Geothermal Well Casing Buckling

c ' ! b d b i A REPORT SAND82-0863 Unlimited Release UC-66c Printed February 1983

Euler Buckling of Geothermal Well Casing

SAND--82-0863

DE83 010292

Robert P. Rechard, Karl W. Schuler

Prepared by Sandia National Laboratories Albuquerque, New Mexico 87 185 and Llvermore, California 94550 for the United States Department of Energy under Contract DE-AC04-76DP00789

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U C - 6 6 ~

SAND82-0863

EULER BUCKLING OF GEOTHERMAL WELL C A S I N G

R. P. Rechard K. W. S c h u l e r

A p p l i e d Mechan ics D i v i s i o n S a n d i a N a t i o n a l L a b o r a t o r i e s

A lbuquerque , New M e x i c o 87185

HOTICE PORTIONS OF THIS REPQRT ARE ILLEGIELE. tt has been rcprosfuced from the best available copy to permit the broadest possible avatfability, - . -. _ . .- - .

ABSTRACT

Geo the rma l w e l l o p e r a t o r s have e x p r e s s e d c o n c e r n o v e r t h e v u l n e r a b i l i t y of u n s u p p o r t e d c a s i n g t o b u c k l i n g f r o m t h e r m a l e l o n g a t i o n . I n t h i s r e p o r t , we p r e s e n t p r e l i m i n a r y n u m e r i c a l and t h e o r e t i c a l c a l c u l a t i o n s , w h i c h i n d i c a t e t h e b u c k l i n g phenomenon s h o u l d n o t be s e r i o u s i n N-80 c a s i n g i f t h e s t r i n g i s t e n s i o n p r e l o a d e d . B u c k l i n g w o u l d b e d e t r i m e n t a l f o r ' K-55 c a s i n g . The e f f e c t o f w a l l c o n t a c t was f o u n d t o b e b e n e f i c i a l f o r c l o s e l y c o n f i n e d p i p e s t r i n g s and o f no d e t r i m e n t when h o l e gaps a r e l a r g e . The weakness o f A P I sc rew j o i n t s i n b e n d i n g a p p e a r s t o be t h e s t r u c t u r a l l i m i t a t i o n . The a n a l y s i s assumed s t r e s s e s above y i e l d c o n s t i t u t e d f a i l u r e , t h a t t h e r m a l e x p a n s i o n was s t r a i n c o n t r o l l e d , and t h a t t h e c a s i n g was c o n t i n u o u s . E x c e s s i v e i n t e r n a l p r e s s u r e i n s t a b i l i t y was i g n o r e d . The t e m p e r a t u r e v a r i a t i o n c o n s i d e r e d was be tween c e m e n t i n g c o n d i t i o n s o f 100-200°F (40-95°C) and s h u t - i n c o n d i t i o n s o f 425-450°F (220-230°C) .

CONTENTS

INTRODUCTION. . . . . . . . . . . . . . . . . . . 1

Geothermal W e l l C o n s t r u c t i o n . . . . . . . . . 1 W e l l Cas ing D e s i g n . . . . . . . . . . . . . 5 T e m p e r a t u r e Env i ronment . . . . . . . . . . . . 6 C a s i n g I n s t a b i l i t y . . . . . . . . . . . . 11

ANALYSIS. . . 1 7

T h e o r e t i c a l Model . . . . . . . . . 1 7 N u m e r i c a l Model . . . . . . . . . . . . . . . 30 A d d i t i o n o f C o n s t a n t S t r e s s . . . . . 3 3 A n a l y t i c Summary. . . . . . . . . . . . . . . . 3 3

RESULT IMPLICATIONS . . . . . . . . . . . . . . . 35

T h e r m a l l y Induced E u l e r B u c k l i n g . . . . . . . . 35 J o i n t B e h a v i o r . , . . . . . . . . . . . . . . . 37

SUMMARY AND CONCLUSIONS . . . . . . . . . . . . 3 9

REFERENCES. . . . . . . . . . . . . . . . . . . . 41

A P P E N D I X A - N o m e n c l a t u r e . . . . . . . . . . . . . 43

A P P E N D I X B - D e r i v a t i o n o f E q u a t i o n s . . . . 45

I L L U STR AT IO NS

F i g u r e Page

1. T y p i c a l Geo the rma l W e l l C o n s t r u c t i o n and C a s i n g T e m p e r a t u r e P r o f i l e . . . . . . . . . . . . 3

2. I d e a l i z e d C o n d i t i o n s C a u s i n g C a s i n g B u c k l i n g w i t h T e m p e r a t u r e E x c u r s i o n . . . . . . . . . . . . 7

3. P r e l i m i n a r y GEOTEMP C a l c u l a t i o n s o f T e m p e r a t u r e C o n d i t i o n s D u r i n g Cement ing . . . . . 9

4. P o s t u l a t e d B u c k l i n g F a i l u r e Modes: a ) L o c a l , P l a s t i c D e f o r m a t i o n , b ) E u l e r b u c k l i n g , c ) E u l e r B u c k l i n g w i t h Subsequen t W a l l C o n t a c t , and d ) H e l i c a l B u c k l i n g . . . . . . . . . . . . . 12

5. Q u a l i t a t i v e P l o t o f T e m p e r a t u r e Change Versus

6. D e f i n i t i o n o f Terms: a ) L i n e S k e t c h and

7 . Locus D e l i n e a t i n g E u l e r B u c k l i n g Reg ion : ? l o t

U n s u p p o r t e d L e n g t h D e p i c t i n g B u c k l i n g R e g i o n s . 1 4

. . . . . . . . . . . . . . b ) F r e e Body Diagram. 1 9

o f T e m p e r a t u r e Change ( A T ) V e r s u s N o r m a l i z e d U n s u p p o r t e d L e n g t h ( L / D ) . . . . . . . . . . . . . 21

8. Maximum S t r e s s ( u ) V e r s u s T e m p e r a t u r e Change ( A T ) f o r 1 3 - 3 / 8 i n c h 54.5 p p f C a s i n g Assuming U n s u p p o r t e d L e n g t h s ( L / D ) o f 50, 100, and 200 . 2 5

9. Maximum D e f l e c t i o n Versus T e m p e r a t u r e Change ( A T ) for 13-3 /8 i n c h 54.5 p p f C a s i n g Assuming U n s u p p o r t e d L e n g t h s ( L / D ) o f 50, 100, and 200 26

10. Deformed C a s i n g Shapes w i t h W a l l C o n s t r a i n t P r e d i c t e d b y MARC and T h e o r e t i c a l Models ,at a ) AT = 80'F, b) MARC R e s u l t s a t A T = 300 F, and c ) A n a l y t i c R e s u l t s a t AT = 300°F . . . . . . 27

11. Maximum S t r e s s ( u ) V e r s u s A T f o r 13 -3 /8 i n c h 54.5 p p f C a s i n g f o r U n s u p p o r t e d L e n g t h ( L / D ) o f 100 w i t h W a l l C o n t a c t : a ) A n a l y t i c Model, b ) MARC Computer Code, and c ) C o n s t a n t S t r e s s A d d i t i o n . . . . . . . . . . . . . . . . . . . . . 29

INTRODUCTION

D r S l l i n g f o r g e o t h e r m a l e n e r g y began as e a r l y as t h e 1 9 2 0 ' s i n t h e Geysers f i e l d i n n o r t h e r n C a l i f o r n i a , b u t a s e r i o u s e f f o r t t o h a r n e s s g e o t h e r m a l e n e r g y f o r power g e n e r a t i o n was n o t begun i n t h e U n i t e d S t a t e s u n t i l t h e 1 9 7 0 ' s . On a n a t i o n a l s c a l e , t h e r e i s t h e g e o l o g i c p o t e n t i a l t o d e v e l o p 20,000 MW o f e l e c t r i c a l e n e r g y . The g e o t h e r m a l e n e r g y i n d u s t r y p e r f o r m a n c e i n t h e l a s t 1 0 y e a r s and t h e g e o l o g i c p r o s p e c t s i n d i c a t e t h e i n d u s t r y has t h e p o t e n t i a l f o r g r o w t h and can make a c o n t r i b u t i o n i n s u p p l y i n g t h e e n e r g y needs o f t h e n a t i o n .

T h e r e a r e numerous s i m i l a r i t i e s be tween c o n v e n t i o n a l o i l and gas w e l l s and g e o t h e r m a l w e l l s i n c o n s t r u c t i o n and o p e r a t i o n . However, i m p o r t a n t d i f f e r e n c e s do e x i s t ( w h e t h e r f r o m d r y steam, d r y h o t r o c k , h o t w a t e r , o r g e o p r e s s u r i t e d f l u i d r e s e r v o i r s ) . F l u i d f l o w r a t e s a r e an o r d e r o f m a g n i t u d e l a r g e r t h a n i n t h e p e t r o l e u m i n d u s t r y . The h i g h t e m p e r a t u r e s e n c o u n t e r e d a f f e c t t h e d r i l l b i t , d r i l l i n g mud and t h e cement p e r f o r m a n c e . R e s e r v o i r c a l c u l a t i o n s must i n c l u d e an e n e r g y b a l a n c e as w e l l as a mass b a l a n c e . F i n a l l y , d i f f i c u l t g e o l o g y , c o r r o s i v e e n v i r o n m e n t s , and t h e r m a l s t r e s s e s i n d u c e d i n t h e w e l l c a s i n g p r e s e n t t h e c a s i n g d e s i g n e r w i t h a new s e t o f f a i l u r e modes t o c o n s i d e r .

Geo the rma l W e l l C o n s t r u c t i o n

T h i s i n t r o d u c t i o n i s i n t e n d e d t o p r o v i d e t h e r e a d e r u n f a m i l i a r w i t h g e o t h e r m a l w e l l c a s i n g d e s i g n and c o n s t r u c t i o n n e c e s s a r y b a c k g r o u n d i n f o r m a t i o n . However, i t a l s o s e r v e s t o r e m i n d t h e r e a d e r t h a t a l t h o u g h t h e c a s i n g s e l e c t i o n i s based on t h e w o r s t c a s e d e s i g n c r i t e r i a f r o m b u r s t , c o l l a p s e , t e n s i o n , e t c . , s t r e s s e s f rom many d i f f e r e n t l o a d s can be

p r e s e n t s i m u l t a n e o u s l y -and c o n t r i b u t e t o c a s i n g f a i l u r e . More comprehens ive d i s c u s s i o n s o f t h e v a r i o u s f a c t s o f Geothermal w e l l s a r e a v a i l a b l e (e.g. Edwards e t a l . , 1982) .

b e used f o r d i s c u s s i o n . T e m p e r a t u r e p r o f i l e s o f t h e c a s i n g and u n d i s t u r b e d f o r m a t i o n a r e a l s o shown. F i g u r e 1 c o n t a i n s w e l l f e a t u r e s f r o m s e v e r a l t y p e s o f g e o t h e r m a l f i e l d s and t h u s c a n n o t t r u l y be c l a s s i f i e d as l l t yp ica l . l l

a r e d i r e c t i o n a l l y d r i l l e d . A p p r o p r i a t e d r i l l s i t e s a r e d i f f i c u l t t o l o c a t e i n t h e r o u g h t e r r a i n o f t e n f o u n d above g e o t h e r m a l f i e l d s . I t i s a l s o d e s i r a b l e t o d r i l l a t an a n g l e t o i n t e r s e c t more f r a c t u r e s . Geothermal r e s e r v o i r s f r e q u e n t l y o c c u r i n f r a c t u r e d r e s e r v o i r s b e l o w 3000 f t ( 9 0 0 m ) ; hence f r a c t u r e s a r e p r i m a r i l y v e r t i c a l .

Most g e o t h e r m a l r e s e r v o i r s a r e b e l o w t h e d r i l l mud h y d r o s t a t i c p r e s s u r e w h i c h causes l o s t c i r c u l a t i o n p r o b l e m s d u r i n g d r i l l i n g and cemen t ing . A l s o , l o w g e o t h e r m a l r e s e r v o i r p r e s s u r e s make d e t e c t i o n o f s team or h o t w a t e r b e a r i n g f r a c t u r e s d i f f i c u l t . The use o f a i r r e d u c e s t h e d r P l l f l u i d d e n s i t y and g r e a t l y speeds up d r i l l i n g . However, t h e d r i l l b i t l i f e i s g r e a t l y r e d u c e d because o f t h e h i g h t e m p e r a t u r e s e n c o u n t e r e d . The n e a r s o n i c v e l o c i t i e s p r o d u c e d w h i l e c a r r y i n g t h e c u t t i n g s up t h e o u t s i d e o f t h e d r i l l p i p e a l s o causes e x c e s s i v e e r o s i o n o f t h e d r i l l p i p e .

c o n d u c t o r p i p e , s u r f a c e c a s i n g , i n t e r m e d i a t e c a s i n g and p r o d u c t i o n c a s i n g . The p r o d u c t i o n c a s i n g i s o f t e n s e t as a p r o d u c t i o n l i n e r w i t h a t i e b a c k . P r o d u c t i o n c a s i n g and p r o d u c t i o n l i n e r s a r e d e s i g n e d w i t h t h e same c r i t e r i o n as i n t e r m e d i a t e c a s i n g and d r i l l i n g l i n e r s e x c e p t t h a t c o n s i d e r a t i o n o f d r i l l i n g wear i s n o t r e q u i r e d . The w e l l c o n s t r u c t i o n d i f f e r s s l i g h t l y f r o m c o n v e n t i o n a l o i l w e l l s i n t h a t each c a s i n g i s cemented t o t h e s u r f a c e .

i n s t a l l e d . I t a i d s i n p r e v e n t i n g washouts a round t h e d r i l l

F i g u r e 1 shows a s c h e m a t i c o f a g e o t h e r m a l w e l l w h i c h w i l l

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

The s t a n d a r d components o f t h e w e l l c a s i n g p r o g r a m a r e

C o n d u c t o r p i p e i s t h e f i r s t s t r i n g o f p i p e t o be

GEOTHERMAL WELL SCHEMATIC AND CASING TEMPERATURE

F i g u r e 1.

THE SURFACE

INTERMEDIATE

13-3/8 INCH

BUTTRESS JOINTS

Q526'F

- 350°F-

TEMPERATURE PROFILES

0 -UNDISTURBED FORMATION

-CEMENT-SET I' TEMPERATURE

0 -0PERATINa CASING TEMPERATURE

-SHUT-IN CASING TEMPERATURE

L 100 200 300 400

L l 500

J TEMPERATUREOF

T y p i c a l Geo the rma l We1 1 C o n s t r u c t T e m p e r a t u r e P r o f i l e .

on and C a s i n g

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

and h e l p s s u p p o r t w e l l head equ ipmen t . C o n d u c t o r p i p e i s s e t s h a l l o w and i s n o t u s u a l l y c o n s i d e r e d a p r e s s u r e s t r i n g .

The s u r f a c e c a s i n g i s t h e f i r s t t r u e c a s i n g s t r i n g . As a p r i m a r y s t r u c t u r a l member i t p r o v i d e s s u p p o r t f o r subsequen t c a s i n g s t r i n g s . To a v o i d b u c k l i n g p r o b l e m s f r o m t h e c o m p r e s s i v e l o a d s a p p l i e d , i t i s o f t e n cemented t o t h e s u r f a c e even i n c o n v e n t i o n a l w e l l s . S u r f a c e c a s i n g must a l s o p r o v i d e s u f f i c i e n t h o l e s t a b i l i t y , p r o t e c t i o n t o a q u i f e r s , s o l i d s u p p o r t f o r t h e r e s e r v o i r p r e s s u r e , and p r e s s u r e i n t e g r i t y i n t h e e v e n t o f a b r u p t p r e s s u r e i n c r e a s e s ( b l o w o u t s and k i c k s ) . S u r f a c e c a s i n g i s s u b j e c t e d t o d r i l l i n g wear w h i c h r e q u i r e s h e a v y c a s i n g . Common s e t t i n g d e p t h s a r e be tween 1000 and 2500 f t (300 t o 760 m).

I n a c o n v e n t i o n a l w e l l , i n t e r m e d i a t e c a s i n g can be exposed t o h i g h b o t t o m h o l e p r e s s u r e s w h i c h r e q u i r e s s u b s t a n t i a l b u r s t r e s i s t a n c e . H i g h c o l l a p s e r e s i s t a n c e i s a l s o r e q u i r e d f o r t h e deeper c a s i n g . Heavy muds and cement s l u r r i e s r e q u i r e d f o r deep d r i l l i n g can c r e a t e h i g h c o l l a p s e l o a d s s h o u l d l o s t c i r c u l a t i o n zones empty t h e p i p e . These c o n d i t i o n s d i c t a t e h e a v y c a s i n g . As w i t h s u r f a c e s t r i n g s , i n t e r m e d i a t e c a s i n g and d r ill i n g 1 i n e r s a r e s u b j e c t e d t o m e c h a n i c a l damage f r o m d r i l l i n g wear.

v a l u e s i n many g e o t h e r m a l f i e l d s . However, a s t a n d a r d c a s i n g p r o g r a m i n t h e p r o m i n e n t Geysers g e o t h e r m a l f i e l d c o n s i s t s o f 26 a n d / o r 20 i n c h ( 6 6 0 o r 508 mm) d i a m e t e r c o n d u c t o r p i p e , 1 3 - 3 / 8 i n c h ( 3 4 0 mm) s u r f a c e c a s i n g , and 9 - 5 / 8 i n c h ( 2 4 4 mm) i n t e r m e d i a t e c a s i n g or l i n e r ( w i t h o r w i t h o u t a t i e b a c k s t r i n g o f e i t h e r 9 - 5 / 8 o r 10 -3 /4 i n c h (244 o r 273 mm) c a s i n g )

(Capuano, 1979) . Because s u p e r h e a t e d s team i s p roduced, a p r o d u c t i o n c a s i n g i s n o t needed. An open h o l e i n t h e r e s e r v o i r i s u s u a l l y s t a b l e .

The c a s i n g s i z e s shown i n F i g u r e 1 a r e commonly s e l e c t e d

W e l l C a s i n g Des& - --

The p r o p e r s e l e c t i o n o f t h e t y p e , s i z e , and s e t t i n g d e p t h o f t h e w e l l c a s i n g i s based on t h e e x p e c t e d w e l l o p e r a t i o n c o n d i t i o n s and t h e d r i l l i n g s i t e g e o l o g y . The u s u a l p r a c t i c e i s t o c o n s i d e r t h e w o r s t case o r maximum l o a d i n d e t e r m i n i n g t h e r e q u i r e d c a s i n g c o n f i g u r a t i o n . n e s t i n g a r e u s u a l l y i g n o r e d . A l i s t o f c a s i n g f a i l u r e modes i n c l u d e s (Snyder , 1 9 7 9 ) :

C o m p l i c a t i o n s due t o c a s i n g

* M e t a l f a i l u r e : b u r s t , c o l l a p s e , t e n s i o n , o r c o r r o s i o n ,

* M e c h a n i c a l damage: d r i l l p i p e wear, w e l d i n g p rob lems , t h r e a t damage, o r l e a k a g e and p e r f o r a t i o n ,

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

*Cement f a i l u r e s : v o i d s f r o m l o s t c i r c u l a t i o n zones' o r cement t o o l p rob lems , cement d i s s o l u t i o n and c o r r o s i o n p e r m i t t i n g f l u i d movement between c a s i n g and f o r m a t i o n , o r p o o r h i g h - t e m p e r a t u r e s l u r r y b e h a v i o r ,

f a i l u r e s ( t e l e s c o p i n g ) , l e a k a g e i n c o u p l i n g s f r o m c y c l i c l o a d i n g , e x c e s s i v e b e n d i n g l o a d s i n dog l e g s , s t r a i n beyond u l t i m a t e .

T h i s t a b u l a t i o n p r e s e n t s p o s s i b l e f a i l u r e modes.

*Thermal s t r e s s f a i l u r e s : c o m p r e s s i o n a n d / o r t e n s i o n

U n f o r t u n a t e l y l i t t l e d e t a i l e d p u b l i c i n f o r m a t i o n e x i s t s on g e o t h e r m a l w e l l c a s i n g f a i l u r e s . The a n a l y s t can o n l y p o s t u l a t e t y p e s and f a i l u r e mechanisms and t h u s t h e danger e x i s t s t h a t an i m p o r t a n t o r more l i k e l y f a i l u r e mechanism has been o v e r l o o k e d .

i n d e p e n d e n t . For--example, a cement f a i l u r e c o u l d cause i n s u f f i c i e n t l a t e r a l s u p p o r t and r e s u l t i n c a s i n g i n s t a b i l i t y when h i g h i n t e r n a l p r e s s u r e s o c c u r r e d . The r e s u l t i n g l a t e r a l d e f l e c t i o n c o u l d i n t u r n r e s u l t i n e x c e s s i v e d r i l l p i p e wear d u r i n g t h e d r i l l i n g o p e r a t i o n and subsequen t b u r s t o f t h e c a s i n g d u r i n g t h e p r o d u c t i o n o p e r a t i o n .

I t s h o u l d be n o t e d t h a t t h e f a i l u r e modes l i s t e d a r e n o t

I n o i l o r gas w e l l c a s i n g d e s i g n , t h e m a j o r c o n c e r n a d d r e s s e d i s m e t a l f a i l u r e f r o m b u r s t , c o l l a p s e , o r t e n s i o n . However, t h e p r e s e n c e o f t h e r m a l l o a d s i n g e o t h e r m a l w e l l

c a s i n g g r e a t l y i n c r e a s e s t h e o p p o r t u n i t y f o r c a s i n g i n s t a b i l i t y . C a s i n g s t a b i l i t y can b e i m p r o v e d by: 1 ) c e m e n t i n g t h e e n t i r e s t r i n g t o p r o v i d e l a t e r a l s u p p o r t o r 2 ) a p p l y i n g a t e n s i o n l o a d i n t h e uncemented s e c t i o n s . F u l l y c e m e n t i n g t h e c a s i n g s t r i n g i s t h e u s u a l c h o i c e . U n f o r t u n a t e l y , p o o r f o r m a t i o n c o n d i t i o n s f r e q u e n t l y e x i s t i n g e o t h e r m a l a reas . The r e s e r v o i r i s u s u a l l y b e l o w h y d r o s t a t i c p r e s s u r e and can be h i g h l y f r a c t u r e d . C o n s e q u e n t l y , l o s t c i r c u l a t i o n w h i l e d r i l l i n g w i t h mud o r c e m e n t i n g c a s i n g i s common. I t i s t h u s i m p o s s i b l e t o e n s u r e a c o m p l e t e cement j o b i n many i n s t a n c e s . F a i l u r e o f s t a g e c e m e n t i n g t o o l s i n g e o t h e r m a l w e l l s i s f r e q u e n t and a l s o c r e a t e s u n s u p p o r t e d t u b u l a r s e c t i o n s (Snyder , 1979). B u c k l i n g f a i l u r e s o f t h e c a s i n g f r o m t h e r m a l e x p a n s i o n where cement f a i l u r e s h a v e o c c u r r e d i s t h e s u b j e c t o f t h i s r e p o r t ( F i g u r e 2 ) .

TemDera tu re E n v i r o n m e n t

The t e m p e r a t u r e e n v i r o n m e n t i s i m p o r t a n t i n f o r m a t i o n f o r t h e t h e r m a l a n a l y s i s . f i g u r e 1 p r e s e n t s a h y p o t h e t i c a l t e m p e r a t u r e e n v i r o n m e n t . The s u r f a c e and b o t t o m h o l e t e m p e r a t u r e s a r e as s u r m i s e d b y t h e w e l l o p e r a t o r s i n The Geysers f i e l d (Pye, 1980; J e n k i n s and Snyder , 1979) , b u t t h e a c t u a l t e m p e r a t u r e p r o f i l e s t h r o u g h o u t t h e s t r a t i g r a p h y and c a s i n g a r e unknown. I n F i g u r e 1 c a s i n g t e m p e r a t u r e s a r e assumed t o v a r y l i n e a r l y . The u n d i s t u r b e d f o r m a t i o n p r o f i l e i s shown w i t h one e lbow. A f e w p r o f i l e s a v a i l a b l e f r o m The Geysers f i e l d c o n t a i n t w o k i n k s : t h e second e lbow o c c u r s w i t h i n t h e f i r s t 500 f t (150 m ) .

F o r w e l l s c o m p l e t e d i n l o w - p r e s s u r e h o t - w a t e r o r s team r e s e r v o i r s , t h e c a s i n g s a r e t h o u g h t t o b e cemented a t a t e m p e r a t u r e be tween 100-2OO'F (40-95°C). T h i s assumes t h e

HOT WATER, STEAM

CEMENT SHEATH

ENLARGED HOL

PIPE DIAMETER

ure 2 . Ideal ized Conditions Causing Casing Buck Temperature Excursion.

, l i n g with

c a s i n g i s n o t p u r p o s e l y a l l o w e d t o h e a t up b e f o r e c e m e n t i n g . Upon c o m p l e t i o n , t h e w e l l i s t e m p e r a t u r e c y c l e d be tween p r o d u c i n g c o n d i t i o n s o f a p p r o x i m a t e l y 325-400°F (160-205°C) and s h u t i n c o n d i t i o n s o f 425-450°F (220-235°C). The c y c l i n g i s due t o a i r p o l l u t i o n s t a n d a r d s w h i c h l i m i t t h e v e n t i n g o f g e o t h e r m a l w e l l s . C y c l i n g can o c c u r 2 t o 3 t i m e s p e r week if t h e steam c o n t a i n s a p o l l u t a n t such as h y d r o g e n s u l f i d e ( H 2 S ) . When t h e w e l l r e q u i r e s r e m e d i a l work, t h e c a s i n g t e m p e r a t u r e i s r e d u c e d t o a r o u n d 100°F (40°C) w i t h c o o l w a t e r . These a r e a p p r o x i m a t e v a l u e s o n l y .

A t e m p e r a t u r e p r o f i l e i s v e r y u s e f u l i n v i s u a l i z i n g t h e t e m p e r a t u r e change t o w h i c h each t y p e o f c a s i n g i s s u b j e c t e d . A c c u r a t e i n f o r m a t i o n o f t h i s t y p e w o u l d g r e a t l y a i d t h e d e s i g n and a n a l y s i s o f t h e c a s i n g i n t e g r i t y . As seen i n F i g u r e 1, t h e c a s i n g can be s u b j e c t e d t o l a r g e t e m p e r a t u r e changes. C o n s e q u e n t l y l a r g e t h e r m a l s t r e s s e s must b e a n t i c i p a t e d . I t i s seen t h a t t h e more s e v e r e t e m p e r a t u r e changes o c c u r n e a r t h e s u r f a c e d u r i n g t h e c y c l i n g be tween p r o d u c t i o n and s h u t - i n . However, t h e w h o l e c a s i n g s t r i n g can be s u b j e c t e d t o l a r g e

t e m p e r a t u r e changes a f t e r c e m e n t i n g and whenever t h e w e l l must b e quenched.

An a c c u r a t e cemen t -se t t e m p e r a t u r e i s e s s e n t i a l t o t h e t h e r m a l s t r e s s a n a l y s i s because t h i s i s t h e t e m p e r a t u r e t h e c a s i n g becomes c o n s t r a i n e d . The GEOTEMP compu te r p r o g r a m (Wooley, 1980; M i t c h e l l , 1982) b e i n g d e v e l o p e d under c o n t r a c t t o S a n d i a w i l l be h e l p f u l i n more c a r e f u l l y d e f i n i n g t h e t e m p e r a t u r e r e g i m e o f t h e w e l l c a s i n g . P r e l i m i n a r y GEOTEMP t e m p e r a t u r e c a l c u l a t i o n s a r e shown i n F i g u r e 3. R a d i a l t e m p e r a t u r e s a t 200 f t ( 6 0 m ) d e p t h u n d e r t h r e e g e o t h e r m a l f l u i d f l o w c o n d i t i o n s a r e d e p i c t e d f o r a G e y s e r s w e l l . The c e m e n t i n g c o n d i t i o n s a r e l o w e r t h a n g e n e r a l l y assumed b y o p e r a t o r s . V e r i f i c a t i o n o f t h e GEOTEMP p r o g r a m i s n o t c o m p l e t e , b u t t h e t e m p e r a t u r e d i f f e r e n c e shown c o u l d be s i g n i f i c a n t and needs t o be more c a r e f u l l y examined.

PROFILES TAKEN FROM GEOTEMP ANALYSIS

100 1 I t I I

20 INCH 13 3/8 INCH

B 5/8 INCH u

90 t -

06 h INJECTION COOLING 250 gal/rnin 0 3 h SHUT - IN AFTER INJECTION

A 5 h CONDITIONING 400 gal/rnin

2 h CEMENTING A n 10' I I I I

r (FEET)

RADIAL TEMPERATURES AT 200 FOOT DEPTH

Figure 3 . Preliminary GEOTEMP Calculat ions o f Temperature Conditions During Cementing.

c a s i n g i s n o t p u r p o s e l y a l l o w e d t o h e a t up b e f o r e cemen t ing .

Upon c o m p l e t i o n , t h e w e l l i s t e m p e r a t u r e c y c l e d be tween p r o d u c i n g c o n d i t i o n s o f a p p r o x i m a t e l y 325-400°F (160-205°C) and s h u t i n c o n d i t i o n s o f 425-450°F (220-235°C). The c y c l i n g i s due t o a i r p o l l u t i o n s t a n d a r d s w h i c h l i m i t t h e v e n t i n g o f g e o t h e r m a l w e l l s . C y c l i n g can o c c u r 2 t o 3 t i m e s p e r week i f t h e s team c o n t a i n s a p o l l u t a n t such as h y d r o g e n s u l f i d e (H2S). When t h e w e l l r e q u i r e s r e m e d i a l work, t h e c a s i n g t e m p e r a t u r e i s r e d u c e d t o a r o u n d 100°F (40'C) w i t h c o o l w a t e r . These a r e a p p r o x i m a t e v a l u e s o n l y .

A t e m p e r a t u r e p r o f i l e i s v e r y u s e f u l i n v i s u a l i z i n g t h e t e m p e r a t u r e change t o w h i c h each t y p e o f c a s i n g i s s u b j e c t e d . A c c u r a t e i n f o r m a t i o n o f t h i s t y p e w o u l d g r e a t l y a i d t h e d e s i g n and a n a l y s i s o f t h e c a s i n g i n t e g r i t y . As seen i n F i g u r e 1, t h e c a s i n g can be s u b j e c t e d t o l a r g e t e m p e r a t u r e changes. C o n s e q u e n t l y l a r g e t h e r m a l s t r e s s e s must be a n t i c i p a t e d . I t i s seen t h a t t h e more s e v e r e t e m p e r a t u r e changes o c c u r n e a r t h e s u r f a c e d u r i n g t h e c y c l i n g be tween p r o d u c t i o n and s h u t - i n . However, t h e w h o l e c a s i n g s t r i n g can be s u b j e c t e d t o l a r g e t e m p e r a t u r e changes a f t e r c e m e n t i n g and whenever t h e w e l l must be quenched.

An a c c u r a t e cement -se t t e m p e r a t u r e i s e s s e n t i a l t o t h e t h e r m a l s t r e s s a n a l y s i s because t h i s i s t h e t e m p e r a t u r e t h e

c a s i n g becomes c o n s t r a i n e d . The GEOTEMP compu te r p r o g r a m (Wooley, 1980; M i t c h e l l , 1982) b e i n g d e v e l o p e d under c o n t r a c t t o Sand ia w i l l be h e l p f u l i n more c a r e f u l l y d e f i n i n g t h e t e m p e r a t u r e r e g i m e o f t h e w e l l c a s i n g . P r e l i m i n a r y GEOTEMP t e m p e r a t u r e c a l c u l a t i o n s a r e shown i n F i g u r e 3. R a d i a l

t e m p e r a t u r e s a t 200 f t . ( 6 0 m ) d e p t h under t h r e e g e o t h e r m a l f l u i d f l o w c o n d i t i o n s a r e d e p i c t e d f o r a G e y s e r s w e l l . The c e m e n t i n g c o n d i t i o n s a r e l o w e r t h a n g e n e r a l l y assumed b y o p e r a t o r s . V e r i f i c a t i o n o f t h e GEOTEMP p r o g r a m i s n o t comp le te , b u t t h e t e m p e r a t u r e d i f f e r e n c e shown c o u l d be s i g n i f i c a n t and needs t o be more c a r e f u l l y examined.

W h i l e f a i l u r e s i n cemented s t r i n g s such as c o m p r e s s i o n a n d / o r t e n s i o n f a i l u r e s and c o n n e c t i o n f a i l u r e s a r e o f c o n c e r n , o p e r a t o r s h a v e e x p r e s s e d g r e a t e r c o n c e r n o v e r c a s i n g buck1 i n g i n p a r t i a l l y cemented s t r i n g s (Pye, 1980; Kumataka, 1981, Snyder , 1979) . As r e g a r d s p a r t i a l l y cemented s t r i n g s , work i n t h e a r c t i c o i l f i e l d s has shown t h a t t h e cement a n d l o r f o r m a t i o n s u p p o r t needed t o a v o i d b u c k l i n g f r o m s u b s i d e n c e i s q u i t e s m a l l ( W i l s o n e t a l . , 1980) . ( B o t h subs ' idence and t h e r m a l s t r e s s l o a d s a r e s t r a i n c o n t r o l l e d . ) , Because l i t t l e l a t e r a l s u p p o r t i s n e c e s s a r y , b u c k l i n g i s l i m i t e d t o a reas where f o r m a t i o n c o n d i t i o n s cause e n l a r g e d h o l e s t o f o r m w i t h s u b s e q u e n t v o i d s i n t h e cement s h e a t h such t h a t a c o m p l e t e l y u n s u p p o r t e d s e c t i o n o c c u r s ( F i g u r e 2 ) .

p a r t i a l l y cemented s t r i n g s can be d i v i d e d i n t o f o u r c a t e g o r i e s . The f a i l u r e t y p e i s dependent on t h e u n s u p p o r t e d c a s i n g l e n g t h and i n t e r n a l - e x t e r n a l p r e s s u r e i n t e r a c t i o n ( F i g u r e 4) . The c a t e g o r i e s a r e :

C a s i n g i n s t a b i l i t y f a i l u r e s f r o m a t h e r m a l l o a d i n

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

~ E u l e r buck1 i n g ,

C o n s t r a i n e d E u l e r b u c k l i n g f o l l o w e d b y p l a s . t i c d e f o r m a t i o n

*He1 i c a l b u c k l i n g .

o r c o l l a p s e due t o o v a l a t i o n , ' .

I t i s i m p o r t a n t t o emphas ize t h e d i f f e r e n c e between s t a n d a r d co lumn b u c k l i n g under an a p p l i e d . l o a d a b u c k l i n g f r o m t h e r m a l f a r c e s where s u p p o r t o f - a f o l l o w e r a x i a l l o a d i s n o t r e q u i r e d . R a t h e r t h a n c a t a s t r o p h i c . 5 f a i l u v e f r o m a c r i t i c a l t e m p e r a t u r e change, t h e c a s i n g s l o w l y de fo rms e l a s t i c a l l y i n t o t h e de fo rmed shape f o r l a r g e + u n s u p p o r t e d

l e n g t h s . Thus co lumn " b e n d i n g " i s a more a p p r o p r i a t e

d e s c r i p t i o n o f t h e phenomenon. The p i p e s t r i n g i n s t a b i l i t y m a n i f e s t s i t s e l f as a l a t e r a l d e f l e c t i o n .

7 SHORT

UNSUPPORTED

a) LOCAL, PLASTIC DEFORMATION

c) EULER BUCKLING WITH SUBSEQUENT WALL CONTACT (PLASTIC DEFORMATION OR COLLAPSE DUE TO OVALATION POSSIBLE)

I LONG

LENGTH UNSUPPORTED

EULER BUCKLING

HELICAL BUCKLING

Figure 4 . Postulated Buckling Fai lure Modes: a ) Local, P l a s t i c Deformation, b ) Euler Buckling, c ) Euler Buckling with Subsequent Wall Contact, a n d d ) Hel ica l Buckling.

The r e s u l t i n g d e f o r m a t i o n may n o t impede o p e r a t i o n s i f t h e d e f o r m a t i o n i s s l i g h t . The l a r g e s t t h e r m a l s t r e s s e s a r e i n t r o d u c e d d u r i n g s h u t - i n a f t e r t h e w e l l i s comp le ted , t h u s t h e danger o f e x c e s s i v e p i p e wear d u r i n g d r i l l i n g has passed. However, even s l i g h t b e n d i n g a t c o n n e c t i o n s can r e s u l t i n j o i n t f a f l u r e because s t a n d a r d Amer i can P e t r o l e u m I n s t i t u t e ( A P I ) j o i n t s a r e n o t d e s i g n e d t o w i t h s t a n d b e n d i n g s t r e s s e s .

F i g u r e 5 q u a l i t a t i v e l y i n d i c a t e s where v a r i o u s buck1 i n g modes o c c u r . I t i s i m p o r t a n t t o n o t e t h a t i n t e r n a l - e x t e r n a l p r e s s u r e i n t e r a c t i o n has been i g n o r e d . O n l y u n s u p p o r t e d l e n g t h and t e m p e r a t u r e was c o n s i d e r e d .

d e f o r m a t i o n a n d / o r c o l l a p s e w o u l d be e x p e c t e d . A t l o n g e r u n s u p p o r t e d l e n g t h s , E u l e r b u c k l i n g w o u l d o c c u r . W i t h c o n t i n u e d t e m p e r a t u r e i n c r e a s e , t h e c a s i n g c o u l d d e f l e c t enough t o c o n t a c t t h e d r i l l h o l e s i d e s . P l a s t i c d e f o r m a t i o n o r p i p e c o l l a p s e f r o m t h e weaken ing e f f e c t s o f c r o s s - s e c t i o n o v a l a t i o n c o u l d f o l l o w .

t h e o i l w e l l i n d u s t r y , t h e c o r k s c r e w i n g i s due p r i m a r i l y t o e x c e s s i v e , d e s t a b i l i z i n g , i n t e r n a l p r e s s u r e s ( L u b i n s k i e t a1 ., 1962) . F r e q u e n t l y , t h e d e f o r m a t i o n i s n o t s e v e r e enough t o cause pe rmanen t d e f o r m a t i o n ( T e x t e r , 1955).. Because l o n g u n s u p p o r t e d l e n g t h s a r e much l e s s l i k e l y and t h e u l t i m a t e f a i l u r e mechanism i s s i m i l a r t o t h a t e n c o u n t e r e d w i t h s i n g l e o r d e r E u l e r b u c k l i n g , t h i s r e g i o n i s o f l e s s i n t e r e s t .

t h e r m a l b u c k l i n g and l o c a l i z e d p l a s t i c d e f o r m a t i o n o f g e o t h e r m a l c a s i n g . F i r s t , a n a l y s i s o f t h e E u l e r b u c k l i n g r e g i m e assuming b u i l t - i n ends and s u b s e q u e n t e l a s t i c - p l a s t i c b e n d i n g needs t o be examined. A n a l y s i s o f n e s t e d c a s i n g

b e h a v i o r when c o n s t r a i n e d b y cement a n d / o r f o r m a t i o n s c o u l d a l s o be i n v e s t i g a t e d more t h o r o u g h l y . Second, a n a l y s i s o f

l o c a l i z e d p l a s t i c d e f o r m a t i o n s such as s y m m e t r i c a l b u c k l i n g and w r i n k l i n g i n s t a b i l i t i e s needs t o be examined. Smal l s c a l e

F o r s h o r t u n s u p p o r t e d l e n g t h s o n l y l o c a l i z e d p l a s t i c

H e l i c a l b u c k l i n g o c c u r s i n l o n g u n s u p p o r t e d l e n g t h s . I n

Two b a s i c s u b j e c t a r e a s need t o b e i n v e s t i g a t e d c o n c e r n i n g

QUALITATIVE DESCRIPTION OF VARIOUS BUCKLING MODES

I- <I FAILURE ZONE

FAILURE WITH WALL CONTACT

FAILURE WITHOUT BUCKLING ZONE WALL CONTACT

HELICAL I- BUCKLING ZONE

NO ADVERSE DEFORMATION

UNSUPPORTED LENGTH, L

F i g u r e 5. Q u a l i t a t i v e P l o t o f T e m p e r a t u r e Change Versus U n s u p p o r t e d L e n g t h D e p i c t i n g Buck1 i n g Reg ions .

laboratory t e s t s of thermally-induced buckling should a l s o be conducted t o enhance the understanding o f the phenomenon.

This report q u a n t i t a t i v e l y de f ines the Euler buckling regime f o r a c a s i n g with f i x e d ends with a n d without subsequent hole wall contac t .

1 5 : :

ANALY S IS

T h e o r e t i c a l Model -4 - -- - 6

----- ASsumpt ions. The down h o l e e f f e c t s o f i m p r o p e r c e m e n t i n g may be m a n i f e s t e d i n many ways. However, w i t h o u t d e t a i l e d f i e l d o r l a b o r a t o r y d a t a on g e o t h e r m a l c a s i n g b e h a v i o r o r c a s i n g f a i l u r e s , t h e r e i s l i t t l e need t o s h a r p l y f o c u s on one s u b j e c t a r e a . T h e r e f o r e a s i m p l e a n a l y t i c and n u m e r i c a l model i s p r e s e n t e d i n o r d e r t o g a i n i n s i g h t i n t o t h e prob lem. The a n a l y s i s assumed:

1 ) t h e c a s i n g was i n i t i a l l y v e r t i c a l ( b o d y f o r c e s i g n o r e d ) and c e m e n t i n g above and be low t h e u n s u p p o r t e d c a s i n g p r o v i d e d f i x e d - e n d c o n d i t i o n s ,

2 ) c o m p l i c a t i o n s f r o m c o u p l i n g s such as changes i n moment o f i n e r t i a ( I ) and weakness i n b e n d i n g were un i m p o r t a n t ,

3 ) c a s i n g s t r e s s e s r e m a i n e d i n t h e e l a s t i c r e g i o n and t h e modu lus o f e l a s t i c i t y (E) was i n d e p e n d e n t of t e m p e r a t u r e and e q u a l t o 2 9 x l o 6 p s i (200 GPa),

4 ) t h e l i n e a r t h e r m a l e x p a n s i o n c o e f f i c i e n t ( a ) was c o n s t a n t and e q u a l t o 6.5 x 10-6'F (1 .2 x lO-5'C),

5 ) c o m p l i c a t i o n s due t o c a s i n g n e s t i n g were n e g l i g i b l e ,

6 ) t h e i n t e r n a l and e x t e r n a l t i t b u l a r p r e s s u r e s were e q u a l ,

7 ) c r o s s - s e c t i o n a l shape changes ( o v a l a t i o n ) due t o l a t e r a l and b e n d i n g f o r c e s were u n i m p o r t a n t .

A s s u m p t i o n s 5, 6, and 7 were j u s t i f i e d b y t h e f o l l o w i n g f a c t s : F i r s t , c a s i n g n e s t i n g i s r a r e l y c o n s i d e r e d i n a c t u a l d e s i g n a p p l i c a t i o n s . Second, c a s i n g o v a l a t i o n p rob lems a t t h e w a l l c o n t a c t were t h o u g h t m i n o r i f s t r e s s e s r e m a i n e d b e l o w y i e l d . The n u c l e a r r e a c t o r i n d u s t r y has been a d d r e s s i n g t h i s

p r o b l e m t o some e x t e n t . F i n a l l y f o r l o w p r e s s u r e , h o t w a t e r , o r s team r e s e r v o i r s , t h e c a s i n g i n t e r n a l p r e s s u r e i s l i k e l y t o b e l e s s t h a n e x t e r n a l f o r m a t i o n p r e s s u r e s ; hence n e g l e c t i n g i n t e r n a l - e x t e r n a l p r e s s u r e i n t e r a c t i o n was f e l t j u s t i f i e d * ( L u b i n s k i e t a1 ., 1962; H a m m e r l i n d l , 1978; J e n k i n s and Snyder, 1979) .

These a s s u m p t i o n s p e r m i t t e d t h e a p p l i c a t i o n o f E u l e r beam t h e o r y . The o u t l i n e o f t h e e q u a t i o n deve lopmen t i s p r e s e n t e d i n t h e f o l l o w i n g s e c t i o n . More d e t a i l e d d e r i v a t i o n s a r e p r e s e n t e d i n Append ix B.

C r i t i c a l Tempera tu re . A f r e e body d i a g r a m f o r a de fo rmed beam i s shown i n F i g u r e 6. The d i f f e r e n t i a l e q u a t i o n t h a t d e s c r i b e s t h e beam i s as f o l l o w s : (Terms a r e d e f i n e d i n F i g u r e 6 and i n Append ix A ) .

M = MR - Py + V X = E Iy "

T h e r e a r e f o u r b o u n d a r y c o n d i t i o n s :

x = o , y = o x = 0, y ' = 0

x = R, y ' = 0 x = R , y = e

The f i r s t two b o u n d a r y c o n d i t i o n s e s t a b l i s h t h e e q u a t i o n :

* As a check on t h i s a s s u m p t i o n , one can compare t h e m a g n i t u d e o f t h e t h e r m a l a x i a l l o a d w i t h t h e l o a d c a l c u l a t e d f r o m t h e i n t e r n a l - e x t e r n a l p r e s s u r e d i f f e r e n c e t i m e s t h e p i p e c r o s s - s e c t i o n a l area. The l a t t e r l o a d s h o u l d be much s m a l l e r i n c o m p a r i s o n t o t h e t h e r m a l l o a d .

DEFINITION OF TERMS

HOLE WALL ~

4 ENDS

F i g u r e 6. D e f i n i t i o n of Terms: a ) L i n e S k e t c h and b) F u l l Body D i a g r a m .

The f o u r t h b o u n d a r y c o n d i t i o n e s t a b l i s h e s a r e l a t i o n s h i p

be tween MR, V, and P ( o r K ) . as seen f r o m t h e f o l l o w i n g e q u a t i o n :

T h e r e a r e t h r e e s p e c i a l cases

MR V y ' = p ( K s i n K R ) - ( 1 - cos K R )

The t h r e e cases a r e :

2 ) 1 - C O S K R = 0 and s i n K R = 0

3 ) M = 0 a..d 1 - c o s K R = 0 R

Case 1 i s t h e s i t u a t i o n b e f o r e w a l l c o n t a c t (y < e ) . c a s i n g t a k e s t h e shape o f a c o s i n e f u n c t i o n :

MR y = p ( 1 - C O S K X )

K = 2 n / L

The c r i t i c a l l o a d (Per) i s t h e r m a l l y i n d u c e d and

t h u s

e q u a l t o AEaAT. The r e s u l t i n g e x p r e s s i o n f o r t h e c r i t i c a l t e m p e r a t u r e change AT^,) w h i c h i n i t i a t e s b u c k l i n g i s :

2 4 n 1 AT^^ = -2 L Aa

F i g u r e 7 p l o t s e q u a t i o n ( 5 ) . The u n s u p p o r t e d l e n g t h ( L ) i s n o r m a l i z e d b y t h e o u t s i d e c a s i n g d i a m e t e r (D). A l t e r n a t e l y one can use t h e r a d i u s o f g y r a t i o n ( r ) ( N e l s o n , 1975) .

13 3/8 tNCH 54.5ppf - ACTUAL \ A 9 5/8 INCH 36.OPPf -

BEHAVIOR - \ rn 9 5/8 INCH 40.0PPf - \

\ ~EFORMATION 100

%ooo g 900- z 2 800-

700- W 2 600-

2 500-

E 300-

Figure 7 .

- - APPROXIMATE LIMIT -

OF EULER BUCKING - FORMULATION -

1580 CASING

I I I I I I I I I

Locus Del ineating Euler Buckling Region: P l o t o f Temperature Change ( A T ) Versus Normal i ted Unsupported Length ( L / D ) .

T h r e e t y p i c a l c a s i n g s were examined: 9 - 5 / 8 i n c h 36 p p f , 9 - 5 / 8 i n c h 40 p p f , and 13 -3 /8 i n c h 54.5 p p f ( 2 4 4 mm 54 kg/m, 244 mm 60 kg/m, and 340 mm 80 k g l m ) . L i t t l e d i f f e r e n c e between t h e c a s i n g s e x i s t s . The moment o f i n e r t i a ( I ) d e c r e a s e s s l i g h t l y f o r t h e 9 - 5 / 8 i n c h 40 p p f (244 mm 60 k g / m ) p i p e because t h e o u t s i d e d i a m e t e r r e m a i n s c o n s t a n t . T h i s e x p l a i n s t h e s l i g h t d e c r e a s e i n AT^^ f o r t h i s s u p p o s e d l y s t r o n g e r p i p e . i m p o r t a n t t o n o t e f rom e q u a t i o n ( 5 ) t h a t l a r g e r d i a m e t e r p i p e w i l l i n c r e a s e t h e c r i t i c a l b u c k l i n g t e m p e r a t u r e w h e t h e r o r n o t t h e p i p e s t r e n g t h i n c r e a s e s .

b y n o t i n g t h a t A I S C ( 1 9 8 0 ) recommends t h e s l e n d e r n e s s r a t i o ( K L / r ) r e m a i n above r ( 2 E / a )ll2.

9 Y e s t a b l i s h e d because co lumn f a i l u r e modes such as l o c a l i z e d p l a s t i c d e f o r m a t i o n o r k i n k i n g became i m p o r t a n t f o r s m a l l e r v a l u e s . The above c r i t e r i o n e s t a b l i s h e s a minimum l e n g t h o f 22 f t (6.8 m ) o r 27.5 D f o r E u l e r b u c k l i n g o f N-80 9 - 5 / 8 i n c h 40 p p f ( 2 4 4 mm 60 k g l m ) c a s i n g .

when E u l e r b u c k l i n g w i l l i n i t i a t e . Whether t h e b u c k l i n g r e s u l t s i n p l a s t i c d e f o r m a t i o n s mus t b e examined f r o m t h e s t r e s s s t a n d p o i n t . The t o t a l s t r e s s ( a t ) a f t e r b e n d i n g a t a maximum f i b e r i s e q u a l t o t h e a x i a l s t r e s s (a,) p l u s t h e maximum b e n d i n g s t r e s s ( a b ) m a x ( o v a l a t i o n s t r e s s e s n e g l e c t e d ) :

It i s

The r a n g e o f a p p l i c a b i l i t y o f e q u a t i o n ( 5 ) can be e s t i m a t e d

T h i s c r i t e r i o n was

Maximum s t r e s s b e f o r e w a l l c o n t a c t . F i g u r e 7 i n d i c a t e s

‘t = lT a + (‘b)max

u = P c r / A = E ~ A T ~ ~ = c o n s t a n t a

= M r /I ( ‘b )max R o

The end moment r e a c t i o n ( I d R ) i n t h e e x p r e s s i o n f o r ( a b ) m a x i s f o u n d f r o m t h e c o n d i t i o n e s t a b l i s h e d b y t h e f i x e d ends; t h e

t h e r m a l d i s p l a c e m e n t ( s ( T ) ) mus t e q u a l t h e sum o f t h e a x i a l l o a d d i s p l a c e m e n t ( s ( P ) ) and t h e c a s i n g d e f l e c t i o n d i s p l a c e m e n t ( s ( y ) ) ( B o l e y and Weiner , 1960) :

The d i s p l a c e m e n t s a r e e x p r e s s e d as:

( 1 1 ) 6 ( P ) = PcrL/AE = c o n s t a n t

a (y ) = ( y l ) ' dx (T imoshenko, 1961) ( 1 2 )

2 =($) L

Once b u c k l i n g has o c c u r r e d t h e t h e r m a l l y i n d u c e d l e n g t h change s ( T ) i s e n t i r e l y abso rbed b y t h e co lumn d e f l e c t i o n t e r m s (y ) ; hence, s ( P ) r e m a i n s c o n s t a n t . C o n s e q u e n t l y u a r e m a i n s a t t h e c r i t i c a l b u c k l i n g s t r e s s as n o t e d above. I n s e r t i n g t h e d i s p l a c e m e n t e x p r e s s i o n s i n t o e q u a t i o n ( 9 ) and s o l v i n g f o r MR

r e s u l t s i n ( A T > AT^,):

( 1 4 ) 1 / 2 MR = [ a ( A T - AT^^)] lr

A s s e m b l i n g t h e e x p r e s s i o n s f o r ua and ( a b ) m a x and

i n s e r t i n g i n t o ( 6 ) y i e l d s :

4 L r 0 1 + --51--T-' [a ( AT-AT,, ) 3 I2 1 ( 1 5 ) f ~ ( r o + r i )

at = EaATcr

E q u a t i o n ( 1 5 ) i s p l o t t e d f o r 13 -3 /8 i n c h 54.5 p p f (340 mm 80 kg /m) c a s i n g w i t h L / D = 50, 100, 200 i n F i g u r e 8. The u p p e r r a n g e o f a p p l i c a b i l i t y f o r f i g u r e 8 i s when t h e s t e e l r e a c h e s

i t s y i e l d p o i n t ( 8 0 k s i ( 5 5 0 MPa) f o r N-80 c a s i n g ) o r t h e c a s i n g d e f l e c t s enough t o c o n t a c t t h e h o l e s i d e s . The l a t t e r c o n d i t i o n i s a d d r e s s e d be low. F i g u r e 9 p l o t s maximum d e f l e c t i o n (ymax) v e r s u s t e m p e r a t u r e change ( A T ) and e n a b l e s one t o p r e d i c t when w a l l c o n t a c t w o u l d o c c u r .

c o n d i t i o n s l e a d s t o t h e r e s u l t t h a t M R = -Mb ( F i g u r e 6 ) . The de fo rmed c a s i n g shape i s n o t c o m p a t i b l e w i t h t h e s e end moments. Hence case 3, a c o n c e n t r a t e d s h e a r l o a d ( V ) a c t i n g a t t h e p o i n t o f c o n t a c t , was used t o m a t h e m a t i c a l l y d e s c r i b e t h e c a s i n g a t w a l l c o n t a c t . * The shape o f t h e c a s i n g be tween p o i n t s and " b t l i n F i g u r e 6 i s :

Maximum s t r e s s a f t e r w a l l c o n t a c t . Case 2 o f t h e b o u n d a r y --- -------------- ----

s i n 2Kx) V Y = p ( X - Z i i ;

N o t e t h a t t h e end moment ( M R ) i s r e p l a c e d w i t h t h e s h e a r f o r c e ( V ) . U n t i l w a l l c o n t a c t , t h e co lumn shape i s d e s c r i b e d b y a c o s i n e f u n c t i o n ( e q u a t i o n 2 ) . An i n s t a n t a n e o u s change i n c a s i n g shape i s r e q u i r e d . An i m p o r t a n t consequence i s t h a t s t r e s s v a l u e s a r e n o t c o m p a t i b l e when w a l l c o n t a c t o c c u r s u s i n g t h e two d i f f e r e n t shapes. E q u a t i o n s ( 4 ) and ( 1 6 ) a l o n g w i t h compu te r r e s u l t s t o be d i s c u s s e d l a t e r a r e p l o t t e d i n F i g u r e l o a .

A s o l u t i o n t o t h e p r e d i c a m e n t ( s u b j e c t t o c o n f i r m a t i o n b y n u m e r i c a l a n a l y s i s ) was t o assume t h e co lumn shape a f t e r c o n t a c t was t h e same as b e f o r e . The o n l y d i f f e r e n c e was t h a t i t was " s p l i t . " T h i s a s s u m p t i o n r e q u i r e d i m a g i n a r y end moments

. i T X m o s h E o (1959) d i s c u s s e s t h e case o f a f i x e d end beam u n i f o r m l y l o a d e d w i t h f o u n d a t i o n c o n t a c t b u t no r e f e r e n c e was found d i s c u s s i n g a x i a l l y - l o a d e d members w i t h w a l l c o n t a c t .

GEOTHERMAL WELL CASING EULER BUCKLING MAXIMUM STRESS

I I I I 1

TEMPERATURE CHANGE, AT ( O R

Figure 8 . Maximum Stress ( a ) Versus Temperature Change ( A T ) f or 13-318 inch 5 4 . 5 ppf C a s i n g Assuming Unsupported Lengths ( L / D ) of 50, 100, a n d 2 0 0 .

GEOTHERMAL WELL CASING EULER BUCKLING

MAXIMUM DEFLECTION I I I

13 3/8 INCH S4.Sppf CASING UNSUPPORTED LENGTH

0 V D = l O O (111 ft) L/D=200 (223 ft)

A uD=60(58ff)

F i g u r e 9. Maximum D e f l e c t i o n Versus T e m p e r a t u r e Change ( A T ) fo r 13 -3 /8 i n c h 54.5 p p f C a s i n g Assuming U n s u p p o r t e d L e n g t h s ( L / D ) o f 50, 100, and 200.

100 200 300 400

COMPARISON OF DEFORMED CASING SHAPES

AT AT EQUAL 80 OF* 10

l a W6 I C H Mdppf CA6lG

HOLE QAP. 010=060 (6.60 NCMS) 0 . UNSUPPORTEO L€NGTn.UD=lOO

6 - 0 NARCRESULTS --- y,=(lIZ) (1 - corn Ka), K=Wl 6 w 7 - ___ yr=(o/U (a 41121o.ln 2x11) ,

0 6 - 5

t: 3. n

,g=- - a) z

fCP g 4 -

*?a O F WAS AT NECESSARY FOR WALL CONTACT ~PREDICM BY MARC) - Af 2 -

400 600 800

1 - j=, / 5 ° OO- 200

LENGTH (INCHES)

YARC RESULTS 0 AT=lOOeF A AT = 200.F

~ ~ = a o o * ~ ANALYTIC RESULTS (AT=SOO*F)

LENGTH (INCHES)

1 15 W6 INCH 64.5 ppf CASING UNSUPPORTED LENGTH. U D = 100 HOLE QAP. o/O=O.SO (6.60 INCHES) YARC RESULT

ANALYTIC RESULT AT AT= SOOT ~ ~ = a o o * ~

LENGTH (INCHES)

F i g u r e 10. Deformed C a s i n g Shapes w i t h W a l l C o n s t r a i n t P r e d i c t e d by MARC and T h e o r e t i c a l Models a t a ) A T = 80°F, b ) M A R C r e s u l t s a t A T o 300°F, and c ) A n a l y t i c R e s u l t s a t AT = 30OoF.

( M R ) e x i s t e d e q u a l t o Pe/2 . The m a t h e m a t i c a l model was s i m i l a r t o t h a t used p r i o r t o t h e w a l l c o n t a c t w i t h t h e e x c e p t i o n o f a v a r i a b l e co lumn l e n g t h ( 2 ) .

between t h e f i x e d end and t h e p o i n t o f w a l l c o n t a c t . The l e n g t h ( 2 ) s h o r t e n s as t h e c a s i n g segment a g a i n s t t h e w a l l l e n g t h e n s due t o i n c r e a s e d d e f o r m a t i o n . An e x p r e s s i o n f o r was f o u n d f r o m t h e c o n d i t i o n t h a t t h e - t h e r m a l d i s p l a c e m e n t ( 6 ( T ) ) e q u a l e d t h e sum o f t h e a x i a l l o a d d i s p l a c e m e n t ( s ( P ) ) and t h e beam d e f l e c t i o n d i s p l a c e m e n t ( a ( y ) ) . . U s i n g t h e same d i s p l a c e m e n t e x p r e s s i o n s as b e f o r e ( e q u a t i o n s ( l o ) , (ll), and ( 1 2 ) ) r e s u l t e d i n :

F i g u r e 6 d e p i c t s t h e v a r i a b l e l e n g t h ( 2 ) . I t i s t h e l e n g t h

R = {(,e)2 A + A2 + 256 L 2 2 1 I A a A T I 1 / 2 ) /16LAaAT ( 1 7 )

The t o t a l s t r e s s ( a b ) a t a maximum f i b e r i n t h e c a s i n g e q u a l s t h e a x i a l s t r e s s ( u a ) p l u s t h e maximum b e n d i n g s t r e s s

( “b )ma x ( e q u a t i o n 6 ) where:

u a = EaATcr

= M r /I (“b)max R o

2 2 MR = P e / 2 = T E I e / = 2

The e x p r e s s i o n f o r t h e maximum s t r e s s i s t h u s :

( 1 9 ) 2 2

at = EaATcr + 1 Eero /2R

F i g u r e l l a p l o t s e q u a t i o n ( 1 9 ) f o r a 1 3 - 3 / 8 i n c h 54.5 p p f (340 mm 8 0 k g / m ) c a s i n g 100 d i a m e t e r s l o n g . W a l l c o n t a c t

100 13 518 INCH S4.SPPf CASING UNSUPPORTED LENGTH, LID =lo0 - 90 - HOLE GAP, */D - 0 0.26 (3.34 INCH)

- 5 0 - cn

ANALYTIC MODEL OF EULER BUCKLING WITH SUBSEQUENT

WALL CONTACT I I

isa /8 INCH s4 . sppf CASING loo - UNSUPPORTED LENGTH, L/D = 100

90 - HOLE GAP, */D 0 0.25 (5.34 INCH)

m 1.00 (13.58 INCH)

Q 8 0 - 0 O I O (6.69 INCH) A 0.71 (10.03 INCH)

0 - a) 0 0

TEMPERATURE CHANGE, AT ( O F )

MARC FE ANALYSIS USING THIN WALLED BEAM ELEMENT

I I I I

F i g u r e 11.

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

( e / D ) i s l e s s t h a n 0.50 and t e m p e r a t u r e changes ( A T ) a r e w i t h i n t h e n o r m a l 200-3OO'F (95-15O'C) r a n g e .

-- N u m e r i c a l Model

The a n a l y s i s o f c a s i n g b u c k l i n g i s d i f f i c u l t f o r a s t a t i c f i n i t e e l e m e n t code t o h a n d l e . I n a d d i t i o n , t h e n u m e r i c a l model f o r m u l a t i o n r e q u i r e s a s l i d i n g i n t e r f a c e c a p a b i l i t y t o s i m u l a t e w a l l c o n t a c t . The n u m e r i c a l model d e v e l o p e d used t h e t h i n - w a l l ed beam e l emen t ( E u l e r t h e o r y ) w i t h c i r c u l a r c r o s s - s e c t i o n and t h e f r i c t i o n gap e l e m e n t f r o m t h e MARC f i n i t e e l e m e n t p r o g r a m (1979) . As a check, a n o t h e r model was a l s o r u n u s i n g a s t a n d a r d beam-column e l e m e n t w i t h an i d e n t i c a l m o m e n t - o f - i n e r t i a a r e a r a t i o ( I / A ) . The r e s u l t i n g s t r e s s e s were p r a c t i c a l l y i d e n t i c a l u n t i l n e a r t h e y i e l d p o i n t .

t o t h o s e o f t h e a n a l y t i c model. C o m p l i c a t i o n s due t o c a s i n g n e s t i n g and c r o s s - s e c t i o n a l shape changes ( o v a l a t i o n ) due t o l a t e r a l and b e n d i n g f o r c e s were n o t i n c l u d e d . C a s i n g i n s t a b i l i t y f r o m i n t e r n a l - e x t e r n a l p r e s s u r e i n t e r a c t i o n was

n e g l e c t e d , F i n a l l y , o n l y 13 -3 /8 i n c h 54.5 p p f ( 3 4 0 mm 80 k g l m ) c a s i n g 100 d i a m e t e r s l o n g was examined. H i g h t e m p e r a t u r e changes were a n t i c i p a t e d n e a r t h e s u r f a c e i f w e l l shutdown o c c u r r e d and t h u s l a r g e d i a m e t e r p i p e was t h o u g h t a p p r o p r i a t e . The u n s u p p o r t e d l e n g t h s e l e c t e d was a r b i t r a r y . F o r t h i s i n i t i a l i n v e s t i g a t i o n , i n e l a s t i c a n a l y s i s above t h e y i e l d p o i n t was o m i t t e d ( 8 0 k s i ( 5 5 2 MPa) f o r N-80 c a s i n g ) .

n u m e r i c a l l y u n s t a b l e ) a t A T ~ ~ e q u a l t o 58°F (14°C) . v a l u e compares p o o r l y w i t h t h e a n a l y t i c a l l y computed 72°F (22'C) c r i t i c a l t e m p e r a t u r e change AT^,). f u n c t i o n o f t h e e c c e n t r i c i t y i n i t i a l l y i n t r o d u c e d i n t h e c a s i n g model .

r u n . The c a s i n g w o u l d n e v e r r e g a i n s t a b i l i t y a f t e r b u c k l i n g

Assumpt ions . --- The n u m e r i c a l model l i m i t a t i o n s a r e s i m i l a r

D i s c u s s i o n . .- -- The n u m e r i c a l model b u c k l e d (became T h i s

However, A T i s a s t r o n g

I t was n o t p o s s i b l e t o p l o t t h e e n t i r e s t r e s s p a t h i n one

even though t h e l o a d i n g was t e m p e r a t u r e ( s t r a i n ) c o n t r o l l e d ; hence i t was n e c e s s a r y t o e s t a b l i s h the c a s i n g deformat ion a t wal l c o n t a c t and then proceed w i t h t h e c a l c u l a t i o n s . Force e q u i l i b r i u m o c c u r r e d a t A T equal t o 7 8 ° F (26°C) which i n d i c a t e d wal l con t a c t .

Snyder (1979) c la imed thermal b u c k l i n g i n an unsupported h o l e i s no t s t r a i n c o n t r o l l e d . Fur thermore , he hypo thes i zed sudden h o r i z o n t a l movement could cause c r a c k i n g o f work-hardened, h i g h g rade s t e e l . A t f i r s t g l a n c e , the i n s t a b i l i t y p r o b l e m w i t h MARC (1979) a p p e a r s t o s u p p o r t t h e s u d d e n h o r i z o n t a l d i sp l acemen t v i ewpo in t .

v e r y h i g h ( h i g h s t r a i n r a t e ) f o r t h e above hypotheses t o be t r u e . T h e a n a l y t i c s o l u t i o n (F igu re 8 ) suggests l a r g e h o r i z o n t a l movement i s no t p o s s i b l e unless c a s i n g s t r e s s has i n c r e a s e d wel l beyond AT^^ b e f o r e buck l ing . w i t h the M A R C model may have been d u e t o t he a r t i f i c i a l t e m p e r a t u r e i n c r e a s e r a t e , b u t t h e l a r g e computer t ime r e q u i r e d t o r u n t h e model prec luded t h e use o f s m a l l e r r a t e s .

f i e l d , Baja , C a l i f o r n i a , the i n i t i a l t e m p e r a t u r e r i s e t o p roduc t ion t e m p e r a t u r e s is c a r e f u l l y monitored t o ensure s t r a i n c o n t r o l . T h e procedure requi res 30-60 days (Snyder , 1979) . T h i s rate w o u l d b e impractical i f many temperature cyc le s were

r e q u i r e d .

( 4 ) , t h e t h e o r e t i c a l curve ( 1 6 ) , and the MARC d i sp l acemen t v a l u e s f p r t h e deformed c a s i n g shape a t t h e time of wal l c o n t a c t w i t h a h o l e gap ( e / D ) equal t o 0.50 a r e compared i n F igure 10. F igure 10b p l o t s t h e MARC deformed shapes f o r t e m p e r a t u r e changes ( A T ) o f 100, 200, and 300°F (38 , 93, and 149°C) and a l s o e q u a t i o n (16) a t A T = 30OoF. A t AT equal t o 300"F, the d i f f e r e n c e i n shape between e q u a t i o n ( 1 6 ) a n d MARC i s g r e a t and w o u l d presumably get worse a t h i g h e r t e m p e r a t u r e changes .

However, t h e r a t e o f t e m p e r a t u r e i n c r e a s e w o u l d have t o b e

T h e d i f f i c u l t i e s

I t i s worth ment joning t h a t i n t h e C e r r o P r i e t o geothermal

As i p d i c a t e d e a r l i e r , t h e s p l i t - c o s i n e b e l l approximation

F o r t h e n u m e r i c a l c a l c u l a t i o n s , t h e c a s i n g c o n t a c t l e n g t h

w i t h t h e w a l l was q u i t e s m a l l . A t A T = 300°F and e/D = 0.75, MARC p r e d i c t e d a c o n t a c t l e n g t h o f 1 0 p e r c e n t o f t h e t o t a l l e n g t h u s i n g a t o l e r a n c e o f 0.05 i n c h e s (1.27 mm). T h i s compares w i t h 47 p e r c e n t f r o m a n a l y t i c c a l c u l a t i o n s .

(149°C) d e v e l o p e d f r o m e q u a t i o n ( 4 ) and ( 1 6 ) u s i n g t h e f r e e , u n s u p p o r t e d l e n g t h ( R ) d e t e r m i n e d f r o m M A R C r a t h e r t h a n a n a l y t i c a l l y . The r a i s e d - c o s i n e e q u a t i o n ( 4 ) c o i n c i d e d w i t h

F i g u r e 1Oc shows two shape p r e d i c t i o n s a t A T e q u a l t o 300°F

t h e MARC d a t a when t h e n u m e r i c a l v a l u e o f R was used. E q u a t i o n ( 1 6 ) p r e d i c t e d s l i g h t l y more c u r v a t u r e .

I f t h e n u m e r i c a l l y d e t e r m i n e d R i s used i n t h e e x p r e s s i o n f o r s t r e s s ( a b ) ( e q u a t i o n 19) , a v a l u e o f 64 k s i ( 4 4 1 MPa) i s p r e d i c t e d when e/D = 0.50 and AT = 300°F (149°C) . The maximum s t r e s s c a l c u l a t e d n u m e r i c a l l y was 72.5 k s i (500 MPa)

( F i g u r e l l b ) . Thus, t h e t h e o r e t i c a l e q u a t i o n s p r o v i d e a good shape and s t r e s s v a l u e p r e d i c t i o n i f t h e l e n g t h (R) i s known.

t h a t e q u a t i o n ( 4 ) d i c t a t e s u t o c c u r s a t t h e w a l l c o n t a c t w h i l e e q u a t i o n ( 1 6 ) d i c t a t e s u t o c c u r s a t t h e q u a r t e r p o i n t o f t h e f r e e l e n g t h . The MARC model showed a t g r a d u a l l y s h i f t i n g f r o m t h e w a l l c o n t a c t p o i n t a t A T e q u a l t o 78°F ( 2 6 ° C ) t o t h e q u a r t e r p o i n t a t 300°F (149°C) when e/D e q u a l s t o 0.50. The s h i f t o f u t t o t h e q u a r t e r p o i n t o c c u r r e d w i t h s m a l l e r changes i n A T as e d e c r e a s e d i n m a g n i t u d e .

F i g u r e l l b p l o t s a t v e r s u s AT f o r MARC v a l u e s . a n a l y t i c c a l c u l a t i o n s f r o m F i g u r e l l a a r e p l o t t e d f o r compar i son . The a n a l y t i c and n u m e r i c a l s o l u t i o n s compare f a v o r a b l y when t h e w a l l d i s t a n c e ( e / D ) i s 0.50. A t a h o l e gap s i z e o f 0.25, t h e a n a l y t i c s o l u t i o n shows a d i p b e l o w t h e EaAT s t r e s s l i n e . T h i s i s u n r e a l i s t i c and i n d e e d t h e n u m e r i c a l s o l u t i o n does n o t c r o s s t h e EaAT l o w e r l i m i t . The n u m e r i c a l r e s u l t s i n d i c a t e t e m p e r a t u r e changes u p t o 310°F (154°C) c o u l d be t o l e r a t e d f o r e/D = 0.75.

I n mak ing c o m p a r i s o n s be tween e q u a t i o n ( 4 ) and ( 1 6 ) , r e c a l l

A d d i t i o n o f & o n s t a n t S t r e s s

I t i s i m p o r t a n t t o n o t e t h e n u m e r i c a l r e s u l t s p a r a l l e l t h e E a A T l i n e i n d i c a t i n g t h e a d d i t i o n o f a c o n s t a n t s t r e s s t o t h e E a A T s t r e s s w o u l d model t h e maximum c a s i n g s t r e s s r e a s o n a b l y w e l l . Assuming t h e s t r e s s change ( A U ) i s a l i n e a r f u n c t i o n o f t h e h o l e gap (e ) , a l e a s t squares f i t r e s u l t s i n (r2 = 0.78) ( i n c h e s and p s i ) :

A U = ae + b

a = 1960; b = 2060 (20)

u = EaAT + Au

The M A R C a n a l y s i s and c o n s t a n t s t r e s s change a s s u m p t i o n a r e compared i n F i g u r e l l c . No te t h a t t h e s i m p l i f i c a t i o n i s v a l i d as l o n g as t h e c a s i n g c o n t a c t l e n g t h r e m a i n s s m a l l and t h e h o l e

gap ( e ) i s s m a l l .

A n a l y s i s Summary - ---- I n summary, t h r e e p r e d i c t i v e s t r e s s methods were examined:

n u m e r i c a l , e m p i r i c a l , and t h e o r e t i c a l . The n u m e r i c a l f o r m u l a t i o n f o r m o d e l i n g t h e c a s i n g a f t e r w a l l c o n t a c t was c o n s t r u c t e d u s i n g M A R C beam e l e m e n t s . The e m p i r i c a l m e t h o d

c o n s i s t e d of a d d i n g a s t r e s s w h i c h was a f u n c t i o n o f t h e h o l e gap d i s t a n c e ( e ) t o t h e t h e r m a l l y i n d u c e d EaAT s t r e s s . The e q u a t i o n f i t t h e n u m e r i c a l r e s u l t s r e a s o n a b l y w e l l f o r s m a l l e v a l u e s . The t h e o r e t i c a l model made use o f s i m p l e beam t h e o r y . B u c k l i n g w i t h o u t w a l l c o n t a c t was e a s i l y examined t h e o r e t i c a l l y . However, t h e c a s i n g shape a f t e r w a l l c o n t a c t as d e r i v e d s t r i c t l y 3 r o m E u l e r beam t h e o r y was n o t c o m p a t i b l e w i t h t h e i n i t i a l b u c k l i n g phase; hence t h e shape and s t r e s s p r e d i c t i o n s were o n l y i n f a i r agreement w i t h t h e n u m e r i c a l

r e s u l t s . The t h e o r e t i c a l f o r m u l a t i o n (16 and 1 9 ) p r o v i d e d a good shape and s t r e s s v a l u e p r e d i c t i o n i f t h e n u m e r i c a l l y d e t e r m i n e d w a l l c o n t a c t l e n g t h ( R ) was used.

A d e s c r i p t i o n o f t h e de fo rmed c a s i n g as a s p l i t c o s i n e b e l l ( e q u a t i o n 4 w i t h MR = P e / 2 ) was c o n t r a s t e d w i t h t h e n u m e r i c a l r e s u l t s . The s p l i t - c o s i n e f o r m u l a p r o v i d e d a good a p p r o x i m a t i o n o f t h e de fo rmed shape. The shape d e s c r i p t i o n was e x c e l l e n t if t h e n u m e r i c a l l y d e t e r m i n e d w a l l c o n t a c t l e n g t h was used. The e x p r e s s i o n f o r t h e maximum c a s i n g s t r e s s was i d e n t i c a l t o t h a t d e r i v e d i n t h e t h e o r e t i c a l f o r m u l a t i o n . O n l y t h e e x p r e s s i o n f o r t h e l e n g t h d i f f e r e d .

RESULT IMPLICATIONS

T h e r m a l l y I n d u c e d E u l e r Buck1 i n g ---- - ---1-_1_

L o o k i n g a t F i g u r e 8, N-80 1 3 - 3 / 8 i n c h 54.5 p p f (340 mm 80 k g l m ) c a s i n g w i t h a y i e l d s t r e s s ( U ) o f a b o u t 75 k s i

( 5 1 7 MPa) a t 500°F (260°C) * and 100 d i a m e t e r u n s u p p o r t e d l e n g t h w o u l d s l i g h t l y exceed y i e l d c o n d i t i o n s i f a 300°F (150°C) t e m p e r a t u r e e x c u r s i o n f r o m cement c o n s t r a i n e d c o n d i t i o n s o f 100-200°F (40-95°C) o c c u r r e d . These c o n d i t i o n s r o u g h l y c o r r e s p o n d t o t y p i c a l i n t e r m e d i a t e c a s i n g i n g e o t h e r m a l w e l l s ( F i g u r e 1 ) . The i n t r o d u c t i o n o f a s m a l l t e n s i l e s t r e s s w o u l d e a s i l y r e d u c e t h e s t r e s s t o b e l o w y i e l d . C o n s e q u e n t l y , s i m p l e E u l e r b u c k l i n g s h o u l d n o t cause a s e r i o u s p r o b l e m i f AT i s l e s s t h a n 275°F (135°C) f o r u n s u p p o r t e d l e n g t h s ( L / D ) above 100. Note, however , t h a t t h e r e was no a l l o w a n c e f o r a d e s i g n

f a c t o r . A p p l i c a t i o n o f e i t h e r d i r e c t t e n s i o n o r i n t e r n a l p r e s s u r e d u r i n g t h e c e m e n t i n g p r o c e s s w o u l d e s t a b l i s h an a x i a l t e n s i l e l o a d a l t h o u g h t h e m i c r o a n n u l u s be tween t h e c a s i n g and cement w o u l d b e e n l a r g e d u s i n g i n t e r n a l p r e s s u r e . O p e r a t o r s h a v e a v o i d e d t h e r m a l w e l l c a s i n g f a i l u r e s i n p a r t i a l l y cemented s team i n j e c t i o n w e l l s b y u s i n g N-80 o r P-110 g r a d e c a s i n g

( H o l l i d a y , 1969) . T h i s t e n d s t o c o n f i r m t h e l o w p r o b a b i l i t y o f

p r o b l e m s w i t h N-80 c a s i n g . The u s e o f l a r g e r d i a m e t e r p i p e w o u l d i n c r e a s e t h e c r i t i c a l

b u c k l i n g t e m p e r a t u r e b u t n o t s i g n i f i c a n t l y . T h i s i s

* One i s s a f e i n assuming no s i g n i f i c a n t r e d u c t i o n i n u l i i m a t e c a s i n g s t r e n g t h o c c u r s f o r t e m p e r a t u r e r i s e s b e l o w 660 F (350°C), b u t t h e y i e l d s t r e n g t h d e f i n i t e l y d e c r e a s e s w i t h an i n c r e a s e i n t e m p e r a t u r e . M a n u f a c t u r e r s g e n e r - a l l y do n o t make t e n s i l e t e s t s a t e l e v a t e d t e m p e r a t u r e s and t h u s s t a t i s t i c a l l y r e l i a b l e i n f o r m a t i o n i s l a c k i n g . K a r l s s o n (1978) s u g g e s t s t h e u s e o f D I N St. 45.8. The maximum p o s s i b l e r e d u c t i o n i n y i e l d s t r e n g t h f r o m t h i s f o r m u l a f o r t h e above c o n d i t i o n s i s 19 p e r c e n t . The above r e d u c t i o n o f 6 p e r c e n t c o r r e s p o n d s t o t h e minimum o b t a i n e d f r o m s e v e r a l p r i v a t e t e s t s .

d e m o n s t r a t e d b y F i g u r e 7 where commonly used c a s i n g d i a m e t e r s

p l o t on t o p o f each o t h e r . I n a d d i t i o n , t h e b e h a v i o r a f t e r b u c k l i n g m i g h t be i m p a i r e d because t h e minimum u n s u p p o r t e d l e n g t h f o r E u l e r b u c k l i n g t o o c c u r w o u l d i n c r e a s e .

I t has been t a c i t l y assumed t h a t s t r e s s e s above y i e l d c o n s t i t u t e d f a i l u r e . T h i s i s a r e a s o n a b l e f a i l u r e c r i t e r i o n because permanent d e f o r m a t i o n c o u l d h i n d e r r e m e d i a l work on t h e w e l l . P l a s t i c d e f o r m a t i o n was p e r m i t t e d a t Prudhoe Bay (Goodman, 1978) where p e r m a f r o s t thaw p roduced s t r a i n c o n t r o l l e d c o m p r e s s i v e f o r c e s , b u t t h e u s u a l l y s u c c e s s f u l cement j o b p r e c l u d e d t h e b u c k l i n g f a i l u r e mode.

c o n c e r n s c a s i n g c o l l a p s e . The t h e o r e t i c a l a rgument ( e q u a t i o n 1 6 ) d i c t a t e s an end r e a c t i o n ( V ) a t t h e w a l l s e p a r a t i o n p o i n t . The end r e a c t i o n i n c r e a s e s w i t h h o l e gap s i z e . A t AT = 300°F and e/D = 0.75, an end r e a c t i o n o f 11 900 l b (52 .9 k N ) was c a l c u l a t e d n u m e r i c a l l y . U s i n g t h i s v a l u e as a p o i n t l o a d on a c y l i n d e r , one c a l c u l a t e s v a l u e s be tween 51.9 and 77.2 k s i (358-532 MPa). The mean i s s l i g h t l y b e l o w t h e 7 5 k s i (517 MPa) y i e l d p o i n t . B u t t h e c a s i n g has d e f i n i t e l y r e a c h e d y i e l d a t t h e q u a r t e r p o i n t ( x / R = 1 / 4 ) . T h e r e f o r e , y i e l d i s more l i k e l y t o o c c u r a t t h e f r e e - l e n g t h q u a r t e r p o i n t f i r s t r a t h e r t h a n a t t h e w a l l c o n t a c t .

A g a i n r e f e r r i n g t o F i g u r e 8, t h e i m p o r t a n c e o f b u c k l i n g changes d r a m a t i c a l l y f o r weaker K - 5 5 c a s i n g . The c a s i n g w o u l d n o r m a l l y y i e l d a t A T = 295°F (145°C) w i t h f u l l l a t e r a l s u p p o r t assuming u = 55 k s i (380 MPa), b u t s i m p l e E u l e r b u c k l i n g wou ld p r o d u c e s t r e s s e s above y i e l d w i t h o n l y a 155°F ( 7 0 ° C ) t e m p e r a t u r e change. I f a no rma l 230-3OO'F (95-15O'C) t e m p e r a t u r e e x c u r s i o n o c c u r r e d , t h e K-55 c a s i n g w o u l d y i e l d when b u c k l i n g o c c u r r e d . L a t e r a l s u p p o r t w o u l d be a n e c e s s i t y . (The c a s i n g m i g h t , y i e l d even w i t h f u l l l a t e r a l s u p p o r t . ) The a n a l y s i s has i g n o r e d t h e s t r e n g t h e n i n g o f c a s i n g n e s t i n g w i t h c o m p l e t e cement j o b s . T h i s s i t u a t i o n m i g h t w a r r a n t e x a m i n a t i o n .

From F i g u r e l l b i t can be g e n e r a l l y s t a t e d t h a t w a l l c o n t a c t a f t e r b u c k l i n g l o w e r e d c a s i n g s t r e s s e s i n c o m p a r i s o n t o s i m p l e

The c a s i n g l o a d s a l o n g t h e w a l l c o n t a c t a r e i m p o r t a n t as

b u c k l i n g . The n u m e r i c a l r e s u l t s i n d i c a t e d 13-318 i n c h 54.5 p p f ( 3 4 0 mm 8 0 k g l m ) N-80 c a s i n g w i t h u n s u p p o r t e d l e n g t h s ( L / D ) above 100 w i t h gaps ( e / D ) b e l o w 0.75 s h o u l d n o t e x p e r i e n c e d i f f i c u l t i e s i f t e m p e r a t u r e e x c u r s i o n s a r e w i t h i n t h e n o r m a l 200-300°F (95-150°C) r a n g e . f i e l d e v i d e n c e t h a t p r o d u c t i o n t u b i n g , w h i c h i s c l o s e l y c o n f i n e d , s e l d o m p e r m a n e n t l y d e f o r m s when b u c k l e d ( T e x t e r ,

1955) . h o l e s e x c e e d i n g e /D > 0.75. The t r e n d o f t h e maximum s t r e s s ( u t ) i n F i g u r e l l b s u g g e s t t h a t l i t t l e d e t r i m e n t w o u l d o c c u r a t w a l l c o n t a c t w i t h g r e a t l y e n l a r g e d h o l e s . The c a s i n g w o u l d a l r e a d y b e n e a r y i e l d f o r t h e t e m p e r a t u r e change ( A T ) n e c e s s a r y f o r t h e l a r g e h o r i z o n t a l d i s p l a c e m e n t s .

These r e s u l t s a r e s u k p o r t e d b y t h e

L o s t c i r c u l a t i o n zones c o u l d p o t e n t i a l l y i n v o l v e e n l a r g e d

J o i n t B e h a v i o r ---- F o r N-80 c a s i n g , a r e l a t i v e l y h i g h a l l o w a b l e AT was p o s s i b l e

b e f o r e t h e c a s i n g s t e e l y i e l d e d . The w o r s t case o c c u r r e d d u r i n g w e l l s h u t - i n r e q u i r e d because o f a i r p o l l u t i o n s t a n d a r d s . If no e q u i p m e n t must pass t h r o u g h t h e c a s i n g d u r i n g t h i s p e r i o d , t h e t u b u l a r m a t e r i a l s h o u l d p e r f o r m s a t i s f a c t o r i l y . However, A P I j o i n t s a r e n o t d e s i g n e d t o w i t h s t a n d b e n d i n g s t r e s s e s and w i l l f r a c t u r e n e a r t h e l a s t engaged t h r e a d ( G r e e n i p , 1978) . B u t t r e s s t h r e a d c o n n e c t i o n s ma tch t h e p i p e body t e n s i l e s t r e n g t h and a r e c a p a b l e o f w i t h s t a n d i n g between 2.3 and 3.4 p e r c e n t s t r a i n ( d e p e n d i n g on w h e t h e r i n c o m p r e s s i o n o r t e n s i o n ) (Woo ley e t a l . , 1977) , b u t t h e y a r e s t i l l weak i n b e n d i n g . C o n s e q u e n t l y , j o i n t b e h a v i o r under l o a d i n g c o u l d p o s s i b l y b e t h e l a r g e s t f a c t o r i n c a s i n g f a i l u r e s f r o m b u c k l i n g . B e h a v i o r o f s h o u l d e r - t y p e j o i n t s u n d e r b e n d i n g s h o u l d be examined t o s e e i f a d e q u a t e improvemen ts a r e o b t a i n e d . A 100 p e r c e n t j o i n t such as t h e e a s i l y a n a l y z e d , f a b r i c a t e d , and assembled t a p e j o i n t d e v e l o p e d a t S a n d i a may b e u s e f u l . ( H u e r t a and B l a c k , 1976; Rechard e t a l . , 1982)

.-----

SUMMARY AND CONCLUSIONS

L i t t l e i n f o r m a t i o n i s a v a i l a b l e on g e o t h e r m a l w e l l c a s i n g b e h a v i o r o r f a i l u r e s ; t h u s e f f o r t s s h o u l d b e made t o o b t a i n d e t a i l e d d e s c r i p t i o n o f f i e l d b e h a v i o r and t h e c r i t e r i a on w h i c h f a i l u r e i s based. O p e r a t o r s h a v e e x p r e s s e d c o n c e r n o v e r c a s i n g i n s t a b i l i t y as a p o s s i b l e f a i l u r e mechanism. If a c a s i n g s t r i n g i s p l a c e d i n a x i a l compress ion f r o m t h e r m a l e l o n g a t i o n and i f t h e r e a r e s i z a b l e s e c t i o n s where n o l a t e r a l s u p p o r t i s p r o v i d e d due t o i n a d e q u a t e cement o r washout zones, t h e s t r i n g can l a t e r a l l y d e f l e c t ( b u c k l e ) .

T h i s r e p o r t p r o v i d e s p r e l i m i n a r y c a l c u l a t i o n s on t h e b u c k l i n g phenomenon. I t can g e n e r a l l y b e s t a t e d t h a t t h e r m a l l y i n d u c e d b u c k l i n g i n N-80 c a s i n g w o u l d n o t be s e r i o u s i f m i n o r a d j u s t m e n t s t o i n c r e a s e c a s i n g s t a b i l i t y were made such as a p p l y i n g a t e n s i o n p r e l o a d o r a d d i t i o n a l i n t e r n a l p r e s s u r e w h i l e cemen t ing . However, t h e s i t u a t i o n i s f a r worse f o r t h e weaker K-55 c a s i n g . F u l l l a t e r a l s u p p o r t w o u l d b e a n e c e s s i t y .

The e f f e c t o f w a l l c o n t a c t was f o u n d t o be b e n e f i c i a l f o r c l o s e l y c o n f i n e d p i p e s t r i n g s and o f no g r e a t d e t r i m e n t when h o l e gaps were l a r g e p r o v i d e d p i p e o v a l a t i o n was u n i m p o r t a n t . The weakness o f a l l Amer ican P e t r o l e u m I n s t i t u t e ( A P I ) sc rew j o i n t s i n b e n d i n g a p p e a r s t o be t h e s t r u c t u r a l l i m i t a t i o n .

The above c o n c l u s i o n s a r e based on t h e a s s u m p t i o n t h a t s t r e s s e s above y i e l d c o n s t i t u t e d f a i l u r e . I t was a l s o assumed t h e t h e r m a l e x p a n s i o n was s l o w enough t o p roduce s t r a i n c o n t r o l l e d l o a d s and t h a t t h e c a s i n g s t r i n g c o u l d be c o n s i d e r e d c o n t i n u o u s . I n t e r n a l p r e s s u r e i n s t a b i l i t y was i g n o r e d . The t e m p e r a t u r e v a r i a t i o n c o n s i d e r e d was be tween c e m e n t i n g c o n d i t i o n s o f 100-2OO'F (40-95°C) and s h u t - i n c o n d i t i o n s o f 425-450°F (220-230°C)

A n o t h e r r e g i m e o f thermal b u c k l i n g w h i c h needs t o b e i n v e s t i g a t e d i s s y m m e t r i c a l b u c k l i n g and w r i n k l i n g i n s t a b i l i t i e s

i n s h o r t , u n s u p p o r t e d c a s i n g l e n g t h s . F u r t h e r m o r e , i t w o u l d be b e n e f i c i a l i f l a b o r a t o r y e x p e r i m e n t s were c o n d u c t e d t o : 1 ) o b s e r v e t h e a b r u p t n e s s and e x t e n t o f h o r i z o n t a l d i s p l a c e m e n t a t b u c k l i n g , 2 ) e v a l u a t e s t r e s s r e d u c t i o n i n t h e c a s i n g f o l l o w i n g w a l l c o n t a c t , 3 ) e v a l u a t e weaken ing e f f e c t s o f p i p e o v a l n e s s , and 4 ) d e t e r m i n e t h e l i k e l i h o o d o f o t h e r f a i l u r e modes.

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E t i t u t e o f T e e l C o n s t r u c t i o n , New York .

-- S t r e s s e s , John W i l e y and Sons, Inc . , N e w Y o r k . 7

3. Capuano, L. E., 1978, "How Geysers Steam W e l l s A r e D r i l l e d and Equ ipped, " W o r l d O i l , F e b r u a r y 1, pp. 69-72.

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5. Edwards, L. M., C h i l i n g a r , G. V., R i e k e 111, H. H., and F e r t l , W. H., eds. 1982, Handbook o f Geothermal Energy , - G u l f Pub1 i s h i n g Company,

6. Goodman, M. A., 1978, ' 'Wor ld O i l ' s Handbook o f A r c t i c W e l l C o m p l e t i o n s , " W o r l d O i l , Hous ton , Texas.

7. Green ip , J. T., 1978, "Opt imum C a s i n g Program D e s i g n S t r e s s e s Economy," The O i l and Gas J o u r n a l , O c t o b e r 16, pp. 76-86.

8. H a m m e r l i n d l , D. J., 1978, " B a s i c F l u i d and P r e s s u r e F o r c e s on O i l W e l l T u b u l a r s , " 5 3 r d Annual T e c h n i c a l C o n f e r e n c e and E x h i b i t i o n o f S o c i e t y o f P e t r o l e u m E n g i n e e r s o f A I M E , SPE 7594, O c t o b e r 1-3.

9. H o l l i d a y , G. H., 1969, " C a l c u l a t i o n o f A l l o w a b l e Maximum C a s i n g Tempera tu res t o P r e v e n t T e n s i o n F a i l u r e s i n Therma l W e l l s , " Paper 69-Pet-10, ASME P e t r o l e u m M e c h a n i c a l E n g i n e e r i n g Con fe rence , T u l s a , OK, September 21-25.

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C a s i n g Des ign : B a s i c D e s i g n C o n s i d e r a t i o n s , " R e p o r t t o S a n d i a N a t i o n a l L a b o r a t o r i e s b y C o m p l e t i o n T e c h n o l o g y Company, August .

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Kumataka, M. K., 1981, l e t t e r t o James R. K e l s e y , S a n d i a N a t i o n a l L a b o r a t o r i e s , f r o m A m i n o i l USA, J u l y 28.

L u b i n s k i , A., W. J. A l t h o u s e , J. L. Logan, 1962, " H e l i c a l B u c k l i n g o f T u b i n g S e a l i n g i n Packe rs , " J. o f P e t r o l e u m Techno logy , pp. 655-670.

M A R C G e n e r a l P u r p o s e F i n i t e E l e m e n t Program, 1979, M A R C A n a l y s i s R e s e a r c h Corp., P a l o A l t o , C a l i f o r n i a 94306.

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Wooley, G. R., 1980, " W e l l - B o r e and S o i l The rma l S imu l a t i o n f o r Geo the rma l We1 1 : Compar i son o f GEOTEMP P r e d i c t i o n s t o F i e l d D a t a and E v a l u a t i o n o f F l o w V a r i a b l e s , " SAND79-7116, Sand ia N a t i o n a l L a b o r a t o r i e s , A lbuquerque , New Mexico, March.

A P P E N D I X A

Nomencl a t u r e

Mb, MR

- c a s i n g c r o s s - s e c t i o n a l a r e a ( L * )

- o u t s i d e c a s i n g d i a m e t e r ( L )

- modu lus o f e l a s t i c i t y ( M / t L )

- gap be tween c a s i n g and d r i l l h o l e ( L )

- moment o f i n e r t i a ( ~ 4 )

- t o t a l u n s u p p o r t e d l e n g t h ( L )

- u n s u p p o r t e d l e n g t h between f i x e d end and

- moment a t p o i n t b and r e a c t i o n moment ( M / t )

p o i n t o f c o n t a c t w i t h w a l l ( L )

P, P C r r Pe - a x i a l l o a d , a x i a l l o a d when E u l e r b u c k l i n g

r i , r o

i n i t i a t e d , and r e a c t i o n l o a d ( M / t L )

- i n s i d e and o u t s i d e c a s i n g r a d i u s ( L )

- r a d i u s o f g y r a t i o n ( L )

v, VR - shear f o r c e a p p l i e d a t w a l l c o n t a c t and r e a c t i o n s h e a r f o r c e ( M l t L ) .

, Y S Ymax - c a s i n g d e f l e c t i o n and maximum c a s i n g

d e f l e c t i o n ( L )

Y ' , Y " - f i r s t and second d e r i v a t i v e s , r e s p e c t i v e l y

- t e m p e r a t u r e change and t e m p e r a t u r e change when E u l e r b u c k l i n g i n i t i a t e d ( T )

e x p a n s i o n (T -1

AT, ATcr

a - l i n e a r c o e f f i c i e n t o f t h e r m a l

d P ) , d T ) - a x i a l l o a d and t h e r m a l l o a d d i s p l a c e m e n t s

& l ( Y ) , 6 d Y ) - d e f l e c t i o n d i s p l a c e m e n t s b e f o r e and a f t e r w a l l c o n t a c t ( L )

u a s ( Q b l m a x - axia l s t r e s s a n d maximum bending s t r e s s (MltL)

Q w / s Q w / o - t o t a l casing s t r e s s w i t h a n d without w a l l contact (MltL)

Q t - t o t a l cas ing s t r e s s (MltL)

- y i e l d s t r e s s ( M / t L ) O Y

A P P E N D I X B

D e r i v a t i o n o f E q u a t i o n s

Append ix B p r e s e n t s i n g r e a t e r d e t a i l t h e d e r i v a t i o n s o f p e r t i n e n t e q u a t i o n s i n t h e t e x t . The c o n c e p t s a r e q u i t e s i m p l e , b u t t h e a l g e b r a can be t e d i o u s . The c a s i n g shape d e s c r i p t i o n and t h e e x p r e s s i o n f o r maximum s t r e s s b e f o r e w a l l c o n t a c t a r e d e r i v e d f i r s t . The c a s i n g shape and s t r e s s e q u a t i o n s a f t e r w a l l c o n t a c t f o l l o w . F o r c l a r i t y , a d d i t i o n a l e q u a t i o n s and f i g u r e s were added t o t h o s e p r e s e n t e d i n t h e t e x t . They a r e p r e f i x e d w i t h t h e l e t t e r IIB."

D i f f e r e n t i a1 E q u a t i o n --- -

From t h e f r e e body d i a g r a m i n F i g u r e 6b, t h e f o l l o w i n g e q u i l i b r i u m c r i t e r i a must be met:

I f a s e c t i o n t h r o u g h t h e beam i s t a k e n b e f o r e p o i n t b ( F i g u r e B - l a ) t h e n t h e f o l l o w i n g e q u a t i o n r e s u l t s :

FREE BODY DIAGRAMS OF WELL CASING

HOLE WALL

F i g u r e B-1. S u p p l e m e n t a r y F r e e Body D iagrams

By assuming ( y ' ) ' i s s m a l l compared t o yi' i n t h e m a t h e m a t i c a l e x p r e s s i o n f o r c u r v a t u r e , E u l e r beam t h e o r y e x p r e s s e s t h e moment M as:

M = EIy"

2 T h i s e q u a t i o n a p p l i e s assuming: 1 ) ( y ' ) i s s m a l l , 2 ) Hook's l a w a p p l i e s , 3) E i s t h e same f o r t e n s i o n and compress ion , 4 ) a p l a n e s e c t i o n r e m a i n s p l a n e a f t e r bend ing , and 5 ) l o n g i t u d i n a l f i b e r l e n g t h s do n o t change. S u b s t i t u t i n g f o r M

The s o l u t i o n o f t h e d i f f e r e n t i a l e q u a t i o n i s

K 2 = P / E I

From F i g u r e 6 a i t i s e v i d e n t t h e f i x e d - e n d b o u n d a r y c o n d i t i o n s a t p o i n t a a re :

x = o , y = o

x = 0, y ' = 0

C o n s e q u e n t l y :

V (1 -cos K X ) + ( x - s i n K X ) - M R

C r 5 t i c a l t e m p e r a t u r e . ---a_---- B e f o r e w a l l c o n t a c t , t h e s h e a r f o r c e -- ( V ) i s z e r o . C o n s e q u e n t l y t h e c a s i n g t a k e s t h e shape o f a r a i s e d c o s i n e f u n c t i o n :

*R y = -p (1 - C O S K x ) ( 4 )

The d e f l e c t i o n and s l o p e a t p o i n t c i s a l s o s p e c i f i e d and e s t a b l i s h e s t w o a d d i t i o n a l b o u n d a r y c o n d i t i o n s :

x = L = 22, y = 0

x = L = 2 2 , y ' = o B o t h b o u n d a r y c o n d i t i o n s e s t a b l i s h a f u r t h e r r e s t r i c t i o n on K :

K = 2 ~ m / L , m = 0, 1, 2 ,... E q u a t i n g t h e two e x p r e s s i o n s f o r K and l e t t i n g m = 1 r e s u l t s i n :

2 2 P = 4r E I / L

F o r a t h e r m a l l y i n d u c e d l o a d :

P c r = A E a A T

S u b s t i t u t i n g f o r t h e l o a d P and s o l v i n g f o r AT :

2 4n I K A T =

( 6 - 4 )

----- Maximum S t r e s s . --- The t o t a l s t r e s s ( u t ) a f t e r t h e c a s i n g has b u c k l e d e q u a l s ( n e g l e c t i n g o v a l a t i o n s t r e s s e s ) :

- Qt - 0 + a

The e x p r e s s i o n o f u a i s :

u a = P c r I A = EabTcr = c o n s t a n t ( 7 )

F o r a l o n g , s l e n d e r c a s i n g i t i s r e a s o n a b l e t o assume a b = Mc/I wh i l e s t r e s s e s remain below y i e l d . From e q u a t i o n ( 6 - 2 ) where v = 0:

= MR C O S K X ( u s i n g 4 )

The maximum bending s t r e s s o c c u r s a t t h e end p o i n t s when:

(‘b)max = MRrolI

The r e s t r a i n i n g moment a f t e r buck l ing ( M R ) i s found by t h e p rocedure o u t l i n e d by Boley and Weiner ( 1 9 6 0 ) . The t h e r m a l l y induced l e n g t h change ( s ( T ) ) must equal t h e sum of t h e a x i a l l oad de fo rma t ion ( s ( P ) ) and t h e c a s i n g d e f l e c t i o n ( s ( y ) )

( F i g u r e B-2a) :

The e x p r e s s i o n s f o r t h e thermal and a x i a l d i s p l a c e m e n t s a r e

s ( T ) = a A T L ( 1 0 )

6 ( P ) = PcrL/AE = c o n s t a n t (11 1

T h e d e f l e c t i o n d i s p l a c e m e n t i s e q u i v a l e n t t o (where ds i s measured a long t h e c a s i n g ) :

q ( y ) = / o L ( d s - d x )

U s i n g a T a y l o r e x p a n s i o n o f t h e i n t e g r a n d and n e g l e c t i n g h i g h e r o r d e r t e r m s r e s u l t s i n :

-MRK y ’ = s i n Kx

S u b s t i t u t i n g ( l o ) , ( l l ) , and ( 1 3 ) i n t o ( 9 ) and s o l v i n g f o r M:

A s a rgued i n t h e r e p o r t , once b u c k l i n g has o c c u r r e d , s ( T ) i s e n t i r e l y a b s o r b e d b y s (y ) ; hence s ( P ) r e m a i n s c o n s t a n t . C o n s e q u e n t l y u a r e m a i n s a t t h e c r i t i c a l b u c k l i n g s t r e s s . S u b s t i t u t i n g (6 -4 ) i n t o ( 9 - 5 ) y i e l d s :

I n s e r t i n g e q u a t i o n ( 1 4 ) i n t o ( 8 ) and e v a l u a t i n g I f o r a p i p e

y i e l d s :

(Ob)max = A E a A T c r [ a ( ~ T - AT,^)] 1 ’ 2 4 L r 0 / n ( r o + r i ) 2 7 (6-6)

S u b s t i t u t i n g (9-6) and ( 7 ) i n t o ( 6 ) r e s u l t s i n :

DISPLACEMENT RELATIONSHIPS

6(T) = 6(P) + 262(P) 262(Y)

= 6(P) + 262(Y) . -

F i g u r e 6-2. Therma l , A x i a l , and Beam D e f l e c t i o n R e l a t i o n s h i p s .

D i s p l a c e m e n t

A f t e r W a l l € o n t a c t ---

- C a s i n g s h a p e . A f t e r w a l l c o n t a c t o c c u r s t h e f i x e d end b o u n d a r y c o n d i t i o n s a t p o i n t "an1 s t i l l h o l d and t h u s e q u a t i o n ( 2 ) i s a p p l i c a b l e . Two new b o u n d a r y c o n d i t i o n s a t t h e p o i n t o f w a l l c o n t a c t a re :

x = R , y = e

x = R , y ' = 0

The f o u r t h b o u n d a r y c o n d i t i o n e s t a b l i s h e s :

K' = P / E R

The p l a u s i b i l i t y o f a c o n c e n t r a t e d s h e a r l o a d ( V ) as opposed t o a d i s t r i b u t e d l o a d a t t h e w a l l c o n t a c t i s a r g u e d i n C r a n d a l l and Dah1 (1959) . The a rgumen t i s based on t h e a s s u m p t i o n t h a t t h e beam-column i s s t r a i g h t a t t h e s u r f a c e c o n t a c t . T h i s s i t u a t i o n p r e c l u d e s t h e p o s s i b i l i t y o f a moment Mb.

P ( o r K ) b u t s e v e r a l cases a r e e v i d e n t . F i r s t whenever s i n K R and l - c o s K R e q u a l z e r o t h e e q u a t i o n h o l d s . However, when t h i s case i s i n s e r t e d i n t o t h e s t a t i c moment r e l a t i o n i t i s f o u n d

MR = -Mb. c a s i n g shape a f t e r w a l l c o n t a c t ( F i g u r e 8 b ) .

( A f t e r w a l l c o n t a c t , t h e shear f o r c e ( V ) i s n o n z e r o i n e q u a t i o n ( 3 ) . ) p o s s i b l e t o show M R = 0 a l s o .

E q u a t i o n ( 3 ) e s t a b l i s h e s a r e l a t i o n s h i p be tween MR, V and

T h i s f a c t i s n o t c o m p a t i b l e w i t h t h e de fo rmed

A n o t h e r p o s s i b i l i t y i s where MR and l - c o s K R e q u a l z e r o .

Because Mb i s assumed t o e q u a l z e r o i t i s

The p l a u s i b i l i t y o f t h i s f a c t can be seen b y e x a m i n i n g t h e f r e e b o d y d i a g r a m i n F i g u r e B - l c . F i g u r e B - l c i s s i m p l y t h e m i r r o r image o f F i g u r e B- lb w h i c h was used t o o b t a i n e q u a t i o n (8 -2 ) . O n l y t h e moment MR i s m i s s i n g . A n o t h e r h e u r i s t i c a rgument i s f o u n d b y assuming F i g u r e s B - l b and B - l c a r e d i v i d e d a t t h e change i n c u r v a t u r e . A t t h e p o i n t o f c o u n t e r f l e x u r e , M = 0. A sum o f moments i n F i g u r e B - l c r e s u l t s i n Pe = V R . A p p l y i n g t h i s r e l a t i o n s h i p t o t h e moment r e l a t i o n s h i p o f e q u a t i o n ( B - 1 ) where M = 0 r e s u l t s i n M R = 0. T h e r e f o r e , i f

a s t r a i g h t s e c t i o n e x i s t s i n t h e c a s i n g a t t h e w a l l c o n t a c t , b o t h end moments a r e r e p l a c e d w i t h c o n c e n t r a t e d s h e a r f o r c e s

( V ) . From t h e c o n d i t i o n MR = 0 and 1-cos K ' x = 0 i n e q u a t i o n

( 3 ) , t h e shape o f t h e c a s i n g be tween p o i n t s ' 'a" and ''b" i s :

K ' = 2 n / R

A t t h e moment o f c o n t a c t R = L / 2 o r K ' = 4 r l L . Use o f t h e t h i r d b o u n d a r y c o n d i t i o n r e s u l t s i n Pe = V R .

I n o r d e r t o make c o m p a r i s o n s be tween e q u a t i o n ( 4 ) and (8 -7) i t i s d e s i r a b l e t o d e f i n e K = 2 r l L i n w h i c h c a s e e q u a t i o n ( 6 - 7 ) becomes :

s i n 2Kx ) ( 1 6 ) V

Y = p ( X - z .

Up u n t i l - w a l l c o n t a c t , t h e c a s i n g shape was d e s c r i b e d b y t h e c o s i n e e x p r e s s i o n ( 4 ) . A f t e r c o n t a c t , e q u a t i o n ( 1 6 ) r e q u i r e s an i n s t a n t a n e o u s change i n c a s i n g shape. The p r e d i c a m e n t r e s u l t s f r o m t h e a s s u m p t i o n o f a s t r a i g h t s e c t i o n

a t t h e i n s t a n t o f c o n t a c t . T h e r e i s a c t u a l l y a t r a n s i t i o n zone where b o t h a moment and a shear f o r c e e x i s t , b u t t h e

s i g n i f i c a n c e o f t h e c a s i n g moment s h o u l d g r a d u a l l y d e c r e a s e a f t e r w a l l c o n t a c t . *

moment and a shear ( o r more r e a l i s t i c a l l y , a d i s t r i b u t e d f o r c e ) e x i s t s i m u l t a n e o u s l y , a more d e t a i l e d a n a l y t i c s o l u t i o n i s r e q u i r e d . However, an a l t e r n a t i v e i s t o d e s c r i b e t h e c a s i n g

Because o f t h e e x i s t e n c e o f a t r a n s i t i o n zone where b o t h a

shape as a " s p l i t c o s i n e b e l l " i n t h e t r a n s i t i o n zone. The s p l i t - c o s i n e f o r m u l a t i o n i s d e v e l o p e d i n a subsequen t s e c t i o n .

F i g u r e s B - l a and B-2b. I t i s t h e l e n g t h be tween t h e f i x e d end and t h e p o i n t o f w a l l c o n t a c t . The l e n g t h ( a ) s h o r t e n s as t h e c a s i n g segment a g a i n s t t h e w a l l l e n g t h e n s due t o i n c r e a s e d d e f o r m a t i o n . An e x p r e s s i o n f o r t h e v a r i a b l e l e n g t h ( 2 )

p e r t a i n i n g t o e q u a t i o n ( 1 6 ) i s f o u n d f r o m t h e c o n d i t i o n t h a t t h e t h e r m a l d i s p l a c e m e n t ( a ( T ) ) e q u a l s t h e sum o f t h e a x i a l l o a d d i s p l a c e m e n t ( a ( P ) ) and t w i c e t h e c a s i n g d e f l e c t i o n ( 6 * ( y ) ) ( F i g u r e B-2b):

V a r i a b l e - ---- L e n g t h -- f k ) . - -- The v a r i a b l e l e n g t h ( R ) i s shown i n

The e x p r e s s i o n s f o r d i s p l a c e m e n t s a r e e q u a t i o n s ( l o ) , ( l l ) , and ( 1 2 ) as b e f o r e . S u b s t i t u t i n g e q u a t i o n ( 1 6 ) i n t o e q u a t i o n ( 1 2 ) r e s u l t s i n :

* g i m p o r t a n c e o f t h e moment c o u l d i n c r e a s e a g a i n i f enough t h e r m a l e l o n g a t i o n o c c u r r e d such t h a t t h e c a s i n g began t o b u c k l e i n t h e r e g i o n o f w a l l c o n t a c t . (W i th a t h i n - w a l l e d . p i p e i t i s more l i k e l y t h a t p l a s t i c d e f o r m a t i o n such as k i n k i n g o r c o l l a p s e w o u l d o c c u r b e f o r e assuming a h i g h e r b u c k l i n g mode.) The b u c k l i n g phenomenon i s e a s i l y v i s u a l i z e d i f a f l e x i b l e b a r o r n o t e c a r d i s b u c k l e d a g a i n s t a s o l i d s u r f a c e .

I t i s i m p o r t a n t t o n o t e t h e d i f f e r e n c e i n t h e c a s i n g d e f l e c t i o n be tween e q u a t i o n s ( 1 3 ) and (B-9). By c o m b i n i n g e q u a t i o n s ( 1 3 ) and (B-5) i t i s e a s i l y shown:

al(y) = aL(AT - AT^^) (B-10)

The e x p r e s s i o n f o r c a s i n g d e f l e c t i o n a f t e r w a l l c o n t a c t i s g i v e n b y (B-9) b u t V = eP/R; hence:

3 e2 9 ( Y ) = (B-11)

A t t h e i n s t a n c e o f w a l l c o n t a c t , e = ymax = 2MR/P. E x p r e s s i n g MR b y ( B - 5 ) and s u b s t i t u t i n g i n t o (B-11) y i e l d s :

2 6,(y) = aL(AT - AT^^) 6 / r (B-12)

2 E q u a t i o n s (B-10) and (B-12) d i f f e r b y t h e f a c t o r 6 1 ~ . A r e l a t i o n s h i p f o r t h e v a r i a b l e l e n g t h i s f o u n d b y

s u b s t i t u t i n g ( l o ) , (ll), and (B-11) i n t o (B-8) w h i c h y i e l d s :

(B-13)

- b u t

(8 -1 5 ) 2 2 = 471 E I / L ' c r

. S u b s t i t u t i n g t h i s e x p r e s s i o n i n t o (8-13) , e v a l u a t i n g I and A, and s o l v i n g f o r R ( A T > AT^^ 1:

2 1 = 7 e 2 L / [ a A T t - r2(r i+ r: )] (B-14) R

I t can be shown !tl = ( 1 2 / r L ) L a t t h e i n s t a n t o f c o n t a c t r a t h e r

t h a n 1 / 2 L.

I n t h e above f o r m u l a t i o n , i t was assumed t h e d i s p l a c e m e n t

f r o m t h e a x i a l l o a d r e m a i n e d c o n s t a n t ( P = P c r ) . new c r i t i c a l b u c k l i n g l o a d e x i s t s f o r t h e s h o r t c a s i n g o f l e n g t h ( a ) w h i c h i s 4 t i m e s as g r e a t .

However, a

2 4r E 1 P = ''7 4

F r i c t i o n a l o n g t h e w a l l c o u l d p r o v i d e a mechanism f o r t h i s new t h r e s h o l d t o be reached . A l s o i f t h e c a s i n g was t o assume a h i g h e r b u c k l i n g mode, l a r g e r a x i a l l o a d s t h a n t h e c r i t i c a l l o a d a s s o c i a t e d w i t h t h e f i r s t b u c k l i n g mode seemed p o s s i b l e . C o n s e q u e n t l y , an e x p r e s s i o n f o r R was d e r i v e d w h i c h p e r m i t t e d an i n c r e a s e i n t h e a x i a l l o a d ( P ) , A q u a d r a t i c e x p r e s s i o n r e s u l t s :

(B-16) 5 = (3e + [9e4 + ~ A T L 2 2 r (do+d i ) ]1 '2 } 2 2 /4aATL

The l o a d P was e x p r e s s e d as r 2 E I / R 2 . T h i s i s t h e i n i t i a l v a l u e o f t h e l o a d P. The t h r e s h o l d f o r t h e l o a d P w o u l d b e 4 r E I I R . The two e x p r e s s i o n s f o r R , e q u a t i o n s (B-14) and (B-15) a r e compared i n T a b l e B - I . From t h e d i s c r e p a n c i e s i n T a b l e B-1, i t i s e v i d e n t t h a t t h e l e n g t h ( 2 ) i s a l i k e l y s o u r c e o f e r r o r i n t h e a n a l y t i c a l e x p r e s s i o n f o r t h e maximum s t r e s s d e r i v e d be low. (Good agreement e x i s t s be tween (B-16) and ( 1 7 ) and w i l l be d i s c u s s e d i n a subsequen t s e c t i o n . ) The a p p r o x i m a t i o n f o r beam d e f l e c t i o n d i s p l a c e m e n t ( s ( y ) ) was examined as a p o s s i b l e s o u r c e o f e r r o r i n t h e t h e o r e t i c a l d e t e r m i n a t i o n o f R. The second t e r m i n t h e T a y l o r e x p a n s i o n o f s ( y ) was i n c o r p o r a t e d b u t i t s i n f l u e n c e was i n s i g n i f i c a n t as o r i g i n a l l y assumed.

a f t e r t h e c a s i n g has b u c k l e d and c o n t a c t e d t h e w a l l i s e x p r e s s e d b y t h e sum o f u a and e x p r e s s i o n f o r ( ab )max i s a g a i n M c / I .

Maximum s t r e s s a f t e r wall c o n t a c t . -I_ The t o t a l s t r e s s ( u t )

( e q u a t i o n 5). The The v a l u e f o r M i s

T a b l e B - I . Comparison o f A n a l y t i c R e s u l t s f o r 13-318 i n c h 54.5 D D f N-80 C a s i n q w i t h an 'Unsupportede L e n g t h o f l O O D -

(ATCr = 71.7 F =: 22'C)

temp h o l e unsupported l e n g t h c h an a e d i s D 1 ace me n t qap c o n s € a n t P v a r . F- S P l l t maximum s t r e s s

Yma x - A T n c r D

- . e - 2 a 1 D L

2R3 - L

- awl 0 pwl = Y O Y

1.5 0.978 .250 . 500 . 750 . 978

2.0 1.370 . 250 .500 .750

1 . 000 1.370

. 079 . 318

.714 1.215

.162 . 364 . 649 1.214

.828 - 8 7 0 0944

- 7 1 8 .750 - 8 0 5 .888

.825 . 860 - 9 2 0

1. 000

- 7 1 6 .743 .786 .854

1 . 000

. 300 0 4 1 1 .486 . 519 0745

. 344 - 7 0 9 ,602 . 660

- 6 5 9 ,659

4.0 2.380 . 500 .054 .520 -517 - 8 3 9 . 750 .120 .547 .538 1 . 094 1.268 1. 000 - 2 1 4 . 587 - 5 7 1

1.500 . 048 - 7 1 3 - 6 7 1 1 . 362 2.380 1.214 1.131 1.000 1.466 1.021

f o u n d f r o m e q u a t i o n (B-2) where MR = 0.

e q u a t i o n ( 1 1 ) and s u b s t i t u t i n g i n t o ( 8 - 2 ) r e s u l t s i n :

E x p r e s s i n g y b y

s i n K ' X K' M =

The maximum moment w i l l o c c u r a t t h e q u a r t e r p o i n t s : X / R = 1 / 4 o r X / R = 3 /4 . ( T h i s d i f f e r s f r o m t h e c o s i n e e q u a t i o n ( 4 ) where t h e maximum b e n d i n g s t r e s s o c c u r s a t x = L / 2 . ) Thus:

( 'b) max = V r o / K ' I

1 / 2 K ' = 2n /R = ( P / E I )

h en ce

) = n E e r o / 2 R 2 'b(max

T h e r e f o r e

u t = EaATCr + r E e r o / 2 R 2

where i s d e s c r i b e d b y e q u a t i o n 8-14 o r B-16. The use o f t h e n u m e r i c a l l y d e t e r m i n e d R i n e q u a t i o n ( 1 9 ) d i d i m p r o v e t h e c o r r e s p o n d e n c e w i t h t h e n u m e r i c a l r e s u l t s .

I t s h o u l d b e n o t e d t h a t an i n c o n s i s t e n c y e x i s t s i n t h e above d e r i v a t i o n . t h e e x p r e s s i o n o f R (B-16) w h i l e t h e a x i a l s t r e s s c a n n o t e x c e e d P c r / A i n e q u a t i o n ( 1 9 ) .

r e s u l t s i n t h e t o t a l s t r e s s ( a t ) i n c r e a s i n g much f a s t e r w i t h i n c r e a s i n g AT. The r e s u l t i n g v a l u e s do n o t c o r r e s p o n d w i t h t h e n u m e r i c a l r e s u l t s and t h u s e q u a t i o n ( 1 9 ) was n o t a l t e r e d .

The a x i a l l o a d (P ) i n c r e a s e s beyond P c r i n

R e p l a c i n g AT^^ w i t h A T i n ( 1 9 )

f o u n d f r o m e q u a t i o n ( 6 - 2 ) where MR = 0.

e q u a t i o n ( 1 1 ) and s u b s t i t u t i n g i n t o ( 6 - 2 ) r e s u l t s i n : E x p r e s s i n g y b y

M = { , s i n K

The maximum moment w i

1 o c c u r a t t h e q u a r t e r p o i n t s : X / & = 1 / 4 o r X / R = 3 / 4 . ( T h i s d i f f e r s f r o m t h e c o s i n e e q u a t i o n ( 4 ) where t h e maximum b e n d i n g s t r e s s o c c u r s a t x = L / 2 . ) Thus:

. . = V r o / K I I ( O b ) ma x

V = Pe /R

K ' = 2 r / R = ( P / E I ) 1 / 2

2 = r E e r o / 2

T h e r e f o r e

( 1 9 ) ut = EaAT,, + r E e r 0 / 2 R 2

where ! t i s d e s c r i b e d b y e q u a t i o n 6-14 o r 6-16. The use o f t h e n u m e r i c a l l y d e t e r m i n e d R i n e q u a t i o n ( 1 9 ) d i d i m p r o v e t h e c o r r e s p o n d e n c e w i t h t h e n u m e r i c a l r e s u l t s .

I t , s h o u l d be n o t e d t h a t an i n c o n s i s t e n c y e x i s t s i n t h e above d e r i v a t i o n . The a x i a l l o a d ( P ) i n c r e a s e s beyond Pcr i n

t h e e x p r e s s i o n o f R ( 6 - 1 6 ) w h i l e t h e a x i a l s t r e s s c a n n o t exceed P c r / A i n e q u a t i o n ( 1 9 ) . r e s u l t s i n t h e t o t a l s t r e s s (u t ) i n c r e a s i n g much f a s t e r w i t h

i n c r e a s i n g AT. The r e s u l t i n g v a l u e s do n o t c o r r e s p o n d w i t h t h e n u m e r i c a l r e s u l t s and t h u s e q u a t i o n ( 1 9 ) was n o t a l t e r e d .

R e p l a c i n g AT^^ w i t h A T i n ( 1 9 )

SD 1 i t-Cos i n e F o r mu 1 a t i on

F i g u r e 10a d e m o n s t r a t e s t h e two a n a l y t i c a l e x p r e s s i o n s ( 3 ) and ( 1 6 ) do n o t a g r e e i n shape when c o n t a c t o c c u r s . T h e r e i s a t r a n s i t i o n p e r i o d f r o m e q u a t i o n ( 3 ) and ( 1 6 ) . R a t h e r t h a n r e s o r t t o a more c o m p l i c a t e d m a t h e m a t i c a l d e s c r i p t i o n o f t h e c a s i n g d u r i n g t h e t r a n s i t i o n , t h e c a s i n g shape was d e s c r i b e d as

a s p l i t - c o s i n e b e l l . One mus t assume t h e end moments c o n t i n u e t o a c t a f t e r w a l l c o n t a c t . Summing moments a b o u t an end o f t h e f r e e b o d y d i a g r a m y i e l d s :

MR = P e / 2 (B-17)

The f o l l o w i n g shape d e s c r i p t i o n r e s u l t s when t h e end moment e x p r e s s i o n i s s u b s t i t u t e d i n t o ( 3 ) :

y = 7 e ( 1 - C O S K x ) (B-18)

An i d e n t i c a l e x p r e s s i o n pops o u t f r o m e q u a t i o n ( 2 ) i f V i s s e t e q u a l t o z e r o and t h e b o u n d a r y c o n d i t i o n s a p p l i c a b l e a f t e r w a l l con t a c t a r e a p p l i e d .

d e s c r i p t i o n o f l e n g t h ( a 3 ) f o l l o w s t h e s t e p s o u t l i n e d i n t h e p r e v i o u s s e c t i o n . E q u a t i o n (B-8) d e s c r i b e s t h e t h e r m a l d i s p l a c e m e n t ( s ( T ) ) . E q u a t i o n (8 -13) d e s c r i b e s t h e beam

d i s p l a c e m e n t ( ~ ( y ) ) . The a x i a l d i s p l a c e m e n t t e r m c o r r e s p o n d s t o ( 1 1 ) e x c e p t t h a t t h e a x i a l l o a d i s a l l o w e d t o i n c r e a s e beyond P c r .

L e n q t h d e s c r i p t i o n R 3 . The d e r i v a t i o n f o r t h e

I t f o l l o w s t h a t R 3 i s r e p r e s e n t e d by :

(6-19) 2 P R 3 = ( n e ) / (8LoAT - m) 2 I f t h e e x p r e s s i o n f o r t h e l o a d P = 7 E I / R 2 i s s u b s t i t u t e d i n

( 6 - 1 8 ) and t h e r e s u l t i n g q u a d r a t i c i s s o l v e d f o r R 3 , t h e n :

1 / 2 = ( ( n e ) A'+ [ ( x ~ ) ~ A ' + 256L 2 2 n IAaAT] ) / 1 6 L A t r ~ T R 3

V a l u e s of k3 f o r v a r i o u s t e m p e r a t u r e changes ( A T ) and h o l e

gaps a r e t a b u l a t e d i n T a b l e B - I . The agreement be tween ( 1 7 ) and (B-19) i s v e r y good.

Maximum s t r e s s . e x p r e s s e d b y e q u a t i o n s (6), ( 7 ) , and ( 8 ) e x c e p t t h a t MR i s e q u i v a l e n t t o P e / 2 (B-17) . An e x p r e s s i o n i d e n t i c a l t o ( 1 9 ) r e s u l t s .

The t o t a l maximum s t r e s s ( u t ) i s ----_II

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Geothermal Well Casing Buckling

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