100

VSG

0.01

0.1

1

10

100

VS

L

Dispersed Bubble

Slug

Froth I

Annular-Mist

StratifiedWavy

Stratified Smooth

Elongated Bubble

Figure 7. Flow pattern map for oil/gas system at 0° inclina-tion (horizontal)

0.01

0.1

1

10

VsG

0.01

0.1

1

10

VsL

1E-007

1E-005

0.001

0.1

10

∆∆∆∆Pfr1E-007

1E-005

0.001

0.1

10

∆∆∆∆Pfr

Figure 8. Frictional pressure gradient for oil/gas system at 0°inclination (horizontal)

0.010.1

1

10VsG

0.01

0.1

1

10

VsL

0

0.2

0.4

0.6

0.8

1

EL 0

0.2

0.4

0.6

0.8

1

EL

Figure 9. Liquid volume fraction for oil/gas systemat 0° inclination (horizontal)

13

0.01 0.1 1 10 100

VSG

0.01

0.1

1

10

100V

SL

Annular-Mist

Froth

Elongated Bubble

Slug

Dispersed Bubble

Figure 10. Flow pattern map for air/water system at 10°upward inclination

0.01

0.1

1

10

VsG

0.01

0.1

1

10

VsL

1E-006

0.0001

0.01

1

100

∆∆∆∆Pfr1E-006

0.0001

0.01

1

100

∆∆∆∆Pfr

Figure 11. Frictional pressure gradient for air/water system at10° upward inclination

0.01

0.1

110

VsG

0.01

0.1

1

10

VsL

0

0.2

0.4

0.6

0.8

1

EL

0

0.2

0.4

0.6

0.8

1

EL

Figure 12. Liquid volume fraction for air/water system at 10°upward inclination

0.01 0.1 1 10 100

VSG

0.01

0.1

1

10

100

VS

L

Froth I

Annular-Mist

Elongated Bubble

Slug

Dispersed Bubble

Figure 13. Flow pattern map for oil/gas system at 10° upwardinclination

0.01

0.1

1

10

VsG

0.01

0.1

1

10

VsL

1E-007

1E-005

0.001

0.1

10

∆∆∆∆Pfr1E-007

1E-005

0.001

0.1

10

∆∆∆∆Pfr

Figure 14. Frictional pressure gradient for oil/gas systemat 10° upward inclination

0.010.1

1

10VsG

0.01

0.1

1

10

VsL

0

0.2

0.4

0.6

0.8

1

EL0

0.2

0.4

0.6

0.8

1

EL

Figure 15. Liquid volume fraction for oil/gas system at 10°upward inclination

14

0.01 0.1 1 10 100

VsG (ft/sec)

0.01

0.1

1

10

100

VsL

(ft/

sec)

Annular-Mist

Froth

Slug

Elongated Bubble

Bubble

Dispersed Bubble

Figure 16. Flow pattern map for air/water system at 90°upward inclination

0.01

0.1

1

10

VsG(ft/sec)

0.01

0.1

1

10

VsL(ft/sec)

1E-006

0.0001

0.01

1

100

∆∆∆∆Pfr (psi/ft)1E-006

0.0001

0.01

1

100

∆∆∆∆Pfr (psi/ft)

Figure 17. Frictional pressure gradient for air/water system at90° upward inclination

0.010.1

1

10VsG (ft/sec)

0.01

0.1

1

10

VsL(ft/sec)

0

0.2

0.4

0.6

0.8

1

EL0

0.2

0.4

0.6

0.8

1

EL

Figure 18. Liquid volume fraction for air/water systemat 90° upward inclination

0.01 0.1 1 10 100

VSG

0.01

0.1

1

10

100

VS

L

Froth

Slug

Annular-Mist

Elongated Bubble

Bubble

Dispersed Bubble

Figure 19. Flow pattern map for oil/gas system at 90° upwardinclination

0.01

0.1

1

10

VsG

0.01

0.1

1

10

VsL

1E-007

1E-005

0.001

0.1

10

∆∆∆∆Pfr1E-007

1E-005

0.001

0.1

10

∆∆∆∆Pfr

Figure 20. Frictional pressure gradient for oil/gas system at90° upward inclination

0.010.1

110

VsG0.01

0.1

1

10

VsL0

0.2

0.4

0.6

0.8

1

EL

0

0.2

0.4

0.6

0.8

1

EL

Figure 21. Liquid volume fraction for oil/gas system at 90°upward inclination

15

0.01 0.1 1 10 100

VSG

0.01

0.1

1

10

100V

SL

Dispersed Bubble

SlugFroth

Annular-Mist

StratifiedWavy

Elongated Bubble

Figure 22. Flow pattern map for air/water system at 10°downward inclination

0.01

0.1

1

10

VsG

0.01

0.1

1

10

VsL

1E-005

0.001

0.1

10

∆∆∆∆Pfr1E-005

0.001

0.1

10

∆∆∆∆Pfr

Figure 23. Frictional pressure gradient for air/water system at10° downward inclination

0.010.1

1

10VsG

0.01

0.1

1

10

VsL

0

0.2

0.4

0.6

0.8

1

EL 0

0.2

0.4

0.6

0.8

1

EL

Figure 24. Liquid volume fraction for air/water systemat 10° downward inclination

0.01 0.1 1 10 100

VSG

0.01

0.1

1

10

100

VS

L

Dispersed Bubble

SlugFroth

Annular-Mist

Elongated Bubble

StratifiedWavy

Figure 25. Flow pattern map for oil/gas system at 10° down-ward inclination

0.01

0.1

1

10

VsG

0.01

0.1

1

10

VsL

1E-005

0.001

0.1

10

∆∆∆∆Pfr1E-005

0.001

0.1

10

∆∆∆∆Pfr

Figure 26. Frictional pressure gradient for oil/gas systemat 10° downward inclination

0.01

0.1

1

10VsG

0.01

0.1

1

10

VsL

0

0.2

0.4

0.6

0.8

1

EL

0

0.2

0.4

0.6

0.8

1

EL

Figure 27. Liquid volume fraction for oil/gas system at 10°downward inclination

16

633

68

382

90293

101242

83

407

146671125611846212

2895

0

500

1000

1500

2000

2500

3000

80°

to90

°

70°

to80

°

60°

to70

°

50°

to60

°

40°

to50

°

30°

to40

°

20°

to30

°

10°

to20

°

0°to

10°

-10°

to0°

-20°

to-1

0°

-30°

to-2

0°

-40°

to-3

0°

-50°

to-4

0°

-60°

to-5

0°

-70°

to-6

0°

-80°

to-7

0°

-90°

to-8

0°

Inclination Angle Range

Nu

mb

ero

fD

ata

Po

ints

Figure 28. Distribution of inclination angle for experimentaldata

-15%

+15%

0.00

0.25

0.50

0.75

1.00

0.00 0.25 0.50 0.75 1.00

Experimental

Ca

lcu

late

d

SU-21, 23

SU-24 to 29SU-53 to 56SU-66

SU-96SU-101SU-108

SU-111 to 113SU-114 to 117SU-120 to 124

SU-138SU-175 to 198SU-199 to 209

SU-210 to 2155,951 Data Points

Figure 29. Mechanistic model volume fraction liquid calcula-tions compared with experimental data

-15%

+15%

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

-0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50

Experimental

Ca

lcu

late

d

SU-21, 23SU-24 to29SU-53 to56

SU-66SU-96SU-101SU-108

SU-111 to 113SU-114 to 117SU-120 to 124SU-138

SU-175 to 198SU-199 to 209SU-210 to 215

5,951 Data Points

Figure 30. Mechanistic model pressure gradient calculationscompared with experimental data

27.6%

41.3%

33.0%

31.7%

41.2%

43.2%

22.0%

40.3%

37.9%

38.0%

49.4%

61.6%

0% 10% 20% 30% 40% 50% 60% 70% 80%

Percentage of Experimental Data Predicted to 15%Accuracy

DP

EL

Mechanistic ModelXiao, ShohamandBrillMukherjeeandBrillBeggsandBrillDukler, Wicks andClevelandHomogeneous Model

EL

∆∆∆∆P

Figure 31. Comparison of selected methods’ ability to predictexperimental volume fraction liquid and pressure gradient towithin 15% accuracy

0% 10% 20% 30% 40% 50% 60% 70%

Percentage of Experimental Data Predicted to 15%Accuracy

Mechanistic Model

Xiao, Shohamand Brill

Mukherjee and Brill

Beggs and Brill

Dukler, Wicks and Cleveland

Homogeneous Model

-90° to -30°-30° to +30°+30° to +90°

Figure 32. Comparison of selected methods’ ability to predictexperimental volume fraction liquid to within 15% accuracy(grouped by angle of inclination)

0% 10% 20% 30% 40% 50% 60% 70% 80%

Percentage of Experimental DataPredicted to 15%Accuracy

Mechanistic Model

Xiao, Shohamand Brill

Mukherjee and Brill

Beggs and Brill

Dukler, Wicks and Cleveland

Homogeneous Model

-90° to -30°-30° to +30°+30° to +90°

Figure 33. Comparison of selected methods' ability to predictexperimental pressure gradient to within 15% accuracy(grouped by angle of inclination)