1 borehole instabilities as bifurcation

8/8/2019 1 Borehole Instabilities as Bifurcation

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I n t . J , R o c k M e c h . M i n . S c i . & G e o m e c h . A b s t r . Vol. 25, No. 3, pp. 159-170, 1988 0148-9062/88 $3.00 + 0.00Prin ted in Grea t Bri t a in Perga mon Press p i c

B o r e h o l e I n s t a b i l i t i e s a s B i f u r c a t i o n

P h e n o m e n a

I. VARDOULAKIS*J. SULEMI"

A. GUENOT~

In this pap er a bifurcation analysis o f a deep borehole un der uniform str ess at

infinity is pre se nte d in order to determ ine the critical rupture load and to

predic t the failure mechanism. I t demonstrates the fac t th at failure is not a

stric t materia l prope rty but depends on the stress path a nd boundary conditions

o f the system. R oc k is described b y the constitutive equations o f a deformation

theory o f plastic i ty fo r r igid-plastic pressure sensitive m aterial w ith dilatancy.

The application o f the mod el to real cases shows that this bifurcation analysis

is in goo d agreement with ex perim enta l and f ie ld observations.

1. INTRODUCTION

Borehole breakouts and exfoliations are important

phenomena that influence the engineering design of

drilling hardware and can become critical for the

progress of the drilling process. Breakouts lead in

general to progressive deterioration of the borehole.

Wellbore breakouts are attributed to the existence of

significant deviatoric stresses that act in the horizontal

plane at great depth and to the stress concentration

around the borehole [1].Most of the existing work devoted to the prediction o f

borehole instabi lity is based on elastoplastic models that

are usually calibrated on test data from conventional

triaxial compression experiments. Most frequently it is

assumed that rupture occurs when the stress state is

beyond the elastic limit that is usually corresponding to

the peak of the stress-strain experimental curve; e.g.

Cbeatham [2]. However this procedure presents two

major drawbacks: (a) uncertainty up to 800% in pre-

dicting the rupture load of well-instrumented, hollow

cylinder tests [3]; (b) inadequacy to describe some

surface rupture modes usually referred to as "axial

cleavage fracture" [4] or "extension rupture" [5].

The first drawback is mainly due to the assumption of

linear elastic behaviour up to failure. It should be

understood that the full stress concentration at the

borehole wall (a factor 2 in the isotropic case) is never

reached in practice. Recently, Santarelli et al. [6] have

derived the stress field around the borehole in case of a

hypoelastic law with Young's modulus depending on the

minor principal stress. In this case, the stress concen-

tration factor was indeed found less than 2. Both

* Department of Civil and Mineral Engineering, University ofMinnesota, Minneapolis, MN 55455-0220, U.S.A.f Centre d'Enseignementet de Recherche en M6canique des Sols,

Ecole Nationale des Ponts et Chaus6es, Paris, France.~: La Soci6t6 Nati6nale Elf Aquitaine (Production), Pau, France.

R . M . M . S . 2 5 / 3- - - E

drawbacks are also related to the ad hoc assumption that

rupture should naturally be associated to the elastic-

plastic limit.

The concept of bifurcation used in this paper provides

an alternative to describe rupture. It allows to differ-

entiate the rheological behaviour of the material and the

rupture phenomenon. Furthermore it is used to explain

and predict the occurrence of the various observed

failure modes.

In this paper a bifurcation analysis of the borehole

problem for a rigid-plastic pressure sensitive material

with dilatancy is proposed. First the material behaviour

is described and the constitutive equations are presented.

Then, a solution for the stress field around the borehole

is derived. Final ly, a complete bifurcation analysis of the

borehole problem is proposed in order to predict the

failure mode and the critical value of the stress at infinity

at failure.

2 . P R O B L E M S T A T E M E N T A N D M A T E R I A L

B E H A V I O U R

2 .1 . Problem s ta tementWe consider here a borehole in a deep rock layer, as

illustrated in Fig. 1. Furthermore, we are interested in a

deep section of the borehole. Under these conditions we

may assume that any deformation of the rock is taking

place in a plane normal to the borehole axis. We

are dealing thus with a plane-strain problem and all

kinemat ical and statical quantities will be described with

respect to a fixed-in-space polar co-ordinate system

(r, 0, z) with the z-axis coinciding with the borehole axis.

Let (%) denote the Cauchy stress tensor:

( ~ i j ) = ( ~ 0 ' ( l )

0

and let (%) denote the infinitesimal strain tensor with

159



1 6 0 V A R D O U L A K I S e t a l . : B O R E H O L E I N S T A B I L I T I E S

I-: ' . / l t ~ ' l l l / " i ~ ' " / t ~ l l l t l l

/

, . . , r o . - .

• z > > r o

E : z = 0 ' , .

f/

/ •

// •

/ • ,

I •

II .

I .

I .I

, -.

\

\\ .

\ •

\: . . . . \

• ( r • , \

. i / . ]

• I

. //

/• • f

/

J

Fig. 1 . Geom etr ic layout o f a deep boreho le .

( a )

( b )

r e s p e c t t o a n i s o t r o p i c r e f e r e n c e c o n f i g u r a t i o n o f t h e

b o r e h o l e :

(E~ ) = E0 (2 )

0

L e t o0 a n d o ~ d e n o t e t h e i n t e r n a l b o r e h o l e p r e s s u r e a n d

t h e i s o t r o p i c s t re s s a t i n f i n it e d i s t a n c e f r o m t h e b o r e h o l e ,

r e s p e c t i v e l y :

0 < 0 0 < 0"~ .

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

c o n d i t i o n :

0 0 = 0 ( 3 )

B y a s s u m i n g c o m p r e s s i o n n e g a t i v e w e h a v e :

a r ( r o ) = - a o , a r ( o o ) = - o ~ . (4 )

F o r a n e q u i l i b r i u m s t r e s s f i e l d :

_ _ 1

dar + _ ( a t - 0 0 ) = 0 , ( 5)d r r

a n d c o n s e q u e n t l y f o r

r o < r < c o t he n a 0 < a , < 0 . (6)

W h e n 0 ~ i s s m a l l a s c o m p a r e d t o t h e u n i a x i a l c o m -

p r e s s s t r e n g t h 0 ¢ o f t h e r o c k , t h e d e f o r m a t i o n o f t h e

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

b o r e h o l e r a d i u s , u n i q u e l y d e t e r m i n e d b y t h e b o u n d a r y

s t re s s es ( F i g . 2a ) . T h e d e f o r m a t i o n o f th e b o r e h o l e a t a

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

a x i s y m m e t r i c d i s p l a c e m e n t v e c t o r f i e l d :

= o , 0 } T . ( 7 )

T h e b o r e h o l e i s a s s u m e d t o b e u n s t a b l e a t C a s s o o na s in a d d i t i o n t o t h e a b o v e " t r i v i a l " s o l u t i o n 6i a n o t h e r

n o n - t r i v i a l s o l u t i o n 6 i e x i st t h a t f u lf il h o m o g e n e o u s

b o u n d a r y c o n d i t i o n s . I n t h i s c a s e a n e q u i l i b r i u m b i f u r -

c a t i o n i s s a i d t o b e t a k i n g p l a c e a t C . T h e c o r r e s p o n d i n g

b i f u r c a t i o n m o d e i s t h e n a l i n e a r c o m b i n a t i o n o f t h e

t r iv i a l a n d o f th e n o n - t r i v ia l m o d e :

u~ = clfi~ + c26~ (8)

A p o s s i bl e b i f u r c a t i o n m o d e is w a r p i n g o f t h e b o r e -

h o l e w a l l ( F i g . 2 b ) . A c c o r d i n g t o V a r d o u l a k i s a n d

M f i h l h a u s [ 7] , t h e w a r p i n g m o d e m u s t b e a c c o u n t e d f o r

f o r m a t i o n o f s h e a r b a n d s ( F i g . 3 b ) o r e x f o t i a t i o n a t t h eb o r e h o l e w a l l d u e t o a c t i v a t i o n a n d s u b s e q u e n t u n s t a b l e

p r o p a g a t i o n o f p r e - e x is t i n g l a t e n t s u r fa c e p a r a ll e l c r a c k s

( F i g . 3 a ) . S u r f a c e p a r a l l e l c r a c k i n g a n d s h e a r b a n d i n g

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

wall

2.2• Material behaviour

A s w e h a v e s e e n a b o v e , t h e l i n e a r e l a s ti c t h e o r y i s n o t

s u f fi c ie n t to d e s c r ib e t h e r e a l b e h a v i o u r o f r o c k s . V a r i o u s

p h e n o m e n a l i k e p r e s u r e s e n s i t i v i t y , s t r a i n h a r d e n i n g ,

n o n - l i n e a r v o l u m e t r i c s t r a i n s e v e n i n t h e s o - c a l l e d

" l i n e a r " p a r t o f t h e s t re s s - s t r a i n l a w , m u s t b e t a k e n

i n t o a c c o u n t . T h i s i s p o s s ib l e w i t h a t h e o r y o f p l a s t ic i t y

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

m a t e r i a l

( a )

Fig . 2 . Deformat ion modes : ( a ) ax isymmetr ic d isp lacement , (b)warp ing .



V A R D O U L A K I S e t a l . : B O R E H O L E I N S T A B I L I T I E S 161

ExfoLiation

i i i ! i ( a )

Shear- banding

• • . • , " . •

. . . ~ •

•' . " • " / ~ " . . '.

• " " ~ " " • " " " . " % 1 • " •

( b )

Fig . 3 . Fa i lu re m ode s : (a ) e x fo l ia t ion , (b ) s h e a r -b a nd ing .

T h r o u g h o u t , t h is p a p e r w e w i ll r e f e r t o e x p e r i m e n t a l

d a t a o f a l im e s t o n e ( C a l c a i r e d ' A n s t r u d e ) . A n e x p e r i -

m e n t a l p r o g r a m o f u n ia x i a l a n d t r i a x ia l t e s t s h a s b e e np e r f o r m e d b y F a b r e a n d O r e n g o [8 ] a n d o f t hi c k w a ll

c y l i n d e r s t es t s b y G u e n o t [3 , 9 ]. F o r t h e u n i a x i a l a n d

t r ia x i a l t e s t it is o b s e r v e d t h a t t h e s l o p e o f t h e u n l o a d i n gcu rves i s a lmos t ve r t i ca l P la s t i c s t ra ins a re l a rge ly

p r e p o n d e r a n t o n e l a s t ic s t ra i n s s o t h a t r i g i d -p l a s ti c i ty i s

i n t h i s c a s e a n a c c e p t a b l e a s s u m p t i o n . A v e r y s i m i l a r

a n a l y s i s c a n b e e a s i l y d e v e l o p e d w i t h a n e l a s t o - p l a s t i c

m o d e l

F o r t h e s a k e o f s i m p l i c i t y w e a d o p t h e r e s m a l l s t r a i n

d e f o r m a t i o n t h e o r y . T h e C a u c h y s t r e s s a n d t h e

i n f in i te s im a l s t r a in a r e d e c o m p o s e d i n t o t h e i r d e v i a t o r i c

and sphe r i ca l pa r t s a s fo l lows :cri j = s i j + p r o ; p = akk / 3 , (9 )

¢~j= e 0 + G6o /3; G = Ekk, (10)

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

s t ra in . In th i s pape r , we wi l l d i s cus s the fo l lowing f in i t e

c o n s t it u ti v e e q u a t io n s o f a d e f o r m a t i o n t h e o r y f o r

c o h e s i v e - f r i c t i o n a l , s t r a i n h a r d e n i n g a n d d i l a t a n t

ma te r i a l [ 7] :

1 s 0 ( 1 1 )e~j = 2hs q - p '

G = D(g), ( 1 2 )

where g i s t he shea r ing s t ra in in t ens i ty

g = ~ . . (1 3)

In the cons t i tu t ive equ a t io n (11) q i s a con s tan t

p a r a m e t e r , r e l a te d t o t h e t e n s i le s t r e n g t h o f th e m a t e r i a l ,

and h~ i s a ha rde ning func t ion o f the shea r ing s t ra in

i n t e n s i t y . E q u a t i o n ( 1 2 ) c o n s t i t u t e s a c o n s t r a i n t f o r t h e

s t ra i n s , a n d c o n s e q u e n t l y t h e m e a n p r e s s u r e p i s a

k i n e m a t i c a l l y i n d e t e r m i n a t e q u a n t i t y .

I n o r d e r t o i ll u s tr a t e t h e m e a n i n g o f t h e a b o v e

equa t ions (11) and (12) a s e t o f s tre s s inva r i an t s i si n t r o d u c e d a s f o l l o w s :

Shea r s t re s s in t ens i ty :

Iz = ~ 2 s q s o , (14)

S t re s s ob l iqu i ty :

= - - . ( 1 5 )t a n O ~ q - - P

F r o m t h e c o n s t i t u t i v e e q u a t i o n ( 1 1 ) a n d t h e d e f i n i t i o n s

(14) and (15) we f ind tha t :

tan Ipog - h s ( 1 6 )

B y a d o p t i n g t h e u n i q u e c u r v e h y p o t h e s i s o f th e d e f o r -m a t i o n t h e o r y [ 1 0 ] :

t a n ~ = F ( g ) . (17)

hs in equa t io n (11) can be iden t i f i ed a s a s ecan t mod u lus

o f t h e a p p r o p r i a t e s t r e s s - s t r a in c u r v e t a n @ ~ = F ( g ) :

hs = tan @~ _ F (g ) (18)g g

F i g u r e s 4 a a n d b i l lu s t r a te t h e m e a n i n g o f t h e s tr e s s

o b l i q u i t y a n g l e 4 o a n d o f t h e s e c a n t m o d u l u s h s,

q( a )

tan~u

h s

g

(b )

Fig . 4 . S t r e s s ob l iq u i ty : (a ) de f in i t ion , (b ) s t r e s s ob l iq u i ty s t r a ind ia g ra m , h , = s e ca n t m odu lu s , h t = t a nge n t m o du lu s .



1 6 2 V A R D O U L A K 1 S et al.: B O R E H O L E I N S T A B I L I T I E S

3 . C A L I B R A T I O N O F T H E C O N S T I T U T I V E M O D E L

I n o r d e r t o i d e n t i f y t h e f u n c t i o n s F ( g ) a n d D ( g ) t h a t

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

(11, 12 and 17) on e shou ld ana ly se the p l ane s t ra in

c o m p r e s s i o n t e st . D u e t o t h e l a c k o f e x p e r i m e n t a l d a t a

o n p l a n e - s t r a i n t e s ts w e h a v e t o c a l i b r a te t h e c o n s t i t u t iv e

m o d e l o n t h e u n i a x i a l c o m p r e s s i o n t e s t ( F ig . 5 ) w h i c h is

a c o m m o n l a b o r a t o r y e x p e r i m e n t f o r w h i c h d a t a f o rv a r i o u s r o c k t y p e s a r e r e a d i l y a v a i l a b l e .

3.1 . Stra in-s tre ss law and dilatancy constraint

I n c a s e o f u n ia x i a l c o m p r e s s i o n , t h e C a u c h y s t r e s s

t e n s o r i s g i v e n b y t h e f o l l o w i n g m a t r i x :( 0 0 )( a , j ) = 0 0 , ( 1 9 )

0 (~::

w h e r e

FG z z = - - o ' < 0 ; 0 " =

x R 2 '

i s t he ax ia l s t re s s . In equa t ion (20) F i s t he ax ia l fo rce an d

R i s th e c u r r e n t r a d i u s o f th e s p e c i m e n .

S ta r t ing f rom a s t re s s f ree i so t rop ic in i t i a l conf igu r -

a t i o n C o f th e r o c k s p e c i m e n , t h e d e f o r m a t i o n C o ~ C

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

the in f in i t e s ima l s t ra in t ensor :

0 .

w h e r e

E z : < E rr = EO 0 ,

a n d

H- no R - RoE:: = ~ < 0; E,~ = ~ (2 3)

H0 R0

I n e q u a t i o n ( 2 3 ) H o ( H ) a n d R 0 ( R ) a r e t h e h e i g h t a n d

rad iu s o f the spec im en in C0(C) , r e spec t ive ly .

IF

H

Z T - rI , , ,

i l l l i i • • n i t

J r - O

ab a

2 R

Fig . 5 . A x i s ymmet r i c un iax ia l c ompr es s ion t es t .

(3 "S

Cm

%

Fig. 6. Mohr 's plane of s t resses .

I n t h e c o n s i d e r e d c a s e w e o b t a i n t h e f o l l o w i n g e x p r e s-

s ions for the va r ious s t re s s inva r i an t s :

(2 0 ) p = -o r / 3 ; ~ = ~ r / v '~ , (24)

t a n $ ~ a / x / ~+ a / 3 " (25)

B e t w e e n th e m o b i li z e d M o h r - C o u l o m b f r ic t io n a n g l e

~bm of the ma te r i a l (F ig . 6 ) and the s t re s s ob l iqu i ty an gle

~k~ the fo l low ing re l a t ionsh ip i s ho ld ing:

s in ~bm = (x/~ /2) tan ~o1 + tan ~bo/(2x/~ ' (26)

w h e r e t h e m o b i l i z e d c o h e s i o n C m i s g i v e n b y :

Cm = q ta n ~bm. (27)T h e e x p e r im e n t a l r e s u l ts m a y b e p l o t t e d i n t e r m s o f

(22) shea r ing s t re s s in t ens i ty vs shea f ing s t ra in in t ens i ty g :

= T ( g ) . ( 2 8 )

I n t h e c o n s i d e r e d c a s e o f a u n i a x i a l c o m p r e s s i o n , w e

have :

g = 2 (E , - E=) /x /f3 . ( 29)

F i g u r e 7 s h o w s t h e s t r e s s - s t r a i n d a t a f r o m a u n i a x i a l

c o m p r e s s i o n t e s t o n t h e l im e s t o n e ( C a l e a i r e d ' A n s t r u d e ) .

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

T = C,g + C2Ln(1 + C3g) , ( 30)

0 ~ 0

O0 1 5 O /r n o /

, ,. 4 D o t

~ e ~ F i t

6 / •

J . J . . "5

I I I

O 0 . 1 0 . 2 O . ~

S hear s t r o i n , g ( % )

F i g . 7. S t re s s - s t ra i n c u r v e ¢ = T ( g ) , e x p e r i m e n t a l d a t a a n d f i t t in g f o rt he l imes tone (Ca lc a i r e d 'A ns t r ude) .



V A R D O U L A K I S e t a l . : B O R E H O L E I N S T A B I L I T I E S 1 63

w h e r e

C~ = 1.11390 x I (P M Pa;

C 2 = 11 .8504 M Pa;

C3 = - 242,08 (31)

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

c h a n g e s c a n b e a p p r o x i m a t e d b y a n a p p r o p r i a t e f u n c -t i o n D ( g ) a s i n d i c a t e d b y e q u a t i o n ( 12 ). I n t h e c o n -

s i d e r e d c a s e ( C a l c a i r e d ' A n s t r u d e ; F i g . 8 ) i t w a s f o u n d

tha t :

D = C 4 g + CsL n (1 + C 6 g ) , (32)

w h e r e

C4 = 24.25;

C5 = - 1.0;

C6 = 24.823. (33)

3.1. I. D iscussion on the volumetric flow rule. F o r t h eg i v e n s et o f d a t a , e q u a t i o n s ( 3 2 ) a n d ( 33 ), t h e c o m p u t e d

di l a t ancy func t ion i s found to y i e ld unrea l i s t i ca l ly h igh

v a l u e s f o r t h e r a t e o f v o l u m e i n c r e a s e . T h i s i s c e r ta i n l y

d u e t o t h e f a c t t h a t u s u a l l y (a s in t h i s c a s e w a s a c t u a l l y

d o n e ) , r a d i a l s t r a i n s a r e m e a s u r e d l o c a l l y b y t h e p l a c e -

m e n t o f s t r a in g a u g e s o n t h e sp e c i m e n s s u r f a ce . L o c a l

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

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

s p e c i m e n .

T h e e x p e r i m e n t a l r e s u l t s , h o w e v e r , i n d i c a t e a s t r o n g

t e n d e n c y t o w a r d s d i l a t a t i o n . I t i s a s s u m e d t h a t t h e

expe r ime nta l da t a a re re l i ab le for g < g t --- 1 .9 x 10-3 ;

f o r g > g~ w e a s s u m e t h a t t h e r a t e o f v o l u m e i n c r e a s e

d D / d g i s cons tan t and equa l t o i t s va lue a t g = g~.

I n o r d e r t o d e t e rm i n e t h e m a g n i t u d e o f p a r a m e t e r q

a d d i t i o n a l i n f o r m a t i o n o n y i e l d c o n d i t i o n s i s n e e d e d .

W e w i l l s h o w i n t h e n e x t s e c t i o n h o w a b i f u r c a t i o n

a n a l y s i s o f t h e t r ia x i a l c o m p r e s s i o n t e s t a ll o w s t o p r e d i c t

t h e v a l u e o f th e u n i a x i al c o m p r e s s i o n s t r e n g t h a ¢ a n d o f

the m ob i l i zed f r i c t ion angle a t fa i lu re ~b~. q i s r e l a t ed to

a~ and ~b~ th rou gh the fo l lowing eq ua t ion :

q = ~- co t ~b~ co t [v~ + -~ - ) . (34 )

F i r s t w e a s s u m e a v a l u e q 0 f o r p a r a m e t e r q . W ed e v e l o p a b i f u r c a t i o n a n a l y s i s t o d e t e r m i n e a ~ a n d ~ b ~ .

0 . 1 2

,<0 . 0 9

._=o 0 . 0 6

• ~ 0 .O3

• D a t a

F it

I ,, I0 ' . , , 0 . t 0 / 0 - 0 .2 0 . 3

• s e ~

- - 0 . 0 3 * ~ e e ' e ' l ~ e " e ' J l '° '°

S h e a r s t r a i n , g ( % )

Fig . 8 . Volumetr ic s t ra in -she ar curve ~ v : D (g) , exper imenta l da ta a ndfitting for the l imes tone (Ca ica i re d 'Ans t ru de) .

q l i s t hen g iven by equa t io n (34) . We i t e ra t e the p roces s

unt i l con verge nce o f the s e r ie s q~ to the l imi t q .

3.2. Bifurcation analysis of the axisymmetric uniaxialcompression test

F r o m e x p e r i m e n t a l o b s e r v a t i o n s , i t a p p e a r s t h a t s h e a r

b a n d f o r m a t i o n i s t h e m a j o r f a i l u r e m o d e f o r t h e

c o n s i d e r e d l i m e s t o n e ( C a l c a i r e d ' A n s t r u d e ) . U s i n g t h ed a t a o f t h e s t r e s s- s t r a i n c u r v e a n d o f t h e v o l u m e t r i c

cu rve i t i s pos s ib l e to deve lop a b i fu rca t ion ana ly s i s t o

pred ic t t he va lue of ax ia l s t re s s aB for wh ich shea r

b a n d f o r m a t i o n i s p o s s i b l e u n d e r a x i s y m m e t r i c u n ia x i a lc o m p r e s s i o n .

3.2.1. Incremental constitutive equations. T h e g e n e r a l

f o r m o f t h e i n c r e m e n t a l c o n s t i t u t i v e e q u a t i o n s c a n b e

de r ived f rom the i r f in i t e fo rm, equa t ions (11) and (12) ,

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

[11 , 12] ) . Le t ~u and o3 /j be the s t ra in ra t e and sp intensor , r e spec t ive ly :

g i j = ( V i . j "1 - V j , / ) / 2 ; f . b ij = ( V i . j - - V j . i ) 1 2 ( 3 5 )

w h e r e v / i s t h e v e l o c i ty a n d ( . )j = O/Ox~.

I n c r e m e n t a l c o n s t i t u t i v e e q u a t i o n s a r e w r i t t e n i nt e r e m s o f o b j e c t i v e J a u m a n n s t re s s r a te v / :

vor~~ dis ~ a , j + a ~ 6 J k s . ( 3 6 )

T h e r a t e f o r m o f t h e c o n s t it u t iv e e q u a t i o n s f o r th e

d e v i a t o r i c s t r e s s i s:

V s ~ j = _ [ : + 2 ( q _ p ) ( h _ h t ) S ,,~ , ,1q - P sin,S,m_]

X S/j + 2(q -- p )h ,~ u . (37)I n t h e a b o v e c o n s t i t u t iv e e q u a t i o n ( 37 ), t h e r a t e p o f

t h e m e a n p r e s s u r e is k in e m a t i c a l l y i n d e t e r m i n a t e . T h i s i s

a c o n s e q u e n c e o f th e a s s u m e d d i l a t a n c y c o n s t r a i n t ( 12 ).

S t a r ti n g f r o m a c o n s t ra i n t o f th e f o r m o f e q u a t i o n

( 1 2 ) , a n e x p r e s s i o n f o r t h e s t r a i n r a t e s c a n b e o b t a i n e dby t ime d i f fe ren t i a t ion :

w h e r e

iv =/~g, (38)

dD. 2eu~ij (39)f l= d g " ~ = g

T h e a b o v e i n c r e m e n t a l e q u a t i o n s ( 3 8 ) a n d ( 3 9 ) c a n b e

e v a l u a t e d f o r v a r i o u s s t a t e s o f i n it ia l s t r e s s in o r d e r t ob e u s e d i n a s h e a r b a n d a n a l y s i s .

I n t h e c a s e o f a x i sy m m e t r ic u n i a x i a l c o m p r e s s i o n a s i t

i s desc r ibed by equa t ions (19) and (21) (of . F ig . 5 ) the

cons t i tu t ive equa t ions (37) and (38) become ( s ee [13] ) :

~r,, = (1 - ta n ~k¢/3v/3)/~

+ # , ( ~ , , - ~ 0 0) - / a t e =

~r:. (I + 2 ta n ik=Iv/ 3)/~ 211t~,:

V •

o r ,, - a o o = 2 / ~ , ( e , , - ~ )

Vo ' , : : 2 p s G =

(40 )



1 6 4 V A R D O U L A K I S e t a l . : B O R E H O L E I N S T A B I L I T I E S

w h e r e #~ a n d # , a r e a p p r o p r i a t e s e c a n t a n d t a n g e n t s h e a r

m o d u l i :

# s = ( q - p ) h ~ ; IA = ( q - p ) h ~ , (41)

a n d

6 2 = x ' / '5 + 2 , b ' ( 4 2 )

3.2.2. S h e a r b a n d a n a l y s i s . S h e a r - b a n d a n a l y s i s h a s

b e e n e x p l a i n e d i n m a n y r e c e n t p a p e r s [ 1 1 , 1 3 -1 5 ]. A b r i e f

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

T h e c o n d i t i o n f o r t h e l o c a li z a t io n o f t h e d e f o r m a t i o n

i n t o a s h e a r - b a n d c a n b e d e r i v e d f r o m t h e r e q u i r e m e n t

t h a t t h e i n t e r n a l a n d e x t e r n a l s t r e s s v e c t o r s a c r o s s a

s h e a r - b a n d b o u n d a r y a r e i n e q u i l i b r i u m . G e n e r a l l y , t h e

s t a t i c a l c o m p a t i b i l i t y c o n d i t i o n s r e a d :

[•,j]nj = 0, (43 )

w h e r e I l k ;j ] d e n o t e t h e j u m p o f t h e r a t e o f t h e 1 .

P i o l a - K i r c h h o f f s t r e s s t e n s o r a c r o s s t h e s h e a r - b a n d

b o u n d a r y w i t h t h e u n i t o u t w a r d n o r m a l v e c t o r ( n ; ) .

~ 0 c a n b e e x p r e s s e d i n t e r m s o f t h e c o n s t i t u t i v e s t r e s s

ra t e ~r ;j acco rd ing to the fo l low ing equa t ion :

l~;j = ~r;j + oS;k ~j - a ;~ kj + %~kk. (44 )

T h e s h e a r - b a n d b o u n d a r i e s a r e a s s u m e d t o b e s t a t i o n -

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

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

m e t r i c a l c o m p a t i b i l i t y c o n d i t i o n h o l d :

[v/j] = Vin j . (45)

a x i s y m m e t r i c c o n d i t i o n s e q u a t i o n ( 4 3 )n d e r

b e c o m e s :

[~',r]nr + (~',: + 2t [¢h])n: = 0v

([a=,] + 2 t [ t h ] ) n , + [V=]n: = 0 (46)

where t i s t he s t re s s d i f fe rence :

t = ( a , - a : ) /2 (47)

an d cb is the spin:

69 = (v: . , - v , . : ) /2 . (48)

D u e t o t h e a s s u m e d a x i s y m m e t r i c s t a t e o f in i ti a l

s t re s s th e s o l u t i o n c o r r e s p o n d s t o a f a m i l y o f sh e a r -b a n d s e q u a l l y i n c l in e d w i t h r e s p e c t t o t h e z - a x i s o f t h e

spec imen:

Vo = no = 0. (49)

S u b t i t u t i n g t h e g e o m e t r i c a l c o m p a t i b i l i t y c o n d i t i o n s

( 4 5 ) i n t o t h e s t a t i c a l c o m p a t i b i l i t y e q u a t i o n s ( 4 6 ) a n d

u s i n g c o n s t i t u t iv e e q u a t i o n s ( 4 0) , t w o j u m p c o n d i t i o n s

f o r [ p ] c a n b e d e r i v e d . E l i m i n a t i n g f r o m t h e s e e q u a t i o n s

t h e j u m p [ p ] o f t h e m e a n p r e s s u r e y i e l d s fi n a ll y th e w e l lk n o w n f o r m o f th e l o c a l i z a ti o n c o n d i t i o n s [ 13 ]:

an ~ + b n~n Z : + c n ~ = 0, (50)

w h e r e

a = # s + t

b = (2 + 2 :)( 2 + 62 )#t/3 + ,~262].2s

- - 2 2 ( # s + t ) - - g 2 ( # s _ t ) .

c = , ~ , 26 2 ( # ~ - t )

2 2 = ( 1 + s i n 4 ~ m ) / ( l- s i n q S , ~ ) = t a n ~ + (5 1)

T h e o r i e n t a t i o n a n g l e 0 B ( F ig . 9 ) o f t h e s h e a r b a n d a x i s

wi th re spec t t o the rad ia l d i rec t ion i s t hen g iven by :

0B = arc tan ( - n r / n : ) (52)

D e p e n d i n g o n t h e s t a t e o f s t ra i n , t h e c h a r a c t e r i s ti c

e q u a t i o n ( 5 0 ) w il l p r o v i d e c o m p l e x o r r e a l c h a r a c t e r i s ti c

d i r e c ti o n s . L o c a l i z a t i o n i s a s s u m e d t o o c c u r a t t h e s t a te

C B ( B f o r b a n d ) w h e r e t h e c h a r a c t e r i s ti c e q u a t i o n

chan ges type ( see [14 , 16]) . Th i s c on di t ion i s fi r st ly me t

a t t h e s t a t e C s :

C B ( b / 2 a ) < O a n d d = b 2 - 4 a c = 0 . (53)

A t C s o n l y t w o s y m m e t r i c s h e a r - b a n d d i r e c t io n s e x i st ,

g iven by :

0 s = + a r c ta n ( L _ \ # - ~ J | j . (5 4 )

F o r a n y s t a t e b e y o n d t h e s t a t e C a , t h e r e a r e f o u r r e a l

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

t h e e x p e r i m e n t a l e v i d e n c e , h o w e v e r , th e o b s e r v e d s h e a r -

b a n d s u s u a l l y b e l o n g t o a s in g le f a m i l y o f s y m m e t r i c

so lu t ions ; s ee F ig . 9 . Th i s obse rva t ion ju s t i f i e s the s e l ec -

t i o n o f C a a s t h e c r i ti c a l b i f u r c a t i o n s t a t e f o r s h e a r - b a n df o r m a t i o n . E q u a t i o n ( 5 3 ) i s t h u s c a ll e d t h e " l o c a l i z a t io n

c o n d i t i o n " .I n t h e c a s e o f t h e c o n s i d e r e d l i m e s t o n e ( C a l c a i r e

d ' A n s t r u d e ) t h e c o m p u t e d c r i t i c a s l v a l u e g s f o r s h e a r -

b a n d f o r m a t i o n i s :

gs = 2.46 x 10 -3, (55)

O"

F i g . 9. S h e a r - b a n d b i f u r c a t i o n m o d e .



V A R D O U L A K IS et al.: BOREHOLE INSTABILITIES 165

a n d c o n s e q u e n t l y :

$ s = 3 6 . 2 ° ; a s = 2 8.9 M P a ; q = 1 0 M P a . (5 6)

T h e c o m p u t e d s h e a r - b a n d i n c l i n a t i o n 0B w i t h re s p e c t

t o t h e r a d i a l d i r e c t i o n i s :

0 s = 60 ° . (57)

T h e e x p e r i m e n t a l v a l u e o f t h e u n i a x ia l c o m p r e s s i o ns t r e n g t h ( p e a k o f t h e s t r e s s - s t ra i n c u r v e ) is 31 M P a a n d

t h e e x p e r i m e n t a l v a l u e o f t h e f r i c t i o n a n g l e $ ~ d e t e r -

m i n e d f r o m a s e ri e s o f t r i a x i a l c o m p r e s s i o n t e s ts i s 3 7 ° .

I t i s a l s o o b s e r v e d e x p e r i m e n t a l l y t h a t t h e d o m i n a n t

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

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

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

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

s t r e s s - s t r a i n a n d v o l u m e t r i c s t r a i n c u r v e s .

3 . 3 . G enera l i za t i on t o p lane s t ra i n condi t ions

D u e t o l a c k o f ex p e r i m e n t a l d a t a o n p l a n e s t r a i nu n i a x i a l c o m p r e s s i o n t e s t s w e h a v e t o i n t r o d u c e a d -

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

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

i s t h a t t h e h a r d e n i n g l a w ~b~ = ~ b = ( g ) a n d t h a t t h e

v o l u m e t r i c s t r a i n l a w E~ = e ~ (g ) a r e t h e s a m e i n t h e

a x i s y m m e t r i c u n i a x i a l c o m p r e s s i o n t e s t a n d i n th e p l a n e

s t r a i n t e s t .

4 . S T R E S S F I E L D A R O U N D T H E B O R E H O L E

I n t h i s s e c t i o n w e w i ll d e ri v e a n u m e r i c a l s o l u t i o n f o r

t h e s t r e s s f i e l d a r o u n d t h e b o r e h o l e . A s a l r e a d y m e n -

t i o n e d i n S e c t i o n 2 . 1 . w e a r e d e a l i n g h e r e w i t h a p l a n e

s t r a i n p r o b l e m .

4 .1 . P lane s t ra i n analys i s

F o r p l a n e s t r a i n d e f o r m a t i o n s t h e s t r e s s a n d s t r a i n

t e n s o r s a r e g i v e n b y e q u a t i o n s ( 1 ) a n d ( 2 ) . I n t h i s c a s e ,

t h e s h e a r i n g s t r a i n i n t e n s i t y is g i v e n b y t h e f o l l o w i n g

e x p r e s s i o n :

= ~ 2 / £ 2g , + ( 5 8 )

L e t

"1,, = E r - - E0 ,

f r o m e q u a t i o n ( 5 8 ) w e o b t a i n :

r x / ~ 1 2 (6 0)= -- ~;v.

I n t h e c a s e o f a n i n c o m p r e s s i b le m a t e r i a l :

g = y = £, -- e0. (61)

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

s t r a i n v e c t o r i n s t r a in s p a c e a n d o f t h e d e v i a t o r i c s t r e ss

v e c t o r i n s t r e s s s p a c e , d e f i n e d a s f o l l o w s ( F i g . 1 0 ) :

c o s 3 a , = - x / ~ eqe#ekt(e , ,~ e , . ) 3 /2

co s 3=~ = - x / ~ s q s j i s , i (62)312

(59)

slQ $

f~ . . "/ / /

I// \ ~ .. ~t " /

° 2 \ . . .

/ / ¢>, , , . .~

/)-"

J

Fig. I0. Deviatoric stress space---orientation of the deviatoric stress

vector "s.

D u e t o t h e c o n s t i t u t i v e e q u a t i o n (1 1 ) th e d e v i a t o r i c

s t r a i n t e n s o r i s p r o p o r t i o n a l t o t h e d e v i a t o r i c s t r e s s

t e n s o r a n d c o n s e q u e n t l y t h e s t r a in v e c t o r a n d t h e s t re s s

v e c t o r a r e a s s u m e d t o b e a l w a y s c o a x i a l in t h e d e v i a t o r i c

p l a n e :

e i j = k s q ~ ct~ = *q = ~. (6 3)

B y s e t t i n g :

W e f i n d t h a t :

A = - -~r /~O. (64 )

(A -- l ) (A + l /2 ) (A + 2)co s 3ct~ = - - (A 2 + A + 03/2 (65)

F r o m t h e a b o v e d e f i n i t i o n s w e c a n d e r i v e e x p r e s s io n s

f o r t h e p r i n c i p a l d e v i a t o r i c s t r e s s e s i n t e r m s o f t h e

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

2S I= ~' c COStX

'v -

s : = - ~ T c o s 5 - a , (6 6)

S 3 = - ~ "C COS + ~X

# 3w h e r e

an d

0 ~< a ~ -~, (67)

s2 < s3 < sl . (68)

B y u s i n g t h e a b o v e d e f i n i t i o n s e x p r e s s io n s f o r t h e

m o b i l i z e d f r i c t i o n a n g l e (])., a n d t h e m o b i l i z e d c o h e s i o n



1 6 6 V A R D O U L A K I S t a l . : B O R E H O L E N S T A B I L IT I E S

C m c a n b e d e r iv e d f r o m M o h r ' s d i a g ra m :

c os ( 6 - c ~ ) t a n q/o

t o ( a )

O. 8

13oo

sin ~bm = , (69 ) xb

1 _ _ s in ( 6 _ c t ) ta n ~ / ` / ~ - ~ i ! f " l ° 'e I /~ ' c - -I 0 "O *Cm = q ta n q~m" (7 0) -,~ 4',~

b4 .2 . Equa t ions o f the problem

The s t r e s s f i e l d a r o un d t he bo r e ho l e i s o b t a in e d

thro ug h the in teg ra t ion of the fo l lowing d i f fe ren t ia l t J I J I I I oosystem : °I z 3 4 5 ~ r a

Eq u a t i o n o f s t r a i n c o mp a t ib i li t y : P

des = C - - Eo (71) ~.o tb)d r r /

E q u a t i o n o f e q u i li b ri u m :

da , a , - aod r + r = O, (72)

l a e l / ~ 00.5 }with bo un da ry cond i t ions : *'o (o)

a r ( r o ) = - a o ; a ,( o o ) = - a ~ . (7 3)

By us in g t he d i l a t a n c y c o n s t r a in t ( 1 2 ) a n d a l s o

e q ua t i o n s ( 20 ) a n d ( 59 ), t he e q ua t i o n o f s t r a in c o m p a t i -b i l i ty (71) r e sul t s to the fo l lowing d i f fe ren t ia l equ a t ion t

o 0.5 1.o

fo r the she ar-st ra in inten sity g: I r~l/tr=

d g 2 L (g ), (74) Fig. 11. (a) Stress field aroun d the borehole, (b) stress-path underd r r u n i f o rm s tr e s s a t i n f in i ty r ~ = a d z

w h e r e

L ( g ) = [ g 2 - ½ D 2 ( g ) ] / [ ~ g x / g 2 - ½ D 2 ( g ) s e n t s t he r e l a t i o n s h ip be tw e e n t he r a d i a l a n d t he ho o p

s t re s s ho ld in g a t v a r i o us r a d i a l d i s t a n c e s f r o m the b o r e -

~ g l ho l e c e n t e r i s s ho w n o n F ig . l i b . F r o m th i s f i g u re i t

- g + ½D(g) ( 75 ) f o l lo w s t ha t t he ho o p s t r e s s a t t he bo r e ho l e w a l l i s l o w e r

tha n t he o n e p r e d i c t e d by l i n e a r e la s t ic i ty . Th i s r e s u l t is

By us in g e q ua t i o n s ( 15 ) a n d ( 66 ) t he e q ua t i o n j u s ti f ie d by t he f a c t t ha t a p r e s s u r e s e n s it iv e ma te r i a l i s

o f e q u i l i b r ium ( 72 ) r e s u lt s f i n al l y t o t he f o l l o w in g s t if f er a n d c o n s e q ue n t ly m o r e s t a b l e t ha n a l i n e a r e l a st i c

d i f f e re n t i a l e q ua t i o n f o r t he r a d i a l s t re s s a , : o n e . I t i s o bv io us t ha t t he mo b i l i z ed f r i c ti o n a n g l e i s

d tr r 2 d e c r e a s in g v e r y r a p id ly w i th i n c r e a s in g d i s t a n c e f r o m the= - - ( q - a r ) M ( g ) , (76) bore hole wa l l. Fa i lu re i s then expec ted to occur in the

d r r imm e d ia t e v i c in it y o f t he bo r e ho l e w a ll .

w he r e As a l r e a d y me n t io n e d i n S e c t i o n 2 .1 . t he r e a r e ba s ic -

( 7 ~ ) ( 6 ) a ll y t w o t y p e s o f f a il u re m o d e s : s h e a r b a n d i n g ( F ig . 3 b)t a n ¢ , c o s ~ - ~ F ( g ) c o s - c t a n d " e x t e n s io n r u p t u r e " o r s u r fa c e b u c k li n g a s i t h a s

M ( g ) = 2 = 2 ( 77 ) be e n o bs e r v e d i n ha r d a n d b r i t t le r o c k s ( F ig . 3 a ). The s e

1 - - - - ~ _ co s ta n ~k, 1 7=_-cos , t F ( g ) tw o typ e s o f fa i l u r e m o d e s w il l be a n a lys e d in t he n e x t,/3 ,/3 p a r t by a l i n e a r b i f u r c a t i o n t he o r y .

The d i f f er e n t ia l s y s t e m ( 7 4) a n d ( 77 ) w i th b o u n d a r y

c o n d i t i o n s (7 3 ) c a n be t he n s o lv e d n ume r i c a l l y by us in ga 2 r i d o r d e r R un g e - - Ku t t a i n t e g r a t i o n s c he me .

4 .3 . A numer ica l example

F ig u r e 1 a i l lu s t r a te s t he v a r i a t i o n o f t he s t re s s e s a n d

o f t h e m o b i l i z ed f r i ct i o n a n g le i n t h e n e i g h b o u r h o o d o fthe bo r e ho l e f o r t he c a s e o f t he l ime s to n e " Ca lc a i r ed 'A n s t r ud e " ( s ee S e c t i o n 3 ) w i th t r~ = ac /2 = 1 5 . 5 M P aa n d % = 0 . The c o r r e s p o n d in g s t r e s s p a th w h ic h r e p r e -

5 . B I F U R C A T I O N A N A L Y S I S O F T H E B O R F ,H O L E

I n t h e p r e s e n t s e c t i o n th e m a g n i t u d e o f t h e h o o p s tr es sa t t h e b o r e h o l e w a l l i s d e t e r m i n e d f o r w h i c h f o r m a t i o no f s he a r ba n d a t t h i s l o c a t i o n o r s u r f a c e i n s t a b i l i t y i sp o s s ib l e . V a r d o u l a k i s a n d P a p a n a s t a s io u [ 1 7 ] ha v e r e -

c e n t l y p r o v e d by a n ume r i c a l a n a lys i s t ha t t he c r i t i c a lb i f u r c a t i o n s t r e s s i s n o t a f f e c t e d by t he g r a d i e n t o f t he

s t r e s s a t t he bo r e ho l e . Th i s o bs e r v a t i o n y i e ld s t o as ig n i f i c a n t s imp l i f i c a t i o n i n t he c o mp u ta t i o n a l p r o -



167

ClOo

cedure . By neglec t ing the s t ress gradient we may s imu-la te a smal l domain (d) of the borehole wal l by aninf inite ly long s t r ip loa ded unde r 0.0 and a ° in plan e

s t ra in condi t ions as shown on Fig. 12.

5.1 . I ncremen t a l cons t i t u t i ve equat i ons

Under plane s t ra in condi t ions , the ra te form of the

consti tut ive equations (31) and (38) is the following (seealso [18]):

v = (1 --f~ )p +C,1~1," 1 1

v = (1 - f 2 ) P + C21~1," 2 2

v = (1 -f 3 )P +C31E,1.33

V0.12 = 2gs~ 12

~11 "4- 62(~22 = 0

w h e r e

2fl = ---= cos ~ tan ~o

, / 3

f 2 = 2 - ~ cos ( ; - ~ ) t a n ~b,, /3

-F CI2~22

-F C22 ~22

+ C32~22

f3 = ~2~ cos ( ; + ~ ) tan ~bo,

CI, = 2{2 #s - 2(#, - #t) co s2 ~}/3

C22 = 2{2 , , - 2 (#, - , , ) cos2 ( 3 - ~ )} /3

C:31 = 2 { - # , + 2 ( # , - # ,) c o s ( ; + ~ ) c o s ~ } / 3

(78)

(79)

C32 = 2 { - # , - 2 (# s - # t) c o s ( 3 + ~ ) co s ( ; - a ) } / 3

( 8 0 )

In the above express ions # t and # , a re appropr ia tetangent and shear modul i g iven by equa t ion (41) .

The di la tancy parameter 62 in equa t io n (78) is r e la tedto f l th roug h the fol lowing express ion:

1 + - - - t a c o s - ~ fl

t~2 ----=~" x /3 (81)

1 - - - F c o s = / ~

4 35.2 . Shear band analys i s o f t he suppor t ed boreho le

U n d e r plane s t ra in condi t ions the k inemat ica l andsta t ica l compat ib i l i ty equa t ions ( 4 3 ) a n d ( 4 5 ) a c r o s s a

shear -band boundary become (see [16]) :

[ v , , s ] = ~ , n j

[ ~ , , ] n , + f i g ' , : ] + 2 t [ o ) ] ) n 2 = 0

([~2,] + 2t [o~])n, + [v~a]n = O, (82)

V A R D O U L A K I S e t a l . : B O R E H O L E I N S T A B I L I T I E S

( a )

/ ~ ' ~ ~ (~ro

( b )

F i g . 1 2 , S i mu l a t i o n o f t h e behav i o u r o f a sma l l d o m a i n d a t t hebo reho l e wa l l : ( a ) su r f ace i n s t ab i l i t y , ( b ) shea r -ban d i n g .

where t is the stress difference in the plane of defor-mation, and cb is the spin:

t = (0., -- 0.2)/2

= ( v ~ . , - v , . ~ ) / 2 . ( 8 3 )

Du e to the assumed plane s t ra in condi t ion s the shear-band plane i s perpend icu la r to the plane of deformat ion :

(3 = n 3 = O . (8 4 )

Fr om th e a b ove gove r n ing e qu a t i ons a nd t h e co r r e -sponding const i tu t ive equa t ions (78) , the charac te r i s t icequa t ion for shear -band format ion under plane s t ra incondi t ions becomes [18] :

a n 4 + b n ~ n~ + c n~ = 0, (85)where

a = # s + t

b = C22 + 22c52C,~ - 22(C t2 + #~ + t ) - 62 (C2, + #~ - t )

c = ; t2 , 52 ( # , - t )

1 - f j = t an 2 + " ( 8 6 )

The or ienta t ion angle 0a (Fig . 12b) of the shear -bandaxis with respect to the radial direction is then given by:

0a = arctan ( - n l / n 2 ) . (87)

Depending aga in on the s ta te of s t r a in , the char -acter ist ic equation (85) wil l provide complex or realcharac ter i s t ic d i rec t ions . As m ent io ned in Sect ion 3 .2 .2 . ,loca l iza t ion i s assumed to occur a t a s ta te CB wherecharacter ist ic equation changes type, i .e . a t :

C n : ( b / 2 a ) < O a n d d = b 2 - 4 a c = O . (88)



168 V A R D O U L A K I S e t a l . : B O R E H O L E I N S T A B I L I T I E S

At Ca o n ly tw o s y mm e t r i c s he a r - ba n d d i r e c t io n s e x is t ,

g iven by:

0 8 = +_ a r c t a n { E 2 2 ~ z ( ~ t ~ - t ~ ] ~ / 4 ~ (89)

I t was obta ined tha t the c r i t ica l s t r a in gB a t CB does

no t dep end on the conf in in g pressure 0"0 and con-

sequ ent ly th e m obi l ized f r ic t ion q~B a t CB is inde pen den tof 0"0 . Th e c r i t ica l hoo p s t r e ss a °~ for which s hea r -b and

form a t io n i s poss ib le wil l o f course de pen d on 0"0.

D i f f e r e n t ia t i n g a n d c o m bin in g e q ua t i o n s ( 92 ) s o a s t oe l imin a t e /5 y i eld s t he f o l l o w in g e q ua t i o n f o r t he s t r e a mf un c t i o n :

a~k.,,,, + b l / / 1 2 2 2 --1--C ~ / / 2 2 2 2 - f " 7~ ( ff .,~ + ) - 2 6 2 1 / / . 2 2 ) = 0. (94)

wh ere a , b , c , 2 2 a re g iven b y equ a t ions (86).

Fol lo wing Bio t [19] p lane s t r a in su r face ins tabi l i tie s

c a n be a n a lys e d by s e t t i n g :

~ , = - - H f ( x ) s i n ymlz

5 .3 . S ur fac e i n s tab i l i t y

O n e p o s si b il it y f o r i n h o m o g e n e o u s d e f o r m a t i o n i s t h ef o r ma t io n o f n a r r o w z o n e s o f l o c a l i z e d s he a r , t he s o -

c a l l e d s he a r - ba n d s a s i t i s p r e s e n t e d a bo v e . An o the r

p o s s ib i li t y i s a s u r f a c e buc k l i n g m o d e t ha t w i l l c a us e t he

o p e n in g o f l a t e n t a x i a l c r a c k s a n d w i l l p o s s ib ly l e a d t o

s l a bb in g f a i l u r e mo d e s .As a l r e a d y m e n t io n e d t he c r i t i ca l b i f u r c a t i o n s t r e ss i s

n o t a f f e c te d by t he g r a d i e n t o f t he s t re s s a t t he bo r e ho l ew a l l . W e ma y the n s imu la t e a s ma l l d o ma in ( d ) o f t he

bo r e ho l e w a l l by a n i n f i n i t ely l o n g s t r i p l o a d e d un d e r a 0a n d a ° ( F ig. 1 2 a) . Ac c o r d in g t o t h i s mo d e l , t he " s ho r t

w a v e l e n g th l i m i t " m o d e o r s u r fa c e b u c kl i n g m o d e c o u l dbe a p p l i e d t o p r o v id e t he c r i t i c a l b i f u r c a t i o n s t r e s s f o r

s l a bb in g o f t he bo r e ho l e w a l l ( s ee a l so [ 12 ]) . To s tud y t he

s t a b i l it y o f c o n t i n u e d e q u i l i b r i um in C the e x i s t en c e o f

a s p e c i a l t yp e o f n o n ho mo g e n e o us i n f i n i t e s ima l t r a n -s i ti o n , C~ C ' , i s i n v e s t i g a te d w i th C s e rv in g a s t he

r e f e r e n c e c o n f ig u r a t i o n . Th e v e lo c i t y fi el d f o r s uc h a

m o t i o n h a s t h e f o r m :

Vk = Vk(Xt , x2)e / ' , (90)

w h e r e t d e n o t e s ti m e . I f a n u n b o u n d e d n o n - p e r i o d i cs o lu t i o n e x i st s ( f > 0 ) , t he n t he e q u i l ib r i um in C , un d e r

d e a d l o a d in g c o n d i t i o n s , i s i n he r e n t l y un s t a b l e . Thecr i t ica l s ta te wi th f = 0 i s the b i furc a t ion s ta te .

The f ie l d e q ua t i o n s a r e e x p r e s s e d in t e r m s o f t he r a t e

o f t he 1. P io l a - K i r c hh o f f s t re s s t e n s o r , w i th C be in g t he

r e f e r e n c e c o n f ig u r a t i o n :

~ j , j = X r ~ (91)

w h e r e ( : ) = d / d t ; Z is t he d e n s i t y o f t he m a te r i a l.

Us in g t he c o n s t i t u t i v e e q ua t i o n s ( 1) t he e q ua t i o n ( 1 4 )be c o me s :

(1 --f l)P .I + Cllvl. , j + (Ct2 + O ' I ) / 3 2 . 2 1

= ' ~ e ' ( j [ 'I s 0'' "OI-.2)/3''212

/ 0", - 0"='~+ +- - g- ) / 3= . , , +(1 +A):.=

+ ( C 2 ~ + 0 " 2 ) v , . 1 2 + C 2 2 v 2 , 2 2 = Z 6 2 ( 9 2 )

In o rde r to fu lf il the d i la tan cy con di t ion g,~ + 62d22 = 0a s t r e a m f un c t i o n ~ , ( x ~ , x 2 ) i s i n t r o d uc e d , s uc h t ha t :

61 = 62q/.2; ~2 = --~k.1. (9 3)

X I X 2x = ~ , y = m n ~ ( m = l , 2 . . . ) , (95)

w he r e 2 n i l p l a ys t he r o l e o f a w a v e l e n g th . W i th t h i s

a s s ump t io n e q ua t i o n ( 9 4 ) be c o me s :

a f " - ( m n ) 2( b + k ) f " + ( m n ) 4 ( c + ) , 2 ~ 2 k ) f = 0 , ( 8 6 )

with

k = Z ( f H / m x ) 2 . (97)

The s o lu t i o n o f ( 96 ) i s o f t he f o r m:

4

6(x ) = ~ Cjexp (mrr ~jx) , (98)j = l

where Cj a re in tegra t ion cons tan ts and ~j sa t i s f ie s thec ha r a c t e r i s t i c e q ua t i o n :

a~4 - (b + k) ~] + (c + 2262k) = 0. (99)

The bo un d a r y c o n d i t i o n s e x p re s s t ha t t he c o n f in in g

p r e s s u r e ao r e m a i n s c o n s t a n t a n d a c t s n o r m a l l y o n t h e

s u r f a c e o f t he l a ye r a t x = 0 . M a the m a t i c a l l y t he s ec o n d i t i o n s r e a d :

£~jn: = o ° (nk g, - n ,a~ ) v~ . ,. ( !0 )

Us in g e q ua t i o n s ( 4 4 ) a n d ( 7 8 ) t he bo un d a r y c o n -

d i t i o n s c a n be e x p r e s s e d i n t e r ms o f 6 :

f o r x = 0

" + ( m n ) 2 6 = 0

a f " - (m n)2 (P + k )z3 ' = 0 (101)

w h e r e

P = C z2 - 62(C21 + / 4 - t) + , ~ 2 ( C i 1 ~ 2 - C 1 2 . (102)

Sur face ins tabi l i t ie s a re on ly poss ib le in the e l l ip t ic

reg ime of the cha rac te r i s t ic equa t ion (85) ( see [12] ) :

( EC) E l l i p ti c c o mp le x r eg ime :

d = b 2 - 4ac < 0. (103)

( EI ) E l l i p t ic im a g in a r y r e g ime :

d = b 2 - 4 a c > 0 ; b / a > O ; c / a > O . (104)

The i n t e g r a t i o n c o n s t a n t s C j a r e c a l c u l a t e d by w r i t in gthe bo u n d a r y c o n d i t i o n s ( 1 01 ) a n d a s k in g f o r n o n - t r i v ia l

s o lu t i o n s ( ( 2 :. = 0 ) . A t t h e b i f u r c a t i o n p o i n t ( k = 0) thef o l l o w in g e q ua t i o n i s o b t a in e d :

P ( , , / 7 - ~ - 6 2 )1 a + , 5 2 ( . , / ~ + b / a ) + c l a = O . (105)



V A R D O U L A K I S et aL: B O R E H O L E I N S T A B I L I T I E S 1 69

40

~m

j l O 0 1

JIOrl

30

20

10m

I I I I I I I I I I I I I

2 3 4 5 6 7 B

P

Fig. 13, Cr i t ical s t ress f ield around the borehole at fai lure.

J

4 0

30

( ' P m2 O

1 0

Th e c r i t i ca l s t r a in gs fo r wh ich su r f ace ins tab i l i ty i s

p o s s i b l e i s t h e s o l u t i o n in the elliptic regime (d < 0 o r

d > 0 with b /a > O, c /a > 0 ) o f t h e g o v e r n i n g e q u a t i o n

(105) .

5.4 . I l lustrative example

F o r t h e c a s e o f t h e c o n s i d e r e d l i m e s t o n e ( C a l c a i r e

d ' A n s t r u d e ) , i t w a s f o u n d t h a t t h e r e is n o s o l u t i o n f o r

equ a t ion (105) in th e e l l ip t i c r eg ime . Su r f ace ins tab i l i t i e s

a r e no t pos s ib le fo r th i s ma te r ia l and sh ea r f a i lu r e i s th e

d o m i n a n t f a i l u r e m o d e . I t w a s a c t u a l l y o b t a i n e d t h a t

e q u a t i o n ( 8 8 ) h a s a s o l u t i o n . T h e c o m p u t e d c r i t i c a l s h e a r

s t r a i n g B f o r w h i c h s h e a r f o r m a t i o n i s p o s s i b l e i s:

gB = 3.35 x 10 -3. (106)

T h e c o r r e s p o n d i n g m o b i l i z e d f r ic t i o n a n g l e ~bB a n d

s h e a r - b a n d i n c l i n a ti o n a n g l e 0 3 a r e:

q~3 = 37.1 °; 03 = 62.1 °. (10 7)

Wi th th ese va lu es , i t i s th en poss ib le , a s sh own in

S e c t i o n 4 , t o c o m p u t e t h e c o r r e s p o n d i n g s t r e s s f i e l d

a r o u n d t h e b o r e h o l e w a l l a t t h e s t a t e o f f a il u r e.

I n p a r t i c u la r f o r a n u n s u p p o r t e d b o r e h o l e t h e c o m -

p u t e d c r i t i c a l h o o p s t r e s s i s :

e°03 = 30.5 M Pa , (108)

and th e cor r esponding c r i t i ca l s t r e s s a t in f in i ty i s ( s ee

Fig. 13) :

3 4 3 M P a . ( 1 0 9 )

A n e x p e r i m e n t a l p r o g r a m f o r te s ti n g t he b e h a v i o u r a t

f a i lu r e o f t h i c k - w a l l c y l i n d e r s w i t h t h e s a m e l i m e s t o n e

h a s b e e n p e r f o r m e d b y G u e n o t [3 , 9 ]. T h e e x p e r i m e n t a l

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

E = 4 0 M P a . T h e o b -n t e r n a l f a i l u r e w a s f o u n d a s a ~ c r

s e r v e d f a i l u r e m o d e w a s s h e a r b a n d i n g ( F i g . 1 4 ) . T h e

c r i ti c a l v a l u e a n d t h e f a i l u re m o d e o b t a i n e d w i t h i n

t h e f r a m e o f t h e p r e s e n t t h e o r y i s i n g o o d a g r e e m e n t

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

e m p h a s i z e d t h a t t h e l in e a r e l a s t ic t h e o r y w o u l d g i v e

a ® c r = a ° r / 2 = 1 5 . 2 M P a . A s a l re a dy m e n ti o n ed in

S e c t i o n 4 t h e h o o p s t r e s s a t t h e b o r e h o l e w a l l i s l o w e r

t h a n t h e o n e p r e d i c t e d b y l i n e a r e l a s t i c i t y .

~:~i~ ~ !iii

Fig . 14 . Ex per imen ta l obs er v a t ion o f t h i c k wal l c y l inder f a i lu r e fo r t he l imes tone (Ca lc a i r e d 'A ns t r ude) .



1 7 0 V A R D O U L A K I S e t a / . : B O R E H O L E I N S T A B I L I T I E S

R e m a r k - - I t s h o u l d b e n o ti c e d t ha t f o r o t h e r m a t er i al s

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

m o d e .

A c k n o w l e d g e m e n t s - - T h e author s wi s h to thank La Soc i r t6 Nat iona leE l f A q ui t a ine (P r od uc t ion) fo r s uppor t ing th i s r es ear c h

6 . C O N C L U S I O N

E x p e r i m e n t s o n h o l l o w c y l i n d e r s a n d a c t u a l o b s e r -

v a t i o n s i n u n d e r g r o u n d e x c a v a t i o n s s e e m t o i n d i c a t e

t h a t t h e r e a r e b a s i c a ll y t w o t y p e s o f f a i lu r e m o d e s : s h e a rb a n d i n g a s it is s h o w n i n F i g . 1 4 a n d " e x t e n s i o n

r u p t u r e " o r s u r f a c e b u c k l i n g a s i t h a s b e e n o b s e r v e d f o r

e x a m p l e i n h a r d a n d b r i t tl e r o c k s a s i n t h e S o u t h A f r i c a n

d e e p m i n e s ( q u a r t z i t e ) .

T h e a b o v e b i f u r c a t i o n a n a l y s is is c o n s i s t e n t w i t h t h e s e

o b s e r v a t i o n s . A l t h o u g h i t i s b a s e d o n s e v e r a l s im p l i f y i n g

a s s u m p t i o n s , t h i s a n a l y s i s a l l o w s t o d e t e r m i n e t h e c r i t i -

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

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

a s tr i c t m a t e r i a l p r o p e r t y b u t d e p e n d s o n s t r e s s -p a t h a n d

b o u n d a r y c o n d i t i o n s o f t he s y s te m .

T h e a p p l i c a t i o n o f t h e m o d e l t o r e a l c a s e s h a sp r o v i d e d a s a t i s f a c t o r y e x p l a n a t i o n o f e x p e r im e n t a l

o b s e r v a t i o n s .

D u e t o t h e l ac k o f e x p e r im e n t a l d a t a o n p l a n e s t r a i n

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

u n i a x i a l a x i s y m m e t r i c c o m p r e s s i o n t e st . A c o r r e c t e v a l u -

a t i o n o f th e s t r e s s - s t r a in l a w f o r th e b o r e h o l e p r o b l e m

w o u l d a c t u a l l y r e q u i r e a n a p p r o p r i a t e e x p e r i m e n t a l a n d

t h e o r e t i c a l p r o c e s s a s f o l l o w s : t h e s t r e s s - s t r a i n c u r v e

f r o m u n i a x i a l p l a n e s t r a in c o m p r e s s i o n t e s t m u s t b e f ir s t

k n o w n . T h e n t h e p r e se n t t h e o r y c a n b e a p p l ie d i n o r d e r

t o p r e d i c t a s t r e s s - p a t h f o r t h e b o r e h o l e p r o b l e m ( s e e

Sec t ion 4 , F ig . 1 b ) . Wi th th i s s t re s s -pa th a s a f i r s t

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

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

e q u a t i o n s ( 3 0 ) a n d ( 32 ), s h o u l d b e d o n e . T h i s m e a n s t h a t

t h e c o r r e c t e v a l u a t i o n o f t h e s t r e s s - s tr a i n l a w s h o u l d

f o l l o w a n i t e ra t i ve p r o c e s s o f th e o r e t i c a l p r e d i c t i o n a n d

e x p e r i m e n t a ti o n . T h i s i s o f p r i m a r y i m p o r t a n c e f o r

m o d e l l i n g t h e co n v e r g e n c e o f u n d e r g r o u n d o p e n i ng s . I t

i s t hen pos s ib l e a s i t i s p roposed in th i s pape r ( s ee

S e c t i o n 5 ) t o d e v e l o p a b i f u r c a t i o n a n a l y s i s to d e t e r m i n e

t h e f a il u r e c o n d i t i o n s o f th e b o r e h o l e c o r r e s p o n d i n g t o

t h e r e a l s t r e s s- p a t h o f t h e s y s t e m .

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defo r m at ion in p r es sur e s ens i t iv e d i l a t an t mate r i a l s . J . Mech . Phys .Solids 23, 371-394.

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1 borehole instabilities as bifurcation

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