Investigating Production-Induced Stress Change at Fracture Tips

8/9/2019 Investigating Production-Induced Stress Change at Fracture Tips

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I n ve st ig a ti n g p r od uc ti o n- in du c ed s tr es s c ha n ge a t f r ac tu r e t ip s:

I mp li ca t io ns f or a n o ve l h y dr a ul ic f r ac tu ri n g t ec h ni qu e

M . K . R a h ma n ⁎, A .H . J oa rd er

S c h oo l o f O i l a n d G a s E n gi n e er i n g, U n i v er s i t y o f We s t e rn A u st r a l ia , M B DP : 0 5 2, P e r th , WA 6 0 09 , A u s tr a l i a

R e ce i ve d 7 A p ri l 2 0 05 ; r e ce i ve d i n r e v is e d f o r m 2 9 N o ve m be r 2 0 05 ; a c ce p te d 2 9 D e ce m be r 2 0 05

Abstract

H y d ra u l ic f r a c tu r i ng f r o m v e r t i c al w e l l s s u b j ec t t o r e v er s e f a u l t i n g s t r e s s r e g i m es i s f o u nd t o b e p r o bl e m at i c d u e t o t u r ni n g a n d

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

o f t h i s p a p er i s t o i n v es t i g at e w h e t h e r e f f e c t iv e h y d r a u li c f r a c t u r i ng c a n b e a c h ie v e d i n t h e se r e s e rv o i r s i n a n u mb e r o f s t a ge s w i t h

p ro d u ctio n in terv als. Th e b asic mech an ism th at is en v isag ed to ach iev e a lo n g , p lan ar p ro d u ctiv e fractu re in su ch a man n er is th e

p ro d u ctio n-in d u ced ch an g e o f stress state aro u n d th e cu rren t fractu re tip to b e su itab le fo r fu rth er p lan ar p ro p ag atio n o f th e fractu re

i n t h e n ex t t r ea t me n t s t ag e. T h e t i me - de p en d en t p r od u ct i on - in d uc e d s t re s s s t at e c a n b e f o rm ul a te d b y c ou p le d f l ui d f l ow a nd

d ef o rm a ti o n p r in ci p le s . T h is c on c ep t h a s b ee n a pp l ie d t o a m od e l- s ca l e r e se r vo i r t o p r od uc e s om e i n si g ht f ul r e su l ts . B as e d o n

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

t h e m e n ti o n ed r e s e rv o i r c o n di t i o ns , a n d h i g hl i g ht e d f u r t he r r e s e ar c h d i r ec t i o ns w i t h f u l l - sc a l e r e s er v o i rs .

© 2 0 0 6 E l s e v i er B . V. A l l r i g h ts r e s e rv e d .

Keywo rd s: H y d r a ul i c f r a c t ur i n g ; M u l t is t a ge f r a c t ur i n g ; C o u p l ed f l u i d f l o w a n d d e f o r ma t i o n ; P o r o e l a s ti c m o d e li n g

1. Introduction

H y d ra u l ic f r ac t u ri n g i s a p r oc e ss w h er e by p r op p a nt -

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

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

f ra ct u re d e ep i n to t h e r es er vo i r. O nc e t he i nj e ct io n i sc ea se d, t he p ro pp e d f ra ct u re b e co me s t h e p ri n ci p al

c o nd u i t f o r f l ow o f t h e h y dr o ca r b on f r om t h e r e s e r vo i r

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

petroleum industry has long been applying hydraulic

f ra ct u ri n g a s a p ri n ci pa l t ec h ni q ue t o i mp r ov e o il a nd

g a s p r o d u c t i o n . O f t h e p r o d u c t i o n w e l l s d r i l l e d i n N o r t h

A m e r i c a s i n c e 1 9 5 0 s , a b o u t 7 0 % o f g a s w e l l s a n d 5 0 %

o f o i l w e l l s h a v e b e e n h y d r a u l i c a l l y f r a c t u r e d ( Valko and

E c o n o mi d e s , 1 9 9 5).

D e sp i t e s o me s u cc e ss c a se s , t h e o p e ra t i on a l p e r fo r -

m an c e a nd c os t – benefit accounts of hydraulic fracture

t re at me n ts h a ve n ot b e en p os it i ve i n m an y o c ca si o ns , particularly onshore Australia. Major difficulties en-

c o un t e re d d u ri n g t r ea t me n t s i n c lu d e t h e r e qu i r em e n t o f

h i g h i n je c t io n p r es s u re , h i gh f r ic t i on a l p r es s u re d r op ,

i n ab i l it y t o i n je c t p r op p an t a t r e qu i r ed c o n ce n t ra t io n s

w it hi n t he p ump c ap ac it y, i na bi li ty t o e xt en d t he

i n i t i a t ed f r a c t u re , a n d c o n s e q ue n t l y, p o o r p o s t - st i m u l a -

t io n p ro d uc t iv i ty. A n i n ve st i ga t io n (R ah ma n e t a l.,

2000a ) e st ab l is he d t h at m an y o f t h e A us tr al i an r es er -

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

f a ul t i ng s t re s s r e gi m e s (H i l li s a n d R e yn o l ds , 2 0 00 ) , i . e .

J o ur n a l o f P e tr o l eu m S c ie n ce a n d E n gi n ee r in g 5 1 ( 2 00 6 ) 1 8 5– 196

www. els evier. com/locate/petrol

⁎ C o rr e sp o nd i ng a u th o r. Te l .: + 6 1 8 6 4 88 7 3 37 ; f a x: + 6 1 8 6 4 88

1964.

E - ma il a d d ress: krahman@cyllene. uwa. edu. au (M. K. Rahman).

0 9 20 - 41 0 5/ $ - s e e f r on t m a tt e r © 2 0 06 E l se v ie r B . V. A l l r i gh t s r e se r ve d .doi:10. 1016/j. petrol. 2005. 12. 009

mailto:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.petrol.2005.12.009http://dx.doi.org/10.1016/j.petrol.2005.12.009http://dx.doi.org/10.1016/j.petrol.2005.12.009mailto:[email protected]


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t h e v e r t i c a l s t r e s s i n t h e r e s e r v o i r r o c k i s s m a l l e r t h a n t h e

h o ri z o nt a l s t re s se s . A v e rt i c al f r ac t u re i n i ti a t ed f r om a

v er ti ca l w el l i n t hi s s tr e ss c o nd i ti on (F ig . 1) r e or i en t s

i ts el f t h ro u gh c o mp l ex t ur ni n g a nd t w is t in g ( us ua ll y

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

t h i s i s t h e p r e f e r r e d f r a c t u r e p l a n e i n t h i s s t r e s s c o n d i t i o n(H o ss ai n e t a l. , 2 0 00 ) . T he t wi st e d f ra ct ur e g eo me t ry

c au se s p ro pp an t b ri dg in g i n t he c on st ri ct ed w id th

r es ul t in g i n p re ma t ur e s cr ee n -o u ts ( in ab i li ty t o i n je ct

f ur th er w it hi n t he p um p c ap ac it y) a nd l ow p ro pp ed

f r a c tu r e c o n d u i t f o r p r o d u c t i on . A l t h o u g h a p p r o p r i a t el y

o ri e nt ed w el l s a nd p er fo r at io n s a re r ec om me nd e d t o

o ve rc om e s om e o f t he se d i ff ic u lt i es (R ah ma n e t a l. ,

2 0 0 0 a ; H o s s a i n e t a l . , 2 0 0 0 ) , d r i l l i n g a n d c o m p l e t i o n o f

s u ch w e l ls i n v ol v e s a d v an c ed t e ch n o lo g i es , h i g he r c o s t

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

l e ad t o o t h er f r ac t u re c o mp l e xi t i es .O n s i mi l a r j u st i fi c a ti o n s, Wa r pi n s ki a n d B r an a ga n

(1989) r e c o mm e n d e d a l t e r e d - s t r es s f r a c t ur i n g a n d d e m -

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

s tu di e s. T he b a si c c on c ep t w as t o p re ss ur iz e a f ra c tu re b y

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

pressure, and in turn the stress state, around the fracture

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

f r ac t ur e e x t en s i on . A s i mi l a r c o nc e p t i s a l so m o de l l ed

a nd s tu d ie d r ig o ro u sl y b y B e rc h e nk o a n d D e t ou r na y

(1997). A pa rt f ro m a dd it io na l c os ts a ss oc ia te d w it h

d ri l li n g a nd t re at i ng s uc h o f fs et w e ll s, t he re a re m an y

s ys te m p ar am e te rs t ha t n e ed t o b e o p ti m iz ed t o d er iv ea n y b e n e f i t s f r o m s u c h o p e r a t i o n s . T h e p r i n c i p l e o f p o r e

pressure change due to injection/productio n, and result-

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

h ow ev er, h as b ee n t he t op ic o f f ur th er r es ea rc h f or

d i f f er e n t a p p l i c a t io n s (D e t o u r n a y e t a l . , 1 9 8 6 ; C h e n g e t

a l ., 1 9 9 3; E l b el a n d M a c k , 1 9 9 3 ; C h en a n d T e uf e l, 2 0 0 1;

S av i ts ki e t a l ., 2 0 00) . T he p or oe l as ti c t he or y h as b ee n t h e

basic foundation for most of these investigations. A

particular effort is noted to investigate the creation and

propagation of a secondary fracture orthogonal to the

primary fracture to derive production benefits from

permeability anisotropy in the reservoirs (S i e b r i t s e t a l . ,

1998 ) . T h e s e c o n d a r y f r a c t u r i n g t r e a t m e n t w a s p r o p o s e dt o c ar ry o ut a ft er a c er ta in p er io d o f p ro du ct io n t im e

f o ll o wi n g t h e p r im a ry f ra c tu r in g t re a tm e nt . F u rt h er

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

t o h y d ra u l ic f r ac t ur i n g h a ve b e en c o nt r i bu t e d b y Dong

a n d d e P a t e r ( 2 0 0 1 , 2 0 0 2 ) .

I n e s s e n c e, t h e a b o v e -d e s c r i be d p r o d u c t i on - i n d u ce d

a l te re d- st re ss s ec on d ar y f ra ct u ri n g i s j u st c re a ti ng a

s t re ss e nv i ro n me n t b y n o n- un i fo rm f lu i d e xt ra c ti o n

f ro m t h e r es er vo i r t o i nf lu en c e t he s ec on d ar y f ra ct u re

g ro wt h a lo ng a d ir ec ti on w hi ch w as i ni ti al ly t he s o

c a l l e d n o n - p r e f e r re d d i r e c t io n f o r f r a c t u r e p r o p a ga t i o n .T h i s m o t i v at e d t h es e a u t ho r s t o i n v es t i ga t e f u rt h e r t h e

production-induced stress change around the wellbore

a nd t he p ri ma ry f ra ct ur e t ip a s a f un ct io n o f m ai nl y

production time and some fracture parameters. One

particular feature of our study is that the primary

f ra ct ur e i s c re at e d a lo n g t h e n on -p r ef er re d d i re c ti o n,

a n d t h en t he a lt e re d -s t re ss es a nd p or e p re ss u re s a t t h e

f ra ct ur e t i p a nd a t a n e ar -w el l bo re r eg io n a lo n g o rt h o-

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

a s a f un ct io n o f p ro du ct io n t im e. U si ng t hi s a lt er ed

s t re ss f ie l d, t h e p ro p ag a ti o n p re ss ur e o f t h e e xi st i ng

f ra ct ur e t ip a nd t h e i n it ia t io n p re ss u re o f a p o te nt i al lys ec o nd a ry f ra ct u re a t t h e o rt h og on a l l oc at i on o n t h e

w e ll b or e w al l a re e st i ma te d a s a f un ct i on o f t i me . We

h a ve t hu s e st ab l is he d a p ro du ct io n- in du ce d s tr es s

c o nd i ti o n w h ic h i s s ui t ab le f or f ur th e r p ro p ag a ti o n o f

t h e e x is t i ng f r ac t ur e a l o ng i t s c u rr e nt d i re c t io n , w h ic h

w a s i n it i al l y n o n- pr ef er re d d ir ec ti on , w i th ou t a ny

t ur ni ng /t wi st in g a nd w it ho ut c re at in g t he p ot en ti al

s e co n d ar y f r ac t u re a l o ng t h e o r th o g on a l l o c at i o n. T h e

i mp li ca ti on o f t hi s s tu dy o n h yd ra ul ic f ra ct ur in g i n

r e se r vo i r s s u bj e c t t o t h e r e ve r s e f a ul t i ng s t re s s r e gi m e

c a n b e d e sc ri b ed a s f ol l ow s.A l i mi t e d e x t en t v e rt i c al f r ac t u re a l o ng t h e p r ef e rr e d

d i r e c t io n ( i . e . a l o n g σH d i r e c t i o n ) w i l l b e c r e a t e d i n t h e

v er ti ca l w el l a t t he f ir st s ta ge . I t i s e xp er ie nc ed t ha t

s uc h p ar ti al ly p en et ra te d f ra ct ur e g eo me tr y c an b e

c re at e d w i th o ut a ny m aj o r d i ff ic u lt ie s i rr es pe ct i ve o f

t h e i n s it u s tr es s c on d it i on . T he t op a nd b ot t om t i ps o f

t h i s v e rt i c al f r ac t u re a r e, h o we v e r, a l o ng n o n -p r ef e rr e d

d ir ec ti on f or p ro pa ga ti on , s in ce σvbσh. W it ho ut

a t te mp t in g t o e x te nd t h e f ra ct u re f ur th e r, t he w el l w i ll

t h er ef or e b e a ll o we d t o p ro d uc e f or a c er ta i n p er io d o f

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

r e v e r se , i . e . σhbσv. H a l ti n g p r od u ct i o n, a s e co n d -s t ag eF ig . 1 . T ur ni ng o f t op a nd b ot to m t ip s o f a v er ti ca l f ra ct ur e f r om av e r t ic a l w e l l i n a r e v e r se f a u l t in g s t r e ss r e g i me .

186 M . K . R a hm a n, A . H . J o a rd e r / J o u rn a l o f P e t ro l e um S c i e nc e a n d E n gi n e e ri n g 5 1 ( 2 00 6 ) 1 8 5 – 1 9 6


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t re at me nt c an t he n b e c arr ie d o ut t o p ro pa ga te t he

f ra ct ur e u p t o t he d is ta nc e h av in g t he r ev er se s tr es s

c on di ti on . A n um be r o f s uc h s ta ge s c an f in al ly

a cc om p li sh a v e ry p ro du c ti ve f ra ct ur e. A l so a s ig ni f-

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

fa r m ay o ffe r a s ig ni fi ca nt f in an ci al b en ef it t o t heoperator.

A s c o p i n g s t u d y w a s u n d e r t a k e n a t t h e U n i v e r s i t y o f

W e s t e r n A u s t r a l i a ( U W A ) w i t h a v e r y l i m i t e d s u p p o r t b y

t h e F a c u l t y a n d U W A g r a n t s . W i t h i n t h e p r o j e c t ' s s c o p e ,

a m od el -s ca le r es er vo ir s ys te m h as b ee n s tu di ed b y

c ou pl ed f lo w a nd d ef or ma ti on a na ly si s. T he m ai n

o bj ec ti ve o f t hi s p ap er i s t o p re se nt t he m od el , t he

a s so c i at e d f o rm u la t i on s a n d t h e r e su l t s t h a t s u pp o r t t h e

c o nc e p t a n d e n c ou r a ge e x t en d e d f u rt h er i n v es t ig a t io n s

i n t h e a r ea , p o ss i b ly w i t h a f u ll - sc a l e r e s e r vo i r.

I n t he r es t o f t hi s p ap er , w e p re se n t t h e c on st i tu t iv ef or mu l at i on s o f t he c o up l ed p ro bl e m, a n al yt i ca l f or -

m u la t i on s f o r f r ac t u re i n i ti a t io n a n d p r op a g at i o n p r es -

s u re s , t h e m o de l d e sc r ip t i on a n d s p ec i a l t e c h n i qu e s t o

r ep re s en t a f ra ct ur e, n u me ri ca l r es ul t s a n d t h ei r i nt e r-

pret atio ns. Fina lly, we have conc lude d the pape r

s um ma ri s in g m aj o r f in di n gs a nd t he w a ys f or f ur th er

research.

2. Coupled fluid–mechanical formulation

T he p ro du ct io n- in du ce d s tr es s s ta te a ro un d t he

w el lb or e a nd t he f ra ct ure t ip re gi on i s a c omp le xf un ct i on o f f or ma ti on p ro pe rt i es a n d c um ul at i ve

production, and should be investigated as a three-

d i me n si o na l f u ll y c o up l ed h y d ro - po r o- m ec h an i c al

prob lem. FLAC3 D (Fast Lagra ngia n Anal ysis of

C on t in u a i n 3 D i me n si o ns ) c od e, a l i ce ns ed p ro d uc t

f ro m I ta sc a C on s ul ti n g G ro u p, h a s b ee n u se d f or t hi s

purpose. FLAC3D provides an option for coupled

fluid–m ec ha ni ca l a na ly si s i n w hi ch t he m ec ha ni ca l

r es po ns e o f t he p or ou s m at er ia l i n t he p re se nc e o f a

s i ng l e -p h a se t r an s i en t f l u id f l ow c o nd i t io n i s m o de l le d

w it h in t he f ra m ew or k o f t h e q u as i- st a ti c B i ot t h eo r y.T he p ar am et er s i nv ol ve d i n t he d es cr ip ti on o f f lu id

fl ow t hr ou gh p oro us m ed ia a re t he p ore p re ssu re ,

s at ura ti on a nd t hr ee c omp on en ts o f t he sp ec if ic

d i sc h a rg e v e ct o r. T h es e p a ra m e te r s a r e r e la t ed t h ro u g h

t h e D a rc y 's l a w f o r f l ui d t r an s p or t , f l ui d m a ss - ba l a nc e

e qu a ti o n, a c on st i tu t iv e e qu a ti o n s pe c if yi n g t h e f lu i d

r es po n se t o c ha ng e s i n p or e p re ss u re , s at u ra t io n a nd

v ol um et ri c s tr ai ns a nd a n e qu at io n o f s ta te r el at in g

pore pressure to saturation in the unsaturated range.

T he p o re p re ss ur e i n fl u en c e i s i nv o lv ed i n t he m ec ha -

n ic al c on st it ut iv e l aw s t o c om pl et e t he f lu id f lo w–

m e c h a n ic a l c o u p l i n g.

T he r es po ns e o f p or e f lu id c an b e f or mu la te d a s a

f u n c t i o n o f t i m e - d e p e n d e n t c h a n g e s i n p o r e p r e s s u r e , p p,

saturation, s, a n d m e ch a n ic a l v o l um e t ri c s t ra i n s, ε as

f o l l o w s (F L A C 3 D , 2 0 0 2 a , b ):

1 M

A p p

At þ /

sA sAt

¼ 1 sAfAt

−aAeAt

ð1Þ

w h e r e M i s t he B io t' s m od ul us , α i s t he Bi ot 's

coefficient, t i s t he f lo w t im e, ϕ i s t he p or os it y i n t he

s o li d a n d ζ i s t h e v a ri a t io n o f f l u i d c o n te n t o r v a ri a t io n

o f f l u i d v o l u m e p e r u n i t v o l u me o f p o ro u s m a t e ri a l d u e

t o d i ff u si v e f l u id m a ss t r an s p or t .

T h e t e rm ∂ζ / ∂t i n E q. ( 1) c an b e o bt ai ne d b y f lu id

m as s b al an ce f or s ma ll d ef or ma ti on c on di ti on a s

follows:

−qi ;i þ qv ¼ AfAt

ð2Þ

where qv i s t h e v o l u m e t r i c f l u i d s o u r c e i n t e n s i t y i n [ 1 / s ]

and qi i s t h e s p e c i f i c d i s c h a r g e v e c t o r ( i = 1 , 2 , 3 i n a 3 - D

s p ac e t o r e pr e se n t 3 s p at i a l c o m p o ne n ts ) o f t h e f l o w i n g

fluid.

T h e f o ll o w in g e x pr e ss i o n o f D a rc y 's l a w d e s cr i be s

t h e s p ec i fi c d i sc h a rg e v e ct o r, qi f or a c on st an t d en -

s it y f lu i d t hr ou g h a h om og e ne o us , p o ro u s, i so t ro p ic

medium:

qi ¼ −k il k̂ ð sÞ½ p p−qf x j g j ;l ð3Þ

where k i s t h e t e n so r o f a b so l u te m o bi l i ty c o e ff i ci e n ts

( F LA C 3D p e rm e ab i l it y t e ns o r ) o f t h e m e di u m , k ˆ ( s) is

t he r el a ti ve m ob i li ty c oe ff ic ie nt w h ic h i s a f un ct i on o f

f l u i d s a t u r at i o n s, ρf i s t h e f l ui d d e n si t y, g j ( j = 1 , 2 , 3 )

r ep re s en t s t h e t hr ee s pa ti a l c o mp o ne n ts o f t he g ra vi t y

v e c t o r, a n d s i m i l a rl y, x j i s t h e p o s i t i o n v e c t o r i n a t h r e e -

d i me n si o n al s p ac e . T h e g e ne r al c o nv e n ti o n t o i n t er p re t

o th er s ub sc ri pt s i n E qs . ( 2) a nd ( 3) i s a s f ol lo ws : t he

symbol ai d e n o te s c o m p o n en t i o f t he v ec to r [a] i n aC a r t e s i a n s y s t e m o f r e f e r e n c e a x e s ; Aij i s c o m p o n e n t (i,

j ) of ten sor [ A]; f ,i d e no te s t he p ar ti a l d e ri v at iv e o f f

w it h r es pe ct t o xi where f d en ot es e it he r a s ca la r

v ar ia b le o r a v ec to r c om po n en t. T hu s, qi,i i n E q. (2 )

r ep re se nt s t he p ar ti al d er iv at iv e o f t he s pe ci fi c

discharge qi w i th r es pe c t t o x1, x2 and x3 f or 3 s pa ti al

c om po n en t s. A l so , t h e t er m [ p p−ρ f x j g j ],l i n t he r ig ht

h an d s id e o f E q. ( 3) r ep re se n ts t he p a rt i al d er iv a ti ve o f

t he f un c ti on ( p p−ρf x j g j ) w it h r es pe ct t o x1, x2 and x3f o r 3 s p at i a l c o mp o n en t s .

T he f un ct io n ( p p−ρ

f x

j g j

) r ep re se nt s t h e p re ss ur e

h e a d a n d t h e q u a n t i t y ( p p−ρf x j g j ) / (ρf g ) ( w h e r e g i s t h e

187 M.K. R a h ma n , A .H. Jo a rd er / Jo u rn a l o f P etro leu m S cien ce a n d E n g in eering 5 1 ( 2 0 0 6 ) 1 8 5 – 1 9 6


4/12

m od ul us o f t he g ra vi ty v ec to r) c an b e d ef in ed a s t he

h e ad . T h e m o bi l i ty c o ef f ic i en t , k i s r el at ed w it h t he

i n t r i n si c p e r m ea b i l i t y, κ as k =κ / μ, i n w hi ch μ i s t he

d y n a m i c v i s c o s it y.

F u r t h er d e t a i l s o f f o r m ul a t i o n s , m o d e l d i s c r et i z a t i o n

a nd s ol u ti on s ch em e s w i th t im e- st e ps c a n b e f ou nd i nF L AC 3 D m a nu a l s (F L A C 3 D , 2 0 0 2 a , b) . T he re a re t wo

d if fe re nt s ch em es i n F LA C3 D t o s ol ve t he c ou pl ed

f o rm u la t i on s : ( 1 ) e x p li c i t f i n it e d i ff e re n c e s c he m e a n d

( 2) i mp li ci t f in it e d if fe re nc e s ch em e. T he f or me r

s ch em e i s u se d t o r un t he m od el i n t hi s w or k. D ur in g

s i mu la ti on , t he s o lu t io n s t ar ts f r o m a s ta t e o f

m e ch a n ic a l e q ui l i br i u m a n d p r og r es s es t h ro u g h a s e ri e s

o f s te ps . E ac h s te p i nc lu de s o ne o r m or e f lo w l oo ps

f ol lo we d b y e no ug h m ec ha ni ca l l oo ps t o m ai nt ai n

q u as i -s t a ti c e q u il i b ri u m. T h e c h a ng e i n p o r e p r es s ur e

d ue t o f lu id f lo w i s e va lu at ed i n t he f lo w l oo p; t hec o nt r ib u t io n f r om v o l um e t ri c s t ra i n i s e v al u a te d i n t h e

mec ha ni c a l l o op as a z o n e v a l u e w h i c h i s t h e n

d i s t r i b u t e d t o t h e n o d e s . F o r e f f e c t i v e s t r e s s c a l c u l a t i o n ,

t he c h an ge i n t o ta l s tr es s d ue t o p o re p re ss ur e c ha ng e

a ri si n g f ro m m ec h an i ca l v o lu m e s tr ai n i s e va lu a te d i n

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

f l ow l o o p.

3. Analytical formulations

A s m en ti on ed e ar li er, t he f ra ct ur e p ro pa ga ti on

pressure at the fracture tip and the fracture initiation pressure at the wellbore wall at 90° phased with the

c u rr e nt f r ac t u re d i r ec t i on a r e c h ar a ct e r iz e d w i t h t i m e a s

f u nc t io n s o f t i me - de p e nd e n t s t re s se s a n d p o re p r es s ur e

a t t h e se p o i nt s . S i mp l e a n a ly t i ca l f o rm u l at i o ns a r e u s e d

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

o r t h o g o n a l h o r i z o n t a l s t r e s s e s , σxx and σyy, t h e w e l l b o r e

f lu i d p re ss u re t ha t i ni t ia te s a f ra ct u re a t A along σxxd i r e c ti o n ( a l o n g x- ax is i n F ig . 2) c an b e c al cu la te d a s

(H o s s a i n e t a l . , 2 0 0 0 ):

p bw ¼ 3ryy−rxx

− p p þ T ð4Þ

where p p i s t h e p o r e p r e s s u r e a n d T i s t h e t e n s i l e s t r e n g t h

o f r o c k . T h e e q u a t i o n i s w i d e l y r e f e r r e d t o a s T e r z a g h i ' s

cri teri on and i s vali d onl y in t he case o f no flui d

penetration assuming that a perfect non-penetrating

m ud c ak e h a s b e e n f or me d o n t he w el l bo re w al l b y t h e

f ra ct ur in g f lu i d a dd i ti ve s. A l so t he e ff ec t o f f ra ct u re

along y- a xi s i s n e gl e c te d , b e c a u se t h e e x is t e nc e o f t h i s

f r ac t ur e r e ta r ds t h e i n i ti a t io n o f f r ac t u re a t A b y s t re s s

r el i ev in g . I n o t he r w o rd s, t h e m in i mu m p re ss ur e t h at

ma y i ni ti at e a f ra ct ur e a t A c an b e o bt ai ne d fr om

E q. ( 4) .

T he m in im u m p re ss ur e i ns i de t he e xi st i ng f ra ct u realong σyy d i re c t io n r e qu i re d t o p r o pa g a te t h e f r ac t u re

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

f ra ct ur e i s d i re ct l y a lo n g t h e p ri n ci pa l s tr es s σyy, th e

s h e a r e f f e c t a n d h e n c e t h e m o d e - I I s t r e s s i n t e n s i t y f a c t o r

( K II) v an is he s. T he re fo re , t he c ri te ri on f or f ra ct ur e

propagation simply becomes:

K Iz K I C ð5Þwhere K IC i s t h e f r a c t u r e t o u g h n e s s , a m a t e r i a l p r o p e r t y

t h a t r e f l e c t s t h e r o c k r e s i s t a n c e f o r a n e x i s t i n g f r a c t u r e t o

propagate. The mode-I stress intensity factor ( K I) f o r o u r

f r a c t u r e c o n f i g u r a t i o n c a n b e d e r i v e d a s ( R a h m a n e t a l . ,2 0 0 0 b):

K I ¼ C ffiffi

l p

ð pf −rxxÞ ð6Þwhere pf i s t h e p re ss ur e i ns i de t he f ra c tu re , l i s t he

f r ac t ur e l e n gt h o n o n e s i d e o f t h e w e ll , u s ua l l y r e f e rr e d

t o a s h al f- le ng t h a n d C e q ua l s t o π f o r l i n e a r- s h a p e d

fracture.

S o lv i n g t h e a b o ve f r ac t u re p r op a g at i o n c r i te r io n , t h e

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

be written as:

pf ¼ rxx þ K

I CC ffiffi l

p : ð7Þ

4. Model description

D u e t o r e so u r ce c o ns t ra i n ts , p a rt i c ul a rl y c o mp u t a-

t i o na l t i me , a s c al e d p r ot o t yp e r e se r v oi r o f r e ct a n gu l a r

s ha pe ( 10 m l en gt h, 1 0 m w id th a nd 2 m h ei gh t) h as b ee n

u s e d ( s ee F i g . 3) . F u rt h e r s i m p l if i ca t i on h a s b e e n m a de

by taking quarter-symmetry of the model reservoir.

F ra ct u re i n t h e r es er vo i r i s c on si d er ed a s r ec ta n gu la r

s ha pe t hr ou gh ou t i ts l en gt h. A v er ti ca l w el lb or e i s

c o n s i d e r e d a t t h e c e n t r e o f t h e r e s e r v o i r .

F ig . 2 . F ra ct ur e a lo ng n o n- p re fe rr ed d ir ec ti on s ub je ct t o t wo

orthogonal s tres s es , σxx and σyy.



5/12

T he o ri gi n o f t he C ar te si an c oo rd in at e s ys te m i s

c o ns i d er e d a t t h e b o tt o m c e n tr e o f t h e w e l lb o re . X and

Y or th o go n al a xe s a re p er pe n di c ul ar t o t he w e ll bo rea xi s a nd Z - ax is i s p ar al le l t o t he w el lb or e a xi s. F or

t he c on v en i en c e o f m od el li n g a n d u nd er st a nd i ng , t h e

i nv es ti ga ti on i s c ar ri ed o ut a t t he l at er al t ip o f t he

f ra ct ur e o ri en te d i ni ti al ly a lo ng t he n on -p re fe rr ed

d ir ec ti on f or p ro pa ga ti on . S in ce t he t op a nd b ot to m

t ip s o f a v er ti ca l f ra ct ur e i n a r ev er se f au lt in g s tr es s

r e gi m e a r e a l so a l o ng t h e n o n -p r ef e rr e d d i re c t io n , a n d

propagation of these tips are also very much similar

t o t he l at er al t ip i n t er ms o f s tr es s f il ed , t he r es ul ts

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

i nd ic at iv e of t he b eh av io ur o f t he v ert ic al ti ps .

H ow ev er, t hi s w il l b e d es cr ib ed f ur th er a ft er t he presentation of results.

4 . 1 . B o u nd a r y c o nd i t io n s

A s t h e m o d e l h a s b e e n t a k e n o n e q u a r t e r o f t h e t o t a lr es er vo ir s iz e d ue t o s ym me tr y, a ll t he b ou nd ar y

c on d it i on s a re a pp l ie d o n t h is q u ar te r o f t h e r es er v oi r.

A l l b o u n d a r i e s ( e . g . f a c e s A B a n d B C ) a r e f i x e d a g a i n s t

m e ch a n ic a l d i s pl a ce m e nt s . F a ce A E i s f r ee f o r m e ch a n -

i c al d i sp l a ce m en t a l o ng X - ax is o nl y a nd t he f ac e C D

along Y - ax is o nl y. T he w el lb or e w al l i s f re e t o m ov e

a l o n g b o t h a x e s .

E xc ep t t he w el l bo r e w al l , a ll t he b ou n da r ie s o f t h e

r e se r v oi r a r e c o n si d e re d a s i m p er m ea b le f o r f l ow, i . e.

t h er e i s n o e x ch a n ge o f f l u i d b e t we e n t h e r e se r v oi r a n d

i ts s ur ro u nd i ng r oc k s t ra t a. T he f ac es A E a nd C D a re

c on si d er ed i mp e rm ea b le f or f lo w a lo n g t h ei r p e rp e n-d i c u l a r d i r e c t i o n s d u e t o s y m m e t r y . A c o n s t a n t w e l l b o r e

F i g . 4 . F r a c tu r e r e p r e se n t a t io n w i t h a r e c t a ng u l a r s t r i p .

F i g . 3 . M o d e l -s c a l e q u a r t e r r e s e r v o ir w i t h a h y d r a ul i c f r a c t u r e .



6/12

pressure which is lower than the reservoir pore pressure

i s m a i nt a i ne d i n t h e w e l lb o r e.

T h e m o d e l i s f i r s t i n i t i a l i s e d w i t h i n s i t u s t r e s s e s , σxxand σyy a t a l l n o de s ( su b -z on e s) , a pp l ie d f or ce s a t t h e

o u te r b o u nd a r y, a u n if o rm p o re p r es s u re t h ro u g ho u t t h e

re se rv oi r a nd a f ix ed p re ss ur e o n t he w el lb or e t o

r e p r e s e n t a c o n s t a n t w e l l b o r e p r e s s u r e . S t r e s s e s σxx and

σyy a nd p o re p re ss u re i n a ll s ub -z o ne s a re a ll o we d t o

c h an ge i n r es po n se t o f lu id f lo w e xc ep t t h e w el l b o re

pressure.

4.2. Fracture representation

F LA C3 D i s a c od e f or F as t L ag r an gi a n A na l ys is o f

C o n t i n u a i n 3 D i m e n s i o n s a n d r i g h t l y i t d o e s n o t h a v e a

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

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

w i th d if fe re nt p er me ab i li t y a nd p o ro si ty b ei n g p a ck ed b y

proppants with respect to the other portion of the

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

a h y dr a ul i c f r ac t u re i s t h e u s e o f n o n -d i me n s io n a l f r a c -

t u r e c o n d u c t i v i t y (C i n c o - L e y e t a l . , 1 9 7 8 ) , d e f i n e d a s :

F C D ¼ jf wjl

ð8Þ

where κf i s t h e p r op p e d f r ac t ur e p e rm e ab i l it y, w i s t he

propped fracture width, κ i s t he i n tr in s ic r es er vo i r

permeability and l i s t h e f r ac t ur e h a l f- l en g t h ( l en g th o f

f ra ct ur e o n o ne s id e o f t he w el l) . T he v al ue o f κf is

s i g n i f i c a n t l y g r e a t e r t h a n κ. T h i s d e s c r i p t i o n o f p r o p p e d

f ra ct ur e a ll ow s u s t o r ep re se nt t he f ra ct ur e i n t he

F LA C3 D m od el b y a s ub -z on e o f w id th w along X -

a x i s o v e r l e n g t h l along Y - a x i s w i t h p e r m e ab i l i t y κf (see

F i g s . 3 a n d 4 ). F i g . 4 s h o w s t h e n e a r - w e l l b o r e p o r t i o n o f

t h e f ra ct u re d r es er vo i r. C l ea rl y, t he f ra ct u re i s r ep re -s en t ed b y a r ec t an g ul a r s tr ip w h ic h i s a ss i gn ed w i th a

v er y h ig h p er me ab il it y c om pa re d t o t he r es er vo ir

permeability. Due to symmetry, a half of the total

Ta b l e 1

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

Parameter s SI units Field units

S i ze o f t h e r e se r vo i r

Length 10 m 32.8 1 ft

Width 10 m 32. 81 ft Payzone height 2 m 6.56 ft

Well bore r adius 0.1 m 0.33 ft

Fracture width 0.01 m 0.03 3 ft

Fracture half length 1.25 m, 0.6 m 4.1 f t, 1 .97 ft

Fluid bulk mo dulus 3.83 e5 Pa 55. 55 ps i

F lu id c om pr es si bi li ty 2 .6 1 × 1 0−6 1/Pa 1.8 × 10−2 1/ps i

Euler' s constant 1.78 1.78

Poros ity 0.15 0.15

Biot modulus 2.55 × 106 Pa 3.7 × 102 ps i

Biot coefficient 0.21, 1 0.21, 1

I n i t ia l r e s e r vo i r

pore pres s ure

2 5 × 1 06 Pa 3.6 × 103 ps i

Well b ore pr es sur e 107 Pa 1.45 0 × 103 ps i

R es er vo ir p er m ea bi li ty 9 .9 × 1 0−16 m2 1 mdF ra ct ur e p er me ab il it y 9 .9 × 1 0−14 m2 1 0 0 m d

Fluid vis cosity 10−3 Pa s 1 cp

Bulk modulus 2.758 × 108 Pa 1.8 5 × 105 ps i

Shear modulus 1.15 × 108 Pa 1.67 79 × 104 ps i

M in im um in si tu st re ss 3 4 × 1 06 Pa 4.9 27 × 103 ps i

M ax im um i n s it u st re ss 5 1 × 1 06,

3 4 . 6 8 9 5 × 1 06

o r 3 4 . 3 4 4 7 5 × 1 06 Pa

7 . 4 × 1 03,

5 . 0 2 7 × 1 03

o r 4 .9 77 5 × 1 03 ps i

Tens ile strength 5.7 × 107 Pa 8.12 × 103 ps i

F i g . 5 . C o n t o u r p l o t o f p o r e p r e s s u r e a f t e r 1 9 0 h o f f l o w w i t h l = 1 . 2 5 m , Δσ= 0 . 3 5 M P a , α = 0 . 2 5 .



7/12

propped fracture width was modelled in the quarter

r es er v oi r m od el i n Fig . 4. I so t ro p ic p er me ab i li t y i s

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

5. Presentation of results and discussions

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

been studied. Two different fracture lengths have been

u s e d t o a n a l y s e t h e e f f e c t o n f l o w r a t e , p o r e p r e s s u r e a n ds t re s s d i st r i bu t i on w i th i n t h e r e se r vo i r c o mp a re d t o a n

u n fr a ct u r ed r e se r vo i r. A p a rt f r om t h e f r ac t u re l e n gt h ,

B i o t c o e f f i c i e n t , a n d t h e m a g n i t u d e o f t h e i n s i t u s t r e s s e s

a re a ls o c ha ng ed t o o bs er ve t he a bo ve m en ti on ed

e f fe c ts . Va l u es f o r d i ff e re n t p a ra m et e rs t o d e sc r ib e t h e

r e se r v oi r c o n di t i on a r e s u mm a ri s ed i n Ta b l e 1.

Va lu es u se d f or t wo p ar am et er s d es er ve s om e

e x pl a n at i o ns h e re . T h e a c tu a l f r ac t u re p e rm e ab i l it y, o r

t h e f r ac t u re c o nd u c ti v i ty, w i l l b e m u ch h i g he r t h a n t h e

v al u e u se d f or t he m od e l h er ei n. A l th ou g h t he m od e l

v al u e i s c ho se n a rb i tr ar il y, t he r el at i ve ly l o w f ra ct u re

permeability was used to avoid the infinite conductivityc on di ti on i n t he s ma ll m od el r es er vo ir. T he f lu id

c o mp r es s ib i l it y i s a b o ut 1 0 0 t i me s t h e c o mp r e ss i bi l i ty

o f n at ura l ga s wi th a p art ic ul ar c omp osi ti on a t a

F i g. 6 . C o n t o ur p l ot o f σxx a f t e r 1 9 0 h o f f l o w w i t h l = 1 . 2 5 m , Δσ= 0 . 3 5 M P a , α = 0 . 2 5 .

F i g. 7 . C o n t o ur p l ot o f σyy a f t e r 1 9 0 h o f f l o w w i t h l = 1 . 2 5 m , Δσ= 0 . 3 5 M P a , α = 0 . 2 5 .



8/12

t e mp e ra t u re a n d t h e r e se r v oi r p r es s u re u s ed . T h e h i gh

c o mp r e ss i bi l i ty w a s u s e d j u s t t o a c ce l er a te t h e c o mp u -

t at i on b y a d ju s ti n g t h e t i me s te p i n t he e xp l ic i t f in it e

d if fe re nc e s ol ut io n s ch em e. T hi s h as h el pe d u s t o

d em on st ra te t he k ey c on ce pt o f t hi s w or k i n a t im e-

e ff ic ie n t m an ne r. I n a r ea l c as e w h er e o th e r r es er vo i r parameters will also be different, the actual fluid

c om pr es si bi l it y m us t b e u se d a n d t h e c o mp u ta ti o na l

t im e i ss ue h as t o b e m an ag ed i n s om e o th er w ay s, a s

d i s c u ss e d l a t e r.

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

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

d ir ec ti o n. To s im u la te t h is s ce na ri o , a f ra ct u re h al f-

l en gt h o f 1 .2 5 m w as m od e ll ed a lo n g Y - d i r e ct i o n w i t h

v al ue s o f i n s it u s tr es se s, σxx = 3 4. 35 M Pa a nd σyy =

3 4 M P a , c r e a t i n g a d e v i a t o r i c s t r e s s o f 5 0 p s i ( 0 . 3 5 M P a ) .

A c ou pl ed fl ow p ro ce ss w as si mu la te d fo r 1 90 h .C o nt o u r p l o ts f o r p o re p r es s u re , a n d a b s ol u t e s t re s se s σxxand σyy a ft er 1 9 0 h o f f l o w a re p re se n te d i n F i g s . 5 , 6

a nd 7, r e s p ec t i v e l y.

A s c a n b e s e e n i n F i g . 5 , t h e p o r e p r e s s u r e d e p l e t i o n

i s e l o ng a t ed a l o ng t h e f r a c t ur e l e n gt h , a n d t h e f l u i d h a s

been extracted both through matrix to the wellbore, and

f ro m t he m at r ix t h ro u gh t h e f ra ct ur e t o t h e w e ll b or e.

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

f l o w p a t t e r n a s e x p e c t e d . N o n - u n i f o r m p a t t e r n s o f s t r e s s

c o n t o u r s i n F i g s . 6 a n d 7 a r e o f p a r t i c u l a r i n t e r e s t , w h i c h

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

i n t e r es t e d i n .To d e v el o p c o n fi d e nc e i n o u r a p p ro a c h f o r p r op p ed

f ra ct ur e m od el li ng i n F LA C3 D, w e s im ul at ed f lo w

processes for two other cases with the same stress

c o n d i t io n : f r a c t ur e h a l f - le n g t h , l = 0 .6 a nd l = 0 m, i.e.

w it h ou t a n y f ra ct u re . F lo w p ro fi l es f or t he t hr ee c as es

(l = 1 . 2 5 , l = 0 .6 a nd l = 0 m) d uring th e first 13 h are

plotted in F i g . 8 . S e n s i b l y , t h e p r e s e n c e o f f r a c t u r e s h a s

i n cr ea se d t h e f lo w r at e s ig n if i ca nt l y, a nd t h e o rd e r o f

i nc re as e w as a ls o c lo se t o a p re di ct ed v al ue b y a n

a p pr ox i ma te a na l yt ic a l m et h od i n w h ic h a f ra ct u re i s

r e p re s e n t e d b y a s k i n a s a f u n c t i on o f t h e n o n -

d i m en s i on a l f r ac t u re c o nd u c ti v i ty (E co n om i de s e t a l. ,

1994 ) . A p p a r e n t l y , a l m o s t t h e s a m e f l o w r a t e f o r t h e t w o

d i f fe r en t f r ac t u re l e n gt h s d u ri n g t h e f i rs t f e w h o u rs c a n

be perc eive d as inco nsist ency. Howev er, rigo rous

i n v es t i ga t io n s r e ve a l ed t h a t t h i s i s v e ry m u ch s e ns i bl e

a s n o a d di t io n al f lo w b en e fi t i s r ea li se d f or t h e l on g er

f ra ct ur e u n ti l t he d i ff us i on p ro ce s s g oe s b ey o nd t h e

l e n g t h o f t h e s h o r t e r f r a c t u r e , a f t e r w h i c h t h e f l o w r a t e i so b v io u s ly h i g he r f o r t h e l o n ge r f r ac t u re .

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

t h e f r a c t u r e t i p a r e p l o t t e d i n F i g . 9 w i t h r e s p e c t t o f l o w

t im e. S en si bl y, t he p or e p re ss ur e h as d ec li ne d a s a

f un ct io n o f p ro du ct io n t im e. T hi s d ec li ni ng p or e

press ure creat es an effect of stress relie f on the

s u rr ou n di ng r oc k m at ri x , a nd t he re fo re t h e a bs o lu te

F i g. 8 . F l ow r a te c o mp a ri s on f o r d i ff e re n t f r ac t ur e c o nd i ti o ns w i thΔσ= 0 . 35 M P a, α = 0 . 2 5 .

F i g. 9 . A b so l ut e s t re s se s a n d p o re p r es s ur e a t t h e f r a ct u r e t i p v e rs u s

production time with l = 1 . 2 5 m , Δσ= 0 . 3 5 M P a , α = 0 . 2 5 .

F i g. 1 0 . Va r ia t io n o f s t re s se s a n d p o re p r es s ur e a l on g t h e f r ac t ur ed i r e c ti o n w i t h l = 1 . 2 5 m , Δσ= 0 . 3 5 M P a , α = 0 . 2 5 .



9/12

s tr es se s h av e a ls o d e cl in e d a s a f un ct i on o f f lo w t i me .T h e m o s t i m p o r t a n t a n d i n t e r e s t i n g i n f o r m a t i o n t h a t c a n

be derived from this figure is that the stress reversal

( st r es s c o nd i ti o n f ro m σxxNσyy to σyyNσxx) h a s

o cc u rr e d a f t er a bo u t 4 8 h o f f lo w, a nd a ls o t h e r e v er se

d ev ia to ri c s tr es s a t t he f ra ct ur e t ip h as i nc re as ed

a ft er wa rd s w it h f lo w t im e. I t i s i nt er es ti ng t o k no w

h ow f ar f ro m t he f ra ct ur e t ip t hi s s tr es s r ev er sa l h as

m o v e d a f t e r 1 9 0 h o f f l o w . F o r t h i s p u r p o s e , s t r e s s e s a n d

pore pressure at different points from the fracture tip

a l o n g t h e f r a c t u r e d i r e c t i o n a r e p l o t t e d i n F i g . 1 0, w h i c h

c l ea r l y s h o w s t h at t h e s t re s s r e v e r sa l r e gi o n e x t en d s u p

t o a bo ut 3 5 c m f ro m t he f ra ct ur e t ip . I mp li ca ti on s o f r es ul ts i n t he se t wo f ig ur es o n m ul ti st ag e h yd ra ul ic

f r ac t u ri n g w i l l b e d i sc u s se d l a t er.

S t r e s s e s a n d p o r e p r e s s u r e a t a p o i n t v e r y n e a r t o t h e

w e l l b o r e w a l l a l o n g X - a x i s a r e p l o t t e d i n F i g . 1 1 against

f l o w t i m e . N o t e t h a t t h e p o i n t i s n o t t a k e n e x a c t l y o n t h e

w el l bo r e w al l t o a v oi d t h e e ff ec t o f c o ns ta n t w e ll bo re

pressure. The figure shows that the total stresses and the

pore pressure decrease very sharply during the initial

f lo w p er io d a nd t he c ha ng es a re v er y n eg li gi bl e

a ft er wa rd s. A n i nt er es ti ng n ot e t o b e t ak en f ro m t he

f i g u r e t h a t t h e d i f f e r e n c e b e t w e e n t h e t o t a l s t r e s s e s , σxx−σyy, h a s i n c r e as e d s i g n i fi c an t l y d u e t o t h e f l o w e f fe c t

f ro m i ts i ni t ia l v e ry n eg li g ib l e v al u e. A ls o t h er e i s n o possibility of stress reversal at this point due to the flow

e f fe c t. H o w ev e r, t h i s d o es n o t c l ea r l y a n sw e r w h et h e r

t he e x is ti ng f ra ct u re w il l p ro p ag a te f ur th er a lo n g i ts

plane (Y - a xi s ) w i t ho u t i n it i a ti n g a n o th e r f r ac t ur e a t t h is

point (along X - a xi s ) i f t h e w e l lb o r e i s p r es s ur i ze d a f te r

1 9 0 h . F o r t h i s p u r p o s e , t h e f r a c t u r e i n i t i a t i o n p r e s s u r e a t

t hi s p o in t a nd t h e p ro pa g at i on p re ss ur e a t t he e xi s ti ng

f ra ct u re t ip p ro p ag a ti o n a re c al c ul at e d a nd p l ot te d i n

F ig . 1 2. T h e t e n s i l e s t r e n gt h T h a s b e en a ss um ed z er o

w it h w h ic h t he l ik e li h oo d o f f ra c tu re i n it i at i on a t t h is

point before the tip propagation is maximum. In other w or ds , i f t he s ec on da ry o rt ho go na l f ra ct ur e i s n ot

i n i ti a t ed b e fo r e t i p p r op a g at i o n w h e n T = 0 , it is very

u nl ik el y t ha t w il l h ap pe n i f t he f or ma ti on h as s om e

r e a s o n ab l e t e n s i l e s t r e n g th . F i g . 1 2 c l e a r l y s h o w s t h a t i f

t h e i n i t i a l f r a c t ur e i s n o t c r ea t e d a l o n g X - a x i s a n d i f t h e

w e l l b o r e i s p r e s s u r i z e d a g a i n a f t e r a b o u t 2 0 h o f f l o w , a

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

point. In fact, the declining nature of fracture propaga-

t i o n p r es s ur e i n d ic a t es t h a t t h e f r ac t u re w i l l p r op a g at e

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

T he o pp os it e i s t ru e w it h t he s ec on da ry o rt ho go na l

f r a c tu r e i n i t i a t i on .B e f o r e d i s c u s s i n g t h e i m p l i c a t i o n s o f t h e s e r e s u l t s o n

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

o f r es ul ts w it h c ha ng e o f s om e s tu dy p ar am et er s a re

s u mm a ri s e d i n Ta bl e 2 w i th o u t p r es e nt i n g d e t ai l ed

plotting to shorten this presentation.

It i s c le ar f ro m Ta bl e 2 t h at w i th i nc re a se d s tr es s

d if fe re nc e a l on ge r f lo w t im e i s n ec es sa ry f or t he

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

s h or t er f r ac t u re . F o r a h i g he r v a lu e o f B i ot c o ef f ic i e nt ,

t h e f l o w p e r i o d f o r s t r e s s r e v e r s a l i s s h o r t e r . T h i s i s a l s o

s e ns i bl e , b e ca u se a h i g he r B i o t c o ef f ic i en t r e pr e se n t sl o w er g r ai n c o mp r e ss i b il i t y o f t h e r o ck . T h er e fo r e, t h e

F i g . 1 2 . F r a c t u r e i n i t i a t i o n p r e s s u r e ( a t w e l l b o r e w a l l a l o n g X - a x i s ) a n d

t i p p r op a ga t io n p r es s ur e f o r t h e c u rr e nt f r a ct u re ( a lo n g Y - a x i s ) w i t hl = 1 . 2 5 m , Δσ= 0 . 3 5 M P a , α = 0 . 2 5 .

Ta b l e 2

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

s t r e ss r e v e r sa l t i m e

S t r e ss d i f f er e n c e

σxx−σyy (ps i)

Biot

coefficient, α

Fracture

half-length, l f (m)

S t r e ss r e v e r sa l

t i me ( h )

50 0. 25 1.25 4 8

50 0. 25 0.60 1 2.5

100 0. 25 1.25 9 4

50 1. 0 1.25 4 6

100 1. 0 1.25 8 3

F i g . 1 1 . A b s o l u t e s t r e s se s a n d p o r e p r e s su r e a t t h e n e a r - w e l lb o r e w a l l

along X - a x i s w i t h l = 1 . 2 5 m , Δσ= 0 . 3 5 M P a , α = 0 . 2 5 .



10/12

m ec h an i ca l d i sp l ac em en t i s a bs o rb e d l es s b y t h e r oc k

w it h a h i gh e r B io t c o ef fi ci e nt , a n d i s m or e e ff ec ti v e t o

s q u e e z e p o r e s p a c e s r e s u l t i n g i n a h i g h e r f l o w r a t e . T h i s

h ig h er f lo w r at e a cc el er a te s t h e p or oe la st i c e ff ec t a n d

h en ce a cc el e ra te s t h e s tr es s r ev e rs a l. I t i s i mp or ta n t t o

m en t io n h e re t ha t t h e p er me ab i li t y c h an ge d ue t o f lo we f f e c t i s n o t m o d e l l e d i n t h i s w o r k . F o r r o c k s w i t h a l o w

B i ot c o ef f ic i en t ( e .g . α= 0 .2 5 ), t h e s tr es s r ev e rs a l t im e

w ou l d b e l o ng er t ha n r ep o rt e d d ue t o t i me -d ep e nd en t

per mea bil ity red uct ion res ult ing from hig h gra in

c o m p r e s s i o n . F o r B i o t c o e f f i c i e n t , α = 1 ( i n c o m p r e s s i b l e

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

t o g r ai n c o mp r e ss i bi l i ty a l on e i s z e ro a n d t h er e fo r e t h e

s tr es s r ev e rs a l t i me w o ul d r em ai n u nc h an g ed . W he n

plotted all the graphs for other cases in Ta bl e 2, th e

g e n e r a l n a t u r e s w e r e v e r y s i m i l a r t o t h o s e f o r l = 1 . 2 5 m ,

Δσ= 0 . 3 5 M P a , α = 0 . 2 5 i n F i gs . 5

–

12 .

5 .1 . I m pl i ca ti o ns o f r es u lt s f o r m ul t is t ag e h yd r au li c

fracturing

A s w e m e n t i o n e d e a r l i e r , t h e m a i n m o t i v a t i o n b e h i n d

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

(σv and σh) a t t he t op a nd b ot to m t ip s o f a v ert ic alh y d r a u l i c f r a c t u r e u n d e r a r e v e r s e f a u l t i n g s t r e s s r e g i m e

(σvbσh) r ev er s e i n t e rm s o f t h ei r m ag n it u de s i n r es po n se

t o p ro du c ti on o v er t i me . I f t h ey r ev er se , i .e . t h e l oc a l

s tr es s c o nd i ti on a ro un d t h es e t ip s b ec o me s s uc h t h at

σvNσh, t h e s e t i p s c a n t h e n e a s i l y b e p r o p a g a t e d f u r t h e r v e rt i ca l l y w i t h o ut t u rn i n g a n d t w i st i n g u p t o t h e e x te n t

t h is c on d it io n h a s b e en e st a bl i sh e d. N e gl e ct i ng t h e g r av i ty

e f fe c t o f h y d ro c a rb o n f l ow, t h e h y d ro c a rb o n e x tr a c ti o n

pattern around the top and bottom tips of the fracture is

v er y m uc h s i mi la r t o t ha t a t t he l at e ra l t i p. T he re fo re ,

r e f e rr i n g t o F i g . 9 a n d m a ki n g t h e c o r re s po n d en c e b e t -

ween σv and σyy, a nd σh and σxx w e c an n o w c o nf id e nt l y

a ns we r t h at t h e s ug g es t ed s tr es s r ev er sa l a t t h e t o p a n d

bottom tips is possible, and also possible without

i ni t ia ti n g a n y s ec on da r y f ra ct u re f ro m t he w e ll b or e.

F o r o u r m o de l -s c a le s m al l r e se r vo i r, t h e s t re s s r e ve r sa lh as o cc u rr ed i n t h e m ag n it u de o f h o ur s a nd t he e xt e ns io n

o f t h is s tr es s- re ve rs ed z on e h a s b e en i n c e nt i me t re s. F or a

f u ll - sc a l e r e al r e se r vo i r w i t h r e al i st i c s t re s s d i ff e re n ce ,

t he f ra ct ur e d i me n si o n w i ll b e i n h u nd r ed s o f f ee t a n d t h e

t im e w il l b e i n m on th s, o r y ea rs . H ow ev er, w it h a n

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

t h e p r o po s ed m u lt i s ta g e h y d ra u li c f r ac t u ri n g b y s i mu -

l at i ng t h e p ro c es s f or t he v er ti c al f ra ct u re t ip s o v er a l on g

period of time to extend the stress reversed zone

s ig ni fi ca nt ly. T hi s h as t o b e r ep ea te d i n a n um be r o f

s ta g es a nd t h e p r od u ct i on t i me a n d t h e f ra c tu re d i me n -

s io n s h a ve t o b e r ec o rd e d f o r a l l t h e s t ag e s. R e fr ac t ur i ng

t re at me nt s f or a ll t he s ta ge s h av e t o b e o pt im iz ed a nd

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

It i s i mp or ta nt t o n ot e h er e t ha t t he p ro po se d

m u l ti s ta g e h y d ra u l ic f r ac t u ri n g t e ch n i qu e i s d i ff e re n t

f ro m o th e r m ul ti s ta ge h y dr au l ic f ra ct u ri ng j ob s d i s-

c u s s e d i n m a n y l i t e r a t u r e s a n d o f t e n p e r c e i v e d c u r r e n t l y by the petroleum industry. Currently, there are two

practices that are often referred to as multistage hydraulic

f r ac t ur i n g: ( 1 ) p u mp i ng t h e p r op p an t - la d e n f r ac t u ri n g

f lu i d i n a n u mb e r o f s ta g es w i th s ho rt i nt e rv a ls d u ri n g t h e

f ra ct ur in g j ob c ar ri e d o ut b ef or e t h e p ro du c ti on p h as e

s t a r t s , a n d ( 2 ) a s e c o n d f r a c t u r i n g j o b t h a t i s c a r r i e d o u t

a ft er a f ew y ea rs o f p ro du ct io n, o ft en k no wn a s

r e fr a ct u r in g . T h e p r op o se d m u lt i s ta g e f r ac t u ri n g t e c h-

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

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

t o b e c a r r i e d a t d i f f e r e n t s t a g e s o f p r o d u c t i o n ; h o w e v e r ,t h e n u m b e r o f s t a g e s a n d t h e i r t i m i n g a r e p r o p o s e d t o b e

o p ti mi z ed b as ed o n s im il a r a na l ys es p re se n te d i n t h is

paper, but for the full-scale reservoir. The treatment for

e a ch s ta g e c a n b e o p ti mi z ed i n a w ay t h at a l le v ia t es o th e r

f r a c t u r i n g c o m p l e x i t i e s a s w e l l ( R a h m a n e t a l . , 2 0 0 1 ).

P e rf o rm i n g t h e p r op o s ed m u l ti s ta g e h y d ra u l ic f r ac -

t u r in g s i mu l a ti o n a n d t r ea t m en t o p t im i za t i on f o r a r e al

r e se r vo i r w i ll b e c e rt a i nl y c o mp u t at i o na l l y c h al l e ng i n g

a nd t he re fo re f ur th er e xt en si ve r es ea rc h h as t o b e

u n d er t a ke n t o o v er c om e b o th m o de l l in g a n d c o mp u t a-

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

w h i ch h a s, w i t hi n i t s s c op e , d e mo n s tr a te d t h e p o t en t i -a l it y o f t h e c on c ep t . I n a dd i ti on t o a pp r op ri at e m es h in g i n

d i ff er en t z on es o f t he m od el , t h e u se o f i mp l ic i t f in it e

d i f fe r en c e s o lu t i on s c he m e m i g ht a l l ow a s i gn i fi c a nt l y

l a rg e r t i me s t e p w i t ho u t g r ea t l y c o mp r om i s in g w i t h t h e

a cc ur ac y, a nd h en ce h as a g re at p ot en ti al t o h el p i n

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

i n f o rm u l at i o ns a r e a l so n e ce s s ar y t o c o n si d e r m u lt i p ha s e

f lo w a n d p se ud o -s te ad y s t at e f lo w c o nd it i on s . F or t h e

c on ve ni en ce o f m od el li ng a nd u nd er st an di ng , t he

phenomenon has been studied at the lateral tips of the

f r ac t ur e s w i t h a h i g hl y c o mp r es s i bl e h y p ot h e ti c a l p o r ef l u id . W i th a d e qu a t e r e so u r ce s , a c o mp r e he n si v e i n v es -

t ig at io n w it h a f ul ly 3 -D s ol id m od el f or a f ul l- sc al e

r es er vo ir c on ta in in g a r ea l h yd ro ca rb on f lu id m ay

produce further stimulating insights directly for the top

a nd b ot to m t ip s o f t he f ra ct ur e. I nt ui ti ve ly, t he

production-induced stress alteration process is inherent-

l y d e p en d e nt o n t h e r e s er v o ir p e r me a b il i t y. T h e a p p li c a -

t i on s o f t h is c o nc e pt t o l o w t o e xt r em el y l o w p e r me a bi l it y

f o rm a ti o ns n ee d t o b e s t ud i ed f ur t he r, a n d t h e o p ti m iz a ti o n

o f c o n tr o l la b l e p a r am e t er s l i k e f r ac t u re d i m en s i on s a n d

c o nd u ct i vi t y m ay b e c ru ci a l t o i t s f ea si b le a pp l ic at io n s t o

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



11/12

I de a ll y, t h e d e ve l op m en t o f a s t an d a lo n e c o de w i th

s l i gh t l y s i m pl i f ie d a n d c o m pu t a ti o n al l y e f fi c i en t t h e or i e s

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

s u b je c t o f f u r th e r i n v es t i ga t i on s t o i m p le m e nt c o n ce p t t o

a ct ua l r es er vo ir s. T he F LA C3 D c od e c an b e a v it al

v a li d at i on t o ol i n t h at d e ve l op m en t p r oc e ss . I n s h or t, ag re at d ea l o f w or k t o b e d on e i n t he f ut ur e c ov er in g

f o r mu l a ti o n s, m o d el l i ng a s w e l l a s c o m pu t a ti o n a l a s p ec t s .

O n r e a li s a ti o n o f s u c h a t e c hn o l og y, h o w ev e r, A u s tr a l ia

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

r e s er v o ir s a r e s u b je c t t o r e v er s e f a u lt i n g s t r es s r e g im e s ,

a re e nv is ag ed t o d er iv e h ug e b en ef it s. S uc h a p ro je ct

therefore warrants significant long-term investments.

6. Conclusions

T he ma in mo ti va ti on b eh in d t hi s s tu dy w as t oi n v e s t i ga t e w h e t h e r t h e p r o d u c t io n - i n d u c ed p h e n o m e na

c an e st ab li sh a s tr es s s ta te a t a n e xi st in g h yd ra ul ic

f r ac t u re t i p , w h i ch w a s i n i ti a l ly a l o ng t h e n o n -p r e fe r re d

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

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

d i r e c t io n w i t h o u t t u r n i n g a n d t w i s t i n g. T h e s i g n i f i c a n c e

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

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

i n a n um be r o f s ta ge s t o i ts f in al p la na r, p ro du ct iv e

c o n f i g u ra t i o n b y e x p l o i t i ng t h i s p h e n o m e n o n .

W it h in t h e f ra me w or k o f F LA C 3D , a l i ce n se d p ro -

d u ct o f I t as c a C o ns u l ti n g G r ou p f o r m o de l l in g c o u pl e df lu id f lo w a nd m ec h an i ca l d ef or ma ti o n, a s ol i d m od el

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

producing well and a propped fracture from the well. A

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

v ar yi n g i n s it u s tr es se s, f ra ct u re s iz e a n d p o ro e la st i c

parameters to characterize the time-dependent stress

s ta te . R e su l ts d e mo n st ra t e t h at t h e s tr es s s ta t e a t t h e

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

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

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

f ra ct u re t ip . F or a g iv e n p ro p pe d f ra ct ur e s iz e , i n s it us t re s se s a n d r e se r v oi r p r op e r ti e s, t h i s s u it a b le t i me a n d

t h e t i m e t o e x t en d t h i s f a v o u ra b le s t re s s c o n d it i o n u p t o a

t a rg e t r e gi o n c a n b e e s ta b l is h ed f r om s u c h s i mu l a ti o n s.

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

t he n b e o pt i mi z ed a n d e xe cu t ed t o e xt e nd t h e f ra c tu re

t hr ou g ho ut t h at r eg i on . R ep e at i ng t h is i n a n um be r o f

s t ag e s , a f i na l p r o du c t iv e f r ac t u re c a n b e c r ea t e d. F r om

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

t im e i s n e ce ss ar y f or t he s tr e ss r ev er sa l t o o cc u r i f t h e

d if fe re n ce b e tw ee n t h e o rt h og on a l s tr es se s a t t he t ip i s

h ig he r, t he s ta ge o ne f ra ct ur e i s l on ge r a nd t he r oc k

g r ai n s a r e m o re c o mp r es s ib l e ( i .e . α i s l o w e r ) . I t i s a l s o

e s ta b l is h ed t h a t t h e r e fr a ct u ri n g o f t h e c u rr e n t f r ac t u re

c a n b e d o n e w i t h o ut c r ea t i ng a s e co n d ar y f r ac t u re f r om

t h e w e l l b o r e.

F i na l l y, t h e p a p er h a s c o n cl u d ed b y e m ph a s iz i n g t h e

n e ed f o r f u rt h er r e se a r ch w i t h f u ll - sc a l e r e se r v oi r s, a n d

f ut ur e r es ea rc h d ir ec t io n s t o o p ti mi z e t h e p ro po s edm u l t i s ta g e f r a c t u ri n g p r o c e s s.

Acknowledgements

T h e f i rs t a u th o r a c k no w l ed g es t h e F a cu l t y S t ar t -u p

G r a n t a t U W A t o d e v e l o p t h e b a s i c f a c i l i t i e s f o r r e s e a r c h

i n h y d r a u l i c f r a c t u r i n g , a n d b o t h a u t h o r s t h a n k U W A f o r

U W A R e s e a r c h G r a n t t h a t h a s l e d t h e p r o j e c t u p t o t h i s

point. The two anonymous reviewers must be credited

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

F i n a l l y , C h r i s t i n e D e t o u r n a y o f I t a s c a C o n s u l t i n g G r o u pd e se r v es t h a nk s f o r h e r a d v ic e o n ‘t h e u s u al p r oc e d ur e’

t o f ol l ow w he n p ub l is hi ng a w or k d o ne u si n g I ta sc a' s

s of tw ar e p ro du c ts b as ed o n w hi ch t h is r ev is e d m an u-

s c ri p t i s p r ep a r ed .

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