ON THE LINKING UP BETWEEN BINGHAM FLUID AND PLUGGED FLOW(1)

7/28/2019 ON THE LINKING UP BETWEEN BINGHAM FLUID AND PLUGGED FLOW(1)

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A p p l ie d M a t h e m a t i c s a n d M e c h a n i c s(Eng l ish Ed i t i on , Vo l .8 , No .3 , M ar . 1987) P u b l is h e d b y S U T ,S h an g h a i , C h i n a

O N T H E L I N K I N G U P B E T W E E N B I N G H A M F L U I D A N DP L U G G E D F L O W

T s a i S h u - t a n g ( ~ i )(Department o f M ode rn Mechanics, C hina Univerity of Science and Technology, He fei;

Shanghai Inst itute o f App lied Ma thema tics and Mechanics, Shanghat)Jiang Yi-an ( ~ )

(Department o f Physics, Hangzhou No rm al College, Hangzhou)(Rece ived De c. 4, 1985)

A b s t r a c tWhen Bingham f luid is in motion: plugged f lo w often occurs at places far fr om the

bowutary walls. A s there is not a decisive orm ula o f constitutive relation fo r plugg ed lo w,.in some problems the solutions obtained ma y be indefinite. In this paper, annular lo w andpipe lo w are discussed, and unique solution is obtained in each case by utilizing the analyticprop erty o f shear stress. The solutions are identical in fo rm with the comm only usedform ula for the pressure drop o f mud f lo w in petroleum engineering.

I . I n t r o d u c t i o nM u d is a n eces s a r y m ed i u m i n p e t r o l eu m en g i n ee ri n g . S o m e t i mes s u ch mu d ca n b e t r ea t ed a s

B i n g h am f lu i d. W h en B i n g h am f lu i d is i n mo t i o n , p l u g g ed f lo w o f t en o ccu r s a t p l ace s f a r f r o m t h eb o u n d a r y w a ll s. Th e p r o b l em o f li n k in g - u p b e t w een B i n g h am f u i d an d p l u g g ed f lo w a ri se s t h en. A sthere i s no dec i s ive fo rm ula o f cons t i t u t i ve r e l a t i on fo r p lugged f low, t he so lu t ion m ay be indef in it e .I n t h is p ap e r , u n i q u e s o l u ti o n is o b t a i n ed f o r t h e ca s e o f an n u l a r f lo w an d o f p i p e f l o w b y u t il iz i n gthe cond iti~ )n th a t a t t h e i n t e r face be twee n Bingha m f lu id and p lugged f low the tw o s t r esses shou ldb e eq u a l an d t h e t w o v e l o ci ti e s o f f lo w s h o u l d b e eq u a l . F r o m t h e u n i q u e s o l u t io n , t h e f o r m O a o f th ep r e s s u r e d r o p i n mu d f lo w can b e f o u n d u n d e r t h e ap p r o x i m a t e co n d i t i o n t h a t t h e y i e l d in g s tr e ssan d t h e an n u l a r r ad ii d i f f e r e n ce a r e s ma l l . Th e f o r m u l a o b t a i n ed is i den t ica l in f o r m w i t h t h eco m m o n l y u sed f o r m u l a in p e t r o l eu m en g i n ee r in g .H . S e v e r a l K i n d s o f t h e P r o b l e m o f L i n k l n g u p b e t w e e n B i n g h R m F l u i d a n d P l u g g e d

F l o w( a ) A n n u l a r f l o w

Th e eq u a t i o n o f mo t i o n i n t h is c a s e is

wh ere p i s t he p ressu re , o r , ,i n t eg r a ti o n w e h a v e

1 8 _O _ r c r , . ) _ . 0 ( 2 . 1 )r 8 r

i s the shear s t ress , and . ( r , O , z ) a r e cy l ind r i ca l coo rd ina tes . A f te r1 ~ )P r + c- 7 ( 2 . 2 )

197


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198 Tsa i Shu - t ang and J i ang Yi -anW h er e c is t h e co n s t an t o f in t eg r a ti o n .

F or Bingha m f lu id , l e t r0 be the y ield ing s t ress , r / be the r ig id i ty m odu lus , v~be the axialveloci ty , the n

a . . = ( r o + r T l d v .I d v ~ / d r l d r ( 2 . 3 )A t t h e i n n e r cy l i nd r ica l b o u n d a r y s u r face ,

r = R l , v z = 0A t t h e o u t e r cy l in d r ica l b o u n d a r y s u r face ,

r----Rz, v z = OAt the i nner l i nk ing po in t o f p lugged f l ow and B ingham f lu id ,

r = R ~ , v~--v.-- 0 , d r . /dr - - - -O , t r . r f=cr .~A t t h e o u t e r l in k i n g p o i n t o f p l u g g ed f lo w an d B i n g h am f lu id ,

' ~ O'P= R ~ v ~ - - - - v ~ - - - - 0 , , , ~ - ~ , ~ - - ~ mW here v ~ i s the veloci ty o f p lugge d f low, o '. r~ i s the shea r s t ress o f p lugg ed f low at the in ter face,cr,az s t he shea r s t ress o f B ingham f lu id a t t he i n t e r f ace As t he p ressu re g rad i en t i n B ingham f lu id i st he same as t h a t i n p lugged fl ow, and the r ad i i a re t he sam e a t t he i n t e r face , f rom the equa l i t y o fshear s t r ess a t t he i n t e r f ace we can co nclude t ha t t he cons t an t s o f i n t eg ra t i on c i n t he t h ree r eg ionsa r e t h e s ame , Th u s w e h av e

1 Op (RO)Z = o 1 O p ( R O ) 2r o R ~ 2 O z - - r o R ~ - - ~ - O z

Simpl i fy ing , we ge tR~ - -R~ ----- r0 / ( - -O p/ zO ) (2 .4 )

So lv ing equat ions (2 .2 ) and (2 .3 ) , we have , i n t he B ingham f lu id r eg ion ad j acen t t o t he i nnercy l inder ,

( 1 0 , )- - 1 O p ( R l _ r 2 ) + R ~ . ~ 2 O z ( R ~ r7Vz= 4 aZ l n '- ~ - , - r o ( r - R l ) ( 2 . 5 )an d i n t h e B i n g h am f lu i d r eg io n ad j acen t t o t h e o u t e r cy l in d e r ,

- - (O p 1 O p l n ~ _ r 0 ( R ~ _ r )TIv,---- 4 -0 -~ w R l - r 2 ) + - -R ~ ~ 17 6 2 O z ( 2 . 6 )The pos it i ons o f t he two li nk ing po in t s R ,~ an d R ~ a r e u n k n o w n y e t , b u t t h e r e is an o t h e rre l a t i on be tween them o the r t han (2.4) , v iz ,

flu o. ~ - - 4 - ~ O p / O z ( R ] - - ( R ~ ) ~ -- 2 ( R ~ ) ~ l n [ R ~ / R ~ ] ) - - ro ( R ~ - - R ,~ - R o In [ R ~ / R ~] )9= - - 4 - * S p / a z ( R ] - - (R ~ 2 ( R ~ ) ~ l n [ R ~ - - r o ( R ~ - R ~ - - R ~ l n [ R ~ / R , ] )

(2 .7 )F r o m (2 .4 ) an d (2 .7 ), t h e t w o u n k n o w n R ~ an d R ~ can b e s o lv ed . Th en t h e q u an t i t y o f f lo w Qan d t h e av e r ag e v e l o c i t y v ffi are g iven b y

I ? '- - ( u R I - - ~ R ; ) ~ , - - - - R , 2 u r d r v ~ + u [ ' ( R ~ ) z - - ( R ~ ) * ] v ~ + R~. 2 u r d ro , ( 2 . 8 )


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O n t h e L i n k i n g u p B e t w een B i n g h am F l u i d an d P l u g g ed F l o w 1 99Simpl i fy ing , we g e t

r / Q = :r # p I _ ( R ] _ ( R O ) 2 ) z _ ( ( R O ) Z R ])2 .18 0 z0 $ 0 2 0 2+ ~ - -~ ~ + 3 R , R , + 3 R a R 1 ]

( b ) P i p e f l o wTh e eq uat ion o f m ot ion in t h is case is sti ll (2 .1 ), nam ely

O p 1 0 ( r G , , ) -----0O z t r O--~After in tegrat ion 'we have (2 .2) again , i .e . '

( 2 . 9 )

1 0 p co 'r ,= - - r +2 rT he expres sion for o ' , , i s s ti ll

J rh e b o u n d a r y co n d i t i o n s a r e

G,= = ( v ~ d r ,[d v = / d r [ + n ) d r "

r = R , v . = 0A t t h e l i n k in g p o i n t o f B i n g h am f lu i d an d p l u g g ed f lo w ,

r = R ~ v , - - v , - - o , d v , / d r = o , a / , = a , m ,Th erefo re t he-con s t an t s o f i n t eg ra t ion in t he two reg ions a re t he same. Bu t i n t he p lugged f lowreg ion , c r ,,~ m us t be f in i te , hence c m us t be zero , Th at i s,

Th u s , w e g e t

In t he B ingham f lu id r eg ion ,

1 O p lOo 9 c 1 Op RO"" az- r 0 : 2 a z ( 2 . 1 0 )

Ro= 2 o/- (op/o=) ( 2 . 1 0 ) '

. 1 O p (rZ R Z ) + r o ( r _ R ) ,~ v , = T O zTh e q u an t i t y o f f lo w Q an d t h e av e r ag e v e l o ci ty o = a r e g iv en b y

z R 1 O p ( r Z _ R = ) + r o ( r _ R ) ] d r

(8.11)

1 o p ( R ~ - - R 2 ) + r o ( R ~+ u ( R ~ O z8 # z ( R Z - - ( R ~ 1 7 6 1 7 6

( e ) P l a n e p i p e f l o wT h e e q u a t i o n o f m o t i o n i s n o w


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200 Tsai Shu-tang and J iang Yi-an

OxOP {._8 . ~ _ o . = 0 ( 2 . 1 3 )Integrat ing, w e get

o r , , = - - - ~ - - y + c 12.1.4)In the Bingh am fluid region, the expression for o ' ,~ is

a , , = ( - v0 ( 2 . 1 5 )[ v ` / d y I + ~ 7 d O ,,dyThe b ounda ry cond i t ions a reA t t he w alls, y = + h , v , = 0At the in terface wi th plugged f low,

y=+t,; o . - v . - o , av, /ay=o, ~ , = ~ , . , , ;Therefore in the two Bingham fluid regions and the plugged flow region, the constants ofintegrat ion c are the same. Thus,

Thereby- r o = - ~ - b+c, r0

c = O , b = r0--Oq/OxT h e v e l oc i t y i s t ri b u ti o ns n t h e B i n g h a m f l u id e g i o n s a r e

1 0 p ( h ~ - - y ) + - ~ x b ( h - - Y )y ~ b , ~ v , ` = 2 0 x

(2.16)

I ap (p_y~) + ~_~_pxb(h+yy


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O n t h e L i n k i n g u p B e t w e e n B i n g ha m . F l u i d a n d P l u g g i d F l o w 2 01T h e e q u a t io n s d e t e r m i n in g R ] a n d R ~ n o w b e c o m e

" R ~ ]_ , o ~ o ~ R ~ o R o,- - R , - - ( R , ) + 2 ( R , ) I n R L - - 4 A ( - - R ~ - - R i R ~ ]., i n ( 3 . 2 )

L R ] - - R , =2;t (3.3)Before solving (3.2) and (3.3), Let us consider the case of Newtonian fluid. In this case R t coincideswith R2 and equals the radius for maximum velocity,Rm. The equation satisfied by Rm is

S o l v i n g R . , ,w e g e t

L e t

R mR '2 - - R - ' - - 2 R'~ l n ~ = R I - R ' - + 2 R ' ~ l n R ,tm ( 3 . 4 )

- R , , , = , , [ I R I - - R~ ( 3 . 5 )u 2 1 n ( R z / R t )

R ~ = R m + k z A , . R ~ = R - - - h i 2 ( 3 . 6 )T h u s

k i + k z = 2 ( 3 . 7 )W h e n A i s v e r y s m a l l , w e m a y n e g l e c t h i g h e r o r d e r t e r m s i n 2 9 S o l v i n g ( 3 . 2 ) a n d ( 3 .3 )R , - i , R , }R . - 2 + 1 " - E [ ) / ' " - ~ 7

k , = ( i n _ ~ _ + 2 R , + R ~ ) / I n R ,Rm( 3 . 8 )

I n o r d e r t o c o m p a r e w i t h th e c o m m o n l y u s e d f o r m u l a in p e t r o l e u m e n g i n e e ri n g , w e h a v e fo rt h e c a se w h e n R ~ - - R t ( ( R ~ ,

r l Q - - ~ r ( R I r - - R I ) o ~ ~r O p [ ( R { _ R , ~ ~ ( R m - - R l ) ' - l '8 0z m; --+ 6 r o [ - - 2 R ; - - 2 R ; - - R ~ ( 2 - - 3k2 "-- 3k t )+ 3 R . , R ; ( l - - k 2 ) + 3 R ,,, R I ( 1 - - k ~ ) -I ( 3 . 9 . )

A n d w e h av e 9 f R ' - - R a ) R t( .R .~ + R ~) 1 R z - - R ,4 - )1 Rz--Rt~ . R l ( 1 + -~ - Ri ) ~ ' i ( R l + R z )

2 R ~ I ( R t + R z) z, k , ' ~ k z ~ lR ~ + R ] - - 2 R ~ . ~ ( R , - - R , ) z

S i m p l i fy i n g ( 3 .9 ) b y m e a n s o f (3 .1 0 ), w e g e t

(3.10)


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202 T s a i S h u - t a n g a n d J i a n g Y i - a n

R ep l ac in g R b y d i am e t e r D ,

Op 16r/~9 4%- - - ~ , = (R z_ _R ,)~ t R , - - R ,

8 p 64~10 9 8ro- - a- --Z ( D f - - D , ) ~ + D ~ - - D ,

( b) P i p e f l o wNeg lec t ing h ighe r o rd er t e rms .in A , we h a v e n o w

(3 .11)

( 3 . 11 ) '

~ Q - - ~ z R * o , ~ or O p R ' - - z8 0 - 7 T r~ ( 3 . 1 2 )Af t e r s impl i fi ca ti on we g e t

Op 8 ~ , . 8 ro ( 3 ' 1 3 )a z = ~ 3 . RR ep l ac in g R b y d i ame t e r D ,

8 p 3 2 ~ = 1 6 ~.Oz : ~ 3 D " ( 3 . 1 3 ) "( c) P l a n e p i p e f l o w

Neg lec t ing h igher o rd er t e rms in A , we have

Simpl i fy ing, we get

~Q - - -2hr /~ ,= ap -1 2 - , 0p- " " ' J=-T

- 3%p a j ~ h~ . + _ _0 x 2 h ( 3 . 1 5 )I V . C o m p a r is o n w i t h t h e C o m m o n l y U s e d F o r m u l a

I n t h e p e t r o l eu m i n d u s tr y , t h e r i g id i ty mo d u l u s a n d t h e y i e ld i n g ~ tr e s s a r e o f t en d en o t ed b y t h er ead in g s o f v i s co s i me t e r , 0 . 0 e0 o me an s t h e v i sco s ime t e r r ead i n g a t 6 00 r p m, 0 8 0 o me an s t h ev i scos imeter r ead ing a t 300 rpm. T he r i g id i t y m odu lus (Pv)=0600- -08oo , t he y i e ld ing s t ressy -- --03~0- -(Pv) . Fo r ann u lar f l ow, the com m on ly used fo rm ula fo r p ressu re d rop iU i s

(pv)~l ylP= 60000(D~--D,) z + 200(D~--D,) (4.1)

where D~ is the diameterof the outer tube, D, the diameterof the inner tube, ~ is the averageveloc ity , I is t he well dep th , P i s t he p ressu re d rop . Fo rm ula (4.1) i s iden t i ca l i n fo rm wi th (3 .11 ) :

F o r t h e ca s e o f p i p e f lo w , t h e co m m o n f o r m u l a f o r p r e s s u r e d r o p i s

P= 90000D' 22-2-2-5~ (4.2)Formula (4.2) is also identical in form with (3.13)'.R e f e r e n c e[ 1 ] M oore , P .L . , Drill ing Practices Manual The Pet ro l eum P ub l i sh ing Co . , U .S .A . (1974).

ON THE LINKING UP BETWEEN BINGHAM FLUID AND PLUGGED FLOW(1)

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