spe-20630-ms two phase flow in well bore

7/24/2019 SPE-20630-MS Two Phase Flow in Well Bore

1/15

S

o iety o Petroleum Engine

SPE 63

A

omprehensive Mechanistic

Model for Upward Two Phase

Flow

in Wellbores

A.M. Ansari, Pakistan Petroleum Ltd.; N.D. Sylvester, U of Akron; and O Shoham and

J.P. Brill, U of Tulsa

SPE Members

Copyright 1990, Society of Petroleum Engineers, Inc.

Thispaper was prepared for presentation at the 65th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held

in

New Orleans,

LA

September23-26, 1990.

This paper was selected for presentation by

an

SPE Program Committee following review of information contained

in an

abstract submitted by the author s . Contents of the paper,

as

presented, have not been reviewed by the Society of Petroleum Engineers and are sUbject to correction by the author s . The material,

as

presented. does not necessarily reflect

any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society

of Petroleum Engineers. Permission to copy is restricted to an abstract of not more than 300words. Illustrations may not be copied. The abstract should contain conspicuous acknowledg

ment of where and by whom the paper is presented. Write Publications Manager, SPE, P.O. Box 833836, Richardson, TX 75083-3836 U.S.A. Telex, 730989 SPEDAL.

ABSTRACT

A comprehensive model is formulated to predict the

flow

behavior for upward two phase flow

The

comprehensive model is composed of a model for flow

pattern prediction and a set

of

independent models for

predicting the flow characteristics such as holdup and

pressure drop in bubble, slug and annular flows.

The comprehensive model is evaluated by using a

well databank that is composed of 1775 well cases covering

a wide variety of field data. The performance of the model

is also compared with the six commonly used empirical

correlations.

The overall performance of the model is in good

agreement with the data.

In

comparison with the empirical

correlations, the comprehensive model performs the best,

with the least average error and the smallest scattering of

the results.

INTRODUCTION

Two-phase flow is commonly encountered in

petroleum, chemical and nuclear industr ies. The frequent

occurrence of two-phase flow presents engineers with the

challenge of understanding, analyzing and designing two-

phase systems.

Due to the complex nature of two-phase flow, the

problem was first approached through empirical methods.

References and illustrations at end of paper.

Recently the trend has shifted towards the modelin

approach. The fundamental postulate of the modelin

approach is the existence

of

flow patterns or flo

configurations. Various theories have been developed for th

prediction of flow patterns. Separate models we

developed for each flow pattern

to

predict the flo

characteristics such as holdup and pressure drop.

considering f low mechanics, the result ing models can

applied to flow conditions other than used for the

development with more confidence.

The on ly studies pUblished on comprehensiv

mechanistic modeling of two-phase flow in vertical pip

are by Ozon et al.

l

and Hasan and Kabir

Nevertheles

more work is needed in order to develop models whi

describe the physical phenomena more rigorously.

The purpose of this study is

to formulate a detai

comprehensive mechanist ic model for upward two-pha

flow. The comprehensive model f irst predicts the existi

flow pattern and then calculates the flow variables by taki

into account the actual mechanisms of the predicted flo

. pattern. The model is evaluated against a wide range

experimental and field data available in the updated TUFF

well databank. The performance of the model is al

compared with six empirical correlations used in the field

FLOW PATTERN PREDICTION

The basic work on mechanistic modeling flow

pattern transitions for upward two-phase flow w

presented by Taitel et al.

They identified four distinct fl

patterns, and formulated and evaluated the transiti


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2

A

COMPREHENSIVE MECHANISTIC

MODEL FOR UPWARD

TWO- PHASE FLOW IN WELLBORES

SPE 63

boundaries among them. The four flow patterns are bubble

flow, slug flow, churn flow and annular flow, as shown

in

Fig.1. Later, modifications o f transitions were made by

Barnea et al.

4

to extend the applicability of the model to

inclined flows as well.

In

a relatively recent work, Barnea

s

combined flow pattern prediction models applicable to

dif ferent incl inat ion angle ranges into one unified model.

Based on these different works, flow pattern can be

predicted by defining transition boundaries among bubble,

slug and annular flows.

Bubble-Slug Transition: The minimum diameter at

which bubble flow occurs is given by Taitel et al.

3

as,

For pipe sizes larger than this, the basic transit ion

mechanism for bubble to slug flow is coalescence of small

gas bubbles into large Taylor bubbles. Exper imentally this

was found to occur at a void fract ion of approximately 0.25.

Using this value of void fraction, the transition can be

expressed in terms of superficial and slip velocities as,

V

SG

= 0.25

V

s

0.333 VSL

2

where Vs is the slip or bubble rise velocity given by

Harmath

y

6 as,

V

s

= 1.53 [ g O ~ - P G r 4

3

This is shown as transition A in Fig.

2.

At high liquid rates, turbulent forces break down

large gas bubbles into small ones, even at void fractions

greater than 0.25. This yields the transition to dispersed

bubble flow given by Barnea et al.

2

as,

2 [ 0.4 0 ] 2

PL 3/5 [CL Q n]2/S

PL-PG g 0 0 VL

VSL

VSGf 3.nYs

=

0.725

4.15

VSG

o.s ...... 4

VSG VSL

This is shown as transition B in Fig.

2.

At high gas velocit ies this transi tion is governed by

the maximum packing of bubbles to give coalescence. This

occurs at a void fraction of 0.52, giving the transition for

no-slip dispersed bubble flow as,

VSG

= 1.08 VSL .. 5

This is shown

as

transition C

in

Fig.

2.

5

Transition to Annular Flow: The transition

criterion for annular flow is based on the gas phase velocity

required to prevent fall back of the entrained liquid droplets

in the gas stream. This gives the transition as,

VSG =3.1 [ g O ~ - P G r / 4 6

and is shown

as

transition D in Fig.

2.

The same transition was modified by Barnea

s

by

consider ing the effects of film thickness on the transition.

One effect is bridging of the gas core by a thick liquid film at

high liquid rates. The other effect is instabil ity of the liqUid

film causing downward flow of the film to occur at low

liqUid rates. The mechanism of bridging is governed by the

minimum liquid holdup required to form a liquid slug,

HLF> 0.24.. ..

7

where HLF is the fraction of pipe cross-section occupied by

the liquid film, assuming

no

entrainment in the core.

The mechanism of film instability can be expressed

in terms of the Lockhart and Martinell i parameters, X and Y,

y _

2 - 1.5 HLF

-

8

K LF 1 - 1.5 HLF

where HLF can

be

expressed in terms of minimum film

thickness, as,

HLF =

4

Qrnn

1 -

Q.mn 9

To

account for the effect of the liquid entrainment

in the gas core, Eq. 7 is modified in this study as,

HLF

ALCAc/A

> 0.24 10

In Eq. 8,

X and Y must

be

redefined

in

terms of the

core parameters instead of the gas parameters to account

for the entrainment.

FLOW BEHAVIOR PREDICTION

Following the prediction of flow patterns, the next

step is to develop physical models for the flow behavior in

each of the flow patterns. This resulted in separate models

for bubble flow, slug flow and annular flow. Churn flow has

not yet been modeled due to its complexity, and is

treated

as a part of slug flow. The models developed for other flow

patterns are discussed in the following sections.

Bubble Flow Model:


3/15

SPE

20630

A M,

Ansari

N

D, Sylyester

0

Shoham and

J

P

Bri l l

3

The bubble flow model is based on the work by

Caetano? for flow in an annulus, The two bubble flow

regimes, viz, bubbly flow and dispersed bubble flow, are

considered separately

in

developing the model for the bubble

flow pattern,

Due to the uni form distr ibution of gas bubbles in the

liquid, and no slippage between the two phases, dispersed

bubble flow can be approximated as a pseudo-single phase,

Due to this simplif ication, the two-phase parameters can be

expressed as,

PTP

=

PL

t +

pG 1

-

AL)

..

. . . . . , ..

,,,,, . . . . . . . . ,, 11 )

IlTP = ilL L

+

IlG 1

- Ad.. .

.. .

..

.

. . . . . , , ,

12)

VTP =VM =

VSL

+ VS . . . . . .

. . . . . . .

13)

where,

f L

=

VSL

..

. . . . .

. , .

. . . . . .

14)

(VSL + VSG)

For bubbly flow, the slippage is considered by

taking into account the bubble rise velocity relative to the

mixture velocity. By assuming a turbulent velocity profile

for the mixture with the rising bubble concentrated more at

the center than along the wall of the pipe, the slip velocity

can be expressed as,

Vs = V - 1

,2VM

.... ,

..

..

..

.... , , , ,

. . . .

15 )

An

expression for the bubble rise velocity was given by

Harmath

y

6,

To account for the effect of bubble swarm, this

expression was modified by Zuber and Hench

8

as follows

where the value of n var ies from one study to another.

In

the present study, a value of

0.1

for n was found to give

the best results. Thus, Eq 15 yields,

This gives an implicit equation for the actual holdup for

bubbly flow, The two-phase flow parameters can now be

calculated from,

PTP = PL H

L

+

pG 1 - HL . . . . 18)

IlTP = ilL HL + flG 1 - H d ....

......

,,

19 )

5

The two-phase pressure gradient is comprised of

three components, Thus

(

dP

=

dP + dP +

d

P

) 20)

d t d e d f d a

The elevation pressure gradient is given by,

= PTP g sine

,

,

21)

The friction component is given by,

= f

TP

Vfp , . , ..

,

..

22)

The explicit expression given by Zigrang and Sylvester

9

can

be

used

to

define fTP as,

_1_

= - 2 log

Yhp

10

_

5,02

log /0

+

13,0. t ...... , 23)

3.7 R Tp 3,7 R Tp

where,

R _ PTP VTP 0

Tp - 24)

IlTP

The accelerat ion pressure gradient is negligible

compared to the other pressure gradients.

Slug Flow Model:

The f irst thorough physical model for slug flow was

developed by Fernandes et al.

lO

A simplif ied version of this

model was presented by Sylvester

11

.

The basic

simplification made was the use of a correlation for slug

void fraction, An important assumption of fUlly developed

slug flow was used by these models. The concept of

developing flow was introduced by McQuillan and Whall

ey

12

during their study of flow pattern transitions, Due to the

basic difference in the geometry of the flow, fUlly developed

and developing flow are treated separately in the model.

For a fUlly developed slug unit, as shown in Fig.

3(a), the overall gas and liquid mass balances, respectively,

give,

1

- V LS 1 -

HLLS) . .

..

,

25)


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4

A COMPREHENSIVE MECHANISTIC MODEL

FO R UPWARD

TWO PHASE

FLOW

IN WELLBORES

SPE20630

VSL = 1 - VLLSHLLS - VLTBHLTB 26

where,

~ L T 27

lsu

Mass balances for liquid and gas from liquid slug to Taylor

bubble, respectively, give,

VTB - VLLS)HLLS

= [VTB

- -

VL

TB)]HLTB ........... 28

(VTB - VGLS) HLLS =

(VTB - VGTB) HLTB 29

The Taylor bubble rise velocity is equal to the centerline

velocity plus the Taylor bubble rise velocity in a stagnant

liquid column, Le.,

2

9 D (PL - pG

VTB =

1.2 VM

+

0.35

PL

.......

30

Similar ly, the velocity of the gas bubbles in the liquid slug is

where the second term on the right hand side represents the

bubble rise velocity as defined earlier in Eq (16).

The

velocity

vL B T of the falling film can be

corre lated with the film thickness

OL

using the Brotz 3

expression,

VLTB

=

196.7 g CL 32

where OL is the constant film thickness for developing flow,

and can be expressed

in

terms of Taylor bubble void fraction

to give,

VLTB = 9.916 [9 D 1 y ~ r 2 33

The liquid slug void fraction can be obtained by the

correla tion developed by Sylvester

from Fernandes

et

al.

10

and Schmidt

14

data,

Has

= VSG

34

0.425

+

2.65 VM

Equations 25-26, 28-31, and 33-34 can be solved

iteratively to obtain all eight unknowns that define the

developed slug flow model.

5

To model developing slug flow, as shown in Fig.

3(b) it is necessary to 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, as developed by McQuillan and

Whall

ey

1o is given as,

l :

= -1-[VTB + VNGTB

1

- HNLTB) _ ~ (35)

2g H l.TB HNLTB

where vNGTB and HNLTB are calculated at the terminal film

thickness

ON

(called Nusselt f ilm thickness) given by,

ON = [a D VNLTB d1 - HNLTB ]1/3 (36)

4

g PL-PG

The geometry of the film flow gives HNLTB in terms of ON

as,

HNLTB

=

1

1

....................... 37

To determine vNGTB, the net flow rate at

ON

can be

used to obtain,

v - v - (v _v )

HLLS)

NGTB - TB TB GLS ( ) (38)

1 -

H

NLTB

The length

of

the liquid slug can be calculated

empirical ly from,

LLS

=

C D

39

where

C

was found by Dukler et al. 15 to vary from 16 to

45. It is taken as 30

in

this study. This gives Taylor bubble

length as,

bB =

40

From the comparison of Lc and LTB, if

Lc

>

LTB, the

flow is developing slug flow. This require new values for

LTBo, HLTBo and vLTBo calculated earlier for developed

flow.

For L

TBo,

Taylor bubble volume can be used,

.

t

TB

VGTB

=

Jo

L)dL 41


5/15

SPE 20630

A. M. Ansari- N p. Sylvester. 0 ShQham. and J . P. ri l l

5

where ATB* L) can be expressed in terms

Qf

local holdup

h LT B L), which in turn can be expressed in terms of

velQcities by using Eq. 28. This gives,

(L) = [1 -

VTB

~ H L L S ] A. . . . . (42)

The vQlume VGTB* L) can be expressed in term of

f QW

geQmetry as,

, ,

VGTB

=

V

su

-

V

LS

.. ..

. . . ...

.......

.......

... .

(43a)

or

V ~ T

= VSG J L ~ B +

LLS) -

VTB

VGLS A 1 - HLLS ks ..

(43b)

VTB

SubstitutiQn of Eqs.

42

and

43

into Eq

41

gives,

LTB +

LLS H LLS

VSG -

VGLS 1 -

LLS -

=

VTB

VTB

EquatiQn 44 can be integrated and then simplif ied to

give,

2 [-2ab _ 4c

2

] ,

b

2

_

LTB + L

TB

+ - - 0 .. .. ...... ...

45)

a

2

a

2

where,

a

= 1 - VSG .. .

....

..

..

46)

VTB

b =

VSG

-

VGLS 1

-

HLLS LLS 47)

VTB

c - VTB - VLLS H

- V

2G LLS

..

. 48)

After calculating LTB, the other local parameters

can be calculated from,

VLTB (L)

= V2gL

- VT B

.........

....

.. .. .

.....

(49)

h ~ T B L) =

VTB

-

VLLS HLLS

V2gL . 50)

In calculating pressure gradients, t he effect

Qf

varying film thickness is considered and the effect Qf

frictiQn alQng the TaylQr bubble is neglected.

FQr develQped flQw

the

elevatiQn compQnent

occurring across a slug unit is given by,

.d

P

)

=

[ 1

- 13 PLS +

PG]g

sin9....... .. .... 51)

dL 9

where,

PLS

=PL HLLS +

pG

1 - HLLS

....

... . ... .. ... 52)

The elevation component fQr develQping slug flQW is

given by,

d

P

) = [ 1 - * PLS + 13 PTBA]g sin9 .. ..

.

53)

dL 9

where PTBA is based

Qn

average

VQid

fractiQn

in

the TaylQr

bubble sectiQn with varying film thickness. It is given by,

PTBA = PL

HLTBA

+ pG 1 -

HLTBA) ..

.... . 54)

where HLTBA is obta ined by integrating Eq.

50

and dividing

by LTB* giving,

HLTBA = 2 VTB - VLLS)HLLS . ... ..

.. ... .. 55)

~ g L ~ B

The frictiQn cQmpQnent is the same fQr bQth the

developed and developing slug flQWS as it occurs Qnly across

the liquid slug. This is given as,

=

fLS

1

- L . 56)

where fLS can be calculated by using,

R 0

PLS VM

eLS =

. ....

. .

57)

ilLS

For stable slug flow, the acceleratiQn cQmpQnent Qf

pressure gradient can

be

neglected.

Annular FIQW Model:

A discussion Qn the hydrQdynamics Qf annular flow

was presented by Wallis

16

. Along with this, Wallis also


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6

A COMPREHENSIVE MECHANISTIC MODEL

FOR

UPWARD

TWO- PHASE

FLOW

IN WELLBORES

SPE 20630

presented the classical correlations for entrainment and

interfacial friction as a function of film thickness. Later,

Hewitt and Hal l-Taylor

17

gave a detailed analysis of the

mechanisms involved in an annular flow. All the models that

followed later are based on this approach.

A fully developed annular flow is shown in Fig. 4.

The conservation

of

momentum applied indiv idually to the

core and the film yields,

A

c

d

P

) -

tl

SI - pc

c

9sine = 0 58

dL

c

AF d

P

) + tl S, - tF SF - PL AF 9sine = 0 59

dL F

The core density Pc is given by

pc

=

PL

~

+

pG

1

-

A.Lc)

60

where,

ALc = 1 -

VSG 61

VSG

+ FE VSL

FE is the fraction of the total liquid entrained in the core,

given by Wallis as,

FE = 1 - exp [-0.125 v

crit

- 1.5)]. 62

where,

. _ VSG IlG PG 1/2

VCrit - 10000 r

~ 63

The shear stress in the film can be expressed as

tF = fF PL 64

2

where,

fF =

CF [DH:LVF

r

65

VF _ QL 1 - FE) _

VSL

1 - FE)

- AF - 4 1 _

66

DHF=4.Q. 1 - li D 67

This gives,

6

t - 1 f 1 - FE

)2-n

P [

VSL

J

- -

SL

J L )

68

2 4 ~ ~

where,

fSL

=

C F [ V S ~ L D J n

69

Using the definition of superficial f rict ional pressur

gradient for liquid, Eq.

68

reduces to,

For the shear stress at the interface, exactly th

same

approach can be adopted to give,

t 1 = ~ 1 ~ L 71

where

Z

is a correlating factor for interfacial friction an

the film thickness. Based on the performance of the mod

the Wallis expression for Z works well for thin films or hig

entrainments, whereas the Whalley and Hewitt

18

expressi

is good for thick films or low entrainments. Thus,

Z = 1 + 300 for FE > 0.9 72

PL)1/3

Z = 1 + 24 P for FE < 0.9 73

The pressure gradient for annular flow can

calculated by substituting the above equations into Eqs.

and

59.

Thus,

Z d

P

) + pc 9

sine

1

\5 dL 74

- ~ J S

1

- FErn d

P

)

64

1

_ dL

SL

Z d

P

) + PL 9 sine

)

3

dL

75

4 1 - 1 - sc

The basic unknown in the above equations is t

dimensionless film thickness, An implicit equation for

can be obtained by equating Eqs. 74 and 75. This gives,

Z d

P

)

+

pc 9 sine _

-

dL

sc


7/15

SPE

20630

A

M Ansari

N D

Sylvester 0 Shoham. and J

p

Bri l l

7

1

-

FErn

dP) Z dP)

64

li

3

1 - li f dL

SL +

4

li

1 -

li)

1 - 2lif dL sc

PL

9 sine = 0 ..

. . . . . . . . .

76)

To simplify this equation, the dimensionless

approach developed by Alves et al.

19

is used. This approach

defines the following dimensionless groups,

_

dp/dLht-

- dp/dL)sc

. . . .

77)

Y

M

_ 9 sine

PL

-

pc)

- dp/dL)sc

.. ..

78)

2 dp/dLk

- 9

pc sine

cl c

=

dp/dLhc

. 79)

2

d p d L ~

-

9 PL sine

cl

=

dp/dLh

....

....

.......

80 )

By

using these dimensionless groups,

Eq.

76 reduces to,

1 - FEf - n

[1 - 1 - 2lif]3

Z + YM - 0

[ 1 1 2 ~ f ] 1 2 ~ t

-

..... ..... 81)

The above equat ions can be solved iteratively to obtain

li.

Once li is known, the dimensionless groups

F

and c can be

obtained from the dimensionless form of Eqs. 74 and 75.

By

using the defini tions

of

F and c the total pressure

gradient can

be

obtained as,

= +

9 pc sine

.

..

82)

or

d

P

) = d

P

)

+

9

PL

sine 83)

dL F

dLsL

The two pressure gradients calculated from the above

equations should

be

equal.

It is important to note that the above calculated

total pressure gradient does not include accelerational

pressure gradient. This is based on results found by Lopes

and Dukler

20

indicating that, except for a limited range of

high liqUid flow rates, the accelerational component due to

7

the exchange of liquid droplets between the core and the fil

is negligible.

EVALUATION

The evaluation of the comprehensive model

carried out by comparing the pressure drop from the mod

with the measured data in the updated TUFFP well databan

that comprises 1775 well cases with a wide range of da

as

given in Table

1.

The performance of the model is als

compared with that of the six commonly used correlations

the petroleum industry.

Criteria for Comparison with pata

The evaluation of the model using the databank

based

on

the following statistical parameters,

Average percent Error

E1 =

[l

f eri]

X

100.....

....... ...... ..

..

84)

N

1 1

where,

e . _ L\pi calc - L\pI meas

n -

.

85 )

L\pi meas

E1 indicates the overall trend of the performance.

Absolute Average percentage Error

E2

=

[l fieri

I] X

100

......

...... ......... 86)

N

1 1

E2 indicates how large the errors are on the average.

percent Standard Deviation

E3

=

f

eri -

E1 f 87)

i _ 1

N - 1

E3

indicates the degree of scattering of the error about

average value.

Average Error

E4

= [l

f

el] 88)

N

i 1

where,

ei

=

L\pi calc - L\PI meas

.. . ..

89)

E4

indicates the overall trend independent of the measur

pressure drop.


8/15

8

A COMPREHENSIVE MECHANISTIC MODEL FOR UPWARD

TWO PHASE

FLOW

IN

WELLBORES

SPE 2 63

Absolute Average Error

E5

= f I d 90

N

ja 1

E5 is also independent of the measured pressure drop and

indicates the magnitUde of the average error.

Standard Deviation

E6 = f y

i

- f 91

i

N - 1

E6 indicates the scattering of the results, independent of the

measured pressure drop.

Criteria for Comparison with other Correlations:

The correlations used for the comparison are

modi fi ed Hagedorn

and Brown

26

, Duns and

ROS34

Orkiszweski

3o

with Triggia correction

, Beggs and Brill

36

with Palmer correction

37

, Mukerjee and Brill

6

, and Aziz et

a1.

39

. The comparison is accomplished by comparing the

statistical parameters. The parameter E1 was found to give

very small values for the well cases within the range of the

data used

in

developing empirical correlations.

To

remove

this biasing effect,

E1

is not considered in the comparison of

the model with the correlations. However, its effect is

considered through E4. The comparison involves the use of

a relative performance factor RPE) which is obtained by

dividing each statis tical parameter for each correlat ion and

the model by the minimum values of the respective

parameter and then adding all the fractions together.

Mathematically,

RPF =

E2/E2

MN

+ E3/E3

MN

+ I

E41 /

I

E4

MN

I+

E5/E5MN

+

E6/E6MN 92)

The minimum possible value for

RPE

is 5 indicating the best

performance in all respects.

Oyerall Evaluation:

The overall

evaluation

involves the entire

comprehensive model so as to study the combined

performance of all the independent flow pattern behavior

models together . The evaluation is first done by using the

entire databank. The performance of the model is also

checked for vertical well cases only. To make the

comparison unbiased with respect to the correlat ions, two

dif ferent sets of well cases are considered. One such set

is

composed of all vertical well cases excluding 331 well

cases from the Hagedorn and Brown data. The other set is

composed of all new vertical well cases that were never

8

used before for the evaluation of any correlations. The

results are shown in Tables 2 to 5.

Evaluation of Individual Elow Pattern Models:

The performance of individual flow pattern models

is

based

on

sets of data that are dominant in one particular

flow pattern. Eor the bubble flow model, well cases with

bubble flow over 75 , or more, of well length are

considered in order to have an adequate number of well

cases, whereas for slug and annular flow models, well cases

with

1

slug and annular flows, respectively, are

considered. The performance of the slug flow model

is

also

checked for all vertical well cases as well as for vertical

well cases without Hagedorn and Brown data, which is one

third of all the vertical well cases. The statistical results

are shown in Tables 6 to 9.

CONCWSIONS

From Tables 2 9 the performance of the model

and other empirical correlations indicates that,

-The overall performance of the comprehensive

model is superior to all the correlations. This

superiority is further improved when only

vertical data without Hagedorn and Brown well

cases are considered.

In

fact, for the latter two

sets of data Tables 4 and 5 the performance of

the model is the best in all respects.

-The performance of the bubble flow and the

annular flow models are except ionally better

than all the correlations for all the variety of

data in the databank.

-The performance

of

the slug flow model is

exceeded by the Aziz et al. correlation for non

vertical well cases. This is due to the fact that

the model is valid only for vertical flow, and

does not include the mechanisms related to

directional wells. Indeed, for the vertical well

cases, the performance of the model is improved.

The best performance of the model is obtained

when Hagedorn and Brown data are not included.

RECOMMENDATIONS

Based on the above conclusions, the following

recommendations are suggested.

-The entire comprehensive model should replace all

existing empirical correlat ions used to predict

two-phase flow behavior in wells.

-The slug flow model should be modified to include

flow mechanisms related to directional wells.


9/15

SPE

20630

A M

Ansari .

N D Sylyester 0 ShQham

and

J P Bri l l

9

The

degree

Qf

emplClClsm in the cQmprehensive

mQdel shQuld be reduced tQ further imprQve the

model.

As a final remark, it shQuld be mentiQned that the

present study is Qnly a first step tQwards develQping a

cQmprehensive mQdel that shQuld replace existing pressure

gradient cQrrelations for the ent ire range of operating and

design parameters.

REFERENCES

10

ZQn

, P. M. , Ferschneider, G., and Chwetzoff, A.: A New

Multiphase

FIQW Model

Predict s Pressure And

Temperature Profi les, paper SPE 16535 presented at the

OffshQre EurQpe CQnference, Aberdeen, Sept. 8 -11,

1987.

2Hasan,

A.

8., and Kabir,

C.

S.: A StUdy of Multiphase FIQW

Behavior in Vertical Wells, SPE Prod. Eng. J (May

1988). 263-272.

3Taite l, Y., Barnea, D., and Dukler, A. E.: MQdelling FIQW

Pattern TransitiQns for Steady Upward Gas-Liquid FIQW in

Vertical Tubes, I hE J (1980),

aa

345-354.

4Barnea, D., ShQham, 0. , and Taitel, Y.: Flow Pattern

TransitiQn fQr Vertical DQwnward TWQPhase FIQw,

Chern. Eng. Sci.

1982),li, 741-746.

5Barnea, D.: A Unified

MQdel

fQr Predicting FIQw-Pattern

Transi tion for the Whole Range of Pipe InclinatiQns,

Int.

J

Multiphase Flow (1987), 1-12.

6Harmathy, T. Z.: VelQcity

Qf

Large

DrQPs

and Bubbles in

Media

of

Infinite or Restricted Extent, AIChE

J

1960 ,

2 281.

7Caetano, E. F.: Upward Vertjcal TWQ-phase FIQW ThrQugh

an Annulus, Ph.D. DissertatiQn, The University Qf Tulsa

1985).

8Z

u

ber, N. and Hench,

J :

Steady State and Transient

VQid

Fraction Qf Bubbling Systems and Their Operating Limits.

Part 1: Steady State OperatiQn, General Electric RepQrt,

62GL100 (1962).

9Zigrang, D., and Sylvester, N. D.: Explic it ApprQximatiQn

tQ the SQlutiQn Qf ColebrQok s FrictiQn factQr Equation,

I hE J (1982), 2 a 514.

10Fernandes, R. C., Semait, T., and Dukler, A. E.:

HydrQdynamic MQdel for Gas-Liquid Slug

FIQW

in

Vertical Tubes, I hE

J

(1986), ZQ 981-989.

9

11

Sylvester, N. D.: A Mechanis ti c MQdel fQr TWQ-Phas

Vertical Slug FIQW in Pipes,

ASME

J

Energy Resources

Tech. (1987),

1Q2

206-213.

12McQuillan, K. W., and Whal ley, P. B.: FIQW Patterns in

Vertical Two-Phase FIQw,

Int.

J

Multiphase Flow

(1985),

11

161-175.

13

BrQtz, W.:

Uber die VQrausberechnung de

AbsQrptiQnsgesch- windigkeit VQn Gasen in StrQmende

Flussigkeitsschichten, Chern. Ing. Tech. (1954), 2 2

470.

14S

c

hmidt, Z.: Experimental Study of Gas-Liqujd FIQW

in

a

Pipeline-Riser System. M.

S.

Thesis, The Universit

Qf Tulsa (1976).

.15Dukler, A. E MarQn D. M., and Brauner, N.: A Physica

MQdel

fQr Predicting the Minimum Stable Slug Length,

Chern. Eng. Sci.

(1985), 1379-1385.

16Wallis, G. B.: One-DimensiQnal TWQ-phase FIQw

McGraw-Hili (1969).

17Hewitt, G. F., and Hall-TaylQr, N. S.: Annular TWQ-phas

PergamQn Press (1970).

18Whalley, P. B., and Hewitt, G. F.: The Correlation o

Liquid Entrainment FractiQn and Entrainment Rate

i

Annular TWQ-Phase FIQw, UKAEA RepQrt, AERE

R9187, Harwell (1978).

19A1ves

I.

N. CaetanQ, E. F., Minami, K., and Shoham, 0.

MQdeling Annular

FIQW

BehaviQr for Gas Wells

presented at the Winter Annual Meeting Qf ASME

ChicagQ Nov. 27 - Dec.

2

1988.

20LQpes, J C. B., and Dukler, A. E.: Droplet Entrainment i

Vertical Annular FIQW and its CQntributiQn tQ

MQmentum

Transfer, I hE J (1986), 1500-1515.

21 Brill, J P., and Beggs,

H.

D.: TWQ-phase FIQW in Pipes

The University

Qf

Tulsa, 1988.

22GQvier,

G.

W., and Fogarasi, M.: Pressure Drop in Well

Producing Gas and Condensate, J

Can. Pet. Tech

(Oct.-Dec. 1975), 28-41.

23Asheim, H.: MONA, An Accurate TWQ-Phase Well FIQW

Model Based

Qn

Phase Slippage, SPE Prod.

ng

J (Ma

1986), 221230.

24Reinicke, K. M., Remer, R. J and Hueni, G.: CQmpariso

Qf Measured and Predicted Pressure DrQps in TUbing fQ


10/15

10

A COMPREHENSIVE

MECHANISTIC MODEL

FOR UPWARD

TWO- PHASE

FLOW

IN

WELLBORES

SPE 20630

High-Water-Cut Gas

Wells;

SP Prod ng J Aug.

1987) , 165-177.

Pressure

Gradients in Wells, ASME JERT

111 , 34 - 3 6, M ar ch, 19 89 ) .

25Chierici, G L Cuicei, G M ,and Sclocci, G.: Two-Phase

Vertical Flow in Oil Wells -- Prediction of Pressure

Drop, SP J Pet Tech Aug. 1974), 927-938.

36Beggs, H

D. and Bri l l J.

P.:

A Study o

Two-Phase

Flow in Inc lined Pipes , J. Pet

ilQ..b.

607

- 617, M ay, 1973) .

Descrjption

NOMENCLATURE

38Mukherjee, H. and Bri l l J. P.:

Pressure

Drop Corr

elations for

Inclined Two-

Phas

Row,

Trans. ASME,

JERT

Dec.,

1985).

Pip

Usi n

The

39Aziz, K.,

Gov

ier,

G. W

and

Fogar asi, M.

Pressure Drop in Wells

Producing

Oil an

Gas,

J. Cdn. Pet. Tech .. 38 - 48, Ju ly

September, 1972).

37palmer, C M.: Evaluation of Inclined

Two- Phase Liquid Holdup

Correlations

Experimental

Data, M. S Thesis,

University of Tulsa 1975).

28Hagedorn, A.

R.:

Experimental Study of

Pressure Gradients Occurring during

Continuous

Two-Phase

f low

in

Small

Diameter Vert ical Conduits, Ph.D.

Dissertat ion, The Universi ty of

Texas

at

Austin

1964) .

27Fancher, G

H., and

Brown,

K E.: Predict ion

of Pressure Gradients f or Mul ti phase Flow in

Tubing,

Trans.

AIME 1963 , 2 2 a

59-69.

26Poettmann,

F.

H., and

Carpenter,

P. G.:

The

Multiphase Flow of

G a s O i l and

Water

Thr ough

Vert i cal

Flow

Str ings wit h

Application to

th e Design

of

Gas-Lif t

Instal lations, AP I

Dri l l ing

and Production

Pract ices,

257

- 317

1952).

300rki

szew

ski,

J.: P r

edict

ing Two-

Phase

Pressure Drops in Vert ical Pipes, SPE J.

Pet. Tech. J une 1967) ,

829-

838.

29Baxendell, P B. : The Ca lcu la tion

of

Pressure

Gradients

in

High Rate Flowing

Wells,

SPE

J.

Pet.

Tech.

Oct.

1961) , 1 023.

33Camacho,

C. A.:

Comparison

of Correlations

f or P redic ti ng P re ssur e

Losses

in High Gas

LiQujd

Ratio

Vert ical

wel ls. M.S.

Thesis,

The

Universi ty

of

Tulsa

1970).

31Espanol, H J. H. :

Comparison of

Three

Methods fo r Calculat ing a

Pressure Traverse

in

Vert ical

Multi-Phase

Flow,

M. S.

Thesis

The

Univers i ty of Tulsa

1968).

a coefficient defined in Eq. 46

A cross-sectional area

of

pipe, m

2

b coefficient defined in Eq. 47

c coefficient defined in Eq. 48

C constant factor relating friction factor

to

Reynold

number for smooth pipes

C coeffi cient defined in Eq.

d differential change in a variable

D pipe diameter, m

e

error

function

El

average percentage error ,

E2 absolute average percentage error,

E3 standard deviation,

E4 average error, psi

E5 absolute average error, ps i

E6 standard deviation,

psi

f friction factor

FE fraction

of

liquid entrained in gas core

g gravity acceleration, m/s

2

h local holdup fraction

H average holdup fraction

L length along the pipe, m

n exponent relating friction factor to Reynolds

number for smooth pipes

n exponent

to

account for the swarm effect on bubb

r ise veloci ty

N number of well cases successfully traversed

p pressure, psi [ N/m

2

]

Q flow rate, m

3

/s

Re

Reynolds number

RPF Relative Performance Factor, defined in Eq. 92

S wetted perimeter, m

Discont inui t ies in

th e

Correlat ion for

Predicting

and Ros, N C J.: Vert ical

and

Liquid

Mixtures

in

Wells,

World Pet. Congress, 451,

34Duns, H., J r

Flow of Gas

Proc. 6th

1963) .

35Br i l l J. P. :

Orkiszewski

32Messulam,

S

A.

G. : Comparison of

Correlat ions fo r

Predicting

Multiphase

Flowing

Pressure

Losses

in Vertical

Pipes,

M.S. Thesis,

The

Universi ty

of

Tulsa 1970).

6


11/15

SPE

20630

A M

Ansar i

N

P

Sy

Iy

est

er

Shoham and J

P

Br

v velocity m/s

V volume m

3

X Lockhart and Mar tine ll i p arame te r

Y

Lockhart

and Mar tine ll i parameter

Z empirical factor defining interfacial friction

Greek

letters

p length ratio defined in Eq. 27

1

f ilm thickness m

ratio of film thickness to diameter

1.

difference

E absolute pipe roughness m

l dimensionless groups defined in Eqs. 79 and 80

no-slip holdup fraction

Il dynamic viscosity kg/m s

v kinematic viscosity m

2

/s

8 angle from hortizontal rad

or

deg

p density kg/m

3

surface tension dyne/cm

t

shear stress

m

Subscripts

a acceleration

A

average

c

Taylor bubble cap core

cri t

critical

e

elevation

f

friction

F

film

G

gas

H

hydraulic

ith element

I interfacial

L

liquid

LS liquid

slug

M mixture

mn

minimum

N Nusselt

r relative

s

slip

S

superficial

S slug unit

t total

1B

Taylor bubble

TP two phase

Superscript

developing

slug flow


12/15

TABLE 1

RANGE OF WELL DATA

TABLE 3

STATISTICAL RESULTS USING ALL VERTICAL WELL CASES

0-10150 1.5-10567

MODEL 14.5

AZIZ 14.0

Old TUFFP

Databank

Nom.

Dia

Oil Rate

(in.) (STBO/D)

1-8

Gas

Rate

(MSCF/D)

Oil Gravity

API)

9.5-70.5

HAGBR

E2

( Al)

10.8

E3

( Al)

15.1

19.2

19.3

E4

(psi)

-7.5

-17.7

-18.6

E5

(psi)

95.9

81.3

98.4

E6

(psi)

173.9

144.9

182.5

RFP

)

5.380

6.987

7.542

DUNROS 14.7

BEGBR 16.7

MUKBR

20.9

Govier

Fogarasi

22

Asheim

23

Reinicke

Remer

24

2- 4

8-1600 114-27400

7 20 -2 70 00 7 40 -5 57 00

0.3-5847 448-44980

17-112

35-86

ORKIS 21.1

21.9

23.0

39.5

22.0

23.2

52.0

50.9

78.0

102.0

121.7

154.9

147.2

176.3

199.5

298.8

211.0

8.392

12.913

15.374

17.122

Chierici

et al

25

0.3-69

6-27914

8.3-46

Prudhoe Bay

6 00 -2 30 00 2 00 -1 10 00 0

24-86

TABLE

4

STATISTICAL RESULTS USING ALL VERTICAL

WELL

CASES

WITHOUT HAGEDORN

AND BROWN28 DATA

ORKIS

27.4

MUKBR

20.6

BEGBR 18.2

DUNROS 15.0

RPF

)

5.000

6.801

8.459

19.168

10.814

22.400

21.301

207.2

209.2

235.6

239.5

362.4

E6

(psi)

172.2

216.1

E5

(psi)

97.8

181.9

223.4

130.6

135.2

167.2

126.3

E4

(psi)

-6.4

33.1

92.5

77.5

81.2

-21.9

-12.2

E3

)

22.6

46.8

17.0

14.8

18.0

22.8

23.3

E2

( Al)

12.8

10.1

12.2

AGBR

AZIZ

MODEL

Includes

data

from

Poe ttmann and

Carpenter26.

Fancher

an d Brown27,

Hagedorn an d Brown

2

8, Baxendell and Thomas 29, O rki sz ews ki 30,

Espanol

31

,

Messulam

32

,

and Camach033 field

data from several

oil

companies.

Water

flow

rate

TABLE 2

TABLE 5

STATISTICAL RESULTS USING ENTIRE DATABANK

DUNROS

12.2

MUKBR 17.6

MUKBR

18.2

RPF

)

43.140

58.808

5.000

8.934

9.281

35.685

198.0

191.3

539.1 118.515

E6

(pSi)

164.4

216.7

166.2

280.5

165.8

154.6

176.5

215.9

453.5

122.1

E5

pSi

109.0

E4

(psi)

-3.0

-6.4

13.1

-90.9

152.6

295.6

110.3

E3

( Al)

19.8

25.7

71.9

STATISTICAL RESULTS USING

ALL NEWVERTICAL

WELL

CASES

14.7

14.8

12.3

27.1

E2

)

10.2

8. 6

10.6

24.5

60.7

HAGBR

BEGBR

ORKIS

DUNRO S 18 .1

MODEL

AZIZ

RFP

)

7.101

7.049

8.470

8.653

5.573

14.751

10.102

177.7

178.4

190.4

207.9

217.2

273.3

E6

(psi)

163.9

134.9

159.8

110.9

151.3

102.8

116.6

E5

(psi)

101.3

12.2

33.4

41.3

78.7

E4

(psi)

9. 3

-20.8

-28.5

E3

)

13.6

18.5

20.2

20.2

17.1

16.8

32.2

E2

( Al)

9. 2

16.1

14.4

12.2

12.1

BEGBR

ORKIS

HAGBR

MODEL

AZIZ


13/15

TABLE TABLE 7

STATISTICAL

RESULTS

USING ALL WELL

CASES

STATISTICAL RESULTS

USING ALL

WlTH

OVER 75

BUBBLE

FLOW

WELL

CASES

WlTH

100 SLUG

FLOW

E2 E3 E4 E5

E6

RPF E2 E3 E4

E5

E6 RPF

)

)

si) psi)

psi)

) )

psi)

pSi) psi)

MODEL

3.2 3.7 -25.3 67.0

76.9

5.000 AZIZ

14.8 19.8

5.6 102.9

173.8

6.016

AZIZ

3.2

3.7 -30.3

68.9

79. I

5.286 MODEL 16.2

20.4

13.0 101.2 160.8 7.413

ORKIS

3.3

4.3 -26.9 69.4

90.6

5.493

HAGBR

10.1

14.8

-19.7

90.4

176.8

7.605

DUNROS

3.6

4.0 -47.9 77.5 8.2

6.374 ORKIS 14.6

26.3 17.4 116.3 212.9

8.920

HAGBR

3.8

4.3 -44.9 78.7

90.1 6.511

BEGBR

15.5 21.3

43.7 114.8 184.9

13.181

BEGBR 3.8

4.8 -46.6 79.2 102.6

6.842

DUNROS

15.1

21.4

56.6 108.2 170.7

15.276

MUKBR 7.3 3.8 -154.0 155.6 83.3

12.852

MUKBR

21.5

21.3 99.1 153.2 197.2

24.146

TABLE 8

TABLE 9

STATISTICAL RESULTS USING

AL L

VERTICAL

WELL

CASES

WITH 100 SLUG FLOWWITHOUT STATISTICAL

RESULTS

ALL WELL

CASES

HAGEDORNAND BROWN28 DATA

WITH 100 ANNULAR FLOW

E2

E3

E4

E5 E6 RPF

E2

E3

E4

E5

E6 RPF

) )

psi)

pSi)

psi)

) )

psi) pSi) psi)

MODEL 16.2

20.3

-7.9 10.7

198.7

5.331

MODEL

9.7 12.4

-21.8

90.7 132.9 5.000

AZIZ

19.1 24.1

5.9

126.7 226.3 5.696

AZIZ 12.4 16.5 22.3 106.1

145.4

5.896

HAGBR 17.0

21.1

14.4 140.5

252.6

7.118

HAGBR

15.1

16.4 70.6 128.7

148.2

8.652

DUNROS

24.3

29.3 100.0 169.4 241.9

22.694

DUNROS

20.0 24.8

-79.0

174.9 223.1 11.293

ORKIS

29.6

43.5 101.3 199.8

321.2

24.619

MUKBR

25.5

19.9 202.1 219.9 196.7

17.409

BEGBR

24.7 26.3 118.9

177.0 251.2

25.873

BEGBR

32.2

18.0

250.7

261.9 180.2 20.515

MUKBR

33.2

24.2

152.3 215.4

253.3

32.319

ORKIS

78.7

68.2 504.0

544.9

407.9

45.810


14/15

t

t

t t

O

Oc?080

20

D

1

Ul

-

E

03

>

l

U

0

00

8

BARNEA

I

o

J

BUBBLY

TRANSITION

,

O Oo

00 0

LlJ

I

0

0

>

0

0

0

0

I

ANNULAR

0

0

I

0 boooo

000

0

o

0

:::i 1

D

0

0

0

0

J

I

A

I

a 0 0 0

u

SLUG

OR

CHURN

I

u

I

0 0

00

..

0::

LlJ

I

o ~ o

ll

0.01

I

::>

/)

I

t

t

t t

I

BUBBLE

SLUG CHURN

ANNULAR

2

0

.

2

1

1

1

100

FLOW FLOW FLOW FLOW

SUPERFICIAL

GAS

VELOCITY m/s

Fig Flow patterns

upward

two phase

flow

Fig Typical flow pattern map for wellbores

I

I

VTB

DEVELOPED

TAYLOR

U L E ~

DEVELOPING

TAYLOR

BUBBLE

o DEVELOPED SLUG UNIT

b DEVELOPING

SLUG

UNIT

Fig 3 Schernatic diagram of slug flow

64


15/15

1

+

CALCULATED PRESSURE

9.0

T

x

MEASURED PRESSURE

ANNULAR

I

1

GAS CORE

8.0

LIQUID FILM

I

UJ

.j-

UJ

lJ

7.0

0

ENTRAINED

0

I

0

LIQUID DROPLET

.

....

6.0

V .

C

I

0

SLUG

l

t v

F

5.0

V

F

0

CD

H

LC

UJ

:c

.

4 1 I T

4.0

.

.

i

T

F

0

T

F

c:

lJ

UJ 3.0

u

ic

z

spe-20630-ms two phase flow in well bore

Documents