1994 - stratified three phase flow in pipes.pdf

8/10/2019 1994 - Stratified three phase flow in pipes.pdf

1/8

~ ergamon

I n t J M u l t i p h a s e F l o w V o l . 2 1 , N o . I , p p . 5 3 - 6 0 , 1 9 9 5

Co p y r i g h t ~ (? 1 9 9 5 E ls e v i e r Sc i e n c e L t d

0301-9322(94)00058-1

Pr i n t e d i n G r e a t Br i t a i n . A l l r i g h t s r e s e r v e d

0 3 0 1 - 9 3 2 2 / 9 5 9 .5 0 + 0 .0 0

S T R T I F I E D T H R E E P H S E F L O W I N P I P E S

Y . T A I T E L j, D . B A R N E A l a n d J . P . B R I L L 2

~Department of F luid M echa nics and Heat Transfer F aculty of Engineering, Tel-A viv University,

Ramat-Aviv 69978, Israel

- 'Department o f Petroleum Engineering University of Tulsa, 600 S. College Avenue, Tulsa,

OK 74104, U.S.A.

Received 15 September 1993; in revised for m 14 July 1994)

Abstract--The simultaneous flow of wate r, oil and gas is of practical importance for the oil and gas

industry. M any oil and gas pipelines contain these three phases with varying degrees of concentration,

depending on the particular situation.

In this work the gas/oil /water h oldups for stratified three phase flow are calculated. This information

is usually the first step for analyzing the stability of stratified flow and for develop ing transition criteria.

It is shown that one can obtain three theoretical steady state configurations for stratified flow, but only

the configuration with the thinnest total liquid layer is stable and can actually occur.

Taitel Dukler A IC h E J 1 2 2 , 47-55 , 1976) criterion for transition from stratified flow was applied to

the three phase flow case and w as found to y ield good agreement for low gas flow rates.

Key Words : three phase flow , stratified flow, wate~ oil-gas, flow pattern

I N T R O D U C T I O N

T h r e e p h a s e f lo w o f t w o i m m i s c ib l e l iq u i d s a n d g a s is o f c o n s i d e r a b l e p r a c t i c a l i m p o r t a n c e f o r t h e

o il a n d g a s i n d u s t r y . G a s p i p e li n es o f t e n c o n t a i n w a t e r a n d h y d r o c a r b o n c o n d e n s a t e s a n d o i l p i p el in e s

c a n a l s o c o n t a i n b o t h v a p o r a n d w a t e r . W a t e r p r o d u c t i o n o f t e n i n c re a s e s si g n if i ca n t ly d u r i n g t h e

l a tt e r s ta g e s o f a w el l a n d u s e o f t h e c o n v e n t i o n a l a p p r o x i m a t i o n s o f a t w o p h a s e o i l - g a s s y s t e m

n e g l e ct i n g t h e w a t e r, o r c o m b i n i n g t h e o il a n d w a t e r i n t o a l iq u i d p h a s e , o f t e n b e c o m e s i n a c c u r a te .

I n s p i te o f t h e p r a c t ic a l i m p o r t a n c e , i t is s u r p r is i n g h o w l i tt le w o r k h a s b e e n d e d i c a t e d t o a s t u d y

o f t h e t h r e e p h a s e f l o w c a s e . P r e v i o u s p u b l i c a t i o n s c a n b e d i v i d e d i n t o t w o m a i n c a t e g o r i e s o n t h e

b a s i s o f p i p e i n c l i n a t i o n a n g l e . T h e h o r i z o n t a l a n d / o r s l ig h t ly i n c li n e d c a s e w a s s t u d i e d b y :

A ~ i k g 6 g z e t a l . ( 1 9 9 2 ) , L a h e y e t a l . ( 1 9 9 2 ) , N u l a n d e t a l . ( 1 9 9 1 ) , L e e e t a l . ( 1 9 9 3 ) a n d S t a p e l b e r g

e t a l .

( 1 9 9 0 , 19 91 a b ) ; w h e r e a s t h e v e r t i c a l c a s e w a s c o n s i d e r e d b y C h e n

e t a l .

( 1 9 9 0 ) a n d P l e s h k o

S h a r m a ( 1 9 90 ) . T h e w o r k o f S t a p e l b e r g e t a l . c o n c e n t r a t e d p r i m a r i l y o n s l u g f lo w , w h e r e t h e

f l o w is s u f f i c ie n t l y c h a o t i c t h a t t h e l i q u i d p h a s e i s e s s e n t i a l l y a m i x t u r e o f t h e t w o l i q u i d s u s e d , w a t e r

a n d o i l i n t h i s c a s e .

O f p a r ti c u l a r r e le v a n c e t o th e p r e s e n t w o r k is t h e w o r k b y A q i k g 6 g z e t a l . ( 1 9 9 2 ) a n d L e e e t a l .

( 1 9 93 ) . B o t h r e p o r t e d e x p e r i m e n t a l r e s u l ts f o r a w i d e r a n g e o f f l o w r a t e s a n d d e s c r i b e d i n d e t a il

t h e fl o w p a t te r n s o b s e r v e d f o r t h e c a s e o f h o r i z o n t a l o i l - w a t e r - a i r f lo w . A q i k g 6 g z e t a l . ( 1 9 9 2 )

o b s e r v e d a v e r y c o m p l e x a r r a y o f f l o w p a t t e r n s a n d d e s c r i b e d 1 0 d i f fe r e n t fl o w p a t te r n s . I n t h ei r

w o r k , t h e p i p e d i a m e t e r w a s o n l y 1 9 m m a n d s t r a t i f i c a ti o n w a s s e l d o m a c h i e v e d . T h e L e e e t a l .

( 1 9 9 3 ) e x p e r i m e n t s w e r e c a r r i e d o u t o n a t es t s e c t i o n o f 1 0 c m d i a . T h e y o b s e r v e d a n d c l as s if ie d

s e v e n f lo w p a t t e r n s , w h i c h a r e s i m i l a r t o t h e c a s e o f t w o p h a s e f l o w n a m e l y : ( 1 ) s m o o t h s t ra t if i ed ,

( 2 ) w a v y s t r a t i f i e d , ( 3 ) r o l l i n g w a v e , ( 4 ) p l u g f l o w , ( 5 ) s l u g f l o w , ( 6 ) p s e u d o s l u g a n d ( 7 ) a n n u l a r

f l o w . T h e f i r st t h r e e p a t t e r n s c a n b e c l a s si f ie d a s s tr a t i f ie d f l o w a n d t h e y o b s e r v e d t h a t t h e o i l a n d

t h e w a t e r a r e g e n e r a l l y s e g r e g a t e d , w i t h w a t e r f l o w i n g a s a l iq u i d l a y e r a t t h e b o t t o m o f th e p i p e

a n d o il f l o w i n g o n t o p . E v e n f o r p l u g f lo w , th e w a t e r r e m a i n s a t t h e b o t t o m , b e c a u s e t h e a g i t a t i o n

o f t h e l iq u i d s w a s n o t s u f f i c ie n t to m i x t h e o i l a n d t h e w a t e r . O n t h e o t h e r h a n d , i n s l u g f l ow , p s e u d o

s l u g a n d a n n u l a r f l o w , o i l a n d w a t e r w e r e c o m p l e t e l y d i s p e r s e d .

I n t h is w o r k w e p r e s e n t a t h e o r e t i c a l a p p r o a c h t o s o l v e t h e s t r u c t u r e o f 3 - l ay e r s tr a ti f ie d f l o w .

T h e p u r p o s e o f t h e s o l u t i o n i s t o c a l c u l a te t h e l e ve ls o f t h e b o t t o m l i q u id l a y e r a n d t h e s e c o n d l a y er .

T h e s e c a n b e u s e d t o c a l c u l a t e t h e h o l d u p s o f th e t w o l i q u i d s a n d t h e g a s. O n c e t h e s e v a lu e s a r e

53


2/8

54 Y. TAITEL

et al

c a l c u l a t e d , m a n y o t h e r v a r i a b l e s t h a t p e r t a i n t o t h e t w o l i q u id s a n d g a s v e l o c i ti e s , p r e s s u r e d r o p

a n d s t a b i l it y c o n s i d e r a t i o n s c a n f o l l o w . In p a r t i c u l a r , t h e p o s s i b il i ty o f o b t a i n i n g m u l t ip l e s o l u t i o n s

i s d i s c u s s e d . A l s o c o n s i d e r e d i s t h e t r a n s i t i o n f r o m s t r a t i f i e d f l o w t o s l u g o r a n n u l a r f l o w .

A N A L Y S I S

R e f e r r i n g t o f i g u r e 1 , t h e f l o w o f t h r e e f l u id s is c o n s i d e r e d : w a t e r , o i l a n d g a s. I t i s a s s u m e d t h a t

t h e w a t e r i s h e a v i e r t h a n t h e o i l a n d f l o w s a t t h e b o t t o m . T h e o i l f l o w s i n t h e m i d d l e a n d t h e g a s

a t t h e t o p . A m o m e n t u m b a l a n c e f o r e a c h p h a s e c a n b e w r i t t e n a s f o l l o w s :

_ A w (\ dPdx -) - ~ 'wSw

+ ri S, - p w A w g

sin fl = 0 [11

- A o ( ~ x x ) - r o S o - r , + z j S j - p o A o g s i n f i = O [2]

- - A G d ~x - r G S G - r j S j - O G A c g s i n f l = O

[3]

wh ere A i s a c ro s s s ec t i o na l a r e a , p i s dens i t y , P i s p res s u re a nd f i i s t he i nc l i na t i o n ang l e , pos i t i ve

f o r u p w a r d i n c l i n a t i o n . T h e s u b s c r i p t s a r e W f o r w a t e r , O f o r o il , a n d G f o r g as . F i v e s h e a r s t re s s e s

a r e n e e d e d t o s o l v e [ 1 ] - [ 3 ] : r w , t h e s h e a r s t r e s s a c t i n g o n t h e w a l l w e t t e d b y t h e w a t e r S w ; t o , t h e

s h e a r s t r e s s a c t i n g o n t h e w a l l w e t t e d b y t h e o i l S o ; r G , t h e s h e a r s t r e s s a c t i n g o n t h e w a l l w e t t e d

by t he gas S G; r i , the s hea r s t r e s s ac t i ng o n t he o il wa t e r i n t e r f ac e ; and ~ j , t he s hea r s t re s s ac t i ng

o n t h e o i l - g a s i n t e r f a c e , S j. T h e s e f iv e s h e a r s t re s s e s c a n b e c o r r e l a t e d a s f o l lo w s , w h e r e U is th e

a v e r a g e v e l o c i t y o f t h e f l u i d i n a l a y e r :

- r p w U 2 w

~ w ~ w ~

[ ]

r o = f o P o U ~ [ ]

2

ZG = f ~ PG V ~ [6]

2

p o ( V o -

Uw )l Uo - Uwl [7]

ri = f 2

= r P G ( U G - - U o ) I U 6 -

Uol

[8]

/

J j

2

_ . . - - z - - _

F i g u r e 1 T h e t h r e e l a y e r g e o m e t r y .


3/8

STRATIFIED THREE PHASE FLOW IN PIPES 55

F o r t h e s h e a r s t r e s se s b e t w e e n t h e l i q u i d s o r g a s a n d t h e p i p e s u r f a c e , t h e f r i c t i o n f a c t o r s , f w ,

f o a n d f c c a n b e a p p r o x i m a t e d b y t h e c o r r e l a ti o n

f = C R ; [9]

w h e r e C = 0 .0 4 6 , n = 0 . 2 f o r t u r b u l e n t f lo w , a n d C - - 1 6, n = 1 f o r la m i n a r f lo w . T h e R e y n o l d s

n u m b e r s w e r e d e f i n e d a s R e w = 4 U w A w P w / S w l ~ w fo r t he wa t e r , R ~ o = 4 U o A o P o / S o l ~ o fo r t he o i l

and R eG- - - -4UGAGpG/ (S c + Sj)~G f o r t h e g a s . F o r t h e i n t e r f a c i a l g a s - - o i l s h e a r s t r e s s w e u s e d a

c o n s t a n t v a l u e o f ~ = 0 .0 1 4 , ( C o h e n & H a n r a t t y 1 96 8) b u t i f t h e v a lu e o f f G w a s l a r g e r t h a n f j ,

t h e n f j = f G w a s u s e d . L i k e w i se f o r t h e o i l - w a t e r i n t e r f ac e , f = 0 . 01 4 w a s u s ed o r f = f o w h e n t h e

v a l u e o f f 0 w a s l a r g e r t h a n 0 . 0 1 4 .

S um m i ng [1 ] and [2 ] y i e l d s :

( d P ) TLSL cjSjx A L A L

- - - ~ + - z - - - - p L g s i n f l = O [10]

w h e r e ,

a n d

Z ' L S L = T w S W ~ - ~ o S o

[11]

p w A w + p o A o

PL = [12]

AL

A L = A w + A o [13]

N o t e t h a t [1 0] is t h e c o m b i n e d m o m e n t u m e q u a t i o n f o r th e l iq u i d p h a s e w h i c h is c o m p o s e d o f

t h e w a t e r a n d t h e o i l l a y e r s . E q u a t i o n s [ 1 0 ] a n d [ 3 ] , t h e r e f o r e , h a v e t h e s a m e f o r m a s a t w o - l a y e r

f o r m u l a t i o n f o r l i q u i d a n d g a s .

T h e p r e s s u r e d r o p c a n b e e l i m i n a t e d f r o m [ 3 ] a n d [ 1 0 ] t o y i e l d :

ZL

S

ZGSG / 1 1 )

- - - - - - P L - - PG)g sin fl = 0 [14]

L

/

I n t h e s a m e w a y t h e p r e s s u r e d r o p i s e l i m i n a t e d f r o m [1 ] a n d [2 ] t o y i e ld :

TwSwToSoT~jSjw A o A o ( 1 1 )

~- ziSi -~w + ~ o - pw - Po )g sin f l = O [15]

E q u a t i o n s [ 14 ] a n d [ 15 ] m u s t b e s o l v e d s i m u l t a n e o u s l y t o y i e l d t h e l i q u i d le v e l h E , a n d t h e w a t e r

l e v el , h w . N o t e t h a t f o r t h e c a s e o f t w o l a y e r s , o n e h a s o n l y a si n g le e q u a t i o n w i t h t h e e x a c t f o r m

a s [ 1 4 ] . T h e c a s e w i t h t h r e e l a y e r s , h o w e v e r , i s m u c h m o r e c o m p l i c a t e d . A l t h o u g h [ 1 4 ] h a s t h e

i d e n t i c a l f o r m a t a s f o r t h e t w o - l a y e r e q u a t i o n , i t m u s t b e s o l v e d s i m u l t a n e o u s l y w i t h [ 1 5 ] b e c a u s e ,

un l i ke t he t wo l aye r cas e , ZL , S L , P L and

A L , a r e

g i ven i n t e rm s o f Zw, Vo , S w, S o , Aw and A o a s

c a n b e s e e n f r o m [ 1 1 ] t o [ 13 ].

M e t h o d o f s o lu t io n

E q u a t i o n s [ 1 4 ] a n d [ 15 ] a r e t w o s i m u l t a n e o u s e q u a t i o n s f o r t h e t w o l e v e ls h w a n d h E . T h e

e q u a t i o n s a r e n o t l i n e a r a n d c a n h a v e m u l ti p l e s o lu t i o n s. A m e t h o d o f s o l u t io n t h a t w o r k s q u i t e

wel l i s as fo l lows:

1 . W e s t a r t w i t h a g u e s s o f h E a t s o m e l o w v a l u e .

2 . O n c e h E is k n o w n , [ 1 5 ] is s o l v e d f o r hw b y a o n e - d i m e n s i o n a l s e a r c h p r o c e d u r e . F o r e a c h

c h o i c e o f h E o n l y o n e s o l u t io n f o r h w w a s f o u n d .

3 . E q u a t i o n [ 1 4 ] is t h e n t e s t e d a n d i f n o t s a t is f ie d , a h i g h e r v a l u e o f h L i s u s ed and s t ep 2 i s

r e p e a t e d . C o n v e r g e n c e i s a c h i e v e d u s i n g t h e h a l v a t i o n m e t h o d .

4 . O n c e a s o l u t i o n f o r h E a n d h w i s o b t a i n e d , a s e a r c h f o r a d d i t i o n a l s o l u t i o n s f o r t h e s a m e

f l o w r a t e s i s c a r r i e d o u t b y c o n t i n u i n g t h i s p r o c e d u r e w i t h g u e s s e d v a l u e s o f h E l a r g e r t h a n

t h e o b t a i n e d s o l u t i o n .


4/8

56 Y TAITEL

et al

T h e a d v a n t a g e o f th i s m e t h o d i s t h a t i s c o v e r s s a f e l y t h e w h o l e p o s s i b l e s t e a d y s t a t e s o l u t io n s

t h a t e x i s t a n d c o n v e r g e n c e i s a l w a y s a s s u r e d .

R E S U L T S A N D D I S C U S S I O N

S o m e e x a m p l e s o f t h e c a l c u l a t i o n s a r e s h o w n i n f ig u r e s 2 7 . B e c a u s e o f t h e l a rg e n u m b e r o f

p a r a m e t e r s t h e r e i s n o b e n e f i t t o e x p r e s s t h e r e s u l t s i n a d i m e n s i o n l e s s f o r m a n d t h e r e s u l t s g i v e n

a r e f o r t y p ic a l e x a m p l e s o f w a t e r - o i l - g a s f lo w t o d e m o n s t r a t e t h e c a p a b i l it y o f t he m e t h o d . I n a ll

c a s e s t h e p i p e d i a m e t e r i s 5 c m , a n d t h e p r o p e r t i e s o f w a t e r a n d a i r a r e t a k e n a t s t a n d a r d r o o m

t e m p e r a t u r e a n d a t m o s p h e r i c p r e s s u r e , w i t h t h e v i s c o s it y o f w a t e r b e i n g r o u n d e d t o 1 c P . O il

d e n s i t y i s t a k e n a t 8 0 0 k g / m 3. W e r a n t w o c a s e s f o r o i l v i sc o s i t i e s o f 1 a n d 1 00 c P .

F i g u r e s 2 - 4 s h o w t h e r e s u l t s f o r t h e h o r i z o n t a l c a s e w h e r e t h e l i q u i d l e v e l

hL/D)

a n d t h e w a t e r

l eve l hw/D) a r e p l o t t e d a s a f u n c t i o n o f t h e l iq u i d f l o w r a t e ( U L s = U w s + U o s ) f o r e q u a l f l o w r a t e s

o f o i l a n d w a t e r . F i g u r e 2 i s t h e s o l u t i o n f o r t h e t w o l a y e r c a s e w h e r e o n l y w a t e r a n d a i r a re f l o w in g .

I n f i g u r e 3 , t h e o i l v i s c o s i t y i s i d e n t i c a l t o t h e w a t e r v i s c o s i t y a n d t h e l i q u i d l e v e l s h o w n i n f i g u r e

3 i s a l mo s t i d e n t i c a l t o t h e l i q u i d l e v e l s h o w n i n f i g u r e 2 . N o t e t h a t e v e n f o r t h e c a s e w h e r e w e

c o n s i d e r t h e l i q u i d t o c o n s i s t o f tw o l a y e r s w i t h i d e n t i c a l v i s c o s i t y a n d d e n s i t y , t h e r e s u l t s o f t h e

s o l u t i o n f o r t h e t w o l a y e r a n a l y s i s a n d t h e t h r e e l a y e r a n a l y s i s d o n o t h a v e t o c o i n c i d e . T h a t i s ,

t h e s o l u t i o n f o r hL/D f o r a s i n g l e w a t e r l a y e r i s n o t i d e n t i c a l t o t h e s o l u t i o n w h e r e t h e l i q u i d l a y e r

i s s u b d i v i d e d i n t o t w o e q u a l v i s c o s i t y a n d d e n s i t y l a y e r s . Y e t t h e p r e s e n t r e s u l t s in d i c a t e t h a t t h e

d i f f e r e n c e i s p r a c t i c a l l y n e g l i g i b l e ( a s w e w o u l d e x p e c t ) . T h e b o t t o m w a t e r l e v e l i s s h o w n b y a

b r o k e n l i n e . A s c a n b e s e e n , th e w a t e r l e v el i s u s u a l l y q u i t e h i g h a n d t h e l i q u id c o n s i s t s m o s t l y o f

w a t e r , e s p e c i a l l y f o r l o w f l o w r a t e s . T h i s i s l o g i c a l s i n c e t h e o i l , b e i n g c l o s e r t o t h e f a s t mo v i n g

g a s , is d r a g g e d b y t h e g a s t o h i g h e r v e l o c i ti e s t h a n t h e w a t e r l a y e r a t t h e b o t t o m . B e c a u s e t h e o i l

t r a v e l s f a s t e r t h a n t h e w a t e r , t h e

in situ

h o l d u p o f o il is l o w e r t h a n t h a t o f w a t e r .

W h e n o i l v i s c o s i t y is in c r e a s e d t o 1 00 c P , f i g u r e 4 s h o w s t h a t t h e t o t a l l i q u i d l e v e l i n c r e a s e s ,

e s p e c i a l l y f o r l o w l i q u i d r a t e s , a n d t h e r e l a t i v e l e v el o f t h e w a t e r d e c r e a s e s ( a l t h o u g h n o t

d r a ma t i c a l l y ) . T h i s i s a l s o e x p e c t e d s i n c e t h e mo r e v i s c o u s o i l w i l l mo v e s l o w e r . A s a r e s u l t , i t s

r e l a t iv e h o l d u p w i ll in c r e a s e a s w e l l a s t he t o t a l h o l d u p o f th e l i qu i d ( w a t e r + o il ). T h e p h e n o m e n o n

i s e n h a n c e d f o r l o w l i q u i d f l o w r a t e s b e c a u s e t h e e f f e c t o f v i s c o s i t y is l o w e r f o r h i g h , t u r b u l e n t f l o w

r a t e s c o m p a r e d t o l o w , l a m i n a r f l o w r a t e s .

N o t e t h a t , a s f o r t h e c a s e o f t w o l a y e r s y s t e m s , a s i n gl e s o l u t io n e x i st s f o r th e h o r i z o n t a l c a s e ,

f o r s p e c if ic li q u id s a n d g a s f l o w r a te s . T h u s , f o r a g i v e n s e t o f li q u id p r o p e r t i e s , p i p e d i a m e t e r a n d

w a t e r , o i l a n d g a s f l o w r a te s , t h e r e i s a s i n g le s o l u t i o n f o r t h e t h i c k n e s s o f t h e w a t e r a n d t h e o i l

l a y e r s .

F i g u r e s 5 t o 7 s h o w t h a t m o r e t h a n o n e s o l u t i o n i s p o s s i b l e fo r t h e u p w a r d i n c li n e d c a se . F i g u r e

5 i s t h e s o l u t i o n f o r t h e t w o l a y e r c a s e f o r w a t e r - a i r . T h e p o s s i b i li t y o f o b t a i n i n g t h r e e s o l u t io n s

f o r t h e c a s e o f t w o l a y e r s g a s - l i q u i d f lo w h a s b e e n o b s e r v e d a n d d i s c u s s e d b y o t h e r s ( B a k e r

G r a v e s t o c k 19 87 ; L a n d m a n 1 99 1 a b ; B a r n e a T a i t e l 1 99 2) . F i g u r e s 6 a n d 7 s h o w t h a t a l s o

f o r t h e c a s e o f t h r e e p h a s e f l o w , t h r e e s o l u t i o n s a r e p o s s i b l e f o r t h e l o w l i q u id f l o w r a t e s . I n t h is

r e g i o n o f l o w l i q u i d f l o w r a t e s t h e r e l a t i v e o i l l a y e r i s v e r y t h i n .

]

hL/D

0 5 O~ -

tx 0

O 0 . . . . . . . . . . . , , i , , , . . . . .

Ut s m / s )

Figure 2. Liquid levels for water ai r flow in horizontal pipes.


5/8

STRATIFIED THRE E PHASE FLOW IN PIPES 57

h / D

. L , ; . . . . . . . . . . . . . . . . . . . .

h w / D . . . . . ~ / /

U W S / U L S = 0 . 5 O ' ~

. /

, ~

0 . 5 - O X . . . - -

0

0 0 0 0 1 . 0 0 0 1 . 0 0 1 . 01 .1

U L S ( m / s )

. . . . . . . I .

1 1 0

F i g u r e 3 . L i q u i d l e v e ls fo r w a t e r - o i l - a i r f l ow i n h o r i z o n t a l

p i p e s . O i l v i s c o s i t y 1 c P . W a t e r f l o w r a t e r a t i o 5 0 .

f

h/O hw/O / /

. 0 0 0 0 1 . 0 0 0 1 . 0 0 1 . 0 1 . 1 1 I 0

U k s m / s )

F i g u r e 4 . L i q u i d l e v e l s f o r w a t e r - o i l a i r f lo w in h o r i z o n t a l

p i p e s . O i l v i s c o s i t y 1 0 0 c P . W a t e r f l o w r a t e r a t i o 5 0 .

O n e w o u l d e x p e c t t h a t t h e o i l l a y e r w i l l b e t h i n n e r t h a n t h e w a t e r l a y e r ( f o r e q u a l f l o w r a t e s )

b e c a u s e t h e o i l l a y e r , in b e t w e e n t h e w a t e r a n d t h e g a s l a y e r w i ll h a v e a v e l o c i t y w h i c h i s b e t w e e n

t h e w a t e r a n d t h e f a s t m o v i n g g a s . H o w e v e r , t h e e x t r e m e l y t h i n o i l l a y e r in t h e r e g i o n o f l o w ULS

i s b y n o m e a n s i n t u i t i v e . A n o t h e r , n o n - i n t u i t i v e o b s e r v a t i o n i s t h a t , u n l i k e t h e h o r i z o n t a l c a s e

w h e r e t h e i n c r e a s e o f v i s c o s i ty c a u s e s a s i g n i f i c a n t i n c r e a s e i n t h e l i q u i d l e v e l, f o r t h e c a s e o f u p w a r d

f l ow ( s ee f i gu re 7 ) t he v i s cos i t y has l i t t l e e f f ec t on t he l i qu i d l eve l . On l y fo r gas ve l oc i t y in t he r an ge

o f 8 - 2 0 m / s t h e r e i s a n o t i c e a b l e c h a n g e o f l i q u i d l e ve l d u e t o a n i n c r e a s e i n t h e o i l v i s c o s it y .

T h e m u l t i p l e so lu t i o n s

W h e n m o r e t h a n a s i ng l e s te a d y s t a t e s o l u t i o n e x i s ts , it is i m p o r t a n t t o i d e n t i f y t h e p h y s i c a l l y

r e a l i s t i c s o l u t i o n s a n d w h e t h e r i t i s p o s s i b l e t o h a v e m o r e t h a n o n e s o l u t i o n i n p r a c t i c e . T h i s

p r o b l e m f o r t h e c a se o f t w o l a y e r s w a s t h o r o u g h l y d i s c u s se d b y B a r n e a & T a i t e l ( 1 99 2 ) w h o

a n a l y z e d i n d e t a i l t h e s t a b il i ty o f t h e s t e a d y s t a t e s o l u t i o n s o b t a i n e d f o r t h e c a s e o f tw o l a y e r s . I t

w a s c o n c l u d e d t h a t , w h e n t h r e e s t e a d y s t a t e s o l u t i o n s e x i s t, t h e o n l y r e a li s ti c s o l u t i o n is th e s o l u t i o n

w i t h th e m o s t th i n l iq u i d l e ve l . T h e o t h e r t w o t h i c k s o l u t i o n s w e r e f o u n d t o b e u n s t a b l e . L i k e w i s e

w e a s s u m e h e r e t h a t a l s o f o r t h e t h r e e l a y e r s t h e o n l y v a l i d p h y s i c a l s o l u t i o n i s th e f i r st ( t h in )

s o l u t i o n .

F l o w p a t t e r n t r a n s it i o n

T r a n s i t i o n f r o m s t r a t i f i e d f l o w ( t o s l u g o r a n n u l a r f l o w ) i s b e l i e v e d t o b e t h e r e s u l t o f

K e l v i n - H e l m h o l t z i n s ta b i li ty t h a t c a u s e s w a v e s o n t h e g a s - l i q u id i n t e rf a c e to b e u n s t a b l e a n d g r o w

u p t o t h e p o i n t t h a t t h e y t o u c h t h e t o p p a r t o f t h e p ip e , r e s u l t in g in t r a n s i t i o n t o s l u g o r a n n u l a r

f l o w ( T a i t e l & D u k l e r 1 97 6; L i n & H a n r a t t y 1 98 6: W u e t a l 1987 ; B arnea 1987 ; Ta i t e l 1990 ; B arnea

1 99 1; B a r n e a & T a i t e l 1 9 92 ). I n t h i s w o r k w e u s e d t h e m e t h o d p r o p o s e d b y T a i t e l & D u k l e r ( 1 9 7 6 )

. . . . . . . . I . . . . . . . . . . . . . . . . I . . . . . . . . I . . . . . . . . I . . . . . . . 1 . . . . . . . . . . . . . . . .L / D I I . . . . . . . . [ ' ' . .. . .. ' . . . . . . I . . . . . . . .

h w / D . . . . .

UWS / U LS =0 -5 / / ~

u . < o > o o o . ..........

h L / D

~ f I I

o ~

o o o l o o o l o o l o i

1 ; lo

0 0 0 0 1 0 0 0 1 , 0 0 1 . 0 1 .1 1 I 0

U L $ ( m / s ) U L S ( m / s )

F i g u r e 5 . L i q u i d l e v e l s f o r w a t e r a i r f l o w in p i p e i n c l i n e d F i g u r e 6 . L i q u i d le v e l s f o r w a t e r - o i l - a i r fl o w i n p i p e i n c l i n e d

u p w a r d I . u p w a r d 1 . O i l v i s c o si t y 1 c P . W a t e r f lo w r a t e r a t io 5 0 .


6/8

58 Y T A I T E L

et al

........ I ........ I ........ I ........ r ........ I ..... ..

h L / D - -

h w / D - _ / ' ~

f

h i D U G S = 4 m ls J / / . / / / .

0 5 ...... .6 .... ,.

8

. . : : : - -~

. . - ' .

8 . . . .. . . - : : : - : : # : : :

0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 1 1 0

U

L S (

m l s l

Figure 7. Liquid levels for w ater oil air flow in pip e inclined upward 1'. Oil viscosity 10O cP. Water flow

rate ratio 50 .

t o c a lc u l a te t h e l o c a t io n o f th e t r a n si t io n b o u n d a r i e s . T h e e x t e n s i o n o f t h e m o r e a d v a n c e d m e t h o d s

s u c h a s t h o s e p r o p o s e d b y L i n & H a n r a t t y ( 19 8 6 ), W u ( 19 8 7 ) a n d B a r n e a ( 1 99 1 ) to t h e t h re e p h a s e

g e o m e t r y i s n o t t ri v ia l . B a r n e a ( 1 9 9 1 ) s h o w e d t h a t t h e T a i t e l & D u L l e r ( 1 9 7 6 ) m e t h o d is , i n fa c t,

qui te accura te for v iscosi t ies up to 100 cP. Thu s, in th is wo rk the Tai te l Du Ller (1976) t ransi t ion

f r o m s t r at if i e d f l o w i s c o n s i d e r e d , n a m e l y t h e l i q u id l ev e l is u n s t a b l e w h e n

h L ) ~ p - p G ) g A G c s ~ [16]

U G - U o > 1 - ~ pGSj

a n d o n c e [ 1 6] i s s a t is f i e d , s l u g fl o w w i ll ex i s t f o r h i g h l i q u i d h o l d u p a n d a n n u l a r f l o w f o r l o w l i q u i d

h o l d u p h L / D = 0 . 3 5 , B a r n e a e t a l . 1982) .

F i g u r e 8 s h o w s r e s u l ts f o r t h e h o r i z o n t a l c a s e f o r w a t e r f l o w r a te f r a c t i o n s o f 1 . 0, 0 . 5, 0 .1 a n d

0 . T h e s o l i d l in e r e p r e s e n t s t h e c a s e f o r a w a t e r f r a c t i o n o f 1 . 0 w h e n t h e l i q u i d p h a s e i s a ll w a t e r

( t h e t w o l a y e r m o d e l ) .

I n t r o d u c i n g a n o i l l a y e r o f h i g h e r v i s c o s i ty r e d u c e s t h e l o c al v e l o ci t ie s a n d i n c r ea s e s t h e l i q u i d

l e v el in t h e p i p e . S i n c e t h e t r a n s i t i o n f r o m s t r a t if i e d f l o w is d i r e c t l y r e l a t e d t o t h e l i q u i d l e v e l i n

t h e p i p e ( f o r a g i v e n g a s f l o w r a t e ) , t h e n w h e n t h e w a t e r f l o w r a t e f r a c t i o n d e c r e a s e s , t h e l i q u i d

l e v el i n c r e a s e s a n d t h e t r a n s i t i o n f r o m s t r a t i f i e d f l o w o c c u r s a t l o w e r l i q u i d a n d g a s f lo w r a t es .

A s i m i l a r e f f e c t c a n b e s e e n i n f i g u r e 9 f o r t h e i n c l i n e d c a s e . A s t h e o i l f l o w r a t e f r a c t i o n i s

i n c r e a s ed , t h e r e g i o n o f s tr a ti f ie d f l o w d e cr e a s es . H o w e v e r , t h is s t a t e m e n t n e e d s s o m e e l a b o r a t i o n .

P r e v i o u s l y , t h e o b s e r v a t i o n w a s m a d e t h a t , u n l i k e t h e h o r i z o n t a l c a s e w h e r e t h e l iq u i d l ev e l c h a n g e s

c o n s i d e r a b l y w h e n i n c r e a s i n g t h e o i l f r a c t i o n , t h i s i s n o t t h e c a s e f o r t h e i n c l i n e d c a s e w h e r e t h e

l i q u i d l e v el u s u a l l y i s l it tl e a f f e c t e d b y t h e i n c r e a s e o f o i l f r a c t i o n . H o w e v e r , a s m e n t i o n e d

10

S L U G F L O W

U L s m / s )

U w s/U Ls =1 (w a te r O ~ / / , / ( ~

. . . . : L ' . - - _ _ # ' ? , \

F L O W \ / - \

. 01 . 1 1 10 100

Uc, m/s)

Figure 8. Flow patterns for water oil air flow in horizontal

pipes . Oil viscosi ty 100 cP.

1 0

1

ULS

m / s )

.I

Figure

L UG L O W

/ ~ :

/,

l ; , /

/ / i

.o .

0 . 1 ~

S T R A T I F I E D F L O W - - ' ~ , ~ - ~ t ' t ~

0 1 1 1 1 0 1 0 0

U G S

m / s )

9. Flow patterns for water oil air flow in pipe

inclined upward 1 . Oil viscosity 100cP.


7/8


8/8

60

v TAITEL

et al

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